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Large angle alpha particle scattering from the 3+ unnatural parity states in magnesium
Volume 25B, number 2
LARGE
ANGLE
PHYSICS LETTERS
ALPHA
7 August 1967
PARTICLE SCATTERING FROM P A R I T Y S T A T E S IN M A G N E S I U M
THE
3 + UNNATURAL
J . S . V I N C E N T , E . T . BOSCHITZ and J. R. PRIEST N A S A - L e w i s R e s e a r c h Center. Cleveland. Ohio, USA
Received 20 June 1967
The differential cross sections for inelastic alpha particle scattering to large angles from 3+ states in 24Mg and 26Mg at 42 MeV show a clear diffraction type of pattern and pronounced enhancement compared with forward angles.
The excitation of u n n a t u r a l parity s t a t e s (~r ¢ (-1) J) in even n u c l e i by i n e l a s t i c alpha p a r ticle s c a t t e r i n g has r e c e i v e d c o n s i d e r a b l e a t t e n tion s i n c e E i d s o n and C r a m e r [1] showed it to be forbidden by the usual a s s u m p t i o n s of d i r e c t r e a c t i o n theory. Both 2- and 3 + states have been studied in s e v e r a l nuclei by alpha p a r t i c l e s c a t t e r i n g at v a r i o u s incident e n e r g i e s [1-4]. In m o s t c a s e s the a n g u l a r d i s t r i b u t i o n s cover only the forward hemisphere. Several reaction m e c h a n i s m s have been proposed, e.g. an e x change p r o c e s s or m u l t i p l e phonon excitation. Taken in total, however, the data give no o v e r whelming evidence for the p r e d o m i n a n c e of any one m e c h a n i s m . T h e o r e t i c a l l y , the only a n a l y s i s is that of T a m u r a [5] who c o m p a r e d the 24Mg data (0 < 90) of Kokame [2] with a coupled c h a n n e l s c a l c u l a t i o n , a s s u m i n g a collective, r o t a t i o n v i b r a t i o n model. A r e a s o n a b l e d e s c r i p t i o n of the data was obtained a f t e r the K = 2 m a t r i x e l e m e n t s w e r e modified somewhat a r b i t r a r i l y . The l a r g e angle s c a t t e r i n g from the 3+ state in 24Mg and 26Mg was u n d e r t a k e n to d e t e r m i n e its r e l e v a n c e to any of the p o s s i b l e r e a c t i o n nvechanismso An alpha p a r t i c l e b e a m of the NASA-Lewis cyclotron was m a g n e t i c a l l y a n a l y z e d and f o c u s s e d onto t a r g e t s having isotopic purity g r e a t e r than 99 p e r c e n t . The s c a t t e r e d alpha p a r t i c l e s were detected by s i l i c o n c o u n t e r s . The f i r s t m e a s u r e m e n t s , on 24Mg, a r e p r e s e n t e d in fig. 1. The a n g u l a r d i s t r i b u t i o n for the 3+ (Q = = -5.23 MeY) state a r e c o m p a r e d with the e l a s tic s c a t t e r i n g at Eat = 42 MeV. As was known b e f o r e , the f o r w a r d angle c r o s s section i s g r e a t ly inhibited [3]. However, at l a r g e r a n g l e s , not only is the 3 + c r o s s section enhanced c o m p a r e d with e l a s t i c s c a t t e r i n g , but a p r o n o u n c e d d i f f r a c tion type p a t t e r n r e s u l t s . Thus unique behaviour
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sb
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Ocrn Fig. 1. D i f f e r e n t i a ] c r o s s s e c t i o n s f o r 2 4 M g ( a t , a t ) 2 4 M g . Q = 0 and Q = -5.23 M e V . T h e l i n e is a g u i d e to the eye
only. Insert shows the low lying levels in 24Mg. of the u n n a t u r a l p a r i t y state was not found for other excited s t a t e s (fig. 4). Since l a r g e angle e l a s t i c alpha p a r t i c l e s c a t t e r i n g can u n d e r g o r a p i d fluctuations with changes of a few MeV in the incident beam [6], an i n v e s t i g a t i o n was made to d e t e r m i n e whether this c o m p l i c a t i o n would be p r e s e n t for the 3 + d i f f e r e n t i a l c r o s s section. The l a r g e angle data for Eat = 36.1, 39.4, and 42.0 MeV shown in fig. 2 indicate that the p a t t e r n changes only slightly over this r a n g e . In a r e c e n t study, Reed et al. [4] find that this l a r g e angle p a t t e r n is v e r y s i m i l a r up to at l e a s t 50 MeV. This smooth b e h a v i o u r with energy is e n c o u r a g i n g f r o m the standpoint of an a n a l y s i s such as the coupled channel approach b a s e d on the collective model. 81
Volume 25B. n u m b e r 2
PHYSICS
LETTERS
7 August 1967
I0 °
42 MeV
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Fig. 2. Large angle differential c r o s s sections for 24Mg(ot, ~')24Mg, Q=-5.23 MeV, E~ =36.1, 39.4, 42.0 MeV. T h e p r e d i c t i o n s of t h e c o l l e c t i v e m o d e l a r e n o t s e n s i t i v e t o t h e f i n e d e t a i l s of n u c l e a r s t r u c t u r e . An i n t e r e s t i n g o p p o r t u n i t y t o c o m p a r e t h e b e h a v i o u r of t h e l a r g e a n g l e 3 + c r o s s s e c t i o n s i n n e i g h b o r i n g n u c l e i i s a f f o r d e d by a m e a s u r e m e n t of t h e Q = - 3 . 9 7 M e V l e v e l in 26Mg. T h e d i f f e r e n t i a l c r o s s s e c t i o n s f o r 3 + s t a t e s i n 24Mg a n d 26Mg a r e p r e s e n t e d in fig. 3. T h e c l o s e a g r e e ment which is evident would lead one to believe t h a t t h e m a c r o s c o p i c s t r u c t u r e of t h e s e s t a t e s i s similar. S i n c e t h e 3 + c r o s s s e c t i o n s e e m s to b e s o well behaved in this energy region, a coupled c h a n n e l s c a l c u l a t i o n w a s p e r f o r m e d in 24Mg. T h e p r o g r a m of W i l l s [7], u t i l i z i n g t h e p o t e n t i a l e x p a n s i o n of T a m u r a w a s u s e d . N o a l t e r a t i o n of the K = 2 matrix elements was made, and the adjustable parameters were the optical potentials (same radial shape for real and imaginary parts), the deformation/3, and the zero point vibration
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Fig. 4. Comparison of coupled channel calculations for f i r s t five levels in 24Mg, in the r o t a t i o n - v i b r a t i o n model with unnormalized data; Eot = 42 MeV. Optical potentials and deformations are: for the solid line V = = 90 MeV, W = 15 MeV, R = 4.2 fm, A = 0.57 fro, /3 0.30, 7 o = 0.425; for the dashed c u r v e V=40 MeV, W-13 MeV, R =4.9 fro, A =0.65 fm,/3 =0.28, yo=0.425.
lo
I0
40
80
120
160 (~cm
F i g . 3. C o m p a r i s o n of l a r g e angle d i f f e r e n t i a l e r o s s s e c t i o n s f o r 24Mg(o~,0~)24Mg, Q = - 5 . 2 3 M e V , and
26Mg(O~,~)26Mg, 82
Qj
-3.97 MeV. Both levels have ~ =3 +. Eot=42 MeV.
a m p l i t u d e 7 o . In a p r e l i m i n a r y c a l c u l a t i o n o n l y the ground, first 2+ and 3+ states were computed s i n c e t h e 4 + m e m b e r of t h e g r o u n d s t a t e r o t a tional band and the 2+ band head for the vibration h a d n o t b e e n e x p e r i m e n t a l l y r e s o l v e d (fig. 1 i n s e r t ) . In t h e c o u r s e of a n a l y s i s it b e c a m e e v i d e n t
Volume 25B, number 2
PHYSICS LETTERS
that s o m e a r b i t r a r i n e s s in the choice of Yo could be e l i m i n a t e d if the s e c o n d 2 + state w e r e a v a i l able. C o n s e q u e n t l y p a r t of the e x p e r i m e n t was r e p e a t e d with i n c r e a s e d e n e r g y r e s o l u t i o n . In the following c a l c u l a t i o n , a l l of the f i r s t five s t a t e s of 24Mg w e r e coupled u s in g the potential p a r a m e t e r s of W i l l s [3]. ~ o was d e t e r m i n e d by the magnitude of the d i f f e r e n t i a l c r o s s s e c t i o n f o r 2 + st a te at 4.23 MeV. The dashed l i n e s in fig. 4 a r e the r e s u l t s of this c a l c u l a t i o n . While in f o r w a r d a n g l e s m o s t of the c r o s s s e c t i o n s a r e well d e s c r i b e d , n e i t h e r the magnitude nor the shape is r e p r o d u c e d at l a r g e a n g l e s . If one c o n s i d e r s only the p a r t i a l wave a s p e c t s of the p r o b l e m , the obs e r v e d d i s c r e p a n c y is p o s s i b l y e x p l a i n e d by the work of D r i s k o et al. [8]. They have shown that, for l a r g e a n g l e s , the d i f f e r e n t i a l c r o s s s e c t io n for the e l a s t i c s c a t t e r i n g of s t r o n g l y a b s o r b e d p a r t i c l e s is s e n s i t i v e to the r e f l e c t i o n c o e f f i c i e n t s for l o w e s t p a r t i a l w a v e s which a r e r e f l e c t e d . F u r t h e r m o r e , they s t a t e that the lack of u n i q u e n e s s of the o p t i cal p o t e n t i a l s cannot be r e s o l v e d by a n a l y s i s of e l a s t i c s c a t t e r i n g e x p e r i m e n t s alone. T h e r e f o r e , it s e e m e d r e a s o n a b l e to s e a r c h for s o m e other p o t en t i a l s e t which, in a coupled channels c a l c u l a t i o n , m i g h t give a b e t t e r c o m p a r i s o n with the e x p e r i m e n t a l r e s u l t s . In o r d e r to l o c a t e o t h er p o t e n t i a l s , a s e r i e s of s i m p l e opti c a l m o d e l c a l c u l a t i o n s was m a d e u s in g the f o r w a r d angle e l a s t i c s c a t t e r i n g data. F r o m t h e s e c a l c u l a t i o n s a r e l a t i o n s h i p between the g o o d n e s s of fit (×2) and the r e a l well depth was d e t e r m i n e d . T h e shape of this c u r v e is quite s i m i l a r to the one obtained by D r i s k o f o r e l a s t i c alpha p a r t i c l e s c a t t e r i n g f r o m 58Ni at 43 MeV [8]. The m i n i m a in the ×2 c u r v e for our data o c c u r r e d r e g u l a r l y f o r i n c r e a s e s of about 50 MeV in well depth. Using the r e a l well depth of 90 MeV (second m i n i m a in X2) and its c o r r e s p o n d ing g e o m e t r i c a l p a r a m e t e r s , as d e t e r m i n e d by D r i s k o ' s p r e s c r i p t i o n [8], the coupled channels c a l c u l a t i o n was r e p e a t e d . The r e s u l t s a r e the
7 August 1967
so l i d line of fig. 4. F o r the f o r w a r d a n g l e s this c a l c u l a t i o n r e f l e c t s a m b i g u i t i e s which D r i s k o found f o r e l a s t i c s c a t t e r i n g , but the magnitude of the l a r g e angle data a r e m o r e f a v o r a b l y r e p r o duced than in the cal cu l at i o n for V= 40 MeV (first m i n i m u m in X2). Also the o s c i l l a t o r y b e h a v i o u r of the 3 + l a r g e angle c r o s s s e c t i o n is p r e dicted. F u r t h e r c a l c u l a t i o n s with even d e e p e r p o t en t i al s e t s gave no added i m p r o v e m e n t in comp a r i s o n with the data. In conclusion, a n g u l a r d i s t r i b u t i o n s in the r e a r h e m i s p h e r e for alpha p a r t i c l e s c a t t e r i n g f r o m the 3 + st at e of m a g n e s i u m exhibit s i m p l e r e g u l a r i t y . The e n e r g y v a r i a t i o n of the d i f f e r e n tial c r o s s sect i o n f r o m Ec~ = 36.1 to 42 MeV is s i m p l e and the c r o s s sect i o n p a t t e r n s f o r 24Mg and 26Mg a r e v e r y s i m i l a r . Also, it is evident that a coupled channels a p p r o a c h in t e r m s of the c o l l e c t i v e , r o t a t i o n - v i b r a t i o n m o d e l for 24Mg g i v e s q u a l i t a t i v e a g r e e m e n t with the data at 42 MeV. The a u t h o r s e x p r e s s t h e i r g r a t i t u d e to P r o f e s s o r J. G . W i l l s f o r adapting his coupled channels p r o g r a m f o r this p r o b l e m .
References 1. w . w . Eidson and J. G. Cramer Jr., Phys. Rev. Letters 12 (1962) 497. 2. J. Kokame, K. Fukunaga, N. Inoue and H.Nakamura, Phys. Letters 8 (1964) 342. 3. I.M.Naqib, Thesis, Univ. of Washington (1962), unpublished. 4. M. F. Reed, B.G. Harvey and D. L. Hendrie, UCRL Report 17299 (1967). 5. Taro Tamura, Nuel. Ph y s. 73 (1965) 241. 6. E.T. Boschitz, J.S. Vincent and R. W. Bercaw, Phys. Rev. Letters 13 (1964) 442. 7. J.G. Wills, Thesis, Univ. of Washington (1963), unpublished. 8. R. M. Drisko, G. R. Satchler and R. H. Bassel, Phys. Rev. Letters 5 (1963) 347.
83
LARGE
ANGLE
PHYSICS LETTERS
ALPHA
7 August 1967
PARTICLE SCATTERING FROM P A R I T Y S T A T E S IN M A G N E S I U M
THE
3 + UNNATURAL
J . S . V I N C E N T , E . T . BOSCHITZ and J. R. PRIEST N A S A - L e w i s R e s e a r c h Center. Cleveland. Ohio, USA
Received 20 June 1967
The differential cross sections for inelastic alpha particle scattering to large angles from 3+ states in 24Mg and 26Mg at 42 MeV show a clear diffraction type of pattern and pronounced enhancement compared with forward angles.
The excitation of u n n a t u r a l parity s t a t e s (~r ¢ (-1) J) in even n u c l e i by i n e l a s t i c alpha p a r ticle s c a t t e r i n g has r e c e i v e d c o n s i d e r a b l e a t t e n tion s i n c e E i d s o n and C r a m e r [1] showed it to be forbidden by the usual a s s u m p t i o n s of d i r e c t r e a c t i o n theory. Both 2- and 3 + states have been studied in s e v e r a l nuclei by alpha p a r t i c l e s c a t t e r i n g at v a r i o u s incident e n e r g i e s [1-4]. In m o s t c a s e s the a n g u l a r d i s t r i b u t i o n s cover only the forward hemisphere. Several reaction m e c h a n i s m s have been proposed, e.g. an e x change p r o c e s s or m u l t i p l e phonon excitation. Taken in total, however, the data give no o v e r whelming evidence for the p r e d o m i n a n c e of any one m e c h a n i s m . T h e o r e t i c a l l y , the only a n a l y s i s is that of T a m u r a [5] who c o m p a r e d the 24Mg data (0 < 90) of Kokame [2] with a coupled c h a n n e l s c a l c u l a t i o n , a s s u m i n g a collective, r o t a t i o n v i b r a t i o n model. A r e a s o n a b l e d e s c r i p t i o n of the data was obtained a f t e r the K = 2 m a t r i x e l e m e n t s w e r e modified somewhat a r b i t r a r i l y . The l a r g e angle s c a t t e r i n g from the 3+ state in 24Mg and 26Mg was u n d e r t a k e n to d e t e r m i n e its r e l e v a n c e to any of the p o s s i b l e r e a c t i o n nvechanismso An alpha p a r t i c l e b e a m of the NASA-Lewis cyclotron was m a g n e t i c a l l y a n a l y z e d and f o c u s s e d onto t a r g e t s having isotopic purity g r e a t e r than 99 p e r c e n t . The s c a t t e r e d alpha p a r t i c l e s were detected by s i l i c o n c o u n t e r s . The f i r s t m e a s u r e m e n t s , on 24Mg, a r e p r e s e n t e d in fig. 1. The a n g u l a r d i s t r i b u t i o n for the 3+ (Q = = -5.23 MeY) state a r e c o m p a r e d with the e l a s tic s c a t t e r i n g at Eat = 42 MeV. As was known b e f o r e , the f o r w a r d angle c r o s s section i s g r e a t ly inhibited [3]. However, at l a r g e r a n g l e s , not only is the 3 + c r o s s section enhanced c o m p a r e d with e l a s t i c s c a t t e r i n g , but a p r o n o u n c e d d i f f r a c tion type p a t t e r n r e s u l t s . Thus unique behaviour
I 0 ~2
w
I0" A
jn
•
J =0 g . s .
• ~
• ..
•*
"]
¢=~
2]~--'--] 0" o
~10 °
-o
•
Ja
I E (MeV]
2~=======~
•
I0-'
[
3:1~--~.-15.2~
24Mg
4.23
"
1.37 '0
• •~=
J".3 + ~1%;~+~¢+/~\~ 4~0--
sb
,20
0
Ocrn Fig. 1. D i f f e r e n t i a ] c r o s s s e c t i o n s f o r 2 4 M g ( a t , a t ) 2 4 M g . Q = 0 and Q = -5.23 M e V . T h e l i n e is a g u i d e to the eye
only. Insert shows the low lying levels in 24Mg. of the u n n a t u r a l p a r i t y state was not found for other excited s t a t e s (fig. 4). Since l a r g e angle e l a s t i c alpha p a r t i c l e s c a t t e r i n g can u n d e r g o r a p i d fluctuations with changes of a few MeV in the incident beam [6], an i n v e s t i g a t i o n was made to d e t e r m i n e whether this c o m p l i c a t i o n would be p r e s e n t for the 3 + d i f f e r e n t i a l c r o s s section. The l a r g e angle data for Eat = 36.1, 39.4, and 42.0 MeV shown in fig. 2 indicate that the p a t t e r n changes only slightly over this r a n g e . In a r e c e n t study, Reed et al. [4] find that this l a r g e angle p a t t e r n is v e r y s i m i l a r up to at l e a s t 50 MeV. This smooth b e h a v i o u r with energy is e n c o u r a g i n g f r o m the standpoint of an a n a l y s i s such as the coupled channel approach b a s e d on the collective model. 81
Volume 25B. n u m b e r 2
PHYSICS
LETTERS
7 August 1967
I0 °
42 MeV
,o-
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.,%, , ,, ,, :
s
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160
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Fig. 2. Large angle differential c r o s s sections for 24Mg(ot, ~')24Mg, Q=-5.23 MeV, E~ =36.1, 39.4, 42.0 MeV. T h e p r e d i c t i o n s of t h e c o l l e c t i v e m o d e l a r e n o t s e n s i t i v e t o t h e f i n e d e t a i l s of n u c l e a r s t r u c t u r e . An i n t e r e s t i n g o p p o r t u n i t y t o c o m p a r e t h e b e h a v i o u r of t h e l a r g e a n g l e 3 + c r o s s s e c t i o n s i n n e i g h b o r i n g n u c l e i i s a f f o r d e d by a m e a s u r e m e n t of t h e Q = - 3 . 9 7 M e V l e v e l in 26Mg. T h e d i f f e r e n t i a l c r o s s s e c t i o n s f o r 3 + s t a t e s i n 24Mg a n d 26Mg a r e p r e s e n t e d in fig. 3. T h e c l o s e a g r e e ment which is evident would lead one to believe t h a t t h e m a c r o s c o p i c s t r u c t u r e of t h e s e s t a t e s i s similar. S i n c e t h e 3 + c r o s s s e c t i o n s e e m s to b e s o well behaved in this energy region, a coupled c h a n n e l s c a l c u l a t i o n w a s p e r f o r m e d in 24Mg. T h e p r o g r a m of W i l l s [7], u t i l i z i n g t h e p o t e n t i a l e x p a n s i o n of T a m u r a w a s u s e d . N o a l t e r a t i o n of the K = 2 matrix elements was made, and the adjustable parameters were the optical potentials (same radial shape for real and imaginary parts), the deformation/3, and the zero point vibration
_u=-4.iz V - ~
i
~
.~1 i
.,.:_.
10o ,
,I IJ
10oj
I0"
,..o
J
I0"
,.J;-3~
,-,,' ' ) , 4 * ' f V [
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Fig. 4. Comparison of coupled channel calculations for f i r s t five levels in 24Mg, in the r o t a t i o n - v i b r a t i o n model with unnormalized data; Eot = 42 MeV. Optical potentials and deformations are: for the solid line V = = 90 MeV, W = 15 MeV, R = 4.2 fm, A = 0.57 fro, /3 0.30, 7 o = 0.425; for the dashed c u r v e V=40 MeV, W-13 MeV, R =4.9 fro, A =0.65 fm,/3 =0.28, yo=0.425.
lo
I0
40
80
120
160 (~cm
F i g . 3. C o m p a r i s o n of l a r g e angle d i f f e r e n t i a l e r o s s s e c t i o n s f o r 24Mg(o~,0~)24Mg, Q = - 5 . 2 3 M e V , and
26Mg(O~,~)26Mg, 82
Qj
-3.97 MeV. Both levels have ~ =3 +. Eot=42 MeV.
a m p l i t u d e 7 o . In a p r e l i m i n a r y c a l c u l a t i o n o n l y the ground, first 2+ and 3+ states were computed s i n c e t h e 4 + m e m b e r of t h e g r o u n d s t a t e r o t a tional band and the 2+ band head for the vibration h a d n o t b e e n e x p e r i m e n t a l l y r e s o l v e d (fig. 1 i n s e r t ) . In t h e c o u r s e of a n a l y s i s it b e c a m e e v i d e n t
Volume 25B, number 2
PHYSICS LETTERS
that s o m e a r b i t r a r i n e s s in the choice of Yo could be e l i m i n a t e d if the s e c o n d 2 + state w e r e a v a i l able. C o n s e q u e n t l y p a r t of the e x p e r i m e n t was r e p e a t e d with i n c r e a s e d e n e r g y r e s o l u t i o n . In the following c a l c u l a t i o n , a l l of the f i r s t five s t a t e s of 24Mg w e r e coupled u s in g the potential p a r a m e t e r s of W i l l s [3]. ~ o was d e t e r m i n e d by the magnitude of the d i f f e r e n t i a l c r o s s s e c t i o n f o r 2 + st a te at 4.23 MeV. The dashed l i n e s in fig. 4 a r e the r e s u l t s of this c a l c u l a t i o n . While in f o r w a r d a n g l e s m o s t of the c r o s s s e c t i o n s a r e well d e s c r i b e d , n e i t h e r the magnitude nor the shape is r e p r o d u c e d at l a r g e a n g l e s . If one c o n s i d e r s only the p a r t i a l wave a s p e c t s of the p r o b l e m , the obs e r v e d d i s c r e p a n c y is p o s s i b l y e x p l a i n e d by the work of D r i s k o et al. [8]. They have shown that, for l a r g e a n g l e s , the d i f f e r e n t i a l c r o s s s e c t io n for the e l a s t i c s c a t t e r i n g of s t r o n g l y a b s o r b e d p a r t i c l e s is s e n s i t i v e to the r e f l e c t i o n c o e f f i c i e n t s for l o w e s t p a r t i a l w a v e s which a r e r e f l e c t e d . F u r t h e r m o r e , they s t a t e that the lack of u n i q u e n e s s of the o p t i cal p o t e n t i a l s cannot be r e s o l v e d by a n a l y s i s of e l a s t i c s c a t t e r i n g e x p e r i m e n t s alone. T h e r e f o r e , it s e e m e d r e a s o n a b l e to s e a r c h for s o m e other p o t en t i a l s e t which, in a coupled channels c a l c u l a t i o n , m i g h t give a b e t t e r c o m p a r i s o n with the e x p e r i m e n t a l r e s u l t s . In o r d e r to l o c a t e o t h er p o t e n t i a l s , a s e r i e s of s i m p l e opti c a l m o d e l c a l c u l a t i o n s was m a d e u s in g the f o r w a r d angle e l a s t i c s c a t t e r i n g data. F r o m t h e s e c a l c u l a t i o n s a r e l a t i o n s h i p between the g o o d n e s s of fit (×2) and the r e a l well depth was d e t e r m i n e d . T h e shape of this c u r v e is quite s i m i l a r to the one obtained by D r i s k o f o r e l a s t i c alpha p a r t i c l e s c a t t e r i n g f r o m 58Ni at 43 MeV [8]. The m i n i m a in the ×2 c u r v e for our data o c c u r r e d r e g u l a r l y f o r i n c r e a s e s of about 50 MeV in well depth. Using the r e a l well depth of 90 MeV (second m i n i m a in X2) and its c o r r e s p o n d ing g e o m e t r i c a l p a r a m e t e r s , as d e t e r m i n e d by D r i s k o ' s p r e s c r i p t i o n [8], the coupled channels c a l c u l a t i o n was r e p e a t e d . The r e s u l t s a r e the
7 August 1967
so l i d line of fig. 4. F o r the f o r w a r d a n g l e s this c a l c u l a t i o n r e f l e c t s a m b i g u i t i e s which D r i s k o found f o r e l a s t i c s c a t t e r i n g , but the magnitude of the l a r g e angle data a r e m o r e f a v o r a b l y r e p r o duced than in the cal cu l at i o n for V= 40 MeV (first m i n i m u m in X2). Also the o s c i l l a t o r y b e h a v i o u r of the 3 + l a r g e angle c r o s s s e c t i o n is p r e dicted. F u r t h e r c a l c u l a t i o n s with even d e e p e r p o t en t i al s e t s gave no added i m p r o v e m e n t in comp a r i s o n with the data. In conclusion, a n g u l a r d i s t r i b u t i o n s in the r e a r h e m i s p h e r e for alpha p a r t i c l e s c a t t e r i n g f r o m the 3 + st at e of m a g n e s i u m exhibit s i m p l e r e g u l a r i t y . The e n e r g y v a r i a t i o n of the d i f f e r e n tial c r o s s sect i o n f r o m Ec~ = 36.1 to 42 MeV is s i m p l e and the c r o s s sect i o n p a t t e r n s f o r 24Mg and 26Mg a r e v e r y s i m i l a r . Also, it is evident that a coupled channels a p p r o a c h in t e r m s of the c o l l e c t i v e , r o t a t i o n - v i b r a t i o n m o d e l for 24Mg g i v e s q u a l i t a t i v e a g r e e m e n t with the data at 42 MeV. The a u t h o r s e x p r e s s t h e i r g r a t i t u d e to P r o f e s s o r J. G . W i l l s f o r adapting his coupled channels p r o g r a m f o r this p r o b l e m .
References 1. w . w . Eidson and J. G. Cramer Jr., Phys. Rev. Letters 12 (1962) 497. 2. J. Kokame, K. Fukunaga, N. Inoue and H.Nakamura, Phys. Letters 8 (1964) 342. 3. I.M.Naqib, Thesis, Univ. of Washington (1962), unpublished. 4. M. F. Reed, B.G. Harvey and D. L. Hendrie, UCRL Report 17299 (1967). 5. Taro Tamura, Nuel. Ph y s. 73 (1965) 241. 6. E.T. Boschitz, J.S. Vincent and R. W. Bercaw, Phys. Rev. Letters 13 (1964) 442. 7. J.G. Wills, Thesis, Univ. of Washington (1963), unpublished. 8. R. M. Drisko, G. R. Satchler and R. H. Bassel, Phys. Rev. Letters 5 (1963) 347.
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