- Email: [email protected]
Study of second excited 2+ states in even-even nuclei through β-γ angular correlations
Volume 11, number 4
PHYSICS LETTERS
In table 1 we a l s o l i s t v a l u e s of a d e r i v e d by the u s e of eq. (8). a v a l u e s d e r i v e d u n d e r v a r i o u s conditions a r e i l l u s t r a t e d in fig. 4: The t h e o r e t i c a l c u r v e in fig. 4 is due to Newton 11) and has been modified by Lang 12). It r e p r e s e n t s a = 0.075 I
~
r
i
1
l
ASSUMING (T~• CONSTANT
3<
* a* ~ U.F.,-E ' • o" with U " Eo-E'-,
// A
?
i 3o
.ss..,.G oPT,o...o~-
I0
•b
~
,~o
~o
z~o
24o
ATOMIC WEIGHT F i g . 4. F e r m i - g a s c o e f f i c i e n t s a s d e t e r m i n e d u n d e r the a s s u m p t i o n s : (i) o c = c o n s t . , AD = 0; (ii) c~c = c o n s t . , Z~p = 1.68 ( 1 - A / 4 0 0 ) (2feven + fod'd) ; and (iii) ~c = ( o p t i c a l
model), hp = 0.
15 August 1964
~+J-z + I ) A } , w h e r e j n a n d ] z are averages of spins of neutron and proton states for the nucleus A = N + Z and A 32-is the empirically observed proportionality of single particle level density. It m a y be noticed that not only did Ap not improve the fits in fig. 2, its use introduces increased point scatter in the values of a in fig. 4. F r o m the lower hall of fig. 4 it is clear that the use of optical model ~c improves the agreement between the experimental values of a and the theoretical curve. References 1) H.A.Bethe, Phys. Rev. 50 (1936) 332. 2) V.F.Weisskopf, Phys. Rev. 62 (1937) 295. 3) Wilenzick, Seth, Bevington and Lewis, Nucl. Phys., to be published. 4) T.Ericson, Advan. Physics 9 (1960) 425. 5) K.K.Seth, Bull. Am. Phys. Soc. Ser. 11, 7 (1962) 402; K. K. Seth, Proc. Intern. Conf. on Direct Interactions and Nuclear Reaction Mechanisms (Gordon and Breach Science Publishers, Inc., 1963) p. 267. 6) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, (John Wiley and Sons, New York, 1952) p. 367. 7) D. B. Thomson, Phys. Rev. 129 (1963) 1649. 8) A. M. Lane and R.G.Thomas, Rev. Mod. Phys. 30 (1958) 302. 9) Feshbach, Porter, Weisskopf and Campbell, M.I.T. Tech. Rep. No. 73 (1960), unpublished. 10) F . G . J . Perey, private communication (1962). 11) T.D.Newton, Canad. J. Phys. 34 (1956) 804. 12) D.W. Lang, Nucl. Phys. 26 (1961) 434.
* $ * * $
STUDY
OF
SECOND THROUGH
EXCITED 2 + STATES IN E V E N - E V E N ~-~ ANGULAR CORRELATIONS *
NUCLEI
R. S. R A G H A V A N **, Z.W. G R A B O W S K I and R. M. S T E F F E N Department of Physics, Purdue University, Lafayette, Indiana, U.S.A.
Received 26 Jane 1964 In r e c e n t y e a r s , the e m p h a s i s in the study of f i r s t - f o r b i d d e n ( F F ) ~ - t r a n s i t i o n s has been to gain i n f o r m a t i o n on the s t r u c t u r e of n u c l e a r s t a t e s r a t h e r than on the i n t e r a c t i o n laws of/3-decay. In s e v e r a l c a s e s , it has been p o s s i b l e to d e t e r m i n e e x p e r i m e n t a l l y the v a l u e s of the n u c l e a r m a t r i x * Supported by the U.S.Atomic Energy Commission. ** On leave from Tata Institute of Fundamental Research, Bombay, India.
elements that contribute to ~-transitions to first excited states of even-even daughter nuclei (e.g. Sb124, Eu152, L a 1 4 0 etc.) 1). Attempts are being m a d e to compare the experimental values of the ~-matrix elements with theoretical calculations. The observed relative magnitudes of the ~-matrix elements are usually interpreted in terms of m o del-dependent selection rules (A j-selection rules, A K-selectlon rules) which m a y give some information about the nuclear states involved in the ~311
Volume 11, number 4
PHYSICS LETTERS
t r a n s i t i o n 2). C o n s i d e r a t i o n s of this kind w e r e used by Matumoto et al. 3) to obtain i n f o r m a t i o n on the s t r u c t u r e of f i r s t excited s t a t e s of some even-even nuclei. Very few F F ~ - t r a n s i t i o n s leading to second excited s t a t e s of the daughter n u c l e i have been a n a l y z e d due to the lack of e x p e r i m e n t a l data. The ~ b r a n c h i n g r a t i o s to second excited 2+ s t a t e s a r e u s u a l l y v e r y s m a l l (a few %) and r e l i a ble shape and b - r c i r c u l a r p o l a r i z a t i o n c o r r e l a tion m e a s u r e m e n t s a r e v e r y difficult. Betag a m m a directional c o r r e l a t i o n s , however, can be m e a s u r e d with r e a s o n a b l e a c c u r a c y if p r o p e r p r e c a u t i o n s a r e taken (e. g. c o r r e c t i o n s for 3 -V, V-V and ~- b r e m s s t r a h l u n g c o i n c i d e n c e s , c o r r e c t i o n s for higher o r d e r chance c o i n c i d e n c e s , cons i d e r a t i o n s of e l e c t r o n i c p i l e - u p , etc.). The d i r e c t i o n a l c o r r e l a t i o n of a F F ~-7 c a s c a de is given by 1):
(1)
where the r e d u c e d ~ a n i s o t r o p y factor R depends only on the ~ - m a t r i x e l e m e n t s . The p a r ~ r n e t e r X2(Z , W) i s tabulated in ref. 4. If the ~-approximarion is not applicable, e.g. b e c a u s e of an u n u s u a l l y l a r g e f Bij c o n t r i b u t i o n , the e n e r g y dependence of A 2(W) i s m o r e c o m p l i c a t e d and R defined by eq. (1) b e c o m e s e n e r g y dependent. A c o m p a r i s o n of m e a s u r e d v a l u e s of R for ~t r a n s i t i o n s o r i g i n a t i n g f r o m the s a m e n u c l e u s , but leading to different excited states in the daughter n u c l e u s , m a y give i n f o r m a t i o n about the s t r u c t u r e of excited s t a t e s . We denote the r e d u c e d a n i s o t r o p y factor of the ~-7 cascade involving the zq/1 excited state of a daughter n u c l e u s by R i. If, e.g. R 1 and R 2 differ a p p r e c i a b l y , the r e l a t i v e m a g n i t u d e s of the v a r i o u s m a t r i x e l e m e n t s in the two H-branches m u s t be different i n d i c a t i n g a diff e r e n c e of the s t r u c t u r e of the excited s t a t e s . We r e p o r t h e r e ~-V d i r e c t i o n a l c o r r e l a t i o n m e a s u r e m e n t s involving second excited 2+ s t a t e s of the e v e n - e v e n s p h e r i c a l n u c l e i Se 76 and Te 122 (fig. 1). T h e s e s t a t e s a r e w e l l - k n o w n f r o m v - r a y studies and a r e i n t e r p r e t e d in t e r m s of the v i b r a t i o n a l model. The f i r s t and second excited s t a t e s a r e d e s c r i b e d as one- and two-phonon s t a t e s , r e spectively. It i s i n t e r e s t i n g to see whether the f l - t r a n s i t i o n s leading to the f i r s t two 2+ excited 312
mIE-~'1~2-
,8=. etc.
26 h
,¢'~"~
:i l
- - ~ 1.216
O. 5 6 4
~
t 0÷
0.5,59
t 0÷
T e l = = stoble
34
$e~
42
Fig. 1. Decay of Sb122 and As 76.
Sb~2=
~ to second excited Z + state:
Rz(W)
~ to first excited 2 + state=
o: -o.01
RI(W1
WolB,)
.~ T
wo(/~
i,
--0.03
The v - a n i s o t r o p y factor A2(V), can e a s i l y be exp r e s s e d in t e r m s of the F - c o e f f i c i e n t s . F o r a 2 ~ 0 t r a n s i t i o n , A2(V ) = 0.598. The ~ - a n i s o t r o p y p a r a m e t e r A 2(~) is a function of the fl-energy W (in u n i t s of mc2). If the f - a p p r o x i m a t i o n 4) i s valid: A 2(~) = R X2(Z, W) (W2 - 1/W) ,
2.8 d
E.C
-0.02
W(O) = 1 + A 2 ( ~ ) A 2 ( v ) P 2 ( c o s 0 ) .
15 August 1964
--0.04
RCW)_O.05
!
-- 0.06 --0.07 -0.08 i
15
,
,
,
*
~.0
.
R,(W)
.
.
.
i
.
.
.
.
i
2.5
$.0 W
.
.
.
.
J
.
,
3.5 •
Fig. 2. The reduced anisotropy factors R 1 and R 2 for the Sb122 8-transitions to the first and second excited states of Te 122 respectively. s t a t e s in s p h e r i c a l n u c l e i (which a r e supposed to have the s a m e i n t r i n s i c configuration a c c o r d i n g to the v i b r a t i o n a l m o d e l ) g i v e r i s e to a difference in the fl-decay p a r a m e t e r s R 1 a n d R 2. The m e a s u r e m e n t s w e r e p e r f o r m e d with the m u l t i c h a n n e l ~-v coincidence s p e c t r o m e t e r des c r i b e d in ref. 5. The e n e r g y dependence Of R2 for the 2-(~ 2) 2+(v3}0 + c a s c a d e of As 76 and Sb122 (involving the c r o s s - o v e r v - t r a ' n s i t i o n ) was d e t e r mined. C o r r e c t i o n s for competing fl-r and v - v c a s c a d e s were taken into account. A l l data w e r e c o r r e c t e d for the finite solid a n g l e s of the d e t e c t o r s . The r e d u c e d a n i s o t r o p y f a c t o r R 2 for Sb 122 (dotted line in fig. 2) i s e n e r g y dependent, indicating a b r e a k - d o w n of the ~-approximation. The R 1 f a c t o r , on the other hand, is independent of the H-energy as m e a s u r e d by Steffen 6) and conf i r m e d in the p r e s e n t investigation. The R 1 f a c tor for the f i r s t excited state i s shown a s a solid line in fig. 2. M e a s u r e m e n t s on the ~2-V3 cascade in As 76 yielded a s m a l l and, within l i m i t s of e r r o r , e n e r gy independent a n i s o t r o p y , c o r r e s p o n d i n g to R 2 = + 0.0072 • 0.0025. F i s c h b e c k and Newsome 7) obtained within l i m i t s of e r r o r an i s o t r o p i c ~2-V3 directional correlation.
Volume 11, number 4
PHYSICS LETTERS
15 August 1964
1) See, for example, H. Frauenfelder and R. M. Steffen, in: Alpha-Beta-Gamma Spectroscopy, Ed. K. Siegbahn (to be published). 2) See H.Wiedenm~ller, Revs. Mod. Physics 33 (1961) 574. 3) Z.Matumoto et a l . , Phys. Rev. 129 (1963) 1308. 4) T.Kotani and M.Ross, Phys. Rev. 113 (1959) 622. 5) R.S.Raghavan and R.M.Steffen, Physics Letters 5 (1963) 198. 6) R.M.Steffen, Phys. Rev. 123 (1961) 1787. 7) H.J. Fischbeck and R. W. Newsome J r . , Phys. Roy. 129 (1963) 2231.
It is interesting to note that f o r both nuclei, R I and R 2 are quite different. The cause of this difference seems hard to understand on the basis of the vibrational model alone. Similar Investigations on other F F ~-transitions (e.g. Sb 124, 1126, etc.) axe in progress in an attempt to reveal systematic trends.
* * * * *
OPTICAL
MODEL
ANALYSIS
OF
182 M e V
PROTON
SCATTERING*
G. R. S A T C H L E R Oak Ridge National Laboratory, Oak Ridge, Tennessee and
R. M. H A Y B R O N Oak Ridge National Laboratory, Oak Ridge, Tennessee and Michigan State University, East Lansing, Michigan Received 20 July 1964 R e c e n t l y i n v e s t i g a t i o n s w e r e begun into the u s e f u l n e s s of the i m p u l s e a p p r o x i m a t i o n f o r the i n e l a s t i c s c a t t e r i n g of high e n e r g y p r o t o n s a s a t o o l f o r i n v e s t i g a t i n g n u c l e a r wave functions 1). In t h e s e c a l c u l a t i o n s it i s n e c e s s a r y to t a k e into account d i s t o r t e d - w a v e e f f e c t s , and f o r t h i s we r e q u i r e r e l i a b l e i n f o r m a t i o n on o p t i c a l m o d e l p a r a m e t e r s . F o r t h i s r e a s o n , we have i n i t i a t e d a p r o g r a m of o p t i c a l m o d e l a n a l y s e s of high e n e r g y p r o t o n e l a s t i c s c a t t e r i n g , and we h e r e p r e s e n t the f i r s t r e s u l t s , f o r 182 MeV p r o t o n s on L i , Be, C, A1, Ca, F e , In and Au. T h e only e x t e n s i v e m o d e r n a n a l y s i s of high e n e r g y s c a t t e r i n g w a s m a d e with t h e s e data-2). We find, h o w e v e r , that v e r y g r e a t i m p r o v e m e n t s can be m a d e o v e r the " b e s t f i t s " quoted in r e f . 2. The o p t i c a l p o t e n t i a l u s e d h a s the f o r m U(r) = Vc(r ) - V(e x + 1) -1 - i( W - 4iWDd/dx')(e x' + 1) "1
+ (n/m~c)2 (Vs+iWs)r-i (d/dr)(eX+ I) L.a, where I
x
=
(r -roA~)la,
x' = (r-r'oA~)la'
* Research sponsored by the U.S.Atomic Energy Commission under coatraot with the Union Carbide Corporation.
and Vc(r ) i s the Coulomb p o t e n t i a l f r o m a u n i f o r m c h a r g e of r a d i u s rc.4~. The v a l u e s of r c w e r e t a k e n f r o m e l e c t r o n s c a t t e r i n g r e s u l t s . Effective b o m b a r d i n g e n e r g i e s w e r e u s e d which give the c.m. system relativistic momentum corresponding to 182 MeV l a b o r a t o r y energy. F i t s to the d a t a w e r e obtained through the u s e of an a u t o m a t i c s e a r c h r o u t i n e which m i n i m i s e s the quantity X2 --.~ [aex(0i) - ath(0i)]2/Aaex(0 i)2 , I
where Crex and ~th are the measured and theoretical cross sections at 0i, and A~ex is the error associated with ~ex. The results presented here were obtained by fitting the measured differential cross sections alone; the polarisations are not well fitted, but experience at lower energies suggests that quite small adjustments in the parameters will rectify this. The parameters which give m i n i m u m X 2 are presented in the table 1, and the corresponding fits shown in figs. 1 and 2. Also shown in the figures are the predictions (dotted lines) for the potentials recommended in ref. 2. For the light elements, potentials with "volume" (W D = 0) or ',surface" (W = 0) absorption were tried. A good fit to the Li data with only "volume" absorption 313
PHYSICS LETTERS
In table 1 we a l s o l i s t v a l u e s of a d e r i v e d by the u s e of eq. (8). a v a l u e s d e r i v e d u n d e r v a r i o u s conditions a r e i l l u s t r a t e d in fig. 4: The t h e o r e t i c a l c u r v e in fig. 4 is due to Newton 11) and has been modified by Lang 12). It r e p r e s e n t s a = 0.075 I
~
r
i
1
l
ASSUMING (T~• CONSTANT
3<
* a* ~ U.F.,-E ' • o" with U " Eo-E'-,
// A
?
i 3o
.ss..,.G oPT,o...o~-
I0
•b
~
,~o
~o
z~o
24o
ATOMIC WEIGHT F i g . 4. F e r m i - g a s c o e f f i c i e n t s a s d e t e r m i n e d u n d e r the a s s u m p t i o n s : (i) o c = c o n s t . , AD = 0; (ii) c~c = c o n s t . , Z~p = 1.68 ( 1 - A / 4 0 0 ) (2feven + fod'd) ; and (iii) ~c = ( o p t i c a l
model), hp = 0.
15 August 1964
~+J-z + I ) A } , w h e r e j n a n d ] z are averages of spins of neutron and proton states for the nucleus A = N + Z and A 32-is the empirically observed proportionality of single particle level density. It m a y be noticed that not only did Ap not improve the fits in fig. 2, its use introduces increased point scatter in the values of a in fig. 4. F r o m the lower hall of fig. 4 it is clear that the use of optical model ~c improves the agreement between the experimental values of a and the theoretical curve. References 1) H.A.Bethe, Phys. Rev. 50 (1936) 332. 2) V.F.Weisskopf, Phys. Rev. 62 (1937) 295. 3) Wilenzick, Seth, Bevington and Lewis, Nucl. Phys., to be published. 4) T.Ericson, Advan. Physics 9 (1960) 425. 5) K.K.Seth, Bull. Am. Phys. Soc. Ser. 11, 7 (1962) 402; K. K. Seth, Proc. Intern. Conf. on Direct Interactions and Nuclear Reaction Mechanisms (Gordon and Breach Science Publishers, Inc., 1963) p. 267. 6) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, (John Wiley and Sons, New York, 1952) p. 367. 7) D. B. Thomson, Phys. Rev. 129 (1963) 1649. 8) A. M. Lane and R.G.Thomas, Rev. Mod. Phys. 30 (1958) 302. 9) Feshbach, Porter, Weisskopf and Campbell, M.I.T. Tech. Rep. No. 73 (1960), unpublished. 10) F . G . J . Perey, private communication (1962). 11) T.D.Newton, Canad. J. Phys. 34 (1956) 804. 12) D.W. Lang, Nucl. Phys. 26 (1961) 434.
* $ * * $
STUDY
OF
SECOND THROUGH
EXCITED 2 + STATES IN E V E N - E V E N ~-~ ANGULAR CORRELATIONS *
NUCLEI
R. S. R A G H A V A N **, Z.W. G R A B O W S K I and R. M. S T E F F E N Department of Physics, Purdue University, Lafayette, Indiana, U.S.A.
Received 26 Jane 1964 In r e c e n t y e a r s , the e m p h a s i s in the study of f i r s t - f o r b i d d e n ( F F ) ~ - t r a n s i t i o n s has been to gain i n f o r m a t i o n on the s t r u c t u r e of n u c l e a r s t a t e s r a t h e r than on the i n t e r a c t i o n laws of/3-decay. In s e v e r a l c a s e s , it has been p o s s i b l e to d e t e r m i n e e x p e r i m e n t a l l y the v a l u e s of the n u c l e a r m a t r i x * Supported by the U.S.Atomic Energy Commission. ** On leave from Tata Institute of Fundamental Research, Bombay, India.
elements that contribute to ~-transitions to first excited states of even-even daughter nuclei (e.g. Sb124, Eu152, L a 1 4 0 etc.) 1). Attempts are being m a d e to compare the experimental values of the ~-matrix elements with theoretical calculations. The observed relative magnitudes of the ~-matrix elements are usually interpreted in terms of m o del-dependent selection rules (A j-selection rules, A K-selectlon rules) which m a y give some information about the nuclear states involved in the ~311
Volume 11, number 4
PHYSICS LETTERS
t r a n s i t i o n 2). C o n s i d e r a t i o n s of this kind w e r e used by Matumoto et al. 3) to obtain i n f o r m a t i o n on the s t r u c t u r e of f i r s t excited s t a t e s of some even-even nuclei. Very few F F ~ - t r a n s i t i o n s leading to second excited s t a t e s of the daughter n u c l e i have been a n a l y z e d due to the lack of e x p e r i m e n t a l data. The ~ b r a n c h i n g r a t i o s to second excited 2+ s t a t e s a r e u s u a l l y v e r y s m a l l (a few %) and r e l i a ble shape and b - r c i r c u l a r p o l a r i z a t i o n c o r r e l a tion m e a s u r e m e n t s a r e v e r y difficult. Betag a m m a directional c o r r e l a t i o n s , however, can be m e a s u r e d with r e a s o n a b l e a c c u r a c y if p r o p e r p r e c a u t i o n s a r e taken (e. g. c o r r e c t i o n s for 3 -V, V-V and ~- b r e m s s t r a h l u n g c o i n c i d e n c e s , c o r r e c t i o n s for higher o r d e r chance c o i n c i d e n c e s , cons i d e r a t i o n s of e l e c t r o n i c p i l e - u p , etc.). The d i r e c t i o n a l c o r r e l a t i o n of a F F ~-7 c a s c a de is given by 1):
(1)
where the r e d u c e d ~ a n i s o t r o p y factor R depends only on the ~ - m a t r i x e l e m e n t s . The p a r ~ r n e t e r X2(Z , W) i s tabulated in ref. 4. If the ~-approximarion is not applicable, e.g. b e c a u s e of an u n u s u a l l y l a r g e f Bij c o n t r i b u t i o n , the e n e r g y dependence of A 2(W) i s m o r e c o m p l i c a t e d and R defined by eq. (1) b e c o m e s e n e r g y dependent. A c o m p a r i s o n of m e a s u r e d v a l u e s of R for ~t r a n s i t i o n s o r i g i n a t i n g f r o m the s a m e n u c l e u s , but leading to different excited states in the daughter n u c l e u s , m a y give i n f o r m a t i o n about the s t r u c t u r e of excited s t a t e s . We denote the r e d u c e d a n i s o t r o p y factor of the ~-7 cascade involving the zq/1 excited state of a daughter n u c l e u s by R i. If, e.g. R 1 and R 2 differ a p p r e c i a b l y , the r e l a t i v e m a g n i t u d e s of the v a r i o u s m a t r i x e l e m e n t s in the two H-branches m u s t be different i n d i c a t i n g a diff e r e n c e of the s t r u c t u r e of the excited s t a t e s . We r e p o r t h e r e ~-V d i r e c t i o n a l c o r r e l a t i o n m e a s u r e m e n t s involving second excited 2+ s t a t e s of the e v e n - e v e n s p h e r i c a l n u c l e i Se 76 and Te 122 (fig. 1). T h e s e s t a t e s a r e w e l l - k n o w n f r o m v - r a y studies and a r e i n t e r p r e t e d in t e r m s of the v i b r a t i o n a l model. The f i r s t and second excited s t a t e s a r e d e s c r i b e d as one- and two-phonon s t a t e s , r e spectively. It i s i n t e r e s t i n g to see whether the f l - t r a n s i t i o n s leading to the f i r s t two 2+ excited 312
mIE-~'1~2-
,8=. etc.
26 h
,¢'~"~
:i l
- - ~ 1.216
O. 5 6 4
~
t 0÷
0.5,59
t 0÷
T e l = = stoble
34
$e~
42
Fig. 1. Decay of Sb122 and As 76.
Sb~2=
~ to second excited Z + state:
Rz(W)
~ to first excited 2 + state=
o: -o.01
RI(W1
WolB,)
.~ T
wo(/~
i,
--0.03
The v - a n i s o t r o p y factor A2(V), can e a s i l y be exp r e s s e d in t e r m s of the F - c o e f f i c i e n t s . F o r a 2 ~ 0 t r a n s i t i o n , A2(V ) = 0.598. The ~ - a n i s o t r o p y p a r a m e t e r A 2(~) is a function of the fl-energy W (in u n i t s of mc2). If the f - a p p r o x i m a t i o n 4) i s valid: A 2(~) = R X2(Z, W) (W2 - 1/W) ,
2.8 d
E.C
-0.02
W(O) = 1 + A 2 ( ~ ) A 2 ( v ) P 2 ( c o s 0 ) .
15 August 1964
--0.04
RCW)_O.05
!
-- 0.06 --0.07 -0.08 i
15
,
,
,
*
~.0
.
R,(W)
.
.
.
i
.
.
.
.
i
2.5
$.0 W
.
.
.
.
J
.
,
3.5 •
Fig. 2. The reduced anisotropy factors R 1 and R 2 for the Sb122 8-transitions to the first and second excited states of Te 122 respectively. s t a t e s in s p h e r i c a l n u c l e i (which a r e supposed to have the s a m e i n t r i n s i c configuration a c c o r d i n g to the v i b r a t i o n a l m o d e l ) g i v e r i s e to a difference in the fl-decay p a r a m e t e r s R 1 a n d R 2. The m e a s u r e m e n t s w e r e p e r f o r m e d with the m u l t i c h a n n e l ~-v coincidence s p e c t r o m e t e r des c r i b e d in ref. 5. The e n e r g y dependence Of R2 for the 2-(~ 2) 2+(v3}0 + c a s c a d e of As 76 and Sb122 (involving the c r o s s - o v e r v - t r a ' n s i t i o n ) was d e t e r mined. C o r r e c t i o n s for competing fl-r and v - v c a s c a d e s were taken into account. A l l data w e r e c o r r e c t e d for the finite solid a n g l e s of the d e t e c t o r s . The r e d u c e d a n i s o t r o p y f a c t o r R 2 for Sb 122 (dotted line in fig. 2) i s e n e r g y dependent, indicating a b r e a k - d o w n of the ~-approximation. The R 1 f a c t o r , on the other hand, is independent of the H-energy as m e a s u r e d by Steffen 6) and conf i r m e d in the p r e s e n t investigation. The R 1 f a c tor for the f i r s t excited state i s shown a s a solid line in fig. 2. M e a s u r e m e n t s on the ~2-V3 cascade in As 76 yielded a s m a l l and, within l i m i t s of e r r o r , e n e r gy independent a n i s o t r o p y , c o r r e s p o n d i n g to R 2 = + 0.0072 • 0.0025. F i s c h b e c k and Newsome 7) obtained within l i m i t s of e r r o r an i s o t r o p i c ~2-V3 directional correlation.
Volume 11, number 4
PHYSICS LETTERS
15 August 1964
1) See, for example, H. Frauenfelder and R. M. Steffen, in: Alpha-Beta-Gamma Spectroscopy, Ed. K. Siegbahn (to be published). 2) See H.Wiedenm~ller, Revs. Mod. Physics 33 (1961) 574. 3) Z.Matumoto et a l . , Phys. Rev. 129 (1963) 1308. 4) T.Kotani and M.Ross, Phys. Rev. 113 (1959) 622. 5) R.S.Raghavan and R.M.Steffen, Physics Letters 5 (1963) 198. 6) R.M.Steffen, Phys. Rev. 123 (1961) 1787. 7) H.J. Fischbeck and R. W. Newsome J r . , Phys. Roy. 129 (1963) 2231.
It is interesting to note that f o r both nuclei, R I and R 2 are quite different. The cause of this difference seems hard to understand on the basis of the vibrational model alone. Similar Investigations on other F F ~-transitions (e.g. Sb 124, 1126, etc.) axe in progress in an attempt to reveal systematic trends.
* * * * *
OPTICAL
MODEL
ANALYSIS
OF
182 M e V
PROTON
SCATTERING*
G. R. S A T C H L E R Oak Ridge National Laboratory, Oak Ridge, Tennessee and
R. M. H A Y B R O N Oak Ridge National Laboratory, Oak Ridge, Tennessee and Michigan State University, East Lansing, Michigan Received 20 July 1964 R e c e n t l y i n v e s t i g a t i o n s w e r e begun into the u s e f u l n e s s of the i m p u l s e a p p r o x i m a t i o n f o r the i n e l a s t i c s c a t t e r i n g of high e n e r g y p r o t o n s a s a t o o l f o r i n v e s t i g a t i n g n u c l e a r wave functions 1). In t h e s e c a l c u l a t i o n s it i s n e c e s s a r y to t a k e into account d i s t o r t e d - w a v e e f f e c t s , and f o r t h i s we r e q u i r e r e l i a b l e i n f o r m a t i o n on o p t i c a l m o d e l p a r a m e t e r s . F o r t h i s r e a s o n , we have i n i t i a t e d a p r o g r a m of o p t i c a l m o d e l a n a l y s e s of high e n e r g y p r o t o n e l a s t i c s c a t t e r i n g , and we h e r e p r e s e n t the f i r s t r e s u l t s , f o r 182 MeV p r o t o n s on L i , Be, C, A1, Ca, F e , In and Au. T h e only e x t e n s i v e m o d e r n a n a l y s i s of high e n e r g y s c a t t e r i n g w a s m a d e with t h e s e data-2). We find, h o w e v e r , that v e r y g r e a t i m p r o v e m e n t s can be m a d e o v e r the " b e s t f i t s " quoted in r e f . 2. The o p t i c a l p o t e n t i a l u s e d h a s the f o r m U(r) = Vc(r ) - V(e x + 1) -1 - i( W - 4iWDd/dx')(e x' + 1) "1
+ (n/m~c)2 (Vs+iWs)r-i (d/dr)(eX+ I) L.a, where I
x
=
(r -roA~)la,
x' = (r-r'oA~)la'
* Research sponsored by the U.S.Atomic Energy Commission under coatraot with the Union Carbide Corporation.
and Vc(r ) i s the Coulomb p o t e n t i a l f r o m a u n i f o r m c h a r g e of r a d i u s rc.4~. The v a l u e s of r c w e r e t a k e n f r o m e l e c t r o n s c a t t e r i n g r e s u l t s . Effective b o m b a r d i n g e n e r g i e s w e r e u s e d which give the c.m. system relativistic momentum corresponding to 182 MeV l a b o r a t o r y energy. F i t s to the d a t a w e r e obtained through the u s e of an a u t o m a t i c s e a r c h r o u t i n e which m i n i m i s e s the quantity X2 --.~ [aex(0i) - ath(0i)]2/Aaex(0 i)2 , I
where Crex and ~th are the measured and theoretical cross sections at 0i, and A~ex is the error associated with ~ex. The results presented here were obtained by fitting the measured differential cross sections alone; the polarisations are not well fitted, but experience at lower energies suggests that quite small adjustments in the parameters will rectify this. The parameters which give m i n i m u m X 2 are presented in the table 1, and the corresponding fits shown in figs. 1 and 2. Also shown in the figures are the predictions (dotted lines) for the potentials recommended in ref. 2. For the light elements, potentials with "volume" (W D = 0) or ',surface" (W = 0) absorption were tried. A good fit to the Li data with only "volume" absorption 313