142
S. M. Grimes
and the measured level spacing (eV) are provided. The values indicated by a 1 show
the level density parameters inferred by the authors by calculating the level density
correction using Eqs. 7 and 8. This level density is then divided by the rotational
enhancement factor proposed in Refs. [2] and [3]. A new a (labeled a 2 ) is deduced
from the reduced level density. a 1 values are similar to the a values proposed by the
authors for neighboring spherical nuclei.
The same input data were then re-analyzed using Eq. 9 or 10 as appropriate.
A very substantial increase in level density was observed. This yielded the level
density parameters a 3 . Finally, the rotational enhancement factor was inferred by
calculating the level density by utilizing the Bethe spin distribution. This then results
in parameters a 4 . a 4 values are intrinsic level density parameters (those which would
be observed if a nucleus was not deformed) and are consistent with neighboring
spherical nuclei. To use Eqs. 9 and 10 values for σ ⊥ are needed. Values for σ
were obtained from the results of calculations in Ref. [7]. It may be shown that the
relationship between σ ⊥ and σ is
σ 2
⊥
σ 2
=
1 +
1
2 β +
16
7 β 2 + β 3
1 − β +
10
7 β 2 −
2
7 β 3
.
(11)
In this expression the β is the nuclear deformation and β > 0 is the prolate shape,
β = 0 is a spherical shape, and β < 0 is an oblate shape. Terms in β 4 and β 5 are
also present in the exact form of Eq. 11, but they have less than 1% effect.
Note that the values for the rotational enhancement factor in Table 1 range from
a value of about 8 for A ≈ 25 to a of about 60 for A ≈ 240. These are reasonably
consistent with theoretical predictions in Refs. [2] and [3].
Table 1 Level densities for deformed nuclei inferred with spherical and deformed spin distributions
Nucleus
Spin
S n
D
a 1
a 2
a 3
a 4
R
24 Na
1,2
6.96
9.5·10 4
3.49
1.75
4.35
2.76
8.3
25 Mg
1/2
7.33
4.7·10 5
3.67
1.92
6.33
4.14
7.6
26 Mg
2,3
11.073
5.5·10 4
4.16
2.37
4.48
2.85
8.75
159 Dy
1/2
6.83
30
20.81
13.05
30.0
22.1
47.8
161 Dy
1/2
6.45
27.3
22.12
14.06
32.3
23.7
42.9
162 Dy
2,3
8.197
2
21.34
13.58
26.8
18.4
45.3
163 Dy
1/2
6.27
69
21.08
13.01
31
22.4
42.5
164 Dy
2,3
7.63
5
21.2
13.24
26.9
19.2
46.1
165 Dy
1/2
5.71
170
21.05
12.67
31.7
22.8
44
235 U
1/2
5.298
10.6
30.26
19.23
44.8
32.8
63
238 U
0,1
6.15
3.5
30.55
19.58
45.1
32.8
67
S. M. Grimes
and the measured level spacing (eV) are provided. The values indicated by a 1 show
the level density parameters inferred by the authors by calculating the level density
correction using Eqs. 7 and 8. This level density is then divided by the rotational
enhancement factor proposed in Refs. [2] and [3]. A new a (labeled a 2 ) is deduced
from the reduced level density. a 1 values are similar to the a values proposed by the
authors for neighboring spherical nuclei.
The same input data were then re-analyzed using Eq. 9 or 10 as appropriate.
A very substantial increase in level density was observed. This yielded the level
density parameters a 3 . Finally, the rotational enhancement factor was inferred by
calculating the level density by utilizing the Bethe spin distribution. This then results
in parameters a 4 . a 4 values are intrinsic level density parameters (those which would
be observed if a nucleus was not deformed) and are consistent with neighboring
spherical nuclei. To use Eqs. 9 and 10 values for σ ⊥ are needed. Values for σ
were obtained from the results of calculations in Ref. [7]. It may be shown that the
relationship between σ ⊥ and σ is
σ 2
⊥
σ 2
=
1 +
1
2 β +
16
7 β 2 + β 3
1 − β +
10
7 β 2 −
2
7 β 3
.
(11)
In this expression the β is the nuclear deformation and β > 0 is the prolate shape,
β = 0 is a spherical shape, and β < 0 is an oblate shape. Terms in β 4 and β 5 are
also present in the exact form of Eq. 11, but they have less than 1% effect.
Note that the values for the rotational enhancement factor in Table 1 range from
a value of about 8 for A ≈ 25 to a of about 60 for A ≈ 240. These are reasonably
consistent with theoretical predictions in Refs. [2] and [3].
Table 1 Level densities for deformed nuclei inferred with spherical and deformed spin distributions
Nucleus
Spin
S n
D
a 1
a 2
a 3
a 4
R
24 Na
1,2
6.96
9.5·10 4
3.49
1.75
4.35
2.76
8.3
25 Mg
1/2
7.33
4.7·10 5
3.67
1.92
6.33
4.14
7.6
26 Mg
2,3
11.073
5.5·10 4
4.16
2.37
4.48
2.85
8.75
159 Dy
1/2
6.83
30
20.81
13.05
30.0
22.1
47.8
161 Dy
1/2
6.45
27.3
22.12
14.06
32.3
23.7
42.9
162 Dy
2,3
8.197
2
21.34
13.58
26.8
18.4
45.3
163 Dy
1/2
6.27
69
21.08
13.01
31
22.4
42.5
164 Dy
2,3
7.63
5
21.2
13.24
26.9
19.2
46.1
165 Dy
1/2
5.71
170
21.05
12.67
31.7
22.8
44
235 U
1/2
5.298
10.6
30.26
19.23
44.8
32.8
63
238 U
0,1
6.15
3.5
30.55
19.58
45.1
32.8
67