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Gamma-ray spectra from the radiative capture of 14 MeV neutrons in 28Si and 40Ca
Nuclear Physics A159 (1970) 555--560; (~) North-Holland Publishing Co., Amsterdam N o t to
be reproduced by photoprint or microfilm without written permission from the publisher
GAMMA-RAY SPECTRA FROM THE RADIATIVE CAPTURE OF 14 MeV NEUTRONS IN ZSSi AND 4eCa F. CVELBAR and A. H U D O K L I N
Institute "d. Stefan" and University of Ljubljana, Yugoslavia Received 5 October 1970 Abslract: Gamma-ray spectra from the radiative capture of 14 MeV neutrons in 2°Si and 4°Ca were calculated according to the direct-semi-direct and statistical model and compared with the experimental values. In the case of 2aSi the agreement between the experimental spectrum and the one calculated according to the direct-semi-direct model agree within about 10 ~0 in the region of the excitation energies of the final nucleus from 0 to about 4 MeV. In the case of *°Ca the experimental value is about two times lower than the calculated one. This discrepancy is discussed. The intensity calculated according to the statistical model is dominant in the region of higher excitation energies.
1. Introduction
In the following the ~-ray spectra from the radiative capture of 14 MeV neutrons in 28Si and 4°Ca, calculated according to the theory of statistical and the direct-semidirect capture, are presented and compared with the experimental values measured by a special telescopic scintillation pair spectrometer 1). As discussed in ref. 1) there are three different approaches 2-4) for the calculation of the cross section according to the direct-semi-direct model. The approach of Clement et al. *) and that of Lusnikov and Zaretski 3) yield practically the same result. However the final formula of later authors is simpler. The cross section oJns for the capture of neutron from the initial state li) to the final state (fl is the product of the cross section for the direct capture ~n and the enhancement factor FL containing the effect of the semi-direct capture process: with 2
EL----
2
2
2
(Eo-E~)+FoEo 2 22 2 2~ (ER-E~) +F E R
where Eo is the averav,e energy of the unperturbed dipole particle-hole excitation, Fo is the width of the corresponding unperturbed dipole giant resonance, E~ is the y-ray energy, ER and F are the experimental energy and the width of the dipole giant resonance for the initial nucleus. The direct capture cross section ¢r~ is discussed in detail in ref. 1). Let us recall here only its proportionality to the square of the dipole matrix element (flrli) where li) 555
556
F. CVELBAR AND A. HUDOKLIN
is the optical'-model wave function of initial state and (fl is the single-particle wave function of the final bound state. The method of the statistical model calculation is also outlined in ref. 1). 2. Calculation The initial state wave function li) is the solution of the optical-model Hamiltonian. The potential used is of the Saxon-Woods type, with the addition of the L S coupling term. Parameters used with this potential are the same as those used in ref. ~). The final state wave functions (fl were also calculated by the use of the SaxonWoods potential. Its diffuseness parameter a was the same as used for the calculation o f the initial state wave function. Its depth was determined for each single-particle level so that experimentally observed single-particle excitations were reproduced. If the single-particle level was shared among real states, the energy of the center of gravity of these states was taken as the single-particle excitation. For this excitation the direct capture cross section was calculated and the value obtained was then divided among the real states according to the spectroscopic factor values. If the sum of the TABLE 1 Giant resonance parameters Sample
Et (MeV)
Eo (MeV)
/" (MeV)
-F'o (MeV)
2eSi 4°Ca
20.5 ") 19.8 b)
13.1 12.1
5.2 ") 4.5 b)
3.3 2.8
") The average of the results from refs. 7.s). b) The average of the results from refs. 7, 9). Symbols have the same meaning as in ref.
1).
TABLE 2 Single-particle level structure of the nucleus 29Si and the direct cross section for the radiative capture of neutrons into particular states Singleparticle level 2s~ Id~ Idi If~ 2p t lf~ 2p~
_.¢F (I) (MeV)
0.00 1.28 2.03 3.07 3.62 4.93 6.20 6.38
S (°
Ecx, (MeV)
~ S(O '
~r.z JD ~b)
0.58 0.62 0.14 0.06 1.00 1.00 0.44 1.00
0.00 1.28 2.34
0.58 0.62 0.20
6.52 17.33 4.98
3.62 4.93 6.20 6.38
1.00 1.00 0.44 1.00
87.34 31.54 42.83 12.43
Spectroscopic factors are the average values of the data from refs. to-12). Spin assignments are taken from refs. a3.14). Symbols have the same meaning as in ref. 1).
GAMMA-RAYSPECTRA
557
TABLE3 Single-particle level structure of the nucleus 41Ca and the direct cross section for the radiative capture of neutrons into particular states Singleparticle level
E~.' (MeV)
S (°
E**° (MeV)
~ S") t
a~# (/~b)
lf.]. 2p~.
0.00 1.95 2.47 3.62 3.95 4.20 4.62 4.77 4.98 4.89 5.66 5.81
1.00 0.77 0.23 0.09 0.62 0.01 0.09 0.18 0.07 0.12 0.25 0.11
0.00 2.07
1.00 1.00
42.81. 30.57
4.13
1.00
13.12
4.98 5.50
0.07 0.48
103.26 42.48
2p~
lg~r lft
Spectroscopic factors were taken from ref. 15). The meaning of the symbols is the same as in ref. 1). experimental spectroscopic factors was higher than 1, their values were renormalized to yield 2~S~ = 1. Available spectroscopic data for the bound states in 2sSi and 4°Ca are presented in tables 2 and 3. Giant resonance parameters can be seen from table 1. The parameters used in the statistical theory calculation are shown in table 4.
3. Results and discussion
3.1. DIRECT-SEMI-DIRECT APPROACH Cross-section values for the direct capture of neutrons into particular singleparticle states are presented in tables 2 and 3. These data, divided among real nuclear states, multiplied by the Lushnikov and Zaretsky enhancement factor and smeared out for the experimental resolution yield the final 7-ray spectra presented in figs. 1 and 2, where the experimental values 5) are also shown. For silicon the agreement between the calculated spectrum and the experimental one is within about 10 ~o in the region of the excitation energies of the final nucleus from 0 to about 4 MeV. At higher excitation energies the calculated intensity is much lower than the experimental one. This is mainly due to the destructive interference between the direct and semidirect matrix element resulting in the "enhancement" factor being smaller than 1. However, the spectroscopic data on the energy levels at these energies are also very scarce. In the case of calcium the agreement between the calculated and the experimental spectrum is not so good. The calculated intensity is about a factor of 2 too high. In our previous analysis such an effect was not observed. Looking for the explanation of this discrepancy one should start from the following
558
F. CVELBAR A N D A. H U D O K L I N
28Si(n.,y) 29Si ,,,,,.,.
l&O
=
12o
EXPERIHENT LUSHHIKOV
-.-
100 =
STATISTICAL
++++++
80 60
40 •
.
o
,
.
.
o
,
20
9 13
8 14
I
7 15
I
6 16
I
I
I
5
]
17
,
I
18
3
I
19
2
! 21
0 EXCITATION[HeV] 22 y-RAY ENERGY[HeV]
I
20
I
Fig. 1. Spectrum ofF-rays from the radiative capture of 14 MeV neutrons in silicon.
l'Oca(n,~ ) l'ICa ~ ---
100 "~'~-~'80
~,.t J ~# 4
EXPERIHENT LUSXNIKOVx 0.5
--.-- STATISTICAL
~-, &O ¢_~
2O 0 10 I
12
I 9 I
13
IB~ 6 I
l&
i 7 I
15
"'l 6 I
15
5
4
3
17
10
19
I
I
I
"~, 2 I
20
l 1 I
21
l 0 EXCITATION I I
[HeV]
22 "y-RAY ENERGY[HEY]
Fig. 2. Spectrum ofT-rays from the radiative capture of 14 MeV neutrons in calcium. Spectrum obtained (dashed line) taking the value of I instead of the experimentally observed value of 0.07 for the spectroscopic factor of the lg~ level. facts: (i) the experimental cross section for the capture of 14 MeV neutrons in Ca is not different from that for the neighbouring nuclei; (ii) the width F of the giant resonance of 4°Ca, as used in the calculation, is only about 4 MeV in comparison with the value of 6--8 MeV for neighbouring nuclei, and (iii) the agreement between the experimental and the calculated spectrum is obtained if the F-value for the calculation is taken to be about 6 MoV. Taking into account these facts one puts the question on the supposition, implicitly included in the direct-semi-direct model, that the width of the core giant resonance does not change with the presence of the external
GAMMA-RAY SPECTRA
559
nucleon. It seems that this supposition in many cases, when F ~, 6-8 MeV, is not crucial but fails at the closed-shell nuclei where the width of the giant resonance is about 4 MeV. However, to be sure that such an effect really exists further experimental and theoretical work is necessary. 3.2. STATISTICAL MODEL
Results of the calculation according to the statistical model are also sht)wn in figs. 1 and 2. It seems that in calcium the spectral intensity at excitation energies higher than about 5 MeV could be described by this model. On the other hand in silicon TABLE 4
Parameters used for the calculation of the capture spectra according to the statistical theory Sample
2sSi 4°Ca
Eit
/~ (MeV)
(MeV)
S (MeV.b)
ax (MeV - t )
AA (MeV)
aA+t (MeV -1)
5.2 4.5
20.5 19.8
0.62 0.85
4.5 6.3
--2.1 --1.5
4.6 6.4
d.4+t (MoV)
o¢(E) Co)
--4.3 --3.2
1.00 1.18
Values for A were taken from ref. x6); values for a are the average of the results from refs. xT-t9). The meaning of the symbols is the same as in ref. 1).
the statistical spectrum exhausts only about one half of the experimental value at final nucleus excitations where the direct-semi-direct approach fails.
Note added in proof: Recently Longo and Saporetti (Nucl. Phys. A154 (1970)243) calculated the spectrum of ~,-rays from the radiative capture of 14 MeV neutrons in 28Si using the model of Clement et al. 4). The result obtained was not different from that presented in this paper. References 1) F. Cvelbar, A. Hudoklin, M. V. Mihailovi6, M. Naj~er and V. Ram~ak, Nucl. Phys. A130 (1969) 401; F. Cvelbar, A. Hudoklin, M. V. Mihailovi6, M. Naj~er and M. Petrigi6, Nucl. Phys. AI30 (1969) 413 2) G. E. Brown, Nucl. Phys. 57 (1964) 339 3) A. A. Lushnikov and D. F. Zaretsky, Nucl. Phys. 66 (1965) 35 4) C. F. Clement, A. M. Lane and J. R. Rook, Nucl. Phys. 66 (1965) 273; 293 5) F. Cvelbar, A. Hudoklin and M. Potokar, Nucl. Phys. A138 (1969) 412 6) L. Rosen, 3. G. Beery, A. S. Goidhaber and E. H. Auerbach, Ann. of Phys. 34 (1965) 96 7) J. M. Wyckoff, B. Ziegler, H. W. Koch and R. Uhling, Phys. Rev. 137 (1965) B576 8) G. Kernel, Dissertation, NIJS Report P-175 (1965) 9) D. Jamnik et al. private communication 10) M. H. Macfarlane and J. B. French, Rev. Mod. Phys. 32 (1960) 567 11) N. V. Alekseev, K. I. Zherebcova, V. F. Litvin and Ju. A. Nemilov, ZhETF (USSR)39 (1960) 1508
12) A. G. Blair and K. S. Quisenberry, Phys. Rev. 122 (1961) 869 13) O. F. Nemec, M. B. Pasechnik and N. N. Pucherov, Atom. Energ. 14 (1963) 159
560 14) 15) 16) 17) 18) 19)
F. CVELBARAND A. HUDOKLIN M. Betigcri, R. Bock, H. H. Duhm, S. Martin and R. Stock, Z. Naturf. 21a (1966) 980 T. A. Belote, A. Sperduto and W. W. Buechner, Phys. Rev. 139 (1965) B80 A. G. Cameron, Can. J. Phys. 36 (1958) 1040 E. Erba, U. Facchini and E. Saetta-Menichella, Nuovo Cim. 22 (1961) 1237 E. Saetta-Menichella, F. Tonolini and L. Tonolini-Severgnini, Nucl. Phys. 51 (1964) 449 U. Facchini, E. Saetta-Menichella, F. Tonolini and L. Tonolini-Severgnini, Nucl. Phys. 51 (1964) 460
be reproduced by photoprint or microfilm without written permission from the publisher
GAMMA-RAY SPECTRA FROM THE RADIATIVE CAPTURE OF 14 MeV NEUTRONS IN ZSSi AND 4eCa F. CVELBAR and A. H U D O K L I N
Institute "d. Stefan" and University of Ljubljana, Yugoslavia Received 5 October 1970 Abslract: Gamma-ray spectra from the radiative capture of 14 MeV neutrons in 2°Si and 4°Ca were calculated according to the direct-semi-direct and statistical model and compared with the experimental values. In the case of 2aSi the agreement between the experimental spectrum and the one calculated according to the direct-semi-direct model agree within about 10 ~0 in the region of the excitation energies of the final nucleus from 0 to about 4 MeV. In the case of *°Ca the experimental value is about two times lower than the calculated one. This discrepancy is discussed. The intensity calculated according to the statistical model is dominant in the region of higher excitation energies.
1. Introduction
In the following the ~-ray spectra from the radiative capture of 14 MeV neutrons in 28Si and 4°Ca, calculated according to the theory of statistical and the direct-semidirect capture, are presented and compared with the experimental values measured by a special telescopic scintillation pair spectrometer 1). As discussed in ref. 1) there are three different approaches 2-4) for the calculation of the cross section according to the direct-semi-direct model. The approach of Clement et al. *) and that of Lusnikov and Zaretski 3) yield practically the same result. However the final formula of later authors is simpler. The cross section oJns for the capture of neutron from the initial state li) to the final state (fl is the product of the cross section for the direct capture ~n and the enhancement factor FL containing the effect of the semi-direct capture process: with 2
EL----
2
2
2
(Eo-E~)+FoEo 2 22 2 2~ (ER-E~) +F E R
where Eo is the averav,e energy of the unperturbed dipole particle-hole excitation, Fo is the width of the corresponding unperturbed dipole giant resonance, E~ is the y-ray energy, ER and F are the experimental energy and the width of the dipole giant resonance for the initial nucleus. The direct capture cross section ¢r~ is discussed in detail in ref. 1). Let us recall here only its proportionality to the square of the dipole matrix element (flrli) where li) 555
556
F. CVELBAR AND A. HUDOKLIN
is the optical'-model wave function of initial state and (fl is the single-particle wave function of the final bound state. The method of the statistical model calculation is also outlined in ref. 1). 2. Calculation The initial state wave function li) is the solution of the optical-model Hamiltonian. The potential used is of the Saxon-Woods type, with the addition of the L S coupling term. Parameters used with this potential are the same as those used in ref. ~). The final state wave functions (fl were also calculated by the use of the SaxonWoods potential. Its diffuseness parameter a was the same as used for the calculation o f the initial state wave function. Its depth was determined for each single-particle level so that experimentally observed single-particle excitations were reproduced. If the single-particle level was shared among real states, the energy of the center of gravity of these states was taken as the single-particle excitation. For this excitation the direct capture cross section was calculated and the value obtained was then divided among the real states according to the spectroscopic factor values. If the sum of the TABLE 1 Giant resonance parameters Sample
Et (MeV)
Eo (MeV)
/" (MeV)
-F'o (MeV)
2eSi 4°Ca
20.5 ") 19.8 b)
13.1 12.1
5.2 ") 4.5 b)
3.3 2.8
") The average of the results from refs. 7.s). b) The average of the results from refs. 7, 9). Symbols have the same meaning as in ref.
1).
TABLE 2 Single-particle level structure of the nucleus 29Si and the direct cross section for the radiative capture of neutrons into particular states Singleparticle level 2s~ Id~ Idi If~ 2p t lf~ 2p~
_.¢F (I) (MeV)
0.00 1.28 2.03 3.07 3.62 4.93 6.20 6.38
S (°
Ecx, (MeV)
~ S(O '
~r.z JD ~b)
0.58 0.62 0.14 0.06 1.00 1.00 0.44 1.00
0.00 1.28 2.34
0.58 0.62 0.20
6.52 17.33 4.98
3.62 4.93 6.20 6.38
1.00 1.00 0.44 1.00
87.34 31.54 42.83 12.43
Spectroscopic factors are the average values of the data from refs. to-12). Spin assignments are taken from refs. a3.14). Symbols have the same meaning as in ref. 1).
GAMMA-RAYSPECTRA
557
TABLE3 Single-particle level structure of the nucleus 41Ca and the direct cross section for the radiative capture of neutrons into particular states Singleparticle level
E~.' (MeV)
S (°
E**° (MeV)
~ S") t
a~# (/~b)
lf.]. 2p~.
0.00 1.95 2.47 3.62 3.95 4.20 4.62 4.77 4.98 4.89 5.66 5.81
1.00 0.77 0.23 0.09 0.62 0.01 0.09 0.18 0.07 0.12 0.25 0.11
0.00 2.07
1.00 1.00
42.81. 30.57
4.13
1.00
13.12
4.98 5.50
0.07 0.48
103.26 42.48
2p~
lg~r lft
Spectroscopic factors were taken from ref. 15). The meaning of the symbols is the same as in ref. 1). experimental spectroscopic factors was higher than 1, their values were renormalized to yield 2~S~ = 1. Available spectroscopic data for the bound states in 2sSi and 4°Ca are presented in tables 2 and 3. Giant resonance parameters can be seen from table 1. The parameters used in the statistical theory calculation are shown in table 4.
3. Results and discussion
3.1. DIRECT-SEMI-DIRECT APPROACH Cross-section values for the direct capture of neutrons into particular singleparticle states are presented in tables 2 and 3. These data, divided among real nuclear states, multiplied by the Lushnikov and Zaretsky enhancement factor and smeared out for the experimental resolution yield the final 7-ray spectra presented in figs. 1 and 2, where the experimental values 5) are also shown. For silicon the agreement between the calculated spectrum and the experimental one is within about 10 ~o in the region of the excitation energies of the final nucleus from 0 to about 4 MeV. At higher excitation energies the calculated intensity is much lower than the experimental one. This is mainly due to the destructive interference between the direct and semidirect matrix element resulting in the "enhancement" factor being smaller than 1. However, the spectroscopic data on the energy levels at these energies are also very scarce. In the case of calcium the agreement between the calculated and the experimental spectrum is not so good. The calculated intensity is about a factor of 2 too high. In our previous analysis such an effect was not observed. Looking for the explanation of this discrepancy one should start from the following
558
F. CVELBAR A N D A. H U D O K L I N
28Si(n.,y) 29Si ,,,,,.,.
l&O
=
12o
EXPERIHENT LUSHHIKOV
-.-
100 =
STATISTICAL
++++++
80 60
40 •
.
o
,
.
.
o
,
20
9 13
8 14
I
7 15
I
6 16
I
I
I
5
]
17
,
I
18
3
I
19
2
! 21
0 EXCITATION[HeV] 22 y-RAY ENERGY[HeV]
I
20
I
Fig. 1. Spectrum ofF-rays from the radiative capture of 14 MeV neutrons in silicon.
l'Oca(n,~ ) l'ICa ~ ---
100 "~'~-~'80
~,.t J ~# 4
EXPERIHENT LUSXNIKOVx 0.5
--.-- STATISTICAL
~-, &O ¢_~
2O 0 10 I
12
I 9 I
13
IB~ 6 I
l&
i 7 I
15
"'l 6 I
15
5
4
3
17
10
19
I
I
I
"~, 2 I
20
l 1 I
21
l 0 EXCITATION I I
[HeV]
22 "y-RAY ENERGY[HEY]
Fig. 2. Spectrum ofT-rays from the radiative capture of 14 MeV neutrons in calcium. Spectrum obtained (dashed line) taking the value of I instead of the experimentally observed value of 0.07 for the spectroscopic factor of the lg~ level. facts: (i) the experimental cross section for the capture of 14 MeV neutrons in Ca is not different from that for the neighbouring nuclei; (ii) the width F of the giant resonance of 4°Ca, as used in the calculation, is only about 4 MeV in comparison with the value of 6--8 MeV for neighbouring nuclei, and (iii) the agreement between the experimental and the calculated spectrum is obtained if the F-value for the calculation is taken to be about 6 MoV. Taking into account these facts one puts the question on the supposition, implicitly included in the direct-semi-direct model, that the width of the core giant resonance does not change with the presence of the external
GAMMA-RAY SPECTRA
559
nucleon. It seems that this supposition in many cases, when F ~, 6-8 MeV, is not crucial but fails at the closed-shell nuclei where the width of the giant resonance is about 4 MeV. However, to be sure that such an effect really exists further experimental and theoretical work is necessary. 3.2. STATISTICAL MODEL
Results of the calculation according to the statistical model are also sht)wn in figs. 1 and 2. It seems that in calcium the spectral intensity at excitation energies higher than about 5 MeV could be described by this model. On the other hand in silicon TABLE 4
Parameters used for the calculation of the capture spectra according to the statistical theory Sample
2sSi 4°Ca
Eit
/~ (MeV)
(MeV)
S (MeV.b)
ax (MeV - t )
AA (MeV)
aA+t (MeV -1)
5.2 4.5
20.5 19.8
0.62 0.85
4.5 6.3
--2.1 --1.5
4.6 6.4
d.4+t (MoV)
o¢(E) Co)
--4.3 --3.2
1.00 1.18
Values for A were taken from ref. x6); values for a are the average of the results from refs. xT-t9). The meaning of the symbols is the same as in ref. 1).
the statistical spectrum exhausts only about one half of the experimental value at final nucleus excitations where the direct-semi-direct approach fails.
Note added in proof: Recently Longo and Saporetti (Nucl. Phys. A154 (1970)243) calculated the spectrum of ~,-rays from the radiative capture of 14 MeV neutrons in 28Si using the model of Clement et al. 4). The result obtained was not different from that presented in this paper. References 1) F. Cvelbar, A. Hudoklin, M. V. Mihailovi6, M. Naj~er and V. Ram~ak, Nucl. Phys. A130 (1969) 401; F. Cvelbar, A. Hudoklin, M. V. Mihailovi6, M. Naj~er and M. Petrigi6, Nucl. Phys. AI30 (1969) 413 2) G. E. Brown, Nucl. Phys. 57 (1964) 339 3) A. A. Lushnikov and D. F. Zaretsky, Nucl. Phys. 66 (1965) 35 4) C. F. Clement, A. M. Lane and J. R. Rook, Nucl. Phys. 66 (1965) 273; 293 5) F. Cvelbar, A. Hudoklin and M. Potokar, Nucl. Phys. A138 (1969) 412 6) L. Rosen, 3. G. Beery, A. S. Goidhaber and E. H. Auerbach, Ann. of Phys. 34 (1965) 96 7) J. M. Wyckoff, B. Ziegler, H. W. Koch and R. Uhling, Phys. Rev. 137 (1965) B576 8) G. Kernel, Dissertation, NIJS Report P-175 (1965) 9) D. Jamnik et al. private communication 10) M. H. Macfarlane and J. B. French, Rev. Mod. Phys. 32 (1960) 567 11) N. V. Alekseev, K. I. Zherebcova, V. F. Litvin and Ju. A. Nemilov, ZhETF (USSR)39 (1960) 1508
12) A. G. Blair and K. S. Quisenberry, Phys. Rev. 122 (1961) 869 13) O. F. Nemec, M. B. Pasechnik and N. N. Pucherov, Atom. Energ. 14 (1963) 159
560 14) 15) 16) 17) 18) 19)
F. CVELBARAND A. HUDOKLIN M. Betigcri, R. Bock, H. H. Duhm, S. Martin and R. Stock, Z. Naturf. 21a (1966) 980 T. A. Belote, A. Sperduto and W. W. Buechner, Phys. Rev. 139 (1965) B80 A. G. Cameron, Can. J. Phys. 36 (1958) 1040 E. Erba, U. Facchini and E. Saetta-Menichella, Nuovo Cim. 22 (1961) 1237 E. Saetta-Menichella, F. Tonolini and L. Tonolini-Severgnini, Nucl. Phys. 51 (1964) 449 U. Facchini, E. Saetta-Menichella, F. Tonolini and L. Tonolini-Severgnini, Nucl. Phys. 51 (1964) 460