THE UNIVERSITY CF M?:J -IGAN ENGINEERING LIBRARY

This is an authorized facsimile and was produced by microfilm-xerography in 1967 by University Microfilms, A Xerox Company, Ann Arbor, Michigan, U.S.A. * * *

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Negative microfilm 64-6660 received from university CARPENTER, John Marland, 1935 -PREDICTION AND MEASUREMENT OF NEUTRON CHOPPER BURST SHAPES. The University of Michigan, Ph.D., 1963 Engineering, nuclear University Microfilms, Inc., Ann Arbor, Michigan

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PREDICTION AND MEASUREMENT OF NEUTRON CHOPPER BURST SHAPES by John Marland Carpenter Applying an original method* for the graphic representation of the angular and spatial density of neutrons in a collimated beam, a technique is developed for calculation of the number c. neutrons passed through one or more rotating elements in such a beam. The P-gram construction is briefly discussed. The effects of moving elements in a beam of finite-velocity neutrons are described. The apparent configuration of the rotating element is shown to differ from that of a stationary element and the apparent collimator formed by it is represented in terms of the P-gram construction. Consideration of the joint probability of transmission of a neutron through both a stationary collimator which defines a beam, and through a rotor, which interrupts the beam, leads to an expression for the time-dependent neutron transmission probability. An experiment is described in which a monochromatic neutron beam was selected by means of a crystal monochromator and the method is applied to its analysis. Formulae are presented which can be used for hand calculation of results to be expected in the experiment. Results of the measurements are displayed and these demonstrate the validity of the calculations. *Carpenter, John M.; "A Construction for the Investigation of Collimator Performance," work unpublished. The method of this paper will be referred to as the "P-gram construction."

PREDICTION AND MEASUREMENT OF NEUTRON CHOPPER BURST SHAPES by John Marland Carpenter A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1963 Doctoral Committee: Associate Professor Dietrich H. Vincent, Chairman Professor Albert E. Heins Professor William Kerr Professor John S. King Professor Paul F. Zweifel

TABLE OF CONTENTS Page LIST OF FIGURES iv LIST OF TABLES viii ACKNOWLEDGEMENTS ix CHAPTER I. INTRODUCTION AND DESCRIPTION OF THE PROBLEM 1 II. USE OF THE P-GRAM CONSTRUCTION FOR THE CALCULATION OF CHOPPER BURST SHAPES 6 A. Introduction of the P-gram Construction 6 B. The P-gram for a Chopper Rotor 10 C. Apparent Slit Boundaries 13 D. The P-gram for the Apparent Boundaries of a Rotating Slit 16 E. The Burst Shape Function, Definition and Derivation of an Expression 21 III. A CORRESPONDENCE THEOREM 28 IV. ANALYSIS OF THE BURST SHAPE MEASUREMENT EXPERIMENT 34 V. CALCULATION OF THE BURST SHAPE 44 VI. DESCRIPTION OF THE EXPERIMENTAL ARRANGEMENT 57 A. Collimator 62 B. Rotor 1 62 C. Rotor 2 63 D. Crystal Spectrometer 65 E. Detector 65 F. Shaft Angle and Frequency Signal Source 66 G. Phase Monitor 67 H. Time-Delay Generator 68 I. Time Analyzer 68 ii

TABLE OF CONTENTS (Concluded) CHAPTER Page VII. COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS 70 VIII. DISCUSSION AND CONCLUSION 88 APPENDIX EXPANSION OF THE FUNCTION Z (T) ABOUT p = 0 92 BIBLIOGRAPHY 95 iii

LIST OF FIGURES Page Figure 1. A schematic representation of a simple chopper system. 3 Figure 2. The neutron intensity as a function of time at three successive points in a chopped neutron beam containing neutrons of many velocities. 3 Figure 3. A schematic representation of a mechanical neu — tron monochromator used in a scattering experiment. 3 Figure 4. The coordinate system in terms of which collimator transmission probabilities are separable for Soller Slit collimators. The x-direction is the direction of the collimator centerline. The y-direction is taken parallel to the vertical dimension of the- rectangular collimator aperture. The z-direction is parallel to the horizontal dimension. 7 Figure 5. A section through two successive collimators in the resolution plane, showing a transmitted neutron trajectory and its intercept on a screen at position x. 7 Figure 6. Section through a collimator and the trajectory plane (z-r plane) showing the region of transmitted trajectories (P-gram). 9 Figure 7. Section through a rotor turned at angle 0 f.om the reference axis, showing the region of transmitted trajectories on a trajectory plane at distance x from the rotor centerplane, 12 Figure 8. Positions of the P-gram for a spinning rotor at several times, 12 Figure 9. Coordinate systems (x,z) fixed in space and (x',z') fixed in a rotor turned at angle 0, and the boundaries of a straight-slit rotor. 14 iv

Page Figure 10. The apparent collimator boundaries of a straightslit rotor turning at angular velocity a, as seen by a neutron of speed v. 14 Figure 11. The apparent collimator boundaries for a spinning rotor, and the construction by means of which the corresponding P-gram boundaries are calculated for a screen at distance L from the rotor centerplane. 18 Figure 12. The ~-gram for a straight-slit rotor for the condition v 4 4R w/h, c counterclockwise. 19 Figure 13. 2The P-gram f2r a straight-slit rotor for the condition R co/h < v 4 4R co/h, X counterclockwise. 19 Figure 14. Section through a collimator and a rotor in the resolution plane, showing the rotor turned at a negative angle. 22 Figure 15. The P-gram picture of the formation of a burst by the rotor-collimator system, for a negative time t. The shaded area is proportional to the number of neutrons transmitted at time t. ' 22 Figure 16. The P-gram picture of the formation of a burst by a rotor in a unidirectional beam. 22 Figure 17. Apparent rotor boundaries F.(a,3) for y and the transformed apparent boundaries Fi (x,zj for Y, illustrating the comparison to be made between the two functions. 32 Figure 18. Schematic representation of the system used in the burst shape measurements, showing collimator, fastneutron rotor, thermal-neutron rotor, monochromating crystal at Bragg angle 0B, and detector. 35 Figure 19. The collimator P-gram for a case in which the P-gram boundaries are parallel lines in the region where the rotor P-gram is not identically zero. 35 Figure 20. The P-gram for a rotor for the general case of finite velocity. 35 V

Page Figure 21. The P-gram picture showing the collimator P-gram undistorted, and the rotor P-gram at two instants of time; tin at which the burst just begins, and tout at which the burst just ends. 41 Figure 22. The P-gram picture showing the collimator P-gram distorted by the facoy a 1, and the rotor P-gram at two irsyants of time; tia at which the burst just begins, and t at which the burst just ends. 41 out Figure 23. The rotor P-gram PR(x,z,77 + a z/S,v/c) showing the effect of the argument i + a z/S in distorting the P-gram. Drawn for 0 < C < 1/4 ( = R w/hv). 47 Figure 24. The distorted rotor P-gram P (xz,n + a z/S,v/a)) drawn for 1/4 4 1 ( = Ro/hv). 47 Figure 25. Schematic representation of the machine configuration during the burst shape measurements. 58 Figure 26. Block diagram showing the routing of signals from various components of the machine during the burst shape measurements. 58 Figure 27. Time scale of events during one cycle of the machine during the burst shape measurements. 61 Figure 28. The physical configuration of the fast-neutron rotor. The rotor is constructed of fiberglass-resin. 61 Figure 29. The physical configuration of the thermal neutron rotor. The rotor is of Mg-Zn-Cd alloy. 64 Figure 30. Shape of the timing pulse from the magnetic pickup. 66 Figure 31. Block diagram of the phase monitor circuit. 67 vi

Page Figure 32. Horizontal foil strip activity at a position 132" from the collimator exit. 71 Figure 33. Assumed form of the deformation of the collimator.72 Figure 34. Horizontal foil strip activity at a position 66" from the collimator exit. 73 Figure 35. The collimator P-gram at the position of rotor 2 centerplane, 54" from the collimator exit. Drawn for z = 0, * = 0 on the rotor support rail centerline. 75 Figure 36. The measured burst shape for rotor angular speed w = 440 cycles per second counterclockwise, and neutron speed VB = 2650 meters per second. 78 Figure 37. The measured burst shape for rotor angular speed = 280 cycles per second counterclockwise and neutron speed B = 1780 meters per second. 80 Figure 38. The measured burst shape for rotor angular speed a = 280 cycles per second clockwise and neutron speed v = 1780 meters per second, with the detector 1.51 meters from toe rotor centerplane. 84 Figure 39. The measured burst shape for rotor angular speed ) = 280 cycles per second clockwise and neutron speed = 1780 meters per second, with the detector 1.45 meters from the rotor centerplane. 85 Figure 40. The measured burst shape for rotor angular speed 0 = 280 cycles per second clockwise and neutron speed vB = 1780 meters per second, with the detector 1.27 meters from the rotor centerplane. 86 Figure 41. The measured burst shape for rotor angular speed W = 280 cycles per second clockwise and neutron speed VB = 1780 meters per second, with the detector 1.07 meters from the rotor centerplane. 87 vii

LIST OF TABLES Page Table V-la The Functiions z (7) and z(rT,) for 0 1 ~ 1/4 51 Table V-lb The Functions z (7) and zt(T7) for 1/4 3 < 1 52 Table V-2a The Function Z (n) for 0 < <3 1/4 53 Table V-2b The Function ZZ(7n) for 1/4 < 3 ~ 1 54 Table V-3a The Function Z (n) for 0 1 4 1/4 55 Table V-3b The Function Z (n) for 1/4 f 3 1 56 U Table VII-1 Intersections.in and Slopes pin out out fcr the Collimator P-gram 77 Table VII-2 Conditions for the Measurement 440-1 79 Table VII-3 Conditions for the Measurement 280-5 81 Table VII-4 Conditions for the Measurements Using Clockwise Rotation 83 viii

ACKNOWLEDGEMENT The author expresses his sincere gratitude to Professor Dietrich H. Vincent, doctoral committee chairman, for his help in bringing this work to fruition. Dr. Vincent's suggestions, guidance and encouragement have been invaluable both in the performance of the work presented here, and as co-worker in all phases of the program to construct the mechanical neutron monochromator. The valuable discussions with Professors John S. King and Paul F. Zweifel are gratefully acknowledged, as is the encouragement offered by them and Professor William Kerr during the period of graduate study. During the first three years of study, the author was supported by fellowships granted by the Oak Ridge Institute of Nuclear Studies. This support is recognized with thanks. The United States Atormic Energy Commission, under contract No. AT(ll-1)-317, provided funds for construction of the mechanical neutron.-onochromator, and for the author's salary as research associate on that project. Use of the facilities of the University of Michigan and the Phoenix Memorial laboratory is appreciated. The efforts of Mrs, Owen Scott in deciphering and rendering into readable form the author's manuscript are recognized. Finally and most heartily, the author acknowledges his gratitude to his wife for her inspiration and forbearance during the student years, ix

CHAPTER I INTRODUCTION AND DESCRIPTION OF THE PROBLEM The development of nuclear reactors and other devices as plentiful sources of neutrons led to experiments involved with determining the energy spectra of neutrons. In addition to spectral examinations of the sources themselves, these experiments include use of the neutrons as a probing radiation for studies of properties of nuclei and systems of nuclei. This is in just the same sense as that in which sources of electromagnetic and acoustical, and also other particle radiations have been studied and used as probes. A branch of spectroscopy has developed with the use of each of these radiations in this way, and sub-branches according to wavelength or energy. Each is supplied with its own kit of tools, and each has been found useful for studying certain classes of problems. One of the tools in each branch of spectroscopy is a device for analyzing spectra according to energy (velocity, frequency or wavelength.) In neutron spectroscopy one of the basic tools is a mechanical shutter. This is a device by means of which a beam of neutrons is opened for a short interval of time, then closed, producing a short burst of neutrons. The neutrons in the beam, having a distribution of velocities, will arri-ve at a point some distance downstream at different times according to their velocities. If one detects and sorts out these neutrons according to the - 1 -

- 2 - time of their arrival, then this sorting according' to times of flight is a velocity analysis. Figure 1 shows schematically a time-of-flight spectrometer, and 2 a plot of neutron intensity as a function of time at several downstream points from the shutter. Velocity analysis by this method is not new, nor is it confined to neutron, nor even to particle analyzers. Fermi, Marshall, and Marshall(1) operated the first such neutron "chopper" system in 1943. Erected at the University of Michigan Ford Nuclear Reactor is a spectrometer which operates upon this same principle. It is intended, however, for investigation of neutron scattering events, and the change of speed and direction experienced by the neutron in the scattering interaction. Thus in this device the velocity analysis is done twice, once before scattering, again after scattering. Figure 3 shows the University of Michigan machine diagrammatically. Two shutters perform the analysis of neutrons incident on the scatterer. The first produces a short burst containing neutrons of many speeds which travel a distance L and then are incident on the second shutter. The neutrons arrive at the second shutter at different times according to their speeds, If one briefly opens the second shutter at some (1) Fermi, E.; Marshall, J.; and Marshall, L.; Phys. Rev. 72, p. 193, (1947) *The University of Michigan device in many respects is patterned after that of Egelstaff and co-workers at Chalk River, for example, see Egelstaff, P. A.; Cocking, S. J.; and Alexander, T. K.; IAEA Proc. of the Symposium on Inelastic Scattering of Neutrons in Solids & Liquids, Vienna, Oct., 1960, p. 165, IAEA (1961:. We shall refer to this paper as reference 6.

- 3 -COLLIMATOR SOURCE PLANE COLLI / ^SHUTTER DETECTOR PLANE L i~L Figure 1. A schematic representation of a simple chopper system. I ~_z L 0 L=L1>O L: L2 >L, WIt t t Figure 2. The Neutron intensity as a function of time at three successive points in a chopped neutron beam containing neutrons of many velocities. COLLIMATOR SHUTTER# SCATTERER LZ~Z2ir/SUTTER i V/7/////m i. REACTOR L ' / DETECTOR ~ Figure 3. A schematic representation of a mechanical neutron monochromator used in a scattering experiment.

- 4 - time t later, a burst of neutrons of well-determined speed v = L will pass. The neutrons then strike the scatterer and in general, change their speeds. The second velocity analysis is done by time of flight, that is, the neutrons are allowed to travel a distance L' after scattering, and are then detected. A neutron detected at time t' after the I,' burst strikes the scatterer, thus had a final speed v' =. In the University of Michigan machine, the shutters are rotating discs with several parallel, curved slits across the diameter; as the slits pass parallel to the beam of neutrons, neutrons pass through in a short burst. The duration of the burst. and also the number of neutrons in the burst, is determined by the size and angular divergence of the beam, and by the angular velocity and slit configuration of the rotor. It is the purpose of this paper to develop a method for predicting as a function of time the number of neutrons passed by a system of rotors of this kind. Other notable efforts to this end have been made by Stone and Slovacek2) and Marseguerra and Pauli (3) The work of these authors was directed toward prediction of the performance of single-rotor systems and is not readily generalized to multirotor systems. Also, these authors treated somewhat idealized, albeit useful, models of chopper systems; Marseguerra and Pauli a uniform, unidirectional beam, and Stone and Slovacek (2) Stone, R. S,; and Slovacek, R.E.; KAPL 1499 (1956) (3) Marseguerra, M,; and Pauli, G.; Nuclear Instruments and Methods 4, pp, 140-150, (1959)

-5 -a beam distributed uniformly in angle and uniformly in space. The method of this paper may be used to calculate the performance of a multi-rotor system, and properly accounts for the spatial and angular distribution of the beam passed through the system. It is based upon the author's "P-gram" (4) construction (, which was-developed to investigate the angular and spatial distribution of a beam defined by an arbitrary collimating system of rectangular geometry. (4) Carpenter, J. M.; work unpublished

CHAPTER II USE OF THE P-GRAM* CONSTRUCTION FOR THE CALCULATION OF CHOPPER BURST SHAPES A. Introduction of the P-gram Construction Let us begin by discussing briefly the idea behind the P-gram construction. We shall speak in the context of the following diagrams, figure,i and figure 5. The neutron trajectory, in direction Q is specified in terms of angles T and 9. The angle p is the angle between the x - axis and the projection of 0 against the x-y plane, while 9 is correspondingly measured in the x-z plane. These are not the angles of the usual polar coordinate representation but are appropriate here because most collimating elements are Soller Slits, that is, they have rectangular cross sections. In this case the probability that a neutron which crosses a downstream point x, y, z in direction cp, be transmitted through the collimating system is separable into the product of two factors. One is dependent upon x, y, qc and the other is dependent upon x, z, 9. Here we have assumed y and z to be measured along the rectangular cross-sections of the collimator(s) while x is measured as the distance downstream from some reference point. The probabilities dependent upon x, y, p can thus be considered independently of those for x, z,, This separation is the more significant because in *See footnote below - 6 -

- 7 - Z in j Figure 4. The coordinate system in terms of which collimator transmission probabilities are separable for Soller Slit collimators. The x-direction is the direction of the collimator centerline. The y-direction is taken parallel to the vertical dimension of the rectangular collimator aperture. The z-direction is parallel to the horizontal dimension. COLLMATO IXCOL COLLIMATOR i COLLIMATOR 2 Figure 5. A section through two successive collimators in the resolution plane, showing a transmitted neutron trajectery and its intercept on a screen at position x.

- 8 - most cases only one plane is important for calculating chopper resolution. The rotor turns in this plane. We chose it to be the x, z plane and consider angles V in this plane. Having restricted our attention to effects in the "resolution" plane, we erect an imaginary screen at an arbitrary downstream point x, and at that point draw z, * axes as in figure 6. Each point on the z, 9 plane specifies a trajectory. The collimator will allow neutrons to cross the screen at x for some trajectories z, * and will not allow those others for which the trajectory passes through the walls of the collimator. We draw a diagram in the z, A plane, the interior of which corresponds to "allowed" trajectories for which the transmission probability is unity. The exterior corresponds to those trajectories which intersect collimator boundaries and for these, the transmission probability is zero. We call this diagram a P-gram.* We define a function P(x,z,*) which is unity inside the P-gram, zero outside, i.e., P = 1 if z, * satis.fy certain conditions, and P = 0 otherwise. The function P is the transmission probability. The function P depends upon the downstream distance, x, at which the beam is examined, to the extent that this distance establishes the shape of the P-gram. We shall carry *The word "P-gram" is used here to designate the region in the z, i plane containing transmitted trajectories. For simple collimators, this region has the shape of a parallelogram, while it also represents the region of unit transmission probability, thus the name "P-gram."

- 9 - Figure 6. Section through a collimator and the trajectory plane (z-f) plane showing the region of transmitted trajectories (P-gram).

- 10 - x as an argument in P therefore to indicate this fact. In those instances where formulae are presented, the origin of that coordinate will be specified. The utility of the construction is further enhanced if one notes that if there is more than one collimator in the beam, then the event of tranmission through all consists of transmission through each individually. Thus the probability of transmission through all is the product of probabilities for transmission through each, since these are independent events. The P-gram picture then consists of several overlapping P-grams, each appropriate to one collimating element, and the overlapping areas are those in which the joint transmission probability is unity. Thus if Pl(x,z,9) is the probability of transmission of neutron x, z, * through collimator 1 and P2(x,z,!) is for collimator 2, then P12(x,z,r), the probability of transmission through both is P12(x,z,) = Pl(X,z,') P2(xz,*). B. The P-gram for a Chopper Rotor The rotating slits of a chopper rotor may be considered to constitute a collimator. There is the added complication that the orientation of the rotor is not constant with time~ However, consider carefully the following argument for a stationary collimator oriented with its slit centerline at some angle 0 with respect to the beam centerline (x - axis) as in figure 7. The slit centerline projects to a point

- 11 - xe on the z - axis; the maximum angle transmitted is 0 +; max. the minimum is 0 + min. A P-gram for this slit may be drawn but it is centered about z = xO, and * = 0. For small angles the P-gram for this slit is the same when the slit is oriented off-axis by an angle 0 as when it is oriented with the beam axis, except for this displacement. If this angle 0 is time dependent, Q = wt, the rotor P-gram moves in the z, * plane with its center along the locus z = xwt, = wt (or z = x$) as in figure 8. For counterclockwise rotation, as assumed, motion is in the direction of the arrow; if rotation is clockwise, then the rotor P-gram moves in the opposite direction. The P-grams of figures 7 and 8 represent regions of allowed trajectories for neutrons of infinite speed, which cross the rotor centerplane when the rotor is oriented at angle 0 = wt. For neutrons of finite speed the slit will appear curved, and the P-grams will have a different shape. We reserve until later the problem of calculating this effect. For now, let us address the problem of assigning some meaning to the P-gram as seen in a coordinate system moving along with it as it traverses the z, 9 plane. Recall that the time reference t = 0 is the time at which the slit centerline is parallel to the beam axis. At any later time t the slit centerline will be oriented at an angle wt from the beam axis. The region of allowed trajectories (P-gram) for neutrons which cross the rotor centerplane when its slit is parallel to the beam axis is

- 12 - Z=X ~9 0+*rn in ~ X Figure 7. Section through a rotor turned at angle 0 from the reference axis, showing the region of transmitted trajectories on a trajectory plane at distance x from the rotor centerplane A-110 Z= X* wt _wt t _ t<O Figure 8. Positions of the P-gram for a spinning rotor at several times.

- 13 - centered at z = 0, O = 0. The P-gram for neutrons which cross the rotor centerplane when its slit is at angle cut to the beam axis, is centered at z = xct, w = cut. This is to say, the P-gram displaced to the point z = xut, 9 = out is for neutrons which cross the rotor centerplane at time t. The P-gram retains its shape for each time t (angle ct). Thus we need only consider the shape of the P-gram for neutrons which cross the rotor centerplane at time zero, at which time the slit centerline is parallel to the x axis. The shape will depend upon the physical configuration of the rotor, the neutron speed and the rotor angular speed. We shall work here with a straight slit rotor, but we will later show that the results thus derived are applicable to the entire class of curved slit rotors with Archimedes' Spiral centerlines, or those which approximate this condition. The equations for the apparent slit boundaries describe the walls of a collimator which we may now consider to be stationary. From these the boundaries of a P-gram may be determined. C. Apparent Slit Boundaries Let us derive the equations for the apparent slit boundaries as seen by a neutron crossing the rotor centerplane at time zero. Refer to figure 9, where symbols used here are defined. In a coordinate system fixed in the rotor (x',z) the straight slit boundaries are given by z' = - h/2 (choose + for upper, - for lower boundary). Here, h is the

- 14 - \\<~> 0 X -0R Figure 9. Coordinate systems (x,z) fixed in space and (x',z') fixed in a rotor turned at angle e, and the boundaries of a straight-slit rotor. v 2 w 2 h z='V 2 Figure 10. The apparent collimator boundaries of a straightslit rotor turning at angular velocity C, as seen by a neutron of speed v.

- 15 -width of the rotor slit. Coordinates in the x', z' system are given in terms of those in the x, z, system through the transformation x' = x cos 0 + z sin 0 z' = -x sin 0 + z cos S (11-2) Thus in terms of the coordinates of the stationary system, the boundaries are given by + h/2 = -x sin 9 + z cos 9 (11-3) For a spinning rotor, 0 = wt, so that at any instant t the slit boundaries are + h/2 = -x sin. tt + z cos cot (11-4) The apparent slit boundaries are given by the intersections of a neutron plane wave moving at some speed v, and crossing the rotor centerplane x = 0 at t = 0, which is represented by the line x = vt (II-5) The apparent boundaries are thus parametrically ( x = vt -x sin wt + z cos wt = + h/2 (II-6) The parameter t may be eliminated rigorously but it is adequate for our purposes to restrict our attention to small times (wt << 1) which correspond to small angles; then we have

- 16 - and eliminating parameter t, X2 + z = - h/2; v w- 2 + h Z+ v 2x (11-8) These equations (+ for upper, and - for lower) represent the apparent boundaries of a slit rotating at angular speed w, as seen by a neutron moving at speed v.* They pertain of course only for -R < x - R, and are derived for neutrons crossing the rotor centerplane at t = 0, when the slit is parallel to the beam axis. The boundaries are parabolic, and are symmetric functions of x. Moreover note that the boundary equations are dependent only upon the ratio t, and that for v-.- oo or c = 0 the equations are those of a straight slit, z = - h/2. D. The P-gram for the Apparent Boundaries of a Rotating Slit On figure 10 is drawn the collimator formed by the apparent slit boundaries for finite v. We wish to derive a P-gram for this collimator. For this purpose we may note that the re-entrant surface (lower boundary on figure10) is not effective as a collimating surface, and it is the edges (c) and (d) which are important. The upper surface establishes a limiting condition on trajectories z, *, which are tangent to it for -R ( x < R, ard the edges (a) and (b) may also *These results correspond to those of Marseguerra and Pauli, who used the notation y* (x) and (x) for our z (x) and z (x).

- 17 - limit the trajectories. The P-gram boundaries on a screen at a distance L from the rotor axis are established by several conditions. For trajectories z, V which just touch an edge, say (c), there is a unique relation between z and * which gives the P-gram boundary, see figure 11. The value of z on the screen at L is shown for a trajectory with angle X, which grazes the edge (c). The relation thus established between z and * is the boundary of the rotor P-gram due to edge (c). For each edge therefore we have the boundary equations: Edge (a): z = h/2 + R2 + (L + R) V (II-9a) V Edge (b): z = h/2 + R+ (L - R) * (II-9b) V Edge (c): z =-h/2 + R2 + (L + R) * (II-9c) v cn2 Edge (d): z =-h/2 + R + (L - R) * (II-9d) which are referred to as lines (a), (b), (c), and (d) respectively on figures 12 and 13, which are drawn for the simplifying case L = 0. In addition to these boundaries due to the edges of the slit, the upper and lower surfaces may bound the rotor P-gram. The limiting trajectory * is one which is tangent to the slit surface (say the upper surface) at some point x*, z*. At the tangent point, d z 2

- 18 - 18 b _=R2~ h ~ — L fo th rtren Z_ R2,v 2 fL+ R)^ V2-^ L+ R Figure 11. The apparent collimator boundaries for a spinnin rotor, and the construction by means of which the corresponding P-gram boundaries are calculated for a screen at distance L from the rotor centerplane.

- 19 -' z=- -41 4 h Rw 4w 2. h (a)\ c (b h (C) 2+ R 2 v z v 2_ h 4w 2 Figure 12. The I-gram for a straight-slit rotor for the condition v ~/ 4R w/h, X counterclockwise. h.R~ 2-' (b) z=_ v. 2 h 4w 2 (d h _h4 Ru h - -o-~ 2 ~.~' 2R Figure 13. 2The P-gram fir a straight-slit rotor for the condition R )/h 4 v, 4R O/h, 0 counterclockwise.

- 20 - from which x*= (II-10) 20 and * - *2 + h v 2 h v 2 40 + 2 (1I-11) On the screen at L the value of z for this trajectory is z = z* + (L- x*) i - 2 h v 2 v 2. h ~4v + 2 (II-12) In further work, we shall uniformly assume that the screen is erected at the rotor centerplane, i.e., L = 0. The boundary due to the upper surface intersects the line (b) at the intersection z = + h/2 + C R2 R * "z = v 2+ h/2 (II-1 3 from which v -2 03 R2 0 R- R 4 R v = 2 R o/ (4w-14) ~v ~ ~ section of (b and (c,.lines (a) and (b) no longer bound the2 2 R w/v (11-14) For -- such that this intersection coincides with the intersection of (b) and (c),. lines (a) and (b) no longer bound the

- 21 - P-gram. The P-gram of figure 12. thus is for the condition 2Ro < h v 2R or 4R W v >/ 4R2a @ h (11-15) For the condition 4R ) v Q h (II-16) the picture appears as in figure 13. From this last diagram we can see that the P-gram completely closes when the intersection of the lines (c) and (d) meets the parabola, i.e., when - h/2 + R2 /v = + h/2 v = R2 w/h (II-17) No neutrons are transmitted when v ( R2 /h. E. The Burst Shape Function, Definition and Derivation of an Expression We now complete the description of the method of this paper. To this point, we have developed all the results needed to proceed to the calculation of the burst shape. The first is the product rule, equation (II-1). P12(xz,') = P1(X'z,') P2(x'z,') (II-1) Using this, we shall say that P1(x,z,i) is the probability of transmission through the collimator of figure 14, and rename it Pc. P2 applies to the rotor and we rename it PR.

- 22 - COLLIMATOR ROTOR Figure 14. Section through a collimator and a rotor in the resolution plane, showing the rotor turned at a negative angle. P,=l UPt<0 -Figure 15. The P-gram picture of the formation of a burst by the rotor-collimator system, for a negative time t. The shaded area is proportional to the number of neutrons transmitted at time t. ~^t _ (wt<O ) Figure 16. The P-gram picture of the formation of a burst by a rotor in a unidirectional beam.

- 23 - PR is a function not only of x, z, '; but also of the time t and the neutron speed v, rotor speed w, and rotor parameters. Thus the probability P(x,z,A,v,ou,t) may be written P(x,z,*, v,W,t) = (XI P(x,) P(x,z,,v,W(,t). (11-18) Second we appeal to the assertion of section B which stated that time dependence of the rotor transmission probability is simply described as translation of the P-gram in the z -, plane, as a function of t, the shape being unaffected. This may be indicated mathematically by the statement PR(x,z,f,v,o,t) = PR(x,z -Lwt, - ct, v, a) (II-19) where for PR(x,z,,,v,c) we take the function derived in R section D, see figure 12, and L is the distance from the rotor centerplane to the screen. Hereafter we shall assume this distance to be zero. Third but less crucial is the note made at the end of section C, that the shape of the P-gram is a function of the ratio c)/v only: PR(x,z, t V) P ( - GCt, v,) P( - t, v). (II-20) Let us now make a more precise definition of the burst shape function. Consider NI(v) - the number of neutrons incident upon the

- 24 - chopper system with speed v, per unit length z per unit time t, per unit angle V. G(v,t) - the number of neutrons of speed v which cross the rotor centerline in time dt about t, which are transmitted by the chopper system. We shall assume N to be independent of A, z, t. Then G(v,t) = Ni(v) | P (x,z,') PR(xz,' - t, w/v) d z d A (11-21) At first glance it would appear that G should be a function of the position x along the beam, but more careful thought indicates that the probability that a neutron be transmitted does not change with the position x at which it is observed. The position x appears in PC and PR only to the extent that it establishes the shapes of the P-grams at whatever position is chosen to examine them. After the integrations are performed the results are independent of x. The position x can be chosen at will for convenience, and here we pick it to be the rotor centerplane. We define the burst shape function (v,t) (Ni (v) (1I-22) = P (x,z,4) PR(X,z, - wt,w/v) d z d I (11-23)

- 25 - P then has dimensions of length times angle. The significance of P(v,t) in reference to the z-f plane of the P-gram construction is now clear: It is the total overlapping area of the collimator P-gram, PC = 1 and the rotor P-gram. PR = 1. This area is shaded in figure 15 and can be seen to change with time t, while it is a function of o/v because the shape of the PR boundaries is dependent on c/v. The function P(v,t) is related to the burst shape functions of Stone and Slovacek (2) and Marseguerra and Pauli (3) in a simple fashion. Both pairs of authors considered a unidirectional beam (that is, a perfectly collimated beam) high enough to cover the rotor slit (height h as in figure 9.) In the same sense as in (11-22), let us define the burst shape function for the unidirectional beam, P (v,t). The collimator transmission probability for a perfectly collimated beam containing At neutrons traveling in direction 0 = 0 may be represented Puc(X'Z' ) = A5 *6(); -h 2 < z < h12 0; otherwise This function is represented as an infinitesimally narrow strip in figure 16. Then, as in (11-23), h/2 Pu(vt) = Pc(x, z,9) PR(xz, -cot,w/v) d z d ' -h/2 (h/2 = | Af6(3) PR(x,z,4 -wt,w/v) d z d d I -h/2

- 26 - h/2 = h PR(x,z,-3t, //v) d z -h/2 (11-24) PR is an even function in many cases, so we shall replace -wt by wt in the argument of PR. Since PR is unity inside the P-gram and zero outside the integral (II-24) is just the difference between z on the upper boundary of the P-gram and z on the lower boundary, P (v,t) = A(Z(max(Wt,/v) - Zin (t,'/v ) (II-25) This is the result of references (2) and (3), except for the normalization constant h, inserted by both pairs of authors. Using the notation of Stone and Slovacek for the burst shape function, f(vt) = (Z (Wut,a /v) - (min(t,)/ ) h max min hA P(v,t) (II-26) Use of the P-grain construction for the calculation of burst shapes has the advantage that it enables the spatial and angular distributions to be handled in a conceptually simple way. The authors of references (2) and (3) confined themselves to calculations for unidirectional beams*, *It must be noted that Stone and Slovacek did proceed to develop a Fourier series for the burst shape produced by a rotor in a beam distributed uniformly in angle and uniformly in space. Their procedure, although giving a good approximation to the correct result in many cases, did not account for spatial effects at all.

- 27 - presumably because of their difficulty of including the effects of distributed beams. The P-gram construction as applied here has the advantages that spatial and angular effects are well accounted for and easily understood. Moreover, the method can be generalized to cases involving several rotors and more complex collimators. In those cases it could be the basis for a very efficient digital computer code, or in fact could lead to development of simple optical analogies.

CHAPTER III A CORRESPONDENCE THEOREM In this chapter we undertake to prove a correspondence theorem, through which the performance characteristics of similar* rotors are related, as a function of rotor angular speed, neutron speed, and rotor design parameters. Invoking this theorem will enable us to justify making calculations for simplified cases and generalizing them to more complicated and realistic cases, As a criterion for judging similarity of performance, we shall fix upon the "apparent rotor boundaries" discussed in sections IIB and IIC, (see especially figure 10). That is, we shall exhibit conditions relating rotor speed, neutron speed and rotor parameters, under which the apparent boundaries of different rotors are identical. Let us say that we wish to construct a rotor for which the apparent boundaries are described by equations Fi(x,z) = 0 (III-1) where i indexes the boundary surfaces of the rotor, and (x,z) is the stationary coordinate system of figure 9. In section IIC, we displayed the transformation through which the boundary equations written in the rotor coordinate system (x,'z of figure 9, are related to the apparent boundaries. The transformation is a function of neutron speed v and rotor speed w: *What is meant by "similar" will be explicitly stated below. - 28 -

-29 -x = vt x = X cos Wt - z sin cot z = x' sin wt + z' cos wt (III-2) This may be rewritten in terms of the single parameter 7 = v, when the dummy time t is replaced by the parameter e = ct: x = 0/7 x = x' cos 9 - z' sin 9 z = x' sin 0 + z' cos s (III-3) Now, if the slit boundaries are described by equations fi (x',z')= O (III-4) in the rotor system, then the function Fi(x,z) describing the apparent boundaries is given parametrically through the system f (x',z') = 0 Fi(x,z) 0 x = 0/7 x = x' cos e - z' sin e z = x' sin e + z' cos 9 (III-5) or inversely, the rotor boundary functions are given in terms of the apparent boundary functions through Fi (x,z) = 0 fi(x',z') = 0 x = 8/7 13 x = x' cos e - z' sin e z = x' sin 9 + z' cos 0 (111-6)

- 30 - Having now established the required transformations, let us proceed to the proof. Suppose we design a rotor to produce apparent boundaries Fi(x,z) = 0 for 7 = yo; then the rotor slit boundaries will be given by F (a,3) = 0 fo (x',z') = 0 a = O/70 f a = x' cos e - z' sin e = x' sin 0 + z' cos e (III-7) in which we have replaced dummy parameters (x,z) by (a,O). Further let us inquire what are the apparent boundaries for this rotor for r = 7l. They will be given by the system ft(,n) = 0 10 x = cp/7l Fi 0(x,z) = 0 F _ xx = cos p - 1 sin p z = ~ sin p + 77 cos p (III-8) Here the dummy parameters (x',z') have been replaced by (,77), and the equation fi. (,7) = O may be replaced by the system (III-7), to form /Fi(a,3) =O a =- /0 F10(xz- a = cos 0 - n sin F = sin0 + cos = 4 sin e + 77 cos e x = cos 7 - 7 sin p z = s sin p + 77 cos q x = P/71 (III-9)

- 31 - It is only the transformation that we are interested in, that is that part of the system which for a given (a,3) gives (x,z). We may easily eliminate (,717) to obtain x = a cos (p - 0) - 3 sin (cp - ) z = a sin (qp - 0) + 3 cos (p - 0) x = (p/l a = 6/7o (III-10) This system may also be viewed as one which for given (a,x) gives (3,z). Refer to figure 17. For each value of x = a, we wish to compare the value of f from the equation F~(a,3) = 0 (on the original apparent boundary) with the value 10 of z given by F. (x,z) = 0 (on the transformed boundary). That is for each abscissa x = a, we compare 3 from o0 10 F. (a,) = 0 with z from F. (x = a,z) = 0. The system (10) for the choice x = a, gives the values of P, z under the stated transformation. If P lies on the original apparent boundary, for a choice of a, then z lies on the transformed apparent boundary for the choice x = a. The transformation (10), for x = a reduces to a = a cos (cp - ) + 3 sin (p - 0) z = a sin (q - e) + C cos (p - 0) (P- = a (l - o) (IIl-11) which is a function of the parameter (y1 - yo) only. We may then state explicitly the following theorem:

- 32 - zt F (a,,B)=O 10 FX1 z)=O X. =a x,a Figure 17. Apparent rotor boundaries Fi (a,) for 7 and the transformed apparent boundaries F. x,z) for 7, illustrating the comparison to be made between the two functions.

- 33 - If rotor A produces apparent boundaries Fi(x,z) = 0 for 7 = 7 and rotor B produces apparent boundaries Fi(x,z) = F(x,z) = O B for 7 = y0, then the apparent boundaries A produced by A for y = 71 are identical to B those produced by B for 7 = 71 if A A B B 71 7o = 7l -1 o (III-12) This result is to be applied in the following manner. Recall that, equations describing apparent slit boundaries were developed in Chapter II, for a rotor with straight slits. In the language of the theorem above, the straight slits are the apparent boundaries for infinite speed neuA trons (y = 0). We have developed expressions for the apparent boundaries for finite neutron speeds, that is for 7= >O. 0 The rotors which were used in the measurements described below have curved slits, such that their apparent boundaries B are straight slits for a certain value of 7, 7 = 7o. The theorem states that for neutrons with =, the apparent theorem states that for neutrons with 7 = yl, the apparent boundaries of the curved slit rotors are identical to those for the straight slit rotor, if (III-13) is satisfied. Thus we shall use the formulae of Chapter II to describe curved slit rotors by replacing v everywhere by v3 v D o -1 B - ~ -- B (III-13) v v

CHAPTER IV ANALYSIS OF THE BURST SHAPE MEASUREMENT EXPERIMENT For the purpose of these measurements, the FNR chopper was set up in the configuration shown in figure 18. The beam is formed in the collimator and strikes the thermal neutron rotor, which is assumed to turn counterclockwise. The pulsed beam emerging from the chopper rotor contains neutrons of many speeds, and we select those having a welldefined speed by placing a monochromating crystal in the beam. Neutrons travel to the detector and are counted, having traversed a flight path of length L2D from the rotor centerplane to the detector. We assume the detector to be thin, (it is ~ 3mm, thick) and that no appreciable uncertainty is introduced into the flight path in this manner. That is, we assume that neutrons are detected instantaneously after crossing the detector aperture. Further explanation of the details of the experiment is given in Chapter VI, below. We wish to develop an expression for the counting rate at the detector, as a function of time. Let us make the following definitions: H(t) dt - the number of neutrons crossing the detector aperture in time dt about t. (IV-1) PC(x,z,) the probability that a neutron with trajectory x,z,* is transmitted through the - 34 -

- 35 - FAST-NEUTRON THERMALCOLLIMATOR ROTOR NEUTRON ROTOR _____ _ ^\7E CRYSTAL 7////7 e 7 DETECTOR Figure 18. Schematic representation of the system used in the burst shape measurements, showing collimator, fastneutron rotor, thermal-neutron rotor, monochromating crystal at Bragg angle B, and detector. t/iP~ ~ i ~ ~ -~ e -REGION WHERE -F- +PR f ~ Figure 19. The collimator P-gram for a case in which the P-gram boundaries are parallel lines in the region where the rotor P-gram is not identically zero. /PR (X, Z, t) I Figure 20. The P-gram for a rotor for the general case of finite velocity.

- 36 - collimator. (IV-2) PR(x,z,*,t',T) - the probability that a neutron with trajectory x,.z,f crossing the rotor centerplane at time t', with flight time T, shall be L2D transmitted through the rotor, where T = and L2D is the flight path from rotor 2, the thermal neutron rotor, to the detector. (IV-3) N (v)dv - the number of neutrons incident on the chopper system with speed v in dv, per unit angle I, per unit length z, per unit time t. (IV-4) X(,T) -= probability that a neutron traveling at angle ' and with flight time T shall be reflected from the crystal. (IV-5) D(t',t,T)dt - the probability that a neutron of flight time T which crosses the rotor centerplane at time t' shall cross the detector aperture in time dt about t. (IV-6) P (x,z,'): PC is unity within a region of the z, 9 plane, zero outside as in figure 19. The shape is determined as in reference (4).

- 37 - PR(X,Z,,tt ',): R PR is unity within a region of the z, ' plane and zero outside. The region PR = 1 moves in the z - a plane, PR(XIZI,,t',T) = PR(x,z,;-Ot',T) and has a shape determined by the method of chapters II and III. See figure 20 x(i,T): A neutron is reflected if it satisfies the Bragg condition: A = 2d sin (OB + ) (IV-7) We may write this as a condition on the flight time as well as on the wavelength: A = K; K = a constant (IV-8) L L 2D 2D (IV-9) T =; = A (IV-9) V K 2D = 2d sin (eB + I) (IV-10) K B Recall that * is small, so that sin _ I, cos 1; we write therefore 2dL T 2D [ sin 9B cos a + cos 0 sin 2dL 2D c K sin B+ cos Csj K B = sin B 1 + cot (IV-11) The mean angle of the V distribution is V = 0, and this gives the mean flight time T reflected from the crystal, = 2D sn B2) TB = K sin Bg (IV-12)

-38 -thus (11) may be written T = B I 1 + * cot B (IV-13) The neutrons are reflected with an efficiency R(T), (5) related to the crystal reflectivity of Bacon) and having the dimensions of flight time. Thus X(~,T) = R(T) 6(T - TB 1l + cot OB] ) (IV-14) D(t',t,T): Neutrons crossing a point at time t' always reach a downstream point at distance L2D at a time t given by t = t' + T, so D(t', t, T) = 5(t - t' - T) (IV-15) As defined, these functions combine to give the measured counting rate H(t)dt= i NO(T) Pc(xz,')R(T)5(T-TB l1 + V cot B1 )() (x) PR(x,z,*-wt',T)5(t-t'-) d z d z d d dt' dt (IV-16) The first 5-function under the integral takes care of the T-integration, replacing T wherever it appears by T = TB(1 + V cot 0B). However, we may consider R(T) N(T) to be slowly-varying functions of T, and therefore evaluate T at its mean value, TB, in those functions. Similarly, (5)Bacon,"Neutron Diffraction," p. 64, 82., Oxford University Press, London (1955)

- 39 - PR varies slowly with respect to T in its last argument and the same is done there H(t)dt = ( N(TB) PC(x,z,')R(T- )PR(x,z, -3t',TB ) b(t-t'- B [1 + cot RB] ) d z d t' dt (IV-17) The t'-integration is easily performed, requiring that we evaluate t' at t' = t - TB( + cot 0 B): H(t)dt = I N0(B)PC(X'Z,')PR(X,'Z, - u(t - TB [1 + cot 0B,TB) d z d * dt (IV-18) = No(TB)Pc(x,Z,*)R(TB)P (x,z, rl+a3TBcoteB]-w(t-TB),TB) d zd V (IV-19) In the same spirit as in the case of the burst shape function P(v,t) of Chapter II, equation (11-23), we normalize this function to unit incident intensity and unit reflectivity, H(t) (O B''t N o(r ) R(-B) = PC(X,Z, )PR(X,Z,E (1 + W TBCOt eB)- (t-T) ),T) d zd d (IV-20) The function M(eB,t) will be referred to as the "measured function." Note the factor (1 + a TB cot 9B) multiplying? in PR R%

- 40 - The rate of motion of the rotor P-gram across the z - a plane can be obtained by following a point on the P-gram, f(1 + OTB cot B) - W(t - TB) = constant d V w (IV-21) d t (1 + yrB cot 0B) B B Say the factor (1 + A B cot 6B) is greater than unity. Then the rotor P-gram moves across the z - i plane more slowly in this case than in the case of the burst shape function. Thus the rotor P-gram moves more slowly through the collimator P-gram; the effect is to broaden the burst by broadening the contribution of the collimator P-gram. What does the multiplier on V do to the P-gram picture of the formation of the burst? Figure 21 shows the case for a perfectly monochromatic beam. Figure 22 shows how this picture is modified by the presence of the factor (1 + (rB cot 0B) arising due to the fact that a crystal monochromator was used. We make a minor change in the-form of (IV-20), in terms of which to discuss this modification. For this, define a new dummy variable = V(1 + By rcot d 1 + Wct B B M(OBt) =Pc(x'z',/(l + rBKcot B))c PR(XBZ - |(t CB -' ) (1 + - cBot B) ( IV(IV-22) In a z - -plane, the rotor P-gram is undeformed by the factor (1 + CrT cot 0B) = a, although the collimator P-gram D - B

- 41 - z Pc(X,z,)= I PR (X q/-U(tin- )'.B )= I ~' Xi I Jtt/ *,/PR(X'-Z ( W(tOut- TB)TTB ) W(ti -T) ( tut B ) Figure 21. The P-gram picture showing the collimator P-gram undistorted, and the rotor P-gram at two instants of time; tin at which the burst just begins, and out at which the burst just ends. F. xz,(Xt0Z, )=I distot / a t oo o.. ^^,,.( tout- T.) - ~ ~ ' ~ ^ -(outB ~ Figure 22. The P-gram picture showing the collimator P-gram distorted by the factor a > I, and the rotor P-gram at two itstants of time; t.' at which the burst just begins, and t at which the burst just ends. outt

- 42 - is now modified. Also, the rate of motion of the rotor P-gram across the z - c plane is d t ) w (IV-23) d t In addition, for a = 0, Ma (eB,t) = (x,zb)PR(xz,-o(t-B),TB)d z d V = Ir[Pc(x,z, Q)d 1 PR(X,Z,-Oa(t-TB),TB) d z (IV-24) If [Pc(x,z, )d j is constant with respect to z in the range where PR / 0, then M =0(0Bt) is proportional to the function P (v,t) of equation (11-24). Further, the total number of neutrons per burst for M(OB t) is (M(B,t)dt = ( Pc(Xt,))P R(XZ',-(t-TB) TB)d z d ' d t (IV-25) where a (1 + c) TB cot eB). Put af - cu(t-TB) = p, dp. = wdt then M(iB,t)dt = - | P (x,z,*)d k PR(x, z,,TB) d nt d z (IV-26) which is independent of a, and therefore remains finite for all values of a. (One might expect that as a-'-0, as it may, the neutron intensity would approach zero.) This result corresponds to that for the monochromatic beam, that is for P (vt). Still further to illustrate the effect of the presence

- 43 - of the factor a. refer to figure 21, for o=l (for P(v,t)). The burst begins at the time tin such that (tin-rB) -(/2 + in; tin = TB -(5/2 + in (IV-27) and ends at time to such that u(tot- B) = (5/2 + out out =rB +(5/2 + in) (IV-28) The width of the burst is thus tout in = ( + in + out (IV-29) where 5 is due to the rotor, and (in + )out is due to the collimator. Referring to figure 22, for a > 1, we see that the collimator P-gram on a z - g plane is broadened. Thus, proceeding as above, the width is (a) -(a) out tin ( +a(in + out)) (IV-30) illustrating the broadening effect as one which increases the collimator contribution to the width by the factor a. Further and more complete results will be developed in the following chapter.

CHAPTER V CALCULATION OF THE BURST SHAPE In the preceding, expressions were developed for the functions P(v,t) and M(!B,t), exhibiting them as integrals involving the collimator and rotor transmission probability. Equations describing the apparent boundaries of the P-gram for a straight-slit rotor were derived. It was shown how to apply those results to the description of a class of similar curved-slit rotors. Little has been said concerning the collimator P-gram itself, since its shape can be determined by the methods of reference (4). The effect of the monochromating crystal has been shown to be one which broadens the collimator P-gram in the V dimension by the factor a = (1 + wu cot 0B). B B Two approaches could be followed from this point: To perform the actual calculations using a computer, or to develop formulae for hand calculation. For more complex analyses than the present, the computer is the only reasonable choice, and the techniques described above should form the basis for a very efficient program. However, the experiment which we propose to analyze is fairly uncomplicated and we can expect hand calculations to be quite satisfactory, as well as more general. We proceed in the following to develop formulae suited to hand calculation. It is the case in many systems that the collimator P-gram has the special property illustrated in figure 19. - 44 -

- 45 - Here the collimator P-gram boundaries are particularly simple in the important region traversed by the rotor P-gram: ( 1 if - +z/S + z/S P(x, z, ) = 0; otherwise (V-l) Where o is the maximum angle transmitted by the collimator through the point z = 0, and S is the distance from the screen to the aperture in the collimator defining the collimator P-gram boundary. In terms of the variable 4 = at, 1; if -ac 0 + a z/S + a z/S P (x,z, /a) = C 0; otherwise (V-1') For the range of z over which (V-l) holds, PC may be represented therefore -Ro 0 (v-2) -o or P(x,z,0/a) 6!(oCz/S - ~ + ~') d i' a ( (v-2') Recalling equation (iV-22), M(B',t) a= | Pc(X:Z',/a) PR(XZ' - ct, TB)d z d2 (IV-22)

- 46 - Applying (V-2'), we have (o M(eBt) = \ (a z/S - + g')d'PR(x,z, -ct,TB)d z d -0o (V-3) Putting g = 7 +a z/S +wt, dg = d7, this becomes 0 M(OB,t) = ) 5((' -7 —wt)PR(x,z,7 + az/S,TB)dS 'd7 d z -a*0 (V-4) and further changing ~" = ' - ct, dg" = d ' M(0Bt) = 1\ ("-7)PR(x,z,7 + az/S,TB) d" dd d z -a0- 0t (V-5) The function PR(xz,n +az/S,TB) looks much like the previous rotor P-gram, but is deformed, as in figures 23 and 24. The integral (PR(x, z,? + az/S, B)5 ("-7) d71 d z (V-6) is interpretable as the length of a vertical line passing through 77 = g", which lies inside the deformed rotor P-gram. This interpretation is the same as that given in section IIE, in showing the correspondence between the P-gram burst shape calculation, and the calculations of Stone and Slovacek,

-47 - Sb) z=R(-L- 42RA +7p (c) ZR (n7 + //2 +R8 (C) Z= R (-778 ( (-'+ 8,2 +,08) L8(-2$+PP-P/2).8(2p +,PP/2 ~ S(Op +1/2) 8(P/2 -~8 (-,P+'I/2) '7? (d) Z=-R (7-8/2+ ) I+P (e) z= rp( /2 + Figure 23. The rotor P-gram PR(xIzz, + a -, v/X) showing the effect of the argument + a z/S in distorting the P-gram. Drawn for 0 < < 1/4( = R /hv). Z - -I (,) z=R(- 778+ 2( 3)7 p P2 p + + 8(-2v/'+2- -2pe +P/2 +.p) S (2 -2- 2 pv + P2 + P) (h) z=- (+ p( 2 (9) z - = — -+2 Figure 24. The distorted rotor P-gram PR(x,z,? + a z/S,v/cw) drawn for 1/4 ~ 1 ( = R c/hv).

- 48 - and of Marsegue-ra and Pauli. The difference here is in the deformation of the rotor P-gram, and arises due to the correct inclusion of the effect of the collimator. In similar fashion to the interpretation of section IIE, we may write this length as the difference between z on the upper boundary of the deformed P-gram, and z on its lower boundary: PR(Z'7 + a S"' /v) 5("-n)d7n d z = Zu(") - z R (") (V-7) The burst shape function, from (4) is then aJo-cot M(OBt) = \ (") - (V") JdC" -a o-wt (V-8) Further, let us define these integrals z u(') dT' = Zu(n) u (7n') d7' = ZR(n) (V-9) acxL Here au and a^ are irrelevant limits, chosen to simplify the work. We shall calculate these last-defined functions Z (7) u and Z (7), in terms of which M(eBt) = Z (ac -cot) - Zu(-aO-iot)- Z (a Z -o t)+Z (-Ca-ct) = AZ (a - cot) - AZ (-a wo - at) (V-10) 0 0 The calculation is performed as outlined above. The boundary equations of the deformed P-gram, PR(x,z,T+aS,TB)= 1

- 49 - are first computed. These equations are displayed in tables 2 2 2 Ia (for -^h L v oo ) and Ib (for- < v h -,and Figures 23 and 24. The results are presented in terms of reduced variables R = S h R2 hv (V-ll) The functions Z (n) and Z (7) are calculated from z (n) and zA (n) on these boundaries. The work of the actual calculation is algebraically involved but altogether trivial, requiring integration of elementary forms. Therefore, only the results are presented. Table IIa contains the function 4~(7R2 R R203 Table Zu(7) for 4R2 v cO<, IIb for D < v Table IIIa contains Z2 (77) for h C v4O, IIIb for h v h h v~OO, II~b for —v h As it happens, a removable singularity exists at p = 0 in the functions (V-30) and (V-34). As a consequence, for small values of p, these equations do not lend themselves to easy calculation. To simplify the numerical calculations it is appropriate to approximate these equations with an expansion in powers of p. The development of the expansion is straightforward and is presented as an Appendix,where the notation is also clarified. The result is (1) 4R P _2 4 In/21 (-3/2)n-k (ljn ^ 8 3 p3 n = 3 k 0 kk (n-2k)' __ k n-2k n)

- 50 - the expansion being the same for both (V-30) and (V-34). The ranges of applicability are the same as for the original formulae, of course.

(dr - z/d)g 9 au' (Z/T + di)g- (Qg + Z/9 - L-)' + = (z (a) (9T-A) ~, (Z/T + dg-)9 I > (d9 - z/d)g (gg + Z/9 - a)d I T az (P) (ST-A) Ll f.^.. (S/T + dZ-)1~ l) L (Z/d - d + gz)g (9- + T/9 = z (3) (tT-A) (z/d - dd + )z) d z d + - _ -)I = z (q) (~T-A),i,. (./d d. +..-)g........... + __ -.. (Z/d - dg + gz-)9 a 1 (i + dg)g- +(l + + / )d - = nz (ZT-A) a(|9T* bTA' ON aBuBH (a) 2z zo (L)0z eaS)S A-e ba -punog ( o, ^ 3 ~z ' o0) 0 3 o eT - A ~iUL (Pz I

TABLE V - lb For ^ 1 or ( v 4 h h BoundEq. ary(See z (77) or z(7n) Range No. Fig.16b) 7 26@ p2 + ~ + + r;) (V-17) (f) zu R(- p 2 + 2 q 7q2 (V-18) (g) zp = 1R(7 - /2 + ) 6(p/2 - p) < 7 q2 (V-19) (h) z= 1 + p(-7 - 6/2 + 36) q1 n 6(p/2 - Pp) ql = 6(-2 '-+ 23 - 2p +-+ p/2 + 3p) q2 = 6(2"'- 23 - 2pJ'+ p/2 + Bp)

4R2 TABLE V- 2a For 0 B ( 1/4 (Or $ v oo ) a = 6(p/2 - Pp) Equation Zl (7) Range Number R52 (V-20) - 2R (1/4 - B) (1 - p) 7 - 5(1p + 1/2) 2 2 02 R 1772, (V-21) 1 p - + 6(1/2 - )7 - (p2 + 2p)(2 - 5(i p + 1/2)Wr | 6(p/2 - 3p) (V-22) 1 g- I- + 5(1/2 - )7 + (p2 _ 2p) 6(p/2 p) (/ (1/2 )2(1/2 - p) 2 2 (V-23) - (1/4 -) (1-p) 6(1/2 - Pp)

R2w 2 4Rwc TABLE V-2b For 4 < P (Or - v<) a, 6(p/2 - p) Equation Z (77) Range Number _________________________________________________ (V-24) - 2R62 (1 + p)4(V7 - l)( f'- 1/2) 77 N q^ (V-25) R I +b(1/2-P)7?-l-(p2 + 2p)(l/2 - )q,1 77< $ (p/2 - Pp) R 772 6 P 2 2 (V-26) ~ ~ 7 6(1/2 - ^)77 + ~( 6(p/2 -Pp) ^ 7 ^ q^ (2 2 (>(p - 2p)(l/2 -) (V-27) 2R6 (1- p) ( -l)( -1/2) 7>7 q q^ = 5(-2 + 23 - 2p + p/2 + Bp) q = 5(2 - 23 - 2p{'+ p/2 + pp)

(Z/T + dg-)Qg L d^ - 1:- g+ (d + T) - (ZC-A) (Z/T + g-)9 + + ~ - -!(d/Z + 7U cl~ z s1 1 (TC-A) ( /- d+ z)g - +(Z/T + d9 -) d((/d - (./+ -)z - i (d 8 d)Qdd~ E d 8 s ( Z+a d d Fe d Z 1 OE tL ~ (z/d - d9f - gt-) ) 9 -W + + uj) p - -A) 95UGH Id Z uoit ~u~ (Z/~ +dE))9- 8 g ZvZE / - U, L, (~~+ d~ - ~-)g 8 ~' ~ z ci I z(//T + dg) - l(- -( + d z -:- + uO) g o - U T 9+ g - (6A-A) (z/T + d9 T a 5we GO nzT I 4 UenbS +/dgd - e coo m 10) 9 o xg) r(i 'eE A OT flV 3 co zI~~tr

2 2 TABLE V-3b For i 1 (Or v 4 Rw For T (o h h ~ aU 5 &p/2 Equation Z (7?) Range Number Rb2 (2{ B- B)( `- - 1) + j- '-2 2 /271 77 (V-33) R6 I (2 (2 1) + (2 (V-34) R 72 + 2 4 5L 65 3/2 2p 2 3p p p f p 2 l p - (p/8 4- )-] 3 3 1 p. (V-35) R (2 -) (2 -3 - 1) — ( -) (2 3+1/2 q = 6 (-2 + 23-2p + p/2 + p) q = 6(22^ - 23 -2p + p/2 + p)

CHAPTER VI DESCRIPTION OF THE EXPERIMENTAL ARRANGEMENT In this section are described several experiments performed at the University of Michigan machine which are subject to analysis by the methods developed above. The physical configuration of the machine during these measurements is depicted in figure 25; the signal flow system is shown in figure 26. A brief description of the significant aspects of each component is given below, but we now undertake to discuss the operation of the entire system. Two rotors were actually placed in the beam; nearest the collimator (A)* exit was a fiberglass-resin rotor (B), referred to hereafter as rotor l,and further downstream was the chopping rotor (C) referred to as rotor 2. Rotor 1 produces a very broad burst, that is a burst of relatively very long duration, four times per revolution and was run at one-fourth the speed of rotor 2, in phase with the chopping rotor so as not to interfere with the beam chopped by rotor 2. Its effect was to reduce the background at the detector due to fast (i.e., epicadmium) neutrons, which are not greatly attenuated by the chopping rotor. The neutrons striking rotor 2 are chopped by this high-resolution rotor into a narrow burst, which contains neutrons of many speeds. In the absence of a device to select only one speed *Letters in parenthesis indicate the section below in which the system component referred to is described. - 57 -

- 58 - COLLIMATOR ROTOR I /ROTOR 2 CRYSTAL 7^^^^^^^^~~~~ ~59 V L2D 4.75" __ 2.__00.. r^~//108"~ 54" ~ 8 (NOT TO SCALE) DETECTOR SPECTROMETER SCALE Figure 25. Schematic representation of the machine configuration during the burst shape measurements. Rotor I | Start ____~Phase Display Stop Monitor Recorder Rotor 2 - Time Delay Generator Ter ime of Detector Flight ~I~ -'...... Signal [Ana lyzer~ ~ ' ~Signal * - Figure 26. Block diagram showing the routing of signals from various components of the machine during the burst shape measurements.

- 59 - from this distribution, the faster neutrons would arrive at a downstream detector very soon after the burst was formed, while the slower ones would arrive considerably later. This broadening of the burst would make interpretation of measurements difficult. A single neutron speed was therefore selected by erecting a monochromating crystal (D) in the beam. A thin detector (E) was placed in the deflected beam. The angle through which the beam was deflected (which is equal to twice the Bragg angle) was established by the following technique. On a calibrated circular scale centered at the middle of the crystal face, a small-diameter BF3 detector was placed at the desired deflection angle. The crystal was rotated until the beam fell in this detector. The crystal was left in this position and the BF3 detector was replaced by the large-area, thin detector, which was assumed to intercept the entire reflected beam. In order to bring rotors 1 and 2 into the correct phase relation to one another, a timing signal (F) was derived from the shaft of each rotor. These signals occurred at well-known times relative to the times at which the rotors reached their open positions. The time interval between the pulses from the two rotors was measured by a phase monitor (G). The timing pulse from rotor 2 was also used as a time-reference for the counting system. Since it was necessary to introduce a time delay between the occurrence of the pulse from rotor 2 and the beginning of the counting

- 60 - cycle, the pulse was fed through a time-delay generator (H), the output pulse from which triggered the time analyzer (I). The sequence of events is illustrated in figure 27. Let us say that the first event is the occurrence of the timing pulse from rotor 1. This event precedes the opening of rotor 1 by an angle Ol, through which the rotor, turning at speed 401 3/4, moves in time. The timing pulse from rotor 2 occurs at time t after that from rotor 1, and rotor 2, turning at speed 0, opens a time 2 later. If L12 is the distance C1) between the centerlines of rotor 1 and rotor 2, and the neutron speed selected by the crystal is v, then these neutrons L12 travel from rotor 1 to rotor 2 in the time. For the V two-rotor system to pass this speed optimally (more accurately, to insure that rotor 1 has no part in forming the burst of neutrons) the phase time t must be such that 401 12 t v 1 p 312 - ^41 "2 12 -, C0 v (VI-1) In each case the phase time t was adjusted to satisfy equation (VI-), Neutrons which cross rotor 2 centerplane when the rotor is in its full open position (neutrons in the peak of the L2D burst) travel the distance L2 to the detector in time 2D The time analyzer is triggered (that is, its analyzing cycle is initiated) by a pulse from the delay generator, which. pulse follows the timing pulse from rotor 2 after a delay

-61 - 4,, Li2 L 2D t V V ~to nt at r III I t TIME TIMING ROTOR I TIMING ROTOR 2 DELAY NEUTRON PULSE FROM OPENS PULSE FROM OPENS GENERATOR ARRIVES IN ROTOR I ROTOR 2 OUTPUT CHANNEL n PULSE Figure 27. Time scale of events during one cycle of the. machine during the burst shape measurements.: I _ ~ 5.00 R --; rc r A 2.5<O"Tt-~~~ SO [1.25" TYP. 2.50"TYP 19.44" TYP. Figure 28. The physical configuration of the fast-neutron rotor. The rotor is constructed of fiberglass-resin laminate.

- 62 - of length tD. A pulse from the detector which arrives at time nAt after the analyzer is triggered, is stored in channel n of the analyzer. Here, At is the width of the analyzer. time channel, and n is the channel number. In each case the delay time tD was adjusted so as to cause the neutrons at the peak of burst to be counted at, or near the middle channel of the time analyzer, n = 128. That is, again referring to figure 27, in each case t was adjusted so that e2 L2D t + nAt -, n t 128 (VI-2) We now undertake to describe the function of each of the significant parts of the device. A. Collimator The collimator resolution-plane geometry is what is important for the present purposes. Shown in figure 25 is the collimation system, in which the collimator is represented as designed. This representation of the collimator is not adequate to predict the burst shapes measured in the experiments discussed here, as was found in the initial comparison of theoretical and experimental results. Development of a more adequate representation is discussed below. B. Rotor 1 This rotor is of fiberglass-resin laminate, with slit configuration as detailed in figure 28. The rotor has two

- 63 - identical, symmetric slits, oriented at 900 with respect to one another. It therefore produces four bursts per revolution and is turned at one-fourth the speed of the chopping rotor. Its purpose in this experiment was to reduce the background due to epicadmium neutrons. C. Rotor 2 This rotor is of high-strength Mg-Zn alloy containing 10% Cd for neutron attenuation, and although practically opaque to thermal neutrons, it is quite transparent to neutrons above the cadmium cutoff. Its slit configuration is as shown in figure 29; one burst is produced per revolution. Description of the mechanism of burst formation by this rotor is the primary object of this paper, and has been thoroughly discussed above. It remains to be said of this analysis however, that in the interest of simplicity the three slits in the rotor have been treated as central slits. That is, the formulae derived above have been developed for a slit rotating about a point at its center, and these formulae have been applied in the analysis to all three slits. The approximation involved is of the order of that made in seth h ting cos - = 1 where - =.05. R R Surfaces of the slits are circular arcs concentric with the arc of the central slit centerline. The design ratio of neutron speed to peripheral velov city is t = 8, which leads to the choice of the radius of curvature of the central slit centerline. The arc, of curvature of the central slit centerline. The arc,

- 64 - VIEW A 5" R f |4 ~.125 TP. 20.19" R VIEW A Figure 29. The physical configuration of the thermal neutron rotor. The rotor is of Mg-Zn-Cd alloy.

- 65 -? = 14.3 is an alternative parameter indicating the optimum neutron speed, v pt= 2 Thus to apply the results of Chapter III, the correspondence principle, to the formulae of Chapters II, IV and V, it is necessary to replace 1 appearing there by I - = 1. Vop, v 2Rw D. Crystal Spectrometer The operation of the crystal spectrometer used to select a well-defined energy from the chopped beam has been discussed above, but the details of the device are as follows. The monochromator was a crystal of Copper, cut on the (200) plane. Spacing for this plane is 1.816 i. Mosaic width of this crystal has been measured( to be 4 minutes of arc, and the contribution to the experimental resolution width is correspondingly small. Since t - -A = cot B A 0 A 0 = 4'.0012 rad. and flight times do not exceed 1000 microseconds, this contribution to the resolution is approximately 2.0 microsecond and is neglected. E. Detector The detector used for measurements of the burst shapes was one of the type described in reference 8* The detector thickness contributes to the resolution of the experiment At - AL to a negligible extent, since t t L (7) J. L. Donovan. private communication *See (8) on next page

- 66 - where AL is the thickness of the detector,-.3 cm, while L is 2D the flight path of neutrons from rotor to detector, approximately 1.5 meters. Thus At.3.002 t 150 and the time uncertainty thus introduced for flight times of about 1000 microseconds is 2 microseconds, at maximum. This effect in the measured function is also neglected. F.- Shaft Angle and Frequency Signal Source This device consists of two parts; a disc which is attached to the top of the shaft driving the rotor, and a magnetic pickup mounted in a fixed position perpendicular to the rotating disc. The disc is of aluminum and contains a 1/32" wide slug of iron in its periphery. The pickup is a high-output permanent magnet pickup (Electro-Mation Model 3030 AN). As the iron slug passes the pickup, a timing pulse is produced which appears as in figure 30. 200 to 400 VOLTS Figure 30. Shape of the timing pulse from the magnetic pickup. (8) Vincent, D.H.; Carpenter, J.M.; and Sutton, J.D.; "A Thermal Neutron Detector Using LiF and Scintillating Plastic," Bulletin of the American Physical Society, series II, Vol. 8, No. 4, (1963), Paper UA-12.

-67 For this series of measurements, the lag angles 91 and 9 (see figure 27)were 1 = 135 and 9 = 620 G. Phase Monitor The phase monitor is an electronic device which measures the time interval between the occurrence of two events, in this case the occurrence of a pulse from rotor 1 and 2. Refer to figure 31. I - Megacycle Pulse Generator PULSE FROM ROTOR I n on Gate PULSE FROM ROTOR 2 \ off Binary Scaler Light ___ 4 ~~Display Digital-to- Strip Analog Chart Converter Recorder Figure 31. Block diagram of the phase monitor circuit. This description omits some of the workings of the circuit but serves to indicate its general manner of operation.

- 68 - A pulse frcm rotor 1 closes a circuit so that pulses from a l-Mc clock are counted on a scaler. The scaler counts until a pulse from rotor 2 opens the circuit, thus stopping the scaler in which are registered a number of counts proportional to the time interval between pulses from rotor 1 and 2. The output is read out in two ways: on a binary light display and, through a digital-to-analog converter (which produces a d.c. voltage proportional to the number in the scaler) to a stripchart recorder which continuously monitors changes in the phase time. This device, the phase monitor, is a crucial part of the system inasmuch as the data it provides are used to set and maintain the correct phasing between rotors. H. Time-Delay Generator One of the functions of the Tektronix Model 535-A oscilloscope is to produce an output pulse at a set time after a triggering pulse is received The pulse from rotor 2 was used as trigger and the output pulse from the oscilloscope was delayed by whatever time was required to place the burst of neutrons near the center of the time interval scanned by the time-of-flight analyzer, chosen according to formula VI-2. I. Time Analyzer The time analyzer used was a TMC Model CN-110 computer with Model 211 time-of-flight input. Its function is to store signal pulses in memory channels according to the time

- 69 - at which the signal pulses arrive, relative to the time at which a triggerir.g pulse arrives. The time interval associated with each memory channel can be varied in steps from.25 to 64 pLsec., and a time of 16 isec. is required to perform the storage operation. That is, the dead time of the instrument is 16 microseconds. Delayed pulses from the oscilloscope were used to trigger the analyzer, and pulses from the detector were used as input. For all runs, the channel width was 1 psec. Using the equipment as described, burst shape data were accumulated for five different neutron speeds (crystal angles) and five different rotor speeds for each neutron speed. The results of several of these runs are presented below, and compared with the theoretical predictions.

CHAPTER VII COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS The difficulty with regard to the collimator was mentioned above in Section VI-A. Comparison of burst shapes predicted under the assumption that the collimator was as designed, with measurements, showed poor agreement. To provide a modified representation of the collimator, data from an experiment originally performed to locate the collimated beam, were analyzed for this purpose. This measurement was made in the following manner. At two positions downstream from the collimator, a strip of gold foil was placed perpendicularly across the beam in the resolution plane, and activated by neutrons streaming from the reactor through the collimator. The foil strips were scanned by a small-aperture detector to determine the activation per unit length as a function of position in the beam. This activation at each point is proportional to the neutron flux at that point. This flux may be predicted using the methods of reference '* and is proportional to an integral N(x,z) = P(x,z,f) di over the P-gram which may be interpreted as the i-interval containing allowed trajectories at a point x, z. Figure 32 shows the measured activation density, and the predicted density assuming the collimator to be as designed (the -70 -

o Measurement 1000 Best Fit To 1000 Mesurement co """~Design Prediction LC) h- -~-Prediction For z 800-.....+_- Deformed Coimator - 600- - Figur /.32'. o f 0'" 0 400. = -2.70 C- 200 -5 -4 -3 -2 -1 01 2 3 4 5 z, INCHES — NORTH --- Figure 32. Horizontal foil strip activity at a position 132" from the collimator exit.

- 72 - design prediction) for a position 132" from the collimator exit. The measured activation density deviates from the design prediction in a manner which suggests that the collimator wall is not flat, as shown in figure 25, but rather contains a lump or bulge, which can be represented approximately as an edge, as in figure 33. 4.75" ~ -____a_____, 2.00"__ C - Io0" - Figure 33. Assumed form of the deformation of the collimator. The position and height (two parameters) of the bulge were determined so as to fit the points z and zb at which a b the measured activation.begins to deviate from the design prediction. The activation predicted using this model of the deformed collimator is also shown in figure 32. The analysis produced a value for the position of the bulge, y = 15" from the collimator exit, on the south collimator wall, and a value for the height of the bulge, c =.79" as shown in figure 33. Presented as figure 34 is the result of a similar mea

0 Measurement 2400 RAI RAIL -~ Best Fit To 12000 - ~~~~~~Measurement t^~~~~ 2000-~ ^~ "' ---Prediction For u Deformed Collimator H- / ^\Assuming Co/nimator - 2 //Z Rai p Coincident;E1600 - z ~ 6^00-~ /~ '~ ~ ----Same Displaced.16" South LO W I // \\ '^ ^ 1200 / z D: \ 0 p U~~~~~~~ >-800 " - C. /'\ < 400 - I'/ \ o o C/ ~\ o 0 o -4 -3 -2 -1 0 2 3 4 z, INCHES — NORTH-~Figure 34. Horizontal foil strip activity at a position 66" from the collimator exit.

- 74 - surement at a position 66" from the collimator exit. Also shown is the prediction using the parameters of the deformed collimator, assuming the centerline of the rotor support rails to correspond to that of the collimator. It seems a better assumption to say that the beam centerline does not coincide with the rail centerline, but instead lies.16" south. The predicted activation for the deformed collimator, displaced.16" is also shown in figure 34. Although the prediction thus generated for the activation at the 66" position does not match the measurement exceptionally well, it is an improvement over the design prediction. Deviations are attributed to the fact that such a simple model of the deformation is not completely realistic. Conclusions drawn from this analysis may be summarized. The collimator is deformed and the deformation may be represented as in figure 33. The rotor support rails are not perfectly aligned with the collimator beam centerline, rather the beam centerline passes the 66" position.16" south of the rail centerline and passes the 132" position at the rail centerline. Therefore, we conclude that the collimator is 16" misaligned by an angle of 6 =.00243 rad., and that the collimator exit is misplaced a distance (.00243) (132") =.32", south of the rail centerline. With this modified representation for the collimator, a P-gram for the deformed collimator can be constructed, at the position of the rotor 2 centerplane. This is shown in figure 35, in which the origin of the z-coordinate is taken

3 -Cn W PATH OF UPPER SLuT z PATH OF CENTRAL A/ r iO 4 -:0 3 -: 2 /0 I 0.01 /.02.03 04 /^ / y^^ ^RADIANS~^II -2 - Note that the center of the collimator P"gram is displaced, to z=-.19", -3-0 \.0024 RAD. PATH OF LOWER SL/T\ Figure 35. The collimator P-gram at the position of rotor 2 centerplane, 54" from the collimator exit. Drawn for z = 0, = 0 on the rotor support rail centerline.

- 76 - as the position of the rail centerline, while that of the 9-coordinate is taken as the direction of the rail centerline. Therefore the collimator P-gram must be drawn displaced, its center at z = -.32" + 54" (.00243) = -.32" +.13" = -.19" =.00243 radians Also shown as figure 35 are the boundaries of the collimator as designed, illustrating the effect of the bulge in deforming the collimator P-gram. The paths of the P-grams for the three slits of rotor 2 are also shown on figure 35. The intersections of these paths with the collimator P-gram boundaries determine the angles.i and out which enter the arguments of AZ. Recall the result of Chapter IV M(eB t) PC(X,'Z',/) PR(x,z,I-t)d ~ d z (IV-22) where a= (1 + aT BCOt eB), and for our purposes we may regard 1/a as a constant of normalization. Also recall the function:Z(p,7;) in terms of which the calculation of M( B,t) was cast; M(,t) = Z( Poutaout - Pin in - t) for each slit in the moving rotor; where p's and -'s are to be chosen for the distended P-gram, i.e., Pc(x,z, /a)=l. The observed burst shape function is the sum of the bursts

- 77 - from each slit. From figure 35 the intersection bin and slopes z/Si 3.in in out out are tabulated below in Table 1. Table VII-1 ^ z/S, u z/S in _in ut outout Upper Slit -.0053 5/54.0208 5/172 Central Slit -.0109 5/172.0169 5/69 Lower Slit -.0133 5/172.0114 5/69 Several records have been chosen for comparison with theory: Figure 36 and Table 2 are for rotor speed 440 revolutions/sec. counterclockwise, and e = 24.50. B (vB = 2650 eters), (referred to hereafter as record 440-1). sec. Figure 37 and Table 3 are for rotor speed 280 revolutions/sec., counterclockwise and eB = 38.30 (vB = 1780 M/sec.). (Referred to hereafter as record 280-5).

6001 500 BACKGROUND 400 - 00 O \ w lo, ~ ______ ^___________\ \ ________ ^ o 0^ 0040 300 MEASUREMENT 0 0 W/- KN S9i / 1 /o \o \SUBTRACTED a. / $ \> oC ^ z CALCUL A TED 0 "~ 0~~~ 200 -0 0 0 0 70 800 90 100 110 120 130 140 150 0160 TIME, MICROSECONDS~ ---. Figure 36. The measured burst shape for rotor angular speed w = 440 cycles per second counterclockwise, and neutron speed vB = 2650 meters per second.

- 79 -Table VII-2 (For Record 440-1) 3 = 440 cps = 2760 radians/sec. Counterclockwise = 24.5; A= 2d sin = 2(1.816) sin 24.50 1.51 B B 4000 = = 2650 m/sec B xB cot e = 2.194 L2D 1.58 -4 B- = 2650 = 5.97 x 10 sec. B v 2650 1 + TrBcot E = 1 + (5.97 x 10-4) (2.760 x 103) (2.194) B B 1 + 3.60 = 4.6 R2c 1 1 R o R = - vB opt B 2h (25) (2.54)(2.76 x 10) _ (5) (.25) = 2.64- 2.50 (.25) (2650) 2(.25) =.14 iltn Pin t out Pout Upper Slit -.0243.43.0956.13 Central Slit -.0501.13.0778.33 Lower Slit -.0611.13.0524.33

300 ' o MEAS UREMENT W/TH o o BACKGROUND| BACKGROUND SUBTRACTED -_ 1200^~J/^^>~"~ - z z CALCULATED ----- o bJ 100 - o C/0) 00 WClQn 100 110 120 130 140 150 160 170 1I) 70 o TIME, MICROSECONDS- Figure 37. The measured burst shape for rotor angular speed w - 280 cycles per second counterclockwise and neutron speed VB = 1780 meters per second. B

-81 -Table VII-3 (For Record 280-5) 3 = 280 cps = 1760 radian/sec. Counterclockwise e = 38.30 A = 2(1.816) sin 38.30 = 2.25 R 4000 B =4 = 1780 m/scc. B cot B = 1.265 L2D 1.58 -4 B = 2 1.80 8.88 x 10 sec. B VB 1780 B B (1 + reBcot B) = 1 + (1760) (8.88 x 10 4) (1.265) 1 + 1.98 =2.98 R 44= 5- 2.50 hvB 2h 1780 = 0 a^in Pin aout Pout Upper Slit -.0158.276.0622.0869 Central Slit -.0325.0869.0505.216 Lower Slit -.0397.0869-.0341.216

- 82 - Measurements were also made with the rotors turning clockwise, i.e., for c negative. The effect is to leave all else unchanged except the factor 1 + UTB cot 0B, in which the second term, positive for counterclockwise rotation, becomes negative. With proper choice of parameters, this factor approaches zero, and the effective width of the collimator angular distribution is zero. What actually happens is that the collimator angular spread is exactly compensated by the spread in flight times of neutrons reflected by the crystal. Under such circumstances, the beam might be said to be "time focused", and the burst shape seen is that predicted for a unidirectional, monochromatic beam. Since it was not possible to reach the condition of time focusing because of experimental limitations several measurements were made, for varying flight paths from rotor centerline to detector. In table 4 are exhibited the conditions for each measurement. Results of these measurements are shown in figures 38, 39, 40, 41. Plotted on each figure is the predicted burst shape for the "focussed" condition. In each case, the prediction is the same function, normalized to have its area equal to that beneath the measured peak. Agreement is not expected for any figure, rather it is to be noted that as the length L2D approaches that "focussed" value,.80 meters, the measured burst shapes approach the prediction for the "focussed" condition.

- 82 - Measurements were also made with the rotors turning clockwise, i.e., for c negative. The effect is to leave all else unchanged except the factor 1 + -TB cot 0B, in which the second term, positive for counterclockwise rotation, becomes negative. With proper choice of parameters, this factor approaches zero, and the effective width of the collimator angular distribution is zero. What actually happens is that the collimator angular spread is exactly compensated by the spread in flight times of neutrons reflected by the crystal. Under such circumstances, the beam might be said to be "time focused", and the burst shape seen is that predicted for a unidirectional, monochromatic beam. Since it was not possible to reach the condition of time focusing because of experimental limitations several measurements were made, for varying flight paths from rotor centerline to detector. In table 4 are exhibited the conditions for each measurement. Results of these measurements are shown in figures 38, 39, 40, 41. Plotted on each figure is the predicted burst shape for the "focussed" condition. In each case, the prediction is the same function, normalized to have its area equal to that beneath the measured peak. Agreement is not expected for any figure, rather it is to be noted that as the length L2D approaches that "focussed" value,.80 meters, the measured burst shapes approach the prediction for the "focussed" condition.

- 83 - Table VII-4 Conditions for measurements using clockwise rotation. For all runs: c) = 280 cps. clockwise = 1760 rad/sec. clockwise e = 38.30 B = 2.25 B VB = 1780 cot e = 1.265 B 1 + WB cot 0 =1 + cot B vB =1_ 1760 (1.265) L 1780 = I - 1.250 L 2 R= = 0 hvB 2h For figure 30 L 1.51 meters 1-1.25 L = -.8 2D 2D For figure 31 L = 1.45 meters 1-1.25 L =-.82 2D 2D For figure 32 L 1.27 meters 1-1.25 L =-.58 2D 2D For figure 33 L = 1.07 meters 1-1.25 L =-.34 At "focus", 1 -1.25 L = 0 (L =.80 meters) 2D 2D

1000 t 800 W — J z z PREDICTION FOR < 600 - FOCUSSED CONDITION 60 0~~~~~~~~~~~~ CLii 0~~~~ ~,400 v)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c 0 80 U " 6tr" 100 110 120 130 140 0 160 170 TIM E, M CROSECOKNDSFigure 38. The measured burst shape for rotor angular speed w 280 cycles per second clockwise and neutron speed v 1780 meters per second, with the detector 1.52 meters from tde rotor centerplane.

1000, I t 800 -J 1<600 / o / o\ -~PREDICT/ON FOR FOCUSSED CONDITION c 400 5 z - V BACKGROUND o 200 ' '' \~ _ 7080 o 0 \ 0 - 70 80 0 90 100 110 120 130 140 150 160 TIME, MICROSECONDS-~ Figure 39. The measured burst shape for rotor angular speed w = 280 cycles per second clockwise and neutron speed v = 1780 meters per second, with the detector 1.45 meters from the rotor centerplane.

1000 - o800 ri BACKGROUND z l600- o OOo o2~~8 10 80 ~ o o 110 120 130 140 1500o o 170 0- PREDiCT/ON FOR T 40ME,0 U CONDITCON DS z 0 O 200 0 00 0 0 0000 0 0 80 o 0 o 11Fi 120 130 140 50 fo oo a 70 TIME, M1CROSECONDS — Figure 40. The measured burst shape for rotor angular speed X = 280 cycles per second clockwise and neutron speed v = 1780 meters per second, with the detector 1.27 meters from the rotor centerplane.

PREDICTION FOR 1000- / FOCUSSED CONDTON ( / ^tf 0 0 \ 0._. 600 -_ _r o~~~~~~~~~~~~~~~~~ W z BACKGMICOSCONS~ z O 200-/0 v> 400 - -0 400-~ ~ ~ 0 0~~~~~~~~~~~~~~~~~~~ Figure 41. The measured burst shape for rotor angular speed a) = 280 cycles per second clockwise and neutron speed v == 1780 meters per second, with the detector 1.07 meters fom the rotor centerp00ane. 00 30 400 O 000 "O 110 120~~~~~0 0 30 400 0 0~~~~~~ ~~~~~~ o 60 70 80 90 TIME, MICROS ECONDSS —~ Figure 41. Tne measured burst shape for rotor angular pe wu 280 cycles per second clockwise and neutron speed V 1780 meters per second, with the detector 1.07 meter fpom the rotor centerplane.

CHAPTER VIII DISCUSSION AND CONCLUSION Comparisons of experimental data with predictions from the theory developed above were displayed in the preceding chapter for several situations. With regard to the justification of the theory by comparison with these data, let it first be said that the experiments were designed for another purpose. Therefore, some difficulties are to be anticipated, due to the fact that certain conditions were not well established and certain parameters necessary to the analysis, were not well measured. This was true with respect to one most important length, L2D from the rotor centerplane to the detector. This length establishes not only the magnitude of the factor (1 + CT cot 0), but also the location of the chopping rotor. The length L2D was inferred from measurements of times of flight. A large number of burst shapes were measured, of which those presented are only representative. For each measurement, the neutron velocity was well known as well as time of arrival relative to the time at which the chopping rotor opened (i.e. the flight time.) The length L2D used here is the average value of the product of the neutron velocity and the flight time, from a large number of measurements. It is also true that the knowledge of the shape of the collimator P-gram is not exact, as was already discussed. Further there is the lingering doubt as to the chopping - 88 -

- 89 - effect of rotor 1, (which was assumed to have no effect on the chopped beam.) This effect can be computed by the method of this paper, but the approach undertaken here has been to present formulae and methods adaptable to hand calculation. As a practical matter this almost precludes even undertaking to estimate this effect, or to judge the conditions under which it is present or absent, within a reasonable time. With these apologies, the relatively poor agreement between prediction and measurement shown for figure 36, may not be surprising. This is so because in that case, the effect of the collimator is exaggerated to its maximum extent (because the factor (1 + wB cot Q ) is largest.), UB B and any errors in the same factor (arising due to error in L2D or inaccuracies in the description of the collimator) are therefore similarly at maximum. The situation with respect to figure 37 is in contruast to that of figure 36.* Here, agreement between prediction and measurement is highly convincing. This may be somewhat fortuitous, but it is to be noted that in this case, the effects of the doubtful collimator data are smaller than for the measurement of figure 36. *Here one should note that there are two compensating effects establishing the width of the burst shape, which is given by At -( +cuTcot O )Alp + A* At = (1 + Bc B collimator rotor A -V + Ar coll' rot. ~+ T cot 6 A TB B coll. The first term is larger in the cases of the lower frequency, while the second term, which varies as cosine Q9 is smaller for larger Bragg angles, and tends to compensate.

- 90 - Further substantiation of the theory is exhibited in the sequence of figures 38, 39, 40, 41, for varying lengths L2D In each case the unidirectional-beam prediction is superposed upon the plot of the measured function. To be especially noted is the fact that the measured functions approach this prediction as the length L2D approaches the focussed condition. In the light of these comparisons, it is concluded that the theory is well substantiated by the measurements. However, the theory itself is intrinsically satisfying, because of the insight it provides into the mechanism of burst formation in a chopping device. This is in addition to, or perhaps the basis for the fact that calculations of the burst formation follow readily from the P-gram theory, as it has been applied. Reflecting, we can see those points at which significant clarifications and simplifications have come about due to the use of the P-gram theory. The first is the manner in which expressions for the expected counting rates are formulated, as in equations II-21 and IV-19. Here, the expressions involve functions either subject to straightforward measurement (as in the case of the incident neutron intensity), or available from other theory (as in the case of the crystal reflection) or calculable from the P-gram theory. Inasmuch as these formulae give instantaneous counting rates, they can also be used to give long-time-average counting rates, and resolution

- 91 - functions. Second is the straightforward explanation the theory provides for the effect of the monochromating crystal in a chopped beam. Third is the ease with which the chopping effects of several rotors in sequence can be understood, by merely considering several rotor transmission probabilities, that is, by superposing several rotor P-grams on the same plane. A digital computer code could be written applying the P-gram concept to more complex systems. The code might be one which samples points in the trajectory plane (z - V plane), and makes decisions whether successive points be within the P-grams for each element of a device. It would be fairly easy to include the effects of semi-transparent (as opposed to totally opaque vs. perfectly transparent) components in such a code. A further clarification of chopper dynamics is in the theorem of Chapter III, which doubtlessly is not new, and indeed has been proved approximately in reference 3. However, the proof is without approximation here, and is new in this context. The proof as detailed has the advantage that the method for calculating apparent boundary transformations is a part of the proof. The methods of analysis presented here could be generalized to treat many of the common forms of neutron spectrometers. Indeed these same techniques may possibly also be applied to analysis of systems employing other radiations.

APPENDIX EXPANSION OF THE FUNCTION Z (Xi) ABOUT p = 0 u In order to avoid the difficulties associated with calculating the values of the functions (V-30) and (V-34) near the removable singularity at p = 0, we remove the singularity in an expansion in powers of p. The term on which we focus attention is the radical, p b_ 3/2 3/2 p 3/2 (+ +1) = (P (1 + (1 + )) 2 P P 27 (A-1) in which the largest term is, the smallest, The p 2 binomial expansion oo nW0 (1 - x)P = ( n x (A-2) n= 0 n: where (a)n ( (a)( + 1) (a+ 2)... (a + n- 1) (A-3) (a)o l n applied to A-1 gives 0(0 3/2 = _3/2 5 (-3/2) &n ( )( )n ( P + + ) = P5 2 ' P n'0 n! b 217 (A-4) Another application gives ( k1 (-1) k7 (A-5) (1 + P)n - E - _ - ~27? k: 21) (A-5) k=0 - 92 -

- 93 - so that n 2 )+ ~ + n) = (, _ ) C) n 0 k 0 (x) (P-) (-1)n + k (2 _ o k nk i3= ^ n3/2 (-n3/2) (n)k k n-kn+k P n! k! 2 -^> ^n n. ~n:~k~k p (-1) (A8 n=Ok=0 For k' 0, (-n) = (-n) (-n + 1)... (-n + k- 1) k (-1)k n(n -1)... (n - k + 1) (n - k).. 1 (n - k) (n- k- 1) (-1) nk (n - k) (A-7) Thus nDk 3/2 3/2 (-3/2) (-i op+ ay5 Ne r (n 1) k n-k 2 p p: k! (n - k)f 2 Y ~ Pn = 0 k = 0 k n pn+k (1) n+k (A-8) Replacing n by n - k everywhere leads to the more appropriate form '9} 3/2 3/2 nn/23 (-3/2) 1) + 27) = (.)n ^2 +p ~P "- k (n-2 k) ) n =0 k= 0 1 k _ n-2k n (*) (- -~) ( ) Pb (A-9) (9)Rainvil1,eE.D.; "Special Functions", The Macmillan Company, New York (1960) p. 56

- 94 - The symbol Ln/21 is defined to be the greatest integer in n, that is, for example, if 2= 7, n/2] 3. In full, then the function Z (7) may be represented Zu() = R 2 2 + 2 4 2 22 (-3/2)n-k() 1 )k ) n-2k n 3 3 ~ - k'(n-2k) 2 () ( P P n=3k=0 224 27 2 7?2 2 3 3 p2 2p p 8 P 3 3p (A-10) In this form the negative powers of p vanish, and 5_ 2 2 ~22 >O [n/2] (-3/2) (-1) Zu(?) =R - 48 n-k zi (p) n = 3 k k=0 ' (n-2 k) lk (7 )n-2k n W^ (_ 23P^ ) P (A-11) As is suggested by the form of the generating function, the polynomials appearing in the expansion A-ll are a type of ultraspherical polynomial, of which the well-known Legendre polynomials are a special case. This knowledge could be helpful in performing calculations from A-1i, except for the fact that these polynomials are not commonly found in tabulated form.

BIBLIOGRAPHY REFERENCE (1) Fermi, E.; Marshall, J.; and Marshall, L.; Phys. Rev. 72, p. 193 (1947) (2) Stone, R.S.; and Slovacek, R.E.; KAPL 1499 (1956) (3) Marseguerra, M.; and Pauli, G.; Nuclear Instruments & Methods 4, pp. 140 - 150; (1959) (4) Carpenter, J.M.; Work Unpublished (5) Bacon, "Neutron Diffraction," p. 64, 82, Oxford University Press, London (1955),(6) Egelstaff, P.A.; Cocking, S.J.; and Alexander, T.K.; IAEA Proc. of the Symposium on Inelastic Scattering of Neutrons in Solids & Liquids, Vienna, Oct., 1960, p. 165 IAEA (1961) (7) Donovan, J.L.; private communication (8) Vincent, D.H.; Carpenter, J.M.; and Sutton, J.D.; Bulletin of the American Physical Society, Series II, Vol. 8, No. 4 (1963) Paper UA-12 (9) Rainvile, E.D.; "Special Functions," The Macmillan Company, New York (1960) p. 56 - 95 -

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