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Level density for doubly odd deformed nuclei
Volume 55B, number 1
PHYSICS LETTERS
20 January 1975
LEVEL DENSITY FOR DOUBLY ODD DEFORMED NUCLEI L.A. MALOV, V.G. SOLOVIEV and V.V. VORONOV Joint Institute for Nuclear Research, Dubna, USSR Received 15 October 1974 The level density for doubly odd deformed nuclei near Bn is calculated within the framework of the semi-microscopic approach. The calculated values ate found to be in good agreement with the experimental ones. It is shown that for the I ~ 1 states it is important to take into account rotational motion. A model for describing the structure of highly excited nuclear states and a simple method of calculating the density of levels with fixed spins and parities at different excitation energies were suggested in ref. [1]. This method was used in rofs. [2-4] to investigate the A-dependence of 1/2 + level density and to calculate the density of levels with fixed spins. In ref. [5] the rotational motion effect was taken into account. A good description of the level density at the neutron binding energy was obtained. In:the present paper the semi-microscopic method of calculating the level density is generalized to the case of doubly odd nuclei, and level density calculations are performed for a number of deformed nuclei. The average spacing D between states with f'Lxed I n = K n are calculated in the following manner: in a given energy interval A ~ one finds n states of the type
e(s)+e(v)+o~gl+6og2+...-
0(s0,v0),
(1)
and then one finds D by the formula D = A /n. Here cog is the phonon energy, g = ~,/a/, )~# is the multipolarity and its projection, / the number of the root of the secular equation for phonons, e(s) and e(p) are the neutron and proton quasiparticle energies, respectively, C0(s0, u0) is the ground state energy for a doubly odd deformed nucleus. The method of the calculation of these quantities is suggested in ref. [6]. The wave functions and single-particle energies were calculated with the Saxon-Woods potential. The pairing interaction constants, the multipote-multipole interaction constants and the equilibrium deformation parameters are taken the same as in ref. [3]. All these quantities were fixed in the study of the low-lying states of deformed nuclei, therefore in the calculation of the level density there is not a single free parameter.
The average spacing between the levels with two spin values was calculated by the formula D ( ~ , I 0 + 1/2) = ( p ( d , I 0 - 1/2) + p(~,I 0 + 1/2)} -1 . (2) On each internal state with given K n one constructed a rotational band with an excitation energy Erot = 1 [i(1+ 1) -
K 2] .
(3)
The level density at an excitation energy ~ for states with given I n taking into account rotation was calculated by the formula I P(~,In)=K~=Op ( ~ - I
[I(I+ I ) - K 2 ] , K n )
. (4)
The value of the moment of inertia was taken to be equal to the rigid-rotation moment of inertia Jrig = ~mAR 2, since the calculations show that for small spins the level density depends rather weakly on the moment of inertia. For example, for the nucleus 166Ho the ratio of the density with J = ½Jri- to the density with J = Jri- at ~. = B n for states w i ~ I n --- 2 - , 3-, 4 is 0.86, 0.9~), 0.88, respectively. The results of our calculations and the experimental data from refs. [7-9] are given in the table. They are seen to bein satisfactory agreement. The account of the rotational motion results in an increase of the level density by a factor of 2 or 3 and makes the agreement noticeably better. The results of the statistical calculations of refs. [10, 11] are given in the same table. A systematic difference between the results of ref. [11 ] an&ours can possibly be explained partly by the fact that in ref. [11] the collective vibrational states have not been taken into account. 17
Volume 55B, number 1
PHYSICS LETTERS
20 January 1975
Table 1 The average spacing D between levels with given I n near the neutron binding energy B n. Dexp (eV) Compound nucleus
11r
Bn
Dtheor (eV)
(MeV) Ref. [71
ref. [8]
ref. [91
present paper eq. (4)
ref. [10] ref. [11]
4.3 5.6 8.2 7.2 8.5 8.9 14.9 2.7
1.9 3.4 3.1 7.0 3.5 5.0 0.5
I=K
lS4Eu 16°Tb 166Ho 17°Tm
176Lu 182Ta XS6Re 238Np
2+, 3+ 1+, 2+ 3-,40+, 1+ 3+, 4 + 3+, 4 + 2+, 3+ 2+, 3÷
6.44 6.38 6.24 6.59 6.29 6.06 6.18 5.48
1.4 + 0.4 3.9 + 0.6 6.1 ± 1.2 6 ± 1.5 3.7 ± 0:7 4.4 ± 0.4 3.8 ± 0.8 0.69
1.3 4.3 5.67 6.6 3.61 4.33 3.2 0.72
The semi-microscopic calculations of the level density with the account of vibrational and rotational motion performed in the present paper and in refs. [2, 3, 5] give a satisfactory agreement with the appropriate experimental data at the n e u t r o n binding energy. At intermediate excitation energies, the energy and spin dependence of the level density calculated by us noticeably differs in some cases from the statistical model calculations. At high excitation energies, the semimicroscopic calculations become much more difficult while the accuracy falls because of an insufficiently correct account of the Pauli principle and an ambiguous expression of many-quasiparticle configurations in terms of collective phonons. Therefore it is recommendsed to perform semi-microscopic calculations up to e = B n and, at stiU higher energies, the statistical approach should be used.
18
± 0.4 ± 0.78 ± 0.74 ± 1.3 ± 0.62 ± 0.51 ± 0.8 ± 0.079
1.1 4.2 5.5 7.3 3.0 4.3 3.3 0.67
1.3 3.0 2.4 5.5 2.4 3.5 5.4 0.9
8.9 8.8 8.5 22 10 18 18 1.43
References [1] V.G. Soloviev and L.A. Malov, Nucl. Phys. A196 (1972) 433. [2] A.I. Vdovin et al., Yad. Fiz., 19 (1974) 516. [3] L.A. Malov, V.G. Soloviev and V.V. Voronov, Nucl. Phys. A224 (1974) 396. [4] V.G. Soloviev, Ch. Stoyanov and A.I. Vdovin, Nucl. Phys. A224 (1974) 411. [5] L.A. Malov, V.G. Soloviev and V.V. Voronov, preprint JINR FA-7818, Dubna (1974). [6] V.G. Soloviev, The theory of complex nuclei (Nauka, Moscow, 1971). [7] I.E. Lynn, The theory of neutron resonance reactions (Clarendon Press, Oxford, 1968). [8] N. Baba, Nucl. Phys. A159 (1970) 625. [9] W. Dily, W. Schantl, N. Vonach, Nucl. Phys. A217 (1973) 269. [10] I.R. Huizenga, A.N. Behkami, R.W. Atcher et al., Nucl. Phys. A223 (1974) 589. [11] T.D. Ossing, A.S. Jensen, Nucl. Phys. A222 (1974) 493.
PHYSICS LETTERS
20 January 1975
LEVEL DENSITY FOR DOUBLY ODD DEFORMED NUCLEI L.A. MALOV, V.G. SOLOVIEV and V.V. VORONOV Joint Institute for Nuclear Research, Dubna, USSR Received 15 October 1974 The level density for doubly odd deformed nuclei near Bn is calculated within the framework of the semi-microscopic approach. The calculated values ate found to be in good agreement with the experimental ones. It is shown that for the I ~ 1 states it is important to take into account rotational motion. A model for describing the structure of highly excited nuclear states and a simple method of calculating the density of levels with fixed spins and parities at different excitation energies were suggested in ref. [1]. This method was used in rofs. [2-4] to investigate the A-dependence of 1/2 + level density and to calculate the density of levels with fixed spins. In ref. [5] the rotational motion effect was taken into account. A good description of the level density at the neutron binding energy was obtained. In:the present paper the semi-microscopic method of calculating the level density is generalized to the case of doubly odd nuclei, and level density calculations are performed for a number of deformed nuclei. The average spacing D between states with f'Lxed I n = K n are calculated in the following manner: in a given energy interval A ~ one finds n states of the type
e(s)+e(v)+o~gl+6og2+...-
0(s0,v0),
(1)
and then one finds D by the formula D = A /n. Here cog is the phonon energy, g = ~,/a/, )~# is the multipolarity and its projection, / the number of the root of the secular equation for phonons, e(s) and e(p) are the neutron and proton quasiparticle energies, respectively, C0(s0, u0) is the ground state energy for a doubly odd deformed nucleus. The method of the calculation of these quantities is suggested in ref. [6]. The wave functions and single-particle energies were calculated with the Saxon-Woods potential. The pairing interaction constants, the multipote-multipole interaction constants and the equilibrium deformation parameters are taken the same as in ref. [3]. All these quantities were fixed in the study of the low-lying states of deformed nuclei, therefore in the calculation of the level density there is not a single free parameter.
The average spacing between the levels with two spin values was calculated by the formula D ( ~ , I 0 + 1/2) = ( p ( d , I 0 - 1/2) + p(~,I 0 + 1/2)} -1 . (2) On each internal state with given K n one constructed a rotational band with an excitation energy Erot = 1 [i(1+ 1) -
K 2] .
(3)
The level density at an excitation energy ~ for states with given I n taking into account rotation was calculated by the formula I P(~,In)=K~=Op ( ~ - I
[I(I+ I ) - K 2 ] , K n )
. (4)
The value of the moment of inertia was taken to be equal to the rigid-rotation moment of inertia Jrig = ~mAR 2, since the calculations show that for small spins the level density depends rather weakly on the moment of inertia. For example, for the nucleus 166Ho the ratio of the density with J = ½Jri- to the density with J = Jri- at ~. = B n for states w i ~ I n --- 2 - , 3-, 4 is 0.86, 0.9~), 0.88, respectively. The results of our calculations and the experimental data from refs. [7-9] are given in the table. They are seen to bein satisfactory agreement. The account of the rotational motion results in an increase of the level density by a factor of 2 or 3 and makes the agreement noticeably better. The results of the statistical calculations of refs. [10, 11] are given in the same table. A systematic difference between the results of ref. [11 ] an&ours can possibly be explained partly by the fact that in ref. [11] the collective vibrational states have not been taken into account. 17
Volume 55B, number 1
PHYSICS LETTERS
20 January 1975
Table 1 The average spacing D between levels with given I n near the neutron binding energy B n. Dexp (eV) Compound nucleus
11r
Bn
Dtheor (eV)
(MeV) Ref. [71
ref. [8]
ref. [91
present paper eq. (4)
ref. [10] ref. [11]
4.3 5.6 8.2 7.2 8.5 8.9 14.9 2.7
1.9 3.4 3.1 7.0 3.5 5.0 0.5
I=K
lS4Eu 16°Tb 166Ho 17°Tm
176Lu 182Ta XS6Re 238Np
2+, 3+ 1+, 2+ 3-,40+, 1+ 3+, 4 + 3+, 4 + 2+, 3+ 2+, 3÷
6.44 6.38 6.24 6.59 6.29 6.06 6.18 5.48
1.4 + 0.4 3.9 + 0.6 6.1 ± 1.2 6 ± 1.5 3.7 ± 0:7 4.4 ± 0.4 3.8 ± 0.8 0.69
1.3 4.3 5.67 6.6 3.61 4.33 3.2 0.72
The semi-microscopic calculations of the level density with the account of vibrational and rotational motion performed in the present paper and in refs. [2, 3, 5] give a satisfactory agreement with the appropriate experimental data at the n e u t r o n binding energy. At intermediate excitation energies, the energy and spin dependence of the level density calculated by us noticeably differs in some cases from the statistical model calculations. At high excitation energies, the semimicroscopic calculations become much more difficult while the accuracy falls because of an insufficiently correct account of the Pauli principle and an ambiguous expression of many-quasiparticle configurations in terms of collective phonons. Therefore it is recommendsed to perform semi-microscopic calculations up to e = B n and, at stiU higher energies, the statistical approach should be used.
18
± 0.4 ± 0.78 ± 0.74 ± 1.3 ± 0.62 ± 0.51 ± 0.8 ± 0.079
1.1 4.2 5.5 7.3 3.0 4.3 3.3 0.67
1.3 3.0 2.4 5.5 2.4 3.5 5.4 0.9
8.9 8.8 8.5 22 10 18 18 1.43
References [1] V.G. Soloviev and L.A. Malov, Nucl. Phys. A196 (1972) 433. [2] A.I. Vdovin et al., Yad. Fiz., 19 (1974) 516. [3] L.A. Malov, V.G. Soloviev and V.V. Voronov, Nucl. Phys. A224 (1974) 396. [4] V.G. Soloviev, Ch. Stoyanov and A.I. Vdovin, Nucl. Phys. A224 (1974) 411. [5] L.A. Malov, V.G. Soloviev and V.V. Voronov, preprint JINR FA-7818, Dubna (1974). [6] V.G. Soloviev, The theory of complex nuclei (Nauka, Moscow, 1971). [7] I.E. Lynn, The theory of neutron resonance reactions (Clarendon Press, Oxford, 1968). [8] N. Baba, Nucl. Phys. A159 (1970) 625. [9] W. Dily, W. Schantl, N. Vonach, Nucl. Phys. A217 (1973) 269. [10] I.R. Huizenga, A.N. Behkami, R.W. Atcher et al., Nucl. Phys. A223 (1974) 589. [11] T.D. Ossing, A.S. Jensen, Nucl. Phys. A222 (1974) 493.