Channel analysis of cross sections and angular distributions for fission induced by neutrons and photons

2.1:2.J [

Nuclear Physics AI01 (1967) 460---472; (~) North-Holland Pubh~hln9 Co, Amsterdam Not to be reproduced by photoprmt or microfilm without written permission from the pubhsher

C H A N N E L ANALYSIS OF C R O S S S E C T I O N S AND A N G U L A R D I S T R I B U T I O N S F O R F I S S I O N I N D U C E D BY N E U T R O N S AND P H O T O N S R. V A N D E N B O S C H

Uni~'erstty of Washtnyton, Seattle, Washington t Received 3 A p r d 1967 Abstract: Excitation functions and angular d~stnbut~on data for neutron-reduced fission of '-'34U, z3sU and 2s~Pu and photofission of 23~Pu have been analysed to extract reformation about the fission barrier. The fission b a m e r heights and curvatures deduced from the z39Pu photofisston data predict a s~gnificant contribution o f the (n, ?'f) reaction for low-energy, neutron-induced fission o f ~3~Pu m agreement w~th a recent measurement. The rap~d varmt~on w~th energy o f the fission fragment angular a m s o t r o p y for z3'U reqmres an effective fission b a m e r with a curvature smaller than that gwen by previous estimates. This may be due to a single-particle effect of the odd nucleon. It is also s h o w n that K-state assignments made m p r e w o u s analyses of a m s o t r o p y measurements are not umque.

1. Introduction

The primary motivation for this analysis arose from an observation of an apparent inconsistency between two features of neutron-induced fission, namely the rounded shape of the total fission cross-section excitation function i - 3) in the vicinity of the barrier and the rapid variation of the anisotropy of the fragment distribution with neutron energy 2-4) (see full curves in fig. I). The former observation implies a fission barrier with large enough curvature so that appreciable penetration occurs below the barrier and also appreciable reflection occurs above the barrier. The angular anisotropy however fluctuates quite rapidly with neutron energy, which would seem to imply that new barrier states come in rather suddenly. A further motivation for this analysis came from a desire to explore the uniqueness of previous K-band assignments to saddle-point states and to assess the possibilities for further assignments in other nuclei. A preliminary report covering part of this work has been reported previously 5). These remarks are primarily concerned with odd-mass-number, compound nuclei. The compound nucleus may de-excite by emission of a neutron or a photon or may fission through one of the available nuclear states at the saddle point. The saddlepoint states of the highly distorted, odd-mass-number nucleus can be identified as the Nllsson states of a deformed potential. The first detailed analysis of this type of angular distribution data in terms of ,~addle-pomt K-states was performed by Wdets and Chase 6) For purposes of discussing cross sections and angular distributions, * Work supported in part by the Umted States Atomic Energy Comm~sslon 460

~,FUTRON-INDUCED

461

REACTIONS

these states are characterized by the total angular momentum I, the projection of the angular momentum I along the symmetry axis K and the parity n. Each Nilsson state 1 16

i

i

i

234U +rl

-o)

12

×z

% (b)

3d ~4

z

LAMPHERF O8 Zt

x 7/2

~7,-"

04

o

0

!%/

b)

20

WlO*) W tgo*)

2:08 o ~'~. .tOdpN[l~ u r/{ 3,,~

15

IC

--

O5

I

I

I

03O 050 NEUTRON ENERGY

084 (MeV)

Fig 1. T h e full curves are t h e fission excttatton function (a) a n d a n g u l a r anisotroptes (b) reported by L a m p h e r e =,3). T h e ~arlous s y m b o l s are the results o f the calculations described in the text Each s y m b o l c o r r e s p o n d s to a calculatton for a sequence o f three K-bands, two o f which have K -- ~, a n d one o f which has the K-value indicated by the symbol.

20

..

J8 ~ I

I

.... ~

[ ~

......

W(t/Z, 7/2}1

T

--

~

I

-tl

16 I

- - - ~ t

--

L

w,,/z,,~z, t2

.

o~'

I

0

.

.

NORMALIZ ATJON

.

'~Y

.

/_b-~_.-. ~0

.

~

.

l . . . . .

I

I

W(K,/)d(COSS)=t

it-

....

1 .

--, .

.

t

--~"-...~

~_.-.----~S 20

~

30

.

i

.

I .

.

,~':.~..- r$

-----~,''' "

40

.

50

.

.

.

.

.

.

'~'-

~

60

z

--";-

=,

70

7/

~"~'~"~

I

80

q;',~.~', 90

8 tOegl

Fig 2. Theoretical fission fragment a n g u l a r d l s t r l b u t m n s for fission t h r o u g h p a m c u l a r rotational m e m b e r s o f various K-bands [after L a m p h e r e ,.,,3)].

462

R VAN!I)| NBOSCH

with a particular K (or f2 in Nilsson's notation) presumably has built on it a rotational band. The fragment angular distributions for various I, K combinations are illustrated in fig. 2. The most important qualitative dependence of the anisotropy on K can be summarized by the fact that W ( O ' ) / W ( 9 0 0 is 1 0 or larger for K = ~, and zero for K # ½. The dependence of the anisotropy on energy for neutron-induced fission of 234U, illustrated by the full curve in fig. 1, indicates that at the lowest bombarding energy fission is occurring primarily through a K = ~ band, at a 0.5 MeV neutron energy a band with K > ½ is contributing importantly and at 0.84 MeV neutron energy K = ½ bands dominate again Lamphere z, 3) has analysed these data at several incident neutron energies assuming a sequence of K, n band: ~-, ] - , t - . Neutron competition from each I, n state of the compound nucleus was included m this anal:ysis, and the relative contribuuons of the different K-bands were deduced from the anisotropy at each incident neutron energy. No attempt however was made by Lamphere to relate the K-band fission probabilities at one inctdent neutron energy to that at another incident energy. This we have attempted to do by defining the barrier height/:'f and the barrier curvature h~o for each fission channel.

2. Statistical-model calculation

A statistical-model calculation of the average fission cross section and of the fragment angular distributions has been performed using the Hauser-Feshbach formahsm v). The partial fission cross section through a particular K-band is given by a} = ~n~2 ~ 7".(1, E.) 1=0

cjs,(2J+ 1)T~(J, E)

x

~.R~T.(J,E) J =

t

+ ~sJTT.(I,,E,)+T.:(j,E)'

(l)

E'l"

following the notation of Rae etal. 8). The fission barrier was assumed to be parabolic. The transmission through (or reflection above) such a barrier is given by the HillWheeler 9) penetrability expression Tf(E) = 1/(I + exp [2n(E,-E)/fiog]).

(2)

Each K-state at the fission barrier is assumed to have a rotational band built on xt with spacings determined by a rotational energy constant h2/2.9 ° = 4 keV. The calculations are not sensitive to small variations of this constant. The neutron transmission coeffÉcients employed were those of Emmerich ~o) as given by Lamphere 2.3). These optical-model coefficients are very similar to those given by Campbell et al. ~1). Neither of these calculations includes a spin-orbit term. Transmission coefficients from an optical-model calculation including a spin-orbit term 12) differ by less than 1 0 ~ for J = l + s and J = l - s , except for l = 3. Radiative capture has been neglected in the analysis of fast neutron-induced fission, as the radiation width is only

NEUTRON-INDUCLDRLACIIO',S

463

10% or less of the total width 13). The width fluctuation 14) R has been neglected in most of the calculations reported as it has been shown that this correction (varying between 0.5 and 1.0) can usually be compensated for by slight shifts of the fission b a m e r heights. 2 1. TIIE23zU r-n REACTION A n exploratory analysis of the 234U fission cross-section and anisotropy data, a s s u m i n g a ~ ~, 3 - , .~- sequence and equal barrier curvature for all states, showed that the varmtion of cross section with energy implied a value of he) > 400 keV, whereas the a m s o t r o p y data implied a curvature of he) < 250 keV. The c o m p u t e r codc used in the analysis was then generahzed to allow other K-states, inclusmn of a d e c o u p h n g c o n s t a n t for describing the r o t a t m n a l states for K = ~ bands and the possibility of associating different barrier curvatures ~ l t h &fferent K-bands. Various types of calculations were then performed in an a t t e m p t to clarify the sensttwity to the various parameters. Five sets of b a r r i e r states ~ h l c h seemed the most promising for explaining the results ~ere chosen for further e x p l o r a u o n s . These are given on the TABI I" 1 Best set of parameters obtained by s~multaneous fitting of cros~-se,.tlon and antsotropy data for the '2"U(n, f) reaction Set 1 K

Ef he,)

~-

]-

Set II ~-

350 740 780 150 : 12 810

~510 350

½-

Set 111 ~+

650 670 <12 630

]-

~

Set 1V ]-

350 660 680 120 750 30

~ 200 --60

]

Set V ]

680 1260 "__-601470

~-

~'

½

60 750 900 125 19 950

The barrier heights Ef (relative to the neutron binding energy of 5 24 MeV) and the curvatures h.~ are expressed m keV. first line of table I. Calculations using more than three K-states did not give significantly better fits to the data available at the time of the analysis. A few calculations were performed with both K = 1 states having negative parity, but the fits were slightly inferior. The strong p-wave capture suggests that K = z~- states should be especially i m p o r t a n t , as was pointed out by Lamphere 2, 3). This effect is largely cancelled at low energies by the fact that the m:tjor c o n t r i b u t i o n to the d e n o m m a t o r of eq. (1) comes from p-wave n e u t r o n emission. The barrier heights, barrier curvatures and decoupling constants were varied individually with a search r o u t m e t to fit simultaneously both the anisotroples and the total fission cross sections at 0.3, 0.5 a n d 0.84 MeV together with an additional cross sectxon value at 0.15 MeV where no anisotropy i n f o r m a t i o n is available. The absolute values of the decoupling constants were restricted to values of 9 or less. The fits to th ~, * The search routine was obtained from J. P. Chandler, Indiana University, through the Quantum Chemistry Program Exchange.

464

R. VANDENBOSCtl

anisotropy and cross-section measurements are shown in fig. 1. The barrier height and barrier curvature parameters obtained are given in table 1. The calculations turned out to be fairly insensitive to the decoupling constants, and therefore the values of these constants are not believed to be significant and are not tabulated. The total fission cross section turns out to be fiurly sensitive to the optical model chosen. If one uses the Auerbach and Perey ~z) potential, which predicts stronger d-wave capture, the calculated cross sections for band sequences including K--- ]+ and K = ~ + bands are much closer to the experimental results. The parameters in table 1 were also used to c o m p u t e complete angular distributions at 0.5 and 0.84 keV neutron energies. The resulting angular distributions are shown in fig. 3. As fig. 3 2o

W(90°'----)

84

o 3,2.

OI

~

l

0

30

610

b) E , : 0 5

0

O

Z ~ ' o

90

"'"*'""...,

I

J

,

30

60

90

Oc m

(degrees)

Fig. 3. Angular dlstrtbuttons predicted for 2~*Uat (a) 0 84 MeV and (b) 0.5 MeV. The full curve m (a) is the experimental dtstrtbutlon reported by Lamphere ..,3).

illustrates, of the five sets chosen only two give an angular distribution qualitatively different than the others. One must remember that the angular distribution is not simply a function o f the K- and /-values of the avadable barrier states, but it also depends on the spin and p a n t y distribution of the c o m p o u n d states formed at a particular neutron energy. The reason that K -- ~ - , ~+ and ~- bands do not show peaking at intermediate angles can be understood by reference to fig. 2 and recognition o f the fact that only I = K members of the above bands are accessible for cornp o u n d states formed by the lowest incoming angular m o m e n t u m / - w a v e . The K = 23 + and ~ - bands however have their I -- K and I = K + I members populated by the

NEU I RON-I NDUCt D RI ACTIONS

405

same i n c o m i n g partial wave. U n f o r t u n a t e l y no c o m p l e t e a n g u l a r d i s t r i b u t i o n for 0.5 MeV neutron energy a p p e a r s in the literature, and wtthout such a d i s t r l b t m o n , K-assignments o f K # zt states c a n n o t be m a d e with any confidence. In a recznt abstract, Bekhami et al. 15) r e p o r t that at s o m e neutron energies, the a n g u l a r distribution exhibits a peak at an m t e r m e d i a t e angle. This implies the presence o f K = ~+ or a K = 2s - band. It can be seen from table 1 that the o r d e r i n g o f K-states is not w h a t might be expected from the a m s o t r o p y pattern. This reflects the fact that at 0 3 MeV n e u t r o n energy the spins and parities o f the c o m p o u n d states formzd with high p r o b a b d l t y favour fission t h r o u g h a K = ½ state even though the K 4: I- state has a lower b a m e r . 22

THF

~U+n

REACTION

A similar analysis to that described for 234U has been p e r f o r m e d for neutroninduced fission o f 236U. In this case, the variation in a n i s o t r o p y 2.3) is (see lig. 4) less than for z 3 a L , but hke 234U there xs an energy where a K # ~ b a n d is a d o m i m m t 05 04 °"F ( b )

03 02

,_-_-°)

Or

15

i'

' '2~!

05

084

I

I

r 05

b) w(o o) W(90 °)

Jo

05

I

05 0 84 I05 NEUTRON ENERGY (MeV)

Fig. 4. Excitation function (a) and anlsotroples (b) for neutron-induced fission '-', is) of-'36U. c o n t r i b u t o r to the cross section. F o r t u n a t e l y a fairly c o m p l e t e a n g u l a r d i s t r i b u t i o n at this energy a p p e a r s in the literature 16). Initial p a r a m e t e r searches indicated that, except for the set with K --- ~ - , a fourth b a n d at higher energy was required to fit the cross-section a n d a n i s o t r o p y d a t a at 1.05 MeV neutron energy. F o r t u n a t e l y this b a n d does not c o n t r i b u t e significantly at 0.84 MeV, where the lower K =~ 2a b a n d is the d o m i n a n t c o n t r i b u t o r to the cross section. T h e r e f o r e the calculated a n g u l a r distrib u t i o n at 0.84 MeV is not sensitive to the p a r t i c u l a r K-value chosen for the higher

466

R. VANDENBOSCH

band. The parameters o b t a i n e d from a fit to the a n i s o t r o p y and cross-section data at 0.5, 0.84 and 1.05 MeV are given in table 2. The calculated values of the anisotropies a n d cross sections are not indicated in fig. 4, as in all cases they were within 5 % of the experimental values. The complete a n g u l a r distribution predicted at 0.84 MeV, where the lowest K 4: ~ state is d o m i n a n t , is c o m p a r e d in fig. 5 with the a n g u l a r d i s t r i b u t i o n reported by S i m m o n s and Henkel ~6). This c o m p a r i s o n indicates that a + or { - state is involved. En=084

'5

#36U

~,~"

W(90 °) O5

I

0

I

30

60 m

90

(degrees)

Fig. 5. Angular distributions predicted for -°36Uat 0 84 MeV. The full curve ~s the experimental dtstrlbution reported by Simmons and Henkel ~6).

TABLE 2

Best set of parameters obtained by s~multaneous fittmg of cross-secuon and amsotropy data for the 'a~'U(n, f) rcactmn Set I K E he,

~'

~

Set 11 ½-

900 1 120 1260 530 480 780

2~-

~

Set III ½-

980 1080 I 400 580 370 970

~.-

~+

Set IV ½-

790 1 I00 I 120 480 690 500

~-

,~-

Set V ~

870 1 100 1 120 590 390 740

~-

~-

J~+

750 I 100 I 150 490 550 740

"1he barrier heights El (relatwe to the neutron binding energy ol 5.27 McV) and the curvatures hu~ arc expressed tn keV. 2.3. I HI. "~9Pu(7, f), '-'asPu(n,f) AND THE 2aspu(n, 7f) REACTIONS A very precise m e a s u r e m e n t of the a n i s o t r o p y for p h o t o - i n d u c e d fission of 239pu has recently been reported 17) a n d is illustrated in fig. 6. The positive anisotropy at the lowest excitation energy indicates that f s s i o n through a K = ½ b a n d is pred o m i n a t i n g . At slightly higher energies, a K # ½ b a n d must be c o n t r i b u t i n g strongly, b u t as the excitation energy increases further the relatwe c o n t r i b u t i o n from fission with K = ~ increases. If we assume that electric d~pole a b s o r p t i o n p r e d o m i n a t e s in the p h o t o - a b s o r p t i o n process, fission through ½- and ] - states is expected. T h e observation 18) that the thermal and low-energy, n e u t r o n - i n d u c e d fission cross sections for 238pu are very small requires that the ~ + fission channel be largely closed,

"%FL T R O N - I N D U C E D Rt A C 1 l O N g

467

s u p p o r t i n g the n e g a t i v e - p a r i t y assignments o f the principal c o n t r i b u t i n g states in the photofission o f 239pu T h e most o b v i o u s i n t e r p r e t a t i o n o f the a n i s o t r o p y pattern presented in fig. 6 would be to assume a sequence o f saddle states starting with a K = ~- b a n d , followed by a K = ~ - b a n d and a second K = ½- band. H o w e v e r we have f o u n d that such a sequence is inconsistent with the observed tission cross section o f 238pu by 0.15 MeV neutrons. If two K = ~.- a n d one K = 3 - bands ate open, the p r e d i c t e d total tission cross section is larger than the o b s e r v e d valu e t,)). Also, if a second K = ~- b a n d was o p e n i n g up one would expect the a n l s o t r o p y to increase significantly a b o v e unity at higher excitation energies, c o n t r a r y to observ a t i o n XT). W e therefore p r o p o s e that the origin o f the photofission a n N o t r o p y p a t t e r n arises from only two K-band~. T h e K = 32- b a n d w o u l d be lowest, but a i

i

i Exp~tlmenl Colcu olea

II

W(9001 W(OO)

~0

09

OB

.52 MEAN

54 EXCITATION

56 ENERGY

Fig 6. The circles represent the experimental fragment anlsotroples (presented as W(90~)/W(0 ~) rather than W(O-)/W(90°) since dipole absorption results in angular momentum along the beam axis rather than perpendicular to the beam ax~s as in neutron-reduced reactions) for photofisston of :J9pu. ]'he trlangle~ represent the calculated fit obtained as described in the text. K = ½- b a r r i e r with larger c u r v a t u r e w o u l d lie slightly higher in energy. A t the lowest p h o t o n energy, fission would occur p r e d o m i n a n t l y by p e n e t r a t i o n o f the m o r e easily p e n e t r a b l e K = ½- barrier, even t h o u g h it were higher than the b r o a d e r K = ~ - barrier. A t higher excitation encrgies, the K = 3 b a r r i e r o f low c u r v a t u r e would open up rather suddenly, allowing fission t h r o u g h the K = _~-b a n d to reduce the a n i s o t r o p y below unity. A t still higher energies the K = ½- b a n d w o u l d be fully open, and an a n i s o t r o p y close to u m t y would result. The suggested e x p l a n a t i o n has been tested by a c a l c u l a t i o n similar to that described a b o v e for n e u t r o n - i n d u c e d fission. F o r m o s t o f the excitation energies o f interest the neutron exit channels are closed, a n d the o n l y c o m p e t i t i o n with fission

468

R. VANDLNBOSCH

is g a m m a emission. The radiation width is known is) at the neutron binding energy, and was assumed to vary with excitauon energy according to the empirical relationship 20) 1"~ ac [D(E)]" E 4 3. As the total photo-absorption cross section is u n k n o w n in this energy region, only anisotropy data have been used in the search to lind the barrier energies and curvatures. A satisfactory tit to the energy dependence of the a m s o t r o p y was obtained with a K = gi- state at 5.35 MeV o f excitation energy (0.27 MeV below the neutron binding energy) and a K = ½- band at 5.5 MeV o f excltauon energy (0.12 MeV below the neutron binding energy). The corresponding values ofho~ are 235 keV and 595 keV. These barriers are illustrated in fig. 7. The barrmr hmghts and curvatures are rather well detined as a consequence of the tight

keV / ,;hw=lO00 K: -~- + ~ ~ : 2 0 0

ke~

Bn=562

5 )¢r hi Z ~3

\"

04

l~w=245 keV~w=SZ95 kev

\

FISSION.-~

Fig 7. Schematic of the °-agPufission barrier heights and curvatures as determined from the analysis of the photofission amsotropies and slow neutron fission resonance widths. The excttatmn energy corresponding to the neutron binding energy is indicated at the left responding to the neutron binding energy is mdmated at the left of the diagram.

constraints imposed by the limitation of two K-bands. The c o m p u t e d anisotropies are indicated in fig. 6. In a recent letter, B o w m a n et al. ~s) have reported evidence for the (n, 7f) reaction in slow neutron capture by 238pu. The fission widths for ten resonances have been measured and have been found to fluctuate less strongly than would be expected for the (n, f) reaction. For the (n, f) reaction, only a few exit channels would be expected to be open, and a wide fluctuation in fission widths would be expected. If fission is actually occurring subsequent to g a m m a emission, the width fluctuations would be greatly d a m p e d due to the large n u m b e r of secondary, compound-nucleus states reached directly by g a m m a radiation from the initial state. As Bowman et al. l s)

NFU FRON-INDUCFD

REACIIONS

469

have pointed out, the 239pu compound nucleus is a good candidate for observation of the (n, 7f) reaction, since sizable photofission cross sections have been observed below the neutron binding energy 17). Furthermore, the K -- J2+ fission saddle state is largely closed at this energy. If one assumes t that the smallest fission width observed by Bowman et al. (F r = 1.2 meV)corresponds to the average value of the nonfluctuating (ll,),f) process, then the average width for the (n, f) process is estimated to be 4 7 5 - 1 . 2 = 3.55 meV. From the observed spacing and the relationship 2 n ( F t / D ) = Tf, a penetrability of 0 002 for the K = x2+ barrier is deduced. This implies that the lowest ½+ barrier is approximately 0.5 MeV above the neutron binding energy assuming a curvature given by hco --- 0.5. Some of the possible barrier curvature-barrier height combinations consistent with this penetrability are illustrated in tig. 7. Since our analysis of the 239pu photofission anisotropy has located the lowest K = ~- and o~- barriers, we have attempted to calculate the expected magnitude of the (n, 7f) reaction for this compound nucleus. Besides the barrier positions and shapes, the other variable upon which the calculation of the (n, 7f) process is most sensitive is that characterizing the energy distribution of the primary gamma-ray spectrum. Lynn has examined this problem in some detail in his discussion of the (n, 7f) process, and we have used his results as given in fig. 3 of his Letter 21). The gamma-ray spectrum used by Lynn has a smaller fraction of low-energy gamma rays than does the energy spectrum assumed by Stavinsky and Shaker 22), and hence leads to a smaller probability for the (n, 7f) reaction. We have also assumed that the primary radiation is all electric dipole, so that the intermediate compound nuclei following the first gamma ray have spin and parity of 12- or ~-. The calculation indicates that 2.5 i/ooof the intermediate compound nuclei fission, yielding an expected contribution to the total fission width of 0.025 F~ = 1meV. This is in good agreement with the value of 1.2 meV derived from experiment by assummg that the smallest fission width observed IS a measure of the non-fluctuating contribution from the (n, yf) reaction. It is interesting to note that most of the contribution to the (n, 7f) reaction is from penetration through the fission barriers rather than fission over the barriers due to the larger fraction of intermediate compound nuclei formed with energies below the barriers. Vorontnikov et al. 23) have reported anisotropy and cross-section results for fast neutron reduced fission of 238pu. We have computed the anlsotropies and cross sections for 0.15, 0.3, 0.5 and 0 83 MeV neutrons using the tission barrier heights and widths determined from the analysis of the phototissIon and low-energy, neutronresonance data. The computed values of the anlsotropies are in satisfactory agreement with experiment. At neutron energies above 0.3 MeV, the computed cross sections are much too low, indicating that additional saddle states have become t No error estimates v,ere g~ven by Bowman et al ~) for the fi~ston widths. They point out howeser that part of the observed fluctuat~on~ may be attributed to uncertainties m determining the fission ~ldths, so that better data may show smaller fluctuations.

470

R. V A N D L N B O S C H

available, as would be expected. Vorontnikov et al. 23) have also performed an analysis similar in principle to ours. Our analysis is in definite disagreement with their analysis, as they conclude that all of the fission barriers are approximately I MeV higher than those obtained in our analysis. This discrepancy appears to arise from their treatment of the relationship between the fission width and the transmission coefficient, as they have allowed the quantity (2nFf/D) for a single, fully open channel to take arbitrary values much larger than unity. They have thus inadvertently reproduced the observed fission cross section at energies below their fission barriers by using many (20-95) channels each of which is only slightly open. 24. OTHER NEUTRON-INDUCED REACTIONS Available data for other doubly even targets were reviewed to assess whether further analyses of the type described here might be proiitable. Vorotnikov et al i~) have rcported angular distribution measurements and an analysis for neutroninduccd fission of 23°Th. Their experimental results unfortunately differ from those reported by Simmons and Hcnkel by larger amounts than the expected differences for various possible K :~ ½ bands. A similarly large discrepancy exists for 232Th at 1.6 MeV. The anisotroples reported 2) for 2"~8U and 24°pu show little character and hence do not provide much promise for K-band analyses. 3. Discussion It appears that no firm, unique K-band assignments for neutron-induced tission of thorium, uranium or plutonium can be made. Either K = ~ + or K = ~-, and probably both, bands are involved near threshold in several cases. A ~2+ or ~r- band must be revolved for 236U. Thc analysis of photo-induced fission of 23°Pti indicates a ½- band and a ] - band are present, although other K ~ -~ bands may have slightly lower barriers which are not appreciably excited due to the predominance of electric dipole absorption 24). The energies and curvatures of the K = ~.- and K = ~bands deduced from photofission anisotropies are consistent with the observcd magnitude of the (n, 7f) reaction. With respect to the queshon of the barrier curvature, one sees from table l that we have been able to account for both the sub-barrier fission behavlour and the rapidly varying anisotropies by having varying curvatures. The states with large curvature account for the tirst effect, and the state with small curvature permits sharper changes in the anisotropy with energy. The indication that the barrier curvatures are different for different K-states does not seem physically unreasonable. As the oddmass number tisslonlng nuclcus is deforming, it must conserve angular momentum and parity, so that many of the crossings on the Nllsson diagram cannot be exploited. This effect can be expected to produce effcctive curwltures which may be either larger or smaller than those characteriTmg the barrier for doubly even fissioning nuclei, depending on the location of forbidden crosoing~ relative to the saddle deformation. The effect of forbidden crossings has been d~scussed by Newton 25) and by Wheeler 26)

NEUTRON-INDUCED REACrlo',.S

471

in connection with spontaneous fission. Two remarks should be made concerning the representation of the fission barrier by a parabola. If the varying widths are due to single-particle effects, the real barriers will be " b u m p y " and cannot be described by a smooth parabola. It is felt that available information does not permit a more specific characterization of the true barrier shapes. It must also be recognized that the effective barrier curvatures derived from anisotropy measurements for energies in the neighbourhood of the barrier are not expected to correctly describe the barrier shape at energies several MeV below the barrier. These curvatures therefore have no relevance for spontaneous fission. We are currently exploring the Nilsson diagram to see if one can understand the barrier curvatures in more detail. If the varying curvatures do indeed rcflect the details of the single-particle states of the deformed nucleus, one can understand why some target nuclei like 234U and 232Th (both of which form compound nuclei with 143 neutrons) exhibit large variations of the anisotropy wah energy, while other target nuclei such as 238U and 24°pu do not. It one attributes the fluctuatmns in hm to a single-particle effect and further assumes that this effect can either increase or decrease the effcctive curvaturc, one can estimate the curvature in the absence of single-particle effects by averaging the individual curvatures. For 234U, one obtains an average curvature of 510 keV (averaged over the tivc sets of assumed K-states) and for 236U one obtains an average curvature of 580 keV (a~erage of the two sets with K = 3+ 2- and ~2- statcs). These values are shghtly lcss than a recent prediction 27) of approximately 620 keV. I wish to thank Professor L. Wdets for several helpful discussions, and Professor I. Halpern, Dr. John Huizenga and Mr. K. Wolf for their comments on portions of the manuscript. References 1) D J Hughes and R. B. Schv, artT, Umtcd States Atomic Energy Commission Report BNL 325, 2nd ed. (19581 2) R. W Lamphere, Nuclear Physics 38 (1962) 561 3) R. W. Lamphcre, m Physics and chemistry of l,sston, Vol 1 (lnternanonal Atomic Energy Agency, Vienna, 1965) p. 63 4) R Vandenbosch, J. P. Unik and J R. ltmzenga, tbM, p 547 5) R Vandcnbosch, Proc. Int. C o n f on nuclear physics, Gathnburg (1966) unpubhshed 6) L '~'dets anti D M. Chase, Plays Re',' 103 (1956) 1296 7) W H a u s e r a n d H Feshbach, Ph~s. Re~ 87 (1952) 366 8) E R. Rae, B. Margohs and E S. Troubetzko~, Phys. Rev 112 (1958) 492 9) D. L Hdl and J A. Wheeler, Plays. Rev 89 (195~) 1102 10) Vv S Eminent.h, Westinghouse Res Lab Report 6-94511-R19, lhttsburgh, Pa. (1958) unpubhshed 11) E. J Campbell, H Feshbach, C E Porter and V. F. ~¢clsskopf, MIT Laboratory for Nu~.lear Sctcnt.e'Icchmcal Report No 73 (1960) unpubhshcd 121 F 11 Auerbach and F. G J. Percy, Brookha~cn Natmnal 1. aboratory Report BNL 765 (July 1962 ) tmpubllshed 13) J R. Stchn et a l , Brookha,,cn National Laboratory Report BNL 325, 2nd, ed., Suppl. No 2 (February 1965) unpubhshed

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