Jakubaßa-Amundsen, Doris H.

English

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Z. Phys ik  268, 409-411 (1974) 9 by Spr inger -Ver lag  1974 Calculation of the 0 + and 2 + Giant Resonances in Ni  58 Doris H. JakubaBa Physik-Department, Technische Universit~it Miinchen, Garching Received February 21, 1974 Abstract. Estimates of the excitation energy, width and sum rule fraction of the collective monopole and quadrupole vibrations indicate that the peak in the 62 A -~/3 MeV region is a superposition of the 0 + and 2 + modes. Experiments on the inelastic scattering of electrons and protons on various target nuclei [-1] have shown a giant resonance with energy around 62 A -1/3 MeV. Angular distributions were consistent with either a 0 + or a 2 + assignment. The form factors of the two multipolarities were found to differ from each other when the beam was polarized. Therefore new experi- ments have been performed on Ni 5s using a polarized proton beam [-2]. In order to identify this giant resonance we present calculations of the collective 0+ and 2 + vibrations in Ni 58 and compare them with experiment. The excitations of a nucleus can be described within the shell model taking into account one part ic le-  one hole excitations. We restricted our configuration space to the levels shown in Table 2. The single particle energies and corresponding wave functions are taken as eigenvalues and eigenvectors of a Woods Saxon potential diagonalized in the space of the harmonic oscillator functions [3]. The potential parameters are chosen to fit the energies of the shells near the Fermi level to the experimental Tab le  1. Parameters  of the Woods  Saxon potent ia l  Vo R o a o 2 Rls al~ R c [MeV]  [ fm] [ fm]  [ fm] [ fm] [ fm] P rotons  -48 .56  5.27 0.660 24.5 4.80 0.5 4.9 Neut rons  -49 .16  4.94 0.535 28.1 4.84 0.5 0.0 excitation energies of the neighbouring nuclei. The parameters defined as in Ref. 3 are given in Table 1. Table 2 shows the single particle energies. The energies of the collective modes were obtained with the RPA method. To simplify the calculations we reduced the problem to that of a closed shell nucleus by interchanging the neutron 2 P3/2 and 2 Pl/2 level. A zero range residual interaction was used V(1, 2)= V o 6(r 1 -r2) 9 (Fo + Fo zl ~2 + Go al a2 + Go a162 "gl "~2) Table  2. Single part ic le energies for Ni  5s Ni  58 Proton  energy Neut ron  energy [MeV]  [MeV]  2 fv/2 9.435 5.169 1 h1~/2 9.088 4.233 1 g7/2 7.512 2.699 2 d3/2 5.803 1.481 3 sl/2 4.848 -0 .080  2 ds/z 4.282 -0 .983  1 gg/z 0.301 -5 .738  1 fs/2 - -2 .497 -8 .659  2 Pl/z - 2.917 - 8.189 2 P3/2 - 3.417 - 10.601 1 f7/2 - 8.177 - 15.498 2 sl/2 - 12.192 - -20.486 1 d3/z - 12.553 -21 .171 1 ds/2 - 16.058 -24 .593 1 Pl/2 - 21.669 - 31.586 1 P3/2 - 23.261 - 33.052 410 Table 3. Parameters of the residual interaction in Ni ss Vo F~  F;x F2" F," a, ~, [MeV fm 3] 380 0.65 -1.39 -0.85 -2.29 1.3 -0.3 Table 4. Energy, width and reduced transition probability of the 0 + and 2 + isoscalar and isovector modes in Ni 5s L '~ T he) F B(EL)'f [MeV] [MeV] [e 2 fm 4] 0 + 0 17.0 0.72 932 0 18.0 0.32 49 1 22.0 1.00 50 1 28.3 4.70 220 0 14.6 3.19 1067 0 14.9 0.30 81 1 24.5 4.28 121 1 26.2 0.87 51 2 + where the parameters Vo, Fo, F;, G o and G o were fitted to the experimental energies and transition probabilities of the lowest 2 + and 3 + modes and the 1- giant resonance. As the Migdal force should be universal we took the parameters which are different inside and outside the nucleus close to the Pb 2~ values [4]. These are shown in Table 3, defined as in Ref. 3. The resulting excitation energies of the isoscalar 0 + and 2 + modes can be approximated by 65 A -1/3 MeV and 56A -1/3 MeV respectively, which is consistent with calculations on other nuclei 1-4]. The width of these collective modes results mainly from their coupling to single particle continuum states, It can be expressed by [5] F 2 re_  I(hco - 8i) X i l  2 where the sum runs over all continuum states of the chosen configuration space; e~ is the single particle energy and AE the energy range of continuum states around the energy he) of the collective mode. X~ is the RPA amplitude. Table 4 shows energy, width and reduced transition probability of the collective vibrations. The collectivity of the excitations i expressed by the fraction of the corresponding sum rules which these modes exhaust. The isoscalar monopole and quadru- pole sum rules are given by [6] (2L+l)2h2AR2 L=0,2 .  ~,h~oj.B(EL)j.T- (L+2)~zm ' f D. H. Jakubaga The sum runs over all final states that can be excited by the multipole operator; A is the mass number, R the rms radius of the charge distribution of the nucleus and B(EL)sT= IL~: i~=lr2y~ ~ 0+) 2 is a modified reduced transition probability, the sum including neutron states. Summing over all isoscalar excitations with energy up to 24 MeV, one finds that most of the sum rule is exhausted: 75.0% for L=0 and 90.5% for L=2.  The scattering experiments [2] show a peak which extends from 13 to 18 MeV. The calculated energies and widths of both multipole resonances are thus compatible with the experimental data. Satchler [2] carried out a DWBA calculation of the differential cross section for the excitation of the monopole as well as the quadrupole mode. For the normalization of the curves he assumed the maximum transition strength allowed by the sum rules, The differential cross section was measured in the energy region 14.6-16,7 MeV. Including the 0 + mode at 17.0 MeV we find in this region only 49.6 and 36.8 % of the 0 + and 2 + sum rules respectively. As the 2 + mode at 14.6 MeV is lying at the edge of the con- sidered region and has a large width we took only ~--~ [mb/sr] Ni58( P, P') 5~ ~ .................................... E = 1/~.6-16.7 MeV ' \ \  ~ 0+and 2 + 0.5 \. ~ 2+ \. I X. \ .  \ .  \ ~',~, i "  0 + 0.1 I I I I 1 =~L 10' 15" 20" 25* 30 ~ 35 o Fig. 1. Cross section for the giant 0 + and 2 + resonances; the data as well as the relative values of the theoretical predictions are taken from Ref. 2. The dotted DWBA curve labelled with (2 +) is normalized as done in Ref. 2 and is shown for comparison. The full line is a superposition ofthe 0 + and 2 + curves with the proper normalization Z. Physik (1974) Calculation of the 0 + and 2 + Giant Resonances in Ni 5s 411 half of its strength when computing the 2 + sum rule fraction. Fig. 1 shows the renormalized DWBA curves com- pared with the experimental cross section. The 2 + curve is in good agreement with the experiment for larger angles while the behavior at small angles is better described by the 0 + mode. The full line denotes the superposition of the 0 + and 2 + cross sections and is in fair agreement with the experimental data in the total angular range considered. If we take the energy region from 12.7-23.7 MeV we find a similar result. Thus is strong evidence that the isoscalar giant reso- nance excited in proton scattering experiments is a superposition of a monopole and a quadrupole mode. I wish to thank Dr. P. Ring and R. Bauer for allowing me to use their numerical programs, and Prof. H. Schmidt and Dr. M. Kleber for stimulating discussions. References 1. Nagao, M., Torizuka, Y.: Phys. Rev. Lett. 30, 1068 (1973) Lewis, M.B., Bertrand, F.E., Horen, D.J.: Phys. Rev. C8, 398 (1973) 2. Kocher, D.C., Bertrand, F.E., Gross, E.E., Lord, R.S., Newman, E.: Phys. Rev. Lett. 31, 1070 (1973) 3. Ring, P., Bauer, R., Speth, J.: Nucl. Phys. A206, 97 (1973) 4. JakubaBa, D.H.: Z. Physik 261, 305 (1973) 5. FlieBbach, T.: Z. Physik 218, 385 (1969) 6. Lane, A.M.: Nuclear Theory, p. 79. New York: W.A. Benjamin 1964 Dipl.-Phys. Doris H. JakubaBa Physik-Department Technische Universit~it Miinchen Theoretisches Teilinstitut D-8046 Garching, Reaktorgeliinde Federal Republic of Germany Z. Physik (1974) 

JakubaBa Physik-Department Technische Universit~it Miinchen Theoretisches Teilinstitut D-8046 Garching, Reaktorgeliinde Federal Republic of Germany Z. Physik

Calculation on the 0+ and 2+ giant resonances in Ni58

Universität München: Elektronischen Publikationen

Z. P h y s i k  268, 4 0 9 - 4 1 1  (1974) 
9 b y  S p r i n g e r - V e r l a g  1974 
Calculat ion of  the 0 + and 2 + Giant Resonances  in Ni  58 
Doris H. JakubaBa 
Physik-Department, Technische Universit~it Miinchen, Garching 
Received February 21, 1974 
Abstract. Estimates of the excitation energy, width and sum rule fraction of the collective 
monopole and quadrupole vibrations indicate that the peak in the 62 A -~/3 MeV region 
is a superposition of the 0 + and 2 + modes. 
Experiments on the inelastic scattering of electrons 
and protons on various target nuclei [-1] have shown 
a giant resonance with energy around 62 A -1/3 MeV. 
Angular distributions were consistent with either a 
0 + or a 2 + assignment. The form factors of the two 
multipolarities were found to differ from each other 
when the beam was polarized. Therefore new experi- 
ments have been performed on Ni 5s using a polarized 
proton beam [-2]. 
In order to identify this giant resonance we present 
calculations of the collective 0 + and 2 + vibrations in 
Ni 58 and compare them with experiment. 
The excitations of a nucleus can be described within 
the shell model taking into account one par t i c le -  one 
hole excitations. We restricted our configuration space 
to the levels shown in Table 2. The single particle 
energies and corresponding wave functions are taken 
as eigenvalues and eigenvectors of a Woods Saxon 
potential diagonalized in the space of the harmonic 
oscillator functions [3]. 
The potential parameters are chosen to fit the energies 
of the shells near the Fermi level to the experimental 
T a b l e  1. P a r a m e t e r s  of  the  W o o d s  S a x o n  po t en t i a l  
Vo R o a o 2 Rls al~ R c 
[ M e V ]  [ fm]  [ f m ]  [ fm]  [ fm]  [ fm]  
P r o t o n s  - 4 8 . 5 6  5.27 0.660 24.5 4.80 0.5 4.9 
N e u t r o n s  - 4 9 . 1 6  4.94 0.535 28.1 4.84 0.5 0.0 
excitation energies of the neighbouring nuclei. The 
parameters defined as in Ref. 3 are given in Table 1. 
Table 2 shows the single particle energies. 
The energies of the collective modes were obtained 
with the RPA method. To simplify the calculations 
we reduced the problem to that of a closed shell nucleus 
by interchanging the neutron 2 P3/2 and 2 Pl/2 level. 
A zero range residual interaction was used 
V(1, 2)= V o 6(r 1 - r2)  
9 (Fo + Fo zl ~2 + Go al a2 + Go a162 "gl "~2) 
T a b l e  2. Single  pa r t i c le  energies  for  N i  5s 
N i  58 P r o t o n  ene rgy  N e u t r o n  ene rgy  
[ M e V ]  [ M e V ]  
2 fv/2 9.435 5.169 
1 h1~/2 9.088 4.233 
1 g7/2 7.512 2.699 
2 d3/2 5.803 1.481 
3 sl/2 4.848 - 0 . 0 8 0  
2 ds/z 4.282 - 0 . 9 8 3  
1 gg/z 0.301 - 5 . 7 3 8  
1 fs/2 - - 2 . 497  - 8 . 6 5 9  
2 Pl/z - 2.917 - 8.189 
2 P3/2 - 3.417 - 10.601 
1 f7/2 - 8.177 - 15.498 
2 sl/2 - 12.192 - -20 .486  
1 d3/z - 12.553 - 2 1 . 1 7 1  
1 ds/2 - 16.058 - 2 4 . 5 9 3  
1 Pl/2 - 21.669 - 31.586 
1 P3/2 - 23.261 - 33.052 
410 
Table 3. Parameters of the residual interaction in Ni ss 
Vo F~ ~ F;x F2" F," a, ~, 
[MeV fm 3] 
380 0.65 -1 .39 -0 .85 -2 .29 1.3 -0 .3  
Table 4. Energy, width and reduced transition probability of the 0 + 
and 2 + isoscalar and isovector modes in Ni 5s 
L '~ T he) F B(EL)'f 
[MeV] [MeV] [e 2 fm 4] 
0 + 0 17.0 0.72 932 
0 18.0 0.32 49 
1 22.0 1.00 50 
1 28.3 4.70 220 
0 14.6 3.19 1067 
0 14.9 0.30 81 
1 24.5 4.28 121 
1 26.2 0.87 51 
2 + 
where the parameters Vo, Fo, F;, G o and G o were 
fitted to the experimental energies and transition 
probabilities of the lowest 2 + and 3 + modes and the 
1- giant resonance. As the Migdal force should be 
universal we took the parameters which are different 
inside and outside the nucleus close to the Pb 2~ 
values [4]. These are shown in Table 3, defined as in 
Ref. 3. 
The resulting excitation energies of the isoscalar 0 + 
and 2 + modes can be approximated by 65 A -1/3 MeV 
and 56A -1/3 MeV respectively, which is consistent 
with calculations on other nuclei 1-4]. 
The width of these collective modes results mainly 
from their coupling to single particle continuum 
states, It can be expressed by [5] 
F 2 r e _  
I(hco - 8i) X i l  2 
where the sum runs over all continuum states of the 
chosen configuration space; e~ is the single particle 
energy and AE the energy range of continuum states 
around the energy he) of the collective mode. X~ is the 
RPA amplitude. 
Table 4 shows energy, width and reduced transition 
probability of the collective vibrations. 
The collectivity of the excitations is expressed by the 
fraction of the corresponding sum rules which these 
modes exhaust. The isoscalar monopole and quadru- 
pole sum rules are given by [6] 
(2L+l)2h2AR2 L = 0 , 2 .  
~,h~oj.B(EL)j.T- (L+2)~zm ' 
f 
D. H. Jakubaga 
The sum runs over all final states that can be excited 
by the multipole operator; A is the mass number, 
R the rms radius of the charge distribution of the 
nucleus and 
B(EL)sT= IL~: i~=lr2y~ ~ 0+) 2 
is a modified reduced transition probability, the sum 
including neutron states. 
Summing over all isoscalar excitations with energy up 
to 24 MeV, one finds that most of the sum rule is 
exhausted: 75.0% for L = 0  and 90.5% for L = 2 .  
The scattering experiments [2] show a peak which 
extends from 13 to 18 MeV. The calculated energies 
and widths of both multipole resonances are thus 
compatible with the experimental data. 
Satchler [2] carried out a DWBA calculation of the 
differential cross section for the excitation of the 
monopole as well as the quadrupole mode. For  the 
normalization of the curves he assumed the maximum 
transition strength allowed by the sum rules, 
The differential cross section was measured in the 
energy region 14.6-16,7 MeV. Including the 0 + mode 
at 17.0 MeV we find in this region only 49.6 and 
36.8 % of the 0 + and 2 + sum rules respectively. As the 
2 + mode at 14.6 MeV is lying at the edge of the con- 
sidered region and has a large width we took only 
~--~ [mb/sr]  Ni58( P, P') 
5 ~  ~ .................................... E = 1/~.6-16.7 MeV 
' \ \  ~ 0+and 2 + 
0.5 \. ~ 2+ 
\. I X. \ .  
\ .  
\ 
~',~, i "  0 + 
0.1 
I I I I 1 =~L 
10' 15" 20" 25* 30 ~ 35 o 
Fig. 1. Cross section for the giant 0 + and 2 + resonances; the data as 
well as the relative values of the theoretical predictions are taken 
from Ref. 2. The dotted DWBA curve labelled with (2 +) is normalized 
as done in Ref. 2 and is shown for comparison. The full line is a 
superposition of the 0 + and 2 + curves with the proper normalization 
Z. Physik (1974) 
Calculation of the 0 + and 2 + Giant Resonances in Ni 5s 411 
half of its strength when computing the 2 + sum rule 
fraction. 
Fig. 1 shows the renormalized D W B A  curves com- 
pared with the experimental cross section. The 2 + 
curve is in good agreement with the experiment for 
larger angles while the behavior at small angles is 
better described by the 0 + mode. The full line denotes 
the superposition of the 0 + and 2 + cross sections and 
is in fair agreement with the experimental data in the 
total angular range considered. If we take the energy 
region from 12.7-23.7 MeV we find a similar result. 
Thus is strong evidence that the isoscalar giant reso- 
nance excited in proton scattering experiments is a 
superposition of a monopo le  and a quadrupole mode. 
I wish to thank Dr. P. Ring and R. Bauer for allowing me to use 
their numerical programs, and Prof. H. Schmidt and Dr. M. Kleber 
for stimulating discussions. 
References 
1. Nagao, M., Torizuka, Y.: Phys. Rev. Lett. 30, 1068 (1973) 
Lewis, M.B., Bertrand, F.E., Horen, D.J.: Phys. Rev. C8, 398 
(1973) 
2. Kocher, D.C., Bertrand, F.E., Gross, E.E., Lord, R.S., Newman, 
E.: Phys. Rev. Lett. 31, 1070 (1973) 
3. Ring, P., Bauer, R., Speth, J.: Nucl. Phys. A206, 97 (1973) 
4. JakubaBa, D.H.: Z. Physik 261, 305 (1973) 
5. FlieBbach, T.: Z. Physik 218, 385 (1969) 
6. Lane, A.M.: Nuclear Theory, p. 79. New York: W.A. Benjamin 
1964 
Dipl.-Phys. Doris H. JakubaBa 
Physik-Department 
Technische Universit~it Miinchen 
Theoretisches Teilinstitut 
D-8046 Garching, Reaktorgeliinde 
Federal Republic of Germany 
Z. Physik (1974) 


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Calculation on the 0+ and 2+ giant resonances in Ni58

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