The study of the level density of excited nuclei of neptunium isotopes

Nuclear Physics AS03 (1989) 461-472 North-Holland, Amsterdam

THE STUDY OF THE LEVEL DENSITY OF EXCITED NUCLEI OF NEPTUNIUM ISOTOPES S.Yu. PLATO NOV, O.V. FOTINA and O.A. YUMINOV Institute of Nuclear Physics, Moscow State University, Moscow 119899, USSR

Received 7 February 1989 Abstract: The lifetime of the induced fission of 235.236.23g.239 Np is measured by the blocking technique. The new method for obtaining the level density in the second potential well, P2( U, J), is used to extract the absolute values of P2( U, J) for the neptunium isotopes in the excitation energy range 3-12 MeV. The obtained P2( U, J) data are analyzed in the level density phenomenological model. NUCLEAR E

235.236,238.

239

REACTIONS 235· 23R U(d , xnf'), E = 7.5-15.5 MeV; measured Tf (Exl; Np deduced level density in the second potential well; level density phenomenological model.

1. Introduction

The study of the basic characteristics of atomic nuclei at high excitation energies, large angular momenta and in the strongly deformed states is an urgent problem in nuclear physics. In particular, in order to describe fission in the framework of the statistical theory of nuclear reactions it is necessary to study the dependence of the level density of the nucleus on its deformation. In most papers, the influence of the nuclear deformation on the magnitude of the level density is studied on the basis of comparison between the level density parameters for nuclei with different deformation of the ground state. This is connected, first of al1, with the fact that there has been accumulated, by now, extensive experimental information about the absolute values of the level density for a rather narrow range, bounded by the values of the equilibrium deformation, by the values of the excitation energies E* ~ 1-2 MeV and by the neutron binding energies. However, the interpolation of the soobtained phenomenological dependences of the level density parameters to the regions of high excitation energies and of large deformations leads to considerable uncertainties in the absolute values of the level density. In this connection, in order to analyze in detail the dependences of the level density on the excitation energy and on the nuclear deformation, it is necessary to have an essential1y larger amount of experimental data on the absolute values of the level density at different excitation energies and deformations of the same nucleus. Comparison between the density of excited states in the first and second potential wel1s of heavy excited nuclei gives a unique possibility for obtaining such information. 0375-9474/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

462

S. Yu. Platonov et al. / Study oj the level density

The present paper suggests a method for obtaining the energy dependences of the level density of heavy excited nuclei in the second potential well, pi U, J). The method is used to extract the absolute values of P2( U, J) for the neptunium isotopes 239 235.236,238, Np in the excitation energy range of 3-12 MeV. The obtained pi U, J) data are analyzed in the level density phenomenological model 1). 2. Theory

The statistical approach to the fission of heavy excited nuclei in the double-humped fission barrier model, with allowance for the existance of two classes of excited nuclear states in the first and second potential wells, was developed in refs. 2,3) and generalized to the time characteristics of decay in ref. 4). The existence of quasistationary excited states in the second potential well has the result that the deexcitation-time of heavy fissionable nuclei depends on the decay mode. That is, both the classes of quasi-stationary states are successively populated during the fission of the excited nucleus, and the de-excitation through any other channel proceeds mainly from the states under equilibrium deformation. The decay time of heavy excited nuclei through the fission channel, 7e, is larger than the de-excitation time via any other channel, 7i' It was shown in refs. 5,6) that in the weak-coupling approximation of two classes of excited states and, if the emission of particles and y-quanta from the excited states in the second potential well are neglected, the time delay in the decay of an excited nucleus via the fission channel is determined by the expressions: (1) L!7 """.!!.- = 21Th P2( U, J) r, N 2(U, J)'

(2)

where r 2 is the total decay width in the second potential well; N 2 is the effective number of decay channels of these states; P2 is the density of excited states in the second potential well. Experimental determination of the time delay of the induced fission, L!7, permits one to obtain information about the characteristics of the strongly-deformed excited state of nuclei (the fission barrier parameters, the density of excited states in the second potential well, etc.). This is the difference between the time delay and the traditional characteristics of fission (cross sections, angular distributions of fragments, etc.), which are insensitive to the structure of excited states in the second well for excitation energies beyond the barrier region. 3. Experimental details

The delay in the mean decay time of heavy nuclei through the fission channel was experimentally revealed when we studied the time of the induced fission of actinide Np isotopes, by using the method based on the blocking effect 6,7). Fig. 1

S. Yu. Platono» et al. / Study of the level density

463

shows the experimental values of the fission time of the excited nuclei 235,236,238, 2 39 Np produced in the 235, 2 3 8 U(d, xnf) reactions. The experimental arrangement and the processing of experimental data on the lifetime of concrete nuclei were traditional for the blocking technique 8). The experimental values of the fission time were compared with both the fission times calculated in the traditional approach (in which the fission time is determined by the lifetime of excited states under equilibrium deformation 7f= 11/ T\) and the calculated decay times of the corresponding nuclei via the neutron channel. The present values were calculated in the statistical theory of nuclear reactions using the GFOT code 5), which enables one to describe the time characteristics of the decay of excited nuclei in the traditional approach and with allowance for the lifetime of excited states in the second potential well. In the latter case the experimental values of 7f were described by varying the level density P2' 7,(5)

10- 16

10

9

8

II

12

13

7,(5)

E*,(MeV)

(b)

10- 16

5

6

7

8

9

10

II

E*,(MeVI

Fig.!. The mean fission time of 2J6Np (a), 2J5Np (b), 239Np (c) and 238Np (d) nuclei versus the mean excitation energy. The dots show the experimental values. The curves show the variants of calculations: the dot-dashed line, without allowance for the lifetime of states in the second potential well; the solid curve, with allowance for the lifetime of states in the second potential well. The broken curve is the decay time via the neutron channel with the parameters the same as for the solid-line curve.

464

S. YU. Platonov et al. / Study of the level density

4. The method of extraction of the level density in the second potential well

Using the GFOT code, we have extracted the absolute values of the level density in the second well of 235,236,238, 239 Np from the experimental values of the induced fission time. The level density pi U, J) was determined by the relations (0 and (2). In this case we used for 7f the experimental values of the fission time measured by the blocking technique, 7; was calculated in the statistical theory of nuclear reactions using the double-humped fission barrier model, with allowance for the existence of two classes of excited states 6): (3) where the level density in the second well refers to the experimental values of 7f. The total effective numbers of decay channels of first and second-well states are (4)

S. Yu. Platonov et al. / Study of the level density

465

(5) where i is the index of the emitted particle (neutron, proton, a-particle) or the v-quantum. The determination of N li is given in ref. 9), N 2 i was calculated by the same formulae but with the parameters of the level density in the second well. The parameters of the level density in the first well were taken from the known systematics 10,11). The effective number of channels of the transition from the first to the second potential well was determined by the expression: E-B'

A

N =

I ) 'PA ( E-Bf-EI,J dE 1+exp(-21TE/ hwI) '

f-B~

It was assumed in the present paper that this is equal to the effective number of channels of the transition from the second to the first potential well. Analogously, for the transitions through the outer fission barrier: E

NB =

-

B II

f-B:'

r

PB(E - B f11 -

E,

J) ds

1+ exp (-21TE/ h(2) ,

(7)

where PA and PB, the level density in the first and the second saddle points, were determined to provide the best description of experimental data on the fissionability of the corresponding nuclei 12,13). The double-humped fission barrier parameters were taken from the review in ref. 14), Fig. 2 shows the absolute values of the level density in the second well of 235,236,238, 239Np nuclei, the values were extracted from the experimental data on the induced fission time. The P2( U, 0) values were extracted from the values of the mean angular momentum of the corresponding nuclei produced in the 235, 238U(d, xnf) reactions and they were recalculated for J = O. Fig. 2 presents also the experimental data on PI in., 0) [ref. 15)] and P2(Bn -.1 E, 0) [ref. 16)] for the 238Np nucleus (where .1E is the energy difference between the ground state in the first and second potential wells).

5. Analysis of the results Comparing the present values of the level density in the second well with the values of the level density in the first well, measured in experiment and also calculated in the Fermi-gas model, shows that the absolute values of P2( U, 0) are much larger than the PI (U, 0) values, for all the investigated nuclei. Analysis of the energy dependence of the present P2( U, 0) values was made in the level-density phenomenological model I), which differs from the Fermi-gas model in that it includes the correlation effects of the superconducting type and the effects of collective nature. To calculate the total level density in the phenomenological model with allowance for the collective-excitation contributions we use in accordance with ref. 17) the

i i i

6 7

p

I

9

,

U,(MeV)

3

;

4

,

5

,

6

i

7

9

U,(MeV)

i i I

8

The curves represent the calculations in the level-density phenomenological model in the approximation of axial and mirror symmetry (dash-double dotted); of axial symmetry and mirror asymmetry (dash-dotted); of elipsoidal (0 2 ) symmetry (dashed) and of total asymmetry (full line) of the nuclear shape. For 238Np the calculation of PI (U, 0) with the parameters from table 1 (dash-three dots) is presented.

P2(E* - tiE, 0) values. In fig. 2c, the experimental values of P2(B n - tiE, 0) and PI(B n , 0) for 238Np are shown by a triangle and open circle, respectively.

Fig. 2. The level density of 235Np(a), 236Np (b), 238Np (c) and 239Np (d) as a function of the internal excitation energy. The dots are the experimental

5

ro3 11 ({{

4

10t2

~

ro4

ro 3

I i i

ro5

101

IT (

10 6

J (b)

ro 5

roE

109

P2(U,O), (MeV- 1)

10 7

I

J (a)

ra6

10 7

roP

P2(U,OJ, (MeV- 1)

-1>0

~

ce

::

f}

'"~

i:: ~ ;;;-

Vl

-<,

:-

l:>

'" ~

:: c

C

l:>

~

~

~

0'< 0'<

S.Yu. Platonov et a1/ . Stu dy OJ.r the level density

467

S. Yu. Platonov et al. / Study of the level density

468

approximation:

p( U, J)

=

(8)

Pin( U, J)K co l ( U) ,

where Pin( U, J) is the density of internal states; K co l ( U) is the collective enhancement coefficient. The density of internal nuclear states was calculated by the formula suggested in ref. I): Pin( U, J) =

(2J + 1)h /

24.)2 (T~ff'" a t

3 5

( (J +~)2) exp 2at - - - ,

(9)

2(T~ff

t t.

where (T~ff = 12j3I l For the nuclear momenta of inertia the solid-state values are used, in which the dependence on the parameter e, the quadrupole nuclear deformation, is explicitly expressed 18): III =-;(m 2)a(1-e

2

(10)

) 1/ 3 ,

7T"

(11) The dependences (10) and (11), distinct from the analogous dependences of ref. I), are suitable for any values of the quadrupole deformation. For the dependence of the mean square of the nuclear momentum projection the parametrization used was (m 2 ) = gA2/3, where g = 0.19 [ref. 18)]. The quadrupole deformation e for the ground state was taken as 0.2 and for the second potential well as 0.6 [ref. 19)]. At nuclear temperatures above the critical point the internal nuclear energy is of the form: U = at' -.1- E c o n d where the condensation energy, characterizing the decrease of the ground-state energy due to correlation interaction, was determined from the parameter .1, which determines the odd-even differences in the nuclear binding energies (masses), in the ground state it was chosen on the basis of the semi empirical estimate: .10 = 12/-vA (MeV). In the second potential well the corresponding parameter .12 was calculated by the indirect semi empirical method proposed in the review 14). The values of .1 used in the calculations, are listed in table 1. The phenomenological model parameter a, which accounts for the shell effects in the behaviour of the level density in the framework of the shell correction method, TABLE 1 The parameters of the phenomenological level-density model, used in calculations First-well states Isotope

235Np 236Np 238Np 239Np

Second-well states

al

.1,

(MeV-I)

(MeV)

8WI (MeV)

1'1

(MeV-I)

a2

.12 (MeV)

8W2 (MeV)

82

21.6 21.6 21.8 21.9

1.56 0.78 1.56 0.78

-1.5 -1.5 -1.5 -1.5

0.2 0.2 0.2 0.2

24.0 24.0 24.2 24.3

1.30 0.65 1.30 0.65

-2.5 -2.5 -2.5 -2.5

0.6 0.6 0.6 0.6

469

S. Yu. Platonov et al. / Study of the level density

was taken at temperatures above the critical point in accordance with ref. 1):

),

(12)

(u) = 1 - exp {- y( U - Eco n d + L1)}

(13)

a(u) =

a(1 + f(u)

8W

U - E co n d + L1

where

f

is the dimensionless universal function determining the energy dependence of the level density parameter. For the parameter y we choose the traditional value y = 0.064 obtained from the systematized data on the approximation of the density of neutron resonances for heavy nuclei 20). The values of the shell correction were taken from ref. 14). The asymptotic value ofthe level density parameter at high excitation energies was calculated with allowance for the influence of the diffusivity of the near-surface layer of nuclei, in accord with the paper 21), as

a

(14) where

s, =

t.:r du/47TR

2 ,

(15)

s, = t.:r dUk/87TR

(16)

are the surface area of the deformed nucleus and its integral curvature that are normalized to the corresponding values for the spherical nucleus; '0 (= 1.16 fm) is the "scale" parameter corresponding to the charge radius which is determined from the experiments on elastic electron scattering; the numerical values of the parameters (x, (3 and yare taken from ref. 21). The numerical values of the coefficients B; and B k for different values of the quadrupole and hexadecapole deformation are taken from the calculations of ref. 22). The obtained values of used in calculations are listed in table 1 along with the corresponding values of the shell correction. Below the phase-transition point the parametrization proposed in ref. 1) is used to describe the above-listed thermodynamical functions. The collective enhancement coefficient is of the form:

a

(17) where KyibJ U) and K rot( U) are, respectively, coefficients of the vibrational and rotational enhancement. In the calculation for the coefficient of the vibrational enhancement the liquid-drop estimation was made, i.e. _

A ~

Cl.d.

KyibJ U) - exp { 1.69 ( 47TUl.d. C )

2/ 3

4/3 }

t

,

(18)

470

S. Yu. Platonov et al. / Study of the level density

where (Tl.d = 1.2 MeV' fm- 2 is the surface tension in the liquid-drop model, which corresponds to the analogous phenomenological parameter of the mass formula. The ratio C / Cl.d. characterizes the difference of the rigidity coefficient of the excited nucleus from the corresponding coefficients of the liquid-drop modeL In our calculations the liquid-drop value of rigidity coefficient was taken (i.e. C = Cl.dJ For the coefficient of the rotational enhancement we used the expression which depends strongly on the nuclear deformation: (T~

K ro t = 2(T~

.J vs:;;: (T~ (Til I271' (T.L2 (Til

for spherical nuclei; for axially- and mirror-symmetric nuclei; for axially-symmetric and mirror-asymmetric nuclei; for elipsoid ( D 2) symmetry; for nuclei possessing no symmetry.

Here (T.L = I.Lt/h2 and (T11=J11t/h 2. Fig. 2 shows the results of the calculation of P2( U, 0) made with different coefficients of rotational enhancement, depending on the type of symmetry of the nuclear shape. The same figure presents the calculation for PI ( U, 0) of 238Np in the approximation of axial and mirror symmetry of the nuclear shape; the PI ( U, 0) calculation describes satisfactorily the values of Pl(B n , 0) extracted from experimental data on the density of neutron resonances. Comparison of the calculated data with the experimental P2( U, 0) values for the 235,236,238, 239Np nuclei permits the observed increases of the level density in the second well of the investigated nuclei, as compared with the level density in the first well, to be attributed to the increased contribution of the effects of a collective nature, resulting from the symmetry breakdown of the nuclear shape at large values of the deformation. A satisfactory description of the experimental data on P2( U, 0) is achieved if the axial and mirror symmetry of the nucleus is assumed to be broken in the second potential welL 6. Conclusion

Because of the lack of the experimental data on the level density in the second potential well there exist no stable opinions of the behaviour of P2( U, J). Many authors (see, for example refs. 23,24)) are included to the, at first sight, most simple assumption that the behaviour of Pl( U, J) and P2( U, J) is the same. At the same time there is some evidence for the difference in values between PI (U, J) and P2( U, J) [see, for example, ref. 25) which reports the ~ 10 fold excess of P2( U, J) over PI ( U, J) at the same intrinsic excitation energy. This excess is especially great for nuclei odd in proton or neutron number]. A large quantity of the experimental findings indicates that the basic nuclear characteristics (deformation, moment of inertia, shell correction, correlation function), which affect essentially the level density are different in the first and second potential wells.

S. Yu. Platonov et al. / Study of the level density

471

The method of extraction of the absolute values of the level density in the second well, suggested in the present paper, could shed further light on this problem. It is necessary to estimate the accuracy of the present method, which is determined by the statistical accuracy of the results of the induced fission time measurements and by the uncertainty of the theoretical analysis. The statistical accuracy of the fission time measurement is ~20-100% and can, in principle, be raised. When extracting the data on P2( U, J) from Tr there can arise systematic errors due to the approximate form of (2) and the inacuracy of the parameters used in the method. The expression (2) was obtained with the neglect of the decays of the nuclei in the second well states via the channels with the emission of particles and v-quanta, which holds true within ~20% in the excitation energy region > B r . The accuracy of calculation of T; and N 2 is determined by the accuracy of the experimental data on the density of neutron resonances in the region of the neutron-binding energy and by the accuracy of the nuclear fission data used to determine the model parameters. If all the statistical errors of the experimental data on Tr and uncertainties of the theoretical analysis are taken into account and error in P2( U, J) will not be greater than 200% . Hence, it is safe to state that the values of pA U, J) are greater than the values of PI (U, J), at the same internal excitation energy. The theoretical calculation in the generalized superfluid model of the level density with allowance for the coherent effects and for the effects of a collective nature, which uses the model parameters, determined from the independent experimental data, describes uniformly the data on PI( U, J) and P2( U, J) (see fig. 2). To describe the values of PI( U, J) it is sufficient to use the generally accepted assumption of the axial and mirror symmetry of the nuclear shape, and to describe the data on P2( U, J), even with allowance for the factor of 2-3, it is necessary to assume a more complex shape of the nucleus in the second-well excited states. Thus, the measurements of the fission time of excited actinide nuclei give experimental information about the absolute values of the level density in the second well in a wide range of excitation energies which permits drawing conclusions about the nuclear shape.

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