Spin-spin effect in the 165Ho total neutron cross section at energies of 0.4 and 1.0 MeV

I

2.L

I I

Nuclear Physics A130 (1969) 609 --623; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

SPIN-SPIN EFFECT I N T H E 165Ho T O T A L N E U T R O N C R O S S S E C T I O N AT E N E R G I E S O F 0.4 A N D 1.0 MeV * T. R. FISHER

Department of Physics, Stanford University, Stanford, California 94305 and Lockheed Palo Alto Research Laboratory, Palo Alto, California D. C. HEALEY

Department of Physics, Stanford Unit'ersity, Stanford, CaliJornia 94305 and J. S. McCARTHY

High Energy Physics Laboratory, Stanford University, StanJord, CaliJornia 94305 Received 24 February 1969 Abstract: Measurements of the spin-spin effect, a,~ in the total cross section for polarized neutrons on polarized t 6 5 H o are reported at neutron energies of 0.4 and 1.0 MeV. The absence of an observable effect at 1.0 MeV is in contradiction with a recently reported experimental result. The results of an improved theoretical calculation of a , for neutron energies below 8 MeV are presented and used with the new experimental data to obtain the new limit [V,] < 300 keV for the strength of the spin-spin l~otential. J E

NUCLEAR REACTIONS polarized ~6SHo(polarized n, n), E = 0.41, 0.98 MeV; measured a°T(E). Deduced spin-correlation effect tr~ and strength of spin-spin potential V~,.

1. Introduction The application of polarized targets to the study of direct interactions makes it possible to search for correlation effects which depend on the relative orientation of the spin of the projectile a n d the spin of the heavy nucleus. Several recent attempts t - 3) have been made to observe a change in the total n e u t r o n cross section of 16Silo resulting from the reversal of the relative orientation of the n e u t r o n a n d t65Ho spins. If a¢,* and a~,: denote the measured total cross sections for the parallel a n d antiparallel spin orientations, the spin correlation effect or spin-spin effect ass m a y be defined as tt ass = ( a ~ ¢ - a ~ ) / 2 P , P,, where P , a n d Pt are the vector polarizations of the n e u t r o n a n d nuclear spin systems. The simplest p h e n o m e n o l o g i c a i potential which could give rise to such a spin-spin effect is of the form 4) _ V~f(r) [I" a/1], where ! is the nuclear spin a n d ~ the n e u t r o n t Work supported in part by the National Science Foundation, the U.S. Office of Naval Research (Nonr 225(67)) and Lockheed Independent Research Funds. HEPL-594. ** Slightly different notation is used in refs. 1-3). 609

610

T.R. FISHERet al.

spin. For convenience, it is customary to assume a radial dependence, f ( r ) , which is identical to that of the real part of the optical potential. If the other optical parameters are known, a measurement of a~, can be related to the strength of the spin-spin potential * V , . Wagner et al. 2) and Fisher et al. 2) have reported measurements of ~s~ at 0.35 and 8 MeV. Both measurements are consistent with zero. The implied limit on the magnitude of Vss was 500 keV, a limit consistent with the prediction (Vs~ ~ 60keV) of a simple microscopic model 2). More recently, Kobayashi et al. 3) reported a large non-zero value for a~ at a neutron energy of 0.92 MeV. The results of a DWBA calculation, which assumed a spherical distorting potential, were also presented, and it was shown that the experimental data on ~ could be consistently explained by the calculation if V~ ~ 2.5 MeV. The large phenomenological value for V~ favored by Kobayashi et al. 3) as the explanation for their data would create some problems. It differs by almost two orders of magnitude from the prediction of the simple model of ref. 2) and raises again the question of why effects due to V~s cannot be seen in experiments which do not employ polarized targets 5). The clarification of this situation formed part of the motivation for the present work, but the extension of previous work at 8 MeV to lower neutron energies was a natural undertaking in any case. We have therefore measured as~ at neutron energies 0.4 and 1.0 MeV using essentially the same experimental arrangement as was employed previously 2) at 8 MeV. The detection of a spin-spin effect an order of magnitude smaller than that actually reported by Kobayashi et al. 3) should have been possible. The results of new calculations of ass for neutron energies less than 8 MeV are also presented. The calculations employ optical parameters which give agreement with recent data 6,7) on the deformation effect ,t Atrdef in the 165Ho total cross section.

2. Experimental arrangement

2.1. POLARIZED leSHo TARGET A diagram of the experimental arrangement is shown in fig. 1. The polarized ~6Silo target has been described in detail 2), and only a brief summary of pertinent information will be provided here. The cylinder of polycrystailine holmium metal 1.6 cm diam. x 7.4 cm long was oriented with its axis parallel to the neutron beam direction and magnetized to 71 ~ of saturation in the 18 kOe field of a superconducting solenoid. The Ho metal was thermally anchored to a bath of liquid aHe and cooled to 0.30 ° K. This produced a vector polarization of 0.58_+0.03 parallel to the direction of the neutron beam. The set of population densities of magnetic substates which t Different normalizations for V. are used in refs. t. 3). *t The deformation effect is usually defined as Atraa = erotic,ten--O'unor|cnted,where tro,,,°t,d is the total cross section when nuclear orientation is present and t~unotlcat¢a the total cross section when no nuclear orientation is present.

16Silo TOTAL NEUTRON CROSS SECTION

611

TAnLE 1 Population densities PM for the nuclear orientations o f the present experiment and o f refs. 1. a)

Present experiment (Pt = 0.58) Ref. 1) (Pt = 0.15) ~) Ref. 3) (Pt = 0.29) u)

0.386

0.239

0.149

0.091

0.056

0.037

0.026

0.016

0.216

0.147

0.120

0.109

0.104

0.102

0.101

0.101

0.212

0.188

0.155

0.135

0.115

0.095

0.062

0.038

a) z-axis perpendicular to neutron beam direction. The population densities were calculated assuming 8 1 % o f the nuclei were randomly oriented and 19 ~o had the orientation correspondingly to a magnetically saturated single crystal at 0.34 °K. This is not a completely accurate representation since the 8 1 % ensemble had a net alignment, although it had no net polarization. For purposes o f the present calculation this is unimportant, but the set o f population densities given here could not be used in the calculation o f any quantity (such as Zhrda) which depends on the even m o m e n t s of the nuclear orientation. b) z-axis perpendicular to neutron beam direction. The population densities were calculated by assuming a polarization o f 0.29 with all higher-order m o m e n t s (B2, B3 etc.) equal to zero. Because a polycrystal was used and the temperature was above 1 °K, this latter assumption is a reasonable one.

.

.

~

.

.

.

.

.

.

.

.

.

/i

/

\

Fig. I. Experimental arrangement. 1 - LiF target, 2 - paraffin moderator, 3 - BFs proportional counter, 4- iron magnet, 5 - paraffin shielding, 6 - outline o f cryostat, 7 - superconducting solenoid, 8 - h o l m i u m sample, 9 - paraffin collimator, 10- magnetic shielding, 11 - stilbene crystal, 12 - lucite cone, 13 - p h o t o multiplier. The magnet pole pieces are 37 cm long. The figure is drawn approximately to scale.

612

T . R . FISHER e t al.

describes completely the nuclear orientation is given in table 1. A discussion of how these are obtained from a knowledge of the temperature and magnetization of the Ho metal has already been given 2). 2.2. POLARIZED NEUTRON SOURCE The reaction 7Li(p, n)TBe was used as the source of polarized neutrons, the neutron beam making an angle of 45 ° with the incident proton beam. Targets of LiF were prepared by evaporation of the salt onto tantalum blanks. The targets, when gyrated and cooled by blowing a stream of air across the back surface, were stable under the bombardment of a 5/~A proton beam for up to 100 h. The neutron beam passed between the pole faces of an iron magnet and emerged with its spin parallel or antiparallel to its direction of flight depending on the direction of the magnetic field. This spin-precessing magnet and the associated collimator have been described 2). The neutron polarization in the reaction 7Li(p, n)TBe has been studied extensively 8,9), but measurements at the exact angle and energy employed in the present experiment are not available. For this reason and to check for depolarizing effects due to the neutron collimator and method of spin precession, the neutron polarization was measured directly under conditions closely simulating those of the polarized target measurements. The 1 6 5 H o target was replaced by a suitable analysing scatterer, and neutrons scattered at 90 ° were detected. The spin-precessing magnet rotated the neutron spins by 0 ~, 180 ° or 360 °, and the corresponding changes in the scattered intensity were observed. At 0.4 MeV, 160 in the compound S n O 2 w a s used as the analysing scatterer and the measured neutron polarization was 0.38+_0.07; at 1.0 MeV, 9Be served as analyser and a neutron polarization of 0.25+_0.08 was measured. The analysing powers of beryllium and oxygen were taken from the data of Elwyn et al. 9). At 1.0 MeV, the small variation of neutron polarization with energy and angle makes possible a reliable interpolation between existing data a, 9). The neutron polarization thus obtained was 0.28+0.04 in good agreement with our measured value. 2.3. NEUTRON DETECTION SYSTEM The neutron flux monitor was a paraffin-moderated BF 3 proportional counter. A stilbene fast neutron spectrometer employed previously 6) in measurements of the deformation effect Ao'dc f w a s used to detect the transmitted neutrons. The 1.9 cm diam. x 2.5 cm long stilbene crystal was polished and mounted in a well inside a polished lucite cone to improve the efficiency for light collection. The lucite cone was optically coupled to the face of a 6810 photomultiplier tube using Dow Corning 200 silicon grease and was surrounded by a magnesium oxide reflector. The entire assembly was wrapped with several layers of magnetic shielding, the number of layers being increased until no observable shift in the neutron spectrum resulted from turning on the 18 kOe field of the superconducting solenoid. Gamma-ray discrimination was provided by a type NE 5553 pre-amplifier furnished by Nuclear Enterprises Ltd.

613

105Ho TOTAL NEUTRON CROSS SECTION

When half the neutron spectrum was included in the analysis, the gamma-ray rejection ratio was 400 to 1 at 1.0 MeV and 30 to 1 at 0.4 MeV. Without discrimination, the number of counts from g a m m a rays was considerably smaller than that from neutrons, therefore the discrimination was quite adequate even at 0.4 MeV. Since the spectrometer resolution was capable of resolving the neutron group corresponding to the first excited state of 7Be from the ground state group, it was possible to avoid a time-of-flight detection system. The quality of the neutron spectra is indicated in fig. 2, which shows typical spectra at 0.4 and 1.0 MeV taken during the actual polarized target measurements. Each spectrum represents about 30 min of running time.

0.41 MeV

400

,,,.I t=.l Z Z

eeeeeeeeoe•°••••••

200

o

...... o........

••

a. (,n 1-

• •o•

400 0

•

• •

•

098

.......•.

MeV

"'........:.

200

• ••

I 50

I(30

I 150

CHANNEL NUMBER Fig. 2. Typical neutron spectra from the reaction 7Li(p, n)TBe taken with the stilbene fast ncutron spectrometer during the polarized target measurements. The cut-off below channel 50 is due to the gamma-ray discrimination coincidence requirement. The curves were obtained from a c o m p u t e r search as described in the text. The suggestion of a b u m p at channel 50 in the lower curve may be due to the excited state neutron group, which is weakly excited at this energy and angle.

It was anticipated that the stilbene spectrometer might be more susceptible to gain shifts than the Si(Li) detector employed previously at 8 MeV, and a consistent procedure for correcting such shifts was developed. The curves shown in fig. 2 were generated by the function

N, = A(l +iB) (l +exp [~-°]) -',

(1)

T.R. FISHERet al.

614

where Nt is the number of counts in the ith channel. The best values of the parameters A, B, C and i o were determined by a non-linear last-squares search 10) utilizing the Stanford PDP-9 computer, and it was intended to correct for gain shifts based on changes in the parameter i0. Such corrections actually proved unnecessary because the spectrometer exhibited a stability of 0.5 ~o or better over periods of several hours, and because data-recording procedures were chosen to minimize the effects of gain shifts on the final results.

3. Experimental procedure and results Data readouts were taken at approximately 15 min intervals, and the field of the spin-precessing magnet was reversed at the end of each interval. The field of the superconducting solenoid was reversed at longer intervals of several hours. This procedure is similar to that followed in the previous measurement of tr,~ at 8 MeV. The instrumental asymmetry was measured by repeating the procedure with the 16SHo sample at 4.2 °K (nuclear polarization 0.06). The measured instrumental asymmetry was consistent with zero as expected, but the results were statistically combined with the results at 0.3 °K to obtain the final statistical limits on as,. The data at each neutron energy consisted of roughly 100 individual runs, and appropriate statistical tests for internal consistency were imposed on the data sets. The experimental results are summarized in table 2. A small background correction (the background amounted to 6 ~o of the transmitted neutron flux) is included in the final results; inscattering corrections were negligible in the present geometry. TABLE 2 Summary of experimental zesults

t~. (MeV)

AE.

P.

½(o÷÷--a+÷)

(keV) (rob)

Instrumental asymmetry a) (mb)

0.41

40

0.38 +0.07

--0.5 ±2.5

-- 1.0!3.5

0.98

40

0.28±0.04

1.5-t-3.0

--2.0i4.0

~TuP~ b) (mb)

1.5 ± 12 14 -t-20

i) The instrumental asymmetry is the quantity ½(a÷4,--a++) measured at 4.2 °K. b) Obtained by statistically combining columns 4 and 5, dividing by P. and multiplying by 1.I 1 to take into account the 6 ~ nuclear polarization still present in the instrumental symmetry measurement.

The experimental results for a , are consistent with zero at both neutron energies. There is little possibility that such a negative result was obtained erroneously. That a large nuclear polarization was indeed present in the ~65H0 target could be directly verified by the observed change in the transmitted neutron intensity when the target was cooled to 0.3 °K [deformation effect 6, 7)]. The polarization of the neutron beam

165Ho TOTAL NEUTRON CROSS SECTION

615

and the effects of the spin-precessing magnet were independently verified in the scattering measurements on 9Be and 160. Since the 165H0 sample was not magnetically saturated, the neutron beam experienced a slight depolarization in its passage through the sample. The net depolarization was estimated 2) to be less than 1 ~ for 8 MeV neutrons. This limit should probably be increased to 3 ~ because of the lower neutron energy in the present experiment (part of the effect is offset by the shorter sample length) but remains too small to be significant in the final results. 4. Theoretical calculations In a previous paper 2), a calculation of tr,s as a function of neutron energy over the energy range from 0.3 to 15 MeV was given. However, the optical parameters used in that calculation do not give too good a fit to recent data 6, 7) on the deformation effect Atrdcf for neutron energies less than 8 MeV. We describe new calculations of as, using optical parameters which give a satisfactory fit to all available data on the 165I-{0 cross section (refs. 1-3)). The calculation was performed in the adiabatic approximation 11,12) utilizing the computer code of Barrett 13). The form assumed for the main optical potential describing the interaction of the neutron with the deformed 165H0 nucleus was

V(R, 0') = - Uf[r, R(O'), a] + iW [4~ Of (r, R(O'), a)l fir, R(O'), a]

= {exp [ ( r -

R(O'))/a] + 1}-',

R(0') = ro A [1 +#Y2o(0')], /~(0') = P0 a~[ 1 +/~Y2o(0')] •

(2)

Consensus values 1,, 1s) were adopted for all parameters except U and W r 2 = 1.25fm,

a =0.65fm,

= 1.25 fm,

~ = 0.47 fm,

(3)

and the value 0.3 was chosen for the nuclear deformation parameter 16) fl in the expressions for R and ,~. The values assumed for U and W (in MeV) were U = 47--0.25 E,

W = 6.5.

(4)

At 8 and 15 MeV, these values of U and W do not differ significantly fromthose used in ref. 2), therefore the fits to the differential cross-section measurements presented there remain unaltered. The fit to Aadcr and a t is shown in fig. 3, where the data have been taken from refs. 2.8). [In the region of overlap above 2 MeV, these data are reasonably consistent with those of ref. 9).] The choice (4) of U and W improves considerably the fit to Aad,t (see ref. 8), fig. 1). The fit to the 0.35 MeV differential cross-section

616

T.R. FISHER et al.

measurement of Wagner et al. t) is shown in fig. 4, where the calculation of Davies and Satchler 17) has been used to estimate the compound elastic contribution. The E

(MeV)

0.5,'5

[0

2.0

4.0

8.0

1,5.0

I

I

I

I

I

I

5

200

I00

E

0

<3 -I00

-200

0

r

I

I

0.1

0.2

0.3

I

I

0.4 0.,5 K ( f r o -1)

I

I

I

0.6

0.7

0.8

Fig. 3. Fits to the total cross section at and the deformation effect ZJedct in the ~6SHo cross section. The data are from refs. 2.6), and the curves were calculated using the optical potential given in sect. 4 o f the text. The nuclear orientation is approximately described by the population densities in table I with B2/B2(max) = +0.25.

fits to all data are satisfactory, and we conclude that the optical potential described by (2)-(4) is a reasonable one to use in our calculation of a~s. To calculate the spin-spin cross section, we assume the usual form for the spin-spin

165Ho TOTAL NEUTRON CROSS SECTION

617

term in the optical potential

- V~,f(r, r o A ~, a)a" I/I.

(5)

If the possibility of angular momentum transfer by the 1 • a potential is neglected [otherwise the adiabatic approximation cannot be applied 2)], the spin-spin effect is the change in cross section which results from adding - V~sf(r, ro A¢, a) to the real part of the optical potential

6ssPt = ~. ( P , , - P - u ) M>o

M

i-

(a..)M = ½[a.( U + V..)- a~a(U - V..)].

(6)

The PM are the population densities of the nuclear magnetic substates, P, the magnitude of the nuclear polarization, a~ the total cross section for a nucleus in substate

En

= 0.35 MeV

1200

] 1 I

I000 800 b

600

400 [ 200 i

-

20

40

60

80 I00 120 140 160 ec.M.(deg)

180

Fig. 4. Fit to the angular distribution of unpolarized neutrons elastically scattered from unpolarized t65Ho at 0.35 MeV. The experimental points are from ref. t), and the estimate of c o m p o u n d elastic scattering shown by the dashed curve from ref. ,7). The Z 2 for the fit is approximately 12 for 12 data points.

M and a u ( U + V~) the calculated value of this cross section when - V~,f(r, r o A ~, a) is added to the real optical potential. Eq. (6) is identical to eq. (14) of ref. 2) and is limited to the case where the orientation of both spin systems is symmetric with respect to the beam direction. If both spin systems are symmetric with respect to an axis

T . R . FISHER e t aL

618

which makes an angle 0 with the direction of the neutron beam, the appropriate modification of (6) is

a,~ P, =

~

( _ ) u - M'(M,/I)(PM,- P - M,)(IMI -- MI KO)

M'>O,M,K

x (IM'I - M'I KO)PK(COS O)(a~,)M, (7) where the unprimed reference frame has its z-axis along the beam direction and the primed reference frame in the direction of the symmetry axis for the oriented spin systems. E (MeV} 0.35

0.5

1.0

2.0

4.0

8.0

I

I

I

i

i

i

Vss = 0.5 MeV,

,8 = 0.3

I00

////

%~%%

A aEl

//

%% %%%

/ /

V

tM =

-I00

%

- -

ORIENTATION PARALLEL TO BEAM 1 ORIENTATION PERPENDICULARI TO BEAM /

¸-

..... I

I

o.i

02

I

I

0.5 0.4 K (fm-ll

1

0.5

I

0.6

I

Fig. 5. Calculation of or,, for complete orientation parallel and perpendicular to the neutron beam direction. The optical parameters and method of calculation are described in sect. 4 of the text. The curves may be scaled linearly with the magnitude of V,,.

The manner in which the nuclear deformation enters the calculation is important. For a spherical nucleus, the quantities (a,,)~t are all equal, but they can be quite different for a nucleus as strongly deformed as 165Ho. As a result, the observed change in cross section will depend on the angle between the direction of the neutron beam and the direction of Pt. In fig. 5, the magnitude of this dependence is illustrated. The curves for orientation parallel and perpendicular to the neutron beam direction would coincide for ~ = 0 but are completely out of phase for/~ = 0.3. It is for this reason that a calculation which neglects nuclear deformation such as the DWBA calculation of Kobayashi et al. 3) cannot be trusted to give meaningful results for 16~Ho" The optical potential described by (2)-(4) has been used to calculate values of

619

t65Ho TOTAL NEUTRON CROSS SECTION

( a , , ) u o v e r the e n e r g y r a n g e 0 . 3 - 8 M e V . T h e s e results h a v e b e e n c o m b i n e d w i t h the o r i e n t a t i o n p a r a m e t e r s listed in t a b l e 1 to g e n e r a t e c u r v e s o f tr~,P t for the n u c l e a r E (MeV) 0.35 1

I.O I

2.0 I

4.0 6.0 8.0 I l I Vss = 4.0 MeV ORNL ORIENTATION

IOO E ,50

-I00 Vss = 0,4

MeV

25 E b= ~

0

-25 • I0 MeV

Vss TOKYO

200 I

, o~

"100 I-

:ooI I O.I

<

I 0.2

I I 0,3 0.4 K ( f r n -~ )

I 0..'5

N -

I 0.6

Fig. 6. Calculation of tr,~Pt for the nuclear orientations given in table 1. The first and third curves are similar, since for both the nuclear polarization is perpendicular to the neutron beam. The curves may be scaled linearly with the magnitude of V,. The experimental point on the first curve is from ref. '); the points on the second curve are from ref. 2) and the present experiment, and the point on the third curve is from ref. 3). o r i e n t a t i o n s o f refs. 1-3). T h e c u r v e s a r e d i s p l a y e d in fig. 6, a n d scale l i n e a r l y w i t h V~s as l o n g as t h e c o n d i t i o n

VsJU <<

1 is fulfilled.

T h e three s i m p l i f y i n g a p p r o x i m a t i o n s w h i c h h a v e b e e n i n t r o d u c e d i n t o the p r e s e n t

620

T.R. FISHERet al.

calculation are the omission of a spin-orbit term from the main optical potential, the neglect of angular momentum transfer by the I" a potential and the use of the adiabatic approximation. While it cannot be rigorously shown that these approximation ~. do not significantly affect the calculations, certain plausibility arguments can be given to support this contention. In the appendix, it is shown that in the limit of zero deformation, the present treatment is equivalent to the DWBA approach 17) without spin-orbit coupling. This somewhat accidental feature depends on the extremely simple choice made for the form of the I • ~r interaction, but it is significant since the DWBA treatment should give accurate results when fl = 0 and is affected only slightly by the inclusion of spin-orbit coupling 17). (A plausible explanation for this fact is also discussed in the appendix.) The present method may then be regarded as the simplest consistent technique for including the effects of nuclear deformation in the problem. Although the adiabatic approximation itself might be questioned for neutron energies as low as 0.3 MeV, the fits obtained to the data in this energy region suggest that it is probably all right to use it in the calculation of total or elastic cross sections. Finally, the only real alternative to the present approach lies in an exact coupling calculation 18) in which the dynamical effects of all angular momentum couplings are included. Such a calculation is feasible but would be extremely time-consuming at the higher neutron energies. 5. Conclusions

An examination of fig. 6 reveals that the result of Wagner et al. l ) is consistent with EVssl < 4 MeV, whereas the results of the present experiment require I Vs~l < 300 keV if the 0.4 MeV point is to lie within one standard deviation of the calculated curve. This limit is a slight improvement over that quoted in ref. 2)([ Vs~l < 500 keV) but is still considerably greater than what would be expected if V~s arises from the interaction of the incident neutron with a single extra-core proton (V~ ~ 60 keV). On the other hand, the result of Kobayashi et al. 3) falls more than one standard deviation from the calculated curve unless V~ > 10 MeV. In view of the other experimental evidence, it seems most probable that this departure from zero should be attributed to counting statistics. As has been pointed out 2), the proper statistical combination of the instrumental asymmetry measurement with the polarized target measurement brings the point of Kobayashi et al. 3) to within one standard deviation of zero. Although original interest in the spin-spin effect centered around the possibility 4) that a term similar to (5) might be important in ordinary optical-model calculations, the magnitude of Vs~ is too small for this to be the case. However, the possibility of studying directly the spin dependence in the interaction of the incident neutron with one or more of the outermost nucleons of the 165Ho nucleus remains very interesting. In this picture, a potential of the form of (5) is merely a convenient way of simulating the grossest features which would arise from such a spin dependence.

165Ho

TOTAL

NEUTRON

CROSS SECTION

621

The effective change in refractive index for the neutron wave modifies the " R a m s a u e r Effect" pattern 19, 20) in the total cross section, giving rise to curves of the type shown in fig. 6. The application of the formalism of Madsen 2 ~) to the problem would enable the interaction of the neutron with the extra core nucleon to be treated explicitly in the D W B A approximation and would be a considerable improvement over the phenomenological potential approach. Unfortunately, the formalism cannot be applied to t65Ho in a straightforward manner because of the large nuclear deformation. Further experimental and theoretical investigations related to the present work are in progress. The authors thank R. Barrett for the use of his computer code. Helpful discussions with W. E. Meyerhof, S. S. Hanna, and J. D. Walecka are gratefully acknowledged. The authors acknowledge their continued indebtedness to R. S. Safrata for development of the polarized 165Ho target. Appendix

DWBA EXPRESSIONS FOR THE SPIN-SPIN CROSS SECTIONS IN THE ABSENCE OF SPIN-ORBIT COUPLING Following the notation of Davies and Satchler 17), we write the scattering amplitude in the form Mfof.~iO'l

_- - J¢(o) x ~¢(t) 6'fOl VMfMi I J M f o - f M i o .

1

(A.1)

where Ma, Mr, ai and af represent the initial and final projections of the nuclear spin I and the neutron spin a. The amplitude.f (°) is independent of M, and it is assumed t h a t f ( t ) <
dO

Tr

4n

[ptp"f +(O)f(O)],

a t = -- lm Tr k

[ptp"f(O°)],

(A.2)

where p' and p" are the density matrices for the target and incident neutron and f+ the Hermitian conjugate o f f . We have assumed k e = k i = k. If the polarization vectors describing the neutron and target ensembles are parallel, the spin-spin cross sections may be written

I ~ (O)], P,,Pt = 2 Re {do(O)Tr[ptpnf(t)'(O)]}, a~P, Pt

= 4_n lm Tr [p'p"f(t)(0°)], k

(A.3)

where for the sake of completeness, we have extended our definition of ass to the differential cross section as well. The quantity [(da/df2)(O)]~,P,Pt is identical to

622

T . R . FISHER et al.

[dtr/dt2]t . as defined in eq. (3.9 d) of Davies and Satchler 17) when P,× P, = 0. The quantity d o is the amplitude fro) in the case of no spin flip, i.e. d o = f , to). The inclusion of spin-orbit coupling in the calculation can affect a , only through its effect on the amplitudef ct), which is contained in the terms A~q, Akk, A,, and (Akq+ A,~k) as defined by Davies and Satchler a7). These authors showed by actual calculation that the preceding terms were small, but an examination of the DWBA amplitude ft~) reveals a plausible explanation for this fact. In the simple case of a spherical I • a interaction of the form given by (5) of the main text, the expressions in appendix I of Davies and Satchler ~7) may be combined to give fM,I,

,.,M,.,

= (_)t+M~+,f+½ V~, I2(I + 1)(2I + 1)1 ½

E

3I

E~ (2J + 1)2

x ( I M i I - Mr[ 1/0(Ja fJ - af[l - la)(Jai ½- tr~[LO)(Ja t ½- trflL0) x W(JJ42~; IL)PL(COS O)gu, gLS =

z ~ ( k r ) f ( r , ro A ~, a)dr.

(A.4)

Setting the spin-orbit potential equal to zero removes the J-dependence in the radial distorted waves ZLJ and enables a summation to be performed over J f ~,i,r - , o, = (_),+M~+~f+~ VS~E[2(1+ 1)(2131+ 1)1' x (IMi I - Mfl lu)(½trl ½- afl l - / 0 ~ (2L + 1)Pt.(cos 0)gL, L

gL =

fo'

(A.5)

x[(kr)f(r, ro .4 ~, a)dr.

The neglect of spin-orbit coupling then corresponds to setting g L. L+÷ = gL. L-½ = gZ' Although the radial distorted waves g ~ will differ in phase when spin-orbit coupling is present so that the point-to-point correspondence between XL.L+¢, XL.L-t and Xt. may be poor, we might expect the condition 9L. L+¢ ~" 9L. L-¢ "~ 9L to be fairly well satisfied iff(r, r o A~, a) remains approximately constant over the region of integration. Even if gL.L+~ and 9L, L-~ differ appreciably from 9L, there will be a tendency toward cancellation of this difference when they are summed. Combining eqs. (A.5) and (A.3) leads to the following simple expressions for the spin-spin cross sections:

Ida (0)] . _ 2VE s Re {do(0 ) ZL (2L+ I)PL(cos O)g*}, 47z

a,, = k E V,s Im { ~ (2L+ 1)gL}. L

(A.6)

1651-to TOTAL NEUTRON CROSS SECTION

623

It will be noted that the expressions in (A.6) are the changes in cross section which would result from a change V,, in the real part of the optical potential and are thus equivalent to eq. (6) of the main text in the case fl = 0 (up to terms of second order in V~s/U). The expression for trs~is identical with eq. (AII-13) of Kobayashi et al. 3). References 1) R. Wagner, P. D. Miller, T. Tamura and H. Marshak, Phys. Rev. 139 (1965) B29 2) T. R. Fisher, R. S. Safrata, E. G. Shelley, J. McCarthy, S. M. Austin and R. C. Barrett, Phys. Rev. 157 (1967) 1149 3) S. Kobayashi, H. Kamitsubo, K. Katori, A. Uchida, M. lmaizumi, K. Nagamine and A. Mikuni, J. Phys. Soc. Japan 22 (1967) 368 4) H. Freshbach, in Nuclear spectroscopy, Part B, ed. by F. Ajzenberg-Selove (Academic Press, New York, 1960) p. 1046 5) L. Rosen, J. E. Brolley, Jr. and L. Stewart, Phys. Rev. 121 (1961) 1423 6) J. S. McCarthy, T. R. Fisher, E. G. Shelley, R. S. Safrata and D. Healey, Phys. Rev. Lett. 20 (1968) 502 7) H. Marshak, A. Langsford, C. Y. Wong and T. Tamura, Phys. Rev. Lett. 20 (1968) 554 8) S. M. Austin, S. E. Darden, A. Okazaki and Z. Wilhelmi, Nucl. Phys. 22 (1961) 451 9) A. J. Elwyn and R. O. Lane, Nucl. Phys. 31 (1962) 78 10) T. R. Fisher, Proc. Fall Decus Symposium, Anaheim, Calif. (1967) 11) D. M. Chase, L. Wilets and A. R. Edmonds, Phys. Rev. 110 (1958) 1080 12) S. J. Drozdov, Z h E T F (USSR) 36 (1959) 1875; [JETP (Sov. Phys.) 9 (1959) 1335] 13) R. C. Barett, Nucl. Phys. 51 (1964) 27 14) F. Bjorklund and S. Fernbach, Phys. Rev. 109 (1958) 1295 15) F. G. Perry and B. Buck, Nucl. Phys. 32 (1962) 353 16) M. Danos and W. Greiner, Phys. Lett. 8 (1964) 114; E. Ambler, E. G. Fuller and H. Marshak, Phys. Rev. 138 (1965) BlI7; B. Elbek, Determination of nuclear transition probabilities by Coulomb excitation (Ejnar Munksgaard, Copenhagen, 1963) 17) K. T. g. Davies and G. R. Satchler, Nucl. Phys. 53 (1964) 1 18) T. Tamura, Res. Mod. Phys. 37 (1965) 679 19) J. M. Peterson, Phys. Key. 125 (1962) 955 20) K. W. McVoy, Phys. Lett. 17 (1965) 42 21) V. A. Madsen, Nucl. Phys. 80 (1966) 177