Neutron Optics and Neutron Interferometry - Indiana University

Neutron Optics and Neutron Interferometry Helmut Kaiser IUCF

Outline:

Introduction Refraction Reflection Diffraction Interferometry

Appeared in “The New Yorker”, 1940

Message • Neutron Optics ⇒ Analogy to Light Optics ⎛ h⎞ λ = • Neutron is a de Broglie Matter Wave ⎜ ⎟ p ⎝ ⎠

• Neutron Interferometry ⇒ Unique technique for Probing and Elucidating Fundamental Quantum Mechanical Principles on a Macroscopic Scale

INTRODUCTION

Properties of the Neutron

Bound Coherent Scattering Lengths

b : phenomenological constant, determined by experiment

REFRACTION Strong nuclear interaction: 2 2 2π 2 k Vn = N bc , E = m 2m Vn λ 2 N bc n = 1− = 1− 2E 2π

Including absorption: 2π 2 Vn → V0 − iV1 =

m

N bc − i

2

Nσ r

λ Nσ r λ2N 2 ⎛ σr ⎞ n 1− bc − ⎜ + i , ⎟ λ π 2π 2 4 ⎝ ⎠ 2

with

σr = σa +σi

Snell’ law: n

K v sin φ0 = = , neutrons k v0 sin φ

n

K v0 c sin φ0 = = = , light k v v sin φ

I Transmission : T = = exp( − N σ t d ) I0

REFRACTION (cont.) Magnetic interaction: Vm = VZ + VS + VF = − μ ⋅ B −

mc

μ ⋅ (E × k ) −

μ 2mc

divE ,

Vm = VZ = − μσ ⋅ B , V + Vm b ± p(0) n± = 1 − n = 1− λ2N c , E 2π with p(0) = p = −

μA μ mB γ = − r e 2π N 2 2μ B

p : average magnetic scattering in forward direction B : mean magnetization ( magnetic field of the unpaired electrons ) μ = γμ N : neutron magnetic dipole moment

γ = −1.913 : gyromagnetic ratio e : nuclear magneton 2mc e2 = 2.818 fm : classical electron radius re = me c 2

μN =

Examples:

bc ± p

Fe : 9.54 ± 5.98 fm

μB =

e : Bohr magneton 2me c

Co : 2.54 ± 4.64 fm Ni : 10.31 ± 1.62 fm

μA =

B : average magnetic dipole moment 4π N per atom

REFLECTION Glancing incidence : θ → 0 Θ = 0 ↔ θ = θ‘

cos θ = cos θ c , cos Θ θ c : critical angle of total reflection n=

Rewriting Snell’s law

θ c = 1 − n2 Nbc

=λ

π

⇒ θ ≤ θc

Reflectivity and Transmissivity: I r = RI 0 and I t = TI 0 , with R + T = 1 sin θ − n sin Θ R= sin θ + n sin Θ Since n is close to 1 : R = 1 , 2

R=

for θ ≤ θ c

1 − 1 − (θ c θ )

2

1 + 1 − (θ c θ )

2

2

,

for θ > θ c

DIFFRACTION Branches of neutron scattering theory

• Diffraction from macroscopic objects • Diffraction from perfect crystals

Diffraction ⇒ Interference of coherent waves

Diffraction from macroscopic objects • • • • • •

Single and double slit diffraction (Frauenhofer) Edge diffraction (Fresnel) Grating diffraction Diffraction by planar structures → reflectometry Fresnel zone plates and supermirrors ............

Bragg planes

Diffraction from perfect crystals

Laue diffraction

Bragg planes

Bragg diffraction

Kinematic Bragg Diffraction




Dynamical Diffraction Theory (Laue case)

H = Bragg vector 2π n H = d K 0 = internal forward scattered wave K H = external forward scattered wave

Bragg condition:

K H − K0 = H Solve Schrödinger Eqn. inside crystal:

(∇

2

+ k02 ) Ψ (r ) = v(r )Ψ (r ) with v(r ) = 4π ∑ bi δ ( r − ri ) = i

∑v n

Hn

eiH n ⋅r

Dynamical Diffraction Theory

α

β

α

β

internal wave function: Ψ ( r ) = ψ 0α eiK0 ⋅r + ψ 0β eiK0 ⋅r + ψ Hα eiK H ⋅r + ψ Hβ eiK H ⋅r

Dynamical Diffraction Theory ψ 0α =

1⎡ y ⎤ ⎥ A0 ⎢1 − 2 ⎢⎣ 1 + y 2 ⎥⎦

ψ 0β =

1⎡ y ⎢1 + 2 ⎢⎣ 1 + y2

⎤ ⎥ A0 ⎥⎦

ψ αH = − ⎢

1⎡ 1 2 ⎢⎣ 1 + y 2

⎤ ⎥ A0 ⎥⎦

1⎡ 1 ψH = + ⎢ 2 ⎢⎣ 1 + y 2

⎤ ⎥ A0 ⎥⎦

β

y=

k0 sin 2θ B δθ 2ν H

misset parameter

α

β

α

β

internal wave function: Ψ ( r ) = ψ 0α eiK0 ⋅r + ψ 0β eiK0 ⋅r + ψ Hα eiK H ⋅r + ψ Hβ eiK H ⋅r

Dynamical Diffraction Theory


Ψ trans (r ) = ψ tr 0 eik0 ⋅r + ψ tr H eikH ⋅r

Transmitted wave:

⎡

⎤ sin Φ ⎥ ei (φ1 −φ0 ) A0 1+ y2 ⎥⎦ ⎤ − i (φ +φ ) −iy sin Φ ⎥ e 1 0 A0 2 1+ y ⎥⎦

ψ tr 0 = ⎢cos Φ − ψ tr H

⎢⎣ ⎡ =⎢ ⎢⎣

iy

φ0 =

with

ν0D cos θ B

, φ1 =

⎛ 1 Φ = ⎜ν H ⎜ 1+ y2 ⎝

νHD cos θ B

⎞ D ⎟ ⎟ ⎠ cos θ B

“Pendellösung interference”

I 0 = ψ tr 0 Transmitted intensities:

2

⎡ 2 ⎤ y2 2 = A ⎢cos Φ + sin Φ ⎥ 1+ y2 ⎣ ⎦ 2 0

⎡ 1 ⎤ 2 2 sin Φ I H = ψ tr H = A02 ⎢ ⎥ 2 ⎣1 + y ⎦

Transmitted Intensities

For the 〈111〉 reflection in Si at λ=2.70 Å:

y = 1 ⇒ 0.9 arcsec

Angle Amplification

For small δ (~10 -3 arcsec):

Ω

δ

≈ 10 6


Dynamical Diffraction Theory (Bragg case)

y = −2πΔ e Δθ cos(θ B ) / λ Dispersion Equation:

Pendelloesung length ( = extinction length) Δ e

approximate:

Δe =

π sin(θ B ) λ Nbc e w

NEUTRON INTERFEROMETRY

Phase of the neutron wave function is directly accessible to experiment

Applications: Measurement of bc Optics experiments Quantum mechanics experiments

NEUTRON INTERFEROMETRY Basic Principle

I ∼ ψ I + ψ II

2

Phase shift : 1 χ = ∫ p ds =

2

2

= ψ I + ψ II + 2 ψ I ψ II

∫ k ds

Perfect-crystal interferometry: X-ray interferometry: Bonse and Hart (1965) Neutron interferometry: Rauch, Treimer and Bonse (1974)

NEUTRON INTERFEROMETRY Michelson Interferometer

Mach-Zender Interferometer

Practical Neutron Interferometer

Perfect Crystal LLL Neutron Interferometer

Bragg condition: nλ = 2d sin θ

d = lattice spacing




Nuclear Phase Shift

Nuclear Phase Shift

Nbλ 2 index of refraction: n = 1 − 2π relative phase shift: Δχ = k0 − nk0 = Nbλ

D cos θ

Interferogram

Interferogram

O beam: I O = A [1 + f cos( χ 2 − χ1 ) ] IH

IO

H beam: I H = B − A f cos( χ 2 − χ1 )

contrast f =

Cmax − Cmin Cmax + Cmin

Precision Phase Shift Measurement

Δχ = Nbλ

D cos θ

Example: aluminum sample, λ = 2.70 Å, 〈111〉 reflection D = 100 μm ⇒ Δχ = 2π

Non-Dispersive Geometry

path length

=

D sin θ

Δχ = 2Nb d D independent of λ


NIST perfect crystal silicon interferometers

Scattering length bc measurements

4π Rotational Symmetry of Spinors Rotation operator:

Rnˆ (α ) = e

Spin-1/2 particle:

S=

Rotations about z-axis:

1 2

σ

i − α nˆ ⋅ S

so

Rnˆ (α ) = e

⎛ e − iα / 2 Rz (α ) = ⎜ ⎝ 0

0 ⎞ ⎟ eiα / 2 ⎠

Rz (2π ) χ = − χ

Symmetry: Rz (4π ) χ = χ

α

− i nˆ ⋅σ 2

4π spinor symmetry Quantum mechanical principle:

ψ (α ) = ℜ (α )ψ (0)

spin ½

= e − iσ ⋅α / 2ψ (0) ⇒ ψ (2π ) = −ψ (0)

ψ (4π ) = ψ (0) ψ (t ) = e

− i Ht /

ψ (0) = e

− i μ ⋅ Bt /

ψ (0) = e

H = −μ ⋅ B = −μ σ ⋅ B

ψ (0) = ψ (α )

− iσ ⋅α / 2

Larmor precession angle:

α=

2μ

∫ B dt ≅

2μ B ds v∫

Larmor precession phase:

Δφ = ± 2πμn mn λ B /

2

Experimental result: α = (715.87±3.8)° Rauch et al., Phys.Lett. 54A, 1975 Werner et al., PRL 35, 1975

Spin Superposition Quantum mechanical principle: Quantum mechanical spin superposition (Wigner, Am J. Phys. 31 (1963)

Results:

Summhammer at al., PRA27, 1983

Quantum Phase Shift Due To Gravity (COW Experiments)

Δφ =

2πλ gA min mgrav h2

A = H = area of parallelogram

min = neutron inertial mass mgrav = neutron gravitational mass

test of weak equivalence principle at the quantum limit

Gravitationally induced quantum interference Quantum mechanical principle: Neutron moving in the gravitational field of the rotating Earth

mg M p2 −G −Ω⋅L H (r , p) = r 2 mi L=r×p g 0 (r ) = ΔΦ =

1

GM rˆ 2 r

β

∫ p ⋅ dr

p → −i ∇ → WKB → p = k Phase shift due to gravity

ΔΦ COW = Φ II − Φ I = −2πλ

mi mg h

2

gH 0 S sin β

α

ΔΦ (α ) = ΔΦ grav (α ) + ΔΦ bend (α ) ΔΦ bend (α ) = −(2π / λ )ΔL0 sin α

Colella, Overhauser, Werner, PRL 34, 1975 Staudenmann et al., PRA21, 1980

Results:

ΔΦ (α ) = q sin α 2 2 qgrav = (qexp − qSagnac )1/ 2 − qbend

qgrav (exp) = (60.122 − 1.452 )1/ 2 − 1.42 = 58.72 ± 0.03rad qgrav (theory ) = 59.2 ± 0.1rad Werner, Kaiser, et al., Physics B151, 1988

0.8%

Floating COW Experiment

D2O+ZnBr2

ρ(Si)=2.33g/cm3 ρ(D2O)=1.11g/cm3 ρ(ZnBr2)=4.20g/cm3

References to Neutron Optics: •

Neutron Optics, Varley F. Sears, Oxford University Press, Oxford (1989).

•

Neutron Interferometry – Lessons in Experimental Quantum Mechanics, Helmut Rauch and Samuel A. Werner, Oxford University Press, New York (2000).

•

“De Broglie wave optics: neutrons, atoms and molecules,” Helmut Kaiser and Helmut Rauch, in Optics (ed. H. Niedrig), Walter de Gruyter, Berlin (1999).