Triassico: A Sphere Positioning System for Surface ... - Science Direct

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ScienceDirect Physics Procedia 90 (2017) 22 – 31

Conference on the Application of Accelerators in Research and Industry, CAARI 2016, 30 October – 4 November 2016, Ft. Worth, TX, USA

Triassico: A Sphere Positioning System for Surface Studies with IBA Techniques Cristiano L. Fontanaa*, Barney L. Doyleb a

Physics Department “Galileo Galilei,” University of Padua, Via Marzolo 8, I-35131, Padova, Italy b Sandia National Laboratories, Albuquerque, NM, USA

Abstract We propose here a novel device, called the Triassico, to microscopically study the entire surface of millimeter-sized spheres. The sphere dimensions can be as small as 1 mm, and the upper limit defined only by the power and by the mechanical characteristics of the motors used. Three motorized driving rods are arranged so an equilateral triangle is formed by the rod's axes, on such a triangle the sphere sits. Movement is achieved by rotating the rods with precise relative speeds and by exploiting the friction between the sphere and the rods surfaces. The sphere can be held in place by gravity or by an opposing trio of rods. By rotating the rods with specific relative angular velocities, a net torque can be exerted on the sphere which then rotates. No repositioning of the sphere or of the motors is needed to cover the full surface with the investigating tools. An algorithm was developed to position the sphere at any arbitrary polar and azimuthal angle. The algorithm minimizes the number of rotations needed by the rods, in order to efficiently select a particular position on the sphere surface. A prototype Triassico was developed for the National Ignition Facility, of the Lawrence Livermore National Laboratory (Livermore, California, USA), as a sphere manipulation apparatus for ion microbeam analysis at Sandia National Laboratories (Albuquerque, NM, USA) of Xe-doped DT inertial confinement fusion fuel spheres. Other applications span from samples orientation, ball bearing manufacturing, or jewelry. byby Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2017 2017 The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of the Conference on the Application of Accelerators in Research Peer-review under responsibility of the Scientific Committee of the Conference on the Application of Accelerators in Research and Industry Industry. and Keywords: sphere orientation; surface investigation of spheres.

* Corresponding author. Tel.: +39-049-827-5933; fax: +0-000-000-0000 . E-mail address: [email protected]

1875-3892 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of the Conference on the Application of Accelerators in Research and Industry doi:10.1016/j.phpro.2017.09.013

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1. Background The Triassico was initially developed for the National Ignition Facility (NIF), of the Lawrence Livermore National Laboratory (LLNL), as a sphere manipulation apparatus for R&D of the DT inertial confinement fusion fuel spheres (Shaughnessy, 2012). This was part of the Radiochemical Analysis of Gaseous Samples (RAGS) project, between LLNL and Sandia National Laboratories (SNL), where 124Xe was to be implanted into the surface of the plastic ablator shell of these fuel spheres. During a DT shot on NIF the 124Xe could experience two nuclear reactions with the fusion neutrons: (n,γ) producing 125Xe and (n,2n) producing 123Xe. Following cryogenic collection, the gamma rays produced when these two unstable isotopes decay provide an indirect measurement of the important Lawson criteria parameter ρR. The Triassico was to be used to perform microbeam Heavy Ion Backscattering Analysis (Doyle, 1989) on the 2mm diameter NIF target spheres to measure the uniformity of the 124Xe implants performed at LLNL (Shin, 2012). For these measurements the HIBS microbeam (250 keV N) would be fixed, and the Triassico would rotate these targets to all angles on the unit sphere. There were also plans to perform the implantations with a microbeam where the Triassico would target the poles or equator of these tiny spheres. Unfortunately, the use of implantation doped NIF targets has been put on hold. Nevertheless, the main purpose of this paper is to document the Triassico mechanism and the mathematics that goes into the details of how the unit can rotate spheres to any arbitrary polar and azimuthal angle. A second purpose is to stimulate the reader to propose other applications of this unique, small and vacuumcompatible tool for performing these spherical rotations.

2. Introduction Again, the project’s aim is to investigate the surface of a sphere employing Ion Beam Analysis (IBA) techniques. Common approaches employ, for instance, carbon tape to attach the sphere on a sample holder and orientate the sample holder itself. Obviously only a portion of the sphere can be analyzed, depending on the capabilities of the sample holder manipulator. In order to be able to analyze the whole surface, the sphere has to be subsequently detached and turned; this has the problem of measuring the precise orientation before and after the procedure.

Figure 1. Simplest geometry with a sphere sitting on three rods in an equilateral triangle geometry. The sphere can be held in place by gravity (single layer configuration) or by an opposing trio of rods (double layer configuration).

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We propose here a specific sphere holder that can rotate the sample in any direction, without the need of a manual repositioning. We propose to use three rods, arranged so that an equilateral triangle is formed by the rod’s axes. The sphere sits on such triangle. The sphere can be held in place by gravity or by an opposing trio of rods. Movement is achieved by rotating the rods with precise relative speeds and by exploiting the friction between the sphere and the axes surfaces. By rotating the rods with specific relative angular velocities, a net torque can be exerted on the sphere causing it to rotate. Since the sphere is mechanically held in place, and it is not attached, it can rotate indefinitely. Therefore, no repositioning of the sphere, or of the motors, is needed to expose the full surface to the investigating beam. There are no fixed positions on the sphere so a continuous movement with no blind spots can be achieved. The rods angular velocities have to be carefully controlled: to maximize the exploited friction components that induce the movement, and to minimize the components that oppose the movement (see section 3). 3. Rods geometry and velocities

Figure 2. Some examples of regular rod dispositions. Irregular polygons may be employed as well, but they have to circumscribe a circle.

In principle the rods may form any convex polygon that circumscribes a circle, in which the sphere sits. Obviously, regular polygons (Figure 2) simplify the calculations of the rods angular velocities, and a regular triangle is the simplest case. To achieve a movement, one of the rods has to be elected as the driving rod, while the other rods will be defined as the auxiliary rods. The sphere will rotate with its angular velocity parallel to the driving rod’s axis. From now on we will consider the case of three rods in a triangular geometry. On Figure 3 we can see a sketch of ̰௜ , which form the angles ߙ, ߚ and ߛ in between the pairs. The rods angular the rods axes, indicated by the unit vectors ‫ݔ‬ ሬሬԦ௜ The rods unit vectors are defined as velocities are indicated by ߱ െ…‘•ሺߙሻ …‘•ሺߙ ൅ ߚሻ ͳ ̰ଶ ൌ ൥ •‹ሺߙሻ ൩ ǡ ̰ ̰ଵ ൌ ൥Ͳ൩ ǡ ‫ݔ‬ ‫ݔ‬ ‫ ݔ‬ଷ ൌ ൥െ•‹ሺߙ ൅ ߚሻ൩Ǥ Ͳ Ͳ Ͳ

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Figure 3. Vectors indicating the rods directions and velocities, in the case of a regular triangular geometry.

Figure 4. Angular velocities of the driving rod and the sphere

Assuming that there is no slippage between the sphere and the driving rod, we can calculate its relative angular velocities (Figure 4). Defining the rod’s radius as ‫ݎ‬, the sphere radius as ܴ, the rod’s angular velocity as ߱୰ and the sphere’s angular velocity as ߱ୖ we have the relationship between the modules: ‫ݎ‬ ߱௥ ‫ ݎ ڄ‬ൌ ߱ௌ ‫ܴ ڄ‬᩷ ֜ ᩷ ߱ௌ ൌ ߱௥ ‫ ڄ‬Ǥ ܴ There is a flip of direction of rotation between the rod and the sphere. There is a second flip of direction, if the auxiliary rods are then moved by the friction with the sphere. The double flip allows us to directly compare the rod’s angular velocities. Let us take rod number 1 as the driving rod, its angular velocity is ሬሬԦଵ ൌ ߱௥ ̰ ‫ݔ‬ଵ ߱ ୄ

We can decompose ߱ ሬሬԦଵ (Figure 4) onto the auxiliary rods axes (߱ ሬሬԦ௜ ) and their orthogonal directions (߱ ሬሬԦ௜ ):

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ሬሬԦଶ ߱

̰ଶ ൯Ԝ‫ݔ‬ ̰ଶ ൌ ܿ‫ݏ݋‬ሺߨ െ ߙሻ ߱ଵ ̰ ൌ ൫߱ ሬሬԦଵ ‫ݔ ڄ‬ ‫ ݔ‬ଶ ൌ െ ܿ‫ݏ݋‬ሺߙሻ ߱ଵ ̰ ‫ݔ‬ଶ ǡ

ୄ

ൌ߱ ሬሬԦଵ െ ߱ ሬሬԦଶ ǡ ̰ ̰ଷ ൌ ܿ‫ݏ݋‬ሺߨ െ ߛሻ ߱ଵ ̰ ൌ ൫߱ ሬሬԦଵ ‫ ݔ ڄ‬ଷ ൯Ԝ‫ݔ‬ ‫ ݔ‬ଷ ൌ െ ܿ‫ݏ݋‬ሺߛሻ ߱ଵ ̰ ‫ݔ‬ଷ ǡ

ሬሬԦଶ ߱ ߱ ሬሬԦଷ

ୄ

ሬ߱ ሬԦଷ

ൌ߱ ሬሬԦଵ െ ߱ ሬሬԦଷ Ǥ

E.g. for an equilateral triangle the angular velocities are ߱ ሬሬԦଵ ൌ ߱௥ ̰ ‫ݔ‬ଵ ǡ᩷߱ ሬሬԦଶ ൌ െ

߱௥ ߱௥ ̰ ‫ ݔ‬ǡ᩷߱ ሬሬԦଷ ൌ െ ̰ ‫ ݔ‬Ǥ ʹ ଶ ʹ ଷ

ୄ

ሬ߱ ሬԦ௜ and ሬ߱ ሬԦ௜ have an associated physical meaning: ሬሬԦ௜ is the rotation parallel to the rod’s axis, thus is the rotation that is transferred to the rod by the sphere; x ߱ ୄ x ߱ ሬሬԦ௜ is the rotation orthogonal to that direction, so it is a slippage of the sphere’s surface along the rod direction. In order to ease the sphere movement, the auxiliary rods may be motorized as well. Their angular velocity should be equal to the velocity induced by the sphere parallel to the axis. The slippage along the rods direction cannot be compensated, though. Therefore, if the driving rod is the number 1 and the geometry is an equilateral triangle, the rods angular velocities should be ߱ଵ ൌ ߱௥ ǡ᩷߱ଶ ൌ െ

߱௥ ߱௥ ǡ᩷߱ଷ ൌ െ Ǥ ʹ ʹ

The slippage velocities along the auxiliary rods are ȁ߱ଶୄ ȁ ൌ ȁ߱ଷୄ ȁ ൌ

ξ͵ ȁ߱௥ ȁ ൎ ͲǤͺ͹ȁ߱௥ ȁǤ ʹ

It should be noted that this slippage could damage the surface of the sphere. This is an issue that will be investigated.

4. Sphere movements

Figure 5. Examples of point orbits. Orbits are circumferences perpendicular to the rotation axis and passing through the considered point, therefore they are not great circles.

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In general, the orbit of a point on the sphere, driven by one of the rods, is a circumference with its axis parallel to the rod axis and passing through the point itself (Figure 5). Therefore, orbits are not great circles. The locus of the positions reachable by a chosen starting point ‫ݏ‬ሬԦ is an orbit. To reach any other position a second move is required, electing as the driving rod one of the two auxiliary rods. Depending on the first rotation’s orbit, not all other positions on the sphere are reachable. Some positions are reachable with a second rotation along any of the two auxiliary rods, or with only one of the two. In general, two moves, with two different axes, are required for a point to reach any destination. Not all combinations of axes can be suitable for a given start and target. Being two movement required, a ሬሬሬԦ for the movement has to be defined, by intersecting the circles that pass through the starting point ‫ݏ‬ሬԦ and midpoint ݉ the target point ‫ݐ‬Ԧ.

Figure 6. Reachable destinations for a point with only two rotations (top orthographic projection). The highlighted areas represent the locus of the points reachable, with the subsequent movement of one or two auxiliary axes. The purple region can be reached with a combination of a rotation along the first axis (black line) and any of the other two axes. The red and blue regions can be reached with only one axis.

5. Midpoint determination Let us take a movement from ሬ‫ݏ‬Ԧ ൌ ሺܴǡ ߴ௦ ǡ ߮௦ ሻ ൌ ሺߴǡ ߮ሻ௦ ᩷‘˜‡–‘᩷ ‫ݐ‬Ԧ ൌ ሺܴǡ ߴ௧ ǡ ߮௧ ሻ ൌ ሺߴǡ ߮ሻ௧ ǡ ሬሬሬԦ (see section 4): We have to split the movement in two phases and stop at a midpoint ݉ ሬሬԦ ൌ ሺܴǡ ߴ௠ ǡ ߮௠ ሻ ൌ ሺߴǡ ߮ሻ௠ ᩷‘˜‡–‘᩷ ‫ݐ‬Ԧ ൌ ሺߴǡ ߮ሻ௧ Ǥ ‫ݏ‬ሬԦ ൌ ሺߴǡ ߮ሻ௦ ᩷‘˜‡–‘᩷ ሬ݉ Since not all combinations of rods movements are able to reach the target point, a computationally efficient and reliable algorithm is required. A naive search for a midpoint involves the root finding of a 4-dimensional periodic function (the Euler’s angles of the first rotation and of the second rotation). We propose here an efficient algorithm for the determination of the rotation angles. ̰, ̰ Let us define a new coordinate system with axes ܽ ܾ and ̰ ܿ that are defined as: ̰ൌ̰ ̰ൌ̰ ̰ Ǥ ᩷(Equation ͳ) ̰ ‫ ݖ‬ǡܾ᩷ ܿ ൈܽ ܿൌ̰ ‫ ݔ‬௜ ǡ᩷ܽ A generic point in this system can be determined using the associated spherical coordinates: ̰ ൅ •‹ߠ•‹߶Ԝ ̰ ሬԦ ൌ ܴ ቂ…‘•ߠ•‹߶Ԝ ܽ ܾ ൅ …‘•߶Ԝ ̰ ܿ ቃ Ǥ ᩷(Equation ʹ) ‫݌‬ In this system the rod rotation angle directly corresponds to ߠ. The inverse relationships are:

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ܴ –ƒߠ …‘•߶

ൌ ȁ‫݌‬ ሬԦȁǡ ̰ ሬ‫݌‬Ԧ ‫ܾ ڄ‬ ǡ ൌ ̰ ሬ‫݌‬Ԧ ‫ܽ ڄ‬ ሬ‫݌‬Ԧ ‫̰ ڄ‬ ܿ Ǥ ൌ ܴ

The orbit of one rotation can be determined by intersecting the spherical surface with the rotation plane, which is perpendicular to the rod’s axis: ̰௜ ൌ ͲǤ ᩷ሺ“—ƒ–‹‘͵ሻ ሾ‫݌‬ ሬԦ െ ‫ݏ‬ሬԦሿ ‫ݔ ڄ‬ We can expand ᩷ሺ“—ƒ–‹‘͵) remembering the definition of ሬ‫݌‬Ԧ ᩷(Equation ʹ), and ᩷(Equation ͳ): Ͳ

̰௜ ሬԦ െ ‫ݏ‬ሬԦሿ ‫ݔ ڄ‬ ൌ ሾ‫݌‬ ̰௜ ൌ‫݌‬ ሬԦ ‫̰ ڄ‬ ܿ െ ‫ݏ‬ሬԦ ‫ݔ ڄ‬ ̰௜ ൌ ܴ…‘•߶ െ ሬ‫ݏ‬Ԧ ‫ݔ ڄ‬

thus we can obtain the value of ߶: …‘•߶ ൌ

̰௜ ሬ‫ݏ‬Ԧ ‫ݔ ڄ‬ Ǥ ܴ

We see that ߠ is undefined, thus we have only one free variable left to determine. If we imagine performing a ̰௝ we have: second rotation along ‫ݔ‬ ݂ሺߠሻ ൌ ൣ‫݌‬ ሬԦ ሺߠሻ െ ‫ݐ‬Ԧ൧ ‫̰ ڄ‬ ‫ݔ‬௝ ǡ where we expressed the dependency on ߠ. In order to find the midpoint we can find the roots ߠ௡ҧ of the equation ݂ሺߠሻ ൌ ͲǤ Therefore, we have reduced the complexity of the root finding problem, from a 4-dimensional to a 1-dimensional, just by choosing opportune coordinates systems. The midpoint(s) would be ሬሬԦ௡ ൌ ‫݌‬ ሬԦ ሺߠ௡ҧ ሻǤ ݉ The algorithm then tries all the possible rods combinations and determines the relative midpoints. Finally a midpoint is selected, in order to minimize the travel distance between the starting point and the target point.

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6. Steps determination

Figure 7. Errors due to stepwise movements.

Assuming that the rods are motorized by stepper motors, we can calculate the error that is associated with a movement. If the distance (i.e. arc length) between two points in an orbit is ‫ܦ‬, then the number of steps between the two is ‫ܦ‬ ߂ ൌ ”‘—† ൬ ൰ ߜ where ߜ is the step size. Therefore, the maximum error of a movement is ߪ௠௔௫ ൑ ߜ and the associated standard deviation for a uniform distribution is ߪൌ

ߜ

Ǥ ξͳʹ

If we imagine carrying out a second movement (Figure 7) the total standard deviation would be ߪ ൌ ටߪ௜ଶ ൅ ߪ௝ଶ ൌ

ߜ

Ǥ ξ͸

In our prototype the rods have ‫ ݎ‬ൌ ͳ and the number of steps per rod turn is ܰ ൌ ͶͲͲͲ, therefore ߜൌ

ʹߨ‫ݎ‬ ൎ ͳǤ͸Ɋǡ ܰ

that is not dependent of the sphere radius ܴ. The standard deviation of a series of two rotations is ߪൌ

ߜ ξ͸

ൎ ͲǤ͸Ɋ

It is worth reminding that, in general, two movements are required to go from a start position to a target position. Moreover, this result is valid only with the assumption that there is no slippage between the sphere and the driving rod.

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7. Prototype

Figure 8. Stage of the prototype that holds the sphere in the center. Six stepper motors move the rods, which are on two layers to hold the sphere with any orientation of the stage. The stage is approximately 10×10×2 cm on a side and was designed to fit and operate in the target vacuum system of the microbeam of the IBL Pelletron accelerator.

Figure 9. Close-ups of the rods holding the sphere, top-view and side-view.

The first prototype was engineered by a joint venture between FMB Informatica s.r.l. (Bassano del Grappa, Italy) and Inel Elettronica s.r.l. (Mussolente, Italy). It was sent to the Ion Beam Laboratory at the Sandia National Labs (Albuquerque, NM) for testing. It is composed of a stage (Figure 8 and Figure 9) with six stepper motors that control the sphere movements. The stage can be oriented in any direction because the sphere is trapped in between two layers of rods. The upper layer was designed in order to trap the sphere and regulate the pressure that the rods exert on the sphere. The stage can be configured to hold spheres that have dimensions varying from 1 mm to 2 mm. Bigger spheres

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can be manipulated, with a modification of the stage. The slippage between the sphere and the auxiliary rods might damage the sphere’s surface, and that needs to be investigated. The system is controlled by a controller with an embedded computer that performs the calculations needed for the functioning (Error! Reference source not found.).

Figure 10. Left, controller of the stage, with the stepper motor drivers and an embedded computer. Right, brand and logo of the prototype.

8. Conclusions The concept of the Triassico allows the arbitrary orientation of tiny spheres, and allows an analysis of its surface with microbeam IBA techniques or ion implantations at any polar or azimuthal angle. It is versatile, as it accommodates different sphere sizes, geometries and orientations of the stage. The Triassico has been miniaturized as much as possible and was designed to fit on the xyz stage of the Pelletron microbeam in the Ion Beam Laboratory at Sandia National Labs. Other applications of this technique span from general samples orientation to jewelry manufacturing, and readers of this paper are encouraged to communicate other ideas to the authors. A new algorithm was developed in order to control the sphere orientation. The theoretical resolution of 0.6 μm is very good, compared to the classical beam sizes used in IBA. Relying on friction the prototype needs testing, in in order to assess its real movement resolution and reproducibility. 9. Acknowledgements Engineering and development costs were sustained by FMB Informatica s.r.l. (Bassano del Grappa, Italy) and Inel Elettronica s.r.l. (Mussolente, Italy).

10. References Shaughnessy, D. A., C. A. Velsko, Jedlovec, D. R., Yeamans, C. B., Moody, K. J., Tereshatov, E., Stoeffl, W., and Riddle, A., 2012. The Radiochemical Analysis of Gaseous Samples (RAGS) apparatus for nuclear diagnostics at the National Ignition Facility , Review of Scientific Instruments 83, 10D917 . Shin, S. J., Kucheyev, S. O., Orme, C. A., Youngblood, K.P., Nikroo, A. et al., 2012. Xenon doping of glow discharge polymer by ion implantation, J. Appl. Phys. 111, 096101 . B.L. Doyle, B.L., Knapp, J.A., Buller, D.L. 1989. Heavy ion backscattering spectrometry (HIBS) — An improved technique for trace element detection, NIM-B, Volume 42, Issue 2, 295.

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