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Deep hole states
Nuclear Physics A396(1983)39c-60c. ©North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher.
39c
DEEP HOLE STATES Jean MOUGEY DRF/CPN, Centre d'Etudes Nucl~aires de Grenoble 85X, 38041 Grenoble Cedex, France A b s t r a c t : Our present knowledge on d e e p l y - l y i n g hole s t a t e s from one nucleon pick-up and knock-out r e a c t i o n s is reviewed. Results on n e u t r o n - h o l e s t a t e s in the Zr, Sn and Pb region from recent pick-up measurements are discussed, w i t h emphasis on the use o f p o l a r i z e d beams f o r spin assignments. The s i t u a t i o n concerning proton hole s t a t e s in medium-mass (6~I00 MeV) nucleon pick-up and knock-out r e a c t i o n s . We s h a l l see t h a t our present knowledge o f nuclear hole s t a t e s i s s t i l l poor. Inner hole s t a t e s do not appear as simple as would be bare holes in a Hartree-Fock nucleus, and a more o p e r a t i o n a l p i c t u r e would be the one of a "dressed h o l e " , or "quasi hole ''1) l i k e sketched in f i g . i . Due to the r e s i d u a l i n t e r a c t i o n which makes t h a t the hole s t a t e is not an e i g e n s t a t e f o r the r e s i d u a l nucleus and also to f i n i t e escape widths when i n n e r holes are consid~re~ the s t r e n g t h d i s t r i b u t i o n can be broad (F ~ 10 Mev) and the centroTd energy E~ s h i f t e d w i t h respect to the p o s i t i o n o f the s i n g l e - p a r t i c l e energy - c~. In f a c t , T, Ec and the peak energy cannot be unambiguously obtained in general,~as : ( i ) the true d i s t r i b u t i o n can be more complicated than the one sketched in f i g . I , ( i i ) the s t r e n g t h d i s t r i b u t i o n corresponding to d i f f e r e n t s h e l l s are found to s t r o n g l y o v e r l a p ( i i i ) s h o r t range c o r r e l a t i o n s can produce a background which cannot be a t t r i b u t e d to a s p e c i f i c s h e l l , and could make r be i n f i n i t e , i f defined as the second moment o f the s t r e n g t h d i s t r i b u t i o n =) In t h i s t a l k , a f t e r a s h o r t formal i n t r o d u c t i o n to the s i n g l e nucleon s p e c t r a l f u n c t i o n as obtained from knock -F-.o, E c out arld pick-up r e a c t i o n s , I s h a l l discuss b r i e f l y ( s e c t i o n 3) some recent r e s u l t s on deep hole s t a t e s in heavy (Z ~ 4 0 ) n u c l e i . Section 4 w i l l concern deep holes in medium (3 < Z ~ 30) nuclei s t u d i e d through ( e , e ' p ) and (p,2p) r e a c t i o n s In s e c t i o n 5, I s h a l l b r i e f l y review recent attempts to compute deep hole s t r e n g t h d i s t r i b u t i o n s . E F i g . 1. S k e t c h o f a h o l e s t r e n g t h d i s t r i b u . t i o n f o r a given s h e l l model o r b i t , exhibiting a resonance-like structure on a flat background (hatched) resulting from short range correlations.
40c
J. MOUGEY 2. The spectral function
Assuming in what follows that both knock-out and pick-up reactions r e s u l t in a sudden removal of a nucleon, the essential nuclear q u a n t i t y which is measured is the diagonal spectral function 3'4) S(X,E)
=
a~ 6(H-E-c~)
%IA> = Z I
A>I 2 6 ( E - ~ _ l + g ~)
(1)
f
where II> is an orthonormal basis of single p a r t i c l e states. Energy and momentum conservation in ( e , e ' p ) (or in (pz2p)) reactions determine the separation energy E and the r e c o i l momentum p~ 1' which, in pla~e wave impulse approximation (PWIA) is opposite to the i n i t i a T - p r o t o n momentum p. Then one obtains a plane wave representation of the spectral function S(p,E), a f t e r d i v i d i n g the experimental cross section by some " o f f - s h e l l " e l e c t r o n - ( o r proton-) proton cross-section. Equation ( i ) becomes
S(~,E) = ~ [#fi(~)l%(E-~Af 1 + c~)
(2)
where ~ f i ( P ) = < A - l , fta(~)lA > is the Fourier transform of the overlap i n t e g r a l betweenIthe i n i t i a l and f i n a l nuclear states. In IPSM, ~fi is i d e n t i c a l to the single p a r t i c l e wave function for an occupied o r b i t ~ = { n l j } and the spectral function takes the form S(~,E) = ~ P ( E ) I ~ ( ~ ) I ~ with P (E) = 6(E+~ ) (3) G
DEEP HOLE STATES
41c
t~_1 which make up the Born approximation reaction cross section in i t s most general form has been investigated by Boffi et al. 9) for the ~60(e,e'p)15N^. reaction. I f a factorized approximation holds, the four functions must be ~ i d e n t i c a l to the spectral function. One sees t h a t , when the d i s t o r t i n g potential contains a spin o r b i t term, both to~ and t 1 _ 1 a r e d i f f e r e n t from the spectral function, and the factorization is destroyed Io), although t01 and t1_~ give rather small contributions in usual kinematical conditions.
o-,f
(tin 3) 0.4
0.2
0.2
0.0
0.0
/
/
/ //X
-0.2
.~"
/
1
/'
I ! 5
I
-0.2
/ "Ix -5
l~
i I
-0.4
- 0.4
(o)
-0.6
i /
//
(b)
,, -0.6
!/' I
0°
15 °
30°
4 5°
0°
I
30°
15 =
Y
4 5°
F
Fig. 3. Structure functions for the pl/2 hole in 160 entering in the (e,e'~) cross section as a function of the angle ¥ between the outgoing proton momentum p' and momentum transfer q. The kinematics is such that q = 2.3 fm-I and p'2/2M = 100Mev. Full l i n e too, dotted l i n e t1~, dashed l i n e to~, dot-dashed l i n e t~-~. In (b) the optical potential contains a s p i n - o r b i t term, which is absent in (a) (from r e f . 9 ) ) Nevertheless, PWIA, or the factorized DWIA represent good f i r s t order approximationsespecially for the (e,e'p) reaction and have been used in most of the analysis of knock-out experiments. Then instead of (3), the quantity deduced from the data is a "distorted spectral function" sD(~A_f,E) : ~
(E)I,~(~A_I)I
2
(4)
where the i i distorted momentum d i s t r i b• u t i o. n s j l ~D are obtained from bound and optical model scattering state wave functions, Pick-up experiments are much easier to perform and have a superior ($100 keY) energy resolution. On the other hand, they are more strongly affected by distortion, and the strong absorption induces a high angular momentum s e l e c t i v i t y through matching conditions. Thus, instead of (2), the spectral function is obtained in an orbital basis through the spectroscopic factors C2Snlj = I < A-1,fl anl j [A>I2
(5)
for excitation energies below~,15 MeV, and under matched conditions. However, i t is known that uncertainties exist in the absolute values of the
42c
J. MOUGEY
spectroscopic f a c t o r s as determined in standard DWBA a n a l y s i s . They have several o r i g i n s , namely : the c o n t r i b u t i o n s from m u l t i s t e p processes, the energy dependence o f the overlap i n t e g r a l s , the incomplete knowledge o f the r a d i a l bound s t a t e wave functions at l a r g e d i s t a n c e , the high level density at high e x c i t a t i o n energy and the experimental energy cut o f f . I t has been shown r e c e n t l y t h a t the knowledge of r a d i a l wave f u n c t i o n f o r the valence nucleon in odd-A nuclei from e l a s t i c magnetic e l e c t r o n s c a t t e r i n g can g r e a t l y improve the d e t e r m i n a t i o n of absolute spectroscopic f a c t o r s ' 1 ) . In the case o f the nuclei 51V and 87Sr f o r which the valence wave functions @~(r) and the diagonal one body d e n s i t i e s , d i r e c t l y r e l a t e d to the spectroscopic f a c t o r s , are known from e l e c t r o n s c a t t e r i n g . i t has been found t h a t the spectroscopic f a c t o r s from pick-up r e a c t i o n s , when c o r r e c t e d so as to be coherent w i t h the valence o r b i t r a d i i , are 30 to 40 % below the s h e l l model sum r u l e . Moreover, at high i n c i d e n t energies which are necessary to i n v e s t i g a t e deeper s h e l l s 12'~3) the angular d i s t r i b u t i o n s are less c h a r a c t e r i s t i c due to l a r g e r angular momentum mismatch, and the e x t r a c t i o n o f C2S is more p r o b l e m a t i c . However, pick-up r e a c t i o n s are, f o r the moment, almost the unique source o f i n f o r m a t i o n on neutron hole s t a t e s . 3. Recent r e s u l t s in pick-up reactions : Deep hole states in heavy nuclei Pick-up r e a c t i o n s - ( p , d ) , ( d , t ) , ( 3 H e , ~ ) - h a v e been w i d e l y used in the l a s t few years to i n v e s t i g a t e neutron hole states in the f i r s t i n n e r s h e l l o f medium-mass and heavy n u c l e i . Moreover, using p o l a r i z e d beams i t has been p o s s i b l e to d i s t i n g u i s h between the j = ~ + 1/2 and j = ~ - 1/2 component o f a given o r b i t a l momentum d o u b l e t . Since several d e t a i l e d reviews have been made r e c e n t l y by e x p e r t s 1 4 ' 1 ~ ) , I s h a l l give only a b r i e f summary of the present status and discuss a few examples. In p a r t i c u l a r , f o r lack o f t i m e , I s h a l l omit the discussion on t w o - p a r t i c l e pick-up r e s u l t s (see refs 14,16-21). 3.1 STATUS OF DEEP NEUTRON HOLE STATES FROM PICK-UP REACTIONS Fig. 4 shows as an example e x c i t a t i o n energy spectra in 115Sn and 2°TPb from the (3He,~) r e a c t i o n at 283 MeV i n c i d e n t energy22). This high value allows to extend the e x c i t a t i o n energy range up to 50 MeV f o r the f i r s t time. I t involve~ a high momentum mismatch which s t r o n g l y reduces the small h - t r a n s f e r c o n t r i b u t i o n s . In the iZSSn spectrum, the Ig9/2 hole s t a t e manifests i t s e l f as a sharp l i n e at 5.3 MeV, as expected from previous works , but also as a broader s t r u c t u r e peaked at 7.5 MeV (~ 5 MeV FWHM). This d i s t r i b u t i o n agrees reasonably well w i t h the p r e d i c t i o n s of Soloviev et a i . 2 3 ) , who consider the coupling o f the hole s t a t e w i t h I h o l e - I phonon and I hole-2 phonon s t a t e s . The broad s t r u c t u r e around 15 MeV, below the i s o b a r i c analog s t a t e s (9/2 + ) observed at 13.26 and 14.76 MeV, has been a t t r i b u t e d to I f 7 / 2 - h o l e s t r e n g t h by analysis o f the angular d i s t r i b u t i o n . In 2°sPb, 53 % of the lh 11/2 s t r e n g t h is found in the bump between 6.6 and 9.8 MeV but the experimental cross s e c t i o n up to 20 MeV may account f o r the missing lh 11/2 and t o t a l Ig 7/2 and l g 9/2 s t r e n g t h s , although no concentration of s t r e n g t h is observed f o r these s t a t e s . Using Hartree-Fock p r e d i c t i o n s to l o c a t e a l l i n n e r neutron s t a t e s down to I s , t h e i r c o n t r i b u t i o n s have been evaluated. A f t e r s u b s t r a c t i n g a f l a t background as shown in f i g . 4, the i n t e g r a t e d experimental cross s e c t i o n in both nuclei is found to exceed the p r e d i c t i o n f o r the t o t a l sum r u l e by ~ 27 %. For heavy n u c l e i , the experimental s i t u a t i o n can be summarized as f o l l o w s : (a) in the Zr r e g i o n 2 ~ ' 2 ~ ) , 50 to 70 % of the i f 7/2 s t r e n g t h is found at E ~ 5.2. MeV, which gives a s p i n - o r b i t s p l i t t i n g f o r the I f o r b i t o f ~ 3 MeV. (b) in the Sn region (Cd, Sn, Te) d e t a i l e d r e s u l t s ( s e e l " ' I s ) and references there i n ) have been obtained on the fragmentation o f the l g 9/2 s t r e n g t h , o f which 30 to 60 % is found around 5.5. MeV, and 20 to 50 % around 8 MeV. The f 5/2 s t r e n g t h is centered around I i MeV22'33) and the f 7/2 s t r e n g t h c o n t r i b u t e s mainly in the range 12-19 MeV22)
DEEP HOLE STATES
2°8pb(3He,cx)2°Tpb
ll6Sn(3He, o()115Sn Einc: 283PleV ,
o -
,,q ~
8°
E inc =2831"4eV m
,,q o
43c
0:8
:~°5
°
0:2
°
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°
,!
i'
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;
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, ,
',
:::, D~
05
B
I
I
i
,
i
20
10
N
2-
L E x (HEY)
.'.0
3'0
~10
10
E x (MeV)
.'.0
30
Fig. 4 E x c i t a t i o n energy spectra o f the zZ6Sn (3He,~) and 2°epb(~He,~) reactions at 283 MeV i n c i d e n t energy. Hatched peaks are from contaminants. The p o s i t i o n of some subshells as given by Hartree-Fock c a l c u l a t i o n s from Beiner et a l . are i n d i c a t e d by arrows (from r e f . 2 2 ) . (c) in the Sm isotopes Is) a concentration of ig 9/2 strength has been observed at 7.6Mev in ~"3Sm. But in the heavier isotopes, a large spreading of the ig 9/2 strength and i t s overlap with the 2d 5/2 and Ig 7/2 c o n t r i b u t i o n s seem to occur. (d) in the Pb region, the p o s i t i o n of the lh 11/2 discussed above and confirmed by other results 26,27) leads to a s p i n - o r b i t s p l i t t i n g for the lh o r b i t of 4.9 MeV as compared to 5.6 MeV from Hartree-Fock c a l c u l a t i o n s 2 8 ) . 3.2 NEUTRON HOLE STATES STUDIES USING POLARIZED BEAMS In s i n g l e nucleon t r a n s f e r r e a c t i o n s , the angular d i s t r i b u t i o n s allow to determine the t r a n s f e r r e d o r b i t a l angular momentum ~, but not the t o t a l spin j , which has to be i n f e r r e d using shell-model arguments. However, DWBA c a l c u l a t i o n s and (p, d) measurements on l i g h t nuclei 29) have shown some years ago t h a t the analyzing power Ay(O) could e x h i b i t a strong j-dependence . This has been confirmed by recent experiments, and proved to be very useful for spin assignment to deep hole fragmented states. As an example f i g . 5 shows energy spectra and analyzing powers f o r the ~2°Sn(~,d) r e a c t i o n measured with a 90 MeV p o l a r i z e d beam from the Indiana Univers i t y Cyclotron3°). One sees already from f i g . 5a t h a t the structures above 5 MeV e x c i t a t i o n energy have d i f f e r e n t behaviours f o r proton spin up and spin down. The analyzing power f o r the region from 4.3 to 6.6 MeV which is assumed to correspond to the 9/2 hole s t a t e , is shown in f i g . 5b compared to j = 9/2 and j = 7/2 behaviours, e i t h e r empirical or from DWBA c a l c u l a t i o n s . One sees t h a t the 9/2 s p i n a s s i g n m e n t is unambiguous. S i m i l a r l y the xZ6Sn(d,t) reaction at 40 MeV31) has been used to study the
44c
J. MOUGEY
120Sn (~',d) 119Sn
8:14 o ' m,,-~
Spm Up
I
1
•
IZOSn(i5, d )ll9Sn Ex:4.3~66MeV
Ay x4
06
.
Ill
I
I
Ex(MeV)
15
Z
"n 0 o
ii
:
~n
I0
5 _-~.> x4
Spin Down
0"4f -06
- -
90Z,(~,d)SSZr(9/2+ I
--
.....
'ZOsn(~,d)"9Sm( 7/2+ )
.....
DWBA ( g g / z ) DWBA (97/2)
0 8cm.ideg}
(b)
I
(a)
Fig. 5 E x c i t a t i o n energy spectra (a) and analyzing power (b) from the z2OSn(~,d)119Sn r e a c t i o n a t 90 MeV i n c i d e n t energy. The analyzing power data correspond to the broad peak observed between 4.3 and 6.6 MeV e x c i t a t i o n energy. The s o l i d curves correspond e i t h e r t o the e m p i r i c a l Ay f o r the known j ~ = 9/2 + ground s t a t e of 89Zr, o r to a DWBA c a l c u l a t i o n assumlng j = 9/2. S i m i l a r l y , the dashed durve is from e m p i r i c a l Av f o r the j = 7/2, E~ = 0.79 MeV s t a t e of z19Sn or from a DWBA c a l c u l a t i o n assuming j = 7/2 (from re@3°). 5 MeV region in 11SSn, but w i t h a r e a c t i o n which gives less r e d u c t i o n o f the l~wer C components. A strong j-dependence o f Av(e) is also observed f o r the (d,t) r e a c t i o n , reasonably well reproduced by DWBA c a l c u l a t i o n s on l e v e l s of known j - v a l u e . In p a r t i c u l a r , spin assignment o f j = 9/2 is obtained f o r the 3.67 MeV l e v e l in zlSsn, as shown in f i g . 6. In the bump region only a 9/2++ 1/2combination can account f o r the behaviour o f Ay (O),corresponding to c o n t r i b u t i o n s from the ig 9/2 and 2p 1/2 i n n e r hole s t a t e s . Tragments belonging to the 2p 1/2 subshell have also been i d e n t i f i e d a t 2.6 and - 4.2 Mev, wheras 2p 3/2 components have been found a t 4.50 and 4.77 Mev. 3.3 DEEP LYING PROTON HOLE STATES Much less data are a v a i l a b l e on deeplE-bound proton hole s t a t e s 32) than on neutron hole states in heavy nuclei. The (d, 3 He) reaction on 9 0 Zr and 1 4 4 Sm has been used r e c e n t l y 3'+) to i n v e s t i g a t e the i f and Ig p r o t r o n holes, extending the method u t i l i z e d on l i g h t e r nuclei3S). Fig. 7 shows a (~,3He) spectrum at 52 MeV i n c i d e n t energy measured in a maximum o f a ~ = 3 angular d i s t r i b u t i o n . From the v e c t o r - a n a l y z i n g power measurement, i t has been found t h a t the "giant-resonance l i k e " bump (shaded area) which peaks at Ex = 6.8 MeV contains mainly I f 7/2 s t r e n g t h , although the narrow peak a t 5 Mev could also be a t t r i b u t e d to i f 5/2 hole
DEEP HOLE STATES
45c
o~
r 367
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-J1 03 02 01
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ANALYZING
POWERS
I
I
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'
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90Zr(d, 3He)89y
o!I H.I
llSSn (d,t)11 SSn
I
Ed : 52 MeV QLab : 11 °
o u4
1000
/
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....... ......
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'-'1
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;/2-
(b)
8c.
10
20
30
Fig.6. Analyzing Dowers and DWBApredictions for the ~t6Sn(~,t)11SSn reaction at 40 MeV. (a) characteristic j = 9/2 and j = 7/2 behaviours; (b) analyzing powers measured in the bump region and compared with DWBA calculations for different+spin assumptions. The association j = 9/2 + 1/2- is the best suited to reproduce A (@) in the range 4.9 < E < 5.9 MeV Y(from ref.
31).
z
0
H-] 02 [49-S91
___
+
X
25
20
15
10
5
0
Ex(MeV)
F i g . 7. E x c i t a t i o n energy spectrum of the 9°Zr(~,3He)89Y reaction a t 52 MeV i n c i dent energy (from ref.34).
46c
J. MOUGEY
However, the 1.75 MeV peak, f o r which a j = 5/2 spin assignment is n i c e l y confirmed, exhausts already the sum rule f o r t h i s s u b s h e l l , (C2S = 8.9) as in previous work (C2S = 7.8, from3~)). The s t r u c t u r e at 3 . 3 . < Ex < 11.3, i f e n t i r e l y considered as a I f 7/2 component exhausts also the sum r u l e (C=S = 9 . 2 ) , in c o n t r a s t to neutron pick-up reactionswhere only ~ 50 % of the shell model s t r e n g t h is found in corresponding inner holes. This r e s u l t , and a s i m i l a r one f o r the Ig s t r e n g t h in i,.3p allows the f i r s t meaningful d e t e r m i n a t i o n o f the spin - o r b i t s p l i t t i n g eso = EC~+~/2 -E #-~/2 f o r inner s h e l l s in heavy n u c l e i . The r e s u l t s are given in t a b l e 1 compared to the value obtained in the Woods-Saxon well used f o r the DWBA f i t s to the data and to Hartree-Fock p r e d i c t i o n s , showing a good agreement. TABLE i Spectroscopic r e s u l t s and s p i n - o r b i t s p l i t t i n g eso(MeV) f o r deeply bound hole s t a t e in heavy nuclei (from r e f . ) nlj 89y
143pm
Ex(HeV)
C2S
I f 5/2
I .75
8.9
i f 7/2
3.3 - 11.3
9.2
lg 7/2
0.27
8.6
e-so
WS ~so
eHFref36) so
5.1
4.0
5.6
4.7 Ig 9/2
3.9-
6.9
4.1
9.4
500 90ZdS.3He)8~ Ed= 52 MeV
~.~ee~
13.0-17oMeV
eL= 12.5"
400 u~ z
3OO
/
o (._)
i 200
(a)
°'!
b .~
100
~ I t 0.
(b) m
2~.
:m.
e.
@c.m. ( G R A D )
2b
1'o
6
E x (MeV) Fig. 8 E x c i t a t i o n energy spectrum (a) and angular d i s t r i b u t i o n in the energy range 13< Ex< 17 MeV (b) f o r the 9°Zr(~, 3He)~gY r e a c t i o n , showing evidence f o r 2 s i / 2 proton pick-up (dark area in ( a ) ) . An amount o f s t r e n g t h corresponding to the c o n t r i b u t i o n s from background ( f u l l l i n e in (a)) and the broad s t r u c t u r e peaked at 6.8 MeV (dashed l i n e ) has been subtracted in the angular d i s t r i b u t i o n shown in (b) (from r e f 3 ~ ) ) .
DEEP HOLE STATES
47c
At the h i g h e r e x c i t a t i o n energy o f ~ 15 MeV in the 8~y e x c i t a t i o n energy spectrum, a s t r u c t u r e have been observed a t Ex = 15.3 MeV having ~ 5 MeV FWHM, see f i g . 8 from37). The a n g u l a r d i s t r i b u t i o n is compatible w i t h ~ = 0 t r a n s f e r , which would be the f i r s t evidence f o r p r o t o n p i c k - u p from the 2s I / 2 s h e l l in such heavy n u c l e i . With t h i s assumption, and s u b s t r a c t i o n o f o t h e r c o n t r i b u t i o n s l i k e in f i g . 8, the s p e c t r o s c o p i c f a c t o r is found to be C2S = 3 . 7 . 4.
Deep hole s t a t e s in medium n u c l e i
: the e x p e r i m e n t a l s i t u a t i o n
Deep hole s t a t e s in the l p and 2 s - l d s h e l l n u c l e i have been r a t h e r e x t e n s i v e l y s t u d i e d from (p, 2p) and ( e , e ' p ) r e a c t i o n s at high i n c i d e n t energy. However, v e r y few e x p e r i m e n t a l s p e c t r o s c o p i c s t u d i e s using these r e a c t i o n s have been r e p o r t e d since the l a s t r e v i e w p a p e r s 3 8 ' 3 9 ) . Most o f the r e c e n t experiments have been devoted to the d i s i n t e g r a t i o n o f H and He i s o t o p e s 4 ° ' " 1 ' " z ) , and to the r e a c t i o n mechanism s t u d i e s mentioned above. N e v e r t h e l e s s , as new e x p e r i m e n t a l facilities f o r c o i n c i d e n c e experiments are now o p e r a t i o n a l , namely a t TRIUMF, SIN, MIT-Bates and NIKHEF, and as s e v e r a l high energy, high duty f a c t o r e l e c t r o n a c c e l e r a t o r s are i n p r o j e c t , i t may be useful to r e v i e w b r i e f l y the p r e s e n t e x p e r imental s t a t u s . 4.1 1p-SHELL NUCLEI
S(p,E)
9Be (e,e'p) P(MeV/c) 10-50 50-100 ,
I
I
I
20
40
60
~
r
100-150
I 150-200 80
E (MeV) Fig. 9 Proton separation energy spectra for the 9Be(e,e'p) reaction within different recoil momentumbins. The energy resolution of ~ 0.9 MeV allows to separate different excited states of SLi at low excitation energy (from ref. 39)). A l o t of experiments have evidenced the ls proton knock-out from lp shell nuclei. As a typical result, f i g . 9 shows separation energy spectra within different recoil momentumbins for the 9Be(e,e'p) reaction. One sees clearly that the lp hole strength is mainly distributed over a few states in the residual 8Li,
48c
J. MOUGEY
w i t h r e l a t i v e i n t e n s i t i e s in reasonable agreement w i t h spectroscopic f a c t o r s as p r e d i c t e d by c a l c u l a t i o n s in the i n t e r m e d i a t e coupling scheme43). The ls s t a t e shows up on the low momentum spectrum as a wide and asymmetrical bump peaked at an energy of 24 MeV. S i m i l a r r e s u l t s have been found in the t2C and 160 n u c l e i , w i t h almost no overlap of the Ip and is hole d i s t r i b u t i o n s . The spreading of the is d i s t r i b u t i o n c l e a r l y i n d i c a t e s the f a i l u r e o f a pure IPSM d e s c r i p t i o n o f such nuclei. However, the shape of the momentum d i s t r i b u t i o n s analysed w i t h i n DWIA are well reproduced by s i n g l e p a r t i c l e wave functions g i v i n g good f i t s to e l a s t i c s c a t t e r ing data (see f i g . 10).
100
,2 C (e'ep)
~
~ so g
Z 0
DWlA
.~
PWIA
N i
Z w
:sO5O15 _
15<_Em<21,5 MeV
I
100
i
30
J
200
i
3
(~)
1
F
100
i
~\I I
200
J
t
300
( MeV/c ) Fig. 10 Momentum d i s t r i b u t i o n s f o r the z2C(e,e'p) r e a c t i o n w i t h i n separation energy regions corresponding to the Ip ( l e f t ) and Is ( r i g h t ) hole s t a t e s . The curves are from Woods-Saxon type d i s t r i b u t i o n s normalized to the data. For the 12C(e,e'p) r e a c t i o n , the experimental d i s t r i b u t i o n s in d i f f e r e n t energy bins in the ls region are found to be a l l compatible w i t h a pure is d i s t r i b u t i o n 45) which enforces the basic d e s c r i p t i o n of the o v e r l a p i n t e g r a l s in terms of a few p a r t i c l e o r b i t a l s . The s t a t i s t i c a l accuracy o f the e x i s t i n g data are b a r e l y s u f f i c i e n t to examine s t r u c t u r e s in the Is s t r e n g t h d i s t r i b u t i o n . Fig. i i shows a 12C(e,e'p) spectrum in the is region compared to neutron pick-up r e a c t i o n spectra-12C(p,d) at 185 MeV47).I2C(3He,~) at 216 MeV46). The two l a s t spectra show s t r u c t u r e s above the proton emission t h r e s h o l d (8.7 MeV) around 11.5 and 15.5 MeV. Whereas the f i r s t one does not seem to have i t s analog in the ( e , e ' p ) spectrum, and would have to be a t t r i b u t e d to m u l t i s t e p processes, the second one corresponds to the p o s s i b l e s t r u c t u r e observed a t E ~ 32.5 MeV. More data from knock-out r e a c t i o n s w i t h high energy r e s o l u t i o n in the is region o f 12C and t~O would be h i g h l y d e s i r a b l e , these nuclei being the best s u i t e d to study the s t r e n g t h d i s t r i b u t i o n of a deep hole. For the Li i s o t o p e s , the r e s u l t s show t h a t s i n g l e - p a r t i c l e models are not the best s u i t e d to describe such l i g h t n u c l e i . Let us consider the eLi case as an example. The separation energy spectrum shown in f i g . 12 as measured w i t h 1.5 Mev r e s o l u t i o n through the (p,2p) r e a c t i o n a t 100 Mev48) e x h i b i t s two main s t r u c t u r e s a t E ~ 4.7 and E ~ 21 MeV a t t r i b u t e d to ip and is proton knock-out in a s h e l l model d e s c r i p t i o n . However, the momentum d i s t r i b u t i o n s (see f i g . 13)
DEEP HOLE STATES
49c
S I'1 t2C(p,d)llC , Ep =185 MeV 0 =2.5 ° G = 194-43 MeV/c
0.3
",.'t.
~L
L
;j:
,iM
" ! '
j,
=E 0.2
(a)
.n.:" .,.4
L
0.1 uJ
5~0
4~0
30
20
~2C(sHo c x ) ~ c/)
1500
10
0
Ex~MeVp
Sl
@lob = 7 °
c 0 L)
IOOC
×±2ot (b)
500 L
60
50
40
30
~0
20
0 ExIM~)
"t 12C(e,e'p)11B
S[
1
'~Eu0.30 70
(c)
0.15
== o u
60
50
4b
40
3b
30
2b
E (MeV)
Ib
E, MW
Fig. 11 Excitation enerqy spectra of the residual nucleus in various nucleon removal reactions from z2C : (a) ~2C(p,d) at 185 MeV47) ; (b) 12C(3He,~) at 216 MeV46) ; (c) ~2C(e,e'p) at 497 MeV45). Structure I at Ex ~ 15.5 MeV in (a) and (b) seems to be present a l s o i n ( c ~ Structure I I at % 12 MeV is not v i s i b l e in (c) and could be a t t r i b u t e d to multistep processes.
50c
J. MOUGEY
]
25
20
15
1
I
I
Ex(MeV) ~0
5
I
0 F-
I
6Li(p' 2P)SHe Binding Energy Spectrum 81 = - e z : 35.4"
ed
tt
:L l a j c~
-a_ 40
"o -~ 20 ~1. b "o
%5
"30
25
20 15 IO ; .... OSeparation Energy (MeV) Fig. 12 Separation energy spectrum for the ~Li(p,2p)SHe reaction at 100 HeY (from ref. "~)).
/~
120
)2Ol / 6Li
lp
.
Data E =I-TMeV ""u
.
.
. 6Li
ls
Data E =16-28MeV
80
+
5" v
80\ k ~ ~ ~
x078
b
E
z,0
e..,
0
"/~ l
x088
I
0
i
100 p
i
I
200 (MeV/c)
_
~
0
100 p
200 (MeV/c)
Fig. 13 Momentum d i s t r i b u t i o n s for the bLi(e,e'p) reaction with separation energy regions including each of the peaks in f i g . 12. The curves are from PWIA and DWIA calculations using ip and is s i n g l e - p a r t i c l e wave functions, and with a r b i t r a r y normalization to the data (from ref. 49)). measured in the Tokyo (e,e'p) experiment 49) cannot be s a t i s f a c t o r i l y reproduced by PW or DW calculations using single p a r t i c l e o r b i t a l s . The experimental d i s t r i b u t i o n s e x h i b i t more important low momentum components than the computed
51c
DEEP HOLE STATES
ones, a behaviour observed in a l l the (p,2p) experiments on L i . This has been q u a l i t a t i v e l y understood in an ~-d c l u s t e r desc r i p t i o n of L i . The r e s i d u a l ~-n system which is unbound a t the r e a c t i o n t h r e s h o l d E = 3.7 MeV, has two resonant s t a t e s at 0.96 MeV ( 3 / 2 - ) and 2.6 MeV ( 1 / 2 - ) superimposed on a continuum s - s t a t e . Thus at low removal e n e r g i e s , both s - a n d p - s t a t e components of the valence proton c o n t r i b u t e to the momentum d i s t r i b u t i o n . Calculations including s c a t t e r i n g states o f the n-~ system s°) have shown t h a t the shape of the d i s t r i b u t i o n is expected to change very r a p i d l y w i t h the s e p a r a t i o n energy when going through the p 3/2 and p 1/2 resonances (see f i g . 14). Thus, a high r e s o l u t i o n (~ 0.5 MeV) knock-out
\
E:
eV)
b 0
L
121
3.7 (~hresholld)
I
100
>
200
p (M ~ v/~) Fig. 14 L i ( e , e ' p ) momentum d i s t r i b u t i o n around the p 3/2 ~-n resonance in the s c a t t e r i n g s t a t e approximation (from r e f . S°)).
T
12C (~+,T:+pi'B
I
T
Eex<9.75MeV
=Z ioo
I
,
50
:
l
q
Eex>9.75MeW t.
¢N ul .D
i
12C (~*,':':+p)11B
P.. ,6 /
v
¢1. IO t.t
"0 50
25
"10 "0
fll
I 140
~
I 180
L p=
1 i 220 (MeV/c)
*40
180
220
p~ (MeV/c)
Fig. 15 12C(~+,~+p) r e a c t i o n cross s e c t i o n as a f u n c t i o n o f the momentum p~ of the s c a t t e r e d pion, a t an i n c i d e n t energy o f E~ = 199 MeV and angles o f e~=-i1775 and On =30 ° . The f u l l (dashed) curve is a DWIA c a l c u l a t i o n using the f i n a l s t a t e (initial s t a t e ) p r e s c r i p t i o n to compute the n-nucleon elementary cross s e c t i o n . The energy regions are chosen so as to separate lp ( l e f t ) from is ( r i g h t ) knockout (from r e f . S l ) ) .
52c
J. MOUGEY
experiment would be very useful to investigate this problem. S i m i l a r l y , the strongly structured peak at high energy would have to be considered in terms of t-d scattering states. Before leaving the Ip-shell n u c l e i , I would l i k e to mention the recent results in (~,~N) reactions. Although mainly designed to study ~-nucleon i n t e r a c t i o n within n u c l e i , they have been shown to be valuable tools f or spectroscopic studies. Fig. 15 shows a DWIA analysis sl) of a recent 12C(~+,~+p) experiment performed at 130-200 MeV incident energy, with I MeV energy resolutionS2). Reasonable f i t s are obtained using the Woods-Saxon wave functions reproducing the (e,e'p) data of f i g . 11, and the deduced occupation numbers (2.9 f o r I p - s h e l l , 1.8 f o r I s - s h e l l ) using f i n a l - s t a t e prescription f or the h a l f - o f f - s h e l l ~-nucleon cross section are close to the corresponding (e,e'p) ones (2.5 and I ) . The 160(~±,~-p) reactions have also been investigated recently with high energy resolutionS3). 4.2 HEAVIER NUCLEI The separation energy spectra obtained on heavier nuclei do not show separated structures above 20 MeV, as expected from the spreading of the Is hole strength d i s t r i b u t i o n observed in Ip-shell nuclei which is comparable to the major shell spacing. Thus, (p,2p) and (e,e'p) results have been analyzed by f i t t i n g an expansion l i k e eq. (4) to the "distorted spectral function" in which only the normally occupied orbits are considered. In the Saclay experiments" ) reasonable f i t s to the momentum d i s t r i b u t i o n s in d i f f e r e n t energy bins have been obtained using Woods-Saxon single p a r t i c l e wave functions in a DWIA analysis, and the procedure was used to extract the hole strength d i s t r i b u t i o n P~(E) shown in f i g . 16a. One sees that most of them e x h i b i t the same dissymmetric~l shape as f or the l i g h t e r nuclei. Similar results have been obtained through the (p,2p) reactionSS), as shown in f i g . 16b. The strength d i s t r i b u t i o n s P^(E) have been used to compute C occupation numbers N~ = % P~(E)dE and mean removal energies E~ = N~I % EP~(E)dE compiled and compare~ in several reviews papersS~,sT,se). One remembers that an overall agreement was claimed, with an observed tendancy of E~ to saturate f o r A ~ 30 at ~ 40 MeV f o r the Ip shell and 60 MeV f or the Is sheTl. However, I would l i k e to i n s i s t on the fact that the i d e n t i f i c a t i o n of the deepest hole states is s t i l l highly unprecise. To disentangle the various hole state contributions, d i f f e r e n t recipees have been used. For the (p,2p) data, the momentum d i s t r i b u t i o n s have been e i t h e r taken from a model in a PWIA analysis s3) or from the experiment i t s e l f , assuming that, in some typical energy bins, only one hole state contributesSg). In the Tokyo (e,e'p) data analysis 6°) gaussian shapes f o r P~(E) have been assumed, whereas Maxwellian shapes were considered by Amaldi et a l ~ ) . Moreover, the background due to m u l t i p l e - c o l l i s i o n processes have not always been removed, although i t s r e l a t i v e importance depends strongly upon the type of reaction, the kinematics, and the detector acceptances. Thus the large dispersion of the published values concerning the position of the Is hol~ in 4°Ca ranging from 50 + 12 (ref. Ss)) to 77 + 14 MeV ( r e f . ~ l ) ) could mainly r e f l e c t the d i f f e r e n t types of analysis and t h e i r d i f f i c u l t i e s as regards to the rather poor s t a t i s t i c s of the experiments. The r e l a t i o n of the experimental mean removal energies to the s i n g l e - p a r t i c l e energies determined by self-consistent calculations and to the t o t a l binding energy have been often discussed (see among others r e f s . 3 ' 4 ~ ' 6 2 ) ) . As an example, one gives in table 2 typical results f o r "°Ca from experiment and theory, refecting the general trends. One sees that in BHF and RBHF theories not enough binding is obtained whereas, in DDHF c a l c u l a t i o n s , the s i n g l e - p a r t i c l e energies tend to be less negative than the experimental removal energies. However, f or the reasons mentioned above, one must be cautious about these conclusions. As momentum d i s t r i b u t i o n s at low ~-values mainly r e f l e c t the o r b i t a l angular momentum ~ of the single-hole state, they cannot be used in general to make j-spin assignments. As f o r pick-up reactions, polarized proton beams have been used to investigate the j-dependence of the analyzing powerS3'67). Such strong dependence has been predicted by Jacob et al 9z) within DWIA as caused by nuclear s p i n - o r b i t
DEEP HOLE STATES
53c
TABLE 2 Experimental proton mean removal energies and t h e o r e t i c a l 4°Ca. Also quoted is the binding energy per nucleon (in MeV). 2s
Id
Ip
HF energies f o r
Is
EA/A
Experiment (E~) (p,2p)
(a)
(e,e'p)
(b)
(e,e'p)
(c)
-8.55 14
+_7
{10.9 _+ 0.7
14.4 _+ O.
19.0 + i . I 111.2 + 0.~ 14.9 + 0.8
34 + 7
50 +- 11
35 -+ I 41
59 +- 3 56
Theory (c a) BHF RBHF DDHF DDHF
(d)
-18.5 -13.3 -9.7 -11
-17.4 -12.8 -8 -10
(d)
(e) (f)
-39.5 -31.5 -25.3 -25
-3.9 -4.5 -7.5 -8.3
-61.7 -50.8 -43.6 -38
References : (a) James et al sS) ; (b) Nakamura et a l , peak energies ~°) ; (c) Mougey et al s") ; (d) Davies et al ~4~ (e) Negele 65) ; ( f ) Nemeth andVautherin66). ,
,
,
,
,
,
,
,
2.0-58Ni
40Ca 0.8
1.5If
0.6 1.00.4 0.5-
0.2
20
/
40
60
40Ca
~tl d
if
(a)
~-~ ld 2s
1, "'¢-k.
20
40
1[, 60
80 E (MEW)
58Ni
(b) '
/~s
lp
ls
10 20 30 40 50 60
Fig. 16
10 20 30 z,0 50 60 EIMeV)
Hole strength energy d i s t r i b u t i o n s (a) the ( e , e ' p ) r e a c t i o n S " ) ; (b) the (p,2p) reactionSS).
in 4°Ca and 5eNi from :
54c
J. MOUGEY
coupling combined with the strong absorption in (p,2p) reactions. However, spino r b i t d i s t o r t i o n e f f e c t s , as discussed in the i n t r o d u c t i o n : ° ) , could change the predicted asymmetries. The experimental results obtained for the !GO and 4°Ca(~,2p) reactions using a 200 MeV p o l a r i z e d beam at TRIUMF have e s s e n t i a l l y confirmed the predicted j-dependence, and the v a l i d i t y of the f a c t o r i z e d DWIA in the chosen kinematics. Some evidence has been obtained f o r lp i / 2 and ip 3/2 strength in "°Ca at ~ 21MeV and ~ 30 MeV separation energy r e s p e c t i v e l y , which would agree with the ip strength d i s t r i b u t i o n as determined from ( e , e ' p ) reactions (see f i g . 16). To conclude t h i s section, I would l i k e to show some spectra of the f i r s t high r e s o l u t i o n (p,pn) experiment which has been performed at IUCF by Watson et a168) using a 149.5 MeV proton beam on a)40Ca(p,pn)39Ca ep = 44.3~ 4°Ca and ~SCa targets. As e n = 36.1° shown in f i g . 17, ld 3/2 1200 2Sl/2 and 2s 1/2 neutron holes, as well as I f 7/2 hole strength in 47Ca are 800 c l e a r l y i s o l a t e d with the I MeV energy resolution, 400 ld3/2 ld5/2 and probable Id 5/2 and Ip strength is seen for both 09 targets. DWIA c a l c u l a t i o n s Z using standard Woods-Saxon bound state wave functions 0 and the "Indiana" o p t i c a l 0 b)48Ca(P,P n)47ca ep = 47.£model Gg) i n c l u d i n g spinen = 36.1o r b i t have been performed 400 (2s1/2+1d3/2) to obtain occupation numbers for the valence ~lf7/2[ ~ ld5/2 shell in reasonable agreement with previous pick-up results (see table 3).These results e s t a b l i s h the v i a b i l i t y o f (p,pn) reactions as a tool 0 10 2o 30 4o for neutron-hole spectroscopy. SEPARATION ENERGY, E s (MeV) . . . . .
o
-
i
r
"
F
--
i
-
Fig. 17 Neutron separation energy spectra "°Ca(p,pn)~gCa and 4SCa(p,pn)47Ca reactions at 149.5 MeV (from r e f . ~ 8 ) . TABLE 3 Spectroscopic factors f o r neutron-hole states from the (p,pn) r e a c t i o n , compared to neutron pick-up results (from r e f . 6 8 ) ) .
Nucleus
nlj
CZS(p,pn) ref~8)
CZS(p,d) refTO)
CzS(~He,~) refTZ)
39Ca
2s 1/2
1.8
1.9
1.74
47Ca
2s 1/2 ld 3/2 I f 7/2
1.3 2.6 8
1.8 3.6 6.7
1.38 2.23 6.94
DEEP HOLE STATES
55c
5. C a l c u l a t i o n s of deep hole s t r e n g t h d i s t r i b u t i o n s As pointed out in the preceding s e c t i o n s , one has to go beyond IPSM to i n t e r p r e t the spreading o f d e e p - l y i n g hole s t a t e s . Several approaches have been made to describe the fragmentation o f one q u a s i - p a r t i c l e s t a t e s at low and i n t e r m e d i a t e e x c i t a t i o n energy 72'73'23'28) showing the r o l e of the coupling o f the hole to c o l l e c t i v e v i b r a t i o n s and to few q u a s i p a r t i c l e s t a t e s . Here, I would l i k e to comment on recent attempts to c a l c u l a t e the hole s t r e n g t h d i s t r i b u t i o n s from s t r o n g l y bound s h e l l s i . e the energy spectra measured in knock-out r e a c t i o n s . One p o s s i b i l i t y is to extend the IPSM approach by r e p l a c i n g the real s h e l l model p o t e n t i a l by a complex one so as to take care o f the f i n i t e l i f e time of the hole s t a t e . This has been 2o< pB< 5o done, using phenomenological l o c a l energy-independent p o t e n t i a l s by Herscovitz 7") and ShantaT5). One finds t h a t the shape o f the imaginary p a r t o f the p o t e n t i a l influences also the momentum distributions. A more d i r e c t approach is to 0 look f o r a wave equation f o r the 50< pB< 100 holes, t h a t is f o r the overlap 4 i n t e g r a l s of eq. I . Their general p r o p e r t i e s have been studied many years ago by Berggren76), JacksonT~ and ClementT8). I t has been shown 0 2 t h a t they are s o l u t i o n s of a set o f coupled i n t e g r o d i f f e r e n t i a l equations having the s t r u c t u r e of a 0 Schr~dinger equation w i t h a noniO0 < pB< 2O0 l o c a l s t a t e dependent e f f e c t i v e p o t e n t i a l . A s i m i l a r r e s u l t has 4 been obtained by B o f f i and Capuzzi 79) in a more fundamental approach i n v o l v i n g the hole mass o p e r a t o r which, close to sharp resonances in the spectrum, plays the r o l e o f a non-local energydependent p o t e n t i a l . S t a r t i n g also 0 20 40 60 E 80(MeV) from the general p r o p e r t i e s o f the mass o p e r a t o r , but w i t h d r a s t i c approximations to reduce i t to a Fig. 18 Separation energy spectra f o r the local energy-dependent s i n g l e " ° C a ( e , e ' p ) r e a c t i o n in various r e c o i l p a r t i c l e complex p o t e n t i a l momentum bins 6°) compared to t h e o r e t i c a l p r e d i c t i o n s b y Gorchakov et a l . (from r e f . 8 ° ) ) . Gorchakov et al 8°) have succeeded in reproducing a t l e a s t q u a l i t a t i v e l y the removal energy s p e c t r a , as shown f o r 4°Ca in f i g . 18. A s i m i l a r o p t i c a l model approach has been f o l l o w e d r e c e n t l y by Klevansky and Lemmere7). F u l l y microscopic c a l c u l a t i o n s o f the s p e c t r a l f u n c t i o n using p e r t u r b a t i o n theory and the Green f u n c t i o n techniques 81), or using a t r u n c a t e d shell-model basis i n c l u d i n g continuum states to d i a g o n a l i z e the mass o p e r a t o r 8 2 ) , have not reached a stage where a meaningful comparison w i t h data can be performed. Another approach i s to s t a r t from the hole propagator in nuclear m a t t e r . The complex s i n g l e - p a r t i c l e f i e l d , i n c l u d i n g the f i r s t o r d e r Hartree-Fock f i e l d and various h i g h e r - o r d e r c o r r e c t i o n s due to the remaining two-body c o l l i s i o n s which w i l l generate the spreading, is computed using a r e a l i s t i c nucleon-nucleon i n t e r a c t i o n . The r e s u l t s are adapted to f i n i t e nuclei by choosing proper r e l e v a n t v a r i a b l e s . Such approach is j u s t i f i e d by the observation t h a t the e s s e n t i a l features o f the hole d i s t r i b u t i o n s , t h e i r widths and shapes are weakly dependent
56c
J. MOUGEY
I
SS
28Si I!,
1.0
1ss
1
58Ni
Z,Oco
lIO
0.5
D,
,,',
i!o
1:
.C t/)
rl
0.6
D~
0.6
,
!l
03
I\
!
o.2 0
o.2 LO
:o
°
0
Ol
40
8O 0
40
80
E (MeV) Fig. 19 Comparison between experimental S") and t h e o r e t i c a l hole strength d i s t r i butions f o r the Is ( t r i a n g l e s and f u l l curves) Ip ( f u l l dots and long dashes) and Id (open c i r c l e s and short dashes) shells in 28Si, 4°Ca and SSNi (from r e f . 8 4 ) ) . upon the s p e c i f i c n u c l i d e considered. The recent c a l c u l a t i o n s by Orland and Schaeffer83), Sartor and Mahaux8") and Antonov et al 8s) show that i t provides a good d e s c r i p t i o n of the observed d i s t r i b u t i o n s . Orland and Schaeffer have shown that the width of the spectral d i s t r i b u t i o n s P~(E) is e s s e n t i a l l y given by the a v a i l a b l e phase-space f o r two-body c o l l i s i o n s in the nuclear medium, which, in the Fermi-gas model, depends upon the distance to the Fermi energy. This also makes the resonance to have more strength on the high energy side. In a less phenomenol o g i c a l way Sartor and Mahaux obtained a s i m i l a r expression f o r the spectral funct i o n associated with the s i n g l e - p a r t i c l e state ~ (energy ca) S (p,E) = ~I Z2(p)
W(p) + (E + E ) R(p) (E + ~ )2 + 1/4(2Z(p) ~(p))2 -i
where Z(p) = ( I - ~I~ V(p,E) 1 E= -c a
and R ( p ) = ( ~ E
-z
(6)
W(p,E))E = -c
V(p,E) and W(p,E) being the ~eal and imaginary part of the mass operator with onshell (E = -e~) expressions V(p) and ~(p). Fig. 19 shows such c a l c u l a t i o n s compared to the strength functions obtained from (e,e'p) experiments. The r e s u l t s are s t r o n g l y dependent on the value of the i n f i n i t e nuclear matter density (or the Fermi momentum) at which the mass-operator is calculated. One notices in f i g . 19 that the agreement is f a i r f o r the Is d i s t r i b u t i o n s , but t h a t the Ip and Id calculated peaks are too narrow i f a normal density of p = 0.17 N/fm 3 is used. This could come from a lower average nuclear density seen by the outer o r b i t s . Antonov et al 8s) obtained reasonable agreement using the coherent f l u c t u a t i o n model 86) in which the f i n i t e nucleus density d i s t r i b u t i o n , as given by electron s c a t t e r i n g , is ~ u i l t by the superimposition of uniform d i s t r i b u t i o n s with weighting f a c t o r s . 6. Conclusions Since a few years the q u a n t i t y and q u a l i t y of experimental data concerning the deep-lying hole states have g r e a t l y improved. Inner neutron-hole strengths have been observed in a v a r i e t y of single neutron pick-up reactions f o r a wide
DEEP HOLE STATES
57c
range o f medium and heavy n u c l e i . High r e s o l u t i o n ( e , e ' p ) and(p,2p) experiments have demonstrated the p o s s i b l i t y to i n v e s t i g a t e the deepest s h e l l s in medium n u c l e i . New s p e c t r o s c o p i c t o o l s l i k e the use o f p o l a r i z e d beams, the (~,~N) and (p,pn) r e a c t i o n s have proved t h e i r v a l i d i t y . E v e n more d e c i s i v e progress is expected in the near f u t u r e w i t h new f a c i l i t i e s l i k e the MEA(NIKHEF-K) a c c e l e r a t o r here in Amsterdam . Fig. 20c shows the extremely good r e s o l u t i o n o f 250 keY achieved in a ~ 2 C ( e , e ' p ) ~ B experiment at 253.5 MeV i n c i d e n t energy.
C'Z ( p , 2 ~ ' } B u
(a)
50MeV
5OMeV
C Iz ( p , d ) C H
e= 65" B"
(b)
-~
qs
. Is
3/2"
5'2 -
Ii I;
I
:i
'lZ
T
11
E (e,e'p) B
Pm: 100 MeV/c I
e = 2?0.5 HeY e e = 90 °
ep = 37.8
(C)
o
E *a
G.S.
2] 2
S.02
',,',,,.,,.,,,;' %..,,,,.,:.;:.:,,.,,.,"'%__,,..,.,,,
.....
5
Excitation Energy [HeV]
>
Fig. 20 High r e s o l u t i o n e x c i t a t i o n energy spectra from (a) t h e 1 2 C ( p , 2 p ) l I B r e a c t i o n a t 50 MeV (b) the12C(p,d)11C r e a c t i o n at 50 MeV, (c) t h e 1 2 C ( e , e ' p ) i I B r e a c t i o n at 253.5 MeV ( ( a ) and (b) from r e f . 89), (c) from r e f . e s ) ) . Such a r e s o l u t i o n , comparable to the best one (350 keV) obtained in a (p,2p) experiment 89) and to the ones obtained in t r a n s f e r r e a c t i o n s , w i l l a l l o w to use the ( e , e ' p ) r e a c t i o n f o r d e t a i l e d s p e c t r o s c o p i c studies w i t h the enormous advantage o f the weak d i s t o r t i o n on e l e c t r o n waves. I t is c l e a r from f i g . 20 t h a t , in the ( e , e ' p ) case, onl~ those l e v e l s in 11B which correspond to the d i r e c t removal o f a I p - p r o t o n ( J =1/2- o r 3 / 2 - ) are e x c i t e d w i t h n o t i c e a b l e s t r e n g t h . In p a r t i c u l a r , the 5/2 , 4.45 MeV s t a t e is absent in the ( e , e ' p ) spectrum, which may i n d i c a t e much s m a l l e r c o n t r i b u t i o n s from m u l t i s t e p processes in t h i s r e a c t i o n . Acknowledgments I would l i k e to thank Profs. H. L a n g e v i n - J o l i o t , G.J. Wagner and C.H.Q. Ingram f o r t h e i r precious advice and comments concerning t h i s t a l k , and, t o g e t h e r w i t h Prof. W.T.N. Van Oers, Dr. J.W. Watson and Dr. L. Lapikas, f o r prov i d i n g me w i t h both published and unpublished data.
58c
J. MOUGEY
References i) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44)
C.A Engelbrecht and H.A WeidenmUller, Nucl. Phys. A184 (1972) 385 ; G.J Wagner, Proc. I n t . Symp. on h i g h l y Excited States in Nuclei, JUlich 1975, ed. A. Faessler, C. Mayer-B~ricke and P. Turek, Vol. 2, p. 175. A.E.L Dieperink and P.J Brussaard, Z. Phys. 261 (1973) 117. A.E.L Dieperink and T. De Forest, Ann. Rev. Nucl. Sci. 25 (1975) I. D.H.E Gross and R. Lipperheide, Nucl. Phys. A150 (1970)~49. ~ Berggren, Nucl. Phys. AI09 (1968) 265. S. B o f f i , C. Giusti and F.D. Pacati, Nucl. Phys. A336 (1980) 416 ; Nucl. Phys. A3~(1980) 437 ; C. Giusti and F.D Pacati, Nucl. ~ s . A336 (1980) 427. ~ Roos et a l , Phys. Rev. Lett. 40 (1978) 1439. M. Bernheim et a l , Proc. Int. ConT-~.on Reaction Mechanisms, BalatonfUred (Hungary), 1977 ; S. B o f f i , C. G i u s t i , F.D Pacati and S. F r u l l a n i , Nucl. Phys. A319 (1979) 461. S. B o f f i , C. Giusti and F.D Pacati, to be published. D.F Jackson, Nucl. Phys. A257 (1976) 221 A.E.L Dieperink and I. Sic~-~--,Phys. Lett. 1098 (1982) I. J. K~llne and B. Fagerstr~m, Proc. F i f t h I n t . Conf. on High Energy Physics and Nuclear Structure, Uppsala, 1973, ed. G. Tibell (North-Holland, Amsterdam, 1974) P. 369. J. Van De Wiele et a l , Nucl. Phys. A297 (1978) 61. G.M Crawley, Proc. Int. Conf. on Structure of Medium-Heavy Nuclei, Rhodes, 1979 ( I n t . Conf. Series 49, The I n s t i t u t e of Physics, Bristol and London) p. 127 ; G.M Crawley, Pro~. I n t . Symp. on Highly Excited States in Nuclear Reactions, Osaka, 1980 (RCNP, Osaka University) p. 590. S. Gales, Nucl. Phys. A354 (1981) 193c. G.M Crawley et a l , Phys. Rev. C23 (1981) 589. G.M Crawley et a l , Phys. Lett. ~ 9 B (1982) 8. T. Nakagawa et a l , Nucl. Phys. A--3-76 (1982) 513. R.S Tickle, W.S Gray and R.D Bent, Nucl. Phys. A376 (1982) 309. E. Gerlic et a l , submitted to Phys. Lett. G. Mairle et a l , Nucl. Phys. A382 (1982) 173. H. Langevin-Joliot et a l , Phys. Lett. 114B (1982) 103 and communication to t h i s conference. V.G Soloviev, C.Stoyanov and A. Vdovin, Nucl. Phys. A342 (1980) 261. G.M Crawley et a l , Phys. Rev. C23 (1981) 1818. G. Duhamel et a l , J. Phys. G7 (Tg81) 1415. S. Gales et a l , Phys. Rev. CT8 (1978) 2475. J. G u i l l o t et a l , Phys. Rev.~21 (1980) 879. Nguyen Van Giai, Proc. I n t . Sym~p. on Highly Excited States in Nuclear Reactions, Osaka, 1980 (RCNP, Osaka University) p. 682. K. Hosono et a l , Nucl. Phys. A343 (1980) 234. G,M Crawley et a l , Phys. Rev. C23 (1981) 1818. G. Perrin et a l , Nucl. Phys. A 3 ~ (1981) 61. P. Doll et a l , Phys. L e t t . B82~979) 357. R.H Siemssen et a l , Phys. Let~. I03B (1981) 323. A. S t u i r b r i n k et a l , Z. Phys. A297 (1980) 307. V. Bechtold et a l , Phys. Lett. 7~2-B-(1977) 169. M. Beiner, H. Flocard, Nguyen Va~Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. G. Seegert, Diplomarbeit, Max-Planck I n s t i t u t e , Heidelberg, 1982 (unpublished) and G.J Wagner, private communication. W.J Mc Donald, Nucl. Phys. A335 (1980) 463. J. Mougey, Nucl. Phys. A335~T980) 35. M. Bernheim et a l , N u c l . ~ y s . A365 (1981) 349. E. Jans et a l , submitted to Phys.---~ev. Lett. W.T.H Vanders et a l , Phys. Rev. C25 (1982) 390 ; D.J Margaziotis et a l , Phys. Rev. C25 (1982) 2873. V.V Ba~shov, A.N Boyarkina and I. Rotter, Nucl. Phys. 59 (1964) 417. D. Royer e t a ] , Phys. Rev. C12 (1975) 327.
DEEP HOLE STATES 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65) 66) 67) 68) 69) 70) 71) 72) 73) 74) 75) 76) 77) 78) 79) 80) 81) 82) 83) 84) 85) 86) 87) 88) 89) 90) 91)
59c
J. Mougey, Thesis, Universit~ de Paris-sud, unpublished. E. Gerlic et a l , Phys. Rev. C12 (1975) 2106. P.E Hodqson, Rep. Progr. Phys__38 (1975) 847. R.K Bhowmik, C.C Chang, P.G RooTand H.D Holmgren,Nucl. Phys. A226 (1974) 36~ K. Nakamura et a l , Nucl. Phys. A296 (1978) 431. K. Nakamura, Report Univ. Tokyo,~PN-36 (1974) ; S. Saito et a l , Prog. Theor. Phys. 39 (1968) 635. N.S Chant, L. RYes and P.G Roos, Phys. Rev. Lett. 48 (1982) 1784. N.J Ziock et a l , Phys. Rev. C24 (1981) 2674. C.H.Q Ingram, private communi~tion. J. Mougey et a l , Nucl. Phys. A262 (1976) 461. A.N James et a l , Nucl. Phys. A-I-3-~ (1969) 145. G. Jacob and Tho A.J Maris, Rev. Mod. Phys. 45 (1973) 6. D.F Jackson, Proc. Int. Conf. on Nuclear Struc-ture and Spectroscopy, Amsterdam, 1974 eds. H.P Block and A.E.L Dieperink, p. 396. J. Mougey, Proc. Int. Conf. on Nuclear Structure, Tokyo, 1977, ed. T. Marumori (Physical Society of Japan, 1978) p. 420. G. Landaud et a l , Nucl. Phys. A173 (1971) 337. K. Nakamura et a l , Nucl. Phys. ~ I (1976) 221. U. Amaldi et a l , Phys. Lett. 22 -(T~66) 593. C. Ciofi Degli A t t i , Progr. in--Part, and Nucl. Phys. 3 (1980) 163. P. Kitching et a l , Nucl. Phys. A340 (1980) 423. K.T.R Davies, R.J Mc Cart~y and P-PTU.Sauer, Phys. Rev. C5 (1972) 1461. J.W Negele, Physo Rev. CI (1970) 1260. J. Nemeth and D. Vauther~n, Phys. Lett. 32B (1970) 561. L. Antonuk et a l , Nucl. Phys. A370 (1981]--389. J.W Watson et a l , to be p u b l i s h ~ i n Phys. Rev. C. A. Nadasen et a l , Phys. Rev. C23 (1981) 1023. P. Martin et a l , Nucl. Phys. AT~5 (1972) 465. S. Fortier et a l , Nucl. Phys. ~ I (1978) 324. T. Koeling and F . - l a c h e l l o , Nuc.l~--Phys. A295 (1978). A. Famery, J. of Phys. G5 (1979) 241. V.E Herscovitz, Nucl. Phys. A161 (1971) 321. R. Shanta, Nucl. Phys. A199 (T~=f3) 624. T. Berggren, Nucl. P h y s . ~ (1965) 337. D.F Jackson, Adv. in Nucl-/-Phys. 4 (1971) I and references therein. C.F Clement, Nucl. Phys. A213 (1973) 469 ; Nucl. Phys. A213 (1973) 493. S. Boffi and F. Capuzzi, Nuc--cT. Phys. A351 (1981) 219. V.V Gorchakov, A.A Goi and B.L R e z n i k , ~ v . J. Nucl. Phys. 22 (1975) 514. U. Wille and R. Lipperheide, Nucl. Phys. A189 (1972) 113. - W. Fritsch, R. Lipperheide and U. Wille, Nuc--cT. Phys. A241 (1975) 79. H. Orland and R. Schaeffer, Nucl. Phys. A299 (1978) 442. R. Sartor and C. Mahaux, Phys. Rev. C21 ( I ~ ) 2613. A.N Antonov, V.A Nikolaev and I . Zh P-e-tk~v, Report Ins. of Nucl. Research and Nucl. Energy, Bulgarian Academy of Sciences, E4-81-715, 1981. A.N Antonov, V.A Nikolaev and I . Zh Petkov, Z. Phys. A297 (1980) 257. S.P Klevansky and R.H Lemmer, Phys. Rev. C25 (1982) 3 1 ~ A.W Bronkhorst et a l , communication to thi-s-conference and L. Lapikas, private communication. H.G Pugh et a l , Phys. Rev. Lett. 14 (1965) 434. B.M. Preedom, E. New,nan and J.C H3e-bert, Phys. Rev. 166 (1968) 1156. G. Jacob et a l , Phys. Lett. 45B (1973) 181 ; Nucl. Phys. A257 (1976) 517.
39c
DEEP HOLE STATES Jean MOUGEY DRF/CPN, Centre d'Etudes Nucl~aires de Grenoble 85X, 38041 Grenoble Cedex, France A b s t r a c t : Our present knowledge on d e e p l y - l y i n g hole s t a t e s from one nucleon pick-up and knock-out r e a c t i o n s is reviewed. Results on n e u t r o n - h o l e s t a t e s in the Zr, Sn and Pb region from recent pick-up measurements are discussed, w i t h emphasis on the use o f p o l a r i z e d beams f o r spin assignments. The s i t u a t i o n concerning proton hole s t a t e s in medium-mass (6
40c
J. MOUGEY 2. The spectral function
Assuming in what follows that both knock-out and pick-up reactions r e s u l t in a sudden removal of a nucleon, the essential nuclear q u a n t i t y which is measured is the diagonal spectral function 3'4) S(X,E)
=
a~ 6(H-E-c~)
%IA> = Z I
A>I 2 6 ( E - ~ _ l + g ~)
(1)
f
where II> is an orthonormal basis of single p a r t i c l e states. Energy and momentum conservation in ( e , e ' p ) (or in (pz2p)) reactions determine the separation energy E and the r e c o i l momentum p~ 1' which, in pla~e wave impulse approximation (PWIA) is opposite to the i n i t i a T - p r o t o n momentum p. Then one obtains a plane wave representation of the spectral function S(p,E), a f t e r d i v i d i n g the experimental cross section by some " o f f - s h e l l " e l e c t r o n - ( o r proton-) proton cross-section. Equation ( i ) becomes
S(~,E) = ~ [#fi(~)l%(E-~Af 1 + c~)
(2)
where ~ f i ( P ) = < A - l , fta(~)lA > is the Fourier transform of the overlap i n t e g r a l betweenIthe i n i t i a l and f i n a l nuclear states. In IPSM, ~fi is i d e n t i c a l to the single p a r t i c l e wave function for an occupied o r b i t ~ = { n l j } and the spectral function takes the form S(~,E) = ~ P ( E ) I ~ ( ~ ) I ~ with P (E) = 6(E+~ ) (3) G
DEEP HOLE STATES
41c
t~_1 which make up the Born approximation reaction cross section in i t s most general form has been investigated by Boffi et al. 9) for the ~60(e,e'p)15N^. reaction. I f a factorized approximation holds, the four functions must be ~ i d e n t i c a l to the spectral function. One sees t h a t , when the d i s t o r t i n g potential contains a spin o r b i t term, both to~ and t 1 _ 1 a r e d i f f e r e n t from the spectral function, and the factorization is destroyed Io), although t01 and t1_~ give rather small contributions in usual kinematical conditions.
o-,f
(tin 3) 0.4
0.2
0.2
0.0
0.0
/
/
/ //X
-0.2
.~"
/
1
/'
I ! 5
I
-0.2
/ "Ix -5
l~
i I
-0.4
- 0.4
(o)
-0.6
i /
//
(b)
,, -0.6
!/' I
0°
15 °
30°
4 5°
0°
I
30°
15 =
Y
4 5°
F
Fig. 3. Structure functions for the pl/2 hole in 160 entering in the (e,e'~) cross section as a function of the angle ¥ between the outgoing proton momentum p' and momentum transfer q. The kinematics is such that q = 2.3 fm-I and p'2/2M = 100Mev. Full l i n e too, dotted l i n e t1~, dashed l i n e to~, dot-dashed l i n e t~-~. In (b) the optical potential contains a s p i n - o r b i t term, which is absent in (a) (from r e f . 9 ) ) Nevertheless, PWIA, or the factorized DWIA represent good f i r s t order approximationsespecially for the (e,e'p) reaction and have been used in most of the analysis of knock-out experiments. Then instead of (3), the quantity deduced from the data is a "distorted spectral function" sD(~A_f,E) : ~
(E)I,~(~A_I)I
2
(4)
where the i i distorted momentum d i s t r i b• u t i o. n s j l ~D are obtained from bound and optical model scattering state wave functions, Pick-up experiments are much easier to perform and have a superior ($100 keY) energy resolution. On the other hand, they are more strongly affected by distortion, and the strong absorption induces a high angular momentum s e l e c t i v i t y through matching conditions. Thus, instead of (2), the spectral function is obtained in an orbital basis through the spectroscopic factors C2Snlj = I < A-1,fl anl j [A>I2
(5)
for excitation energies below~,15 MeV, and under matched conditions. However, i t is known that uncertainties exist in the absolute values of the
42c
J. MOUGEY
spectroscopic f a c t o r s as determined in standard DWBA a n a l y s i s . They have several o r i g i n s , namely : the c o n t r i b u t i o n s from m u l t i s t e p processes, the energy dependence o f the overlap i n t e g r a l s , the incomplete knowledge o f the r a d i a l bound s t a t e wave functions at l a r g e d i s t a n c e , the high level density at high e x c i t a t i o n energy and the experimental energy cut o f f . I t has been shown r e c e n t l y t h a t the knowledge of r a d i a l wave f u n c t i o n f o r the valence nucleon in odd-A nuclei from e l a s t i c magnetic e l e c t r o n s c a t t e r i n g can g r e a t l y improve the d e t e r m i n a t i o n of absolute spectroscopic f a c t o r s ' 1 ) . In the case o f the nuclei 51V and 87Sr f o r which the valence wave functions @~(r) and the diagonal one body d e n s i t i e s , d i r e c t l y r e l a t e d to the spectroscopic f a c t o r s , are known from e l e c t r o n s c a t t e r i n g . i t has been found t h a t the spectroscopic f a c t o r s from pick-up r e a c t i o n s , when c o r r e c t e d so as to be coherent w i t h the valence o r b i t r a d i i , are 30 to 40 % below the s h e l l model sum r u l e . Moreover, at high i n c i d e n t energies which are necessary to i n v e s t i g a t e deeper s h e l l s 12'~3) the angular d i s t r i b u t i o n s are less c h a r a c t e r i s t i c due to l a r g e r angular momentum mismatch, and the e x t r a c t i o n o f C2S is more p r o b l e m a t i c . However, pick-up r e a c t i o n s are, f o r the moment, almost the unique source o f i n f o r m a t i o n on neutron hole s t a t e s . 3. Recent r e s u l t s in pick-up reactions : Deep hole states in heavy nuclei Pick-up r e a c t i o n s - ( p , d ) , ( d , t ) , ( 3 H e , ~ ) - h a v e been w i d e l y used in the l a s t few years to i n v e s t i g a t e neutron hole states in the f i r s t i n n e r s h e l l o f medium-mass and heavy n u c l e i . Moreover, using p o l a r i z e d beams i t has been p o s s i b l e to d i s t i n g u i s h between the j = ~ + 1/2 and j = ~ - 1/2 component o f a given o r b i t a l momentum d o u b l e t . Since several d e t a i l e d reviews have been made r e c e n t l y by e x p e r t s 1 4 ' 1 ~ ) , I s h a l l give only a b r i e f summary of the present status and discuss a few examples. In p a r t i c u l a r , f o r lack o f t i m e , I s h a l l omit the discussion on t w o - p a r t i c l e pick-up r e s u l t s (see refs 14,16-21). 3.1 STATUS OF DEEP NEUTRON HOLE STATES FROM PICK-UP REACTIONS Fig. 4 shows as an example e x c i t a t i o n energy spectra in 115Sn and 2°TPb from the (3He,~) r e a c t i o n at 283 MeV i n c i d e n t energy22). This high value allows to extend the e x c i t a t i o n energy range up to 50 MeV f o r the f i r s t time. I t involve~ a high momentum mismatch which s t r o n g l y reduces the small h - t r a n s f e r c o n t r i b u t i o n s . In the iZSSn spectrum, the Ig9/2 hole s t a t e manifests i t s e l f as a sharp l i n e at 5.3 MeV, as expected from previous works , but also as a broader s t r u c t u r e peaked at 7.5 MeV (~ 5 MeV FWHM). This d i s t r i b u t i o n agrees reasonably well w i t h the p r e d i c t i o n s of Soloviev et a i . 2 3 ) , who consider the coupling o f the hole s t a t e w i t h I h o l e - I phonon and I hole-2 phonon s t a t e s . The broad s t r u c t u r e around 15 MeV, below the i s o b a r i c analog s t a t e s (9/2 + ) observed at 13.26 and 14.76 MeV, has been a t t r i b u t e d to I f 7 / 2 - h o l e s t r e n g t h by analysis o f the angular d i s t r i b u t i o n . In 2°sPb, 53 % of the lh 11/2 s t r e n g t h is found in the bump between 6.6 and 9.8 MeV but the experimental cross s e c t i o n up to 20 MeV may account f o r the missing lh 11/2 and t o t a l Ig 7/2 and l g 9/2 s t r e n g t h s , although no concentration of s t r e n g t h is observed f o r these s t a t e s . Using Hartree-Fock p r e d i c t i o n s to l o c a t e a l l i n n e r neutron s t a t e s down to I s , t h e i r c o n t r i b u t i o n s have been evaluated. A f t e r s u b s t r a c t i n g a f l a t background as shown in f i g . 4, the i n t e g r a t e d experimental cross s e c t i o n in both nuclei is found to exceed the p r e d i c t i o n f o r the t o t a l sum r u l e by ~ 27 %. For heavy n u c l e i , the experimental s i t u a t i o n can be summarized as f o l l o w s : (a) in the Zr r e g i o n 2 ~ ' 2 ~ ) , 50 to 70 % of the i f 7/2 s t r e n g t h is found at E ~ 5.2. MeV, which gives a s p i n - o r b i t s p l i t t i n g f o r the I f o r b i t o f ~ 3 MeV. (b) in the Sn region (Cd, Sn, Te) d e t a i l e d r e s u l t s ( s e e l " ' I s ) and references there i n ) have been obtained on the fragmentation o f the l g 9/2 s t r e n g t h , o f which 30 to 60 % is found around 5.5. MeV, and 20 to 50 % around 8 MeV. The f 5/2 s t r e n g t h is centered around I i MeV22'33) and the f 7/2 s t r e n g t h c o n t r i b u t e s mainly in the range 12-19 MeV22)
DEEP HOLE STATES
2°8pb(3He,cx)2°Tpb
ll6Sn(3He, o()115Sn Einc: 283PleV ,
o -
,,q ~
8°
E inc =2831"4eV m
,,q o
43c
0:8
:~°5
°
0:2
°
~
°
,!
i'
Ig
;
?i
, ,
',
:::, D~
05
B
I
I
i
,
i
20
10
N
2-
L E x (HEY)
.'.0
3'0
~10
10
E x (MeV)
.'.0
30
Fig. 4 E x c i t a t i o n energy spectra o f the zZ6Sn (3He,~) and 2°epb(~He,~) reactions at 283 MeV i n c i d e n t energy. Hatched peaks are from contaminants. The p o s i t i o n of some subshells as given by Hartree-Fock c a l c u l a t i o n s from Beiner et a l . are i n d i c a t e d by arrows (from r e f . 2 2 ) . (c) in the Sm isotopes Is) a concentration of ig 9/2 strength has been observed at 7.6Mev in ~"3Sm. But in the heavier isotopes, a large spreading of the ig 9/2 strength and i t s overlap with the 2d 5/2 and Ig 7/2 c o n t r i b u t i o n s seem to occur. (d) in the Pb region, the p o s i t i o n of the lh 11/2 discussed above and confirmed by other results 26,27) leads to a s p i n - o r b i t s p l i t t i n g for the lh o r b i t of 4.9 MeV as compared to 5.6 MeV from Hartree-Fock c a l c u l a t i o n s 2 8 ) . 3.2 NEUTRON HOLE STATES STUDIES USING POLARIZED BEAMS In s i n g l e nucleon t r a n s f e r r e a c t i o n s , the angular d i s t r i b u t i o n s allow to determine the t r a n s f e r r e d o r b i t a l angular momentum ~, but not the t o t a l spin j , which has to be i n f e r r e d using shell-model arguments. However, DWBA c a l c u l a t i o n s and (p, d) measurements on l i g h t nuclei 29) have shown some years ago t h a t the analyzing power Ay(O) could e x h i b i t a strong j-dependence . This has been confirmed by recent experiments, and proved to be very useful for spin assignment to deep hole fragmented states. As an example f i g . 5 shows energy spectra and analyzing powers f o r the ~2°Sn(~,d) r e a c t i o n measured with a 90 MeV p o l a r i z e d beam from the Indiana Univers i t y Cyclotron3°). One sees already from f i g . 5a t h a t the structures above 5 MeV e x c i t a t i o n energy have d i f f e r e n t behaviours f o r proton spin up and spin down. The analyzing power f o r the region from 4.3 to 6.6 MeV which is assumed to correspond to the 9/2 hole s t a t e , is shown in f i g . 5b compared to j = 9/2 and j = 7/2 behaviours, e i t h e r empirical or from DWBA c a l c u l a t i o n s . One sees t h a t the 9/2 s p i n a s s i g n m e n t is unambiguous. S i m i l a r l y the xZ6Sn(d,t) reaction at 40 MeV31) has been used to study the
44c
J. MOUGEY
120Sn (~',d) 119Sn
8:14 o ' m,,-~
Spm Up
I
1
•
IZOSn(i5, d )ll9Sn Ex:4.3~66MeV
Ay x4
06
.
Ill
I
I
Ex(MeV)
15
Z
"n 0 o
ii
:
~n
I0
5 _-~.> x4
Spin Down
0"4f -06
- -
90Z,(~,d)SSZr(9/2+ I
--
.....
'ZOsn(~,d)"9Sm( 7/2+ )
.....
DWBA ( g g / z ) DWBA (97/2)
0 8cm.ideg}
(b)
I
(a)
Fig. 5 E x c i t a t i o n energy spectra (a) and analyzing power (b) from the z2OSn(~,d)119Sn r e a c t i o n a t 90 MeV i n c i d e n t energy. The analyzing power data correspond to the broad peak observed between 4.3 and 6.6 MeV e x c i t a t i o n energy. The s o l i d curves correspond e i t h e r t o the e m p i r i c a l Ay f o r the known j ~ = 9/2 + ground s t a t e of 89Zr, o r to a DWBA c a l c u l a t i o n assumlng j = 9/2. S i m i l a r l y , the dashed durve is from e m p i r i c a l Av f o r the j = 7/2, E~ = 0.79 MeV s t a t e of z19Sn or from a DWBA c a l c u l a t i o n assuming j = 7/2 (from re@3°). 5 MeV region in 11SSn, but w i t h a r e a c t i o n which gives less r e d u c t i o n o f the l~wer C components. A strong j-dependence o f Av(e) is also observed f o r the (d,t) r e a c t i o n , reasonably well reproduced by DWBA c a l c u l a t i o n s on l e v e l s of known j - v a l u e . In p a r t i c u l a r , spin assignment o f j = 9/2 is obtained f o r the 3.67 MeV l e v e l in zlSsn, as shown in f i g . 6. In the bump region only a 9/2++ 1/2combination can account f o r the behaviour o f Ay (O),corresponding to c o n t r i b u t i o n s from the ig 9/2 and 2p 1/2 i n n e r hole s t a t e s . Tragments belonging to the 2p 1/2 subshell have also been i d e n t i f i e d a t 2.6 and - 4.2 Mev, wheras 2p 3/2 components have been found a t 4.50 and 4.77 Mev. 3.3 DEEP LYING PROTON HOLE STATES Much less data are a v a i l a b l e on deeplE-bound proton hole s t a t e s 32) than on neutron hole states in heavy nuclei. The (d, 3 He) reaction on 9 0 Zr and 1 4 4 Sm has been used r e c e n t l y 3'+) to i n v e s t i g a t e the i f and Ig p r o t r o n holes, extending the method u t i l i z e d on l i g h t e r nuclei3S). Fig. 7 shows a (~,3He) spectrum at 52 MeV i n c i d e n t energy measured in a maximum o f a ~ = 3 angular d i s t r i b u t i o n . From the v e c t o r - a n a l y z i n g power measurement, i t has been found t h a t the "giant-resonance l i k e " bump (shaded area) which peaks at Ex = 6.8 MeV contains mainly I f 7/2 s t r e n g t h , although the narrow peak a t 5 Mev could also be a t t r i b u t e d to i f 5/2 hole
DEEP HOLE STATES
45c
o~
r 367
0
:it÷+,",!
''
',
-J1 03 02 01
'
31~ - ~,"
-" xi
!
rll
3/2
"/
T[ .
-a3t 186 0 4 ~ 7/2+1 10
t 20BCMS0
-0 1 -02
(a)
-O3
i" 1500
E2
I
< 450>
ANALYZING
POWERS
I
I
'
'
4
90Zr(d, 3He)89y
o!I H.I
llSSn (d,t)11 SSn
I
Ed : 52 MeV QLab : 11 °
o u4
1000
/
o
u~
F-
--02
,
912
9/2+
....... ......
9/2~ + !,' 29/~ + 3,,,27 / J , 1/2-
......
7/2~ +
1/2 ,
(3[3
t15
'-'1
5oo
;/2-
(b)
8c.
10
20
30
Fig.6. Analyzing Dowers and DWBApredictions for the ~t6Sn(~,t)11SSn reaction at 40 MeV. (a) characteristic j = 9/2 and j = 7/2 behaviours; (b) analyzing powers measured in the bump region and compared with DWBA calculations for different+spin assumptions. The association j = 9/2 + 1/2- is the best suited to reproduce A (@) in the range 4.9 < E < 5.9 MeV Y(from ref.
31).
z
0
H-] 02 [49-S91
___
+
X
25
20
15
10
5
0
Ex(MeV)
F i g . 7. E x c i t a t i o n energy spectrum of the 9°Zr(~,3He)89Y reaction a t 52 MeV i n c i dent energy (from ref.34).
46c
J. MOUGEY
However, the 1.75 MeV peak, f o r which a j = 5/2 spin assignment is n i c e l y confirmed, exhausts already the sum rule f o r t h i s s u b s h e l l , (C2S = 8.9) as in previous work (C2S = 7.8, from3~)). The s t r u c t u r e at 3 . 3 . < Ex < 11.3, i f e n t i r e l y considered as a I f 7/2 component exhausts also the sum r u l e (C=S = 9 . 2 ) , in c o n t r a s t to neutron pick-up reactionswhere only ~ 50 % of the shell model s t r e n g t h is found in corresponding inner holes. This r e s u l t , and a s i m i l a r one f o r the Ig s t r e n g t h in i,.3p allows the f i r s t meaningful d e t e r m i n a t i o n o f the spin - o r b i t s p l i t t i n g eso = EC~+~/2 -E #-~/2 f o r inner s h e l l s in heavy n u c l e i . The r e s u l t s are given in t a b l e 1 compared to the value obtained in the Woods-Saxon well used f o r the DWBA f i t s to the data and to Hartree-Fock p r e d i c t i o n s , showing a good agreement. TABLE i Spectroscopic r e s u l t s and s p i n - o r b i t s p l i t t i n g eso(MeV) f o r deeply bound hole s t a t e in heavy nuclei (from r e f . ) nlj 89y
143pm
Ex(HeV)
C2S
I f 5/2
I .75
8.9
i f 7/2
3.3 - 11.3
9.2
lg 7/2
0.27
8.6
e-so
WS ~so
eHFref36) so
5.1
4.0
5.6
4.7 Ig 9/2
3.9-
6.9
4.1
9.4
500 90ZdS.3He)8~ Ed= 52 MeV
~.~ee~
13.0-17oMeV
eL= 12.5"
400 u~ z
3OO
/
o (._)
i 200
(a)
°'!
b .~
100
~ I t 0.
(b) m
2~.
:m.
e.
@c.m. ( G R A D )
2b
1'o
6
E x (MeV) Fig. 8 E x c i t a t i o n energy spectrum (a) and angular d i s t r i b u t i o n in the energy range 13< Ex< 17 MeV (b) f o r the 9°Zr(~, 3He)~gY r e a c t i o n , showing evidence f o r 2 s i / 2 proton pick-up (dark area in ( a ) ) . An amount o f s t r e n g t h corresponding to the c o n t r i b u t i o n s from background ( f u l l l i n e in (a)) and the broad s t r u c t u r e peaked at 6.8 MeV (dashed l i n e ) has been subtracted in the angular d i s t r i b u t i o n shown in (b) (from r e f 3 ~ ) ) .
DEEP HOLE STATES
47c
At the h i g h e r e x c i t a t i o n energy o f ~ 15 MeV in the 8~y e x c i t a t i o n energy spectrum, a s t r u c t u r e have been observed a t Ex = 15.3 MeV having ~ 5 MeV FWHM, see f i g . 8 from37). The a n g u l a r d i s t r i b u t i o n is compatible w i t h ~ = 0 t r a n s f e r , which would be the f i r s t evidence f o r p r o t o n p i c k - u p from the 2s I / 2 s h e l l in such heavy n u c l e i . With t h i s assumption, and s u b s t r a c t i o n o f o t h e r c o n t r i b u t i o n s l i k e in f i g . 8, the s p e c t r o s c o p i c f a c t o r is found to be C2S = 3 . 7 . 4.
Deep hole s t a t e s in medium n u c l e i
: the e x p e r i m e n t a l s i t u a t i o n
Deep hole s t a t e s in the l p and 2 s - l d s h e l l n u c l e i have been r a t h e r e x t e n s i v e l y s t u d i e d from (p, 2p) and ( e , e ' p ) r e a c t i o n s at high i n c i d e n t energy. However, v e r y few e x p e r i m e n t a l s p e c t r o s c o p i c s t u d i e s using these r e a c t i o n s have been r e p o r t e d since the l a s t r e v i e w p a p e r s 3 8 ' 3 9 ) . Most o f the r e c e n t experiments have been devoted to the d i s i n t e g r a t i o n o f H and He i s o t o p e s 4 ° ' " 1 ' " z ) , and to the r e a c t i o n mechanism s t u d i e s mentioned above. N e v e r t h e l e s s , as new e x p e r i m e n t a l facilities f o r c o i n c i d e n c e experiments are now o p e r a t i o n a l , namely a t TRIUMF, SIN, MIT-Bates and NIKHEF, and as s e v e r a l high energy, high duty f a c t o r e l e c t r o n a c c e l e r a t o r s are i n p r o j e c t , i t may be useful to r e v i e w b r i e f l y the p r e s e n t e x p e r imental s t a t u s . 4.1 1p-SHELL NUCLEI
S(p,E)
9Be (e,e'p) P(MeV/c) 10-50 50-100 ,
I
I
I
20
40
60
~
r
100-150
I 150-200 80
E (MeV) Fig. 9 Proton separation energy spectra for the 9Be(e,e'p) reaction within different recoil momentumbins. The energy resolution of ~ 0.9 MeV allows to separate different excited states of SLi at low excitation energy (from ref. 39)). A l o t of experiments have evidenced the ls proton knock-out from lp shell nuclei. As a typical result, f i g . 9 shows separation energy spectra within different recoil momentumbins for the 9Be(e,e'p) reaction. One sees clearly that the lp hole strength is mainly distributed over a few states in the residual 8Li,
48c
J. MOUGEY
w i t h r e l a t i v e i n t e n s i t i e s in reasonable agreement w i t h spectroscopic f a c t o r s as p r e d i c t e d by c a l c u l a t i o n s in the i n t e r m e d i a t e coupling scheme43). The ls s t a t e shows up on the low momentum spectrum as a wide and asymmetrical bump peaked at an energy of 24 MeV. S i m i l a r r e s u l t s have been found in the t2C and 160 n u c l e i , w i t h almost no overlap of the Ip and is hole d i s t r i b u t i o n s . The spreading of the is d i s t r i b u t i o n c l e a r l y i n d i c a t e s the f a i l u r e o f a pure IPSM d e s c r i p t i o n o f such nuclei. However, the shape of the momentum d i s t r i b u t i o n s analysed w i t h i n DWIA are well reproduced by s i n g l e p a r t i c l e wave functions g i v i n g good f i t s to e l a s t i c s c a t t e r ing data (see f i g . 10).
100
,2 C (e'ep)
~
~ so g
Z 0
DWlA
.~
PWIA
N i
Z w
:sO5O15 _
15<_Em<21,5 MeV
I
100
i
30
J
200
i
3
(~)
1
F
100
i
~\I I
200
J
t
300
( MeV/c ) Fig. 10 Momentum d i s t r i b u t i o n s f o r the z2C(e,e'p) r e a c t i o n w i t h i n separation energy regions corresponding to the Ip ( l e f t ) and Is ( r i g h t ) hole s t a t e s . The curves are from Woods-Saxon type d i s t r i b u t i o n s normalized to the data. For the 12C(e,e'p) r e a c t i o n , the experimental d i s t r i b u t i o n s in d i f f e r e n t energy bins in the ls region are found to be a l l compatible w i t h a pure is d i s t r i b u t i o n 45) which enforces the basic d e s c r i p t i o n of the o v e r l a p i n t e g r a l s in terms of a few p a r t i c l e o r b i t a l s . The s t a t i s t i c a l accuracy o f the e x i s t i n g data are b a r e l y s u f f i c i e n t to examine s t r u c t u r e s in the Is s t r e n g t h d i s t r i b u t i o n . Fig. i i shows a 12C(e,e'p) spectrum in the is region compared to neutron pick-up r e a c t i o n spectra-12C(p,d) at 185 MeV47).I2C(3He,~) at 216 MeV46). The two l a s t spectra show s t r u c t u r e s above the proton emission t h r e s h o l d (8.7 MeV) around 11.5 and 15.5 MeV. Whereas the f i r s t one does not seem to have i t s analog in the ( e , e ' p ) spectrum, and would have to be a t t r i b u t e d to m u l t i s t e p processes, the second one corresponds to the p o s s i b l e s t r u c t u r e observed a t E ~ 32.5 MeV. More data from knock-out r e a c t i o n s w i t h high energy r e s o l u t i o n in the is region o f 12C and t~O would be h i g h l y d e s i r a b l e , these nuclei being the best s u i t e d to study the s t r e n g t h d i s t r i b u t i o n of a deep hole. For the Li i s o t o p e s , the r e s u l t s show t h a t s i n g l e - p a r t i c l e models are not the best s u i t e d to describe such l i g h t n u c l e i . Let us consider the eLi case as an example. The separation energy spectrum shown in f i g . 12 as measured w i t h 1.5 Mev r e s o l u t i o n through the (p,2p) r e a c t i o n a t 100 Mev48) e x h i b i t s two main s t r u c t u r e s a t E ~ 4.7 and E ~ 21 MeV a t t r i b u t e d to ip and is proton knock-out in a s h e l l model d e s c r i p t i o n . However, the momentum d i s t r i b u t i o n s (see f i g . 13)
DEEP HOLE STATES
49c
S I'1 t2C(p,d)llC , Ep =185 MeV 0 =2.5 ° G = 194-43 MeV/c
0.3
",.'t.
~L
L
;j:
,iM
" ! '
j,
=E 0.2
(a)
.n.:" .,.4
L
0.1 uJ
5~0
4~0
30
20
~2C(sHo c x ) ~ c/)
1500
10
0
Ex~MeVp
Sl
@lob = 7 °
c 0 L)
IOOC
×±2ot (b)
500 L
60
50
40
30
~0
20
0 ExIM~)
"t 12C(e,e'p)11B
S[
1
'~Eu0.30 70
(c)
0.15
== o u
60
50
4b
40
3b
30
2b
E (MeV)
Ib
E, MW
Fig. 11 Excitation enerqy spectra of the residual nucleus in various nucleon removal reactions from z2C : (a) ~2C(p,d) at 185 MeV47) ; (b) 12C(3He,~) at 216 MeV46) ; (c) ~2C(e,e'p) at 497 MeV45). Structure I at Ex ~ 15.5 MeV in (a) and (b) seems to be present a l s o i n ( c ~ Structure I I at % 12 MeV is not v i s i b l e in (c) and could be a t t r i b u t e d to multistep processes.
50c
J. MOUGEY
]
25
20
15
1
I
I
Ex(MeV) ~0
5
I
0 F-
I
6Li(p' 2P)SHe Binding Energy Spectrum 81 = - e z : 35.4"
ed
tt
:L l a j c~
-a_ 40
"o -~ 20 ~1. b "o
%5
"30
25
20 15 IO ; .... OSeparation Energy (MeV) Fig. 12 Separation energy spectrum for the ~Li(p,2p)SHe reaction at 100 HeY (from ref. "~)).
/~
120
)2Ol / 6Li
lp
.
Data E =I-TMeV ""u
.
.
. 6Li
ls
Data E =16-28MeV
80
+
5" v
80\ k ~ ~ ~
x078
b
E
z,0
e..,
0
"/~ l
x088
I
0
i
100 p
i
I
200 (MeV/c)
_
~
0
100 p
200 (MeV/c)
Fig. 13 Momentum d i s t r i b u t i o n s for the bLi(e,e'p) reaction with separation energy regions including each of the peaks in f i g . 12. The curves are from PWIA and DWIA calculations using ip and is s i n g l e - p a r t i c l e wave functions, and with a r b i t r a r y normalization to the data (from ref. 49)). measured in the Tokyo (e,e'p) experiment 49) cannot be s a t i s f a c t o r i l y reproduced by PW or DW calculations using single p a r t i c l e o r b i t a l s . The experimental d i s t r i b u t i o n s e x h i b i t more important low momentum components than the computed
51c
DEEP HOLE STATES
ones, a behaviour observed in a l l the (p,2p) experiments on L i . This has been q u a l i t a t i v e l y understood in an ~-d c l u s t e r desc r i p t i o n of L i . The r e s i d u a l ~-n system which is unbound a t the r e a c t i o n t h r e s h o l d E = 3.7 MeV, has two resonant s t a t e s at 0.96 MeV ( 3 / 2 - ) and 2.6 MeV ( 1 / 2 - ) superimposed on a continuum s - s t a t e . Thus at low removal e n e r g i e s , both s - a n d p - s t a t e components of the valence proton c o n t r i b u t e to the momentum d i s t r i b u t i o n . Calculations including s c a t t e r i n g states o f the n-~ system s°) have shown t h a t the shape of the d i s t r i b u t i o n is expected to change very r a p i d l y w i t h the s e p a r a t i o n energy when going through the p 3/2 and p 1/2 resonances (see f i g . 14). Thus, a high r e s o l u t i o n (~ 0.5 MeV) knock-out
\
E:
eV)
b 0
L
121
3.7 (~hresholld)
I
100
>
200
p (M ~ v/~) Fig. 14 L i ( e , e ' p ) momentum d i s t r i b u t i o n around the p 3/2 ~-n resonance in the s c a t t e r i n g s t a t e approximation (from r e f . S°)).
T
12C (~+,T:+pi'B
I
T
Eex<9.75MeV
=Z ioo
I
,
50
:
l
q
Eex>9.75MeW t.
¢N ul .D
i
12C (~*,':':+p)11B
P.. ,6 /
v
¢1. IO t.t
"0 50
25
"10 "0
fll
I 140
~
I 180
L p=
1 i 220 (MeV/c)
*40
180
220
p~ (MeV/c)
Fig. 15 12C(~+,~+p) r e a c t i o n cross s e c t i o n as a f u n c t i o n o f the momentum p~ of the s c a t t e r e d pion, a t an i n c i d e n t energy o f E~ = 199 MeV and angles o f e~=-i1775 and On =30 ° . The f u l l (dashed) curve is a DWIA c a l c u l a t i o n using the f i n a l s t a t e (initial s t a t e ) p r e s c r i p t i o n to compute the n-nucleon elementary cross s e c t i o n . The energy regions are chosen so as to separate lp ( l e f t ) from is ( r i g h t ) knockout (from r e f . S l ) ) .
52c
J. MOUGEY
experiment would be very useful to investigate this problem. S i m i l a r l y , the strongly structured peak at high energy would have to be considered in terms of t-d scattering states. Before leaving the Ip-shell n u c l e i , I would l i k e to mention the recent results in (~,~N) reactions. Although mainly designed to study ~-nucleon i n t e r a c t i o n within n u c l e i , they have been shown to be valuable tools f or spectroscopic studies. Fig. 15 shows a DWIA analysis sl) of a recent 12C(~+,~+p) experiment performed at 130-200 MeV incident energy, with I MeV energy resolutionS2). Reasonable f i t s are obtained using the Woods-Saxon wave functions reproducing the (e,e'p) data of f i g . 11, and the deduced occupation numbers (2.9 f o r I p - s h e l l , 1.8 f o r I s - s h e l l ) using f i n a l - s t a t e prescription f or the h a l f - o f f - s h e l l ~-nucleon cross section are close to the corresponding (e,e'p) ones (2.5 and I ) . The 160(~±,~-p) reactions have also been investigated recently with high energy resolutionS3). 4.2 HEAVIER NUCLEI The separation energy spectra obtained on heavier nuclei do not show separated structures above 20 MeV, as expected from the spreading of the Is hole strength d i s t r i b u t i o n observed in Ip-shell nuclei which is comparable to the major shell spacing. Thus, (p,2p) and (e,e'p) results have been analyzed by f i t t i n g an expansion l i k e eq. (4) to the "distorted spectral function" in which only the normally occupied orbits are considered. In the Saclay experiments" ) reasonable f i t s to the momentum d i s t r i b u t i o n s in d i f f e r e n t energy bins have been obtained using Woods-Saxon single p a r t i c l e wave functions in a DWIA analysis, and the procedure was used to extract the hole strength d i s t r i b u t i o n P~(E) shown in f i g . 16a. One sees that most of them e x h i b i t the same dissymmetric~l shape as f or the l i g h t e r nuclei. Similar results have been obtained through the (p,2p) reactionSS), as shown in f i g . 16b. The strength d i s t r i b u t i o n s P^(E) have been used to compute C occupation numbers N~ = % P~(E)dE and mean removal energies E~ = N~I % EP~(E)dE compiled and compare~ in several reviews papersS~,sT,se). One remembers that an overall agreement was claimed, with an observed tendancy of E~ to saturate f o r A ~ 30 at ~ 40 MeV f o r the Ip shell and 60 MeV f or the Is sheTl. However, I would l i k e to i n s i s t on the fact that the i d e n t i f i c a t i o n of the deepest hole states is s t i l l highly unprecise. To disentangle the various hole state contributions, d i f f e r e n t recipees have been used. For the (p,2p) data, the momentum d i s t r i b u t i o n s have been e i t h e r taken from a model in a PWIA analysis s3) or from the experiment i t s e l f , assuming that, in some typical energy bins, only one hole state contributesSg). In the Tokyo (e,e'p) data analysis 6°) gaussian shapes f o r P~(E) have been assumed, whereas Maxwellian shapes were considered by Amaldi et a l ~ ) . Moreover, the background due to m u l t i p l e - c o l l i s i o n processes have not always been removed, although i t s r e l a t i v e importance depends strongly upon the type of reaction, the kinematics, and the detector acceptances. Thus the large dispersion of the published values concerning the position of the Is hol~ in 4°Ca ranging from 50 + 12 (ref. Ss)) to 77 + 14 MeV ( r e f . ~ l ) ) could mainly r e f l e c t the d i f f e r e n t types of analysis and t h e i r d i f f i c u l t i e s as regards to the rather poor s t a t i s t i c s of the experiments. The r e l a t i o n of the experimental mean removal energies to the s i n g l e - p a r t i c l e energies determined by self-consistent calculations and to the t o t a l binding energy have been often discussed (see among others r e f s . 3 ' 4 ~ ' 6 2 ) ) . As an example, one gives in table 2 typical results f o r "°Ca from experiment and theory, refecting the general trends. One sees that in BHF and RBHF theories not enough binding is obtained whereas, in DDHF c a l c u l a t i o n s , the s i n g l e - p a r t i c l e energies tend to be less negative than the experimental removal energies. However, f or the reasons mentioned above, one must be cautious about these conclusions. As momentum d i s t r i b u t i o n s at low ~-values mainly r e f l e c t the o r b i t a l angular momentum ~ of the single-hole state, they cannot be used in general to make j-spin assignments. As f o r pick-up reactions, polarized proton beams have been used to investigate the j-dependence of the analyzing powerS3'67). Such strong dependence has been predicted by Jacob et al 9z) within DWIA as caused by nuclear s p i n - o r b i t
DEEP HOLE STATES
53c
TABLE 2 Experimental proton mean removal energies and t h e o r e t i c a l 4°Ca. Also quoted is the binding energy per nucleon (in MeV). 2s
Id
Ip
HF energies f o r
Is
EA/A
Experiment (E~) (p,2p)
(a)
(e,e'p)
(b)
(e,e'p)
(c)
-8.55 14
+_7
{10.9 _+ 0.7
14.4 _+ O.
19.0 + i . I 111.2 + 0.~ 14.9 + 0.8
34 + 7
50 +- 11
35 -+ I 41
59 +- 3 56
Theory (c a) BHF RBHF DDHF DDHF
(d)
-18.5 -13.3 -9.7 -11
-17.4 -12.8 -8 -10
(d)
(e) (f)
-39.5 -31.5 -25.3 -25
-3.9 -4.5 -7.5 -8.3
-61.7 -50.8 -43.6 -38
References : (a) James et al sS) ; (b) Nakamura et a l , peak energies ~°) ; (c) Mougey et al s") ; (d) Davies et al ~4~ (e) Negele 65) ; ( f ) Nemeth andVautherin66). ,
,
,
,
,
,
,
,
2.0-58Ni
40Ca 0.8
1.5If
0.6 1.00.4 0.5-
0.2
20
/
40
60
40Ca
~tl d
if
(a)
~-~ ld 2s
1, "'¢-k.
20
40
1[, 60
80 E (MEW)
58Ni
(b) '
/~s
lp
ls
10 20 30 40 50 60
Fig. 16
10 20 30 z,0 50 60 EIMeV)
Hole strength energy d i s t r i b u t i o n s (a) the ( e , e ' p ) r e a c t i o n S " ) ; (b) the (p,2p) reactionSS).
in 4°Ca and 5eNi from :
54c
J. MOUGEY
coupling combined with the strong absorption in (p,2p) reactions. However, spino r b i t d i s t o r t i o n e f f e c t s , as discussed in the i n t r o d u c t i o n : ° ) , could change the predicted asymmetries. The experimental results obtained for the !GO and 4°Ca(~,2p) reactions using a 200 MeV p o l a r i z e d beam at TRIUMF have e s s e n t i a l l y confirmed the predicted j-dependence, and the v a l i d i t y of the f a c t o r i z e d DWIA in the chosen kinematics. Some evidence has been obtained f o r lp i / 2 and ip 3/2 strength in "°Ca at ~ 21MeV and ~ 30 MeV separation energy r e s p e c t i v e l y , which would agree with the ip strength d i s t r i b u t i o n as determined from ( e , e ' p ) reactions (see f i g . 16). To conclude t h i s section, I would l i k e to show some spectra of the f i r s t high r e s o l u t i o n (p,pn) experiment which has been performed at IUCF by Watson et a168) using a 149.5 MeV proton beam on a)40Ca(p,pn)39Ca ep = 44.3~ 4°Ca and ~SCa targets. As e n = 36.1° shown in f i g . 17, ld 3/2 1200 2Sl/2 and 2s 1/2 neutron holes, as well as I f 7/2 hole strength in 47Ca are 800 c l e a r l y i s o l a t e d with the I MeV energy resolution, 400 ld3/2 ld5/2 and probable Id 5/2 and Ip strength is seen for both 09 targets. DWIA c a l c u l a t i o n s Z using standard Woods-Saxon bound state wave functions 0 and the "Indiana" o p t i c a l 0 b)48Ca(P,P n)47ca ep = 47.£model Gg) i n c l u d i n g spinen = 36.1o r b i t have been performed 400 (2s1/2+1d3/2) to obtain occupation numbers for the valence ~lf7/2[ ~ ld5/2 shell in reasonable agreement with previous pick-up results (see table 3).These results e s t a b l i s h the v i a b i l i t y o f (p,pn) reactions as a tool 0 10 2o 30 4o for neutron-hole spectroscopy. SEPARATION ENERGY, E s (MeV) . . . . .
o
-
i
r
"
F
--
i
-
Fig. 17 Neutron separation energy spectra "°Ca(p,pn)~gCa and 4SCa(p,pn)47Ca reactions at 149.5 MeV (from r e f . ~ 8 ) . TABLE 3 Spectroscopic factors f o r neutron-hole states from the (p,pn) r e a c t i o n , compared to neutron pick-up results (from r e f . 6 8 ) ) .
Nucleus
nlj
CZS(p,pn) ref~8)
CZS(p,d) refTO)
CzS(~He,~) refTZ)
39Ca
2s 1/2
1.8
1.9
1.74
47Ca
2s 1/2 ld 3/2 I f 7/2
1.3 2.6 8
1.8 3.6 6.7
1.38 2.23 6.94
DEEP HOLE STATES
55c
5. C a l c u l a t i o n s of deep hole s t r e n g t h d i s t r i b u t i o n s As pointed out in the preceding s e c t i o n s , one has to go beyond IPSM to i n t e r p r e t the spreading o f d e e p - l y i n g hole s t a t e s . Several approaches have been made to describe the fragmentation o f one q u a s i - p a r t i c l e s t a t e s at low and i n t e r m e d i a t e e x c i t a t i o n energy 72'73'23'28) showing the r o l e of the coupling o f the hole to c o l l e c t i v e v i b r a t i o n s and to few q u a s i p a r t i c l e s t a t e s . Here, I would l i k e to comment on recent attempts to c a l c u l a t e the hole s t r e n g t h d i s t r i b u t i o n s from s t r o n g l y bound s h e l l s i . e the energy spectra measured in knock-out r e a c t i o n s . One p o s s i b i l i t y is to extend the IPSM approach by r e p l a c i n g the real s h e l l model p o t e n t i a l by a complex one so as to take care o f the f i n i t e l i f e time of the hole s t a t e . This has been 2o< pB< 5o done, using phenomenological l o c a l energy-independent p o t e n t i a l s by Herscovitz 7") and ShantaT5). One finds t h a t the shape o f the imaginary p a r t o f the p o t e n t i a l influences also the momentum distributions. A more d i r e c t approach is to 0 look f o r a wave equation f o r the 50< pB< 100 holes, t h a t is f o r the overlap 4 i n t e g r a l s of eq. I . Their general p r o p e r t i e s have been studied many years ago by Berggren76), JacksonT~ and ClementT8). I t has been shown 0 2 t h a t they are s o l u t i o n s of a set o f coupled i n t e g r o d i f f e r e n t i a l equations having the s t r u c t u r e of a 0 Schr~dinger equation w i t h a noniO0 < pB< 2O0 l o c a l s t a t e dependent e f f e c t i v e p o t e n t i a l . A s i m i l a r r e s u l t has 4 been obtained by B o f f i and Capuzzi 79) in a more fundamental approach i n v o l v i n g the hole mass o p e r a t o r which, close to sharp resonances in the spectrum, plays the r o l e o f a non-local energydependent p o t e n t i a l . S t a r t i n g also 0 20 40 60 E 80(MeV) from the general p r o p e r t i e s o f the mass o p e r a t o r , but w i t h d r a s t i c approximations to reduce i t to a Fig. 18 Separation energy spectra f o r the local energy-dependent s i n g l e " ° C a ( e , e ' p ) r e a c t i o n in various r e c o i l p a r t i c l e complex p o t e n t i a l momentum bins 6°) compared to t h e o r e t i c a l p r e d i c t i o n s b y Gorchakov et a l . (from r e f . 8 ° ) ) . Gorchakov et al 8°) have succeeded in reproducing a t l e a s t q u a l i t a t i v e l y the removal energy s p e c t r a , as shown f o r 4°Ca in f i g . 18. A s i m i l a r o p t i c a l model approach has been f o l l o w e d r e c e n t l y by Klevansky and Lemmere7). F u l l y microscopic c a l c u l a t i o n s o f the s p e c t r a l f u n c t i o n using p e r t u r b a t i o n theory and the Green f u n c t i o n techniques 81), or using a t r u n c a t e d shell-model basis i n c l u d i n g continuum states to d i a g o n a l i z e the mass o p e r a t o r 8 2 ) , have not reached a stage where a meaningful comparison w i t h data can be performed. Another approach i s to s t a r t from the hole propagator in nuclear m a t t e r . The complex s i n g l e - p a r t i c l e f i e l d , i n c l u d i n g the f i r s t o r d e r Hartree-Fock f i e l d and various h i g h e r - o r d e r c o r r e c t i o n s due to the remaining two-body c o l l i s i o n s which w i l l generate the spreading, is computed using a r e a l i s t i c nucleon-nucleon i n t e r a c t i o n . The r e s u l t s are adapted to f i n i t e nuclei by choosing proper r e l e v a n t v a r i a b l e s . Such approach is j u s t i f i e d by the observation t h a t the e s s e n t i a l features o f the hole d i s t r i b u t i o n s , t h e i r widths and shapes are weakly dependent
56c
J. MOUGEY
I
SS
28Si I!,
1.0
1ss
1
58Ni
Z,Oco
lIO
0.5
D,
,,',
i!o
1:
.C t/)
rl
0.6
D~
0.6
,
!l
03
I\
!
o.2 0
o.2 LO
:o
°
0
Ol
40
8O 0
40
80
E (MeV) Fig. 19 Comparison between experimental S") and t h e o r e t i c a l hole strength d i s t r i butions f o r the Is ( t r i a n g l e s and f u l l curves) Ip ( f u l l dots and long dashes) and Id (open c i r c l e s and short dashes) shells in 28Si, 4°Ca and SSNi (from r e f . 8 4 ) ) . upon the s p e c i f i c n u c l i d e considered. The recent c a l c u l a t i o n s by Orland and Schaeffer83), Sartor and Mahaux8") and Antonov et al 8s) show that i t provides a good d e s c r i p t i o n of the observed d i s t r i b u t i o n s . Orland and Schaeffer have shown that the width of the spectral d i s t r i b u t i o n s P~(E) is e s s e n t i a l l y given by the a v a i l a b l e phase-space f o r two-body c o l l i s i o n s in the nuclear medium, which, in the Fermi-gas model, depends upon the distance to the Fermi energy. This also makes the resonance to have more strength on the high energy side. In a less phenomenol o g i c a l way Sartor and Mahaux obtained a s i m i l a r expression f o r the spectral funct i o n associated with the s i n g l e - p a r t i c l e state ~ (energy ca) S (p,E) = ~I Z2(p)
W(p) + (E + E ) R(p) (E + ~ )2 + 1/4(2Z(p) ~(p))2 -i
where Z(p) = ( I - ~I~ V(p,E) 1 E= -c a
and R ( p ) = ( ~ E
-z
(6)
W(p,E))E = -c
V(p,E) and W(p,E) being the ~eal and imaginary part of the mass operator with onshell (E = -e~) expressions V(p) and ~(p). Fig. 19 shows such c a l c u l a t i o n s compared to the strength functions obtained from (e,e'p) experiments. The r e s u l t s are s t r o n g l y dependent on the value of the i n f i n i t e nuclear matter density (or the Fermi momentum) at which the mass-operator is calculated. One notices in f i g . 19 that the agreement is f a i r f o r the Is d i s t r i b u t i o n s , but t h a t the Ip and Id calculated peaks are too narrow i f a normal density of p = 0.17 N/fm 3 is used. This could come from a lower average nuclear density seen by the outer o r b i t s . Antonov et al 8s) obtained reasonable agreement using the coherent f l u c t u a t i o n model 86) in which the f i n i t e nucleus density d i s t r i b u t i o n , as given by electron s c a t t e r i n g , is ~ u i l t by the superimposition of uniform d i s t r i b u t i o n s with weighting f a c t o r s . 6. Conclusions Since a few years the q u a n t i t y and q u a l i t y of experimental data concerning the deep-lying hole states have g r e a t l y improved. Inner neutron-hole strengths have been observed in a v a r i e t y of single neutron pick-up reactions f o r a wide
DEEP HOLE STATES
57c
range o f medium and heavy n u c l e i . High r e s o l u t i o n ( e , e ' p ) and(p,2p) experiments have demonstrated the p o s s i b l i t y to i n v e s t i g a t e the deepest s h e l l s in medium n u c l e i . New s p e c t r o s c o p i c t o o l s l i k e the use o f p o l a r i z e d beams, the (~,~N) and (p,pn) r e a c t i o n s have proved t h e i r v a l i d i t y . E v e n more d e c i s i v e progress is expected in the near f u t u r e w i t h new f a c i l i t i e s l i k e the MEA(NIKHEF-K) a c c e l e r a t o r here in Amsterdam . Fig. 20c shows the extremely good r e s o l u t i o n o f 250 keY achieved in a ~ 2 C ( e , e ' p ) ~ B experiment at 253.5 MeV i n c i d e n t energy.
C'Z ( p , 2 ~ ' } B u
(a)
50MeV
5OMeV
C Iz ( p , d ) C H
e= 65" B"
(b)
-~
qs
. Is
3/2"
5'2 -
Ii I;
I
:i
'lZ
T
11
E (e,e'p) B
Pm: 100 MeV/c I
e = 2?0.5 HeY e e = 90 °
ep = 37.8
(C)
o
E *a
G.S.
2] 2
S.02
',,',,,.,,.,,,;' %..,,,,.,:.;:.:,,.,,.,"'%__,,..,.,,,
.....
5
Excitation Energy [HeV]
>
Fig. 20 High r e s o l u t i o n e x c i t a t i o n energy spectra from (a) t h e 1 2 C ( p , 2 p ) l I B r e a c t i o n a t 50 MeV (b) the12C(p,d)11C r e a c t i o n at 50 MeV, (c) t h e 1 2 C ( e , e ' p ) i I B r e a c t i o n at 253.5 MeV ( ( a ) and (b) from r e f . 89), (c) from r e f . e s ) ) . Such a r e s o l u t i o n , comparable to the best one (350 keV) obtained in a (p,2p) experiment 89) and to the ones obtained in t r a n s f e r r e a c t i o n s , w i l l a l l o w to use the ( e , e ' p ) r e a c t i o n f o r d e t a i l e d s p e c t r o s c o p i c studies w i t h the enormous advantage o f the weak d i s t o r t i o n on e l e c t r o n waves. I t is c l e a r from f i g . 20 t h a t , in the ( e , e ' p ) case, onl~ those l e v e l s in 11B which correspond to the d i r e c t removal o f a I p - p r o t o n ( J =1/2- o r 3 / 2 - ) are e x c i t e d w i t h n o t i c e a b l e s t r e n g t h . In p a r t i c u l a r , the 5/2 , 4.45 MeV s t a t e is absent in the ( e , e ' p ) spectrum, which may i n d i c a t e much s m a l l e r c o n t r i b u t i o n s from m u l t i s t e p processes in t h i s r e a c t i o n . Acknowledgments I would l i k e to thank Profs. H. L a n g e v i n - J o l i o t , G.J. Wagner and C.H.Q. Ingram f o r t h e i r precious advice and comments concerning t h i s t a l k , and, t o g e t h e r w i t h Prof. W.T.N. Van Oers, Dr. J.W. Watson and Dr. L. Lapikas, f o r prov i d i n g me w i t h both published and unpublished data.
58c
J. MOUGEY
References i) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44)
C.A Engelbrecht and H.A WeidenmUller, Nucl. Phys. A184 (1972) 385 ; G.J Wagner, Proc. I n t . Symp. on h i g h l y Excited States in Nuclei, JUlich 1975, ed. A. Faessler, C. Mayer-B~ricke and P. Turek, Vol. 2, p. 175. A.E.L Dieperink and P.J Brussaard, Z. Phys. 261 (1973) 117. A.E.L Dieperink and T. De Forest, Ann. Rev. Nucl. Sci. 25 (1975) I. D.H.E Gross and R. Lipperheide, Nucl. Phys. A150 (1970)~49. ~ Berggren, Nucl. Phys. AI09 (1968) 265. S. B o f f i , C. Giusti and F.D. Pacati, Nucl. Phys. A336 (1980) 416 ; Nucl. Phys. A3~(1980) 437 ; C. Giusti and F.D Pacati, Nucl. ~ s . A336 (1980) 427. ~ Roos et a l , Phys. Rev. Lett. 40 (1978) 1439. M. Bernheim et a l , Proc. Int. ConT-~.on Reaction Mechanisms, BalatonfUred (Hungary), 1977 ; S. B o f f i , C. G i u s t i , F.D Pacati and S. F r u l l a n i , Nucl. Phys. A319 (1979) 461. S. B o f f i , C. Giusti and F.D Pacati, to be published. D.F Jackson, Nucl. Phys. A257 (1976) 221 A.E.L Dieperink and I. Sic~-~--,Phys. Lett. 1098 (1982) I. J. K~llne and B. Fagerstr~m, Proc. F i f t h I n t . Conf. on High Energy Physics and Nuclear Structure, Uppsala, 1973, ed. G. Tibell (North-Holland, Amsterdam, 1974) P. 369. J. Van De Wiele et a l , Nucl. Phys. A297 (1978) 61. G.M Crawley, Proc. Int. Conf. on Structure of Medium-Heavy Nuclei, Rhodes, 1979 ( I n t . Conf. Series 49, The I n s t i t u t e of Physics, Bristol and London) p. 127 ; G.M Crawley, Pro~. I n t . Symp. on Highly Excited States in Nuclear Reactions, Osaka, 1980 (RCNP, Osaka University) p. 590. S. Gales, Nucl. Phys. A354 (1981) 193c. G.M Crawley et a l , Phys. Rev. C23 (1981) 589. G.M Crawley et a l , Phys. Lett. ~ 9 B (1982) 8. T. Nakagawa et a l , Nucl. Phys. A--3-76 (1982) 513. R.S Tickle, W.S Gray and R.D Bent, Nucl. Phys. A376 (1982) 309. E. Gerlic et a l , submitted to Phys. Lett. G. Mairle et a l , Nucl. Phys. A382 (1982) 173. H. Langevin-Joliot et a l , Phys. Lett. 114B (1982) 103 and communication to t h i s conference. V.G Soloviev, C.Stoyanov and A. Vdovin, Nucl. Phys. A342 (1980) 261. G.M Crawley et a l , Phys. Rev. C23 (1981) 1818. G. Duhamel et a l , J. Phys. G7 (Tg81) 1415. S. Gales et a l , Phys. Rev. CT8 (1978) 2475. J. G u i l l o t et a l , Phys. Rev.~21 (1980) 879. Nguyen Van Giai, Proc. I n t . Sym~p. on Highly Excited States in Nuclear Reactions, Osaka, 1980 (RCNP, Osaka University) p. 682. K. Hosono et a l , Nucl. Phys. A343 (1980) 234. G,M Crawley et a l , Phys. Rev. C23 (1981) 1818. G. Perrin et a l , Nucl. Phys. A 3 ~ (1981) 61. P. Doll et a l , Phys. L e t t . B82~979) 357. R.H Siemssen et a l , Phys. Let~. I03B (1981) 323. A. S t u i r b r i n k et a l , Z. Phys. A297 (1980) 307. V. Bechtold et a l , Phys. Lett. 7~2-B-(1977) 169. M. Beiner, H. Flocard, Nguyen Va~Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. G. Seegert, Diplomarbeit, Max-Planck I n s t i t u t e , Heidelberg, 1982 (unpublished) and G.J Wagner, private communication. W.J Mc Donald, Nucl. Phys. A335 (1980) 463. J. Mougey, Nucl. Phys. A335~T980) 35. M. Bernheim et a l , N u c l . ~ y s . A365 (1981) 349. E. Jans et a l , submitted to Phys.---~ev. Lett. W.T.H Vanders et a l , Phys. Rev. C25 (1982) 390 ; D.J Margaziotis et a l , Phys. Rev. C25 (1982) 2873. V.V Ba~shov, A.N Boyarkina and I. Rotter, Nucl. Phys. 59 (1964) 417. D. Royer e t a ] , Phys. Rev. C12 (1975) 327.
DEEP HOLE STATES 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65) 66) 67) 68) 69) 70) 71) 72) 73) 74) 75) 76) 77) 78) 79) 80) 81) 82) 83) 84) 85) 86) 87) 88) 89) 90) 91)
59c
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