INVESTIGATING HARD SPHERE INTERACTIONS THROUGH SPIN ECHO SCATTERING ANGLE MEASUREMENT
Adam Washington
Submitted to the faculty of the University Graduate School in partial fulfillment of the requirements for the degree Doctor of Philosophy in the Department of Mathematics, Indiana University August 2013
Accepted by the Graduate Faculty, Indiana University, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Doctoral Committee
Prof. Roger Pynn
Prof. William Michael Snow
Prof. David Baxter
Prof. Garfield Warren
August 8, 2013
ii
c 2013 Copyright Adam Washington
iii
Adam Washington
INVESTIGATING HARD SPHERE INTERACTIONS THROUGH SPIN ECHO SCATTERING ANGLE MEASUREMENT
Spin Echo Scattering Angle Measurement (SESAME) allows neutron scattering instruments to perform real space measurements on large micron scale samples by encoding the scattering angle into the neutron’s spin state via Larmor precession. I have built a SESAME instrument at the Low Energy Neutron Source. I have also assisted in the construction of a modular SESAME instrument on the ASTERIX beamline at Los Alamos National lab. The ability to tune these instruments has been proved mathematically and optimized and automated experimentally. Practical limits of the SESAME technique with respect to polarization analyzers, neutron spectra, Larmor elements, and data analysis were investigated. The SESAME technique was used to examine the interaction of hard spheres under depletion. Poly(methyl methacrylate) spheres suspended in decalin had previously been studied as a hard sphere solution. The interparticle correlations between the spheres were found to match the Percus-Yevick closure, as had been previously seen in dynamical light scattering experiments. To expand beyond pure hard spheres, 900kDa polystyrene was added to the solution in concentrations of less than 1% by mass. The steric effects of the polystyrene were expected to produce a short-range, attractive, “sticky” potential. Experiment showed, however, that the “sticky” potential was not a stable state and that the spheres would eventually form long range aggregates.
iv
CONTENTS
1 NEUTRON SPIN ECHO
1
1.1
SCATTERING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
EARLY SCATTERING . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
NEUTRON SCATTERING . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.4
SESAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
PRECESSING MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.6
BIREFRINGENT MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.7
PATTERSON FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.8
INSTRUMENT TUNING . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.8.1
TUNING MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.8.2
TUNING SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . .
25
2 POLARIZED BEAM-LINE CONSTRUCTION
34
2.1
LAYOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.2
POLARIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.2.1
NEUTRON GUIDES . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.2.2
BENDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.2.3
WIDE ANGLE ANALYZER . . . . . . . . . . . . . . . . . . . . . .
44
2.2.4
3 HE
48
ANALYZER . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
2.3
WOLLASTON PRISMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.3.1
CONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.3.2
POWER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
2.4
EFFICIENCY MEASUREMENTS . . . . . . . . . . . . . . . . . . . . . . .
58
2.5
DETECTOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
2.5.1
BEAM MONITOR . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.5.2
SCINTILLATING DETECTOR . . . . . . . . . . . . . . . . . . . .
62
2.5.3
He3 DETECTOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3 DEVELOPMENT OF SESAME METHOD ON ASTERIX
65
3.1
EQUIPMENT MOUNTING . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.2
TRIANGLE GAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.3
TUNING PROCEDURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.4
WAVELENGTH RANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.5
SAMPLE ENVIRONMENT . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.5.1
SAMPLE CELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.5.2
SAMPLE FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.6
CALIBRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.7
SOLENOID CURRENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
3.8
CADMIUM SLITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.9
DETECTOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3.10 POLARIZATION ANALYZER . . . . . . . . . . . . . . . . . . . . . . . . .
90
4 ANALYSIS OF COLLOIDS
93
4.1
PERCUS–YEVICK CLOSURE . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.2
PURE HARD SPHERES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
vi
4.3
CORE SHELL MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
4.4
TIME DEPENDENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
4.5
SHORT RANGE ATTRACTION . . . . . . . . . . . . . . . . . . . . . . . .
104
4.6
LONG RANGE AGGREGATES . . . . . . . . . . . . . . . . . . . . . . . .
109
5 CONCLUSIONS
113
5.1
INSTRUMENT DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
5.2
HARD SPHERE DEPLETION . . . . . . . . . . . . . . . . . . . . . . . . .
114
A DERIVATION OF EFFICIENCY MEASUREMENTS
116
A.1 MEASUREMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
A.1.1 SOLVABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
A.2 FULL SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
A.3 UNCERTAINTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
124
A.3.1 BENDER EFFICIENCY . . . . . . . . . . . . . . . . . . . . . . . .
126
A.3.2 ANALYZER EFFICIENCY . . . . . . . . . . . . . . . . . . . . . . .
131
A.3.3 SUPER-MIRROR EFFICIENCY . . . . . . . . . . . . . . . . . . . .
135
B DESIGN OF PAPA CONTROL SYSTEM
139
B.1 LANGUAGE CHOICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
B.2 CONTROL DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141
C DESIGN OF PELVIS
146
vii
LIST OF TABLES
1.1
Example Scattering Cross-Section . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Description of measurements taken to find the efficiency of the beamline
4
elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.1
Spin Echo Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.1
The total scattering for each of the three diameters of PMMA measured. These scattering constants were found by taking the asymptotic value of the measured scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
4.2
Scattering length densities for the PMMA samples. . . . . . . . . . . . . . .
101
4.3
Core Shell Fitting Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . .
102
A.1 Intermediate Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
viii
LIST OF FIGURES
1.1
Layout of the SESAME beam-line . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Diagram of magnetic field integral versus height . . . . . . . . . . . . . . . .
9
1.3
Layout for a neutron passing through a single Wollaston prism pair. . . . .
10
1.4
A diagram of the relevant angles for neutron refraction at a magnetic field boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.5
Effect of Abel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.6
Diagram of vectors for the two dimensional approximation. The solid arrow is the trajectory of the neutron. . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.7
Effect of path count on polarization simulations . . . . . . . . . . . . . . . .
26
1.8
Change in simulated polarization versus simulated path count. . . . . . . .
27
1.9
Histogram of the number of bits needed to represent the number of paths needed to achieve 0.005% uncertainty on the polarization. . . . . . . . . . .
29
1.10 Average Tuned Polarization versus error in magnetic field and Wollaston prism position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
1.11 Worst Case Tuned Polarization versus error in magnetic field and Wollaston prism position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
1.12 Map of average tuned polarization versus Wollaston prism position and uncertainty in magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
32
1.13 Map of worst case polarization versus Wollaston Prims position and uncertainty in magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.1
Schematic Layout of SESAME instrument.
. . . . . . . . . . . . . . . . . .
35
2.2
Mounting units of the SESAME beamline . . . . . . . . . . . . . . . . . . .
36
2.3
Schematic drawing of neutron guides on SESAME . . . . . . . . . . . . . .
38
2.4
Polarized neutron reflectivity of guides . . . . . . . . . . . . . . . . . . . . .
39
2.5
A schematic view of the polarizing bender . . . . . . . . . . . . . . . . . . .
40
2.6
Simulated spin dependent reflectivities for possible polarizing super-mirrors.
41
2.7
Simulations of bender transmission . . . . . . . . . . . . . . . . . . . . . . .
43
2.8
The flux of the SESAME beamline at the sample position . . . . . . . . . .
44
2.9
A sketch of the wide angle analyzer . . . . . . . . . . . . . . . . . . . . . . .
45
2.10 Cartoon diagram of the Wide Angle Analyzer. . . . . . . . . . . . . . . . .
46
2.11 Wavelength sensitivity of IPNS analyzer . . . . . . . . . . . . . . . . . . . .
47
2.12 Comparison of 3 He analyzer to wide angle supermirror analyzer when tuning the instrument for spin echo with 5 A in the Wollaston prisms. The flipping ratio is measured by dividing the total number of neutron detected per monitor count between 5 ˚ A and 10 ˚ A in the spin up state by the same measurement in the spin down state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.13 First generation Wollaston prism coil . . . . . . . . . . . . . . . . . . . . . .
50
2.14 Winding of Wollaston Prisms . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.15 Comparison of simulated and measured fields from gapped Wollaston prisms. 52 2.16 Thermal image of Wollaston prism . . . . . . . . . . . . . . . . . . . . . . . 2.17 Picture of
π 2
53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.18 Measurement of field uniformity of Wollaston prisms . . . . . . . . . . . . .
56
2.19 Measurement of field uniformity of gapped Wollaston prisms . . . . . . . . .
56
x
2.20 Map of power supply connections for the SESAME beamline. . . . . . . . .
58
2.21 Schematic for the layout of the SESAME beamline during the polarizing efficiency measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
2.22 Polarizing efficiency of SESAME flippers . . . . . . . . . . . . . . . . . . . .
60
2.23 Polarizing Efficiency of SESAME bender and analyzer . . . . . . . . . . . .
61
2.24 A picture of the beam monitor on the beamline.
. . . . . . . . . . . . . . .
61
2.25 Background noise on the ADC channels of the PAPA Detector. . . . . . . .
64
3.1
Images of the SESAME elements on the ASTERIX beamline before the upgrade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.2
Detector Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.3
Analyzer Elevator
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.4
2007 ASTERIX Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.5
The layout of the beamline while tuning a single magnetic prism pair. . . .
71
3.6
Tuning plots
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.7
Effect of tuning the prism currents . . . . . . . . . . . . . . . . . . . . . . .
75
3.8
Wavelength range for uncertainties . . . . . . . . . . . . . . . . . . . . . . .
76
3.9
Sample Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.10 Effect of Magnetic Sample Holder . . . . . . . . . . . . . . . . . . . . . . . .
78
3.11 No sample guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.12 The first sample guide field . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
3.13 Diffraction Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.14 Laser Alignment System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.15 Cadmium Mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.16 Cadmium Mask Mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.17 He3 Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
xi
3.18 ASTERIX shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3.19 Effect of off detector scattering . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.20 IPNS Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.1
Percus-Yevick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.2
Hard Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.3
Total Scattering Versus Particle Size . . . . . . . . . . . . . . . . . . . . . .
100
4.4
Fits of spin echo data to core-shell model. . . . . . . . . . . . . . . . . . . .
103
4.5
Time dependence of PMMA samples . . . . . . . . . . . . . . . . . . . . . .
104
4.6
Sticky Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
4.7
260 nm sphere + 0.2% 900 kDa PS on ASTERIX . . . . . . . . . . . . . . .
106
4.8
200 nm sphere + 0.3% 900 kDa PS on ASTERIX . . . . . . . . . . . . . . .
106
4.9
200 nm sphere + 0.2% 900 kDa PS on OFFSPEC . . . . . . . . . . . . . . .
107
4.10 200 nm sphere + 0.3% 900 kDa PS on OFFSPEC . . . . . . . . . . . . . . .
107
4.11 200 nm sphere + 0.3% 110 kDa PS on ISIS . . . . . . . . . . . . . . . . . . .
108
4.12 200 nm sphere + 0.3% 110 kDa PS on ASTERIX . . . . . . . . . . . . . . .
109
4.13 260 nm spheres + 0.5% 900 kDa PS
. . . . . . . . . . . . . . . . . . . . . .
110
4.14 300 nm sphere + 1.0% 110 kDa PS on ASTERIX . . . . . . . . . . . . . . .
111
4.15 Microscopy Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
B.1 Organization of PAPA Control Software . . . . . . . . . . . . . . . . . . . .
143
C.1 PELVis flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
147
C.2 PELVis Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
C.3 The Spectrum Dialog for PELVis. . . . . . . . . . . . . . . . . . . . . . . . .
151
xii
CHAPTER 1
NEUTRON SPIN ECHO
Macroscopic phenomena are often controlled by interactions on microscopic scales. However, to call the scales microscopic can be a misnomer. Microscopes are inherently constrained by the wavelength of visible light, preventing features sizes smaller than 500 nm from being resolved. Ever since this resolution limit has been encountered, scientists have searched for new measuring devices to probe these unreachable length scales. The microscope tradition has been most directly followed by electron microscopy and atomic force microscopy. While not based on directly magnifying an image via reflecting or refracting elements, both techniques do produce a real-space image of the sample. These systems are not bound by the diffraction limit of their apparatus and have nanometer scale resolution, to the point where atomic force microscopy can resolve individual atoms in ultra high vacuum[1]. However, these microscopes still come with their own limitations. The samples often have to be measured in vacuum. Furthermore, electron microscopy is usually limited to measuring sample areas on the order of millimeters or smaller, with AFM being even further limited. Finally, both techniques are usually used as surface probes and measuring bulk information is rare. This returns to Feynman’s puzzle that no one has ever seen inside a brick[2]. Any system where the surface is not representative of the main volume will provide misleading results under any form of conventional microscopy imaging. 1
1.1
SCATTERING
Scattering techniques provide a solution to these issues. Most scattering can be performed in situ, allowing for a larger range of samples to be measured and increasing the options for sample environments (e.g. pressure cells, temperature controls). Furthermore, scattering techniques can measure the entire sample simultaneously, providing a statistical average of the structure of the entire sample as opposed to highly detailed results of a small portion. Finally, scattering can interact with any point in the bulk of the sample and provides no special consideration for the surface, making the determination of the interior possible. The basic theory of scattering is that the sample is made of a field of potential which will interact with an incoming particles. The particles will interact with the potentials and change their momentum. By comparing the initial and final momenta, it’s possible to map out the potential of the sample. Mathematically, this can be considered by making the incoming particles a plane wave. By taking the first order of the Born approximation, the expected outgoing wave is:
ψ (r, θ) ∝ eikz +
eikr r
Z
~′ ei~q·r V r~′ d3 r~′
(1.1)
Equation 1.1 is merely the Fourier transform of V (r~′ ) into reciprocal space. In the ideal case, this spherical term, which is denoted I(~q), can again be transformed back into real space, though it is possible to learn much about a sample simply by examining its reciprocal space projection. The definition of I(Q) is the scattered neutron intensity per unit volume ~ as a function of wavevector transfer Q
2
1.2
EARLY SCATTERING
The first scattering measurements on the atomic structure of matter were performed by Max von Laue in 1912 [3], earning Laue the 1914 Nobel prize in Physics. Laue scattered x-rays off of a copper sulfate crystal, which helped provide proof of the lattice structure of crystals. The x-rays had wavelengths equivalent to the inter-atomic spacing in many crystal samples and thus diffracted into a regular pattern. Additionally, the periodic structure creates a regular pattern in the I(~q) which is easy to revert into a V (~x). However, X-ray scattering can be taken beyond simple diffraction to look at larger, non-periodic samples. In these cases, the samples often lack orientation, generating a circularly symmetric scattering pattern. While x-ray scattering has a number of advantages, there are some weaknesses as well. First, while hydrogen makes a large part of most organic molecules, it has a very weak x-ray cross-section and is usually invisible in x-ray scattering measurements. Furthermore, the monotonic relationship between scattering cross-section and means that lighter elements like hydrogen and carbon will always be weaker scatters while heavier elements like silicon and aluminum will scatter more strongly. Finally, the x-rays are ionizing radiation and can damage samples.
1.3
NEUTRON SCATTERING
All of these disadvantages can be avoided by using neutrons as the scattering particle. As the neutron is electrically neutral, the scattering is based on a combination of the nuclear force and magnetic moments. The relationship between atomic number and neutron scattering cross-section is far more complicated than the scattering for x-rays. Examples of these scattering cross-sections can be seen in Table 1.1. While hydrogen only scatters x-rays very
3
Atom Scattering Cross-Section (fm2 )
H 8203
D 764
He 134
C 555
N 1151
O 432
Al 150
Si 217
Gd 15100
Table 1.1: Example scattering cross-sections. These values were taken from the Neutron Data Booklet [4]. weakly, it is a very strong scatterer of neutrons. Additionally, since there is a very large difference between the cross-sections of hydrogen and deuterium, it’s possible to tag parts of a sample with deuterium to focus the measurement on an area of interest. Also, since the common components of organic compounds (e.g. carbon, oxygen, nitrogen) all scatter more strongly that aluminum, it’s possible to pass the neutron beam through metallic sample environments while still having the primary scattering signal result from the scattering off of the sample The neutrality of the neutron also provides advantages over the x-ray. While the x-ray will Compton scatter off of the electrons in the sample, the neutron’s only electromagnetic interaction with the electron is the magnetic moment. The weak force, living up to its name, similarly will not provide a significant coupling between the neutron and the electrons. Thus, the scattering will come entirely from interactions with the atomic nuclei in the samples. While x-ray diffraction must use high energy photons to achieve wavelengths small enough to correspond to inter-atomic spacings, the neutron’s heavy rest mass allows it to achieve a similar de Broglie wavelength while having a much smaller kinetic energy. Therefore, while the x-rays are likely to Compton scatter off of the electrons in the sample, possibly destroying chemical bonds and changing the sample, the neutrons are closer to the temperature of the sample and will not risk destroying the sample. The neutron’s rest mass provides a second advantage since the speed of the neutron is proportional to its wavelength, as opposed to the constant speed of all x-rays. It’s possible to calculate the wavelength of a neutron by knowing it’s position at two points in time, while x-rays require direct intervention. This allows for time of flight techniques which can
4
analyze a sample across a broad spectrum of neutrons simultaneously. Just as the neutron’s mass gives an extra degree of information in the time of flight, the neutron’s spin allows for polarized neutron measurements, in which all of the neutrons are chosen to have the same spin. These polarized neutron measurements are sensitive to the magnetic moments of atomic nuclei, allow for the determination of the magnetic structure of materials. This spin sensitive scattering was used by Fert and Grunberg in their Nobel winning work on giant magneto resistance. If the material is non-magnetic, the spin can be used as an extra degree of freedom for another property which is hard to measure conventionally. This was first discovered by Ferenc Mezei in his development of neutron spin echo. In spin echo, a strong magnetic field causes Larmor precession of a polarized neutron beam. After the sample, a second magnetic field will reverse the precession. If the sample has changed its energy, and thereby velocity, during its interaction with the sample, the neutron will spend different amounts of time in the two fields and the precession will not be exactly reversed. By measuring the change in the neutron’s polarization, it’s possible to determine the change in the neutron’s energy with high precision without needing to control the neutron’s actual energy. The main disadvantage of the neutron is its expense relative to the x-ray. X-ray production is simple enough that commercial x-ray diffraction units are available off the shelf [5– 7]. Comparatively, the production of neutrons requires significant infrastructure. The two main sources are nuclear reactors, which usually provide a continuous neutron wave, and accelerator based source, which provide neutrons in pulses. These provide heavy requirements on both engineering and regulatory levels. Furthermore, both systems produce highly energetic neutrons whose wavelengths are far to small to be useful in condensed matter experiments. The neutrons must be moderated down to thermal energies to achieve useful wavelengths. This moderation is often performed by passing the neutron beam through
5
hydrogen rich materials (e.g. water, methane) kept at extremely low temperatures. As the neutron scatters off of the hydrogen in the sample, it passes part of its energy into the moderation material, decreasing the energy of the neutron and increasing its wavelength.
1.4
SESAME
At the simplest level, the Spin Echo Scattering Angle Measurement (SESAME) technique [8] is an apparatus to ensure that the polarization of a neutron, relative to its initial polarization, is proportional to the trajectory angle. Any instrument capable of coding and decoding neutron paths in this matter can perform SESAME. The common solution is a series of inclined magnetic field boundaries, seen in Figure 1.1. The first pair of triangles linearly encodes the position of the neutron into polarization. The second pair decodes the position of the neutron at the second triangle pair’s location. Any change in the neutron’s position with respect to the hypotenuse between the two coils will result in a net polarization encoding through the setup. The final polarization through the two triangles is proportional to the tangent of the scattering angle, which, assuming that the distance between the triangles is large compared to the distance of the neutron’s travel along the hypotenuses (e.g. greater than seven times larger), is equal to the scattering angle itself. Adding two more triangle pairs to the rear end of the instrument removes the original polarization encoding for all unscattered neutrons. Therefore, the final polarization of a neutron is proportional to to the change in the neutron’s trajectory between the front and rear halves of the instrument. The SESAME technique can be explained via two different models. One model assumes classical mechanics and tracks the Larmor precession of the neutron through the instrument. This model is conceptually simple and makes clear the effects of the various magnetic fields in the instrument. Unfortunately, the precession model is far less intuitive at explaining the
6
L y
⊙
B x
z
α
|z›
|-z›
spin echo length
sample
triangular solenoids
Figure 1.1: Layout of the SESAME beam-line. The alternating tints of the triangular regions represent alternative magnetic field direction. The arrows represent a polarized neutron beam, whose wave states are separated spatially by the first Wollaston prism pair and recombined by the final one. nature of SESAME’s length scale and measurement function. The opposing model considers the birefringence of the different neutron polarization states. The spin echo length and the real space measurement on transmission samples immediately fall out of this model, but the practical operation of the instrument is less clear.
1.5
PRECESSING MODEL
According to Ehrenfest’s theorem, the expected evolution of the spin of a polarized neutron is given by.
d dt
1 ~ˆ ˆ ∂ ~ˆ ˆ ~ S = S, H + S ı¯h ∂t
ˆ = Assuming that the neutron’s Hamiltonian is H simplified.
7
pˆ2 2m
(1.2)
~ˆ · B, ~ the expression can be − γS
d dt
#+ + * ˆ2 ˆ2 p p ˆ ˆ ˆ ∂ ~ ·B ~ ~ ·B ~ ~ − γS + ∂t − γS S, 2m 2m *" # + ˆ2 p 1 Sˆi , − Sˆi , γ Sˆi Bi + Sˆj Bj + Sˆk Bk +0 ı¯ h 2m * + h i h i h i 1 0 − γ Sˆi , Sˆi Bi + Sˆi , Sˆj Bj + Sˆi , Sˆk Bk ı¯h E 1 D γ 0 − ı¯hBj Sˆk + ı¯hBk Sˆj ı¯h D E γ −ǫijk Bj Sˆk ˆ ~ ~ γ S×B
~ˆ = S d Sˆi dt
1 ı¯h
= = = =
d dt
~ˆ = S
*"
(1.3)
The result in Equation 1.3 is the classical expression of Larmor precession. This indicates that a purely classical model of a precessing neutron can safely be used in studying this phenomenon. The differential equation in Equation 1.3 can be greatly simplified if we assume that the neutron’s spin is always perpendicular to the magnetic field.
~ B
≡B~k
~ S
¯ h ¯h ≡ sin Θ~i + cos Θ~j 2 2 ˆ ~ ×B ~ =γ S
ˆ d ~ dt S
γ¯hB 2 ¯h dΘ h ¯ γ¯ h B cos Θ~i − dΘ sin Θ~j= dt dt 2 2 2 =
(1.4)
− sin Θ~j + cos Θ~i − sin Θ~j + cos Θ~i
dΘ dt
=γB
Θ
=γBt + C
(1.5)
The variable Θ used in Equation 1.4 to denote a perpendicular direction for the spin
8
B+
b a c BFigure 1.2: The position through which a neutron passes through a solenoid pair controls the precession angle of the neutron. Neutron a travels an equal distance through prisms 1 and 2 and will develop no net precession angle. Neutron b travels more through prism 1 and will develop a net positive angle. Neutron c travels more through prism 2 and will develop a net negative angle. vector can be solved analytically, as seen in Equation 1.5. Θ is called the precession angle, as it is a measure of how far the neutron’s polarization has precessed within the field. If a neutron passes through a field of length L with wavelength λ, the final precession angle is Θ=
γBLmλ , h
where m is the neutron mass and h is Planck’s constant.
Under this model, each triangle pair on the SESAME instrument encodes the position at which the neutron passed through the triangle pair into the precession angle Θ. Passing through at different points on the hypotenuse changes the path length spent in each triangle, as seen in Figure 1.2. This forms the linear position encoding needed for the SESAME technique. Using the coordinate system in Figure 1.3, a straight through neutron will be encoded as: γmλDB Θ= h
2y −1 D tan θ
If two prism pairs are separated by a distance L and the neutron passes through at an angle φ, the total precession angle will be
9
D B1
θ
y
B2 Figure 1.3: A neutron passing through a single triangle pair. We define the length of the prism as D, the magnetic fields as B1 and B2 , the angle of the hypotenuse face as θ, and the displacement of the neutron from the edge of the prism as y.
γmλDB Θ = h
γmλDB 2(y0 + L tan φ) 2(y0 ) −1 − −1 D tan θ h D tan θ
2γm BLλ cot θ tan φ h 2γm ∆Θ= − BLλ cot θ tan (φ + ∆φ) − tan φ h 2γm BLλ cot θ∆φ ≈− h =−
(1.6)
Neutron scattering normally isn’t measured in terms of scattering angle, but rather the q vector, where q =
2π sin ∆φ λ
≈
2π∆φ λ .
∆Θ= −
Plugging this back into equation 1.6 gives:
γm BLλ2 cot θq hπ
= 1.47 T−1 m−2 BLλ2 cot θ q
(1.7)
The coefficient of q in equation 1.7 is called the “Spin Echo Length” and is normally assigned to the variable Z. The importance of this value become clear when we attempt to find the expected polarization of a scattered neutron beam.
10
Assume that a sample scatters isotropically with the probability of scattering at q given by I(q) such that
H
I(q) = 1. The average neutron polarization is then:
P (Z)= = =
ZZZ
ZZZ
Z
I(q) cos (Zqz ) dqx dqy dqz
(1.8)
I(q) cos (Zq cos θ) q 2 sin θ dφ dθ dq
I(q)J0 (qZ)q dq
(1.9)
Equation 1.9 shows that the polarization at spin echo length Z is the Abel transform of I(q). This transformation will be discussed more in Section 1.7.
1.6
BIREFRINGENT MODEL
ˆ = Again, we’re going to start with the Hamiltonian of H
pˆ2 2m
~ˆ · B. ~ When undergoing a − γS
reversal of the magnetic field direction, we find: 2 p21 ~ˆ · B ~ = p2 − γ S ~ˆ · −B ~ − γS 2m 2m p21 − p22 ~ˆ · B ~ =2γ S 2m ~ˆ · B ~ =4mγ S p21 − p22
~ˆ · B ~ p21 − (p1 + δp)2 =4mγ S − 2p1 δp
~ˆ · B ~ ≈4mγ S
− 2p1 mδv
~ˆ · B ~ =4mγ S
δv
=−2
γ ~ˆ ~ S·B p1
We can then plug equation 1.10 into the formula for index of refraction:
11
(1.10)
φ
θ Figure 1.4: A diagram of the relevant angles for neutron refraction at a magnetic field boundary.
v2 v1
=
sin φ2 sin φ1 sin φ1 cos ∆φ + cos φ1 sin δφ sin φ1
1+
δv v1
=
1+
δv v1
= cos ∆φ + cot φ1 sin ∆φ
δv v1
≈∆φ cot φ1
~ˆ · B ~ −2γmS p21 ~ˆ · Bλ ~ 2 −2γmS h2 ~ˆ · Bλ ~ 2 −2γmS h2
=∆φ cot φ1 =∆φ cot φ1 tan φ1 =∆φ
(1.11)
~ is parallel to B, ~ Looking at Figure 1.4 gives that tan φ = − cot θ. If we assume that S ~ ·B ~ = then S
h 4π B.
Therefore, the total displacement over a distance L is:
12
y=L∆φ h Bλ2 2γ 4π cot θ h2 γm = BLλ2 cot θ 2hπ
=L
(1.12)
~ the displaceIf a second neutron beam passes through whose spin is anti-parallel to B, ment will be of equal magnitude, but in the opposite direction. Thus, the total distance between the two beams will be
γm 2 hπ BLλ cot θ,
which is our spin echo length from equa-
tion 1.7. A second set of solenoids with their fields in the opposite direction to the first pair will take the two diverging beams and refract them to be parallel again. We now have two parallel beams, separated by the spin echo length. The remaining two pairs of solenoids after the sample then recombine these parallel beams back to the original beam. To create the initial two parallel and anti-parallel beams, a polarized, we start with a single polarized beam.. This is then passed through a of the
π 2
π 2
flipper. The perpendicular fields
serves a similar purpose to the Stern-Gerlach apparatus. The single polarization
state |⊙i is combined into a linear superposition |↑i + |↓i with a known phase relation. Any phase difference accrued between the two spin states across the spin echo length difference will change the quantum phase between the states and prevent the pure |⊙i state from being recombined at the final
1.7
π 2.
PATTERSON FUNCTION
The Platonic ideal of neutron instruments would return a measurement of scattering length density versus position, ρ(~r)[9]. Unfortunately, practical matters, such as the scale of Avo-
13
gadro’s constant, interfere with such a direct measurement. The next best option would be to measure the auto-correlation in real-space, γ(~r). This overcomes the limitations of tracking each atom in the sample while still giving the important structural information. However, there is, to date, no direct method of making this measurement. Instead, all ~ which is the Fourier transform of the γ(~r), as shown scattering techniques measure I(Q), in Equation 1.1. ~ For an ideal instrument, it would be possible to take I(Q)and perform an inverse Fourier transform on the data to return to γ(~r), but real instruments have finite resolution which can prevent this from being an effective technique. Taking the inverse Fourier transform ~ Any neutrons which scatter at an requires that the data can be integrated over all Q. extreme enough angle to miss the detector will be missing from the analysis and will prevent the correct reconstruction of γ(~r). In the other direction, the use of beam stops in SANS to block the unscattered beam will set a lower limit on the measurable scattering and again prevent the reconstruction of γ(~r), especially for large samples where significant information ~ Additionally, if the background is not uniform across all of the Q ~ may be stored at low Q. measurements, the transform will still not work. If a SANS measurement was performed without a beam-stop, the inability to separate the scattered and unscattered beam at low Q will prevent the resolution of large scale structures in γ(~r). Finally, even if the the unscattered beam could be removed from the system, the transform of γ(~r)will still be limited by the resolution of the measurement of Q. No structures larger than the inverse of the smallest Q resolution can be resolved. ~ As opposed to measuring I(Q)and taking a Fourier transform of the data, SESAME takes a Fourier transform directly on the scattered neutrons, as seen in Equation 1.8. This bypasses all of the issues with detector background and resolution, giving the real space measurement for all length scales. Unfortunately, the SESAME triangles can only encode
14
~ in one direction, so, while the I(Q)is a three dimensional Fourier transform of γ(~r), the SESAME measurement only returns the inverse transform in one dimension.
G(Z)= = ≈ = = =
Z
s(~q)eiZ·qz d~q
Z
s(qx , qy , qz )eiZqz d~q
Z
Z
Z
Z
G(Z)=π
s(0, qy , qz )eiZqz dqy dqz
0
∞ Z 2π
(1.13)
s(0, q, θ)eiZq cos θ q dθ dq
0
∞
s(q)q dq 0 ∞
2π
eiZq cos θ dθ
0
s(q)q dq ∗ πJ0 (qZ)
0
Z
Z
∞
s(q)J0 (qZ)q dq
(1.14)
0
In Equation 1.13, we take the small angle scattering approximation and assume that there’s no change in the neutron’s momentum along the beamline. As seen in Equation 1.14, taking a one dimensional Fourier transform of a spherically symmetric function returns the Hankel transform of the radial function. The Hankel transform of a Fourier transform is the Abel transform, which is the projection of a radially symmetric function onto a single Cartesian axis. Thus, the SESAME technique measures the projection of γ(~r)onto a single encoding axis. This is the Patterson function, G(Z). To achieve this relation, it’s necessary to take the small angle approximation in Equation 1.13. In the small angle limit, there shouldn’t be any change in the scattering vector along the direction of the neutron beam. An interesting property of the Abel transform is that it magnifies long length scale signals with respect to smaller ones (Figure 1.5). Assume that the radial distribution function γ(r) is the average of two functions γ1 (r) and γ2 (r) with correlation lengths ξ1 ≡ 2
15
R∞ 0
γ1 (r) dr
ΓHrL ΓHrL 1.0
z Γ1 = j0 H
0.8
L 0.3
z Γ2 = j0 H L 3
0.6 0.4
1 Γ= HΓ1 +Γ2 L 2
0.2 r 5
10
15
20
25
30
-0.2
(a)
GHZL GHZL 1.0
z Γ1 = j0 H
0.8
L 0.3
z Γ2 = j0 H L 3
0.6 0.4
1 Γ= HΓ1 +Γ2 L 2
0.2 Z 5
10
15
20
25
30
-0.2
(b)
Figure 1.5: In Figure (a), the solid line is the average of a long range signal (dashed line) and a short range signal (dotted line). In Figure (b), the Abel transform of the two signals is plotted, as well as the Abel transform of the average. While both the long and short range signals are visible in the average in Figure (a), the long range signal clearly dominates the transform of the average in Figure (b)
16
and ξ2 ≡ 2
R∞ 0
γ2 (r) dr, respectively. Trivially, we can define:
ξ≡2 = =
∞
Z
Z
γ(r) dr
0 ∞
γ1 (r) dr + 0
ξ1 + ξ2 2
(1.15) Z
∞
γ1 (r) dr 0
By the definition of an Abel transform [9],
2 G(z)= ξ
Z
∞
γ(r)r
1 dr r2 − z 2 2 Z ∞ Z ∞ γ1 (r)r γ2 (r)r 4 = 1 dr + 1 dr ξ1 + ξ2 2 2 2 2 2 2 z z 2 r −z 2 r −z 4 ξ1 ξ1 = G1 (z) + G1 (z) ξ1 + ξ2 4 4
(1.16)
=
(1.17)
z
ξ1 G1 (z) + ξ2 G2 (z) ξ1 + ξ2
Equation 1.17 shows that, while γ(r) is merely an average of two functions, G(z) is a weighted average of the two functions weighted by their correlation lengths. Therefore, longer range signals will be more visible than short range signals. However, Equation 1.17 assumes that each neutron scatters once and only once. In practice, many neutrons will pass through the sample without any scattering while others will scatter multiple times within the sample. The measured polarization will be given by [10]: P (Z) = eΣ(G(Z)−1)
(1.18)
Where Σ is the total scattering of the sample and P (Z) is the normalized polarization at a given spin echo length. This total scattering is proportional to the total correlation
17
length of the sample. It’s then possible to interpret the results as:
log P (Z)∝ξ G(Z) − 1 ξ1 G1 (z) + ξ2 G2 (z) =ξ −1 ξ1 + ξ2 = ξ1 G1 (z) + ξ2 G2 (z) − ξ1 − ξ2
(1.19)
(1.20)
In the region where Z ≪ ξ2 , G2 (Z) ≈ 1 and the measurement is simply:
log P (Z) ∝ ξ1 (G1 (Z) − 1)
(1.21)
Thus, for measurements much smaller than the length scales in G2 (Z), the results of Equations 1.19 and 1.21 are the same. Thus, the measurement of the short range signal is unchanged by the presence of the long range signal. Similar math will show that the presence of the long range signal merely performs a constant shirt on the short range signal.
1.8
INSTRUMENT TUNING
Tuning the SESAME beamline can often be accomplished exclusively via the adjustment of the currents through the solenoids. First, a single guide field coil should be chosen from the beamline and designated as the “phase coil”. By scaling the current through this coil, the coarse offsets and misalignments of the prism coils on the beamline can be canceled out and spin echo can be achieved for the non-scattered beam. The beam can then fully be brought to echo via fine tuning of the prism coils. I have proved that this tuning is possible and shown, via simulation, that the prism coil fine tuning is not necessary for most experiments.
18
1.8.1
TUNING MODEL
TWO DIMENSIONAL APPROXIMATION To simplify discussion of the prism frames, it would be useful to flatten the three dimensional prism into a two dimensional triangles. However, we need to know that this approximation is valid. The two main structures to model in a prism pair are the neutron itself and the hypotenuse frame. The neutron is assumed to have a starting position l~0 at the front of the instrument and an ending position l~1 at the end (Figure 1.6). The hypotenuse frame has a normal vector ~n and passes through point p~0 . By convention, p~0 is chosen so that it’s centered along zˆ and is at the bottom of the prism in yˆ. Therefore, for a perfectly placed prism pair, p~0 = (0, 0, 0). The place where the neutron intersects the hypotenuse face is given in Equation 1.22: (p~0 − l~0 ) · ~n x~1 = l~0 + (l~1 − l~0 ) · ~n
(1.22)
Two more additional points are needed: The point where the neutron enters the first prism and the point where it leaves the final prism.
x~0 =
(p~0 − l~0 ) · x ˆ ~ ~ l~0 + ( l1 − l0 ) ~ ~ ( l1 − l0 ) · x ˆ
p0x ~ ~ ( l1 − l0 ) l~0 + d (p~0 + ℓˆ x − l~0 ) · x ˆ ~ ~ ( l1 − l0 ) x~2 =l~0 + ~ ~ ( l1 − l0 ) · x ˆ =
=
p0x + ℓ ~ ~ l~0 + ( l1 − l0 ) d
(1.23)
(1.24)
The final field integral is proportional to the distance traveled through each region multiplied by the magnetic field in that region. If the field before the interface has magnitude B1 and the field after the interface has magnitude B2 , the total field integral through the 19
~n
x~0
x~2
x~1
l~0
~l1
p~0
Figure 1.6: Diagram of vectors for the two dimensional approximation. The solid arrow is the trajectory of the neutron. two regions is:
Φ = B1
p p (x~0 − x~1 ) · (x~0 − x~1 ) + B2 (x~1 − x~2 ) · (x~1 − x~2 )
(1.25)
At this point, certain approximations can be made. First, the hypotenuse should only be angled along the encoding direction, so we can declare ~n = (0, cos θ, − sin θ) . Additionally, the neutron’s endpoint can be redefined in terms of the initial position, two divergence angles (α and β), and the length of the instrument, d.
l~0 =
(0, y0 , z0 )
(1.26)
l~1 =(d, y0 + tan β, z0 + d tan α) .
(1.27)
By combining Equations 1.26, 1.27, 1.22, 1.23, and 1.24, we can find the final field integral: 20
Φ=
s
cos[θ]2 sec(β)2 + tan(α)2 (−py + y0 + px tan(β))2 B1 (sin(θ) − cos(θ) tan(β))2 s sec(β)2 + tan(α)2 (ℓ sin(θ) + cos(θ)(py − y0 − (px + ℓ) tan(β)))2 +B2 (sin(θ) − cos(θ) tan(β))2
(1.28)
Equation 1.28 is completely independent of pz , so translating along the zˆ axis causes no changes in the resulting calculation. The divergence in that zˆ direction, however, is still present and must be accounted for. At this point, there’s a couple of logical assumptions to make. First, θ must be between 0 and π2 . Since the neutron must pass through both faces of the prism pair, y0 −py +pz tan β > 0, −ℓ tan θ < py − y0 − (pz + ℓ) tan(β) < 0. For the neutron to pass through the prism pair, the divergence must limited to being less than the angle of the hypotenuse face:
θ>
β
tan θ >
tan β
sin θ >cos θ tan β sin θ − cos θ tan β > Finally, since − π2 < β
0 and sec β > 0. Using these assumptions, equa-
tions 1.28 can be reduced to:
Φ=
cos(β) csc(β − θ)
p sec(β)2 + tan(α)2
∗(B2ℓ sin(θ) + cos(θ)((B1 − B2)(py − y0 0) + (−B1pz + B2(px + ℓ)) tan(β)))
21
(1.29)
This can then be taken as a Taylor series in α, the divergence angle in the zˆ direction.
Φ≈ csc(β − θ) ∗ −B2ℓ sin(θ) + cos(θ)
∗ (B1 − B2)(py − y0 ) + (−B1px + B2(px + ℓ))tan(β) 1 2 2 4 ∗ 1 + cos(β) α + O(α) 2
(1.30)
As can be seen in equation 1.30, the total field integral’s dependence on the divergence in the non-coding direction is mainly to second order. Since the divergence angle for a normal beamline is on the order of 0.01, the contribution of the divergence in the non-encoding direction is negligible and can be safely ignored.
EXISTENCE PROOF In the Platonic ideal of SESAME instruments, the instrument consists of four objects that linearly encode the neutron’s yˆ position into the polarization. Two encoders before the sample encode the neutrons trajectory through the change in the yˆ position. A second set of encoders after the sample decode the trajectory after scattering. While this is normally performed via magnetic fields and spin precession, any device which meets these requirements will perform SESAME. Thus, using this linear model, the total encoding acquired by the neutrons through each encoder can be expressed as:
Φi (y) = mi y + bi
(1.31)
Over the course of the instrument, for a neutron with divergence angle φ and and encoders with positions position (xi , yi ), the total phase acquired can be seen in equation 1.32.
22
Φ=Φ1 (y1 ) + Φ2 (y2 ) + Φ3 (y3 ) + Φ4 (y4 ) =Φ1 (y0 + x1 tan φ) + Φ2 (y0 + x2 tan φ) + Φ3 (y0 + x3 tan φ) + Φ4 (y0 + x4 tan φ) =b1 + b2 + b3 + b4
(1.32)
+ y0 (m1 + m2 + m3 + m4 ) + (m1 x1 + m2 x2 + m3 x3 + m4 x4 ) tan φ
Since the total acquired phase for all neutrons should be zero in the echo condition, the constant terms must cancel out, as well as the coefficients of y0 and tan φ must be made to cancel to zero. There are two tools available to achieve this cancellation. First, a phase coil can be added to the end of the beamline. The phase coil is a solenoid with a constant magnetic field which is used to add a fixed number of precessions to all neutrons that pass through the beamline. In practice, this is usually achieved by scaling one of the existing solenoids on the beamline instead of adding a whole new magnet. This is equivalent to adding another constant term to equation 1.32. The other option is to scale the current through a pair of prism solenoids. This will magnify the entire Φi for that encoding unit. Since there are three terms, we’ll define three parameters. The first, p, is the phase added by the phase coil. Scaling the current in prism pairs 3 and 4, and thereby the field integrals Φ3 and Φ4 , will be parameter α. Finally, scaling prism pair 4 on its own will be parameter β. From here, equation 1.32 becomes:
23
Φ=p + Φ1 (y1 ) + Φ2 (y2 ) + αΦ3 (y3 ) + αβΦ4 (y4 ) =p + b1 + b2 + αb3 + αβb4
(1.33)
+ y0 (m1 + m2 + αm3 + αβm4 ) + (m1 x1 + m2 x2 + αm3 x3 + αβm4 x4 ) tan φ
To tune the whole instrument so that the phase is always zero in equations 1.33, p, α, and β need to be set to:
p=
α=
b4 m1 (−x1 + x3 ) + m2 (−x2 + x3 ) m4 (x3 − x4 ) 1 + b3 m1 (x1 − x4 ) + m2 (x2 − x4 ) − (b1 + b2 ) m3 (x3 − x4 )
m1 (−x1 + x4 ) + m2 (−x2 + x4 ) m3 (x3 − x4 )
m3 m1 (x1 − x3 ) + m2 (x2 − x3 ) β= − m4 m1 (x1 − x4 ) + m2 (x2 − x4 )
Unfortunately, while these forumlae show that it’s possible to perfectly tune the instrument, they are not useful for tuning an instrument in practice. The formulas only work if the exact mi and bi values are known. In a perfect instrument, p = 0 and α = β = 1, so tuning is trivial. However, the very uncertainties in the placements of the triangles or the magnetic field strengths that require the instrument to be tuned mean that the values needed to solve the tuning formulas cannot be solved directly. Additionally, once a single tuning parameter in equation 1.33 has been set to maximize the polarization through the beam, any change in the other parameters will require changing the set parameter to keep 24
the instrument tuned. Thus, short of performing a time consuming global search of all three parameters, the best option available for tuning is to tune the most sensitive parameter and leave the other two parameters where they lie. Using the model of a single triangle in Figure 1.3 and ignoring the minor increase in path length for divergent paths, the field integral through a single triangle pair is:
Φ = (2y tan θ − L)B
(1.34)
In relating equation 1.34 to equation 1.31, m = 2B tan θ and b = −2LB. However, there are several uncertainties that need to be accounted for. If the two triangles don’t have the exact same number of turns, there will be uncertainties on the fields δB. Any rotation of the triangles away from the correct alignment will create an uncertainty δθ. Finally, if the triangle has been translated vertically, the y coordinate will have an offset δy. Thus, equation 1.34 becomes:
Φ=(y + δy) tan(θ + δθ)(B + δB1 ) + (L − (y + δy) tan(θ + δθ))(−B + δB2 )
With the uncertainties included, m = (2B + δB1 − δB2 ) tan(θ + δθ) and b = −L(B − δB2 ) + δy(2B + δB1 − δB2 ) tan(θ + δθ). These values can then be plugged into the tuning parameters to find the uncertainties on the tuning parameters.
1.8.2
TUNING SIMULATIONS
To show that it is possible to tune an instrument using just the phase coil, a Monte-Carlo simulation was written in Python. The simulation worked in two dimensions: the beam direction and the encoding direction. The transverse dimensions, due to its small contri-
25
Seed 1 Seed 2
1.000 0.998 0.996 0.994 0.992
Polarization
0.990 0.988 0.986 0.984 0.982 0.980 0.978 0.976 0.974 0.972 0.970 1
10
100
1000
10000
100000
1000000
Neutron Count Figure 1.7: The polarization on two simulated beamlines versus the number of neutron paths simulated.
26
Seed 1 Seed 2
0.01
Difference
1E-3
1E-4
1E-5 1
10
100
1000
10000
100000 1000000
Neutron Count Figure 1.8: The change in the simulated polarization of the beamline versus the number of neutron paths simulated. This plot is equivalent to the difference between the points in Figure 1.7
27
bution, was ignored for computational simplicity. For each simulated beamline, a series of evenly distributed points were selected at both the front and back of the beamline. Using the precessing model from Section 1.5, the total precession angle of each neutron trajectory was calculated. These precession angles were converted into polarization measurements and then an average polarization would be taken for the entire beam. To ensure that the results of the simulation are stable, each virtual instrument is run iteratively. Each iteration doubles the number of points at both the front and rear ends of the instrument, quadrupling the total number of neutron paths. If the average measured polarization deviates from the previous simulation by more that 0.1%, another iteration is performed, again quadrupling the number of neutron trajectories. The polarization returned by this process can be seen in Figure 1.7. Figure 1.8 shows the change in polarization between the consecutive data points in Figure 1.7. The two different instruments both can both be seen to converge as the number of neutron paths increases. One interesting effect visible in both data sets is that the polarization monotonically improves with the neutron count. This is to be expected as an artifact of the method chosen to select neutron paths. The range over which the neutron paths are chosen is constant, meaning that every simulation always includes the most extreme positions and divergence angles. As more neutrons are added to the simulations, they fill in the paths between the extrema and thus usually have a smaller depolarization. Therefore, as the number of neutron paths increases, the extreme paths weigh less heavily on the average and the overall polarization increases. Figure 1.9 shows the number of neutron points needed at each end of the instrument, log base two, for the simulation to converge to within 0.005% polarization. The most important feature of the graph is the data point at 12, corresponding to 4096 points at both the front and back of the instrument, for a total of over sixteen million neutron paths. Given that most instruments needed less than 512 points, or roughly 250,000 neutrons, running sixteen
28
Occurances 140 120
Occurances
100 80 60 40 20 0 0
5
10
Indices bits Figure 1.9: A histogram of bits needed to represent the number of neutron channels necessary for a 0.005% uncertainty on the simulation. For instance, 8 bits corresponds to 256 channels and a total of 65,536 neutron paths. The error bars come from counting statistics. The curve roughly follows a Gaussian.
29
million for each instrument is clearly overkill. However, it should also be noticed that the graph is roughly Gaussian. By fitting the data to a standard normal distribution, it can be estimated that two percent of simulated beams will need 4096 or more endpoints on each side of the beamline to achieve the desired precision. Using the iterative approach to calculating neutron paths for all instrument setups, as opposed to a fixed number of neutrons for every measurement, ensured that all results could be given to the desired precision. Measuring a single instrument doesn’t necessarily provide representative data of all instrument configurations, so it’s necessary to generate a series of instruments and tune each individually. Once the series is complete, the average polarization of all the instruments can be found. An identical number of new instruments are then simulated and added into the original set. If the change in average polarization is below the cutoff threshold of 0.25% polarization, then the average and minimum neutron polarizations have been found. Otherwise, the number of instruments is again doubled and another iteration is performed. During the simulations, the beamline is tuned purely by adding a phase coil term to the simulated beam. To perform the tuning, an initial set of simulated neutrons are sent through the beamline and their final phase angles measured. The average of these phase angles is measured and subtracted from the beam as a simulation of the tuning coil. The cosine of the new phase angles are then averaged to find the final polarization of the beamline. In actual operation on a real beamline, it wouldn’t be possible to measure this phase angle directly and multiple measurements would be needed to find the correct phase angle, but this was skipped in the simulations to save procesing time. In Figures 1.10 and 1.11, the average tuned polarization has been plotted compared to the uncertainty in the magnetic field and solenoid positions, respectively. The full relation between the average polarization after tuning and the uncertainties in the solenoid position and field intensities is plotted in Figure 1.12. As expected, increases in the uncertainty
30
Effect of Magnetic Field Uncertainty on Average Tuned Polarization No Position 1mm 3mm 4.8mm
1
Polarization
0.95 0.9
Error Error Error Error
0.85 0.8 0.75 0.7 0.65 0.6
0 0.5 1 1.5 2 2.5 3 Percentage Error in Magnetic Field
Figure 1.10: The average polarization achievable after tuning a beamline compared against the maximum percentage error in the magnetic fields for the individual solenoids. The hollow squares were simulated with all of the Wollaston prism pairs placed correctly while the solid squares, hollow circles, and solid circles represent simulations with the Wollaston prisms randomly offset by a maximum of 1 mm, 3 mm, and 4.8 mm, respectively.
Effect of Position Uncertainty on Average Tuned Polarization No 1% 2% 3%
1
Polarization
0.95
Field Field Field Field
Error Error Error Error
0.9 0.85 0.8 0.75 0.7 0.65 0.6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Prism Offset (mm)
Figure 1.11: The average polarization achievable on after tuning a beamline compared against the maximum error in the placement of the Wollaston prisms. The hollow squares were all simulated with the correct fields running through each solenoid pair. The solid squares, hollow circles, and solid circles had random field errors of 1%, 2%, and 3%, respectively.
31
Percentage Field Uncertainty
Average Polarization 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6
3 2.5 2 1.5 1 0.5 0 0
0.99 0.95 0.9 0.85 0.8 0.75 0.7 0.65
0.1 0.2 0.3 0.4 Position Uncertainty (cm)
Figure 1.12: A heat map of the average achiveable tune polarization through with a given uncertainty in the position of the solenoids (X axis) and a given relative uncertainty in the strength of the magnetic fields in the solendoids (Y axis). Contour lines have been added to highlight relevant polarizations. decrease polarization. The system is far more sensitive to offsets in the magnetic fields than it is to the position of the triangle pairs. A 1 mm uncertainty in the positions of the solenoids has a minor effect on the ability to tune the beamline, but a 1% change in the magnetic fields in the solenoid drops the polarization by 4% even in the case where the solenoids are aligned perfectly. While Figures 1.10, 1.11, and 1.12 present the average ploarization achievable through tuning, it’s possible for the instrument enter a pathalogical state where the polarization is significantly lower. In Figure 1.13, these pathological cases have also been plotted against the magnetic field uncertainty and position uncertainty. The worst case polarizations drop off far more quickly with uncertainty than the average cases. While a position uncertainty of 4.8 mm and a field uncertainty of 3% leads to an average polarization of 65%, it can lead to a completely depolarized beam in the worst case. Still, keeping the uncertainty on the position to 1 mm and the field uncertainty to 0.4%, assures that we should have at least 95% beam polarization after tuning and will most likely have 99% polarization. These
32
Percentage Field Uncertainty
Worst Case Polarization 1
3 2.5
0.8
2
0.6
1.5
0.4
1
0.2
0.95 0.9 0.7 0.5 0.3 0.1 -0.1
0
0.5
-0.2
0 0
0.1 0.2 0.3 0.4 Position Uncertainty (cm)
Figure 1.13: A heat map of the worst case polarization after tuning the beamline with a given uncertainty in the position of the solenoids (X axis) and a given relative uncertainty in the strength of the magnetic fields in the solendoids (Y axis). These polarizations were the worst cases after enough beamlines had been simulated to converge the average to within 0.25%. Contour lines have been added to highlight relevant polarizations. simulations were performed with 10 A through the solenoids, so the 1% uncertainty in field strength corresponds to a current that must remain stable within 40 mA.
33
CHAPTER 2
POLARIZED BEAM-LINE CONSTRUCTION
To further the study of the SESAME technique, a spin echo beamline has been built at the Low Energy Neutron Source (LENS) at Indiana University. The instrument itself has been named SESAME. Several important lessons on the design and construction of SESAME beamlines have been learned as this device has been built.
2.1
LAYOUT
All of the spin-echo devices on the SESAME beamline are mounted on one of two Newport x48 optical rails. These rails allow the spin-echo devices on the beamline to be easily and reproducibly dismounted and remounted. The samples are mounted on an independent sample stage between the two rails. The upstream rail, known as the Primary Flight Path (PFP), is mounted on two jacks built by Advanced Design Consulting USA, Inc (ADC) [11]. The jacks are spaced a meter apart and each have a range of 10 cm with a precision of 5 µm. This allows the entire upstream rail to be tilted by up to 5.7◦ allowing the instrument to be used as a reflectometer for horizontal samples (Figure 2.2a).
34
I
Y Z
X
C F ED E G C D
J
D EF E
F
B H A
A
Figure 2.1: Schematic Layout of SESAME instrument. Major beamline components have been labeled: A - Jacks to tilt the incident beamline; B Upstream optic rail; C Slits; D Flippers; E - Wollaston Prism Solenoid; F Guide Fields; G Sample Goniometer; H Tiltable Optic Table; I Polarization Analyzer; J Detector The downstream rail, along with the neutron analyzer and detector, are mounted on an adjustable optic table built by ADC. The table rests on three jacks, each with a range of 10 cm and a precision of 5 µm. The jacks provide vertical translation as well as rotation around the x ˆ and zˆ axes. The zˆ rotation will accommodate reflectometry angles of up to 5.4◦ . Three more degrees of freedom are achieved through three motorized sliders. There are two sliders in the zˆ direction and one in the x ˆ, each with a ±5 cm range and 4 µm precision. Should a larger rotation angle around the yˆ axis be needed (e.g. diffraction), the entire table is mounted on wheels and can be freely moved. A nearby wall limits the final rotation angle to 90◦ . A motorized sample mount with six degrees of freedom is located between the primary and secondary flight paths (Figure 2.2c. The sample mount rests on two Newport x95 rails and can be positioned between 168 cm and 180 cm after the exit of the neutron guides (discussed in section 2.2.1). The base of the stage offers a Huber [12] rotation stage with
35
X
B
Z
A
Y
(a)
(b)
(c)
Figure 2.2: Figure (a) is a photo of the primary flight path of the SESAME beamline. The jacks are marked A and B is the optical rail where the spin echo coils are mounted. Figure (b) is a photo of the second flight path of the SESAME instrument. The three yˆ jacks are labeled, as well as one each in the x ˆ and zˆ directions. A second zˆ slider on a rear jack cannot be seen in the photo due to being blocked by the table. Figure (c) is a photo of the motorized sample stage on the SESAME beamline.
36
a full 360◦ rotation range and a precision of 0.005◦ . Atop the rotation stage rests a two dimensional translation stage in the x ˆ and zˆ directions with a range of ±10 cm and a precision of 15 µm along each dimension. Adjustments in the yˆ direction are controlled by a vertical jack on top of the two dimensional stage that possesses a 10 cm range and 1 µm precision. Finally, the vertical stage is topped with a 2-axis Huber goniometer with a range of ±10◦ and a precision of 0.001◦ . The primary beam profile is set by two motorized slits built by ADC. Each slits consists of four boron carbide blades on individual screw shafts to control the beams size and position in the yˆ and zˆ directions. The slits have a maximum aperture of 60 mm in both the yˆ and zˆ directions and can be positioned with 3 µm precision. Unfortunately, these slits do not have limit switches, so setting software limits is critical. Driving the blades past their physical ends can dislodge and warp the threaded rods that the slits are driven on. The motors for the beamline are controlled by a set of three Parker 6k8 motion controllers. Each motor communicates through a central motor control computer via an RS232 cable with a null modem adapter. The motor computer runs motor control software written by the Spallation Neutron Source (SNS). The software reads a series of XML configuration files that define the individual motors, as well as their corresponding encoders, limit switches, and motor controllers.
2.2 2.2.1
POLARIZATION NEUTRON GUIDES
The neutrons produces by the LENS target are transported to the SESAME instrument by segments of neutron guides (Figure 2.3). The guides are aligned at a 1.5◦ angle around the zˆ axis to eliminate the line of sight between the source and the instrument. Removing the line of sight prevents the high energy neutrons from reaching the detector, since the 37
E A
B
D C
Figure 2.3: A schematic drawing of the bend in the neutron guides at the start of the SESAME beamline. “A” is the neutron moderator. “B” are all neutron guides. “C” is the shutter, which can be raised to block the beam while removing one of the guides and the polarizing bender. “D” is the polarizing bender. “E” is the vault shielding wall. high energy particles from the flash are too energetic to be reflected along the guides and are thus pass through the supermirrors into the walls of the target vault. Each guide has a rectangular cross section which is 25mm in the yˆ direction and 75mm in the zˆ direction. The 25mm sides are coated with Ni/Ti super-mirrors whose critical angle is three times that of nickle, while the 75mm sides have super-mirror with only twice nickel’s critical angle. Each neutron guide segment after the 1.5◦ kink is magnetized to 300G with ceramic magnets. To confirm that the neutron guides did not depolarize the beam, the spin flip reflectivity of the guides were measured on the POSY II instrument [13] at the former Intense Pulsed Neutron Source (IPNS). The guides were tested with longitudinal fields of 16.4 Gauss, 44 Gauss, and 300 Gauss. This longitudinal field was provided by a custom built solenoid I wound from water-cooled conductor. The solenoid would produce a 300 Gauss field at 70A, which was the maximum current that could be reached with the available power supplies. The guides were additionally tested with a 300 Gauss perpendicular field 38
Polarized Reflectivity on m=3 supermirrors Spin Flip Reflections at 0.35 Degrees Non-Spin Flip at 0.35 Degrees Spin Flip Reflections at 1.00 Degrees Non-Spin Flip at 1.00 Degrees
Reflectivity
1
0.1
0.01
1E-3
1E-4 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
-1
Qz (Å )
Figure 2.4: Polarized neutron reflectivity for the walls of the Ni/Ti-coated neutron guide1. A 300G field was applied perpendicular to the guide wall. produced by placing large ceramic magnets on the outside edges of the guides. The guides were measured at incident angles of 0.35◦ and 1◦ for each magnet configuration over a wavelength range from 2 ˚ A to 13 ˚ A. As can be seen in Figure 2.4, the ceramic magnets were found to maintain the polarization of the neutron beam to within 2% after reflection. As this spin transport was both superior to the one provided by the longitudinal field and far simpler from an engineering standpoint to implement, the ceramic magnets were used on the final beamline.
2.2.2
BENDER
Due to the kink between the neutron guides, it’s necessary to use a neutron bender to redirect the beam along the new guide direction. Since the bender was already necessary, it was decided to make it serve a dual purpose as a polarizer as well. The bender is constructed from 170 curved silicon wafers. Each wafer has a thickness of 150 µm and is coated by a super-mirror coating. Between each wafer is a 100 nm thick Gd absorbing layer. To ensure that the neutron is reflected at least once by the super-mirrors in the bender, the length of the bender must be chosen so that the radius of curvature prevents any direct view paths (Figure 2.5). Since there are no direct paths, each neutron must be reflected
39
Figure 2.5: A schematic view of the polarizing bender. R is the radius of curvature, 2β is the bending angle, L1 is the length of the straight line path, and d is the thickness of the silicon channels.
40
Figure 2.6: Simulated spin dependent reflectivities for m = 3 super-mirrors made of Fe/Si (Circles), Fe89 Co11 /Si (X), and Fe50 Co50 /Si (Triangles). The spin up plot is given for only the Fe/Si super-mirror, but all three mirrors are nearly identical. off of at least one super-mirror to pass through the bender. The super-mirrors have a spin-dependent reflectivity (Figure 2.6), so only neutrons with the desired spin state will be bent into the secondary neutron guide. The neutrons in the undesired spin state will pass through the super-mirrors and be absorbed by the Gadolinium layers. Two neighboring channels of the bender can be described by the equations:
x21 + y12 =R2 x22 + (y2 + d)2 =R2
For a straight neutron path of length of L1 through the (x1 , y2 ) circle, the equation is given by: y=
r
R2 − (
L1 2 ) 2
(2.1)
The intersection of equation 2.1 with the (x2 , y2 ) circle is:
L1 =
p √ 8Rd − 4d2 ≈ 8Rd
41
(2.2)
For practical benders, the value of d will be on the order of micrometers while the radius will be on the order of meters, so the the approximation in equation 2.2 holds. Therefore, as long as the bender is longer that
√
8Rd, the neutrons must reflect off of at least one of
the super-mirrors. The reflection angle can then be found from basic trigonometry to be:
L1 sin β ≈ β = = 2R
r
2d R
(2.3)
Again, since L1 will be on the order of centimeters and R will be on the order of meters, the approximate in equations 2.3 holds. Knowing the bend angle of the bender and the critical angle γc of the super-mirrors, it is possible to to find the wavelength cut-off for transport through the bender.
β = λc = γc
s
2d 2 Rm2 γN i
(2.4)
In equation 2.4, γN i is the critical angle for Nickel, 0.0017 rad ˚ A−1 [4], and m is a scaling factor beyond the Nickel cutoff for the super-mirrors in the bender. Simulations of the bender [14] showed that, while an m = 3 super-mirror on the concave face significantly improves the polarization of the neutron beam, the convex mirror has a much smaller effect on the final polarization. This follows intuitively from the fact that every neutron that reflects off the convex face must also reflect off of the concave face to escape the bender. Thus, the polarizing ability of the concave face will be experienced by every neutron that escapes the bender and its efficiency is the primary concern. To minimize the spin down reflectivity, Fe89 Co11 was chosen as the material for the concave faces. Fe89 Co11 has a scattering length density equal to that of silicon, minimizing the reflectivity. For the convex faces, since the effect of the spin down reflectivity is less significant, m = 2 Fe/Si
42
Figure 2.7: VITESS simulations with different cut-off wavelengths and super-mirror reflectivities for a 0.7◦ rotation of the bender. The dashed line is based on a simulation with the no changes in the reflectivity and with an assumed 4 ˚ A cutoff. For the solid line, the reflectivity was set to 85% below m=3 and to the simulated multi-layer reflectivity for m¿3. The dotted line also assumed the same 85% reflectivity, with the wavelength cut-off at 4.5 ˚ A. The hollow circles are the actual transmission measurements taken on the LENS-SANS beamline. super-mirrors were used to save on cost. The bender did not behave as expected from the simulations. To investigate the differences, we started with a measurement of the transmission through the bender on the SANS beamline at LENS and on the SESAME beamline. The full details of these measurements were detailed in [14]. Figure 2.7 shows an example of the deviations we observed. Due to a numerical during the ordering of the bender, the cut-off wavelength did not match the initial simulations. The cut-off wavelength was closer to the 4.5 ˚ A than the originally specified 4˚ A. Additionally, the transmission decreases with wavelength. This drop in transmission could be modeled by assuming that the reflectivity of the super-mirrors within the bender was only 85% below the m = 3 reflection edge. This drop in reflectivity could be cause by diffuse scattering from the super-mirrors, but this diffuse scattering has not been proven. The bender, along with a portion of the upstream neutron guide, is mounted atop a concrete beam shutter. This mounting removes the bender and guide from the beam whenever the shutter is raised into the neutron beam. A kinematic mount on the shutter
43
The integrated neutron flux from 3Å to 12Å is 810 Neutrons/s/cm²/kW
Neutrons/s/cm²/kW/Å
250
200
150
100
50
00
2
4
6
8
10
12
Wavelength (Å)
Figure 2.8: The flux of the SESAME beamline at the sample position assures that the bender and guide have a reproducible alignment once the shutter is lowered out of the beam. Atop the shutter, the bender is mounted within a goniometer. The goniometer rotates the bender around the zˆ axis. The rotation of the goniometer has a precision of 0.05◦ . The bender was aligned by placing a 3 He pencil detector at the exit of the final guide and rotating the bender to maximize the neutron count rate. This 3 He detector was also used to measure the beamline’s flux at the sample position, as seen in in Figure 2.8. The total integrated flux over the 3 ˚ A to 12 ˚ A is 810 Neutrons where s cm2 kW the power is the proton accelerator’s power output. The spectrum exhibits a distinct dip just after 5 ˚ A. This dip arises from the silicon Bragg edge from the sheets of silicon used to make the bender channels. The sheets were all placed into the bender in the same orientation, effectively creating a single crystal and generating this Bragg reflection. Future solid state benders for pulse neutron sources should ensure that the silicon wafers are not aligned to avoid this problem.
2.2.3
WIDE ANGLE ANALYZER
The SESAME beamline has two different options for neutron polarization analysis. The most conventional is a wide angle analyzer of polarizing neutron super-mirrors, seen in
44
Figure 2.9: A sketch of the wide angle analyzer. Figure 2.9. The wide angle is accomplished with eighteen 0.7mm-thick glass plates that focus on a point 175 cm upstream of the analyzer. The 0.2235◦ angle between the glass plates gives the analyzer a total angular acceptance of ±2◦ . Each plate is coated with Gd2 O3 to absorb neutrons. The actual polarization is analyzed with by an m=2.3 polarizing super mirror between each set of glass plates. Each of the super-mirrors is at a 0.95◦ angle to the median path between the plates. The desired spin state passes through the mirrors while the other spin state is reflected into the absorbing coating on the plates (Figure 2.10). The super-mirrors maintain their magnetization with the help of a 0.12T ceramic magnet. Although the ability of the SESAME technique to measure samples with large, divergent neutron beams is a great boon for the flux on the sample, it creates difficulties for measuring the neutron polarization. If the polarization efficiency of the analyzer changes with position or trajectory, then the scattered neutrons will be measured with a different polarization efficiency than the unscattered beam and it will not be possible to normalize out the po-
45
Figure 2.10: A diagram of one of the channels in the Wide Angle Analyzer. The central diagonal line is a polarizing supermirror. The incoming neutron beam (solid arrow) is split into the undesired spin state (dotted arrow), which is reflected into the gadolinium (black bars), and the desired spin state (dashed arrow), which continues through the analyzer. larization efficiency of the beamline from the scattering measurements. Unfortunately, the polarizing efficiency of a super-mirror is dependent on both the angle and the wavelength. Some of the angular dependence can be mitigated through an array of super-mirrors, as seen in Figure 2.10, but this array then adds position dependence, since two neutrons with the same trajectory will be analyzed at different efficiencies depending on which channel the neutron passes through. An example of this problem was seen on the ASTERIX beamline at Los Alamos National Lab (LANL). The IPNS analyzer used on that beamline was found to have an angular dependence for the transmission. The transmission for the scattered beam was greater than the transmission for the unscattered beam, as can be seen in Figure 2.11. Since the transmission changes, the relative contribution between the scattered and unscattered factions of the beam changes, causing the unscattered beam to contribute a disproportionately large amount to the measured polarization. Thus, it’s not possible to normalize out the polarization efficiency of the analyzer using an unscattering blank beam. However, it does remain possible to take a difference measurement between two scattering samples. Assume two samples, whose measured, unnormalized polarizations are P1 and P2 . The
46
0.5 Sample 0.1 Sample -0.2 Sample 0.01 Sample 0.5 Blank 0.1 Blank -0.2 Blank 0.01 Blank
Relative Difference in Intensity from 0 Angle Position
1.5
1.0
0.5
0.0
-0.5
-1.0 2
4
6
8
10
12
14
Wavelength
Figure 2.11: The relative intensities measured on the ASTERIX beamline with and without the sample in the beam. As the angle of the analyzer is changed, a change in intensity can be seen. The level of the change depends on the scattering of the beam, which changes depends on whether the sample or the blank is in the beam. The result is that the polarization of the beam cannot be properly normalized between the measurements. Although the effect is wavelength dependent in this instance, even a constant difference between the unscattered and scattered transmissions would change the measured polarization. measurements of these samples would then be G1 = λ−2 log PP01 and G2 = λ−2 log PP20 , where P0 is the polarizing efficiency of the beamline. The difference between the measurements would then be:
P1 P2 − λ−2 log P0 P0 P1 P 0 =λ−2 log P0 P 2 P1 =λ−2 log P2
G1 − G2 =λ−2 log
(2.5)
Equation 2.5 is completely independent of the polarizing efficiency of the beamline. Therefore, as long as the scattering off of the two samples hit the same region on the analyzer, it is possible to understand the difference in their structures, despite not being able to measure the individual G(Z) values.
47
3
He Supermirror
Flipping Ratio from 5-10 Angstroms
8
Comparison at 4A in triangles
6 4 2 0
4.8
5.0
5.2
5.4
5.6
5.8
Phase Current (Amps) Figure 2.12: Comparison of 3 He analyzer to wide angle supermirror analyzer when tuning the instrument for spin echo with 5 A in the Wollaston prisms. The flipping ratio is measured by dividing the total number of neutron detected per monitor count between 5 ˚ A and 10 ˚ A in the spin up state by the same measurement in the spin down state.
2.2.4
3
HE ANALYZER
An alternative to the wide angle analyzer is the 3 He analyzer. This analyzer uses the difference in the absorption cross section of polarized 3 He to allow only the desired state through to the detector. Since the polarizing efficiency is independent of the trajectory of the neutron, it doesn’t have the issues of the wide angle super mirror analyzers. The 3 He is continuously, optically pumped to maintain the polarization during the experiment [15]. While the SESAME technique does not require a continuously pumped polarizer, since the loss of 3 He efficiency can be measured through the total intensity, it’s still desirable to keep the polarizing efficiency as high as possible through the entire measurement to decrease the uncertainty of the measurement. As can be seen in Figure 2.12, the polarizing efficiency of the 3 He analyzer provides
48
a stronger polarization signal than that of the wide angle supermirror analyzer. A large source of this improvement is that the wide angle analyzer has very poor analyzing efficiency below 5.5 ˚ A, as will be seen in Section 2.4.
2.3 2.3.1
WOLLASTON PRISMS CONSTRUCTION
As discussed in Chapter 1, the SESAME technique requires some method to ensure that the polarization of a neutron is proportional to its trajectory angle. The trajectory angle encoding can be implemented via four linear position encoders. The two encoders before the sample encode the initial trajectory through the difference in position between the two encoders before the sample. A second set of encoders after the sample remove the original encoding, assuming that the trajectory has not changed. The linear position encoders needed for the SESAME technique are in the form of four pairs of triangular solenoids (Figure 2.13). The solenoids in a pair produce anti-parallel fields. The yˆ position of the neutron as it passes through the coils controls the total field integral experienced by the neutron and the total Larmor precession angle. To decrease the divergence of the magnetic field within the solenoids, each pair is yoked in a mu-metal frame. Each solenoid is constructed from aluminum wire wrapped around a triangular frame (Figure 2.14b). While aluminum wire isn’t as electrically conductive as copper wire, the low neutron cross-section allows the neutron beam to pass through the wire without significant neutron absorption. The wire is coated in an enamel oxide whose water has been removed by a baking process. The low hydrogen content of the enamel also allow the neutrons to pass through the insulation with neglible scattering. The beam passes through a total of 14 1 mm fire faces, giving a total transmission of 60%. The frame for the prisms is 10.1cm along x ˆ, 6.9cm along yˆ, and 14cm along zˆ. A 49
Figure 2.13: Two first generation triangular solenoids used as linear position encoders for SESAME. The solenoids are yoked into a mu-metal frame, from which one side has been removed in this photo to showcase the solenoids.
(a)
(b)
Z
X
(c)
Y
(d)
Figure 2.14: Figure (a) is the frame mounted on a magnet winding machine with the extension block mounted into frame for forming the gap. Figure (b) is a schematic of a single prism with a gap in the face through which the neutron beam passes at right angles. The figure omits the flux return paths which ensure that an intact, finite-length solenoid has the same field as an infinitely long solenoid. In practice, the current carrying wires cannot end on the vertical gap edges and have to be bent above or below the gap. Figure (c) is a view of the gap on a Wollaston prism with the wires folded back into another current sheet. Figure (d) is the triangular frame used for wrapping the Wollaston prisms. 50
rectangular hole, 4cm along yˆ and 7cm along zˆ, has been bored through the center of the triangular frame along the x ˆ direction (Figure 2.14d). The triangular frame is covered in a thin layer of Kapton tape. Since the insulation on the wires might be scraped by the winding process, the Kapton serves as a second insulator to ensure that there are no shorts across the aluminum body. To help the solenoid maintain its shape after the coil is wound, an epoxy was added to the corners of the frame before the winding began. Additionally, after the winding was completed, Loctite adhesive was added to the sides to maintain the form of the solenoid. Care was taken to ensure that no Loctite seeped onto the hypotenuse frame, where neutrons might scatter off of the adhesive. Each pair of prism coils is yolked by
1 16 ”-thick
mu-metal.
The first generation of prisms have the wire wrapped along the entire length of the frame. These prisms were tested at IPNS to ensure that the design would linearly encode the neutron’s position into the field integral. A narrow neutron beam was sent through a pair of prisms with fields in opposite directions. The position of this neutron beam in the yˆ direction was varied and the polarization versus wavelength measured at each position. Plotting the polarization versus wavelength will return a cosine curve with a frequency proportional to the field integral. Measuring this frequency as a function of beam height will give the field integral versus position. As seen in Figure 2.18, the field inside the solenoids is highly uniform. Thus, the field integral linearly encodes the polarization. Unfortunately, the closed faces of the solenoids cause problems with stay fields. Inside the solenoid, the field is a combination of the fields projecting in from outside the solenoid plus the field produced by the solenoid. Outside the solenoid, the field is simply the stray field. If the stray field and the solenoid field aren’t parallel, there will be an abrupt change in magnetic field direction when the neutron passes through the wire interface. This change in field direction can cause precession and depolarize the beamline.
51
9
Gauss per Ampere
8 7 6 5 4 3 2
Measured Field Calculated Field Solenoid Gap
1 -12
-10
-8
-6
-4
-2
0
Distance from Field Interface
Figure 2.15: Plot of the measured and simulated magnetic field generated by a prism, versus distance from the interface between two prisms along the neutron beamline (x in Figure 2.14b). To overcome this issue, a second generation of coils was built with a gap in the front wire face (Figure 2.14c). The gap is 6cm along yˆ and 8cm along zˆ. When the magnet is wound, a custom-machined extension block is mounted into the gap of the frame. (Figure 2.14a) The piece is machined so that the wire will be of the appropriate length to fold back onto a flat frame, as shown in Figure 2.14c. To ensure that the new, gapped designs still linearly encode the position, the new coil design was tested at the NIST Center for Neutron Research (NCNR) [16]. A single Wollaston pair was mounted onto a translating sample stage between two
π 2 flippers.
Borated slits
narrowed the beam to 50 µm in the yˆ direction. The spin up intensity the coils was then measured in 0.5 mm intervals along yˆ. Since a 2 mm offset was enough for a full precession of the neutron beam, the narrow slits were necessary to keep the difference in polarization angle down to 10◦ across the beam. The intensity was originally measured in a ± 2 mm range around the center of the Wollaston prism (Figure 2.19). Intensity versus beam displacement was then fit to a sinusoid and extrapolated out to the edges of the gaps through the prism (± 17 mm). The model taken from the center of the Wollaston prism correctly predicted the measurements at the edges, confirming that the gapped solenoids still meet the linear position encoding requirements to be a SESAME instrument. 52
Figure 2.16: Thermal image of Wollaston prism with water cooling In addition to the experiment at the NCNR, Paul Stonaha simulated the fields in Mathematica as an infinite solenoid plus a finite set of current sheets to account for the gaps. These simulations agreed within 2% with the field strengths measured by James Hendrie using a 3D magnet mapper (Figure 2.15) Since the aluminum wire isn’t as conductive as copper, the prisms need water cooling to reach above 5A. Each prism has three 3/8” diameter holes along the zˆ axis. As seen in Figure 2.16, the water temperature has a negligible increase after passing through the solenoid, so the six pipes in each prism pair can all be run in series. Each prism pair is then connected in parallel so as not to create to much resistance to the water flow by running them in series. The return connector for the water line has a flow switch that will activate a safety shutoff on the electrical power for the prisms if the water is disconnected or if a large leak has formed. The water supply at LENS runs at 60psi and is at room temperature. Additionally, the face of each prism pair has a thermal cut-off diode that will signal if the prism reaches a temperature over 85 ◦C. The coils can remain below the temperature cutoff with the current below 14A.
53
In addition to the Wollaston prism solenoids, the SESAME instrument requires two π 2 flippers
to start and stop the neutron’s precession, as well as a π flipper in the middle
to start reversing the precession (Figure 2.17). Both types of flippers are wound with the same wire and gap construction as the prisms. The solenoids are a rectangular prism that is 14cm along zˆ, 7cm along yˆ, and 7cm along x ˆ. Just as with the prisms, the flippers are yolked in mu-metal to approximate being an infinite solenoid. Since the thermal camera tests on the Wollaston prisms showed them to be thermally stable below 5A without water cooling and since the design of the flippers is very similar to the design of the prisms, the flippers aren’t water cooled and are simply kept at low current. Since the flippers are made from two separate solenoids, the coils will need to be brought together to form a spin flipper. If there is empty space between the two coils faces, the neutron will precess around whatever uncontrolled stray field may be within that space, ruining the beam’s polarization. To overcome this problem, we added holes to the frames of the flipper coils through which screw rods could be mounted. Nuts and washes on either end of the screw rod are tightened to force the prism coils tightly together and eliminate any space between the solenoids. The pi flipper consists of two of the rectangular prisms with their wire covered faces butted against each other and their fields aligned anti-parallel. The wire face where the two non-gapped sides touch forms an abrupt field transition that flips the direction of the magnetic field with respect to the neutron’s spin. For the
π 2 flipper,
the prisms are aligned
with their fields perpendicular, so that the neutron would begin precessing around the new field direction after the abrupt transition. Since all of the field transitions are non-adiabatic, the flippers work at all wavelength regardless of the flipper’s current. To maintain the polarization of the neutron beam between the Wollaston prism, there must be a field between the prisms. In the absence of a field, the neutrons will precess
54
Figure 2.17: Photo of a
π 2 flipper
mounted on the ANDR beamline at NCNR
around any stray fields that might be present (e.g. Earth’s magnetic field) and wreck the beam polarization. By creating a strong field for the neutron to remain within, any stray fields only contribute a small part to the total field direction and the neutron continues to precess around the desired direction. Since the uniformity of the field direction is important, ceramic magnets have proved to be problematic. Creating large guides requires either a large, solid ceramic magnet which extends the desired length or a series of smaller magnets with a ferrous yoke to distribute the field. The large, solid magnet is cost prohibitive to acquire and would be brittle. While an appropriate guide field could be built with the smaller ceramics and a yoke, great care is needed to ensure both uniform magnetization of the yoke and uniform strength of the ceramic magnets. Thus, while the ceramic guides can be used for non-precessing parts of the beamline, the places between the Wollaston prisms are best served by electromagnetic guides. The field uniformity in a Helmholtz coil is more than adequate to maintain beam polarization.
55
200
Field Integral (Gauss cm)
150 100 50 0 -50
Equation
y = a + b*x
Weight
Instrumental 0.61249
Residual Sum of Squares
-0.99991
Pearson's r
0.99977
Adj. R-Square
-100
Value Field Integral
Intercept Slope
Standard Error
39.50923
0.54056
-12.00736
0.09068
-150 -15
-10
-5
0
5
10
15
Offset (mm)
Figure 2.18: The field integral through a single closed prism versus the yˆ offset of the beam through the prism. This measurement was taken on the POSY II beamline at IPNS. The prism were running a current of 4A. The linear fit has an r2 value of 0.99977 and all of the residuals are within experimental uncertainty.
Figure 2.19: Neutron intensity transmitted through a pair of Wollaston prism solenoids as a function of the displacement across the neutron beam (y-direction in Figure 2.14b). The sinusoidal curve was only fitted to the data around zero displacement, but can be seen to fit the data at much larger displacements, demonstrating that the field integral through the Wollaston prism varies linearly with displacement in the y direction. Data were obtained at the NCNR [16].
56
2.3.2
POWER
Each Wollaston prism is powered by its own Kepco BOP 20-20 power supply. Each supply provides a maximum of 20 A of current at a maximum of 20 V. Due to this limitation and the prism’s resistance of 1 Ω, the prisms are limited to 20 A and 177 G. This limitation can be overcome only by quadrupling the number of power supplies. To exceed 20 A through the 1 Ω coil, two of the power supplies must be wired in series, to generate a potential drop of greater than 20 V. These two supplies must be wired in parallel to two more supplies in series, so that their combined current exceeds 20 A. The issues with the power supply current are not pressing, however, since the primary limitation on the current is thermal. Without water cooling, the prisms are currently limited to 5 A before they trigger the 85 ◦C thermal safety cutoff diodes. With water cooling, the prisms can be safely taken up to 14 A. Going to 20 A, however, would double the power output of the prisms and is far outside the scope of the current design. The current output of the Kepco supplies are controlled by a National Instruments PCI6703 DAQ card. The card is capable of supplying sixteen separate ±10 V signals with 1 mV accuracy. These signals can then be sent as an input to the Kepco BOP supplies. The Kepcos, conveniently, convert a 10 V signal to their own maximum current output. Thus, a Kepco BOP 20-20 will output20 A when given a 10 V signal and a Kepco BOP 20-10 will output 7 A when given a 7 V signal. The PCI-6703 has a 1 mV accuracy, giving the triangles a 2 mA accuracy on their currents. The card is controlled by a custom written server, discussed in Appendix B. The remaining magnets on the beamline are powered by a set of five Kepco BOP 20-10 power supplies, of which four are currently computer controlled. The computer controlled power supplies are designated “Flipper”, “Guides”, “Phase”, and “Sample”. The “Flipper” provides power solely to the downstream coil on the downstream
57
π 2 coil.
This coil is always
Key Power Supply
F Flipper
G Guides
π/2
P Phase
S Supply
U Unlabeled
π
U G
G
S
U U
π/2 S
P
P F
Figure 2.20: Map of power supply connections for the SESAME beamline. run at either 5 A or −5 A and has a resistance of 1 Ω, so the supply could conceivably be replaced by a KEPCO BOP 20-5 without any change in instrument function. The “Flipper” coil is also the only coil on the beamline that requires the bipolar output capabilities of the Kepco BOP series. By flipping the polarization of the magnet, the neutron’s spin direction with respect to the magnetic field can be reverse, effectively creating a π flipper. This allows the instrument to perform counting measurements on both spin states without using beam line space for an additional, dedicated π flipper. The “Phase” supply controls the downstream guide field and the upstream half of the downstream
π 2
(Figure 2.20).
Adjusting the current on this power supply changes the total Larmor phase acquired by neutrons passing through the beamline and is used to tune the instrument to spin echo. The corresponding upstream guide field and the downstream half of the upstream
π 2
are
controlled by the “Guides” supply. The field in the sample guide field and corresponding compensator field are powered by the “Sample” supply. Finally, the upstream half of the upstream
2.4
π 2
and the central π flipper receive current from the unlabeled power supply.
EFFICIENCY MEASUREMENTS
To better understand the beamline components, a series of measurements (Table 2.1) were performed to find the polarizing efficiency of the polarizer and analyzer, as well as understanding the flipping efficiency of the various spin echo components. In order to separate the efficiency of the polarizer from the efficiency of the analyzer, a third neutron polarizing
58
Measurement Name Bf1S BF1S Bf1S BF2S Bf1f2S BF1f2S Bf1F2S BF1F2S Bf2Sf1A Bf2Sf1A BF2SF1A
Flipper 1 Position Fields Region 1 Parallel Region 1 Anti-Parallel Region Region Region Region Region Region Region
1 1 1 1 1 1 1
Anti-Parallel Parallel Parallel Anti-Parallel Parallel Parallel Anti-Parallel
Flipper 2 Position Fields Region Region Region Region Region Region Region Region Region
1 1 1 1 1 1 2 2 2
Parallel Anti-Parallel Parallel Parallel Anti-Parallel Anti-Parallel Parallel Anti-Parallel Anti-Parallel
Analyzer Out Out Out Out Out Out Out Out In In In
Table 2.1: A listing of the measurements made to determine the efficiencies of the beamline components. The first column is the label of the measurement. The second and third column tell which flippers were in which regions on the beamline. The flippers are also labeled as to whether the fields in the flippers were parallel (maintaining the polarization) or anti-parallel (flipping the spin). The fourth column tells whether the wide analyzer was in the beam. yzer Anal
ctor Dete
on 2 Regi Bender
Region 1
Supermirror
Figure 2.21: Schematic for the layout of the SESAME beamline during the polarizing efficiency measurements. element was needed. A single m = 3 supermirror was added onto the sample stage of the and set at 1◦ angle. The secondary flight path was also raised by 1◦ to keep the reflected neutron beam in the center of the detector and wide angle analyzer. The beamline was kept in this configuration and none of the polarizing element were moved during the experiment, except for sliding the analyzer on its rails. Between the polarizing elements, ceramic guides were added to transport the polarization across the beamline. The ceramic guides were individually labeled to ensure that they were kept in the same positions for each experiment. This setup is diagrammed in Figure 2.21 In addition to measuring the efficiency of the polarizing elements, we also wanted to
59
Flipper 1 Off Flipper 1 On Flipper 2 Off Flipper 2 On
1.10 1.08 1.06
Probability
1.04 1.02 1.00 0.98 0.96 0.94 0.92 0.90 4
6
8
10
12
Wavelength
Figure 2.22: The polarization probability for both of the spin flippers in their parallel (Off) and anti-parallel (On) states. The measured probability is the probability that the neutron will exit the flipper with the desired spin state. Thus, a probability of 1 means all neutrons are flipped in the On state and that no neutrons are flipper in the Off state. understand the flipping efficiency of the spin echo elements. Initial measurements showed that the flippers have a measurable polarization loss in their non-flipping state. The standard four flipper measurement [17] assumes that flippers have perfect efficiency in their off states, so a more complicated measurement was needed. Therefore, measurements needed to be made with the fields parallel, anti-parallel, and with the flippers entirely out of the beam. The full formulae for the measurements are listed in Appendix A. The measured efficiency of the flippers can be seen in Figure 2.22. As can be seen, there is a small loss in polarization when neutrons pass through the flippers with the fields in parallel, though the losses are on the order of 2%. When the fields are anti-parallel and the neutrons are flipping, the efficiency is 100% to within uncertainty. Unfortunately, the efficiency formulae in Appendix A have large uncertainties on the anti-parallel state flippers when their efficiencies are close to 100%, so the exact depolarization wasn’t able to be measured in a reasonable time. The measured efficiency of the polarizing bender and width angle analyzer can be seen in Figure 2.23.
60
Analyzer Bender
1.0
Polarizing Probability
0.9
0.8
0.7
0.6
0.5 4
6
8
10
12
Wavelength
Figure 2.23: The polarization probability for both the polarizing bender and the wide angle analyzer. The y-axis is the probability that a neutron that exits the polarizer has the desired spin state. A perfect polarizer would have a probability of 1 and an iron shim would have a probability of 0.5.
Beam Monitor Figure 2.24: A picture of the beam monitor on the beamline.
2.5
DETECTOR
Neutrons are detected at two points along the instrument. At the downstream end of the final neutron guide is a low efficiency detector (Figure 2.24. This detector is used to monitor the neutron flux coming from the moderator before any interactions with the magnets or the samples. The other detector is the primary data collector and is the final element of the beamline. There are two detectors which have been used as primaries for experiments on the SESAME beamline: the Lexitek PAPA detector[18] and an array of He3 pencils.
61
2.5.1
BEAM MONITOR
There’s a low efficiency (0.5%) 3 He detector at the end of the final neutron guide. This detector records the neutron flux coming from the LENS target. The monitor is 75mm in the zˆ direction, 25mm in the yˆ direction, and 19mm in the x ˆ direction. The monitor completely covers the exit of the neutron guide, so all of the neutron flux passes through the monitor. The monitor records the neutrons in 50 µs time of flight bins to provide a view of the spectrum of the neutrons before interacting with the sample.
2.5.2
SCINTILLATING DETECTOR
For experiments that require a very large neutron collection area, there’s a two dimensional scintillating detector [18] produced by Lexitek [19]. The ZnS:Ag scintillator is doped with Li6 . Six large photomultiplier tubes (PMTs) collect the intensity of each neutron event. In the center of the photomultipliers is an image intensifier, followed by an array of 21 PMTs. Each of the smaller PMTs is masked with a Gray-coded grating that, combined, identifies the position of the neutron event to within 0.49mm in both the horizontal and vertical direction. Events in the smaller PMTs are gated against events in the larger PMTs to eliminate background noise. Between the scintillator and the PMTs is a mirror arranged at a 45◦ to the direction of the neutron beam. This allows the electronics to be out of the beam and protected from the radiation. The total resolution of the detector is 512 pixels by 512 pixels. The detector records individual neutron events with 13ns accuracy, but our collection software bins these events into 250ns bins, which correspond to 0.1 ˚ A wavelength bins. The detector accepts a T0 signal from the main accelerator and the time of each neutron event is recorded relative to the last T0 pulse. The scintillating detector multiplexes the signals from the 21 position PMTs and feeds them through four ADCs before interpreting the neutron events. Unfortunately, one of
62
the ADC channels has a significant amount of noise that arises from within the electronics of the detector itself (Figure 2.25). The noise has an amplitude equivalent to that of the neutron signal and cannot be eliminated with a voltage threshold. Due to the mask and Gray code design of the beamline, these fake signals will result in the neutron being placed in the wrong position. In the worst case, when the noise occurs on the PMT with the coarsest mask, the event can even be redirected to the other side of the detector. This blurs the position resolution in a non-trivial way. Additionally, it makes background subtraction problematic, since any “background” region of the detector may contain neutron events that originated from the main signal and were placed by the PMT noise. As such, the PAPA detector is not useful for position sensitive scattering measures (e.g. SANS). The PAPA is best used for measuring the relative intensity of two different signals, since the absolute neutron count cannot be trusted and the position information is also incorrect. However, since most SESAME measurements are based on a ratio between two spin states, the PAPA detector can still perform useful science. In addition to the position noise, the detector also has a high baseline background. The electronic noise produces 18 background events per second, even when the neutron source is off. Just as the noise on the position PMTs can change the position of a neutron event, noise on the strobe PMT can create a neutron event without any light actually hitting the PMTs. While the larger PMTs are intended to confirm the events on the strobe PMT, the signal to noise ratio on the larger PMTs is less than one, so the fault positive rate of the energy channel renders the results useless. This background can be subtracted out by examining regions of the detector where it’s know that neutron events cannot occur. Since the active area of the detector is far larger than the available area of either analyzer, the regions are guaranteed to exist. Decoupling this background from the misplaced pixels, however, is possible, since the noise on ADC channel B is high enough to create a noticeable
63
Figure 2.25: Averaged pulse information for the ADC channels on the PAPA detector. Green is ADC A, red is ADC B, blue is ADC C, and black is ADC D. The high average background on ADC B is due to the averaging of noisy signal peaks. banding pattern. Neutrons events outside of the bright bands will mostly be background and not misplaced events.
2.5.3
He3 DETECTOR
As an alternative to the PAPA detector, the SESAME beamline also possesses an array of 16 linear 3 He detectors. Each detector tube has a resolution of 2 mm along its length and the detectors are approximately 1 cm across. This detector is controlled via the PyDAS control software written by the SNS. Data is against collected in event mode, with each neutron event being recorded by a time code since the last neutron pulse. These time codes are registered with 100 ns accuracy with the codes again being measured from the last T0 pulse.
64
CHAPTER 3
DEVELOPMENT OF SESAME METHOD ON ASTERIX
While Chapter 1 provides a grounding in the theory of SESAME, we started our experiments of how it would work in practice. Along the way, we learned multiple lessons on the best way of performing SESAME. I also helped in an upgrade of the ASTERIX beamline that both improved its capabilities as a polarized neutron reflectometer as well as setting up the beamline as an optional SESAME instrument.
3.1
EQUIPMENT MOUNTING
For our first few experiments, the spin echo elements were placed on the beamline in an ad-hoc fashion, as seen in Figure 3.1. Both the upstream and downstream halves of the instrument were mounted on Newport x48 rails that were mounted on makeshift supports. Downstream, the beamline length was limited by the short distance between the sample position and the end of the analyzer. Upstream, the beamline would run into a shielding wall. The top half of this shielding wall could be removed, as seen in Figure 3.1b, but there was still little space beyond it for extending the beamline.
65
Figure 3.1: Images of the SESAME elements on the ASTERIX beamline before the upgrade. As part of the ASTERIX upgrade project, funded at IU by the Department of Energy’s (DOE) Office of Basic Energy Sciences, the entire beamline was upgraded over a period of three years with equipment designed to mount spin echo elements. The downstream rotation arm of the beamline was replaced with a longer gantry arm with Thomson rails for the positioning of elements, as seen in Figure 3.2. During reflectometry experiments, the detector and analyzer could slide along the rails toward the sample position. However, when the extra space was needed for spin echo experiments, the detector and analyzer could slide toward the far end, granting over a meter of spacing for adding spin echo elements. A Newport x95 rail was added to the bottom of this rotation arm, low enough to avoid interfering with anything sliding on the Thomson rails. In the SESAME setup, two pillars could be mounted on this rail, as seen in Figure 3.2. Atop these two pillars, a second x95 rail can be mounted on an inverted set of carriages. It is on top of this second rail that spin echo elements are added to precess the neutron spin. To increase the available length on the downstream end of the beamline, a shorter neutron polarization analyzer was added, which will be discussed in further detail in Section 3.10. However, we wanted the older analyzer to still be available for reflectometry measurements. Therefore, both analyzers were mounted on a motorized elevator system, seen in Figure 3.3. This elevator was mounted on the Thomson rails discussed previously 66
Upper Rail Pillars Lower Rail Detector Arm
Figure 3.2: The new detector arm added as part of the ASTERIX upgrade. and could be slid to any position along the detector arm. Each analyzer sat atop its own goniometer on a separate shelf on the elevator. Through a single motor command issued by the user, either analyzer could be moved into the beam, or both analyzers could be moved out to measure a raw neutron count. On the upstream side of the beamline, a preexisting 80/20 frame was modified to mount a second x95 rail.Two inverted x95 carriages were added to horizontal cross bars on the frame. An x95 rail was then mounted inside the carriages. During SESAME experiments, the rail would be slid downstream until the upstream edge of the rail was just inside the upstream carriage. This gives the greatest amount of space for spin echo components during the experiment. When performing polarized neutron reflectometry, the rail can be slid back upstream until it is contained completely underneath the frame. The shielding wall can then be replaced to decrease background. The upstream and downstream rails can be aligned via laser, though this is not a simple
67
Figure 3.3: The elevator added to the ASTERIX beamline to allow users to switch polarization analyzers. procedure. A make-shift alignment post was built with wire cross hairs on which to center the beam. I would begin aligning the upstream rail by extending the rail downstream. The alignment post would then be slid along the rail both to see how far off center the rail was and how much the rail was tilted. Alignment would have to start along the zˆ direction, as accessing the screws to adjust the carriages positions in this direction required completely removing the rail. Once the rail was aligned along zˆ, the rail would be aligned along yˆ. This alignment required loosening the cross bars of the 80/20 frame and re-tightening them at the correct height. The cross bars occasionally required a mallet to move, so this could be a time consuming procedure. When the cross bars and carriages were positioned so that the laser was in the center of the cross hairs on both the upstream and downstream ends of the rail, the rail was declared to be aligned and all of the screws given a final tightening. Thankfully, unless the frame or rail are given a heavy push, this alignment will persist between multiple experiments and should not need to be repeated often.
68
The downstream rail had a similar alignment procedure. Instead of adjusting the height of cross bars, the height of the mounting pillars would be adjusted to bring the rail into alignment. Since each pillar had four mounting screws for the height, it was also necessary to check for rotations of the rail around the x ˆ axis. The x ˆ rotation would be measured via a simple level. The remaining alignment used the same procedure as the upstream rail, with the rail being declared aligned when the laser hit the cross hairs at both the upstream and downstream ends of the rail.
3.2
TRIANGLE GAPS
Our first attempts at spin echo on the ASTERIX beamline used the solid faced prisms discussed Section 2.3.1. As previously discussed, the solid wire cause sudden changes in field direction when any outside field crosses into the solenoid. However, I helped develop a way to construct a SESAME beamline using these solid prisms. Prism pairs 1 and 2 were butted against each other, as were pairs 3 and 4, to avoid the field boundary issues present in solid faced triangles (Figure 3.4). Upstream of the first prism pair, and downstream of the last, V-coils were added with a field perpendicular to the prism field. The V-coils were butted against the triangle coils and the perpendicular field interface served as a
π 2 flipper,
just as in the IPNS experiment mentioned in Section 2.3.1 of Chapter 2. Two more V-coils with field directions parallel to the prism fields were added downstream of prism pair 2 and upstream of prism pair 3 to allow for a gradual field change from the sample position while preventing any stray fields from leaking into the prism pairs. The central π flipper was a standard Mezei flipper with current ramping matched to the beamline’s known time of flight. Since all the wire faces were butted against another solenoid face, there was no way to use any form of air cooling to help maintain the temperature. Therefore, experiments in
69
flipper
Figure 3.4: A schematic of the spin echo coils as used with the solid faced coils on ASTERIX. The star is used to represent the sample position. The innermost boxes on either side of the sample are V-coils with their fields parallel to the l The vector arrows mark the field directions. The “flipper” is a Mezei π flipper with it current ramped to match the neutron time of flight. this configuration were limited to triangle currents below 5 A. These triangles were eventually replaced with a new set of triangles with gapped faces (Section 2.3.1 of Chapter 2. These new coils eliminated the requirement that the solenoid faces be adjacent, which conferred multiple advantages. First, there was now increased airflow between the triangle pairs, which increased thermal regulation and allowed for higher current levels. Additionally, the gaps in the Wollaston prisms removed eight layers of aluminum from the beam, decreasing loses due to scattering. More importantly, the triangles could now be spaced further apart, increasing the hypotenuse separation and, thereby, the spin echo length. Finally, these coils also added water cooling to the triangle frames, improving the thermal regulation. These factors combined to increase the spin echo length at 9.5 ˚ A from 105 nm to 1.9 µm. Since the gapped solenoids could no longer used adjacent field faces to serve as π or π 2 flippers,
it became necessary to build standalone versions of the flippers. The using
identical gaps on the π and π2 flippers as on the Wollaston prisms helps the phase aberrations from the gaps to cancel out [16]. I designed a new solenoid that used the same gapped field faces as the Wollaston prisms, but had a flat field interface on the reverse side, instead of the inclined edges used by the Wollaston solenoids. Placing the solenoids with their faces antiparallel produces a π flip while having the fields perpendicular produced a the solenoids separate allowed for the coils to serve as either π or
70
π 2 flippers
π 2 flip.
Keeping
as needed.
Figure 3.5: The layout of the beamline while tuning a single magnetic prism pair.
3.3
TUNING PROCEDURE
The first experiments at ASTERIX were performed before the tuning procedure explained in Section 1.8.1 of Chapter 1 was developed. The first tuning procedure was designed specifically for the solid-faced prisms. Each prism pair was mounted on a micrometer translation stage. The prism pairs were tuned individually before the entire beamline was mounted. During the tuning of a single prism pair, V-coils were butted against the upstream and downstream sides of the prism pair to forced the neutron to precess within the prisms (Figure 3.5). The motorized slits of the ASTERIX beamline were then used to narrow the beam profile to a 0.5 mm in the yˆ direction. Polarization versus wavelength measurements were then taken with the beam at +2 cm yˆ, +1 cm yˆ, -1 cm yˆ, and -2 cm yˆ. Taking a Fourier transform of the polarization versus wavelength graph returned the total field integral at each height. Since the ASTERIX instrument does not use evenly spaced wavelength bins, the Fast Fourier Transform cannot be used and we were forced to use the Non-Uniform Discrete Fourier Transform. Taking a linear regression of the four measurements gave the field integral versus height. The micrometer stage was then set to move the echo position of the triangle pair into the center of the neutron beam. Since each prism pair produces no net field integral through the center of the beam, there is also no net field integral across the whole instrument and there is no need for a phase coil. The largest drawback to this tuning procedure is that it is very time consuming. Each measurement of the polarization versus wavelength could take half an hour. With four prism pairs each needing four calibration measurements plus an additional measurement to confirm that spin echo had been achieved, it would take ten hours to tune the instrument. Additional 71
time would then be needed after the tuning was completed to put all the spin echo elements onto the beamline before any SESAME measurements could be taken. Additionally, as the translation stages were not motorized, the tuning could not be scripted and required human intervention every half hour. Thanks to the shutter design of the ASTERIX beamline, it also required another five minutes to open and close the shutter each time we went in to adjust the micrometers. Even with motorized stages, experimenters would still need to be present to switch the prisms between experiments. Ultimately, a day of beam time would be lost with this tuning procedure, as waking experimenter hours would be spent tuning in the instrument, leaving a blank polarization measurement for overnight. Even if this procedure didn’t have the large drawback with the timing issues, it wouldn’t work with the gapped coils. The translation stage procedure works under the assumption that it can correctly account for all of the precession of the neutron as being contained within the prism coils. Since the V-coils producing the
π 2 flippers
are adjacent to the prism
fields, this holds true during the closed-coil tuning. However, for the gapped coils, there would be additional precession between the triangle pairs, which would not be accounted for in this tuning procedure. The addition of gaps to the prism faces required the development of a new procedure. With the advent of the gapped coils, I implemented a new, simpler tuning procedure. The entire instrument would be mounted on the beamline before any tuning would begin. We would then use an iterative process of measuring the polarization versus wavelength and adjusting the current in one of the guide fields. The direction of the current adjustment was always chosen to minimize the frequency of the polarization versus wavelength graph (Figure 3.6a). The magnitude of the changes were mostly chosen by trial and error. The beamline was declared to be tuned when the polarization was independent of wavelength. There were significant time savings in this tuning procedure from multiple directions. First,
72
since the entire beamline was built before the tuning began, there was no need to repeatedly reconfigure the beamline between measurements. Once the beamline had been built, there was no reason to close the shutter again until samples needed to be switched. Additionally, while the old procedure had needed twenty measurements to achieve tuning, this procedure had a theoretical minimum of three measurements (an initial measurement, a second phase current to establish magnitude and direction, and a third measurement to confirm the final tune). In practice, six or seven measurements were often required, but this is still a factor of three fewer measurements than required previously. The instrument could now be tuned and ready for measurements in four hours. Not only did this free up beam time for use in measuring samples instead of calibration, but it also made it practical to measure at multiple prism currents during the same run cycle. Since the largest time sink in the process was the polarization versus wavelength measurements, we replaced it with a single figure of merit in the polarization integrated over all wavelengths (Figure 3.6b). The total counts on detector would be measured for a fixed time (≈20 s), then the counts would be measured for the same time in the opposite flipper state. A flipping ratio could be taken from these two counts. The immediate advantage is that forty phase currents can be measured in the amount of time that it takes to gain decent statistics on one polarization versus wavelength measurement. The main disadvantage was that the flipping ratio contained far less information that the wavelength plot. When far away from echo, the wavelength plot produces a small period signal whose frequency is an indication of how far off of echo the signal is. The integrated flipping ratio, on the other hand, would simply approach 1, as the small period of the wavelength plot produces approximately equal numbers of spin up and down neutrons. However, since the integrated measurements are so much faster than the wavelength dependent ones, this limitation can be overcome simply by taking the measurements over a wide range of phase currents (e.g.
73
(a)
(b)
Figure 3.6: Figure (a) is the wavelength versus tuning plots for phase currents of 5.1 A (White), 4.9 A (Red), and 5.0 A (Green). The final tuning position was found to be at 4.98 A. Figure (b) is the neutron count versus phase current. The peak neutron count corresponds to the peak polarization. ±1 A). Additionally, once this tuning procedure is performed once, a baseline current for the tune can be established and later tuning measurements can be taken around this point. In addition to the timing advantages, using the integrated polarization had an advantage in unambiguity. Near the echo point, the flipping ratio grows to a high value and the peak value is very distinct. On the other hand, for the wavelength dependent plots, comparing the frequency of two plots whose frequencies are both near zero is very difficult and error prone. To develop the next procedure, I wrote a Python script to simulate the number of precessions through a beam line built with random offsets (see Section 1.8.2 of Chapter 1). I then tested a variety of tuning procedures through this software. The best results came from setting three parameters: a finite phase, scaling the first two prism pairs, and scaling the first prism pair. The finite phase could be generated by scaling the current in the magnetic guide field between the Wollaston prism pairs. The prism scaling parameters could then be controlled through the currents through the prism pairs. However, in practice, the prism tuning parameters rarely contributed to the final spin echo polarization (Figure 3.7), and
74
Integrated Flipping Ratio
4.6 4.4 4.2 4.0 3.8 3.6 3.4 3.2 5.8
5.9
6.0
6.1
6.2
6.3
Triangle Currents
Figure 3.7: A representative example of the effect of tuning of the prism currents. The baseline triangle current was at 6 A, which corresponds to the peak in the tuning plot. so these steps are skipped. With the tuning procedure largely fixed, I wrote a script for FOURC[20] which would could tune the beamline without any human intervention. If seeded with an expected current for spin echo taken from an earlier scan of currents, this script could tune the instrument in fifteen minutes. It is now reasonable to measure one sample to completion on every desired spin echo range before starting the next sample. This helps to remove time dependent effects from the measurements. Additionally, it is now possible to set a sample to run overnight at multiple spin echo lengths without needing an experimenter to stay and perform tuning.
3.4
WAVELENGTH RANGE
The ASTERIX beamline has a wavelength range from 3 ˚ A to 12 ˚ A. For our initial experiments, we sought to gain maximal use of this wavelength range by focusing on the long wavelength neutrons. The theory was that since the 12 ˚ A neutrons are the rarest, any measurement that has collected enough statistics on 12 ˚ A neutrons will have also collected enough neutrons at all of the shorter wavelengths. Additionally, since the spin echo length is proportional to the wavelength squared, The neutrons between 8.75 ˚ A and 12 ˚ A account
75
Uncertainty in G(Z)
1
5A 10 A
0.1
14 A 0
200
400
600
800
1000
Spin Echo Length (nm) Figure 3.8: The absolute uncertainty in a G(Z) measurement on a colloid sample versus spin echo length. The uncertainties have been normalize to a one hour measurement. The solid line was measured with 5 A in the Wollaston prisms, the dashed line with 10 A, and the dotted line with 14 A. These uncertainties were measured on the Linear PSD (Section 3.9) and not the 3 He3 tubes, so the absolute values are much higher than would be expected from current experiments. for half of the spin echo range on the 3 ˚ A to 12 ˚ A spectrum. When tuning was still a process which took days, this was a reasonable route. However, with the advent of the faster tuning procedures, it became practical to measure a sample at multiple prism coil currents. By quadrupling the prism coil current, the spin A. Figure 3.8 shows A could now be reached at 6 ˚ echo range previously reached by a 12 ˚ the uncertainty versus wavelength for a series of measurements on a colloid sample. At any given spin echo range, the optimum current to use is the one represented by the lowest line on the graph. Features at 100 nm are best measured with 5 A in the coils while features at 200 nm are better measured with 10 A in the coils. The reason that the minimum in the uncertainty versus spin echo length graph increases with current is due to the lower echo polarization at higher currents increasing the overall uncertainty of the measurement.
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3.5 3.5.1
SAMPLE ENVIRONMENT SAMPLE CELLS
For some early experiments, we used demountable sample cells. The demountable cells were chosen both for their short path length and to ease cleaning. Even though the cell was covered with multiple layers of Paraflim, the sample solvent would still leak out during the experiment. This ruined several of the measurements as the sample changed over the course of the measurement. Even once we moved away from the demountable cells to straight rectangular quartz cells (Figure 3.9), we found that even the solid rectangular cells had the same leakage problem with dodecane. The capillary force for the dodecane in the quartz cell was enough to pull the solvent out of the cell, slowly emptying out the sample over the course of the experiment. The Teflon stoppers provided with the cells were not tight enough to prevent the sample from leaking out, though it could be sealed with enough Paraflim. Unfortunately, a reproducible procedure for ensuring that the Parafilm prevented the solvent from escaping was never found. Finally, a cell with a round stopper was found to keep the sample in the cells (Figure 3.9). We don’t know whether the round top stops the capillary action or whether the stopper is simply tight enough to keep the sample contained. Several data runs were ruined by a magnetic sample cell holder. While the sample holder frame was not magnetic, the spring that held the sample in place contained steel. Figure 3.10 shows the loss in polarization from having the magnetic material near the beam. After this experiment, a set of copper-beryllium springs were machined by Titan Waterjet to replace that magnetic spring. This also served as an important object lesson to the sensitivity of the instrument with even small amounts of magnetic materials near the beam in the precessing region.
77
Polarization
Figure 3.9: A photo of the rectangular made by Helma [21] (Left) and stoppered quartz cells made by Starna [22] (Right). The stoppered cells are necessary to prevent some samples from escaping the cell through the capillary force.
Wavelength (Å) Figure 3.10: Polarization versus wavelength (˚ A) for SESAME runs on a d-dodecane blank. The black data used the magnetic sample holder while the red data were taken with a non-magnetic holder.
78
Figure 3.11: A colloid sample mounted on the ASTERIX beamline with no sample guide. Cadmium slits are mounted directly on the sample.
3.5.2
SAMPLE FIELD
Initially, our setups on the SESAME beamline did not have a dedicated magnetic field around the sample. For the solid faced coil experiments, the sample sat in the fields extended by the two V-coils, as seen in Figure 3.11. Once the gapped faced prisms were developed, the sample could be sat in the field between the second prism pair and the π flippers. Besides the obvious advantages in simplicity, having no guide field also allowed for greater spin echo lengths, since beam line space could be used for increasing the prism separation instead of a sample guide field. Additionally, if a sample guide is added to the instrument, an equivalent guide must be added on the opposite side of the π flipper to remove the precessions that the neutron accumulates in the sample guide. This “compensator” guide field means that every centimeter of guide field length costs two centimeters of beam line space. However, the lack of a dedicated sample guide also came with serious drawbacks. If the current through the prism pairs was too low, the field in the sample region would not be 79
Figure 3.12: The first sample guide field strong enough to maintain a constant field direction for precessing polarization. Due to a magnetized rebar in the floor of the ASTERIX cave, there was a 3 G field at the sample region. Adding ceramic magnets under the sample stage could decrease the field to 0.3 G, but even this field, could ruin the precessing polarization if there wasn’t a strong enough guide field to keep the field direction constant. The larger the sample, the more space that would be needed between the spin echo elements and the larger the coil which would be needed in those elements. Adding a dedicated guide magnet decoupled the field strength in the prisms from the maximum sample size. Our initial sample guide was built by Dr. Da Luo (Figure 3.12). This guide field provided a nice, small field environment for SESANS samples. For SERGIS, however, it had a couple of weaknesses. It wasn’t capable of mounting sample wafers much larger than 3”. Furthermore, changing reflection samples was difficult. The top could be flipped open to switch out SESANS samples, but changing reflection samples required taking apart
80
the sample mounting rod and often required removing the entire sample guide from the beamline. To overcome this difficulty, I designed and built a second generation of sample guide with a demountable side. Four small screw holes through the mu-metal allowed one of the two solenoid sides to be completely removed from the guide field. This allowed the sample rod to be mounted from the side, as opposed to being snaked into the guide field from the front. Additionally, reflection samples could be mounted directly, instead of being placed on the sample rod and then moved into the guide. The demounting screws, unfortunately, had far too much stress placed on them from the weight of the magnet sides. This could bend the screws and made mounting the sides difficult and time consuming. I solved this by building a third generation of sample guide. The new design only allowed one side to be dismounted. The demountable side rested in a dovetail notch in the guide frame, instead of being screwed in from the side. Two alignment screws at the top allowed ensured that the side we mounted reproducibly.
3.6
CALIBRATION
In our initial experiments, the spin echo length of the beamline was calibrated via dead reckoning from the measured distance between the prism pairs and the magnetic fields within the solenoids. However, with the development of a dynamical theory for the scattering from gratings by Dr. Rana Ashkar [23], it was possible to use reflection measurement with gratings to find the spin echo length of the instrument (Figure 3.13). As can be seen in Table 3.1, the scaling factors were usually within 15% of the measured values. The dead reckoning consistently overstates the spin echo length. This might be due to the smaller field strength in the guide fields between the prism coils, but this has not been tested. The main problem with this calibration technique is performing SERGIS on the gratings requires lining up the samples for reflection. While the ASTERIX beamline does have a
81
Prism Current (A)
Dead Reckoning (nm ˚ A−2 )
Calibrated (nm ˚ A−2 )
6.08 15.16 21.22
5.06 14.15 20.00
4 10 14
Table 3.1: Calibration measurements on a diffraction grating. The first column lists the current in the prism coils during the calibration. The second column is the scaling factor per neutron wavelength squared expected by dead reckoning on the known distances and field strengths. The final column is the scaling factor found from the calibration on the grating.
1.2 1.0 0.8
P/P0
0.6 0.4 0.2 0.0 -0.2 -0.4 0
100
200
300
400
Spin Echo Length (nm) Figure 3.13: Measurement on the ASTERIX beamline of a 140 nm diffraction grating, as compared with prediction from dynamical theory.
82
laser alignment system, the laser is blocked by the spin echo components on the beamline. Initially, all of the spin echo coils upstream of the beamline had to be removed whenever we wanted to align a grating sample. This was effective, but time consuming, since the spin echo would need to be returned once all of the components were returned to be beamline. Alternately, we attempted to line up the reflection samples via the neutron beam. However, the precision of this method was poor and we often wasted an entire day of beam time without ever getting the desired reflection. To solve this, I built a laser alignment system (Laser B) that can be mounted onto the beamline at will (Figure 3.14). Before the spin echo components are mounted, the Laser B is mounted on the beamline and aligned with the main laser. Laser B is mounted as close to the sample position as is feasible on the upstream rail and this mounting position is marked on the rail with a marker. An optical beam splitter divides the ASTERIX laser into a beam which follows the neutron trajectory and a second beam which hits the ceiling. Laser B is mounted on the alignment unit so that the beam will pass through the splitter, again approaching the ceiling and following the main beam. The splitter is mounted on a vertical translation state to ensure that the ASTERIX laser and the alignment laser pass through the splitter at the same point. Three alignment screws on the splitter control the roll,pitch, and yaw of the splitter. The splitter is rotated so that the two laser beams arrive at the same place both on the detector and on the ceiling. This ensures that the alignment beam is truly parallel to the ASTERIX laser. Laser B can then be removed from the beamline. When it comes time to align a sample, the Wollaston prism pair closest to the sample is removed and the laser alignment system is returned to the marked position. Once the sample is aligned, Laser B is removed and the prism pair can be returned. Finally, the instrument needs to be re-tuned for echo, since the Wollaston prism may have moved. Since the calibration procedure is time consuming, even after the addition of the laser
83
Figure 3.14: A photo of the laser alignment frame used when the spin echo elements are on the beamline. The translation and rotation stage on the far right are used for coarse alignment, then the two split beams are brought parallel by adjusting the rotation screws on the mount for the beam splitter.
84
alignment system, most measurements were taken without a calibration run. Instead, the runs from Table 3.1 were used to create an interpolation betwen the dead reckoning and the measured values. The linear regression gave a fit of y = 0.910x − 0.405. This approximation matches all of the measurements to within 1.5%, as opposed to the 15% from dead reckoning. The measurements of the distance between the prism pairs were accurate to within a millimeter, so the change in the spin echo length measurements were almost certainly within the magnetic field. The spin echo length calculations in Equations 1.7 and 1.12 assumes that the field magnitude within the prisms is constant, which is not necessarily true due to the uncertainties in the field strengths mention in Section 1.8.1. An obvious extension of this research would be to repeat the simulations from Section 1.8.2 and compare the simulated relationship between scattering angle and current with the expected values from 1.6.
3.7
SOLENOID CURRENTS
As part of the upgrade of the ASTERIX beamline, two sets of power supplies were added. Four 3 kW power supplies were added to provide current to the Wollaston prisms and two smaller supplies were added to provide power to the remaining spin echo elements. The power for the prism pairs is provided by Four TDK-Lambda GEN150-22-LAN3P208 power supplies. Each power supply had a maximum voltage of 150 V and a maximum current of 22 A. As the combined resistance of a prism pair is 2.2 Ω, the power supplies are heavily current limited, but they also allow for the development of future generations of Wollaston prisms that may be larger and have more resistance. The power supplies also included computer control over Ethernet and safety cutoff facilities which could connect to a water flow switch. At 22A, the factory stated line regulation was 4 mA and the load regulation was 9 mA. As the coils would be kept at constant current, the load should be constant the load regulation should be less significant than the line, which is well within
85
the safe range for the SESAME tuning. As the power supplies were several meters away from the neutron beamline, 4 gauge wire was used for transporting the current between the supplies and the beamline to ensure that the voltage loss was minimal during transport. In addition to the TDK-Lambda supplies, two Hameg HMP4040 supplies[24] were installed. Each Hameg suppled had four power channels with a maximum current of 10 A and 32 V. Between the four supplies, the maximum power is limited to 384 W. These eight power channels gave individual power control to the seven remaining spin echo units on the beamline: 2 π2 flippers, 1 π flipper, 2 guide fields, a sample guide, and the compensator guide discussed in Section 3.5.2. Since none of these coils was water cooled and would be run at around 5 A, the current limit of this supply was more than adequate. Again, this supply was programmable over Ethernet. One unfortunate quirk of the Hameg supplies was the way in which they enforce their power limits. The current output of each channel is limited to the current which would prevent the power supply from exceeding its maximum power at the set voltage, even if that voltage has not been reached. For instance, if two channels are both set to 30 V and 10 A and given a 1 Ω resistance, then the first channel will output 10 A at 10 V, but the second channel will only output 2.8 A at 2.8 V, since 30 V at 10 A and 30 V at 2.8 A combine to 384 W. Lowering the set voltage on each channel to 15 V will bring the current on the second channel up to 10 A, since the set power of each channel now combine to under 384 W. These changes to the set voltage never effect the actual output voltage of the first channel, which remains at 10 A.
3.8
CADMIUM SLITS
Neutrons which do not interact with the sample obviously do not contribute to the useful signal and just generate background. Therefore, the beam needs to be constrained to ensure that all neutrons which reach the detector have passed through the sample. For our initial
86
Figure 3.15: A diagram of the aluminum backing for the 2 mm tall cadmium mask. The two triangular notches at the bottom mount onto the mask holder to keep a reproducible alignment. measurements, we used ad hoc cadmium masks mounted directly to the sample, as seen in Figure 3.11. However, using these ad hoc slits was imprecise and generated cadmium waste. Replacing the ad hoc cadmium slits with motorized slits wasn’t feasible, as the motorized slits would eat valuable beamline space. Additionally, placing large motorized slits between the second Wollaston prism pair and the sample guide would force the solenoids further apart and require a larger field to maintain precessing polarization within the gap. Therefore, we decided to stick with a set of fixed cadmium masks. Each mask consisted of a 1 mm thick aluminum backing with a slit through the center (Figure 3.15). Two strips of 1 mm thick cadmium are cemented on either side of the slit to provide definition to the beam. Each aluminum backing has two triangular notches at the bottom to ensure that the slits are placed reproducibly. By combining a mask with a horizontal slit with a mask with a vertical slit, the entire beam can be defined while using only 4 mm of beamline space. The original design for the mounting system for the cadmium slits was a simple aluminum bar with two triangular teeth for mounting the slits. Posts on either side of the bar prevented the slits from falling off. These posts then mounted to the sample guide to hold the unit in place. While effective, this design had two problems. First, there was no guarantee that the mask mounting bar was perpendicular to the posts. As a result, the masks could be rotated around the x ˆ axis, allowing neutrons to bypass the sample. Secondly, once
87
Figure 3.16: Second generation mask mounting frame for the cadmium slits on the ASTERIX beamline. the mask mounting frame had been attached to the sample guide, there were no degrees of freedom besides the accidental rotational one. If the sample guide was a millimeter off center, the entire guide would need to be moved to center the masks, despite a millimeter of translation on the sample guide being an otherwise safe tolerance for the beamline. Since the sample guide was not designed with a degree of freedom in this direction, either, this could force the slits off center for the entire experiment. I solved this problems by designing a second generation of slit mount (Figure 3.16). Two orthogonal degrees of freedom were added to the mask mount to allow for the alignment of the slits independent of the alignment of the sample guide. Two slotted holes at the bottom of the mount allowed for translation in the zˆ direction. The mounting bar was screwed into two slotted groves in the yˆ direction for the second degree of freedom. The mounting bar
88
Figure 3.17: The 12 3 He pencils were divided into three channels. The four innermost tubes (Gray) are on a channel labeled “Center”. The two outmost tubes on each side (Checkerboard) are on a channel labeled “Outer”. The tubes between the “Center” and “Outer” channels (White) are labeled “Sandwich”, as they are sandwiched between the two other channels. was also grooved to rest against the posts on either side, ensuring that it could not rotate. To align the mask mount, the 1 mm slits in both the horizontal and vertical directions would be placed on the sample bar. The slit hole would then be aligned against the ASTERIX laser, giving the slits a position accuracy of 1 mm.
3.9
DETECTOR
Early experiments were made with a linear position sensitive detector (PSD). Unfortunately, the PSD could only accept low count rates (≈ 250 neutrons second ) without sustaining damage. This isn’t a significant issue in reflectometry, since the beam flux is limited by the need for high collimation. With SESAME, however, large, divergent beams are well handled, leading to the possibility of using a large flux. To solve this problem, an array of twelve 3 Hepencils were added to the beamline as an alternate detector. Each tube had 0.5” diameter, was 6” long, and contained 3 Heat 10 atm of pressure. The 3 He pencils were capable of accepting far higher fluxes than the PSD, at 89
Figure 3.18: The borated polyethylene shielding for the detector on the ASTERIX beamline. the cost of losing the position sensitivity. Since the SESAME technique depends only on the neutron beam’s final polarization and not the position of neutron events, the trade off was acceptable. The detector tubes were combined into three channels of four tubes each, as seen in Figure 3.17. The 3 He tubes sat inside a shielding array of borated polyethylene (Figure 3.18). The shielding can accommodate both the 3 He pencils and the linear PSD simultaneously. The entire shielding setup is mounted on a rotary track to allow the detectors to be rotated by 90◦ , changing the measuring direction of the PSD or the orientation of the 3 He tubes. An alignment screw prevents the shielding from rotating during the experiment and ensures that the 90◦ rotations are reproducible.
3.10
POLARIZATION ANALYZER
The original polarization analyzer on the SESAME beamline was a single super-mirror analyzer set 863 mm from the sample position. The analyzer was 914 mm long and set at a
90
Spin Echo Length (nm) 200
400
600
800
1000
1200
λ−2 log PP0
-1
-2
-3
-4
Figure 3.19: A comparison of the expected λ−2 log PP0 with both an infinite detector (Green Stars) and with the limits on scattering created by the analyzer (Blue Diamonds). Both simulations were ran on 200 nm diameter hard spheres. The spin echo constant was 10.9 nm ˚ A−2 −2 and the scattering was 0.036 ˚ A 1◦ angle to the beam. This provides a 16 mm horizontal aperture over which the beam could be analyzed. This left a maximum scattering angle of 0.5◦ from the sample to still pass through the analyzer and reach the detector. The assumptions of the SESAME technique require that all of the scattering from the sample is measured on the detector. A simulation of the scattering missing the detector can be seen in Figure 3.19. All scattering in the nonencoding direction keeps the polarization of the main beam. If this scattering cannot reach the detector, the encoded scattering weights more heavily on the final signal and brings down the overall polarization. To avoid this issue, the analyzer from the old POSY beamline at IPNS was installed as part of the ASTERIX upgrade. The analyzer has a total acceptance of 5 cm by 8 cm. The analyzer has a series parallel neutron guide channels at the center (Figure 3.20). Each channel is diagonally crossed by a polarizing supermirror that transmits the desired spin state. On either side of the central channels are angled channels to catch the scattered
91
Figure 3.20: A diagram of the operation of the IPNS analyzer. The solid black lines are neutron guide mirrors. The dashes green lines are polarizing supermirrors. Angles are magnified for illustrative purposes. beam. Unfortunately, this analyzer exhibits the angular and wavelength dependence issues discussed in Section 2.2.3 The exact cause of this dependence is still being investigated, as it did not appear in initial measurements but became prominent in measurements made in 2013. Using the difference measurement protocol discussed in Section 2.2.3, it’s still possible to measure samples with the analyzer in this state, but the analyzer needs to be repaired or replaced before absolute measurements will be possible.
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CHAPTER 4
ANALYSIS OF COLLOIDS
4.1
PERCUS–YEVICK CLOSURE
The interaction of particles in solution can be simplified into the local, two-particle interaction via the Ornstein-Zernicke[25] equation:
h(r) = c(r) + ρ
Z
h r − r′ c r′ dr′
(4.1)
Where h(r) is the “total correlation function” and c(r) is the “direct correlation function”. By construction, h(r) = g(r) − 1, where g(r) is the “radial distribution function”, which is the particle density at a distance r for an arbitrarily chosen point particle, normalized against the average density of the sample. For example, g(r) = 0 in any region which does not contain particles and g(r) = 1 as r → ∞ as density fluctuations are averaged out. The “direct correlation function”, c(r), is defined as the contribution to h(r) that arrises from two particle interactions. Finally, ρ is the average density.
93
Via a Fourier transform, equation 4.1 can be represented as:
c(k) =
h(k) 1 + ρh(k)
(4.2)
h(k)=
c(k) 1 − ρc(k)
(4.3)
Thus, knowing the “direct correlation function” solves for the “total correlation function” and vice versa. However, equations 4.2 and 4.3 don’t contain enough information to solve the values on their own. Therefore, a closure relation is needed to approximate the solution. One such closure relation was proposed by Percus and Yevick[26]:
c(r) = g(r) 1 − eβV (r) Here, V (r) is inter-particle potential and β is
1 kT .
(4.4)
This closure was solved analytically
for the case of hard spheres by Wertheim[27] for a sample with hard spheres in volume fraction ρ.
c(r) =
2 3ρ(ρ + 2)2 (1 + 2ρ)2 3 ρ(1 + 2ρ) − r + r (1 − ρ)4 2(1 − ρ)4 2(1 − ρ)4
(4.5)
~ a Fourier transform needs to be performed to take To find the structure factor, S(Q), c(r) into reciprocal space. Since the hard spheres should be spherically symmetric, only the zeroth order spherical harmonic of the Fourier transform will contribute to the final result.
94
3p c(q)= 4πR3
Z Z Z
c(
|~r| ) expi~q·~r d~r 2R
Z π Z 2π ∞ Z 3p X ∞ r 2 c( )jl (qr)r dr = Ylm (θ, φ) sin φ dθ dφ 4πR3 2R 0 0 l=0 0 ∞ Z r 3p X ∞ c( )jl (qr)r2 dr(4πδl0 ) = 3 4πR 2R l=0 0 Z ∞ r 3p c( )j0 (qr)r2 dr = 3 R 0 2R
(4.6)
Equation 4.5 can be substituted into Equation 4.6 to find the analytical c(q).
3p c(q)= 3 R + =
Z
2R 0
3p R3
Z
j0 (qr) 2 (1 + 2ρ)2 3p r dr − 3 4 2R (1 − ρ) R
2R 0
2ρ)2
j0 (qr) 5 ρ(1 + r dr 2R 2(1 − ρ)4
Z
2R 0
j0 (qr) 3 3ρ(ρ + 2)2 r dr 2R 2(1 − ρ)4
3p sin(2qR) − 2qR cos(2qR) (1 + 2ρ)2 R3 2q 3 R (1 − ρ)4 3p −1 + 1 − 2q 2 R2 cos (2qR) + 2qR sin [2qR 3ρ(ρ + 2)2 − 3 R q4R 2(1 − ρ)4 2 R2 + 2q 4 R4 cos (2qR) + 6qR − 4q 3 R3 sin (2qR) 4 −3 + 3 − 6q 3p ρ(1 + 2ρ)2 + 3 R q6R 2(1 − ρ)4 (4.7)
We can find the structure factor S(Q) from the c(q) in Equation 4.7. This structure factor has been plotted for multiple sphere concentrations in Figure 4.1a.
S(q) =
1 1 − c(q)
(4.8)
The final scattering, I(q), is the product of the structure factor, S(Q), and the form factor, F (q). The form factor is the auto-correlation of a single colloidal particle, which can
95
FHQL 1
0.1 SHQL 2.0 0.01 1.5
Φ=5% 0.001
1.0
Φ=20% 10-4
0.5 Φ=40% QR 2
4
6
8
10
12
QR
14
0
2
4
6
(a)
8
10
12
14
(b)
IHQL
GHZL 1.0
1
0.8
0.1 Φ=5%
Φ=5%
0.6
0.01 0.4
Φ=20%
0.001
Φ=20%
0.2 10
-4
Φ=40%
Z 1
QR 0
2
4
6
8
10
12
14
2
3
Φ=40%
4
-0.2
(c)
(d)
Figure 4.1: Figure (a) is the structure factor predicted by the Percus-Yevick closure for multiple sphere concentrations. The dotted line is 5% volume fraction, the dashed line is 20%, and the solid line is 40%. Figure (b) is the form factor of a perfect sphere. A log scale on the y-axis is needed to show the structure at large Q. Figure (c) is the expected intensity versus Q plot for a system of spheres in the Percus-Yevick closure. Again, the dotted line is 5% volume fraction, the dashed line is 20%, and the solid line is 40%. The majority of the structure information is only visible in the first peak. Figure (d) is the Hankel transform of the results from Figure (c). The higher sphere concentrations lead to oscillations at several multiples of the sphere’s radius.
96
be found by taking the square of the Fourier transform of a single particle with scattering length density contrast ∆ρ [4].
2 F (q)= FSphere(r) 2 ZZZ 3 F (q)= ∆ρei~q·~r d~r 4πR3 Sphere
2 Z π Z 2π ∞ Z R X 3∆ρ = jl (qr)r2 dr Ylm (θ, φ) sin φ dθ dφ 4πR3 0 0 0 l=0 !2 Z R 3∆ρ 2 4π j0 (qr)r dr = 4πR3 0 3∆ρ sin(qR) − qR cos(qR) 2 F (q)= R3 q3
(4.9)
The product of Equation 4.8 and Equation 4.9 is the scattering function for the final hard spheres. By numerically taking the Hankel transform, it’s possible to find G(Z), as seen in Figure 4.1d.
4.2
PURE HARD SPHERES
The Percus-Yevick closure forms a good baseline for testing SESAME on hard spheres. Three different sizes of PMMA hard spheres were prepared by Dr. Andrew Schofield of Edinburgh University. Via dynamical light scattering (DLS), the samples were found to have diameters of 200 nm, 260 nm, and 300 nm. Though these were not the exact diameters, the sphere will be referred to by these nominal diameters to prevent confusion. These spheres were then suspended in deuterated decalin at a 40% volume fraction concentration by Dr. Kunlun Hong of Oak Ridge National Lab. PMMA suspended in decalin was chosen since it was a well established hard sphere system[28, 29]. These spheres were then measured on 97
Figure 4.2a 4.2b 4.2c
Diameter (nm) 200 260 300
Scattering Constant (˚ A−2 ) 4.75 × 10−3 7.25 × 10−3 8.5 × 10−3
Table 4.1: The total scattering for each of the three diameters of PMMA measured. These scattering constants were found by taking the asymptotic value of the measured scattering. the ASTERIX instrument at LANL and on the OFFSPEC instrument at ISIS. The results of these measurements can be seen in Figure 4.2. Each sample has been plotted with the corresponding results expected from PercusYevick, with the total scattering fitted as a free parameter. For Figures 4.2b and 4.2c, the samples behave exactly as predicted. However, as can be seen in Figure 4.2a, the PercusYevick approximation only fits for a 172 nm diameter, as opposed to the expected 200 nm. Similar fits find minor variations for the other two samples, with diameters of 263 nm for the 260 nm sample and 305 nm for the 300 nm sample. It’s possible that the DLS did not report the correct diameter, so it’s useful to check against a second parameter. In Table 4.1, the total scattering constants for all three measurements are given. These are then plotted in Figure 4.3, with data points given for the diameters given from DLS and for the diameters found from the SESAME experiments. The total scattering should depend linearly on the radius [9], so a linear regression of the SESAME diameters has also been plotted. As can be seen in Figure 4.3, a linear relationship can be established for either the expected or fitted diameters. One point of concern, however, is that the x-intercept of the fit should be at zero, since the relationship is a direct proportion. The intercepts have an uncertainty of ±10 nm, so the linear relationship from the diameters fitted from SESAME is within error of passing through the intercept, while using the DLS diameters gives a nonphysical result. However, the slope given to the relationship is 5.9 × 10−5 ˚ A−2 nm−1 , while
98
Total Scattering (˚ A−2 )
0 -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008
ISIS 200nm 40% 200nm spheres by PY 40% 172nm spheres by PY
0
200
400 600 Spin Echo Length (nm)
800
1000
(a) 200nm PMMA
Total Scattering (˚ A−2 )
0
ASTERIX 130nm 40% 260nm spheres by PY 40% 263nm spheres by PY
-0.002 -0.004 -0.006 -0.008 -0.01
0
200
400 600 Spin Echo Length (nm)
800
1000
Total Scattering (˚ A−2 )
(b) 260nm PMMA
0 -0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014
ISIS 150nm 40% 300nm spheres by PY 40% 305nm spheres by PY
0
200
400 600 Spin Echo Length (nm)
800
1000
(c) 300nm PMMA
Figure 4.2: Three different radii of 40% pure PMMA hard spheres. In each plot, the solid squares are the experimental data taken at either ASTERIX or ISIS. The solid, black lines are the predictions made via Percus-Yevick, taking the total scattering strength of the sample (the “a” constant) as the only free parameter. The dashed, blue lines also allowed for the fitting of the radius of the particles in addition to the scattering length. The fitted diameters for the 260 nm and 300 nm matched expectations to within a few nanometers, as seen in Figures (b) and (c). The 200 nm spheres, however, were quite far from their reported values and fit best with a 172 nm model radius.
99
Total Scattering ˚ A−2
Total Scattering versus Particle Size Expected Diameter Fitted Diameter −2 nm−1 (x-05.5 nm) 0.01 y=5.9e-05 ˚ A−2 y=7.9e-05 ˚ A nm−1 (x-39.1 nm) 0.008 0.006 0.004 0.002 0
0
50
100 Fitted Particle Radius
150
200
Figure 4.3: Comparison of the effect of the deviation of the diameter of the “200nm” spheres with respect to the total scattering constant. The solid points assume a radius of 200 nm for the smallest spheres, while the hollow points assume at 180 nm radius. The solid line is a linear fit of the fitted diameter data while the dashed line is a linear fit of the expected diameters. The x intercept for both lines are given in the legend. the hard sphere model predicts a smaller slope of 4.2 × 10−5 ˚ A−2 nm−1 . This contradiction shows that the sample does not match a purely hard sphere model.
4.3
CORE SHELL MODEL
One area where the hard sphere model is known to be inaccurate for these samples is that the spheres are not pure PMMA. A small amount of poly(12-hydroxystearic acid) (pHSA) is added to the PMMA spheres to stabilize the spheres. This pHSA forms a corona around the PMMA core, making the core-shell model a more accurate representation of the samples than the hard sphere model. The scattering length densities of the PMMA, pHSA, and the solvent are listed in table 4.2. The contrast between the pHSA and the PMMA is of the same order of magnitude as the contrast between the PMMA and the solvent, so the pHSA could be a significant contributor to the total scattering. The total scattering a of the core shell can then be calculated for spheres with volume 100
Material SLD (˚ A−2 )
PMMA 1.05 × 10−6
pHSA −6.73 × 10−8
Solvent 2.89 × 10−6
Table 4.2: Scattering length densities for the PMMA samples. fraction φ, sphere volume V , form factor F (q), and structure factor S(q):
φ(1 − φ) a= 2πV
Z
∞
qF (q)S(q) dq
(4.10)
0
The scattering of a core shell can be found via a similar procedure to equation 4.9.
F (q) =
1 ZZZ
V 4π = V 4π = V
2 ρ(r) − ρso ei~q·~r d~r
Z
Rc
0
(ρc − ρso)j0 (qr)r2 dr +
(ρc − ρso )
Z
Rs Rc
2
(ρsh − ρso )j0 (qr)r2 dr
sin (Rc q) − Rc q cos (Rc q) q3
(4.11)
2 sin (Rs q) − Rs q cos (Rs q) − sin (Rc q) + Rc q cos (Rc q) + (ρsh − ρso ) q3
By plugging equation 4.11 into equation 4.10, it’s possible to calculate the expected total scattering for a core shell model and, thereby, the expected measurements for these samples. This model can then be fit to the experimental data by using the core radius, corona radius, and scattering length density of the corona. Since the corona should contain both pHSA and solvent, the SLD could range between 0.046 × 10−6 and 2.86 × 10−6 . Furthermore, the fit on the corona thickness was clamped to a range from 10 nm to 20 nm, based on what was known about the pHSA used in the sphere preparation. All three sphere samples were made using the same pHSA, so the corona thicknesses must be identical. The results of the fitting can be seen in Table 4.3 and the corresponding fits are in figure 4.4. The fitted total fitted diameter of the 260 nm spheres was close to the results 101
Sample 200 nm 260 nm 300 nm
Core Diameter (nm)
Shell Thickness (nm)
128 218 278
20 20 20
Corona SLD (˚ A−2 ) 35.4 × 10−8 4.6 × 10−8 4.6 × 10−8
Table 4.3: Fitting coefficients under the core-shell model. Note that twice the shell thickness needs to be added to the core diameter to achieve the total diameter. The fits have an uncertainty of ±1 nm for the core diameter, ±0.5 nm for the shell thickness, and ±7 × 10−8 ˚ A−2 for the scattering length density. reported by DLS, but the 300 nm samples and 200 nm samples were much further away. It is possible that this was an error in the DLS measurement and not in the SESAME results, especially since a similar radius is reported by the ASTERIX results. The fitted scattering length density shows that the corona is pure pHSA for the two large sphere sizes 89% pHSA for the 200 nm sphere with the rest being solvent.
4.4
TIME DEPENDENCE
As an experiment of the stability of the system, the smallest spheres were measured in May, left for half a year, and remeasured in November. The comparison between the two measurements appears in Figure 4.5. It is clearly evident that some aggregation has occurred, since the depth of the first minimum has filled in. Additionally, the measurement taken in November approach an asymptotically higher value than the earlier measurement. The total scattering is 15% smaller than in the original experiment. Since no material was added or removed from the sample during this time, the volume fraction and neutron scattering length density contrast could not have changed. Both measurements were taken in the same cell, so the thickness could not have changed. The correlation length for the G(Z) found in this measurement is 69 nm, which is 57% larger than the 44 nm correlation length measured in May. The current hypothesis is that some of the spheres may be dissolving in the solution with time. After a year, another SESAME experiment was performed on the
102
Total Scattering (˚ A−2 )
0
-0.004 -0.006 -0.008 -0.01
Total Scattering (˚ A−2 ) Total Scattering (\si{\per\angstrom\squared})
ISIS 300nm 278nm core with 20nm corona
-0.002
0
0
400 600 800 Spin Echo 218nmLength core with 20nm corona
1000
(a) 300 nm ISIS Measurement
-0.002
0 -0.004 -0.001 -0.002 -0.006 -0.003 -0.004 -0.008 -0.005 -0.006 -0.01 -0.007 -0.008 -0.009 -0.012
200
ASTERIX 260nm 218nm core with 20nm corona
00
200 200
400 600 400 600 Spin Length SpinEcho Echo Length
800 800
1000 1000
(b) 260 nm ASTERIX Measurement
Total Scattering (˚ A−2 )
0
ASTERIX 200nm 128nm core with 20nm corona
-0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007
0
200
400 600 Spin Echo Length
800
1000
(c) 200 nm ISIS Measurement
Figure 4.4: Fits of the spin echo data on pure sphere samples to a core shell, Percus-Yevick model. For each sample, the radius of the core, the radius of the shell, and the scattering length density of the corona were all used as free parameters. The total scattering was not taken as a fitting parameter and was calculated from the model. The solid lines are the fits, while the dashed lines are the hard sphere fits from Figure 4.2.
103
Total Scattering
0 -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008
May November
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.5: A comparison of what happens to a pure PMMA sample with time. The solid squares are the 200 nm sample measured at ISIS in May of 2012 (as already seen in Figure 4.2a. The open circles are the exact same sample, measured again in November of 2012. Over the six month interval, aggregation has occurred in sample, as evidenced by the filling in of the first minimum in the spin echo data and the decrease in the total scattering. same sample, and saw no structure, lending credence to this hypothesis.
4.5
SHORT RANGE ATTRACTION
To attempt to understand the depletion interaction, more colloid samples were prepared with the 40% PMMA nanosphere concentrations, but with small amounts of polystyrene (PS). The depletion interaction is an entropic effect due to the available volume of the sample being depleted by the added depletant, in this case polystyrene [30, 31]. If a group of nanospheres have clustered around a volume smaller than the size of the depletant (Figure 4.6a), then there is no partial pressure from the depletant inside the volume, but the pressure remains for the depletant surrounding the group of spheres. This osmotic pressure differential brings the group of spheres closer together, creating an effective attractive potential. Considering that the attractive potential is set by a binary condition on whether or not the depletant can fit between two particles, an intuitive model for this system is the “sticky” potential [32]. The sticky potential is infinite within the radius of the sphere, finitely negative in a small region outside the radius, and zero everywhere else. 104
Potential
0 -u
Sticky potential 0
R R+ǫ Distance
Figure 4.6: Figure (a) is a diagram of a depletion interaction. The hollow circles are the spheres and the solid circles are the depletant. The dashed circle highlights the regions where there is not enough space for the depletant to fit. Random interactions with the depletant outside this region will cause the spheres to move further into the region while the lack of depletant in the region will fail to provide any corresponding pressure to push the spheres apart. Figure (b) is a rough drawing of the stick potential. R is the radius of the spheres, ǫ is the range of the stickiness, and u is the strength of the attractive potential. The two different sizes of PS chain were used: 900 kDa and 110 kDa, which correspond to 36.6 nm and 12.8 nm, respectively in a theta solvent. The first of these measurements was a 0.2% 900 kDa PS added to the nominally 260 nm spheres from section 4.2. This sample was measured on ASTERIX and can be seen in Figure 4.7. There’s a couple of changes that are immediately noticeable in the new sample. First, while both samples show a minimum near 200 nm, the depth of the minimum in the sample with the polystyrene is much smaller than for the pure sphere sample. This minimum filling is indicative of a short range attraction with the spheres spending more time in direct contact than in the pure sphere solution. The other major difference is that the sample with the polystyrene shows greater scattering than the pure sphere solution. Since the increased attraction between the spheres would increase the correlation length of the sample, the total scattering should also increase. Another sample was prepared with the nominally 100 nm spheres and 0.3% 900 kDa PS. This sample, seen in Figure 4.8 shows even stronger correlations at short length scale, with the first minimum not even dipping as deeply as the second order maximum for the
105
Total Scattering (˚ A−2 )
0 -0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014
260nm spheres + 0.2% 900k PS on ASTERIX Pure 40% 260nm spheres
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.7: A measurement on the ASTERIX beamline of the nominally 260 nm spheres with 0.2% 900 kDa PS (solid) as compared to the spheres alone (hollow).
Total Scattering (˚ A−2 )
0
200nm sphere + 0.3% 900k PS on ASTERIX ISIS Measurement of 200nm spheres
-0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.8: A measurement on the ASTERIX beamline of the nominally 200 nm spheres with 0.3% 900 kDa PS (solid) as compared to the spheres alone (hollow). pure sphere sample. However, it’s also noticeable that the long range values appear to be approaching a total scattering smaller than that of the pure spheres, when we would expect it to be larger than the total scatting of even the sample with 0.2% 900 kDa PS. As will be seen in Section 4.6, this issue with the total scattering may merely be an artifact of the limited spin echo range of the measurement. The final few points in Figure 4.8 show a downward trend. If the data continues to drop beyond 1 µm, it’s possible the total scattering is greater than for the pure spheres and that the apparent higher scattering constant is actually the continued presence of correlations out at the micron length scale. A second sample with 0.2% 900 kDa PS was prepared, this time with the nominally 106
Total Scattering (˚ A−2 )
0
200nm spheres + 0.2% 900k PS on ISIS Pure 40% 200nm spheres
-0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.9: A measurement on the OFFSPEC beamline at ISIS of the nominally 200 nm spheres with 0.2% 900 kDa PS (solid) as compared to the spheres alone (hollow).
Total Scattering (˚ A−2 )
0
200nm sphere + 0.3% 900k PS on ISIS ISIS Measurement of 200nm spheres
-0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.10: A measurement on the OFFSPEC beamline at ISIS of the nominally 200 nm spheres with 0.3% 900 kDa PS (solid) as compared to the spheres alone (hollow). 200 nm spheres. This sample was measured at ISIS and again compared with a measurement of the spheres without the PS. As can be seen in Figure 4.9, the sample with the polystyrene is almost identical to the sample without it, in a stark contrast to the 260 nm data measured at ASTERIX. Furthermore, the same sample from Figure 4.8 was remeasured at ISIS, as seen in Figure 4.10. While a small fill in of the first minimum is visible in the ISIS data, the effect of the polystyrene is not nearly as strong in the ISIS measurements as in the measurements at ASTERIX. Since the measurements from Figures 4.10 and 4.8 were taken from the same batch of sample, there shouldn’t be any chemical differences between the two. Our current best 107
Total Scattering (˚ A−2 )
0
200nm sphere + 0.3% 110k PS on OFFSPEC ISIS Measurement of 172nm spheres
-0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.11: A measurement on the OFFSPEC beamline at ISIS of the nominally 200 nm spheres with 0.3% 110 kDa PS (solid) as compared to the spheres alone (hollow). hypothesis is that the difference arises from the sample handling performed at both beam lines. The samples are always sonicated before measurement to ensure that the system is in equilibrium. However, the sonicator at LANL is in poor condition, while the ISIS sonicator is well maintained. It is possible that the ISIS results represent the true equilibrium condition of the samples, which does not deviate very far from the sample without a depletant, while the short range aggregations seen at LANL are a meta-stable state that will eventually collapse back to a largely Percus-Yevick structure factor. Another possibility is that this is another, undetected problem with polarization analyzer, but this seems unlikely, since the polarization analyzer was correctly able to measure the pure PMMA samples in Figure 4.2b. This conclusion is further supported by additional measurements made on the identical samples between the two beam lines. Figures 4.12 and 4.11 are both of a 200 nm sphere sample with 0.3% 110KDa PS, as opposed to the 900 kDa PS used in Figures 4.8 and 4.10. Unfortunately, the angular dependence of the IPNS analyzer on ASTERIX prevented the direct measurement of the sample, but it was possible to measure the difference between the sample with and without the polystyrene. The ASTERIX data shows that the sample should have experienced a fill in of the first minimum on the order of 2 × 10−3 ˚ A−2 , while the ISIS data shows half of that. 108
Total Scattering (˚ A−2 ) Difference
0.003 0.0025 0.002 0.0015 0.001 0.0005 0 -0.0005 -0.001
200 nm sphere + 0.3% 110k PS
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.12: A measurement on the ASTERIX beamline of the difference between the nominally 200 nm spheres with and without the 0.3% 110 kDa PS. It is also worth noting that while neither sample measured at ISIS saw a particularly large change from the addition of the polystyrene, the sample with the 0.3% 110k PS saw far more attraction than the sample with the 900 kDa PS. On the one hand, this might seem unexpected, since the potential range on the 110 kDa samples was one third the range as on the 900 kDa samples. On the other hand, both samples had 0.3% PS added by weight. Since the 900 kDa samples are over eight times heavier than the 110 kDa samples, the 110 kDa samples had eight times the molar concentration of the depletant. This extra concentration could account for the stronger attraction in these samples.
4.6
LONG RANGE AGGREGATES
With the small increase from 0.3% 900 kDa PS to 0.5% 900 kDa PS, the samples undergo a phase change, as can been seen in Figure 4.13. The same sample was measured at both LANL and Delft. Figure 4.13a shows that the ASTERIX and Delft data match on this measurement, so the results are for the equilibrium state and are not an artifact of the poor sonicator at LANL. The total scattering for the depleted sample also approaches a value that is 4.3 times greater than that of the pure spheres.
109
Total Scattering (˚ A−2 )
0 -0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 -0.04
260nm spheres with 0.5% 900k PS at Delft 260nm spheres at ASTERIX 260nm spheres with 0.5% 900k PS at ASTERIX
0
Total Scattering (˚ A−2 )
0
2
4
6 8 10 12 Spin Echo Length (µm)
14
16
18
260nm spheres + 0.5% 900k PS at Delft 260nm spheres at ASTERIX 260nm sphere with 0.5% 900k PS at ASTERIX
-0.002 -0.004 -0.006 -0.008 -0.01
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.13: 40% 260 nm spheres with and without 0.5% 900 kDa PS, as measured at LANL and Delft. Figure (b) is a magnification of the upper left hand corner of Figure (a). The hollow circles are the pure spheres measured at ASTERIX. The solid circles are the ASTERIX measurements of the spheres with the polystyrene. The squares are the same sample, measured on a beamline at Delft. The break in the ASTERIX measurement comes from two different measurements at two different currents. The Delft data was taken in a 2 mm cell, while the ASTERIX measurements were made in a 1 mm cell, so the original Delft values have been divided by two for a more direct comparison.
110
Total Scattering (˚ A−2 )
0
Measurement of 300nm spheres Measurement of 300nm sphere with 1.0% 110k PS
-0.002 -0.004 -0.006 -0.008 -0.01 -0.012
0
200
400 600 Spin Echo Length (nm)
800
1000
Figure 4.14: 1% 110 kDa PS added to 40% 300 nm spheres (solid) compared with the spheres without the PS (hollow). As an extension of the 110 kDa data, another sample with the nominally 300 nm spheres with 1.0% 110 kDa PS was measured. The results, seen in Figure 4.14, again show a very strong short range attraction, as we have seen in all of the ASTERIX measurements. Since the length scales involved are so large, it was possible to probe this sample via optical microscopy. In addition to the SESAME measurements, two of the long range aggregate samples were examined via microscopy. The microscopy was performed by Dr. Xin Li on a microscope provided by Professor Bogdan Dragnea. The 40% 260 nm with 0.5% 900 kDa PS was chosen for microscopy since the long range structures would be visible in the optical range. The long range aggregates can clearly be seen in Figure 4.15a. The aggregates formed a variety of sizes and the right side of the image appears to indicate that single, loose spheres are still present within the solution. Each image is 40 µm across. Microscopy was also used on the 40% 300 nm PMMA sample with 1% 900 kDa PS. From Figure 4.15b, it can be seen that the spheres form together into a network with large excluded regions.
111
Figure 4.15: Fig. (a) is a microscope image of the 40% 260 nm PMMA with 0.5% 900 kDa PS. Fig. (b) is a microscope image of the 40% 300 nm PMMA with 1% 110 kDa PS.
112
CHAPTER 5
CONCLUSIONS
5.1
INSTRUMENT DESIGN
Through our work on the beamlines as LENS and LANL, we have shown that it’s possible to build a simple, modular SESAME instrument which can be mounted onto an existing polarized neutron beamline. I have proven that tuning the beamline exactly is possible in the theoretical sense via the adjustment of three solenoid currents. In simulations and real experiments, I’ve shown that adjusting the current in a single guide field can reach an acceptable spin echo and can do so on a reasonable time scale. An uncertainty in the positions of the spin echo elements of 1 mm and an uncertainty in the magnetic field intensities of 0.5% will provide and average echo polarization of 99% and a worst case polarization of 95%. My work on ASTERIX has shown that the SESAME technique can work as an adaptation to an existing polarized beamline, instead of a as a dedicated instrument. As beamline space is always at a premium, the ability of the technique to be used on non-dedicated instruments is a huge advantage. While SESAME may impose some trade-offs against traditional SANS or reflectivity (e.g. valuing a detector’s maximum neutron flux over its spatial resolution), these decisions can be made at the time of the experiment, allowing 113
both techniques to use their optimum arrangements. The implementation of SESAME can also be performed largely with commercially available equipment (e.q. power supplies) with only the solenoids requiring custom machining. Even these solenoids can be wrapped on a magnet wrapper with no modifications required. An unexpected, but crucial lesson from the development of the beamlines has been the importance of the polarization analyzer. First, the analyzer must have a large enough aperture to collect all of the scattered neutrons from the sample. However, there cannot be any gross changes in the polarizing efficiency versus position or trajectory to ensure that the correct normalized polarization is returned. This ultimately means that super-mirror arrays are not a good candidate for analyzers on SESAME beamlines and that 3 He analyzers will provide better results. A second surprising less is the importance of the wavelength range. The optimum use of beam time is not a single measurement over a long range of wavelength, but multiple measurements at different solenoid currents to best utilize the maximum flux of the instrument.
5.2
HARD SPHERE DEPLETION
Our research on hard sphere colloids has again confirmed that PMMA nano-spheres suspended in decalin do behave as hard spheres under the Percus-Yevick closure. We have also shown that the total scattering constant can also provide important information about the sample, such as the length scale. We were also able to extract the thickness of the corona and information about its scattering length density. When we added small amounts of polymer, we did observe short range attractions in between the spheres, as would be expected by the depletion potential. However, the strength of this attraction varied greatly between measurements made on ASTERIX and
114
on OFFSEC. This indicates that this short range attraction isn’t an equilibrium condition based on the “sticky” sphere potential, but that at least one of the measurements was of a non-equilibrium state with a long lifetime. While an obvious extension of this research is to study the depleted samples on long time scale to observe the transition, we have seen that the PMMA spheres are not stable on the timescale of months even without polystyrene. As the polymer concentration increases, the system will change from having a short range attraction to forming long range aggregates. These aggregates were seen on both instruments and via microscopy, indicating that this is likely an equilibrium system. The aggregates have been seen at length scales beyond twenty times the sphere’s diameter.
115
APPENDIX A
DERIVATION OF EFFICIENCY MEASUREMENTS
We can solve the efficiency of the beamline by assuming that the neutron beam is a classical, two-element vector: ( u d ) where u is the number of neutrons in the spin up state and d is the number of neutrons in the spin down state. The neutron beam out of the moderator can be assumed to be ( 1 1 ). Polarizing elements of the beamline are then 2x2 matrices with a large change in magnitude along the diagonal.
1+p 2a
analyzer =ta
0
1+p 2s
super = ts
0
1+p 2b
bender = tb
0
(A.1)
(A.2)
0
1−pa 2
0
1−ps 2
0 ≈N 1+pb 2
1−pb 2
1−pb 2
(A.3)
In equations A.1 through A.3, the values of pb , ps , and pa are the polarizing efficiencies
116
of the bender, supermirror, and analyzer, respectively. The polarizing efficiency is defined as the measured polarization u−d u+d of a neutron beam which which uses this apparatus as a
polarizer, as well as having a perfect polarization analyzer and a perfect flipper. Thus, p = 1
means an instrument which only allows through spin up neutrons, p = −1 is an instrument which only allows spin down neutrons, and p = 0 performs no polarization descrimination. The probability of aneutron passing through the polarizer being a spin up neutron is
1+p 2 .
The values of t are the transmission coefficients for the polarizing elements. While we are not intending to solve for these values, they are included to make sure that they have cancelled out before we perform any calculations. With equation A.3, we can multiply by the unpolarized beam that comes from the moderator to find the initial polarization of the beam. In addition to polarizing elements, we also have flipping elements, which have non-zero values off diagonal.
f lipper1 =
1+F 1 2 1−F 1 2
∗ t1
(A.4)
∗ t1
(A.5)
∗ t2
(A.6)
∗ t2
(A.7)
1−F 1 2 1+F 1 2
1+f 1 2
1−f 1 2
1+F 2 2
1−F 2 2
1+f 2 2
1−f 2 2
f lipperof f1 =
1−f 1 2
f lipper2 =
1−F 2 2
f lipperof f2 =
1−f 2 2
1+f 1 2
1+F 2 2
1+f 2 2
In equations A.4 through A.7, F represents the flipping efficiency of the flipper in its flipping state (on state) while f is the efficiency of the flipper in its non-flipping state (off 117
state). The t values again represent the transmission coefficients and are again included to insure that any intensity loss due to transmission through the flippers is cancelled out from the final calculation. Finally, we can define a detector simply as detector = both spin states.
A.1
1 1
, since the detector sums over
MEASUREMENTS
Now that the beamline elements have been defined, determining the outcome of a measurement is simply a matter of taking the product of the elements of the beamline in order.
z =bender · f lipperof f1 · f lipperof f2 · super · detector
(A.8)
y =bender · f lipper1 · f lipperof f2 · super · detector
(A.9)
w =bender · f lipperof f1 · f lipper2 · super · detector
(A.10)
v =bender · f lipper1 · f lipper2 · super · detector
(A.11)
u =bender · f lipperof f2 · super · detector
(A.12)
t =bender · f lipper2 · super · detector
(A.13)
s =bender · f lipperof f1 · super · detector
(A.14)
r =bender · f lipper1 · super · detector
(A.15)
n =bender · f lipperof f1 · super · f lipper2 · analyzer · detector
(A.16)
m=bender · f lipper1 · super · f lipper2 · analyzer · detector
(A.17)
k =bender · f lipperof f1 · super · f lipperof f2 · analyzer · detector
(A.18)
The experiments defined by equations A.8 through A.18 have the advantage that the
118
bender and the super-mirror can remain in the same place throughout the measurement. This eliminates any concerns about changes of polarization from the polarizer’s alignment changing between measurements. The analyzer only need to be moved once to be placed in the beam for the measurements in equations A.16 through A.18, so alignment issues are again not an issue there.
A.1.1
SOLVABLE
While the measurements described by equations A.8 through A.18 contain enough information to solve for the polarizing efficiencies and transmission, they are also dependent on extraneous values, such as the transmission of the flippers and the super-mirror. Thus, to simplify solving, a series of intermediate polarization values can be defined to cancel out the undesired values. These polarization values are listed in Table A.1. a
b
c
d
e
f
g
z−y z+y
z−w z+w
w−v w+v
u−t u+t
s−r s+r
n−m n+m
n−k n+k
Table A.1: Intermediate Polarizations
The seven in polarizations in Table A.1 provide enough information to solve for the seven polarizing efficiencies in equations A.4 through A.7.
f1 = − f2 = − F 1= F 1=
b((b − 1)c(d + 1) + a(2cb − db + b − 2cd − d + 1)) b2 − 1 (a − c)d
a(−2be − c(b(e − 1) + e + 1) + a(2cb − eb + b + e + 1)) (a + 1)(b − 1)(a − c)e
(bca + a − b − c)((b − 1)c(d + 1) + a(2cb − db + b − 2cd − d + 1)) (a + 1) b2 − 1 (a − c)(c + 1)d c(−2be − c(b(e − 1) + e + 1) + a(2cb − eb + b + e + 1)) (b + 1)(a − c)(c + 1)e
(A.19) (A.20) (A.21) (A.22) (A.23)
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p2b =2((a + 1)(b2 − 1)2 (a − c)2 (c + 1)2 d2 e
(A.24)
∗ ((b − 1)c(f (g + 1) + e(2f g + g − 1)) + a((2cb + b + 1)f (g + 1) + e(−(2c(f + 1) + 1)g + b(−g + 2c(f g − 1) − 1) − 1))))
(((b − 1)c(d + 1) + a(2cb − db + b − 2cd − d + 1))2 (−2be − c(b(e − 1) + e + 1) + a(2cb − eb + b + e + 1)) (a((e(2(g + 1)c2 + (−f g + 2g + 3)c + g + 1) − (2c + 1)(f (c + g + 1) − cg))b2 + (−2gc2 − f (2gc2 + gc + c + g + 1) + e(2f gc + 3gc + c + g + 1))b − c(e + 1)(f + 1)g) − (b − 1)c(c(e + 1)(f + 1)g + b(−cg + f (c + g + 1) + e(2f g + g + c(f g − 1) − 1)))))
120
p2a =2((a + 1)e
(A.25)
∗ (a(b2 (e(2(g + 1)c2 + (−f g + 2g + 3)c + g + 1) − (2c + 1)(f (c + g + 1) − cg)) + b(−2gc2 − f (2gc2 + gc + c + g + 1) + e(2f gc + 3gc + c + g + 1)) − c(e + 1)(f + 1)g) − (b − 1)c(c(e + 1)(f + 1) + b(−cg + f (c + g + 1) + e(2f g + g + c(f g − 1) − 1)))))
((−2be − c(b(e − 1) + e + 1) + a(2cb − eb + b + e + 1)) ∗ ((b − 1)c(f (g + 1) + e(2f g + g − 1)) + a((2cb + b + 1)f (g + 1) + e(−(2c(f + 1) + 1)g + b(−g + 2c(f g − 1) − 1) − 1))))
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p2s =2((a + 1)e
(A.26)
∗ (a(b2 (e(2(g + 1)c2 + (−f g + 2g + 3)c + g + 1) − (2c + 1)(f (c + g + 1) − cg)) + b(−2gc2 − f (2gc2 + gc + c + g + 1) + e(2f gc + 3gc + c + g + 1)) − c(e + 1)(f + 1)g) − (b − 1)c(c(e + 1)(f + 1)g + b(−cg + f (c + g + 1) + e(2f g + g + c(f g − 1) − 1)))))
((−2be − c(b(e − 1) + e + 1) + a(2cb − eb + b + e + 1)) ((b − 1)c(f (g + 1) + e(2f g + g − 1)) + a((2cb + b + 1)f (g + 1) + e(−(2c(f + 1) + 1)g + b(−g + 2c(f g − 1) − 1) − 1))))
A.2
FULL SOLUTION
Plugging the original values from Table A.1 back into equations A.19 through A.27 gives the polarizing efficiencies in terms of the results of the actual measurements.
f1 = f2 = F 1= F 2=
(w − z)(t(y − z) + u(w − v)) (t − u)(wy − vz)
(y − z)(r(w − z) + s(y − v)) (r − s)(wy − vz) (v − y)(t(y − z) + u(w − v)) (t − u)(wy − vz)
(v − w)(r(w − z) + s(y − v)) (r − s)(wy − vz)
122
(A.27) (A.28) (A.29) (A.30)
The efficiencies of the polarizers are a little more complicated.
p2b =(r − s)(t − u)2 (wy − vz)2 (k(r − s)(v − w) + ms(−v + w + y − z)
(A.31)
+ n(−ry + rz + sv − sw))
(r(z − w) + s(v − y))(t(z − y) + u(v − w))2 ∗ (k(v − w)(r(z − w) + s(v − y)) + ms(w − z)(v − w − y + z) + n(r(w − z)(y − z) − s(v − w)(v − y)))
p2a =((r − s)(wy − vz)2 (k(r − s)(v − w) + ms(−v + w + y − z) + n(−ry + rz + sv − sw)) (A.32) (−k(v − w)(r(z − w) + s(v − y)) + ms(w − z)(−v + w + y − z) + n(r(w − z)(z − y) + s(v − w)(v − y))))
(v − w)2 (k(v − w) + n(z − y))2 (r(z − w) + s(v − y))3
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p2s =((r − s)(−k(v − w)(r(z − w) + s(v − y))
(A.33)
+ ms(w − z)(−v + w + y − z) + n(r(w − z)(z − y) + s(v − w)(v − y))))
(r(z − w) + s(v − y))(−k(r − s)(v − w) + ms(v − w − y + z) + n(ry − rz − sv + sw))
A.3
UNCERTAINTIES
Since measurements A.8 through A.18 are independent, their uncertainties can be assumed to be orthogonal. Therefore, it’s possible to find the uncertainties for the various terms with the formula: v u 2 uX d 2 t f (x1 , x2 , ..., xn ) δf = δxi dxi
(A.34)
i
The uncertainties in the measurements for the flipping efficiencies can be seen below.
δf12
(w − z)(y − z) (w − z)(t(y − z) + u(w − v)) 2 =δt − (A.35) (t − u)(wy − vz) (t − u)2 (wy − vz) (w − z)(t(y − z) + u(w − v)) 2 (w − v)(w − z) 2 + + δu (t − u)(wy − vz) (t − u)2 (wy − vz) 2 u(w − z) 2 z(w − z)(t(y − z) + u(w − v)) + δv − (t − u)(wy − vz)2 (t − u)(wy − vz) 2 y(w − z)(t(y − z) + u(w − v)) u(w − z) 2 t(y − z) + u(w − v) − + + δw (t − u)(wy − vz) (t − u)(wy − vz)2 (t − u)(wy − vz) 2 w(w − z)(t(y − z) + u(w − v)) t(w − z) − + δy2 (t − u)(wy − vz) (t − u)(wy − vz)2 2 t(y − z) + u(w − v) v(w − z)(t(y − z) + u(w − v)) t(w − z) 2 + δz − + − (t − u)(wy − vz) (t − u)(wy − vz)2 (t − u)(wy − vz) 2
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δf22
δF12
(w − z)(y − z) (y − z)(r(w − z) + s(y − v)) 2 − (A.36) =δr (r − s)(wy − vz) (r − s)2 (wy − vz) (y − v)(y − z) (y − z)(r(w − z) + s(y − v)) 2 2 + δs + (r − s)(wy − vz) (r − s)2 (wy − vz) 2 s(y − z) 2 z(y − z)(r(w − z) + s(y − v)) − + δv (r − s)(wy − vz)2 (r − s)(wy − vz) r(y − z) y(y − z)(r(w − z) + s(y − v)) 2 2 + δw − (r − s)(wy − vz) (r − s)(wy − vz)2 2 w(y − z)(r(w − z) + s(y − v)) s(y − z) 2 r(w − z) + s(y − v) − + + δy (r − s)(wy − vz) (r − s)(wy − vz)2 (r − s)(wy − vz) 2 r(y − z) r(w − z) + s(y − v) v(y − z)(r(w − z) + s(y − v)) 2 + − + δz − (r − s)(wy − vz) (r − s)(wy − vz)2 (r − s)(wy − vz) 2
(v − y)(y − z) (v − y)(t(y − z) + u(w − v)) 2 =δt − (A.37) (t − u)(wy − vz) (t − u)2 (wy − vz) (v − y)(t(y − z) + u(w − v)) 2 (w − v)(v − y) 2 + + δu (t − u)(wy − vz) (t − u)2 (wy − vz) u(v − y) z(v − y)(t(y − z) + u(w − v)) t(y − z) + u(w − v) 2 2 + δv − + + (t − u)(wy − vz) (t − u)(wy − vz)2 (t − u)(wy − vz) 2 y(v − y)(t(y − z) + u(w − v)) u(v − y) − + δw2 (t − u)(wy − vz) (t − u)(wy − vz)2 w(v − y)(t(y − z) + u(w − v)) t(y − z) + u(w − v) 2 t(v − y) 2 − − + δy (t − u)(wy − vz) (t − u)(wy − vz)2 (t − u)(wy − vz) 2 v(v − y)(t(y − z) + u(w − v)) t(v − y) + δz2 − 2 (t − u)(wy − vz) (t − u)(wy − vz) 2
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δF22
(v − w)(w − z) (v − w)(r(w − z) + s(y − v)) 2 − (A.38) =δr (r − s)(wy − vz) (r − s)2 (wy − vz) (v − w)(y − v) (v − w)(r(w − z) + s(y − v)) 2 2 + δs + (r − s)(wy − vz) (r − s)2 (wy − vz) z(v − w)(r(w − z) + s(y − v)) r(w − z) + s(y − v) 2 s(v − w) 2 + + + δv − (r − s)(wy − vz) (r − s)(wy − vz)2 (r − s)(wy − vz) r(v − w) y(v − w)(r(w − z) + s(y − v)) r(w − z) + s(y − v) 2 2 + δw − − (r − s)(wy − vz) (r − s)(wy − vz)2 (r − s)(wy − vz) 2 w(v − w)(r(w − z) + s(y − v)) s(v − w) − + δy2 (r − s)(wy − vz) (r − s)(wy − vz)2 2 r(v − w) 2 v(v − w)(r(w − z) + s(y − v)) − + δz (r − s)(wy − vz)2 (r − s)(wy − vz) 2
The individual equations for the uncertainties on the polarizing efficiencies are too large to fit on a single page with proper typesetting, but can be expressed through raw text.
A.3.1
BENDER EFFICIENCY
δp2b = ((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (vw-y+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s)2 (t-u)2 (v-w) (w y-v z)2 )/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (v-w) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δk 2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-wy+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s) s (t-u)2 (-v+w+y-z) (w y-v z)2 )/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r
126
(z-w))))-((r-s) s (t-u)2 (w-z) (v-w-y+z) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δm2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (vw-y+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s) (t-u)2 (s v-s w-r y+r z) (w y-v z)2 )/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (r (w-z) (y-z)-s (v-w) (v-y)) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δn2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (vw-y+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s) (t-u)2 (w y-v z)2 (k (v-w)+n (z-y)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))+((t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (z-w) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w))2 (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (w y-v z)2 (n (w-z) (y-z)+k (v-w) (z-w)) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δr2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-wy+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s) (t-u)2 (-k (v-w)+n (v-w)+m (-v+w+y-z)) (w y-v z)2 )/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-
127
y+z)+k (v-w) (s (v-y)+r (z-w))))-((t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (v-y) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w))2 (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (w y-v z)2 (k (v-w) (v-y)-n (v-w) (v-y)+m (w-z) (v-w-y+z)) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δs2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-wy+z)+k (v-w) (s (v-y)+r (z-w))) ((2 (r-s) (t-u) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-(2 (r-s) (t-u)2 (z-y) (w y-v z)2 (k (r-s) (vw)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))3 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δt2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-wy+z)+k (v-w) (s (v-y)+r (z-w))) (-(2 (r-s) (t-u) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-(2 (r-s) (t-u)2 (v-w) (w y-v z)2 (k (r-s) (vw)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))3 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δu2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (vw-y+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s) (k (r-s)-m s+n s) (t-u)2 (w y-v z)2 )/((s (v-y)+r
128
(z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-(2 (r-s) (t-u)2 z (w y-v z) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) s (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w))2 (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-(2 (r-s) (t-u)2 u (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))3 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (k s (v-w)+n (-s (v-w)-s (v-y))+m s (w-z)+k (s (v-y)+r (z-w))))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δv2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (vw-y+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s) (-k (r-s)+m s-n s) (t-u)2 (w y-v z)2 )/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))+(2 (r-s) (t-u)2 y (w y-v z) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))+(r (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w))2 (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))+(2 (r-s) (t-u)2 u (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))3 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (tu)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (-k r (v-w)+n (s (v-y)+r (y-z))-m s (w-z)+m s (v-w-y+z)-k (s (v-y)+r (z-w))))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4
129
(r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δw2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-wy+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s) (m s-n r) (t-u)2 (w y-v z)2 )/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (zw))))+(2 (r-s) (t-u)2 w (w y-v z) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))+((r-s) s (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w))2 (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))+(2 (r-s) t (t-u)2 (w y-v z)2 (k (r-s) (vw)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))3 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (-k s (v-w)+n (s (v-w)+r (w-z))-m s (w-z)) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δy2 +((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-wy+z)+k (v-w) (s (v-y)+r (z-w))) (((r-s) (n r-m s) (t-u)2 (w y-v z)2 )/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-(2 (r-s) (t-u)2 v (w y-v z) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-(r (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w))2 (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-(2 (r-s) t (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))3 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w))))-((r-s) (t-u)2 (w y-v z)2 (k r
130
(v-w)+n (-r (w-z)-r (y-z))+m s (w-z)-m s (v-w-y+z)) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((s (v-y)+r (z-w)) (u (v-w)+t (z-y))2 (n (r (w-z) (y-z)-s (v-w) (v-y))+m s (w-z) (v-w-y+z)+k (v-w) (s (v-y)+r (z-w)))2 ))2 )/(4 (r-s) (t-u)2 (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))δz2
A.3.2
ANALYZER EFFICIENCY
p2a = -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 (((r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((v-w) (s (v-y)+r (z-w))2 (k (v-w)+n (z-y))2 )+(2 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w) (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))3 )-((r-s)2 (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w) (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ))2 )/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))δk2 -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 (-((r-s) s (w-z) (-v+w+y-z) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-((r-s) s (-v+w+y-z) (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ))2 )/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))δm2 -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ((2 (r-s) (z-y) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))3 )-((r-s) (w y-v z)2 (s (v-w) (v-y)+r (w-z) (z-y)) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((v-
131
w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-((r-s) (s v-s w-r y+r z) (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ))2 )/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))δn2 -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ((3 (r-s) (z-w) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))ˆ4 (k (v-w)+n (z-y))2 )-((r-s) (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((vw)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y)))-((r-s) (w y-v z)2 (n (w-z) (z-y)-k (v-w) (z-w)) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((v-w)2 (s (v-y)+r (z-w))3 (k (vw)+n (z-y))2 )-((w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ))2 )/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))δr2 -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 (((w y-v z)2 (k (r-s) (v-w)+m s (-v+w+yz)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )+(3 (r-s) (v-y) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))ˆ4 (k (v-w)+n (z-y))2 )((r-s) (-k (v-w) (v-y)+n (v-w) (v-y)+m (w-z) (-v+w+y-z)) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-((r-s) (-k (v-w)+n (v-w)+m (-v+w+y-z)) (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ))2
132
)/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))δs2 -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ((2 (r-s) z (w y-v z) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )+(2 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)3 (s (v-y)+r (z-w))3 (k (v-w)+n (zy))2 )+(2 k (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))3 )+(3 (r-s) s (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))ˆ4 (k (v-w)+n (z-y))2 )-((r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (-k s (v-w)+n (s (v-w)+s (v-y))-m s (w-z)-k (s (v-y)+r (z-w))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-((r-s) (k (r-s)-m s+n s) (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ))2 )/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+yz)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))δv2 -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 (-((r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (k r (v-w)+m s (w-z)+m s (-v+w+y-z)+k (s (v-y)+r (zw))+n (r (z-y)-s (v-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-((r-s) (-k (r-s)+m s-n s) (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-(2 (r-s) y (w y-v z) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-(2 (r-s) (w y-v
133
z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)3 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-(3 r (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))ˆ4 (k (v-w)+n (z-y))2 )-(2 k (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (zy))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))3 ))2 )/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))δw2 -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 (-((r-s) (k s (v-w)+n (-s (v-w)-r (w-z))+m s (w-z)) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-((r-s) (m s-n r) (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-(2 (r-s) w (w y-v z) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-(3 (r-s) s (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (zy))))/((v-w)2 (s (v-y)+r (z-w))ˆ4 (k (v-w)+n (z-y))2 )-(2 n (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))3 ))2 )/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))δy2 -((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ((2 (r-s) v (w y-v z) (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )+(2 n (r-s) (w
134
y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))3 )+(3 r (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))ˆ4 (k (v-w)+n (z-y))2 )-((r-s) (n r-m s) (w y-v z)2 (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 )-((r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (-k r (v-w)-m s (w-z)-m s (-v+w+y-z)+n (r (w-z)-r (z-y))))/((v-w)2 (s (v-y)+r (z-w))3 (k (v-w)+n (z-y))2 ))2 )/(4 (r-s) (w y-v z)2 (k (r-s) (v-w)+m s (-v+w+y-z)+n (s v-s w-r y+r z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) δz2
A.3.3
SUPER-MIRROR EFFICIENCY
δp2s = ((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) (((r-s)2 (v-w) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (vy)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 )-((r-s) (v-w))/(-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))2 δk2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) +((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) (((r-s) s (w-z) (-v+w+y-z))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))((r-s) s (v-w-y+z) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 ))2 δm2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) +((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) (((r-s) (s
135
(v-w) (v-y)+r (w-z) (z-y)))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (-s v+s w+r y-r z) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 ))2 δn2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) +((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) ((m s (w-z) (v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))+((r-s) (n (w-z) (z-y)-k (v-w) (z-w)))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (z-w) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w))2 (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (n (y-z)-k (v-w)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 ))2 δr2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) +((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) (((r-s) (-k (v-w) (v-y)+n (v-w) (v-y)+m (w-z) (-v+w+y-z)))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-(m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (v-y) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (vy)+r (w-z) (z-y))))/((s (v-y)+r (z-w))2 (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (k (v-w)+n (w-v)+m (v-w-y+z)) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (zw))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 ))2 δs2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) +((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) (((r-s) (-k
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s (v-w)+n (s (v-w)+s (v-y))-m s (w-z)-k (s (v-y)+r (z-w))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) s (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w))2 (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (-k (r-s)+m s-n s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 ))2 δv2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) +((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) (((r-s) (k r (v-w)+m s (w-z)+m s (-v+w+y-z)+k (s (v-y)+r (z-w))+n (r (z-y)-s (v-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))+(r (r-s) (m s (w-z) (-v+w+yz)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w))2 (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (k (r-s)-m s+n s) (m s (w-z) (-v+w+yz)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 ))2 δw2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) +((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) (((r-s) s (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w))2 (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))+((r-s) (k s (v-w)+n (-s (v-w)-r (w-z))+m s (w-z)))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (n r-m s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 ))2 δy2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y)))) +((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)) (((r-s) (-k r (v-w)-m s (w-z)-m s (-v+w+y-z)+n (r (w-z)-r (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m
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s (v-w-y+z)+n (-s v+s w+r y-r z)))-(r (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (zw))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w))2 (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z)))-((r-s) (m s-n r) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))/((s (v-y)+r (z-w)) (-k (r-s) (v-w)+m s (v-w-y+z)+n (-s v+s w+r y-r z))2 ))2 δz2 )/(4 (r-s) (m s (w-z) (-v+w+y-z)-k (v-w) (s (v-y)+r (z-w))+n (s (v-w) (v-y)+r (w-z) (z-y))))
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APPENDIX B
DESIGN OF PAPA CONTROL SYSTEM
The PAPA detector was provided with software by Lexitech. Unfortunately, this software had several deficiencies which left it unsuitable for use on the beamline. The software could not be scripted, meaning that all measurements would need to the started manually. Additionally, the software was written in an outdated version of LabView and could not be edited by more recent versions, so it was impossible to add this scripting capability. The software was also incapable of reading from the neutron beam monitor to give normalized measurements. Therefore, it was necessary to develop a new control system that would handle the PAPA detector and, eventually, the entire beamline.
B.1
LANGUAGE CHOICE
All of the code written for the PAPA control system is in Python. Alternate systems considered were C, LabView, Racket, IDL, and Mathematica. The three primary concerns with the choice of language were applicability, simplicity, and availability. Since multiple graduate students, post-docs, and undergrad would be working on the SESAME beamline, it would be best if all code could be written in a single lingua franca so that everyone will be able to contribute. To this end, the software needs to be useful across a wide variety of
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problem domains (e.g. numerical analysis, GUI programming, server coding, text analysis). Of special importance is the ability to easily interface with the low level hardware used to speak to detector. While LabView has an obvious advantage at hardware control and GUI programming, it’s too low level for conveniently performing many other functions. Similarly, Mathematica would have trouble with server coding and C would be inconvenient at both GUI programming and text processing. Python’s plethora of libraries provided simple starting points for all of these domains. Of special note is the NumPy library[33], which allows Python to perform the rapid array operations needed for much of the numerical analysis and data collection. Since the chosen language will server as a common language for a large number of scientists, it should be easy to learn, preferably building on prior knowledge. Though they can be programmed imperatively, Racket and Mathematica are primarily functional languages and often stymie immigrants from the more common ALGOL derivatives. Labview’s graphical programming techniques are qite approachable for small programs but can be difficult for neophyte programmers to take into larger domains. C’s usage of pointers is infamous as a stumbling block for new programmers. Python is similar enough to C, C++, Perl, PHP, Java, and Fortran that it will be a quick learn for most users coming from those common backgrounds. Additionally, Python is dynamically typed, which won’t require developers from other dynamic languages (e.g. Mathematica, Perl, Scheme) to learn static typing principles. Finally, since this language will be serving on multiple programs in multiple projects across multiple computers, languages with restrictive licenses for development tools will create unnecessary friction in the work environment. LabView, IDL, and Mathematica would each require a license on every machine used for software development. Not only would this require a site license for the group, but the purchase of copies for home computers,
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if any students wanted to work on code from home. Comparatively, Python is available under an open source license and can be freely installed on all computers as needed. An added advantage from choosing to use Python is that the control system is highly portable. Currently, all three computers in the control system are running Windows. With minimal changes, however, the DAQ computer and PowerSupply comptuer could be switched to Linux. Specifically, the Detector class and the PowerSupply class both read from a Windows DLL file for the National Instruments DAQmx drivers to communicate with their corresponding cards. By changing the single line of code in those two classes which refer to the DLL to refer to the corresponding Linux DAQmx shared library, the remained of the software would run unchanged. The bmonserver, however, must remain on Windows, as the C++ code is heavily tied to the Windows OS.
B.2
CONTROL DESIGN
The control system for the PAPA detector runs in five processes across three computers (Figure B.1). The primary process is simply a Python shell with a single instance of the Control object. Since the instrument is controlled via a Python command prompt, the system is programmable by the user at run time. By defining functions at the command prompt, the user can create arbitrary scripts with the full power of the Python programming language. The Control object itself is only responsible for sending and receiving messages from the controlThunk process, which is responsible for most of the detector functions. Despite the name, controlThunk is an independent process and is not a thunk. The name is simply an artifact from an earlier architecture. Keeping controlThunk in a separate process from the Control object allows the user to continue to run commands at the Python prompt while controlThunk performs any request measurements. The primary disadvantage is that exceptions thrown in controlThunk will not be passed back to the Control object. This
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issue is mainly avoided through copious logging, which allows for the diagnosis of faults. At start-up, controlThunk uses an XMLConfig object to read a configuration file which stores important information about the state of the instrument. This configuration file controls where data will be saved and also maintains the current run number for the instrument. A second configuration file is used for storing e-mail settings. If the count rate on the monitor during a measurement drops below the usual count rate, the “mail.json” file is read and e-mails are sent to the addresses contained in the file warning that data is not being collected. This serves both as warning if the accelerator has tripped and as a sanity check against the user running a measurement with the shutter closed. Once the process has started, controlThunk’s primary purpose is marshaling two objects: Coils and Instrument. The Coils object is responsible for setting and reading the power in the Kepco power supplies used to provide current to the spin echo elements of the beamline. The object itself is largely an Extensible Markup Language Remote Procedure Call (XML-RPC) client, which communicates over the network to the “ps server” program running on the power supply computer. “ps server” is an XML-RPC server which wraps the functionality of the PowerSupply object. PowerSupply controls the voltage output of a National Instruments PCI-6703 card. This analog voltage is then fed into the current control on the Kepco power supplies to control the current output. As the system currently stands, there is no read back capability for the currents, as the current levels reached by the power supplies have been too much for any of the shunts we have inserted into the system. Instead, PowerSupply uses a dead reckoning calibration of the supplied voltage versus the returned current for each power supply. This calibration information is stored in a JSON file on the power supply computer. When the PowerSupply object is queried as to the current, it merely returns the requested current. The Instrument object controls all of the data acquisition on the beamline. The inten-
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PAPA Detector
He3 Beam Monitor Legend Coax
T0 Pulse
Digital
User
Mail File
Physical Hardware
Coax
Coax
RS232
Physical Cable
Computer Card
NI DAQ Card
Process Control
bmonserver
NI DAQ Card
Network Connection
Beam Monitor Computer Computer
TCP
XMLManifest
Monitor
Coils
Detector
detectorProcess XMLRPC
Python Class
ps_server Manifest File
Instrument
Monitor File
PowerSupply
Software Connection
Python Shell File
XMLConfig
PEL File
NI DAQ card
Power Supply Computer Twisted Pairs
Config File
Person
Power Supplies
controlThunk
DAQ Computer
Figure B.1: A structural schematic of the control system of the SESAME beamline while using the PAPA detector. The red boxes represent the computers which control the instrument with the blue, rounded boxes inside them representing the different processes running on these computers. The ellipses are the python classes within each process and the dogeared boxes representing files. The connections between software components are denoted with black arrows. Network communication is annotated via the orange arrows, which are labeled with the network protocol. House shaped boxes are used to represent physical cards, whether ADC, DAC, or DAQ. The physical cards are represented by dashed lines, denoting physical cables, to the inverted houses, which represent physical cables and are labeled with the cable type.
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sity and spectrum on the beam monitor are controlled via the Monitor object. The Monitor object communicates with the bmonserver on the Beam Monitor Computer through a TCP protocol developed by the SNS. The monitor object can start and stop the recording of neutron events on bmonserver and download the current intensity spectrum over the network. This intensity spectrum can then be saved as a tab delimited text file. The bmonserver itself was written in C++ by the SNS to read T0 pulses and discriminated neutron pulses on a National Instruments 5112 card. The Instrument object also communicates with a Detector object in the detectorProcess. Detector is controls the PAPA detector and reads the neutron events. The Detector object must be kept in a separate process from Instrument to assure that capturing data from the detector is not interrupted by other operations performed by controlThunk (eg˙ ˙ reading the beam monitor, sending e-mail). Detector communicates with the PAPA along two different routes. The PAPA is controlled by short ASCII commands sent via an RS-232 cable. The Detector object is aware of all of the PAPA’s status variables and records the detector’s state with each measurement. It can also set these states as needed. Of particular importance is the high voltage power supplies for the PMTs and the intensifier. If the detector were to lose power while these high voltage supplies are active, there could be physical damage to the detector. Therefore, the Detector object is designed to always power down the supplies before the program exits, even if the control system has entered an abnormal state. While the detector is controlled by the RS-232 cable, the neutron data is sent through a second, high speed digital cable. This cable is read via a National Instruments PCI-DIO32HS card. During normal operation, each neutron event is a 64-bit field containing the neutron’s measured position, arrival time, and the energy measured by the PMTs. These events are recorded directly to a binary file with no intermediate processing. This file format (PEL) had three large advantages. First, it is the file format of the original detector
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software written by Lexitek, so files written by either control system are interchangeable for comparison. Secondly, recording the raw events allows for event mode data analysis. The time of flight resolution can be chosen at the time of data analysis, instead of at the time of measurement, allowing for greater flexibility. Finally, the binary event mode files tend to be significantly smaller than a histogrammed file. With a 512 pixel x 512 pixel resolution, and assuming 200 time of flight bins, 100 MB are needed to store a completed histogram with 16-bits in each bin. However, this only allows for a maximum of 65,536 events in any one bin, while still requiring 25 MB for all of the neutron events between 15 ˚ A and 20 ˚ A, which might be less than a thousand events. Comparatively, the event mode structure can store a million neutron events in under 8 MB.
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APPENDIX C
DESIGN OF PELVIS
The PEL files produced by either the PAPA control system or the detector software written by Lexitek can be analyzed through a custom Python application called the PEL Visualizer (PELVis). PELVis was written in Python for the same reasons given for the PAPA Controls System in Appendix B. The GUI components are all developed through the wxPython library to allow the software to run on Windows, Mac, and Linux. Figure C.1 is a flowchart of the structure of PELVis. Users first read data from a file via either the Open or Polarized Set menu items. PELVis can either read the PEL files produced by the PAPA control software of NPZ arrays produced by the Python numpy library. When reading a PEL file, the event mode data is histogrammed by position and divided into wavelength bins. The NPZ arrays contain the already histogrammed data. While the NPZ file are far faster to read, since they do not need to go through a histogramming step, they are always 200MB in size, regardless of the number of events of the measurement. The PEL files, however, have size and load times that are both proportional to the number of neutron events. Regardless of file type, the user is then given an option to load a monitor file to normalize the detector intensity to monitor count. The detector event count is normalized to the total number of incident events on the monitor over all wavelengths. If the user does not load a 146
Open
Polarized Set Legend
Load Flat
PelvisOptionPanel
MenuOption
Simulate Flat
flatrun
Software Component
data
Variable
Region of Disinterest
Scale Menu
ColorBarPanel
Save
Analysis Menu
updateData
ImagePanel
Copy
SpectrumDialog
PositionPanel
GraphPanel
Export Images...
Spectrum
Figure C.1: Map of the menus and data flows of the PELVis application. The ellipses represent menu options on the PELVis GUI. The boxes are Python classes and visual components of the application. The diamonds are variables within the application. The two colored paths show the two primary routes through the application for data analysis. The red route would save an image of the neutron data to a file while the blue route would save the neutron data versus wavelength.
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PelvisOptionPanel
ImagePanel
GraphPanel PositionPanel ColorBarPanel Figure C.2: The graphical user interface for PELVis. The major interface components are labeled.
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monitor file, the raw neutron counts are used. The loaded neutron event data is stored in a variable called data. If the user had used the Open menu item, then data simply contains a three dimensional array of the neutron intensity versus detector position and wavelength. If Polarized Set is used, data is a tuple where the first element is the three dimensional array of the intensity in the up state and the second element is the corresponding intensity in the down state. When the new data is loaded, PELVis will call the current updateData function to display a projection of the data onto the detector in the ImagePanel. The user can clamp this projection to values between a minimum and maximum value set on the PelvisOptionPanel. If the clamping options are not set, the ImagePanel automatically scales between the minimum and maximum values in the projection. Projections of the ImagePanel along the x and y axes will be shown in the two GraphPanel s. The choices in the Analysis menu will change the updateData function to allow for different projections to be performed. Currently, it is possible to project the spin up intensity per pixel, the spin down intensity per pixel, the polarization per pixel, or the flipping ratio per pixel. The projections calculated by updateData can also be modified by the PelvisOptionPanel. The user can use the PelvisOptionPanel to select a wavelength range and a region of interest. The ImagePanel doesn’t include neutron events outside of the selected wavelength range. The GraphPanel s also do not include neutron events outside the wavelength range, as well as only projecting neutrons in the region of interest for the other axis. For instance, the GraphPanel for the y axis will only include a projection of neutrons inside the user’s given range of x positions, but will still project over the entire y axis. While the region of interest can be entered directly in the PelvisOptionPanel, it can also be chosen through the image panel. Left clicking on the image panel picks the minimum x and y values, while right clicking sets the maximum values.
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If the detector has a known, position sensitive background problem, it can be handled through Load Flat. A “flat” file must first be prepared by taking a measurement with no neutron source in order to measure the background versus pixel and wavelength. The user can then select this file through Load Flat and PELVis record the average intensity over all wavelengths at each pixel and record normalize it to the time spent taking the flat measurement. The next time a data set is loaded, PELVis will scale this flat value to the time of the measurement and subtract it from the new data. If no flat file is available, one can be faked through Simulate Flat. It takes the average intensity of pixels outside the region of interest and subtracts it from every pixel in the data. Due to the large data sets and memory limitations, the background is subtracted directly from the data, as opposed to being kept in memory and subtracted when new data is loaded. If there are two regions which both are not representative of the background, it is better to use the Region of Disinterest menu option. It’s behavior is identical to Simulate Flat, except that the user intentionally selects a background region to be averaged. The color scale for the ImagePanel can be chosen through the Color menu. A legend of the current color scale is given in the ColorBarPanel. Currently, the Color menu offers three basic scales: “Greyscale”, “Hue and Value”, and “spectral”. Additionally, the “Map Picker” option allows the user to chose from the full range of color scales provided by the mapping library. Generating all of the scales can be slows, so these scales are not generated unless the user chooses the “Map Picker” menu item. PELVis does not currently have the ability for users to generate custom color scales. The user has several options for exporting the two dimensional projection of the data. The “Save” menu item in the “File” menu will allow the user to save the current projection to an image file as it currently appears. Similarly, the “Copy” menu item in the “Edit” menu puts the current ImagePanel on the clipboard, where it can then by pasted into other
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Figure C.3: The Spectrum Dialog for PELVis. applications. For a more complete picture of the neutron data, the “Export Images” menu item in the “File Menu” will save the two dimensional projection for every wavelength bin in the wavelength range given in the PelvisOptionPanel. This series of images can then be used to get a convenient understanding of the full, three dimensional neutron event histogram. While PELVis was design for exploring the position dependence of neutron data, most measurements will be made versus wavelength. The “Spectrum” menu item in the “File” menu will present a dialog (Figure C.3) of options for wavelength dependent data. The dialog only examines data inside the region of interest defined in the PelvisOptionPanel and uses analysis functions provided by the options in the “Analysis” menu to chose how the data is reduced. All off the analysis options also calculate the corresponding errorbars. The text box and radio buttons of the SpectrumDialog control the wavelength binning for the generated spectrum. The “Raw Data” option performs no binning and ignores the input box. The “Auto binning” options successively combined wavelength bins until the percent error is smaller than the percentage given in the text box. This is useful for understanding the shape of a curve, but often is useless in calculations, since no two runs are guaranteed to have the same binning. The “Fixed Binning” divides the data into the number of wavelength bins given by the user. The larger the number, the finer grained the binning. Once the user has chosen a binning option, the spectrum data can either be saved to a tab delimited file or plotted immediately for viewing. The plot will be constrained by the intensity values in the PelvisOptionPanel if available, but the saved data will always provide the entire spectrum. 151
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Adam Washington Education 2005–2013 PhD, Indiana University. 2005–2010 Masters, Indiana University. 2000–2005 Bachelors, Purdue University.
PhD thesis title Investigating Hard Sphere Interactions Through Spin Echo Scattering Angle Measurement supervisors Roger Pynn
Research Experience 2007–present Research Assistant, Indiana University, Bloomington, IN. Studied and implemented Spin Echo Scattering Angle Measurement (SESAME). Detailed achievements: { Built SESAME beamline at the Low Energy Neutron Source: - Built spin echo elements - Developed motor system - Wrote instrument control and data analysis software - Measured efficiency of beamline components { Upgraded ASTERIX beamline at Los Alamos National Lab to support SESAME - Built spin echo elements - Developed electrical, water, and thermal feedback systems - Developed data collection and analysis scripts
Teaching Experience 2008–2013 Tour Guide, Indiana University, Bloomington, IN. Gave tours of the Cyclotron Facility Handled tour groups from middle school students to foreign scientists
2006–2010 Assistant Instructor, Indiana University, Bloomington, IN. Taught algebra and calculus based introductory physics phyics courses Lead and graded laboratory sections
Computer skills Languages Python, C, Mathematica, Haskell, LabView, IDL OS Linux, Windows Software Wolfram Mathematica, Autodesk Inventor
3209 E. 10th Street, Apt. W12 – Bloomington, IN 47408, USA H +1 (317) 698 7685 • B
[email protected]
Talks { Adam Washington, Xin Li, and Roger Pynn. (2012) SESANS Measurements of Hard Sphere Colloids. American Conference on Neutron Scattering
Publications Rana Ashkar, P Stonaha, A Washington, VR Shah, MR Fitzsimmons, B Maranville, CF Majkrzak, WT Lee, and Roger Pynn. Spin-echo resolved grazing incidence scattering (sergis) at pulsed and cw neutron sources. Journal of Physics: Conference Series, 251(1):012066, 2010. Rana Ashkar, P Stonaha, AL Washington, VR Shah, MR Fitzsimmons, B Maranville, CF Majkrzak, WT Lee, WL Schaich, and Roger Pynn. Dynamical theory calculations of spin-echo resolved grazing-incidence scattering from a diffraction grating. Journal of Applied Crystallography, 43(3):455–465, 2010. Xin Li, Roger Pynn, Adam Washington, Wei-Ren Chen, Kunlun Hong, Gregory Smith, and Yun Liu. Investigation of nanoscale structure using spin-echo small-angle neutron scattering (sesans). Bulletin of the American Physical Society, 57, 2012. SR Parnell, H Kaiser, AL Washington, F Li, T Wang, DV Baxter, and R Pynn. Design of a cryogen free cryo-flipper using a high tc ybco film. Physics Procedia, 42:125–129, 2013. SR Parnell, AL Washington, H Kaiser, F Li, T Wang, WA Hamilton, DV Baxter, and R Pynn. Performance of a polarized neutron cryo-flipper using a high tc ybco film. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2013. Roger Pynn, Rana Ashkar, P Stonaha, and AL Washington. Some recent results using spin echo resolved grazing incidence scattering (sergis). Physica B: Condensed Matter, 406(12):2350–2353, 2011. Roger Pynn, MR Fitzsimmons, WT Lee, VR Shah, AL Washington, P Stonaha, and Ken Littrell. Spin echo scattering angle measurement at a pulsed neutron source. Journal of Applied Crystallography, 41(5):897–905, 2008. Roger Pynn, MR Fitzsimmons, WT Lee, P Stonaha, VR Shah, AL Washington, BJ Kirby, CF Majkrzak, and BB Maranville. Birefringent neutron prisms for spin echo scattering angle measurement. Physica B: Condensed Matter, 404(17):2582–2584, 2009. Roger Pynn, WT Lee, P Stonaha, VR Shah, AL Washington, BJ Kirby, CF Majkrzak, and BB Maranville. The use of symmetry to correct larmor phase aberrations in spin echo scattering angle measurement. Review of Scientific Instruments, 79(6):063901– 063901, 2008.
3209 E. 10th Street, Apt. W12 – Bloomington, IN 47408, USA H +1 (317) 698 7685 • B
[email protected] 