S013IA o02l+o0t?l+o09l Q”13IAo08!
Chapters 3 and 4 of this thesis have outlined the various attempts
27 32
to interpret experimentally measured properties of A1 and S in terms of
nuclear models. Experiments to measure some of the properties of excited
states of these two nuclei have been described and the results obtained have been compared to the predictions of the models. No single model is,
27
at present, capable of providing a complete description of either A1 or
32
S; several may be used with varying degrees of success to account for
the different properties such as spins, parities and excitation energies of these two nuclei.
27 32
It is interesting to consider the nuclear shapes of A1 and S
in relation to their positions in the 2s-ld shell and to the predictions 27
of nuclear models. A1 lies in the transition region of the 2s-ld shell
where the nuclear shapes are changing from a strong prolate deformation
around mass A ^ 23-25 to the oblate deformation found at A a, 28. In the
upper region of the shell, there is a tendency for the nuclei to assume
oblate or prolate shapes with smaller deformations; sphericity is achieved
40
at the shell closure at Ca. It is not certain whether the deformation of
27
'Al (Q = 0.15 barns, Fu 69) is prolate or oblate as this is model-dependent
on the assignment of K = 1/2 or 5/2 for the ground state. Hartree-Fock
27
calculations (Ri 68) suggest an oblate shape for A1 and the Coriolis
band-mixing calculations of Dehnhard (De 72) assuming an oblate deformation 28
close to that of Si, give better agreement with experimental B(E2) values
than do similar calculations performed by Malik and Scholz (Ma 67) with 32
a prolate shape. The measured value of Qq = 0.6 barns (Na 70) for S
128
quadrupole moment is surprisingly large in view of the proposed vibrational
32 t 24
character of S ; it is comparable to that of known rotators (e.g. Mg)
in the 2s-Id shell. This result is consistent with the Hartree-Fock
calculations of Bassichis et al. (Ba 67) but not with those of Banerjee 32
et al. (Ba 69) and Zofka and Ripka (Zo 71) for the ground state of S.
However, as pointed out by Bar-Touv and Goswaini (Ba 69a), the spherical and deformed solutions of the Hartree-Fock Hamiltonians for light closed sub shell nuclei are close in energy and may coexist as two distinct states in the same nucleus.
27
The non-zero deformation of Al suggests possible rotational
behaviour; however, the rotational model has been shown to have limited
success in accounting for the properties of this nucleus. This is somewhat
surprising in view of the successful description of neighbouring nuclei, 29
most recently Si (Vi 73), in terms of the rotational model. This model
has also been applied to the low-lying 0*, 2*, 0* and 4^ levels of ^2S (Ba 69a) with some success on the assumption of a deformed ground state, but no other possible members of the ground-state rotational band have yet been located.
The proposed spherical shape (from Hartree-Fock calculations) for 32
the ground state of S and the observed level scheme suggested that this
nucleus is a possible vibrator; the harmonic vibrator model has been quite
successful in explaining many of the properties of low-lying positive parity
states. The extension of the model to negative parity states around 5-7 MeV
is limited by lack of experimental information on spins, parities and gamma
decay rates. There is, however, a serious difficulty in reconciling such
vibrational behaviour with the large quadrupole moment for the 2^ state;
Olin et al. (01 73) have recently given a preliminary report of a new
measurement of Q^(2j) for S which yields a result ^ 0 . 1 5 barns, i.e., in
better agreement with the vibrational picture.
129
t h e t r a d i t i o n a l h arm o n ic v i b r a t o r model assum es a s p h e r i c a l e q u i l i b r i u m s h a p e . 114
Such a d i f f i c u l t y a l s o e x i s t s w ith Cd and o t h e r n u c l e i a r o u n d A ^ 110,
w hich d i s p l a y v i b r a t i o n a l f e a t u r e s n o t e x p l i c a b l e by t h e r o t a t i o n a l model.
+ 114
A l a r g e number o f m e asu rem en ts o f Q ( 2 p f o r Cd y i e l d v a l u e s r a n g i n g from
n e a r z e r o , a s e x p e c te d f o r a h arm onic v i b r a t o r , t o t h e r i g i d r o t a t o r v a l u e .
A r e c e n t , more p r e c i s e m easurement (Be 71) ( u s i n g t h e r e o r i e n t a t i o n e f f e c t )
i n d i c a t e s Q(2*) i s i n t e r m e d i a t e b etw ee n t h e s e e x tre m e s ( - 0 . 3 b a r n s ) b u t i s
s t i l l i n c o n s i s t e n t w i t h t h e p i c t u r e o f a s p h e r i c a l h arm o n ic v i b r a t o r .
E x t e n s i v e t h e o r e t i c a l e f f o r t h a s b e e n ex p en d e d , w i t h some s u c c e s s ,
t o r e c o n c i l e t h e supposed v i b r a t i o n a l b e h a v i o u r w i t h n o n - s p h e r i c a l s h a p e s
and i t i s n e c e s s a r y t o i n t r o d u c e an h arm o n ic e f f e c t s i n t o t h e c a l c u l a t i o n s .
F o r ex am p le, Tamura and Udagawa (Ta 66) p r o p o s e t h a t t h e m a g n itu d e o f
Q(2*) f o r 114Cd can be r e p r o d u c e d by a l l o w i n g m ix in g o f t h e 2* and 2^ s t a t e s . 114
S o re n s o n (S0 67) h a s c o n s i d e r e d an h arm o n ic v i b r a t i o n s i n Cd i n a
c a l c u l a t i o n w hich r e p l a c e s p a i r s o f f e r m i o n s i n an even f e r m io n s y stem by
an e x p a n s io n i n te rm s o f b o so n o p e r a t o r s . N o n - l i n e a r te r m s up t o t h i r d
o r d e r were i n c l u d e d i n t h e c a l c u l a t i o n . I t i s p o s s i b l e t h a t s i m i l a r 32
c a l c u l a t i o n s a l o n g t h e s e l i n e s f o r S w i l l a c c o u n t f o r t h e l a r g e q u a d r u p o le
4* 4"
moment and a p p a r e n t v i b r a t i o n a l c h a r a c t e r . M ixing o f t h e 2^ and 2 s t a t e s 32
o f S i s a l r e a d y r e q u i r e d (Ga 71, Ha 71) t o a c c o u n t f o r t h e c r o s s - o v e r
t r a n s i t i o n from t h e 2* s t a t e . However, d i f f i c u l t i e s t h e n a r i s e w ith
r e p r o d u c i n g t h e r a t i o E ( 2 * ) /E ( 2 * ) .
X Ca A 1
The l a r g e s c a l e s h e l l model c a l c u l a t i o n s t h a t h av e b een c a r r i e d
27 32
o u t on A1 and S, p a r t i c u l a r l y by W i ld e n th a l and h i s c o l l a b o r a t o r s ,
show t h a t t h i s model may be c a p a b l e o f g i v i n g a c o m p re h e n s iv e a c c o u n t o f
+ 32
t h e i r p r o p e r t i e s . In p a r t i c u l a r , t h e m a g n itu d e and s i g n f o r Q(2^) f o r S
was f a i r l y w e ll r e p r o d u c e d by t h e m odel. P r e s e n t d e f i c i e n c i e s may be p a r t l y
due t o t h e a d o p t i o n o f t r u n c a t e d c o n f i g u r a t i o n s p a c e s f o r t h e c a l c u l a t i o n s . 32
For ex am p le, i g n o r i n g p a r t i c l e e x c i t a t i o n t o t h e l f ^ ^ o r b i t s i n ' S r e s t r i c t s
130.
REFERENCES
(Aj 71) F. Aj zenberg-Selove, Nucl. Phys. A166 (1971) 1. (Ar 68) A. Arima, S. Cohen, R.D. Lawson and M.H. MacFarlane,
Nucl. Phys. A108 (1968) 94.
(Ar 68a) A.J. Armini, J.W. Sunier and J.R. Richardson, Phys. Rev. 165 (1968) 1194.
(Ba 67) W.H. Bassichis, A.K. Kerman and J.P. Svenne, Phys. Rev. 160 (1967) 746.
(Ba 69) M.K. Banerjee, C.A. Levinson and G.J. Stephenson, Phys. Rev. 178 (1969) 1709.
(Ba 69a) J. Bar-Touv and A. Goswami, Phys. Lett. 28B (1969) 391. (Be 30) H.A. Bethe, Ann. Physik 5. (1930) 325.
(Be 69) R.A.I. Bell, J. L ’Ecuyer, R.D. Gill, B.C. Robertson, I.S. Towner and H.J. Rose, Nucl. Phys. A133 (1969) 337. (Be 71) Z. Berant, R.A. Eisenstein, J.S. Greenberg, R. Horowitz,
U. Smilansky, P.N. Tandon, A.M. Kleinfeld and H.G. Mäggi, Phys. Rev. Lett. 27_ (1971) 110.
(Be 73) R.A.I. Bell, Nucl. Instrum. Meth. 106 (1973) 307. (Bh 62) K.H. Bhatt, Nucl. Phys. 39 (1962) 375.
(Bi 68) J. Birkholz and F. Beck, Phys. Lett. 28B (1968) 18. (B1 33) F. Bloch, Ann. Physik, 1_6 (1933) 285.
(Bl 66) A.E. Blaugrund, Nucl. Phys. 88^ (1966) 501. (Bo 13) N. Bohr, Phil. Mag. 25 (1913) 10.
(Bo 52) A. Bohr, Dan. Mat. Fys. Medd. :26 (1952) No.14.
(Bo 53) A. Bohr and B.R. Mottelson, Dan. Mat. Fys. Medd. 27_ (1953) No.16.
131
(Bo 64) E.C. Booth, B. Chasan and K.A. Wright, Nucl. Phys. 57 (1964) 403. (Br 68) D. Branford, Ph.D. Thesis, University of Manchester (1968).
(Br 73) C. Broude, F.A. Beck and P. Engelstein, preprint received from Centre de Recherches Nucleaires, Strasbourg (1973).
(Ca 70) B. Castel and K.W.C. Stewart, Can. J. Phys. 48 (1970) 1490. (Ca 70a) K.W. Carter, Ph.D. Thesis, Australian National University (1970) (Ca 71) B. Castel, K.W.C. Stewart and M. Harvey, Phys. Rev. C4 (1971)
1966.
(Ca 71a) K.W. Carter, D.C. Kean, C.J. Piluso and R.H. Spear, Aust. J. Phys. 24_ (1971) 471.
(Ca 72) J.F. Cairns, G.G. Frank, H. Grawe, M.W. Greene, D. Kelly, J.A. Kuehner and A.A. Pilt, Physics in Canada 28^ (1972) 27.
(Ca 73) P.E. Carr, D.C. Bailey, L.L. Green, A.N. James, J.F. Sharpey- Schafer and D.A. Viggars, J. of Phys. A6^ (1973) 705.
(Co 72) W.F. Coetzee, M.A. Meyer and D. Reitmann, Nucl. Phys. A185 (1972) 644.
(Cr 68) G.M. Crawley and G.T. Garvey, Phys. Rev. 167 (1968) 1070. (Da 58) A.S. Davydov and G.F. Filippov, Nucl. Phys. 8^ (1958) 237. (De 72) D. Dehnhard, Phys. Lett. 38B (1972) 389.
(Do 68) 0. Dorum, Physica Norvegica 3_ (1968) 31. (Ei 64) E. Eichler, Rev. Mod. Phys. 3b (1964) 809.
(El 68) R.V. Elliott, Ph.D. Thesis, Australian National University (1968). (El 68a) R.V. Elliott, T.R. Ophel and R.H. Spear, Nucl. Phys. A115 (1968)
673.
(El 68b) R.V. Elliott, K.W. Carter and R.H. Spear, Nucl. Instrum. Meth.
S9_ (1968) 29.
132
(En 67) P.M. Endt and C. Van der Leun, Nucl. Phys. A105 (1967) 1.
(En 73) P.M. Endt and C. Van der Leun, preprint received from
University of Utrecht (1973).
(Er 66) F.C. Erne, Nucl. Phys. 84 (1966) 91.
(Ev 67) D. Evers, J. Hertel, T.W. Retz-Schmidt and S.J. Skorka, Nucl.
Phys. 91 (1967) 472.
(Ev 68) H.C. Evans, B. Castel, J.M. Montague, W.R. Paulson and
W.M. Zuk, Bull. Am. Phys. Soc. U (1968) 87.
(Fa 59) L.W. Fagg and S.S. Hanna, Rev. Mod. Phys. 31 (1959) 711.
(Fa 66) B. Fastrup, P. Hvelplund and C.A. Sautter, Dan. Mat. Fys. Medd.
35 (1966) N o . 10.
(Fe 65) A.J. Ferguson, "Angular Correlation Methods in Gamma Ray
Spectroscopy", North-Holland (1965).
(Fo 70) I. Forsblom, P. Paukku and S. Penttinen, Comm. Phys. Math.
40 (1970) 1.
(Fu 69) G.H. Fuller and V.W. Cohen, Nucl. Data A5 (1969) 433.
(Ga 71) G.T. Garvey, K.W. Jones, L.E. Carlson, D.A. Hutcheon, A.G.
Robertson and D.F.H. Start, Nucl. Phys. A160 (1971) 25.
(Gl 64) P.W.M. Glaudemans, G. Wiechers and P.J. Brussaard, Nucl. Phys.
56 (1964) 529.
(Gl 71) P.W.M. Glaudemans and C. Van der Leun, Phys. Lett. 34B (1971) 41.
(Gl 71a) P.W.M. Glaudemans, P.M. Endt and A.E.L. Dieperink, Annals of
Physics 6_3 (1971) 134.
(Go 56) S. Goldstein and I. Talmi, Phys. Rev. 102 (1956) 589.
(Gr 68) A. Graue, L. Herland, J.R. Lien and E.R. Cosman, Nucl. Phys.
A120 (1968) 513.
(Gr 72) M.W. Greene, J.A. Kuehner, G.C. Ball, C. Broude and J.S.
133
(Gr 73) H. Grawe, J . E . C a i r n s , M.W. G r e e n e and J . A . K u e h n e r , p r e p r i n t r e c e i v e d fro m M cM aster U n i v e r s i t y ( 1 9 7 3 ) .
(Ha 68) 0 . H ä u s s e r , D. P e l t e and J . F . S h a r p e y - S c h a f e r , C an. J . P h y s . 46 (196 8 ) 1145.
(Ha 71) 0 . H ä u s s e r , T .K . A l e x a n d e r , A .B . McDonald and W.T. Diamond, N u c l . P h y s . A175 (1 9 7 1) 593.
(Hi 69) R.G. H i r k o , P h .D . T h e s i s , Y a le U n i v e r s i t y ( 1 9 6 9 ) .
(Hi 6 9a) R.G. H i r k o and D.A. B ro m le y , P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e on P r o p e r t i e s o f N u c l e a r S t a t e s , M o n t r e a l ( 1 9 6 9 ) .
(Ho 66) J . Hough a n d W.L. M outon, N u c l . P h y s . 76 (1 9 6 6 ) 24 8.
(Hv 68) P. H v e l p l u n d an d B. F a s t r u p , P h y s . Rev. 165 (1 9 6 8 ) 4 0 8 . ( I n 69) F. I n g e b r e t s e n , T .K . A l e x a n d e r , 0 . H ä u s s e r a n d D. P e l t e , Can. J . P h y s . 47 (1 9 69 ) 1295. ( I n 71) F . I n g e b r e t s e n , B.W. S a r g e n t , A . J . F e r g u s o n , J . R . L e s l i e , A. H e n r i k s o n and J . H . M o n ta g u e , N u c l . P h y s . A161 (1 9 7 1 ) 4 3 3 . (Ka 71) J . K ä l l n e and 0 . S u n d b e r g , P h y s i c a S c r i p t a 4 (1 97 1) 243 . (Ke 63) I . K e l s o n , P h y s . R ev. 132 (1963) 2189 .
(Ke 69) D.C. Kean, K.W. C a r t e r and R.H. S p e a r , N u c l . P h y s . A123 (196 9 ) 430 .
(Ke 72) D.C. Kean and R.W. O l l e r h e a d , C an. J . P h y s . 50 (1 9 7 2 ) 1539.
(Ku 67) J . A . K uehner a n d E. A l m q v i s t , C an. J . P h y s . 45 (1 9 6 7 ) 1605.
(L i 53) J . L i n d h a r d an d M. S c h a r f f , Dan. M at. F y s . Medd. 27 (1 9 5 3 ) N o . 15.
(L i 54) J . L i n d h a r d , Dan. M at. F y s . Medd. 28 (1954) N o .8.
(L i 61) J . L i n d h a r d and M. S c h a r f f , P h y s . R ev. 124 (1 96 1) 128.
(L i 6 1 a ) A .E . L i t h e r l a n d a n d A . J . F e r g u s o n , Can. J . P h y s . 39^ (1 9 6 1 ) 788.
134 (L i 63) J . L i n d h a r d , M. S c h a r f f a n d H .E . S c h i s t t , Dan. M a t. F y s . Medd. 33 (19 63) N o . 14. (L i 64) A .E . L i t h e r l a n d , i n " R a d i a t i v e T r a n s i t i o n s i n N u c l e a r S t r u c t u r e a n d E l e c t r o m a g n e t i c I n t e r a c t i o n s " , N. M acD onald, E d . , O l i v e r and Boyd ( 1 9 6 4 ) .
(L i 68) J . L i n d h a r d , V. N i e l s e n and M. S c h a r f f , Dan. M at. F y s . Medd. 36 (1968) N o . 10.
(Lo 64) R. Lom bard, P. K o ssan y i-D em ay and G .R. B i s h o p , N u c l . P h y s .