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Core configuration mixing effects in inelastic electron scattering on spherical nuclei
Volume 44B, number I
PHYSICS LETTERS
CORE CONFIGURATION INELASTIC ELECTRON
2 April 1973
M I X I N G E F F E C T S IN
SCATTERING ON SPHERICAL NUCLEI
B. ANTOINE, V. GILLET, J.M. NORMAND Service de Physique Th~orique, CE.N. Saelay, France
J.B. BELLICARD, PHAN XUAN HO and I. SICK D~partrnent de Physique Nucl~aire, C.E.N. Saclay, BP 2, 91190 - Gif sur Yvette, Prance
Received 7 December 1972 We study the effects of core configuration mixing on inelastic electron scattering for the first 2+ and 3- levels of single closed-shell nuclei such as 92M0 and S2Cr. The theory accounts fairly well for the octupole states where particle hole core polarization is important. This last effect is weak for the quadrupole states and our description fails totally to account for their collectivity. We wish to discuss in the present note how much new information as compared to electromagnetic transition rates is proved by inelastic electron scattering in the region of transfer momenta between 1 and 2 fm--1. At low transfer momentum, say smaller than 1 fm -1 and for nuclei with A <~ 100, it is known that within the experimental errors of a few percent the angular dependence of the cross section is generally determined by the overall characteristics of the nuclear potential, more precisely by its root-mean-square radius. The fits of various models in these regions depend only on a normalization coefficient and bring no new information as compared to electromagnetic transition rates, of B(E;~)'s. In particular, the shape of the angular distribution is totally insensitive to configuration mixing. At much larger transfer momenta the cross section becomes sensitive to all the details of the wave function, like its configurational admixtures, its shape at the nuclear surface, and to the effects of short-range correlations, meson exchanges, electromagnetic transverse contributions, etc.. It is therefore relevant to study numerically where is the region of q values where different configuration mixing models start to yield (apart from normalization) different results, but where we can hopefully neglect the more complicated effects such as short-range correlations, meson exchanges or transverse contributions to the cross section. We shall carry out this discussion by computing the cross section for inelastic electron scattering on two
typical spherical nuclei, 52Cr and 92Mo. We use two successive approximations just for purpose of illustration: a) RPA configuration mixing in the space of the open shell of protons corresponding to two-quasiparticle (2QP) configurations; b) the same space to which is added the particle-hole configurations brought in by the nearest closed major shell as an approximation to core contributions. These two models yield characteristic differences in the transition charge densities which will be used for our discussion. For information we shall also give the contribution of the core alone. Details of the nuclear calculation have been given elsewhere [, 2]. The results are compared to the data of a recent experiment at Saclay [3]. The inelastic cross sections are calculated by using the HEINDEL DWBA code 141. We see first (fig. 1) in the transition charge density Pif of the lowest 2 ÷ and 3 - states of 92Mo that the contributions of the core configurations can be quite different in shape from the ones of the open-shell 2QP configurations. This becomes evident from an examination of the configuration space: for the 2 + state the wave function calculated in the 2QP space is largely dominated by the 2p2/2 configuration which yields no change in sign ofPi f. The same is true for the next important contribution, 2Pl/22P3/2, and for the other four smaller contributions. On the contrary, the core contribution is dominated by the 2d5/2 lg9/2 configuration which, of course, yields one relative node. However, these rapidly varying core contributions have only a small effect on the total Pif as can 55
Volume 44B, number 1
PHYSICS LETTERS '
,
,
,
,
,
92
Mo
.
c
-•
~////~
\\\
-
,
2 April 1973
/ \Mo3-
. . . .
,
2+
/r~9
~_
20P*CORE - CO~E ---2QP
// ~.
•
-
/
~
,
\
.
\
.
---co~E
/,.I.i \ /
,
.
.
.
k
,
/ I I
\ \
\ \
i I
\ \ \
2
6
4
r(Fm)
Fig. I. The transition charge density for the lowest quadrupole and octupole states in 92M0. The curve denoted by 2QP is given by the wave function which mixes all two-quagiparticle configurations in the open shell of 92M0. The core contribution curve is a different configuration mixing calculation including only the particle-hole figurations from the next major shell. The solid line curve is a calculation mixing both open shell and core configurations.
10-2'
",,,
?. .-\
"'-L\.
\
E = 2ogM,v " ~ 20~.CORE .-_- ~o~
',,
~3
[~\
,~
.F \
.... ~o~ ---~°~
%
";\ \ Id'
-
'i\\
\\ \k* '
•
'
~o ~
~ /---------~.. \
16'
10 "
x,.\~
1°9
\-~. ,
06
, .
I
1,0
. . . .
\.
,0I7 I0-8F,
I
1.5
06 Q (rm-')
/
~
, .
1' 7
\,
1
10
.
.
.
.
1
t 5
/
!iIi :
I
1.8
Fig. 2. The inelastic electron scattering cross sections for the lowest octupole and quadrupole states of 92Mo with the various models, as compared to the experimental points [31. The experimental quadrupole results are divided by a factor of 3.
56
Volume 44B, n u m b e r 1
i i
PHYSICS LETTERS
i
i
i
i
i 52Cp
I0
24
--2QP+
"\~
CORIE
. . . . c ORE
\~
.... 2QP
+ E,~/2
"\
E
f
\\o
%
----
2QP
•
~
'
E,cp
\
"\.
10-"
\\ \
d b
~:"
'10
IO-' i. 3
S2Cr 3E : 209M*V
10-2
E = 209 MoV
10 3
2 April 1973
10- t \
~\
÷
,)
'\.\ 10"6
.\.
~.~
-.,,. , , >
10-~
\ \ 10-7
10-'
\ \ \
\\
lO ~
10-'
"~.~.~.
~ . ,%.
'~. 0.6
1.0
qlfrn -~ )
1.5
1.7
0.6
1.0
q ( f m -1 )
1.5
1.7
Fig. 3. Same as fig. 2, but for S2Cr where quadrupole results are divided by a factor of 2.
be seen from fig. 1, since the even-parity core contribution has a small weight in the wave function. The situation is somewhat different for the octupole state for which now the large core contribution has been shown to yield an excellent fit to the electromagnetic rates [2]. In this case the open-shell configurations are few, the only possibility being related to the abnormal parity state 1g9/2' There are two open-shell configurations only, namely the lg9/22d3/2, which is the dominant one, and the much smaller l g9/21f5/2. They both yield a change of sign in the transition charge density. On the other hand, the sum over ten different core contributions with well distributed weights yields no node in the transition charge density. Since the two quasi-particle configuration as well as the core contributions have a large weight, the total transition charge density is strongly enhanced at the surface; in the 2 + case this enhancement is much smaller. We turn now to the inelastic cross sections produced by these various transition charge densities (see figs. 2 and 3).
First, for both the quadrupole and the octupole states the shape of the inelastic cross section below 1 fm -1 follows very closely the predictions of the different models. The only information contained in the cross section is a normalization which is identical to the knowledge of the B(EX). Second, if we examine the various curves for transfer momenta larger than 1 fro- 1 in the case of the bigger nucleus 92Mo, and larger than 1.4 fm -1 for the smaller one, 52Cr, we see that the shapes of the calculated inelastic cross section are now sensitive to the nuclear model. For tile octupole state in 92Mo, the node found in the core transition density gives a cross section minimum at 1.6 fro- I which is lacking in the 2QP curve. For 52Cr the octupole state cross sections given by the various models are more similar since the corresponding inelastic transition densities have the same behaviour. For the quadrupole state in 92Mo the minima of the core model and the 2QP model are displaced. This is not the case for 52Cr. where the core contributions 57
Volume 44B, number 1
PHYSICS LETTERS
are weak and the final curve is similar to the 2QP one. We get, however, noticeable differences between 1.0 and 1.7 f m - t which should allow experimentally to distinguish between the two cases. In summary, one has to go above 1.0 fm-1 at least to get a region of still moderate q values where the shapes of the inelastic electron scattering cross sections are clearly dependent on the detailed assumptions made for the nuclear shell model. There remains to discuss the agreement between theory and experiment [31. Up to 1.0 fm -1 the model is quite successful in accounting for the octupole state, provided both core and two quasi-particle configurations are included. This agreement in absolute value for small q can also be seen from the results for 52Cr (fig. 3). On the other hand for the 2 + state, core contributions across one major shell to the next are very small and not sufficient for enhancing the openshell contributions to reach the experimental values. As seen in the figure we nmst introduce a normalization factor of 3 for 92Mo and 2 for 52Cr corresponding to an effective charge of the proton of 1.7, in order to get an agreement up to 1.0 fm 1. The same results are found in 52Cr. A detailed analysis of the B(EX) for more than 25 single closed shell nuclei [2], within this framework yields similar results: the particle-hole core polarization mechanism accounts very nicely for the odd parity vibrational transition but fails totally for quadrupole states where the particle hole excitations include only two successive major shells. Above
58
2 April 1973
1 f m 1, inelastic electron scattering yields cross sections somewhat larger than the ones given by the 2 QP + CORE description. This deviation happens in regions where the missing transverse contribution [8] as well as edge effects [9], meson exchange [5] and short-range correlations [6, 7] are not yet important. Consequently there are some nuclear structure effects still missing, not only in order to increase the quadrupole strength, but also to explain for the 2 + and 3 states the too small shell model cross sections above 1 fm-.l
References [ 1] V. Gillet, G. Giraud, J. Picard and M. Rho, Phys. Lett. 27B (1968) 483. [2] V. Gillet, B. Giraud and M. Rho, Phys. Rev. 178 (1979) 1695 and to be published. {31 Phan Xuan Ho~ J.B. Bellicard, P. Leconte and L Sick. to be published. [4] J. Heisenberg, private communication. [ 5 ] M. Chemtob and A. Lumbroso, Nucl. Phys. B17 (1970) 401. [6] F.C. Khanne, Phys. Rev. Lett. 20 (1968) 871. [7] C. Ciofi l)egli Atti and N. Kabachnik, Phys. Rev. CI (1970) 809. [8] K. Itoh, M. Oyamada and Y. Torizuka, Phys. Rev. C2 (1970) 2181. [9] L.R.B. Elton, Pro¢. Intern. Conf. on Electromagnetic sizes of nuclei (Ottawa 1967) p. 267.
PHYSICS LETTERS
CORE CONFIGURATION INELASTIC ELECTRON
2 April 1973
M I X I N G E F F E C T S IN
SCATTERING ON SPHERICAL NUCLEI
B. ANTOINE, V. GILLET, J.M. NORMAND Service de Physique Th~orique, CE.N. Saelay, France
J.B. BELLICARD, PHAN XUAN HO and I. SICK D~partrnent de Physique Nucl~aire, C.E.N. Saclay, BP 2, 91190 - Gif sur Yvette, Prance
Received 7 December 1972 We study the effects of core configuration mixing on inelastic electron scattering for the first 2+ and 3- levels of single closed-shell nuclei such as 92M0 and S2Cr. The theory accounts fairly well for the octupole states where particle hole core polarization is important. This last effect is weak for the quadrupole states and our description fails totally to account for their collectivity. We wish to discuss in the present note how much new information as compared to electromagnetic transition rates is proved by inelastic electron scattering in the region of transfer momenta between 1 and 2 fm--1. At low transfer momentum, say smaller than 1 fm -1 and for nuclei with A <~ 100, it is known that within the experimental errors of a few percent the angular dependence of the cross section is generally determined by the overall characteristics of the nuclear potential, more precisely by its root-mean-square radius. The fits of various models in these regions depend only on a normalization coefficient and bring no new information as compared to electromagnetic transition rates, of B(E;~)'s. In particular, the shape of the angular distribution is totally insensitive to configuration mixing. At much larger transfer momenta the cross section becomes sensitive to all the details of the wave function, like its configurational admixtures, its shape at the nuclear surface, and to the effects of short-range correlations, meson exchanges, electromagnetic transverse contributions, etc.. It is therefore relevant to study numerically where is the region of q values where different configuration mixing models start to yield (apart from normalization) different results, but where we can hopefully neglect the more complicated effects such as short-range correlations, meson exchanges or transverse contributions to the cross section. We shall carry out this discussion by computing the cross section for inelastic electron scattering on two
typical spherical nuclei, 52Cr and 92Mo. We use two successive approximations just for purpose of illustration: a) RPA configuration mixing in the space of the open shell of protons corresponding to two-quasiparticle (2QP) configurations; b) the same space to which is added the particle-hole configurations brought in by the nearest closed major shell as an approximation to core contributions. These two models yield characteristic differences in the transition charge densities which will be used for our discussion. For information we shall also give the contribution of the core alone. Details of the nuclear calculation have been given elsewhere [, 2]. The results are compared to the data of a recent experiment at Saclay [3]. The inelastic cross sections are calculated by using the HEINDEL DWBA code 141. We see first (fig. 1) in the transition charge density Pif of the lowest 2 ÷ and 3 - states of 92Mo that the contributions of the core configurations can be quite different in shape from the ones of the open-shell 2QP configurations. This becomes evident from an examination of the configuration space: for the 2 + state the wave function calculated in the 2QP space is largely dominated by the 2p2/2 configuration which yields no change in sign ofPi f. The same is true for the next important contribution, 2Pl/22P3/2, and for the other four smaller contributions. On the contrary, the core contribution is dominated by the 2d5/2 lg9/2 configuration which, of course, yields one relative node. However, these rapidly varying core contributions have only a small effect on the total Pif as can 55
Volume 44B, number 1
PHYSICS LETTERS '
,
,
,
,
,
92
Mo
.
c
-•
~////~
\\\
-
,
2 April 1973
/ \Mo3-
. . . .
,
2+
/r~9
~_
20P*CORE - CO~E ---2QP
// ~.
•
-
/
~
,
\
.
\
.
---co~E
/,.I.i \ /
,
.
.
.
k
,
/ I I
\ \
\ \
i I
\ \ \
2
6
4
r(Fm)
Fig. I. The transition charge density for the lowest quadrupole and octupole states in 92M0. The curve denoted by 2QP is given by the wave function which mixes all two-quagiparticle configurations in the open shell of 92M0. The core contribution curve is a different configuration mixing calculation including only the particle-hole figurations from the next major shell. The solid line curve is a calculation mixing both open shell and core configurations.
10-2'
",,,
?. .-\
"'-L\.
\
E = 2ogM,v " ~ 20~.CORE .-_- ~o~
',,
~3
[~\
,~
.F \
.... ~o~ ---~°~
%
";\ \ Id'
-
'i\\
\\ \k* '
•
'
~o ~
~ /---------~.. \
16'
10 "
x,.\~
1°9
\-~. ,
06
, .
I
1,0
. . . .
\.
,0I7 I0-8F,
I
1.5
06 Q (rm-')
/
~
, .
1' 7
\,
1
10
.
.
.
.
1
t 5
/
!iIi :
I
1.8
Fig. 2. The inelastic electron scattering cross sections for the lowest octupole and quadrupole states of 92Mo with the various models, as compared to the experimental points [31. The experimental quadrupole results are divided by a factor of 3.
56
Volume 44B, n u m b e r 1
i i
PHYSICS LETTERS
i
i
i
i
i 52Cp
I0
24
--2QP+
"\~
CORIE
. . . . c ORE
\~
.... 2QP
+ E,~/2
"\
E
f
\\o
%
----
2QP
•
~
'
E,cp
\
"\.
10-"
\\ \
d b
~:"
'10
IO-' i. 3
S2Cr 3E : 209M*V
10-2
E = 209 MoV
10 3
2 April 1973
10- t \
~\
÷
,)
'\.\ 10"6
.\.
~.~
-.,,. , , >
10-~
\ \ 10-7
10-'
\ \ \
\\
lO ~
10-'
"~.~.~.
~ . ,%.
'~. 0.6
1.0
qlfrn -~ )
1.5
1.7
0.6
1.0
q ( f m -1 )
1.5
1.7
Fig. 3. Same as fig. 2, but for S2Cr where quadrupole results are divided by a factor of 2.
be seen from fig. 1, since the even-parity core contribution has a small weight in the wave function. The situation is somewhat different for the octupole state for which now the large core contribution has been shown to yield an excellent fit to the electromagnetic rates [2]. In this case the open-shell configurations are few, the only possibility being related to the abnormal parity state 1g9/2' There are two open-shell configurations only, namely the lg9/22d3/2, which is the dominant one, and the much smaller l g9/21f5/2. They both yield a change of sign in the transition charge density. On the other hand, the sum over ten different core contributions with well distributed weights yields no node in the transition charge density. Since the two quasi-particle configuration as well as the core contributions have a large weight, the total transition charge density is strongly enhanced at the surface; in the 2 + case this enhancement is much smaller. We turn now to the inelastic cross sections produced by these various transition charge densities (see figs. 2 and 3).
First, for both the quadrupole and the octupole states the shape of the inelastic cross section below 1 fm -1 follows very closely the predictions of the different models. The only information contained in the cross section is a normalization which is identical to the knowledge of the B(EX). Second, if we examine the various curves for transfer momenta larger than 1 fro- 1 in the case of the bigger nucleus 92Mo, and larger than 1.4 fm -1 for the smaller one, 52Cr, we see that the shapes of the calculated inelastic cross section are now sensitive to the nuclear model. For tile octupole state in 92Mo, the node found in the core transition density gives a cross section minimum at 1.6 fro- I which is lacking in the 2QP curve. For 52Cr the octupole state cross sections given by the various models are more similar since the corresponding inelastic transition densities have the same behaviour. For the quadrupole state in 92Mo the minima of the core model and the 2QP model are displaced. This is not the case for 52Cr. where the core contributions 57
Volume 44B, number 1
PHYSICS LETTERS
are weak and the final curve is similar to the 2QP one. We get, however, noticeable differences between 1.0 and 1.7 f m - t which should allow experimentally to distinguish between the two cases. In summary, one has to go above 1.0 fm-1 at least to get a region of still moderate q values where the shapes of the inelastic electron scattering cross sections are clearly dependent on the detailed assumptions made for the nuclear shell model. There remains to discuss the agreement between theory and experiment [31. Up to 1.0 fm -1 the model is quite successful in accounting for the octupole state, provided both core and two quasi-particle configurations are included. This agreement in absolute value for small q can also be seen from the results for 52Cr (fig. 3). On the other hand for the 2 + state, core contributions across one major shell to the next are very small and not sufficient for enhancing the openshell contributions to reach the experimental values. As seen in the figure we nmst introduce a normalization factor of 3 for 92Mo and 2 for 52Cr corresponding to an effective charge of the proton of 1.7, in order to get an agreement up to 1.0 fm 1. The same results are found in 52Cr. A detailed analysis of the B(EX) for more than 25 single closed shell nuclei [2], within this framework yields similar results: the particle-hole core polarization mechanism accounts very nicely for the odd parity vibrational transition but fails totally for quadrupole states where the particle hole excitations include only two successive major shells. Above
58
2 April 1973
1 f m 1, inelastic electron scattering yields cross sections somewhat larger than the ones given by the 2 QP + CORE description. This deviation happens in regions where the missing transverse contribution [8] as well as edge effects [9], meson exchange [5] and short-range correlations [6, 7] are not yet important. Consequently there are some nuclear structure effects still missing, not only in order to increase the quadrupole strength, but also to explain for the 2 + and 3 states the too small shell model cross sections above 1 fm-.l
References [ 1] V. Gillet, G. Giraud, J. Picard and M. Rho, Phys. Lett. 27B (1968) 483. [2] V. Gillet, B. Giraud and M. Rho, Phys. Rev. 178 (1979) 1695 and to be published. {31 Phan Xuan Ho~ J.B. Bellicard, P. Leconte and L Sick. to be published. [4] J. Heisenberg, private communication. [ 5 ] M. Chemtob and A. Lumbroso, Nucl. Phys. B17 (1970) 401. [6] F.C. Khanne, Phys. Rev. Lett. 20 (1968) 871. [7] C. Ciofi l)egli Atti and N. Kabachnik, Phys. Rev. CI (1970) 809. [8] K. Itoh, M. Oyamada and Y. Torizuka, Phys. Rev. C2 (1970) 2181. [9] L.R.B. Elton, Pro¢. Intern. Conf. on Electromagnetic sizes of nuclei (Ottawa 1967) p. 267.