Calculation of realistic level densities with Bethe's formula

10 November1994 PHYSICS LETTERS B

ELSEVIER

Physics LettersB 339 (1994) 7-10

Calculation of realistic level densities with Bethe's formula B.K. Agrawal, A. Ansari Institute of Physics, Bhubaneswar-751005, India

Received 7 July 1994; revised manuscriptreceived 7 September 1994 Editor: G.E Bertsch

Abstract

Spin and temperature dependent level density and the level density parameter for 44Ti have been calculated using the auxiliary field representation for the grand canonical partition function with a quadrupole-quadrupole interaction Hamiltonian within the static path approximation (SPA). We find that when these level density parameters (aSPA) are employed in the standard Bethe's formula for the level density, the values of the latter quantities come out to be quite comparable to those obtained from the SPA approach.

Statistical model codes are useful tools for interpreting and predicting the yields in nuclear reactions. One of the important inputs in these codes is the nuclear level density as a function of excitation energy (or temperature) and angular momentum. Recently, Nanal et al. [ 1 ] and Malaguti et al. [ 2 ] have measured the lifetime of the evaporation residues in the decay of 44Ti formed at the initial excitation energy ranging from 55-75 MeV. The lifetime information obtained from the change in minimum yields and blocking dip volume are consistent and show a weak dependence of lifetimes on the excitation energy. In Ref. [ 1 ], statistical model code CASCADE is used for the theoretical estimation of the lifetime which is given as [ 3 ], ~" ,-., h p c ( E * ' J) N ( E * , J)

( 1)

where Pc is the level density of a compound nucleus at an excitation energy E* and angular momentum J and N is the total number of channels to which the compound nucleus can decay. The level density parameter a and the moment of inertia used to determine the level density is assumed to be independent of spin

and temperature (see Eq. (11)). It is found [1] that one can reproduce only the trend but not the absolute lifetimes. Therefore, it appears that a quantitative agreement with the experimental data can be obtained only if the level density is calculated within a more realistic approach. The evaporation residues of which lifetimes have been measured [ 1 ] correspond to A ,,~ 35 and the excitation energy E* ,-~ 25 MeV (or T ,~ 2.5 MeV). At such a high temperature, it is necessary to include the effects of the statistical fluctuations in various degrees of freedom. Now it is well known that the static path approximation (SPA) [4-6] takes into account these fluctuations in a natural fashion and thereby provides a better alternative to the finite temperature mean field approximations. Recently [7-9], we have studied within the SPA approach, the spin and/or temperature dependence of the level density p, the level density parameter a and the moment of inertia. In this letter we shall discuss about the spin and temperature dependence of p and a for 44Ti including statistical fluctuations in all the quadrupole degrees of freedom

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B.K. Agrawal, A. Ansari / Physics Letters B 339 (1994) 7-10

within the SPA. As we shall see, this study would give an idea about the possible effects of statistical fluctuations on the decay lifetimes. The SPA level density in a saddle point approximation [10,11] is given by

p(E, J) = exp [ f i E + lnZsPA +apNp +cenNn + h i ] (2)

Here, ZSPA is the partition function within the SPA approach and I stands for the expectation value of x-component of angular momentum operator Jx- The quantity 79 in Eq. (2) is a 4 x 4 determinant formed by the values of second derivative, a 2 In ZSpA/t~Xi~Xj, evaluated at the saddle point specified by x -: (fl, ap,an,A), where fl = 1/T, ap,n = -tZp,n/T and A = -to/T. For a quadrupole-quadrupole interaction Hamiltonian, ZsvA takes the following form, ZSp A =

Zr DSPA

(3)

where DSPA is the static path representation of the statistical density operator given as =

(

a

")5/2

to

zr/3

2~

*r

2~

× f fl4dfl f sin3Td~'f d~fsinOdOf d¢/ 0

0

0

0

0

x e-~#212Ve - ( f l ' - " p ~ p - ' ~ ° ) / r

(4)

where 1~I' ~° = ~ i h' ~°( i) with h t ~° a cranked quadrupole deformed one-body mean field Hamiltonian, h '°~ = h ( 3 , ~') -

to{.~x,(cos 0 cos ~bcos ¢ - sin ~bsin ¢)

-

.~y,(sin ~bcos ~b + cos 0 cos ~bsin ~,)

-

g =

Np,n =

a In ZSpA 03 3 In ZSPA

)z' sin 0 cos ~b}

with h(fl, 3/) = h0 r2 1 -t~ooB~-[cos yg2o+ ~ sin y(Y22 + ~_2)]

(6)

and ho representing the spherical basis single-particle (sp) energies defined here with respect to 160 as an

(8) alnZsPh aA

(9)

The level density parameter a can be determined using entropy S as S 2T

aSPA = - -

-Z~Tr =

[DsPA In Dspa ] + In ZSPA 2T (10)

On the other hand, the Bethe's level density formula [ 14] based on the Fermi-gas model, as used in the statistical model codes, is given by p(U,J) =

(5)

(7)

tgOlp,n

I = J x / ' ~ + 1) =

\ ~--~]

DSPA

inert core. The value of hzo0 = 41A -1/3 MeV and ot = (hto0)2/X b4 with X b4 = 70A -1"4 MeV as given by Baranger and Kumar [ 13] (for the rare earth nuclei). Here X is the quadrupole interaction strength and b the usual oscillator length parameter. The integration variables in (4) are the well known quadrupole deformation parameters. We must point out here that, at high temperature, T ~>2.0 MeV, the level density is not very much sensitive to interaction strength [ 8 ]. The partition function ZsPa can now be evaluated by diagonalizing sp Hamiltonian (5) in a suitable model space. In the present case, the model space (same for protons and neutrons) consists of lsu2, 0d3/2, 0d5/2, lpl/2, lp3/2, 0f5/2, 0f7/2 and 0g9/2 with the corresponding sp energies (in MeV) -4.03, 0.08, -5.00, 12.23, 10.33, 11.73, 06.30 and 16.13, respectively [ 12,5]. The average value of energy E, proton (neutron) numbers Np(Nn) and angular momentum used in Eq. (2) is calculated using the canonical relations

2J + 1

( h2 ~ 3/2 exp [2v'-a--u-] 12 x/-d\ ~ - ~ j (U+T)2

x e x p [ . J(J + 1 ) ] 2o-2

(11)

where U = a T 2 - T, o.2 = 0.0888 (aU) 1/2 A 2/3 is the spin cut-off factor and Z is the rigid body moment of inertia. In Fig. 1 we have shown the spin and temperature dependence of the level densities using Eqs. (2) and (11). It may be noted that the x-axis represents the

B.K. Agrawal, A. Ansari / Physics Letters B 339 (1994) 7-10

301

J = 0 J = 8

• ~

J

=

44m. 11

16

9

----J 30

- - J • ~

o=A//~,*

J

=

0

= =

8 16

//. i1

2O

2O

Q - I"t° /r

c~

10

/

~/ / 0

/

J

/..I-

,.5 f -

J.~"

.,,"

'~'"

J':"

10

/*/ 0

¢,

T(MeV) Fig. 1. The upper (lower) hunch of curves represents the level density with a = A/8 (a = A/13) used in Bethe's formula (Eq. ( 11 ) ). As labelled, the middle bunch of curves corresponds to the level density in SPA approach (using Eq. ( 2 ) ) .

temperature, since it determines the value of E and U in Eqs. (2) and ( 11 ), respectively. Looking at the plot one can say that both the expressions yield level density which depends strongly on temperature. The SPA level density which does not explicitly depend on the parameter a, seems to be close to the level density obtained using Eq. ( 11 ) with a = A/8 for T ,,~ 2 MeV. Then with increase in temperature (T = 6 MeV), the SPA level density again becomes closer to that with a = A~ 13. Thus, we may say that the inverse level density parameter K = A/a varies roughly from 8-13 for temperatures 2-6 MeV. Furthermore, SPA level density shows a weak dependence on the angular momentum. On the other hand, with a constant a the Bethe's formula shows a relatively stronger spin dependence (see Fig. 1) with a peak at J ~ o-2. It must be emphasized that the SPA results for J = 0 (or to = 0), correspond to "no cranking" intrinsic results. In the present case, we notice from Eq. (10) that since entropy is the thermal average of -ln(TD/Z,), the level density parameter aSPA would simulate in it the effects of statistical fluctuations. In the present case aSPA accounts for the statistical fluctuations in all the quadrupole degrees of freedom, i.e., shape and orientations (see Eq. ( 4 ) ) . Let us now see what happens if

i

2

I

[

i

i

I

3

4 5 6 T(MeV) Fig. 2. The level density using aSPA in Bethe's formula (Eq. (11)). The inset shows the variation of the inverse parameter KSPA ---- A/aSPA as a f u n c t i o n o f spin and temperature.

aspA is used instead of a constant a in the Bethe's formula for the level density given in Eq. ( 11 ). In Fig. 2 we have shown the level density obtained using aSPA in Bethe's formula (Eq. ( 11 ) ). The insert in the figure shows the variation of inverse level density parameter gsPA = A/aSPA as a function of spin and temperature. We see that the values of the level density are quite close to the SPA values (see Fig. 1 ). Another important feature to notice is that, as the aSPA depends very weakly on spin, the spin dependence of the level density is significantly affected in comparison with the earlier case when a constant value of a was used in the Bethe's formula. Therefore, for practical applications, it should be enough to use Bethe's formula (instead of a full SPA calculation for p) along with a realistic value of the level density parameter such as aSPA. In conclusion, we have studied the spin and temperature dependence of the level density and the level density parameter of 44Ti including statistical fluctuations in all the quadrupole degrees of freedom (i.e. shape and orientations), within the framework of static path approximation. Comparing these results with the one obtained employing a fixed value of level density parameter a in the standard Bethe's formula, we observe large deviations from the average. However, if the level density parameter determined in SPA is used

10

B.K. Agrawal, A. Ansari / Physics Letters B 339 (1994) 7-10

in the B e t h e ' s level density formula, it yields quite satisfactory results. Thus, Bethe's formula as such seems quite reasonable provided a spin and temperature dep e n d e n t realistic value o f the level density parameter, viz. asPa, is used in it. In the present context, we m a y also p o i n t out that with the use o f SPA level density or the level density parameter aSPA, the decay lifetimes o f evaporation residues e m e r g i n g from 44Ti should be somewhat reduced relative to that n o r m a l l y obtained using a = A / 8 in the B e t h e ' s formula.

References [ 1] V. Nanal, M.B. Kurup and K.G. Prasad, Phys. Rev. C 49 (1994) 758 [2] E Malaguti et al., Europhys. Lett. 12 (1990) 313 [3] R. Bass, in: Nuclear Reactions with Heavy Ions (Springer Veflag, Berlin, 1980) [4] B. Lauritzen, R Arve and G. Bertsch, Phys. Rev. Lett. 61 (1988) 2835 [5] B. Lauritzen and G. Bertsch, Phys. Rev. C 39 (1989) 2412 [6] Y. Alhassid and B.W. Bush, Nucl. Phys. A 549 (1992) 43 [7] B.K. Agrawal and A. Ansari, Phys. Rev. C 46 (1992) 2319 [8] B.K. Agrawal and A. Ansari, Nucl. Phys. A 567 (1994) 1 [9] B.K. Agrawal and A. Ansari, Nucl. Phys. A (in press) [10] C. Bloch, Phys. Rev. 93 (1954) 1094 [ 11] A. Bohr and B.R. Mottelson, Nuclear Structure (Benjamin, Reading, MA, 1969), Vol. 2, App. B2. [ 12] R.M. Quick, N.J. Davidson, B.J. Cole and H.G. Miller, Phys. Lett. B 254 (1991) 303 [ 13] N. Baranger and K. Kumar, Nucl. Phys. A 110 (1968) 490, 529 [14] R.G. Stokstad, in: Treatise in Heavy Ion Science, Vol. 3, edited by D. Alan Bromely (Plenum Press, New York, 1985) p. 104, 105