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Scattering of neutrons by α-particles
Nuclear Physics 83 (1966) 6 5 - - 7 9 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
SCATTERING
OF NEUTRONS
BY a - P A R T I C L E S
B. H O O P , Jr.t a n d H. H. B A R S C H A L L
Uni~'ersity of Wisconsin, Madison, Wisconsin tt
Received 7 January 1966 Abstract: The differential cross section for the scattering of neutrons by u-particles has been measured at neutron energies between 6 and 30 MeV. In the neighbourhood of the resonance at 22.16 MeV neutron energy, measurements were performed at closely spaced neutron energies. The present data as well as earlier measurements of elastic scattering and total cross sections, and previous polarization measurements were fitted with phase shifts which are based largely on the p--a phase shifts of Weitkamp and Haeberli. Satisfactory fits were obtained for neutron energies up to 30 MeV, including the region of the 22.16 MeV resonance. The analysing power of helium deduced from these phase shifts is presented in the form of a contour plot. E
NUCLEAR REACTIONS. 4He(n, n), E = 6-30 MeV; measured a(E; 0). Deduced phase shifts, resonance parameters, polarization. Natural target.
I
I
1. Introduction H e l i u m is the m o s t widely used a n a l y s e r o f p o l a r i z e d neutrons. The analysing p o w e r o f h e l i u m is u s u a l l y c a l c u l a t e d f r o m p h a s e shifts for the scattering o f n e u t r o n s by a-particles. I t is therefore o f interest to o b t a i n p h a s e shifts which r e p r o d u c e all available n-~ measurements. I n a p r e v i o u s p a p e r 1) the i n f o r m a t i o n on n - ~ scattering available at t h a t time was s u m m a r i z e d . P h a s e shifts for n - • scattering have a l m o s t always been d e d u c e d f r o m the m o r e accurately k n o w n p - ~ p h a s e shifts. The two m o s t often used sets o f n p h a s e shifts b o t h o f which were d e d u c e d f r o m p - ~ p h a s e shifts are the D o d d e r G a m m e l - S e a g r a v e 2) p h a s e shifts ( D G S ) at energies b e l o w 20 M e V a n d the G a m m e l T h a l e r - P e r k i n s a) p h a s e shifts ( G T P ) a b o v e 10 MeV. These two sets o f p h a s e shifts d o not, however, c o n n e c t s m o o t h l y , n o r d o they fit the d a t a 4 - 6 ) a b o v e a n e u t r o n e n e r g y o f a b o u t 15 MeV. N o n e o f the available n - ~ p h a s e shifts include the effects o f the excited state in 5He at an excitation energy o f 16.7 MeV, n o r d o they allow for inelastic processes which occur a b o v e a n e u t r o n energy o f 22 MeV. I n the p r e s e n t p a p e r an a t t e m p t is m a d e to o b t a i n n - ~ phase shifts which include the effects o f this r e s o n a n c e a n d o f inelastic processes a n d which fit e x p e r i m e n t a l results u p to 30 MeV. T h e s e p h a s e shifts are b a s e d again largely on p - ~ p h a s e shifts 7). E x p e r i m e n t s were p e r f o r m e d to i m p r o v e m e a s u r e m e n t s o f the a n g u l a r d i s t r i b u t i o n t Now at the Universit~t Basel, Switzerland. tt Work supported in part by the U.S. Atomic Energy Commission. 65
66
B. HOOP Jr. AND H. H. BARSCHALL
o f neutrons scattered by s-particles carried out earlier at this laboratory 1). The earlier data taken at 10 and 12 MeV extended only over a small angular range because of the presence of break-up neutrons from the d + D reaction. At higher energies the data were very uncertain because of the effect of disintegrations of xenon which had to be mixed with helium. These disintegrations produced a large background which was difficult to subtract reliably. Furthermore, in the present experiments a better method for deducing cross sections from the relative angular distributions was employed.
2. Experimental Method Angular distributions were measured by the method used previously 1). A highpressure helium gas scintillator was bombarded with mono-energetic neutrons, and the pulse-height distribution of recoiling s-particles was observed. This distribution is proportional to the centre-of-mass angular distribution. In the present experiment a gas scintillator 8) was employed which could be operated at higher pressures than had been used for the angular distribution measurements performed previously at this laboratory. The gas scintillator was a thin-walled, approximately hemispherical stainless steel cell of diam 5 cm which contained a mixture of helium and xenon. Gas pressures of 100 to 175 atm helium and 5 to 25 atm xenon were used. The gas pressures were adjusted at each neutron bombarding energy to yield the highest stopping power consistent with low background contributions from neutron disintegrations of xenon and with small multiple scattering effects. In all cases, the maximum range of the recoiling ~-particles was less than 5 mm. A Phillips 56 AVP photomultiplier viewed the scintillations. Previous investigations 8) have shown that the light output of the gas scintillator as a function of ~-energy is linear for 42 arm helium and 8 atm xenon. More recently, however, a non-linear behaviour has been reported in helium-xenon gas mixtures containing more than 30 atm xenon 9). The light output of the scintillator filled with a mixture of helium and xenon at several different pressures was studied as a function of ~-particle energy by bombarding the scintillator with mono-energetic neutrons of several energies and measuring the maximum pulse-height produced by recoiling ~-particles. The gas pressures were approximately those used in the angular distribution measurements. Additional fillings of 10 to 17 atm xenon and 4 to 8 atm helium were also studied. The light output of the scintillator for all fillings used was found to be a linear function of ~-energy within the 4 ~o uncertainty in the measurements over the range of 0~-energies observed in the angular distribution measurements. The procedure for measuring the angular distributions was similar to that used previously i). Neutrons below 16 MeV were obtained from the d + D reaction. Above 16 MeV, the d + T reaction was used as the source of neutrons. Measurements were carried out at about 2 MeV intervals between neutron energies of 6 and 31 MeV, except between 21.8 and 22.5 MeV where measurements were made at about 75 keV
NEUTRON
SCATTERING
67
intervals. Neutron energy spreads for most of the measurements were between 60 and 140 keV. Near 22 MeV the energy spread was about 40 keV. A time-of-flight spectrometer was employed to eliminate contributions to the angular distributions caused by break-up neutrons from the d + D reaction. The pulse-height analyser which recorded the pulses from the helium scintillator was gated for this purpose with pulses from the time-of-flight spectrometer. Only those scintillation pulses were recorded which occurred at times corresponding to the arrival of the mono-energetic neutrons at the scintillator. Since an accurate determination of the scintillation volume and of its effective centre was difficult, the cross sections were measured relative to that at 6 MeV neutron energy. At this energy it was believed to be possible to determine the scattering cross section without knowing the effective number of helium atoms in the cell by using the known total cross section of helium. As will be discussed in sect. 4, at 6 MeV only S- and P-waves are needed to fit all available data. Under this assumption the measured angular distribution can be readily extrapolated to zero pulse-height and the differential cross section can be calculated as was done in the earlier experiment. In order to compare the incident neutron flux at different neutron energies a proton recoil counter telescope similar to that described by Bame et al. 10) was employed for neutron energies up to 25 MeV. Above this neutron energy the number of recoils was compared with the number of 4He(n, d)T disintegrations which appeared as a peak superimposed on the recoil distribution. The disintegration cross section was calculated from the known cross section of the inverse reaction 11) using the principle of detailed balance. The comparison of the scattering cross section and the disintegration cross section of helium is possible without knowing either the number of helium atoms or the neutron flux. The background caused by neutrons from the foils and walls of the gas target which produce pulses in the scintillator was measured with an evacuated gas target. This background was largest for the smallest pulse-heights measured at the highest neutron bombarding energies, but at all energies it decreased rapidly to less than 5 ~o over most of the recoil distribution. Corrections for neutrons scattered into the scintillator by the floor and other surrounding materials except the steel wails of the scintillator were determined by measuring a recoil distribution with a 30 cm brass shadow bar interposed between the scintillator and the gas target. This correction was less than 10 ~o for the smallest pulseheights measured and decreased to less than about 1 ~o over the remainder of the pulse-height spectrum. The effect of scattering of neutrons into the scintillator by the walls of the steel cell was investigated by surrounding the scintillator with a slightly enlarged copy of the cell which increased the effective wall thickness by about a factor of two. The difference between recoil distributions measured with and without the additional steel was less than the statistical uncertainties of the measurements and no correction for this effect was applied.
68
B. HOOP Jr. AND H. H. BARSCHALL
Another background is caused by source neutrons which produce disintegrations or recoils in the walls of the scintillator and in the xenon. This scintillator background is difficult to measure because any change of the gas filling changes both the stopping power and the pulse size caused by particles of a given energy. There is even evidence that the pulse size caused by particles which have different specific energy losses is affected differently by changes in the gas composition 9). If the helium is removed and the background is measured using only xenon in the scintillator, the stopping power of the scintillator is changed and this alters the pulse-height distribution. If, on the other hand, the xenon pressure is increased to keep the stopping power constant, the number of xenon disintegrations increases while the number of particles originating in the wails remains the same. Varying the xenon pressure produced changes both in the shape of the pulse-height distribution and the number of pulses divided by the xenon pressure. It appears, therefore, that a substantial fraction of the background was due to disintegrations in the walls, in the coatings, or in the glass window. For the determination of the scintillator background the scintillator was filled with pure xenon to a pressure such that the stopping power was somewhat less than that used in the foreground measurement. The background so determined was divided by the ratio of the xenon pressure in the background measurement to that in the foreground measurement. This procedure underestimates the background caused by disintegrations produced in the walls and overestimates the effect of disintegration particles, particularly protons, which are intercepted by the walls. There was in addition considerable uncertainty in the gain adjustment of the amplifier when pure xenon was used. A Po ~-particle source was permanently mounted in the scintillator. It was found, however, as has previously been observed 9), that the pulse-height of Po ~-particles is not a reliable measure of the pulse-heights of recoil ~-particles. The reason for this is not fully understood, but it is most likely caused by absorption of light in the wire on which the source was deposited. Because of all these problems the scintillator background correction is estimated to have an uncertainty of 30 ~o. For neutron energies below 20 MeV this background correction was only about 5 ~o so that its uncertainty does not affect the results very much. Above 20 MeV the correction reached values as high as 30 ~o at the minimum in the differential cross section so that it introduces at this minimum a 10~o uncertainty in the cross-section determination. Corrections for the wall effect were applied as described previously 1). The greatest correction amounted to 11 ~o for the largest recoil energies at the highest neutron bombarding energy. 3. R ~
Examples of the present measurements of the differential cross sections for scattering of neutrons by or-particles are plotted in fig. 1-3. The smallest scattering angle at which data are shown is determined by the pulse-height at which backgrounds were
NEUTRON SCATTERING
69
so large that they could not be reliably corrected for, the largest scattering angle by the pulse-height resolution of the scintillator. The gaps in the data points of the cross sections above 22.07 MeV are due to the presence of charged particles from the 4He(n, d)T reaction. At 28.6 MeV some pulses from the 4I-Ie(n, pn)T reaction may be included at c.m. scattering angles < 80 °. The statistical uncertainty of the data was less than the size of the symbols except where error bars are shown. There are a number of additional uncertainties which 400
q
~-
350 \
\
~|
* En =10,00 MeV
subtract I00 from ordinate
+ En=lS.05MeV
\
subtract 50 from ordinate
----
•
En=2091 MeV
250
E IE
b
150
5C
iX 0.5
0
-0.5
-I.0
co~ q Fig. 1. Differential cross sections in the c.m. system for the scattering of 10.00, 15.05 and 20.91 MeV neutrons by or-particles. The curves are calculated from the phase shifts given in table 1.
are not shown by error bars in the figures. The cross-section scale has an uncertainty of the order of 10 Yo. There are also uncertainties which may produce distortions of the distribution, particularly the uncertainties in the background corrections. These tend to increase with increasing neutron energy and decreasing pulse height. As has been mentioned, the uncertainty in the scintillator background correction reaches about 10 ~o of the cross section at the minima in the cross sections at the highest neutron energies. All other background corrections are believed to introduce smaller uncertainties.
Io
0 1.0
25
50
75
0.5
subtract 75 from ordinate
cos e
0
-Q5
subtract 2 5 from ordinate
* En =22.20 MeV
subtract 50 from ordinate
o En =22.10 MeV
-I0
Fig. 2. Differential cross sections in the c.m. system for the scattering o f neutrons with energies near 22 MeV by =-particles. The curves are calculated from the phase shifts and inelastic parameters given in table I.
v
E
125
15o
175
I o En=22.04MeV
0.5
MeV
0
cos e
-0.5
-I.0
Fig. 3. Differential cross sections in the c.m. system for the scattering o f 22.45, 24.7 and. 28.6 MeV neutrons by ~-particles. The curves are calculated from the phase shifts and inelastic parameters given in table 1.
b
v
=E
E
.JD
En=28.6
subtroct 2 5 from ordinate
- - CALCULATED
•
+ En=24.7 MeV
subtract 50 from ordinate
° En =22.45 MeV
NEUTRON
71
SCATTERING
The results obtained at other neutron energies and additional experimental details are reported elsewhere 12). The present measurements are in excellent agreement with the measurements of Shamu and Jenkin 4) between 16 and 24 MeV and are consistent with the earlier measurements at this laboratory 1) at lower energies. 4. Phase Shifts for n - ~ t Scattering The ground state of the compound nucleus 5He is reached at a neutron energy of about 1 MeV and manifests itself as a resonance in the P{ partial wave. A few MeV above this energy the P, partial wave is affected by a broad state in 5He. The only other excited state of 5He, the existence of which has been established, is a ~+ state at an excitation energy of 16.7 MeV. 1801
i
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120r
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......... I
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. . . .
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LLI 90 I- / ~'~ 6°1-/ g/^ II
i
~..
6C
(..9
i
0]/
I
---DGS -'-GTP
......
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l
L..C?... . . . . . . . . . . . . . . . . . . . . . . . . . I
T :I . . . . co 60 D%
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50 LLI U') [~_
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.................... i t
-3
30o
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-30 30
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,-- . . . . . . . . . . . . . . . . . . . . . . i i
..............................................
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r 20
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35
NEUTRON ENERGY (MeV) Fig. 4. Phase shifts for neutron-alpha scattering as a function of neutron bombarding energy. The dashed (DGS) curves are taken from ref. =), the dot-dashed. (GTP) curves from ref. a). The solid curves represent the phase shifts proposed in this paper. Phase shifts describing the n - c t system should vary smoothly with energy except for the three states just mentioned. Each phase shift should approach zero (or an integer multiple of n) as k 2~+ 1, where k is the wave number and l the orbital angular momentum quantum number.
72
B. HOOP Jr. AND H. H. BARSCHALL
The DGS phase shifts 2) shown by the dashed lines in fig. 4 are known to fit n - ~ data below 10 MeV satisfactorily. These phase shifts include negative D-wave phase shifts. On the other hand, p - ~ results at higher energies show that positive D-wave phase shifts are needed to fit the data. Since none of the published n - • data are accurate enough to decide whether any D-waves are needed below 10 MeV, it was assumed for the present analysis that D-waves could be neglected for neutron energies below 6 MeV. The S-wave phase shift was assumed to vary with the wave number k as - 2 . 4 (k fm - t ) up to 6 MeV. This yields the measured cross section of 0.73 b for thermal neutrons lZ). At zero energy the S-wave phase shift was taken to have the value 7r (ref. 14)). The DGS P-wave phase shifts were used except that the P~t phase shift was slightly modified so that S- and P-waves reproduced the measured total cross section. This modification corresponds to a small increase in the reduced width of the PC resonance. Above 17 MeV the n - ~ phase shifts were taken from the p-0~ phase shifts of Weitkamp and Haeberli (WH) 7). These include partial waves up to l = 3. The W H Coulomb phase shifts were set equal to zero and the energy scale was shifted so as to bring into coincidence the excited state in 5Li at a p - • centre-of-mass energy of 18.73 MeV and the mirror state in SHe at a n - ~ centre-of-mass energy of 17.67 MeV. The W H S- and P-wave phase shifts were connected smoothly to the modified DGS phase shifts, and the W H D- and F-wave phase shifts were extrapolated to lower energies according to k 2t+1 The ½+ state in SHe at 16.7 MeV was first observed in the d + T reaction xs). It has been shown to have approximately the same partial width F d for decay into d + T, as the partial width F , , for decay into n + ~ (ref. 18)). Its effect on n - ~ scattering was first observed by Bonner et al. 16). In the n - u system the state occurs just above the threshold for the 4He(n, d)T reaction and should result in a rapid variation with energy of the complex D~ phase shift. In order to study the effect of this resonance on the scattering amplitude it is useful to define a D~ amplitude f2 -
exp ( 2 i ~ ) - 1 2i '
where t5 is the complex D~ phase shift. Frequently the term phase shift is applied only to the real part of 6 which will be denoted by p. The amplitude f 2 can be written f ~ = (z exp (2i#) - 1)/2i where z = exp ( - 2 Im 6). The quantity x is sometimes called the inelastic parameter. In the neighbourhood of the resonance the amplitude f 2 may be written as 17) e 2i~ --
f 2 --
2i
1
F n
+ F n q- F a
e 21p - - 1
2i
e2i~'
where fl is the resonant and q~ the non-resonant D~ phase shift which in this case is
73
NEUTRON SCATTERING
assumed to have a value of 5 °. The resonant phase shift fl is given by
Fn+Fd tg~ = 2(E~+ A - E ) ' where E~ is the characteristic energy and A the level shift. The widths F depend on energy through the penetrabilities P
F
=
2?2p,
where ?2 is the reduced width. The level shift A depends on energy through the shift functions and boundary condition parameters. These boundary condition parameters were chosen so that the level shift is zero at the resonance energy Er which is defined as the energy at which /~ = ½7r. lad I-b..I
=E o.8 n., 0.6 I-to < o41 ..a w z ~ 0.2 a
m w
0 601
40 I
zc
IM to
am-e=_2(3 " 21.6
21.8
22.0
NEUTRON
2~'12
22.4
22.6
22.8
ENERGY (MeV)
Fig. 5. The real part of the D~. phase shift p and the D~ inelastic parameter T in the vicinity of the resonance.
It was desired to find values of y2, ?2 and E~ or E r which best fit the available data. The cross section for the *He(n, d)T reaction was obtained from the known cross section 18) of the inverse reaction, and it was assumed that it is entirely due to absorption of the D~ wave in the resonance region. The resonance parameters must also fit
74
13. HOOP Jr. AND H. H. BARSCHALL
the measured peak in the n - ~ t total cross section. An interaction radius of 5 fm was used which is the value chosen by W H in the analysis of the corresponding 5Li resonance. The following resonance parameters were found to fit the data: 72 = 2.0 MeV, ~2 = 50 keV, Er = 17.669 MeV. The reduced widths have an uncertainty of 2570, the resonance energy is known to about 10 keV. These values are consistent with those used in the W H analysis of the SLi resonance 7). Using these resonance parameters the real part of the phase shift # and the inelastic parameter z were calculated as a function of energy in the resonance region, and the results are shown in fig. 5. In contrast to the usual behaviour of a resonant phase shift, the D t phase shift decreases rapidly in the resonance region. The cause of this is illustrated in fig. 6, where f 2 is
I~
21.0
Re(fz_ )
Fig. 6. Behaviour o f the amplitude f z - in the complex plane as a function o f energy. The n e u t r o n
bombarding energy scale is indicated along the curve. plotted in the resonance region. The angle between the imaginary axis and a line from the centre of the unitary circle to a position on the path followed by f 2 equals 2 #. The behaviour of # is therefore strongly influenced by whether the path o f f 2 does or does not encircle the center of the unitary circle, i.e., whether Fn > I'd or Fn < Fd. The available data are fitted best by Fn < Fd although they are not accurate enough to exclude the possibility F n > F a. In the latter case # would increase rapidly in the resonance region rather than decrease. A rapid decrease of/~ with energy does not contradict Wigner's 19) theorem which applies only to purely elastic interactions. For the present case where there is a large reaction width a more complicated relation between phase shift and momentum applies 20). Above 25 MeV disintegrations of ~-particles by neutrons resulting in three final
NEUTRON SCATTERING
75
particles are energetically possible. T h e r e is also the possibility o f inelastic scattering o f n e u t r o n s leaving the s-particle in an excited state 21). N o i n f o r m a t i o n is available o n the cross sections f o r any o f these processes, n o r are the c o n t r i b u t i o n s o f partial waves o t h e r t h a n D~ to the 4He(n, d ) T r e a c t i o n k n o w n . This lack o f i n f o r m a t i o n m a k e s it impossible to d et e r m in e the c o m p l e x p h a s e shifts a b o v e 25 M e V with any confidence. W H find t h a t only the P~ a n d D~ inelastic p a r a m e t e r s differ significantly f r o m unity u p to 31 bfeV. U s e o f the W H inelastic p a r a m e t e r s does n o t give a very g o o d fit to TABLE 1 Real part of phase shifts for n--~ scattering (degrees) Phase shifts (degrees) Neutron energy (MeV) 0.50 0.80 1.00 1.20 1.50 2.00 3.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 21.00 22.00 22.15 22.20 23.00 24.00 25.00 26.00 28.00 30.00
S~. 163 159 156 154 151 146 138 132 121 113 106 101 98 96 93 91 90 89 89 89 88 86 85 84 82 79 0.99
P~. 2 4 6 7 10 15 25 35 47 55 60 61 60 58 56 54 53 53 53 53 52 0.99 52 0.99 51 0.99 51 0.99 50 0.99 50 0.98
P~
11 33 61 81 108 118 122 121 115 110 107 103 100 97 95 93 92 92 92 92 90 0.97 89 0.92 89 0.86 88 0.82 86 0.80 84 0.84
D~
0 0 0 0 0 0 0 0 0 0 0 1 2 2 3 4 6 19 57 0.15 --9 0.33 5 6 6 7 7 8
0.81 0.86 0.85 0.82 0.78 0.74
D~
F~
F9
0 0 0 0 0 0 0 0 0 1 2 2 3 5 6 8 9 10 10 10 11 12 13 0.99 14 0.96 16 0.91 18 0.87
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 3 3 4 4
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 3 3 3 3 4 4
5 5 6 6 0.99
5 6 6 7 0.99
The quantities in italics are inelastic parameters. They are given when they are less than unity. the n - c t differential or total cross sections a b o v e 25 M e V . A n i m p r o v e m e n t o f the fit was o b t a i n e d by increasing the a b s o r p t i o n in the D~ wave. O t h e r ch an g es in the pha s e shifts or inelastic p a r a m e t e r s m i g h t well have given e q u a l l y g o o d fits. M o r e m e a s u r e m e n t s a b o v e 25 M e V are needed before a reliable analysis in this energy reg i o n is possible. T h e p res en t p h as e shifts up to 31 M e V are p l o t t e d in fig. 4 an d t ab u l at ed in table 1.
76
B.
HOOP
Jr.
AND
H.
H.
BARSCHALL
The G T P and DGS phase shifts are shown for comparison in fig. 4. In table 1 the inelastic parameters used above 22 MeV are given in italics where they are less than unity. It should be emphasized again that the phase shifts are quite uncertain above 25 MeV. The present phase shifts were used to calculate the differential cross sections shown by the solid curves in figs. 1-3. In fig. 7 the solid curve represents the total cross section calculated from the present phase shifts. Measurements of the total cross section 1, 4, 22) are shown for comparison. .....
T
i
T
HELIUM i ; , , o Vou~hn, et al. ----Los A l o m o l Physics o~1 Cryo(jenic GrOUp Autt~, e / e t
I
'
:
Ii
o
o.e
i
;
-- 1 i
---~ i
i
,
j i
'
'°",
,' ! i
,
i
i
i
'
[
NEUTRON ENERGY 0deV) Fig. 7. T h e total cross section o f h e l i u m as a f u n c t i o n o f n e u t r o n energy. Previous measurements 1, ~, 2~)
are compared with calculations using the phase shifts and inelastic parameters in table 1.
5. The Analysing Power of Helium
Although helium has been extensively used as a polarization analyser for neutrons, it has been known that its analysing power was uncertain above about 15 MeV. This information is based on measurements of the variation of the asymmetry of neutrons scattered by helium as a function of scattering angle 5, 6). It was found that the angular variation was not proportional to the calculated analysing power. If the two quantities had been found to be proportional, the constant of proportionality would be the polarization of the incident neutrons. The analysing power of helium calculated from the present phase shifts is shown in the form of a contour plot in fig. 8. In the region of the 22.16 MeV resonance the analysing power varies rapidly with energy as shown in fig. 9. The calculated analysing power is compared with measured asymmetries s) in fig. 10. In the calculation of the curves the experimental angular resolution has been taken into account. The measured asymmetries were divided by an assumed polari-
180
I
i
i
I
i
I
I
i
I
i
i
~
,
i O
I
\ ~ + ~ 3O
--J
n-
i
1
~ /
~
X",~.~-~ ~-----
÷70
l lll .rof ~
'
---------'~
~
~ 3 0
~
do'l
,' - ~
~'
~
-
S
2~
~ ~ ,; ," ~ ,~ ~ 22 INCIDENT NEUTRON ENERGY (MeV)
A
2~
Fig. 8. Contour plot of the polarization of neutrons scattered by helium as a function of laboratory neutron scattering angle and laboratory neutron bombarding energy. The polarizations were calculated from the phase shifts and inelastic parameters given in table 1. The numbers on the contour lines are polarizations in percent. The black region at back angles near 22 MeV is a region o f large negative polarization. '
I'01 21.30'MeV
.~
, 1.0,
22.14 MeV
i
-I0 { IO
0
, ,
45
, ,
, ,
J-i.O ~ 1.0
_..-....
i
i
_
90 135 180 0 4.5 90 155 180 LABORATORY NEUTRON SCATTERINGANGLE (degrees)
Fig. 9. Polarization o f neutrons scattered by alpha particles in the neighbourhood of the 22.16 MeV resonance. The polarization is plotted as a function o f the laboratory scattering angle at each energy. The curves are calculated from the phase shifts and inelastic parameters in table 1. The solid circles at 23.1 MeV are the measurements o f ref. 6) assuming a polarization of 0.54 for the incident neutrons (table 2).
78
B. HOOP Jr. AND H. H. BARSCHALL
zation of the incident n e u t r o n s which yielded the best agreement between calculation a n d m e a s u r e m e n t . The a s s u m e d n e u t r o n source polarizations are t a b u l a t e d in c o l u m n 6 of table 2 together with their u n c e r t a i n t y within which equally good fits could be
1.0
1.01
,
/,\
!
EN=6.0
-'°.°,,v
MeV
0.5
-iJ
z -0.5 o I,,~ -I.0 N 1.0
I
I
,
,
1,0,
,
,
E N = 23.7
J o t.a
MeV
0.5 0.50 0
-0.5
-0.5
#
-I'01.0
015
0
-0.5i
-I.0
- 1,01.0
i 0.5
0i
-0.5r
-I,0
COS ecru Fig. 10. Polarization of neutrons scattered by alpha particles as a function of the cosine of the c.m. scattering angle. The data from ref. 5) are compared with polarizations calculated from t h e phase shifts and inelastic parameters in table 1. The experimental results were asymmetries and had to be divided by an assumed polarization P1 of the incident neutrons. The values of P1 are given in table 2. TABLE 2 Sources of polarized neutrons used in n--~ asymmetry measurements Ref.
5) 6) 5) 6) 5)
Reaction
T(p, n)SHe D(d, n)SHe T(d, n)*He T(d, n)4He T(d, n)4He
(MeV)
(lab)
(lab) (MeV)
7.8 8.4 6.0 7.0 7.7
40° 32° 90° 30° 30°
6.0 10.0 16.4 23.1 23.7
Eb
0
En
PI --0.20 4-0.02 +0.27 4-0.01 -- 0.47 +0.02 +0.544-0.02 +0.49 4-0.02
obtained. I n table 2 are also listed the sources of the polarized n e u t r o n s , the charged particle b o m b a r d i n g energy E b a n d the l a b o r a t o r y angle 0 of the emission of the partially polarized n e u t r o n s of energy E , . I n fig. 9 the 23.1 M e V m e a s u r e m e n t s o f Perkins a n d Glashausser 6) are c o m p a r e d with the calculations.
NEUTRON SCATTERING
79
A c o m p a r i s o n between the d a t a a n d the calculated analysing p o w e r shows fair a g r e e m e n t at all energies. This a g r e e m e n t is m u c h better than was o b t a i n e d with p r e v i o u s phase shifts at energies a b o v e 20 MeV. M o r e data are needed, however, to d e t e r m i n e the analyzing p o w e r reliably a b o v e 25 MeV. Because o f the r a p id v a r i a t i o n o f the analysing p o w e r o f helium in the n e i g h b o u r h o o d o f 22 MeV, h e l i u m is n o t a useful analyser f o r n e u t r o n s in this energy region.
References 1) S. M. Austin, H. H. Barschall and R. E. Shamu, Phys. Rev. 126 (1962) 1532 2) D. C. Dodder and J. L. Gammel, Phys. Rev. 88 (1952) 520; J. D. Seagrave, Phys. Rev. 92 (1953) 1222 3) J. L. Gammel and R. M. Thaler, Phys. Rev. 109 (1958) 2041; R. B. Perkins, private communication (1960) 4) R. E. Shamu and J. G. Jenkin, Phys. Rev. 135 (1964) B99 5) T. H. May, R. L. Walter and H. H. Barschall, Nuclear Physics 45 (1963) 17 6) R. B. Perkins and C. Glashausser, Nuclear Physics 60 (1964) 433 7) W. G. Weitkamp and W. Haeberli, Nuclear Physics 83 0966) 46 8) R. L. Waiter, W. Benenson, P. S. Dubbeldam and T. H. May, Nuclear Physics 30 (1962) 292; R. E. Shamu, Nucl. Instr. 14 (1961) 297 9) J. G. Jenkin and R. E. Shamu, Nucl. Instr. 34 (1965) 116 10) S. J. Bame, E. Haddad, J. E. Perry and R. K. Smith, Rev. Sci. Instr. 28 (1957) 997; W. E. Wilson, Ph. D. Thesis, University of Wisconsin (1961) 11) S. J. Bame and J. E. Perry, Phys. Rev. 107 (1957) 1616; M. D. Goldberg and J. M. LeBlanc, Phys. Rev. 122 (1961) 164 12) B. Hoop, Jr., Ph. D. Thesis, University of Wisconsin (1965) 13) R. Genin et al., J. de Phys. 24 (1963) 21; S. P. Harris, ANL-4680 (1951) 14) P. Swan, Proc. Roy. Soc. A228 (1955) 10 15) E. Bretscher and A. P. French, Phys. Rev. 75 (1949) 1154 16) T. W. Bonner, F. W. Prosser and J. Slattery, Phys. Rev. 115 (1959) 398 17) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 18) H. V. Argo et aL, Phys. Rev. 87 (1952) 612 19) E. P. Wigner, Phys. Rev. 98 (1955) 145 20) C. J. Goebel and K. W. IV~cVoy, Ann. of Phys., to be published 2l) C. H. Poppe, Phys. Lett. 2 (1962) 171; P. D. Parker, P. F. Donovan, J. V. Kane and J. F. Mollenauer, Phys. Rev. Lett. 14 (1965) 15; J. Cerny, C. Detraz and R. H. Pehl, Phys. Rev. Lett. 15 (1965) 300 22) F. J. Vaughn, W. L. Imhof, R. G. Johnson and M. Walt, Phys. Rev. 118 (1960) 683; Los Alamos Physics and Cryogenic Groups, Nuclear Physics 12 (1959) 291
SCATTERING
OF NEUTRONS
BY a - P A R T I C L E S
B. H O O P , Jr.t a n d H. H. B A R S C H A L L
Uni~'ersity of Wisconsin, Madison, Wisconsin tt
Received 7 January 1966 Abstract: The differential cross section for the scattering of neutrons by u-particles has been measured at neutron energies between 6 and 30 MeV. In the neighbourhood of the resonance at 22.16 MeV neutron energy, measurements were performed at closely spaced neutron energies. The present data as well as earlier measurements of elastic scattering and total cross sections, and previous polarization measurements were fitted with phase shifts which are based largely on the p--a phase shifts of Weitkamp and Haeberli. Satisfactory fits were obtained for neutron energies up to 30 MeV, including the region of the 22.16 MeV resonance. The analysing power of helium deduced from these phase shifts is presented in the form of a contour plot. E
NUCLEAR REACTIONS. 4He(n, n), E = 6-30 MeV; measured a(E; 0). Deduced phase shifts, resonance parameters, polarization. Natural target.
I
I
1. Introduction H e l i u m is the m o s t widely used a n a l y s e r o f p o l a r i z e d neutrons. The analysing p o w e r o f h e l i u m is u s u a l l y c a l c u l a t e d f r o m p h a s e shifts for the scattering o f n e u t r o n s by a-particles. I t is therefore o f interest to o b t a i n p h a s e shifts which r e p r o d u c e all available n-~ measurements. I n a p r e v i o u s p a p e r 1) the i n f o r m a t i o n on n - ~ scattering available at t h a t time was s u m m a r i z e d . P h a s e shifts for n - • scattering have a l m o s t always been d e d u c e d f r o m the m o r e accurately k n o w n p - ~ p h a s e shifts. The two m o s t often used sets o f n p h a s e shifts b o t h o f which were d e d u c e d f r o m p - ~ p h a s e shifts are the D o d d e r G a m m e l - S e a g r a v e 2) p h a s e shifts ( D G S ) at energies b e l o w 20 M e V a n d the G a m m e l T h a l e r - P e r k i n s a) p h a s e shifts ( G T P ) a b o v e 10 MeV. These two sets o f p h a s e shifts d o not, however, c o n n e c t s m o o t h l y , n o r d o they fit the d a t a 4 - 6 ) a b o v e a n e u t r o n e n e r g y o f a b o u t 15 MeV. N o n e o f the available n - ~ p h a s e shifts include the effects o f the excited state in 5He at an excitation energy o f 16.7 MeV, n o r d o they allow for inelastic processes which occur a b o v e a n e u t r o n energy o f 22 MeV. I n the p r e s e n t p a p e r an a t t e m p t is m a d e to o b t a i n n - ~ phase shifts which include the effects o f this r e s o n a n c e a n d o f inelastic processes a n d which fit e x p e r i m e n t a l results u p to 30 MeV. T h e s e p h a s e shifts are b a s e d again largely on p - ~ p h a s e shifts 7). E x p e r i m e n t s were p e r f o r m e d to i m p r o v e m e a s u r e m e n t s o f the a n g u l a r d i s t r i b u t i o n t Now at the Universit~t Basel, Switzerland. tt Work supported in part by the U.S. Atomic Energy Commission. 65
66
B. HOOP Jr. AND H. H. BARSCHALL
o f neutrons scattered by s-particles carried out earlier at this laboratory 1). The earlier data taken at 10 and 12 MeV extended only over a small angular range because of the presence of break-up neutrons from the d + D reaction. At higher energies the data were very uncertain because of the effect of disintegrations of xenon which had to be mixed with helium. These disintegrations produced a large background which was difficult to subtract reliably. Furthermore, in the present experiments a better method for deducing cross sections from the relative angular distributions was employed.
2. Experimental Method Angular distributions were measured by the method used previously 1). A highpressure helium gas scintillator was bombarded with mono-energetic neutrons, and the pulse-height distribution of recoiling s-particles was observed. This distribution is proportional to the centre-of-mass angular distribution. In the present experiment a gas scintillator 8) was employed which could be operated at higher pressures than had been used for the angular distribution measurements performed previously at this laboratory. The gas scintillator was a thin-walled, approximately hemispherical stainless steel cell of diam 5 cm which contained a mixture of helium and xenon. Gas pressures of 100 to 175 atm helium and 5 to 25 atm xenon were used. The gas pressures were adjusted at each neutron bombarding energy to yield the highest stopping power consistent with low background contributions from neutron disintegrations of xenon and with small multiple scattering effects. In all cases, the maximum range of the recoiling ~-particles was less than 5 mm. A Phillips 56 AVP photomultiplier viewed the scintillations. Previous investigations 8) have shown that the light output of the gas scintillator as a function of ~-energy is linear for 42 arm helium and 8 atm xenon. More recently, however, a non-linear behaviour has been reported in helium-xenon gas mixtures containing more than 30 atm xenon 9). The light output of the scintillator filled with a mixture of helium and xenon at several different pressures was studied as a function of ~-particle energy by bombarding the scintillator with mono-energetic neutrons of several energies and measuring the maximum pulse-height produced by recoiling ~-particles. The gas pressures were approximately those used in the angular distribution measurements. Additional fillings of 10 to 17 atm xenon and 4 to 8 atm helium were also studied. The light output of the scintillator for all fillings used was found to be a linear function of ~-energy within the 4 ~o uncertainty in the measurements over the range of 0~-energies observed in the angular distribution measurements. The procedure for measuring the angular distributions was similar to that used previously i). Neutrons below 16 MeV were obtained from the d + D reaction. Above 16 MeV, the d + T reaction was used as the source of neutrons. Measurements were carried out at about 2 MeV intervals between neutron energies of 6 and 31 MeV, except between 21.8 and 22.5 MeV where measurements were made at about 75 keV
NEUTRON
SCATTERING
67
intervals. Neutron energy spreads for most of the measurements were between 60 and 140 keV. Near 22 MeV the energy spread was about 40 keV. A time-of-flight spectrometer was employed to eliminate contributions to the angular distributions caused by break-up neutrons from the d + D reaction. The pulse-height analyser which recorded the pulses from the helium scintillator was gated for this purpose with pulses from the time-of-flight spectrometer. Only those scintillation pulses were recorded which occurred at times corresponding to the arrival of the mono-energetic neutrons at the scintillator. Since an accurate determination of the scintillation volume and of its effective centre was difficult, the cross sections were measured relative to that at 6 MeV neutron energy. At this energy it was believed to be possible to determine the scattering cross section without knowing the effective number of helium atoms in the cell by using the known total cross section of helium. As will be discussed in sect. 4, at 6 MeV only S- and P-waves are needed to fit all available data. Under this assumption the measured angular distribution can be readily extrapolated to zero pulse-height and the differential cross section can be calculated as was done in the earlier experiment. In order to compare the incident neutron flux at different neutron energies a proton recoil counter telescope similar to that described by Bame et al. 10) was employed for neutron energies up to 25 MeV. Above this neutron energy the number of recoils was compared with the number of 4He(n, d)T disintegrations which appeared as a peak superimposed on the recoil distribution. The disintegration cross section was calculated from the known cross section of the inverse reaction 11) using the principle of detailed balance. The comparison of the scattering cross section and the disintegration cross section of helium is possible without knowing either the number of helium atoms or the neutron flux. The background caused by neutrons from the foils and walls of the gas target which produce pulses in the scintillator was measured with an evacuated gas target. This background was largest for the smallest pulse-heights measured at the highest neutron bombarding energies, but at all energies it decreased rapidly to less than 5 ~o over most of the recoil distribution. Corrections for neutrons scattered into the scintillator by the floor and other surrounding materials except the steel wails of the scintillator were determined by measuring a recoil distribution with a 30 cm brass shadow bar interposed between the scintillator and the gas target. This correction was less than 10 ~o for the smallest pulseheights measured and decreased to less than about 1 ~o over the remainder of the pulse-height spectrum. The effect of scattering of neutrons into the scintillator by the walls of the steel cell was investigated by surrounding the scintillator with a slightly enlarged copy of the cell which increased the effective wall thickness by about a factor of two. The difference between recoil distributions measured with and without the additional steel was less than the statistical uncertainties of the measurements and no correction for this effect was applied.
68
B. HOOP Jr. AND H. H. BARSCHALL
Another background is caused by source neutrons which produce disintegrations or recoils in the walls of the scintillator and in the xenon. This scintillator background is difficult to measure because any change of the gas filling changes both the stopping power and the pulse size caused by particles of a given energy. There is even evidence that the pulse size caused by particles which have different specific energy losses is affected differently by changes in the gas composition 9). If the helium is removed and the background is measured using only xenon in the scintillator, the stopping power of the scintillator is changed and this alters the pulse-height distribution. If, on the other hand, the xenon pressure is increased to keep the stopping power constant, the number of xenon disintegrations increases while the number of particles originating in the wails remains the same. Varying the xenon pressure produced changes both in the shape of the pulse-height distribution and the number of pulses divided by the xenon pressure. It appears, therefore, that a substantial fraction of the background was due to disintegrations in the walls, in the coatings, or in the glass window. For the determination of the scintillator background the scintillator was filled with pure xenon to a pressure such that the stopping power was somewhat less than that used in the foreground measurement. The background so determined was divided by the ratio of the xenon pressure in the background measurement to that in the foreground measurement. This procedure underestimates the background caused by disintegrations produced in the walls and overestimates the effect of disintegration particles, particularly protons, which are intercepted by the walls. There was in addition considerable uncertainty in the gain adjustment of the amplifier when pure xenon was used. A Po ~-particle source was permanently mounted in the scintillator. It was found, however, as has previously been observed 9), that the pulse-height of Po ~-particles is not a reliable measure of the pulse-heights of recoil ~-particles. The reason for this is not fully understood, but it is most likely caused by absorption of light in the wire on which the source was deposited. Because of all these problems the scintillator background correction is estimated to have an uncertainty of 30 ~o. For neutron energies below 20 MeV this background correction was only about 5 ~o so that its uncertainty does not affect the results very much. Above 20 MeV the correction reached values as high as 30 ~o at the minimum in the differential cross section so that it introduces at this minimum a 10~o uncertainty in the cross-section determination. Corrections for the wall effect were applied as described previously 1). The greatest correction amounted to 11 ~o for the largest recoil energies at the highest neutron bombarding energy. 3. R ~
Examples of the present measurements of the differential cross sections for scattering of neutrons by or-particles are plotted in fig. 1-3. The smallest scattering angle at which data are shown is determined by the pulse-height at which backgrounds were
NEUTRON SCATTERING
69
so large that they could not be reliably corrected for, the largest scattering angle by the pulse-height resolution of the scintillator. The gaps in the data points of the cross sections above 22.07 MeV are due to the presence of charged particles from the 4He(n, d)T reaction. At 28.6 MeV some pulses from the 4I-Ie(n, pn)T reaction may be included at c.m. scattering angles < 80 °. The statistical uncertainty of the data was less than the size of the symbols except where error bars are shown. There are a number of additional uncertainties which 400
q
~-
350 \
\
~|
* En =10,00 MeV
subtract I00 from ordinate
+ En=lS.05MeV
\
subtract 50 from ordinate
----
•
En=2091 MeV
250
E IE
b
150
5C
iX 0.5
0
-0.5
-I.0
co~ q Fig. 1. Differential cross sections in the c.m. system for the scattering of 10.00, 15.05 and 20.91 MeV neutrons by or-particles. The curves are calculated from the phase shifts given in table 1.
are not shown by error bars in the figures. The cross-section scale has an uncertainty of the order of 10 Yo. There are also uncertainties which may produce distortions of the distribution, particularly the uncertainties in the background corrections. These tend to increase with increasing neutron energy and decreasing pulse height. As has been mentioned, the uncertainty in the scintillator background correction reaches about 10 ~o of the cross section at the minima in the cross sections at the highest neutron energies. All other background corrections are believed to introduce smaller uncertainties.
Io
0 1.0
25
50
75
0.5
subtract 75 from ordinate
cos e
0
-Q5
subtract 2 5 from ordinate
* En =22.20 MeV
subtract 50 from ordinate
o En =22.10 MeV
-I0
Fig. 2. Differential cross sections in the c.m. system for the scattering o f neutrons with energies near 22 MeV by =-particles. The curves are calculated from the phase shifts and inelastic parameters given in table I.
v
E
125
15o
175
I o En=22.04MeV
0.5
MeV
0
cos e
-0.5
-I.0
Fig. 3. Differential cross sections in the c.m. system for the scattering o f 22.45, 24.7 and. 28.6 MeV neutrons by ~-particles. The curves are calculated from the phase shifts and inelastic parameters given in table 1.
b
v
=E
E
.JD
En=28.6
subtroct 2 5 from ordinate
- - CALCULATED
•
+ En=24.7 MeV
subtract 50 from ordinate
° En =22.45 MeV
NEUTRON
71
SCATTERING
The results obtained at other neutron energies and additional experimental details are reported elsewhere 12). The present measurements are in excellent agreement with the measurements of Shamu and Jenkin 4) between 16 and 24 MeV and are consistent with the earlier measurements at this laboratory 1) at lower energies. 4. Phase Shifts for n - ~ t Scattering The ground state of the compound nucleus 5He is reached at a neutron energy of about 1 MeV and manifests itself as a resonance in the P{ partial wave. A few MeV above this energy the P, partial wave is affected by a broad state in 5He. The only other excited state of 5He, the existence of which has been established, is a ~+ state at an excitation energy of 16.7 MeV. 1801
i
120
~
60
LLI III
30 0 Ij~
r'r"
120r
I
,
I
I
I
I
I
......... I
I
~
I
i
. . . .
I
~
,
LLI 90 I- / ~'~ 6°1-/ g/^ II
i
~..
6C
(..9
i
0]/
I
---DGS -'-GTP
......
I
l
L..C?... . . . . . . . . . . . . . . . . . . . . . . . . . I
T :I . . . . co 60 D%
I
I
'
'
50 LLI U') [~_
I
.................... i t
-3
30o
gs@
-30 30
;
-n
,-- . . . . . . . . . . . . . . . . . . . . . . i i
..............................................
.....................
I ~
F%
--L--_:
t
I
I
I
,
,
i
.........................
-3
I
I
I
i
I
-3
I I0
I 15
r 20
I 25
30
I
35
NEUTRON ENERGY (MeV) Fig. 4. Phase shifts for neutron-alpha scattering as a function of neutron bombarding energy. The dashed (DGS) curves are taken from ref. =), the dot-dashed. (GTP) curves from ref. a). The solid curves represent the phase shifts proposed in this paper. Phase shifts describing the n - c t system should vary smoothly with energy except for the three states just mentioned. Each phase shift should approach zero (or an integer multiple of n) as k 2~+ 1, where k is the wave number and l the orbital angular momentum quantum number.
72
B. HOOP Jr. AND H. H. BARSCHALL
The DGS phase shifts 2) shown by the dashed lines in fig. 4 are known to fit n - ~ data below 10 MeV satisfactorily. These phase shifts include negative D-wave phase shifts. On the other hand, p - ~ results at higher energies show that positive D-wave phase shifts are needed to fit the data. Since none of the published n - • data are accurate enough to decide whether any D-waves are needed below 10 MeV, it was assumed for the present analysis that D-waves could be neglected for neutron energies below 6 MeV. The S-wave phase shift was assumed to vary with the wave number k as - 2 . 4 (k fm - t ) up to 6 MeV. This yields the measured cross section of 0.73 b for thermal neutrons lZ). At zero energy the S-wave phase shift was taken to have the value 7r (ref. 14)). The DGS P-wave phase shifts were used except that the P~t phase shift was slightly modified so that S- and P-waves reproduced the measured total cross section. This modification corresponds to a small increase in the reduced width of the PC resonance. Above 17 MeV the n - ~ phase shifts were taken from the p-0~ phase shifts of Weitkamp and Haeberli (WH) 7). These include partial waves up to l = 3. The W H Coulomb phase shifts were set equal to zero and the energy scale was shifted so as to bring into coincidence the excited state in 5Li at a p - • centre-of-mass energy of 18.73 MeV and the mirror state in SHe at a n - ~ centre-of-mass energy of 17.67 MeV. The W H S- and P-wave phase shifts were connected smoothly to the modified DGS phase shifts, and the W H D- and F-wave phase shifts were extrapolated to lower energies according to k 2t+1 The ½+ state in SHe at 16.7 MeV was first observed in the d + T reaction xs). It has been shown to have approximately the same partial width F d for decay into d + T, as the partial width F , , for decay into n + ~ (ref. 18)). Its effect on n - ~ scattering was first observed by Bonner et al. 16). In the n - u system the state occurs just above the threshold for the 4He(n, d)T reaction and should result in a rapid variation with energy of the complex D~ phase shift. In order to study the effect of this resonance on the scattering amplitude it is useful to define a D~ amplitude f2 -
exp ( 2 i ~ ) - 1 2i '
where t5 is the complex D~ phase shift. Frequently the term phase shift is applied only to the real part of 6 which will be denoted by p. The amplitude f 2 can be written f ~ = (z exp (2i#) - 1)/2i where z = exp ( - 2 Im 6). The quantity x is sometimes called the inelastic parameter. In the neighbourhood of the resonance the amplitude f 2 may be written as 17) e 2i~ --
f 2 --
2i
1
F n
+ F n q- F a
e 21p - - 1
2i
e2i~'
where fl is the resonant and q~ the non-resonant D~ phase shift which in this case is
73
NEUTRON SCATTERING
assumed to have a value of 5 °. The resonant phase shift fl is given by
Fn+Fd tg~ = 2(E~+ A - E ) ' where E~ is the characteristic energy and A the level shift. The widths F depend on energy through the penetrabilities P
F
=
2?2p,
where ?2 is the reduced width. The level shift A depends on energy through the shift functions and boundary condition parameters. These boundary condition parameters were chosen so that the level shift is zero at the resonance energy Er which is defined as the energy at which /~ = ½7r. lad I-b..I
=E o.8 n., 0.6 I-to < o41 ..a w z ~ 0.2 a
m w
0 601
40 I
zc
IM to
am-e=_2(3 " 21.6
21.8
22.0
NEUTRON
2~'12
22.4
22.6
22.8
ENERGY (MeV)
Fig. 5. The real part of the D~. phase shift p and the D~ inelastic parameter T in the vicinity of the resonance.
It was desired to find values of y2, ?2 and E~ or E r which best fit the available data. The cross section for the *He(n, d)T reaction was obtained from the known cross section 18) of the inverse reaction, and it was assumed that it is entirely due to absorption of the D~ wave in the resonance region. The resonance parameters must also fit
74
13. HOOP Jr. AND H. H. BARSCHALL
the measured peak in the n - ~ t total cross section. An interaction radius of 5 fm was used which is the value chosen by W H in the analysis of the corresponding 5Li resonance. The following resonance parameters were found to fit the data: 72 = 2.0 MeV, ~2 = 50 keV, Er = 17.669 MeV. The reduced widths have an uncertainty of 2570, the resonance energy is known to about 10 keV. These values are consistent with those used in the W H analysis of the SLi resonance 7). Using these resonance parameters the real part of the phase shift # and the inelastic parameter z were calculated as a function of energy in the resonance region, and the results are shown in fig. 5. In contrast to the usual behaviour of a resonant phase shift, the D t phase shift decreases rapidly in the resonance region. The cause of this is illustrated in fig. 6, where f 2 is
I~
21.0
Re(fz_ )
Fig. 6. Behaviour o f the amplitude f z - in the complex plane as a function o f energy. The n e u t r o n
bombarding energy scale is indicated along the curve. plotted in the resonance region. The angle between the imaginary axis and a line from the centre of the unitary circle to a position on the path followed by f 2 equals 2 #. The behaviour of # is therefore strongly influenced by whether the path o f f 2 does or does not encircle the center of the unitary circle, i.e., whether Fn > I'd or Fn < Fd. The available data are fitted best by Fn < Fd although they are not accurate enough to exclude the possibility F n > F a. In the latter case # would increase rapidly in the resonance region rather than decrease. A rapid decrease of/~ with energy does not contradict Wigner's 19) theorem which applies only to purely elastic interactions. For the present case where there is a large reaction width a more complicated relation between phase shift and momentum applies 20). Above 25 MeV disintegrations of ~-particles by neutrons resulting in three final
NEUTRON SCATTERING
75
particles are energetically possible. T h e r e is also the possibility o f inelastic scattering o f n e u t r o n s leaving the s-particle in an excited state 21). N o i n f o r m a t i o n is available o n the cross sections f o r any o f these processes, n o r are the c o n t r i b u t i o n s o f partial waves o t h e r t h a n D~ to the 4He(n, d ) T r e a c t i o n k n o w n . This lack o f i n f o r m a t i o n m a k e s it impossible to d et e r m in e the c o m p l e x p h a s e shifts a b o v e 25 M e V with any confidence. W H find t h a t only the P~ a n d D~ inelastic p a r a m e t e r s differ significantly f r o m unity u p to 31 bfeV. U s e o f the W H inelastic p a r a m e t e r s does n o t give a very g o o d fit to TABLE 1 Real part of phase shifts for n--~ scattering (degrees) Phase shifts (degrees) Neutron energy (MeV) 0.50 0.80 1.00 1.20 1.50 2.00 3.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 21.00 22.00 22.15 22.20 23.00 24.00 25.00 26.00 28.00 30.00
S~. 163 159 156 154 151 146 138 132 121 113 106 101 98 96 93 91 90 89 89 89 88 86 85 84 82 79 0.99
P~. 2 4 6 7 10 15 25 35 47 55 60 61 60 58 56 54 53 53 53 53 52 0.99 52 0.99 51 0.99 51 0.99 50 0.99 50 0.98
P~
11 33 61 81 108 118 122 121 115 110 107 103 100 97 95 93 92 92 92 92 90 0.97 89 0.92 89 0.86 88 0.82 86 0.80 84 0.84
D~
0 0 0 0 0 0 0 0 0 0 0 1 2 2 3 4 6 19 57 0.15 --9 0.33 5 6 6 7 7 8
0.81 0.86 0.85 0.82 0.78 0.74
D~
F~
F9
0 0 0 0 0 0 0 0 0 1 2 2 3 5 6 8 9 10 10 10 11 12 13 0.99 14 0.96 16 0.91 18 0.87
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 3 3 4 4
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 3 3 3 3 4 4
5 5 6 6 0.99
5 6 6 7 0.99
The quantities in italics are inelastic parameters. They are given when they are less than unity. the n - c t differential or total cross sections a b o v e 25 M e V . A n i m p r o v e m e n t o f the fit was o b t a i n e d by increasing the a b s o r p t i o n in the D~ wave. O t h e r ch an g es in the pha s e shifts or inelastic p a r a m e t e r s m i g h t well have given e q u a l l y g o o d fits. M o r e m e a s u r e m e n t s a b o v e 25 M e V are needed before a reliable analysis in this energy reg i o n is possible. T h e p res en t p h as e shifts up to 31 M e V are p l o t t e d in fig. 4 an d t ab u l at ed in table 1.
76
B.
HOOP
Jr.
AND
H.
H.
BARSCHALL
The G T P and DGS phase shifts are shown for comparison in fig. 4. In table 1 the inelastic parameters used above 22 MeV are given in italics where they are less than unity. It should be emphasized again that the phase shifts are quite uncertain above 25 MeV. The present phase shifts were used to calculate the differential cross sections shown by the solid curves in figs. 1-3. In fig. 7 the solid curve represents the total cross section calculated from the present phase shifts. Measurements of the total cross section 1, 4, 22) are shown for comparison. .....
T
i
T
HELIUM i ; , , o Vou~hn, et al. ----Los A l o m o l Physics o~1 Cryo(jenic GrOUp Autt~, e / e t
I
'
:
Ii
o
o.e
i
;
-- 1 i
---~ i
i
,
j i
'
'°",
,' ! i
,
i
i
i
'
[
NEUTRON ENERGY 0deV) Fig. 7. T h e total cross section o f h e l i u m as a f u n c t i o n o f n e u t r o n energy. Previous measurements 1, ~, 2~)
are compared with calculations using the phase shifts and inelastic parameters in table 1.
5. The Analysing Power of Helium
Although helium has been extensively used as a polarization analyser for neutrons, it has been known that its analysing power was uncertain above about 15 MeV. This information is based on measurements of the variation of the asymmetry of neutrons scattered by helium as a function of scattering angle 5, 6). It was found that the angular variation was not proportional to the calculated analysing power. If the two quantities had been found to be proportional, the constant of proportionality would be the polarization of the incident neutrons. The analysing power of helium calculated from the present phase shifts is shown in the form of a contour plot in fig. 8. In the region of the 22.16 MeV resonance the analysing power varies rapidly with energy as shown in fig. 9. The calculated analysing power is compared with measured asymmetries s) in fig. 10. In the calculation of the curves the experimental angular resolution has been taken into account. The measured asymmetries were divided by an assumed polari-
180
I
i
i
I
i
I
I
i
I
i
i
~
,
i O
I
\ ~ + ~ 3O
--J
n-
i
1
~ /
~
X",~.~-~ ~-----
÷70
l lll .rof ~
'
---------'~
~
~ 3 0
~
do'l
,' - ~
~'
~
-
S
2~
~ ~ ,; ," ~ ,~ ~ 22 INCIDENT NEUTRON ENERGY (MeV)
A
2~
Fig. 8. Contour plot of the polarization of neutrons scattered by helium as a function of laboratory neutron scattering angle and laboratory neutron bombarding energy. The polarizations were calculated from the phase shifts and inelastic parameters given in table 1. The numbers on the contour lines are polarizations in percent. The black region at back angles near 22 MeV is a region o f large negative polarization. '
I'01 21.30'MeV
.~
, 1.0,
22.14 MeV
i
-I0 { IO
0
, ,
45
, ,
, ,
J-i.O ~ 1.0
_..-....
i
i
_
90 135 180 0 4.5 90 155 180 LABORATORY NEUTRON SCATTERINGANGLE (degrees)
Fig. 9. Polarization o f neutrons scattered by alpha particles in the neighbourhood of the 22.16 MeV resonance. The polarization is plotted as a function o f the laboratory scattering angle at each energy. The curves are calculated from the phase shifts and inelastic parameters in table 1. The solid circles at 23.1 MeV are the measurements o f ref. 6) assuming a polarization of 0.54 for the incident neutrons (table 2).
78
B. HOOP Jr. AND H. H. BARSCHALL
zation of the incident n e u t r o n s which yielded the best agreement between calculation a n d m e a s u r e m e n t . The a s s u m e d n e u t r o n source polarizations are t a b u l a t e d in c o l u m n 6 of table 2 together with their u n c e r t a i n t y within which equally good fits could be
1.0
1.01
,
/,\
!
EN=6.0
-'°.°,,v
MeV
0.5
-iJ
z -0.5 o I,,~ -I.0 N 1.0
I
I
,
,
1,0,
,
,
E N = 23.7
J o t.a
MeV
0.5 0.50 0
-0.5
-0.5
#
-I'01.0
015
0
-0.5i
-I.0
- 1,01.0
i 0.5
0i
-0.5r
-I,0
COS ecru Fig. 10. Polarization of neutrons scattered by alpha particles as a function of the cosine of the c.m. scattering angle. The data from ref. 5) are compared with polarizations calculated from t h e phase shifts and inelastic parameters in table 1. The experimental results were asymmetries and had to be divided by an assumed polarization P1 of the incident neutrons. The values of P1 are given in table 2. TABLE 2 Sources of polarized neutrons used in n--~ asymmetry measurements Ref.
5) 6) 5) 6) 5)
Reaction
T(p, n)SHe D(d, n)SHe T(d, n)*He T(d, n)4He T(d, n)4He
(MeV)
(lab)
(lab) (MeV)
7.8 8.4 6.0 7.0 7.7
40° 32° 90° 30° 30°
6.0 10.0 16.4 23.1 23.7
Eb
0
En
PI --0.20 4-0.02 +0.27 4-0.01 -- 0.47 +0.02 +0.544-0.02 +0.49 4-0.02
obtained. I n table 2 are also listed the sources of the polarized n e u t r o n s , the charged particle b o m b a r d i n g energy E b a n d the l a b o r a t o r y angle 0 of the emission of the partially polarized n e u t r o n s of energy E , . I n fig. 9 the 23.1 M e V m e a s u r e m e n t s o f Perkins a n d Glashausser 6) are c o m p a r e d with the calculations.
NEUTRON SCATTERING
79
A c o m p a r i s o n between the d a t a a n d the calculated analysing p o w e r shows fair a g r e e m e n t at all energies. This a g r e e m e n t is m u c h better than was o b t a i n e d with p r e v i o u s phase shifts at energies a b o v e 20 MeV. M o r e data are needed, however, to d e t e r m i n e the analyzing p o w e r reliably a b o v e 25 MeV. Because o f the r a p id v a r i a t i o n o f the analysing p o w e r o f helium in the n e i g h b o u r h o o d o f 22 MeV, h e l i u m is n o t a useful analyser f o r n e u t r o n s in this energy region.
References 1) S. M. Austin, H. H. Barschall and R. E. Shamu, Phys. Rev. 126 (1962) 1532 2) D. C. Dodder and J. L. Gammel, Phys. Rev. 88 (1952) 520; J. D. Seagrave, Phys. Rev. 92 (1953) 1222 3) J. L. Gammel and R. M. Thaler, Phys. Rev. 109 (1958) 2041; R. B. Perkins, private communication (1960) 4) R. E. Shamu and J. G. Jenkin, Phys. Rev. 135 (1964) B99 5) T. H. May, R. L. Walter and H. H. Barschall, Nuclear Physics 45 (1963) 17 6) R. B. Perkins and C. Glashausser, Nuclear Physics 60 (1964) 433 7) W. G. Weitkamp and W. Haeberli, Nuclear Physics 83 0966) 46 8) R. L. Waiter, W. Benenson, P. S. Dubbeldam and T. H. May, Nuclear Physics 30 (1962) 292; R. E. Shamu, Nucl. Instr. 14 (1961) 297 9) J. G. Jenkin and R. E. Shamu, Nucl. Instr. 34 (1965) 116 10) S. J. Bame, E. Haddad, J. E. Perry and R. K. Smith, Rev. Sci. Instr. 28 (1957) 997; W. E. Wilson, Ph. D. Thesis, University of Wisconsin (1961) 11) S. J. Bame and J. E. Perry, Phys. Rev. 107 (1957) 1616; M. D. Goldberg and J. M. LeBlanc, Phys. Rev. 122 (1961) 164 12) B. Hoop, Jr., Ph. D. Thesis, University of Wisconsin (1965) 13) R. Genin et al., J. de Phys. 24 (1963) 21; S. P. Harris, ANL-4680 (1951) 14) P. Swan, Proc. Roy. Soc. A228 (1955) 10 15) E. Bretscher and A. P. French, Phys. Rev. 75 (1949) 1154 16) T. W. Bonner, F. W. Prosser and J. Slattery, Phys. Rev. 115 (1959) 398 17) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 18) H. V. Argo et aL, Phys. Rev. 87 (1952) 612 19) E. P. Wigner, Phys. Rev. 98 (1955) 145 20) C. J. Goebel and K. W. IV~cVoy, Ann. of Phys., to be published 2l) C. H. Poppe, Phys. Lett. 2 (1962) 171; P. D. Parker, P. F. Donovan, J. V. Kane and J. F. Mollenauer, Phys. Rev. Lett. 14 (1965) 15; J. Cerny, C. Detraz and R. H. Pehl, Phys. Rev. Lett. 15 (1965) 300 22) F. J. Vaughn, W. L. Imhof, R. G. Johnson and M. Walt, Phys. Rev. 118 (1960) 683; Los Alamos Physics and Cryogenic Groups, Nuclear Physics 12 (1959) 291