A liquid helium polarimeter for fast neutrons II. Analyzing power calculations

NUCLEAR

INSTRUMENTS

AND

METHODS

62

(t968) I3-I8; ©

NORTH-HOLLAND

PUBLISHING

CO.

A LIQUID HELIUM POLARIMETER FOR FAST N E U T R O N S II. A N A L Y Z I N G P O W E R C A L C U L A T I O N S * G. M. STINSON, S. M. TANG and J. T. SAMPLE

Nuch'ar Research Centre,Physics Department, Universityof Alberta, Edmonton, Alberta, Canada Received 23 January 1968 The effective analyzing power of the neutron polarimeter described in the previous article 1) is calculated by numerical integration. Also presented is a detailed study of double scattering within the cryostat. The dependence of double scattering upon detector angle, size of cryostat, and incident neutron energy is

obtained. It is found that the correction for double scatter is more important than that for finite geometry. With minor modifications, the computer code written for the calculation can be adapted to other polarimeters.

1. Introduction

P0 is incident u p o n the cryostat. The probability that, after n and only n scatterings at n specified points within the cryostat, any one of those neutrons will reach a detector area subtending an angle de) at the last scattering point is

The preceding paper 1) has described in detail a neutron polarimeter employing two side detectors. During an experiment, a beam o f neutrons falls u p o n a liquid helium analyzer f r o m which neutrons are scattered into the detectors. The detectors are placed at the same angle on either side o f the cryostat and with their axes on the reaction plane. F r o m the measured rightleft* asymmetry of the scattered neutrons, and from k n o w n properties of the analyzer, the polarization of the incident neutron beam can be determined. The relatively large sizes of the scattering volume and of the side detectors of the polarimeter are necessary in order that a reasonable counting rate be obtained during an experiment. Those same sizes, however, make necessary the correction o f data for finite geometry effects. Further, the size o f the cryostat m a y be such that a correction is necessary for the multiple scattering of neutrons within the cryostat. This paper presents a detailed study of the double scattering of neutrons inside an analyzer. Higher order scatterings are not considered because their effects are found to be negligible4). The variation of double scattering as a function of incident energy, size o f analyzer, and detector angle is calculated with specific reference to the neutron polarimeter at the University of Alberta. Although similar calculations of multiple scattering corrections in polarization experiments have been reported previously3-7), none have been reported in detail. Those calculations have been primarily of M o n t e Carlo type, and, in all but one of them 3' 5, ~. 7), various assumptions have been made as to the spin direction of a neutron after the first scattering. That direction has been calculated exactly in this work and that of Perkins and Glashausser4).

dpn = exp { - an+ 1Nopl,+ 1}din" • I~I r ( 2exp { - aiNopli}. i=l

•(da/df2)i [1 + P i - 1" Ai]. (l) In this expression, N o is A v o g a d r o ' s number, p is the density of the scattering material, ri is the distance between the ( i - 1 ) th and i,h scattering points, and I i = r i except that 11 and l,+ 1 are, respectively, the distances travelled in the cryostat by the neutron before the first and after the last scatterings. The probability that the neutron is not scattered in the distance li is exp{-aiNopl~}, and the total cross section for neutrons of energy E~_I is al. P i - x is the polarization of the neutron prior to the i th scatter and A i is the analyzing power of the scatterer for such a neutron. (da/d~?)i is the differential cross section for the scattering o f an unpolarized neutron of energy E~_ 1 through an angle 0, relative to the direction o f the neutron prior to the i th scatter• The probability, p,, that a neutron reaches a detector after n and only n scatters within the cryostat is obtained by summing eq. (1) over all distinct combinations of n-scattering within the cryostat and integrating over all detector areas. The total probability that a n e u t r o n reaches a detector, making allowance for multiple scatter in the cryostat, is then

p = ~ p, = ~ (a,+b,Po). n=l

(2)

n=l

* Work supported in part by the Atomic Energy Control Board of Canada. t The Basle convention of positive polarization 2) is used throughout this paper.

2. Theoretical background A beam of neutrons of energy Eo and polarization 13

G.M. STINSON et al.

14

That the p, can be expressed as a linear function of Po is due to the fact that the expectation value of the spin of the neutron, ( a } , after n scatterings is a linear function 6, 7) of Po. Experimentally, one measures the right-left asymmetry e, defined as

= (NR--NL)/(NR+NL) = (PR--PL)/(PR+PL)"

(3)

NR and NL are the number of neutrons detected by the right and left detectors, respectively, and PR and PL are the total probabilities for right and left scatter, respectively. In general, e can be expressed as

(aR, + b~Po)- ~ (a~ + bL,po) g=

n=l

n=l

(aR+bRpo)+ ~ (a~+b~Po) n=l

n=l

[(a,R- a,L) + (b~ - b.L)p0] = .=1

(4)

[(a,R+ a,L) + (b,R+ b,L)po] rt=l

F r o m the symmetry of the polarimeter, a,R = a,.L Also, b.R-~ - b ~ since the direction of Po is only slightly different for first scatterings at different points within the cryostat. Thus, the right-left asymmetry can be approximated by

= ( A } P o ~-

a,, Po.

(5)

tl

in which superscripts have been dropped. The quantity

is called the effective analyzing power of the cryostat. For an analyzer whose dimensions are comparable to or Jess than the mean free path of the neutron in the analyzer, the probability of an incident particle reaching a detector after only i scatterings is greater than that after only ( i + 1) scatterings - that is, ]all > [ ai+ 11 and ]bi[ > [bs+ll. Hence, if only single and double scatterings within the cryostat are considered, the effective analyzing power of the cryostat can be written

( A ) = (bl + b2)/(ae +a2).

(6)

3. M e t h o d o f calculation

The analytic expressions for the a, and b, of eq. (2) are sufficiently complex that their exact calculation is hopeless. Of other possible methods of their calculation, that adapted here is the division of the cryostat

into small elements of equal volume and the division of the detectors into a number of small elements of equal area. The integration over the detector surface then becomes a summation over scatterings to the detector areas, and the summation over scatterings from nucleus to nucleus reduces to a summation over scatterings from the centres of the cryostat volume elements. As the number of divisions in the cryostat and detectors increases, the result of this "numerical" integration approaches that of the "true" integration. For definiteness, we assume a beam of particles of wave vector kb strikes a target, producing neutrons. The neutrons incident upon a specific volume element in the cryostat have energies E0, wave vector k0, and polarization Po. The number of neutrons per unit solid angle is assumed constant across the face of the cryostat. Variation of the magnitude of the polarization across the face of the cryostat is neglected, but variation of direction of polarization is taken into account. After scattering from the first and second cryostat volume elements, the neutrons have wave vectors k~ and k 2 and energies E l and E2, respectively. Angles Oi,i+ 1 and qSs,i+ 1 are defined by the relations C O S 0 i,i+ l = k i ' k s + l / ] k i ' k i + l

and COS ~)i,i + 1 = h i - 1,i' ns,i + 1

with ~p = (k= x kp) / I k~ x kp 1. (7)

The angle Os,i+l is measured relative to kl, and angle ~b~,i+ 1 is the angle between the normals to the planes defined by ki_ 1 and ks, and k~ and ki+ 1. Further, xs is the distance travelled within the cryostat by a neutron with wave vector ks; x~ is equal to 1~+1 of eq. (1). Introducing the detector efficiency e(E), the expressions for al and bl are

a, = ~, (pNo/r~)exp[-- {a(Eo)xo +rr(Ea)x,}Nop]" AVAco

•

bl =

VA o,

Z

( 1 / r ~ ) e x p [ - {ff(Eo)xo-q-ff(E1)Xl}NoD ] .

AVAo)

•(dcr/df2),e(E~)A,(Eo,Ool)cos~olAVAco.

(8)

In eq. (8), we have used quantities defined above and those defined in eq. (1). The summation over A V and Am implies a summation over all volume elements in the cryostat and over all detector areas. Assuming that nuclei are uniformly distributed throughout the scattering volume, all volume elements AV contain the same number of nuclei. All other quantities within the summation will then depend upon the locations of the

A LIQUID

HELIUM

POLARIMETER

scattering centres within the cryostat and of the detector areas• Similarly, a2 and b2 can be written 8)

FOR FAST NEUTRONS, I

1.0

I

I

1I I

15 I

I

I

i

a2 = c~+fl, b2 = 7 + 6 + q ,



with o~= Z ~ ' =

Z

0.9

(pNo)Z(AV/r2)(AV'/r~) Am"

AVAV'A
•exp [ - {

(Eo)Xo +

+

o

}pNo] •

o

I

o,8

fi =

Z

0

cdA,(Eo,Ool)A2(El,012)c°s~°,2,

= Z

- - o -

+~+_____kb)

•e(E2) (da/df2) 1 (da/dQ)2, I 200

I

I

I

400

~

I 600

.

+

_

I 800

Nv Fig. 2. Effective analyzing power for different n u m b e r s o f cryostat elements, Nv, calculated with (a) Nd = 1 ; (b) No = 7.

'A,(e0,00,)cos q ol,

= 2offh2(El,O12)cos~olcos~12,

(9)

rl = - - X ~ ' G A 2 ( E l , 0 1 ~ ) s i n $ o l s i n $ , 2 ,

and

h2).

G = [ ( g 2 _ hZ)cos 0ol - (gh* + g*h)sin 001]/(g2 +

The quantities g and h are the conventional non-spin flip and spin flip wave amplitudes, respectively, of scattering theory. Substitution of eqs. (8) and (9) into eq. (6) yields an explicit expression for the effective analyzing power, corrected for double scatter, of the cryostat. 4. Results of calculations

The polarimeter for which these calculations were performed consisted of a cylindrical cryostat, 10.16 cm in height and 5.08 cm in dia., and two side detectors, each consisting of a NE 213 liquid scintillator* 5.08 cm thick and 8.89 cm in dia. mounted on a Philips 5" XP 1040 phototube. Each scintillator was 25 cm from the centre of the cryostat and each detector's cut-off energy was 0.5 MeV. The efficiency of the neutron detectors I

I

I

I

0.90

{4

085 0.80: 0.40 0.35 "" . . . . o 0.30

o I 10

I 20

(b) I 30

o I 40

Nd Fig. 1. Effective analyzing power for different n u m b e r s o f detector elements, No. Calculation p e r f o r m e d with E0 = 5.4 M e V a n d m e a n detector angle Oe,: ( a ) = 123 ° (lab); ( b ) = 100 ° (lab).

was calculated on the basis of a simple semi-empirical expression9). A complete description of the polarimeter is given in the previous paper1). The phase shifts of H o o p and Barschall ~2) were fitted with a power series in neutron energy. The d~ and d4 phase shifts were included in the calculation. The cryostat was divided into a series of layers of equal number of spherical elements of equal volume. Their total volume equalled that of the cryostat. Since the cryostat was cylindrical, alternate layers were rotated 90 ° about a vertical axis in order to approximate as closely as possible a uniform distribution of the centres of the scattering volumes. Similarly, the scintillator face was divided into a number of elements of equal area. Fig. 1 shows the effect of increasing the number of elements in the detector while keeping that in the cryostat constant. It is apparent that, regardless of the number of cryostat volume elements, one obtains a nearly constant value for the effective analyzing power for all numbers of detector elements equal to or greater than seven. Since accurate results could be obtained in minimum computer time using only seven detector elements, that number of detector elements was used in subsequent calculations. Fig. 2 illustrates the effect of increasing the number of scattering volumes in the cryostat, Nv, with the number of detector elements, Nd, kept constant. The curves drop rapidly from Nv = 1 to Nv = 100, from which point ( A ) decreases slowly with increasing Nv. The change in ( A ) is 2% from Nv = 112 to Nv = 832. In an attempt to choose between accuracy of calculation and minim u m of computer time, 360 cryostat volume elements were chosen for use in the calculation. It is seen that that number of cryostat volume elements provides a * Supplied by N u c l e a r Enterprises Ltd., 550 Berry St., Winnipeg 21, C a n a d a .

16

G.M. STINSON et al.

g o o d estimate of the effective analyzing power of the system. The dependence of the effective analyzing power u p o n the energy of the incident n e u t r o n is shown in fig. 3. Both point-scatterer point-detector and single scatter finite geometry values are shown for comparison. The single scatter finite geometry correction to the analyzing power is less than that of the point-scatterer point-detector by approximately 5% absolute. Including the effect o f double scatter, one obtains a value which is approximately 12% absolute less than that of the point-scatterer point-detector value. The decrease of each curve at low and high energies is mainly caused by the variations of the phase shifts. A l t h o u g h the detector cut-off affects the low energy results slightly, the rapid increase in the value of the average analyzing power reflects the rapid increase in the values o f the phase shifts in that energy region. The lower portion of fig. 4 shows the ratio of double scatter to single scatter as a function o f detector angle for 5.4 MeV incident neutrons. The reason for the higher ratio at large angles can be seen f r o m the upper portion of the figure which is a plot of the angular distributions for single and for double scatterings. The counting rate for single scatter decreases by approximately a factor of ten as the detector angle increases f r o m 0 ° (lab) to 100 ° (lab), and is almost constant at larger detector angles. For double scatter, however, the counting rate varies slowly with detector angle, decreasing by a factor of three f r o m 0 ° to 180 °. Consequently, the ratio of double scatter to single scatter is higher at large detector angles. Fig. 5 shows the variation of analyzing power with mean detector angle 0a. Usually, detectors are placed 1.0

'

~

-

-

(c)

0.8

0,6

0.4

i

.5

I

2.5

,

I

3.5

,

I

4.5

i

I

5.5 Eo(MeV}

,

I

6.5

,

I

7.5

I

I

8.5

,

9.5

Fig. 3. Effective analyzing power calculated with a neutron detector threshold of 0.5 MeV and 04 = 123° (lab): (a) for point scatterer and point detector; (b) corrected for finite geometry; (c) corrected for finite geometry and double scattering.

~6

SCATTER

42 (o)

.~ v ~ 08 "~ "o SCATTER (XlO)

-o4

0 o 2 .~ ,6 ~0- ,2 ~ 8

i

4 o

1

0

I

I 20

R

I 40

i

I 60

I

I 80

I

I I I00

I f I I 120 140

I I 160

I 180

Od (deQ. lab.)

Fig. 4. (a). The angular distribution corrected for finite geometry of singly and doubly scattered neutrons from liquid 4He. (b). The ratio of number of double scattered neutrons to that of singly scattered neutrons as a function of detector angle. In each case, the incident neutron energy is 4 MeV. at angles where the analyzing power is large. However, our calculations indicate that, at such angles, the double scattering effect is also relatively large. Also shown in fig. 5 is a plot of b2/a2, curve (d), which is the analyzing power for those neutrons scattered twice within the cryostat. Alternately, the ratio b2/a 2 is the polarization of doubly scattered neutrons for an unpolarized incident neutron beam. The shape of the curve is similar to that of ( A ) , but has reduced magnitude. The zero crossing point is also shifted toward large angles. While the previous calculations were performed for a particular cryostat detection system, it was t h o u g h t interesting to investigate double scatter within cryostats of different dimensions. Fig. 6 shows the percentage o f double scatter relative to single scatter as a function o f the ratio of cryostat radius to mean free path of a 5 MeV neutron in liquid 4He ( ~ 25 cm). Note that, in order to reduce double scattering to less than 5%

A LIQUID HELIUM POLARIMETER I

I

I a

I-0

"5



I

0

/I

/

I

-'5

-For 0

i

,

,

-i

45

90

135

180

Od ( d e g . lob.)

Fig. 5. Variation o f effective analyzing power with detector angle at Eo = 5.4 MeV (a) for point scatterer and point detector; (b) corrected for finite geometry; (c) corrected for finite geometry and double scatter; (d) for doubly scattered neutrons only.

of single scattering, a considerably smaller cryostat is required than that for which the above calculations were performed. The disadvantage of a smaller cryostat, of course, is that the counting rate is severely reduced. Consequently it appears that, in order to obtain a reasonable counting rate, a relatively large scattering volume should be used. Data must then be appropriately corrected for double scatter. The line in fig. 6 is drawn through the calculated points for small cryostats. Since the number of cryostat volume elements is kept constant in this calculation, it is probable that our summation method is more accurate for smaller scattering volumes. Conversely, one expects the points calculated for a larger cryostat to deviate from that line and that, in fact, is found. As the number of cryostat elements is increased, the calculated values approach that line (cf. the points at r/2=0.102). Approximately, a2/a 1 is expected to be proportional to the radius of the cryostat.

F O R FAST N E U T R O N S ,

17

II

neutron beam polarization after the first scatter, and of finite geometry. The effects of efficiency and the cut-off energy of the detector are also included. It is assumed that the neutron intensity across the cryostat face, the magnitude (but not direction) of the neutron polarization, and incident neutron energy are all constant. Triple and higher order scatterings are neglected. Assuming the phase shifts to be exact, we estimate the accuracy of our calculation to be about 2% absolute for N v = 360. Of that error, 1% is due to the finite number of cryostat volume elements considered. The remainder is estimated to arise from higher order scattering. That value is based upon the statement of Perkins and Glashausser 4) that triple scattering was found negligible and upon the dubious argument that, since double scatter is about 10% of single scatter, then third and higher order scatter should be about 10% of double scatter, or about 1% of single scatter. If 832 cryostat elements are used, error from that approximation is reduced to < 1%, whereas 112 cryostat elements produce an error of ~ 3°// ' o . Our 12% correction for double scattering is in excellent agreement with the references cited previously. Consequently, in any cryostat of dimensions comparable to that described here, data correction for double scattering is necessary. In fact, only for cryostats whose volumes are less than 15% of ours does the double scattering correction become less than 10%. The radial dependence of double scattering is itself significant since it means double scatter is proportional to the cube root of the cryostat volume. Since single scatter and, hence, the experimental counting rate, is proportional to the cryostat volume, a reduction in the size of the scattering volume will reduce the I

0.30

I

I

I

I

• Nv=tl2 x Nv=360 + Nv=832

o

~. 0.20

l

~x ~ X

x

x O.IG

i

0

x

0.02

oo,

°06

°08

o,

o,

r/X

5. Discussion of results

This paper has presented the results of a detailed study of double scattering in a neutron polarization analyzer. The study has included the effects of beam attenuation within the cryostat, of the change of the

Fig. 6. Ratio of the number o f doubly scattered neutrons to that o f singly scattered neutrons as a function o f the ratio o f the cryostat radius to the neutron mean free path in liquid 4He. Calculated with 0(~ = 123 ° (lab) and E 0 = 5.0 MeV for a cryostat whose height is four times its radius. The line is an extrapolation o f that joining the points calculated for small cryostats.

18

G.M. STINSON et al.

c o u n t i n g rate proportionally, b u t will have little effect u p o n the double scattering correction. Thus, it is most efficient to use large scattering volumes, correcting data for double scatter. By modification of one subroutine, the p r o g r a m can readily be modified for n e u t r o n polarimeters different from that considered here. With further relatively simple modifications, the p r o g r a m can be used for the calculation of double scattering corrections for p r o t o n polarization experiments. R u n n i n g time for the existing p r o g r a m is roughly p r o p o r t i o n a l to the square of the n u m b e r of cryostat volume elements. With an IBM 360/67 computer, a calculation for 7 detector elements and 112 cryostat elements requires a b o u t 80 sec, whereas that for the same n u m b e r of detector elements and 360 cryostat elements requires a b o u t 15 min. The a u t h o r s wish to t h a n k Dr. S. T. Lain for providing analytical fits to the phase shifts. The interest of Dr. D. A. Gedcke in this calculation is appreciated.

References 1) S. T. Lain, D. A. Gedcke, G. M. Stinson, S. M. Tang and J. T. Sample, Nucl. Instr. and Meth. 62 (1968) 1. 2) Sign convention for particle polarization, Nuclear Physics 21 (1960) 696. 3) A. J. Elwyn and R. O. Lane, Nuclear Physics 31 (1962) 78. 4) R. B. Perkins and C. Glashausser, Nuclear Physics 60 (1964) 433. 5) B. D. Walker, C. Wong, J. D. Anderson, J. W. McClure and R. W. Bauer, Phys. Rev. 137 (1965) 347. ~) R. W. Jewell, W. John, J. E. Sherwood and D. H. White, Phys. Rev. 142 (1966) 687. 7) M. M. Meier, L. A. Schaller and R. L. Walter, Nuclear Physics 150 (1966) 821. 8) L. Wolfenstein and J. Ashkin, Phys. Rev. 85 (1952) 947. :~) L. Wolfenstein, Phys. Rev. 96 (1954) 1654. ,0) See for example, W. S. C. Williams, An Introduction to Elementary Particles ~Academic Press, New York, 1961) Ch. 8, p. 179. 1~) A. Obst, M. Sc. Thesis (Physics Department, University of Alberta, 1966) unpublished. 12) B. Hoop, Jr. and H. H. Barschall, Nuclear Physics 83 (1966) 65.