Quantum mechanical description of the intrinsic excitation of fissioning nuclei

QUANTUM MECHANICAL DESCRIPTION OF THE INTRINSIC EXCITATION OF FISSIONING NUCLEI

G. SCHUT~E Instittit für Theore:ische Physik der Universilä: Heidelberg. Germany

so

1q51

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICS REPORTh (Review Section of Physics Letters) 80. No.2 (1981) 113-156. NORTh-HOLLAND PUBLISHING COMPANY

QUANTUM MECHANICAL DESCRIPTION OF THE INTRINSIC EXCiTATION OF FISSIONING NUCLEI G. SCHUUE InsWutfür Theoretische Physik der Universitdt Heidelberg, Gerrvtwiy Received May 1981 Contents: 1. Introduction 1.1. Physical situation 1.2. Theoretical approaches 1.3. Experimental information on the scission point 2. Selfconsistent approach 2.1. rime dependent Hartree—Fock method 2.2. Time dependent Hartree—Fock—Bogoliubov method 3. Cranking model 3.1. The single particle potential 3.2. Independent quasi-particles 3.3. Coupling matrix elements 3.4. Harmonic oscillator 4. Zcroth order solution 4.1. Hilbcrt apace and form of the solution 4.2. Proof for the solution 4.3. Correspondence to the Landau—Zcner problem 4.4. The energy expression 5. Broken pairs 5.1. Extended Hilbert space 5.2. Projection formalism 5.3. Initial conditions 5.4. Solution 5.5. Total energy

115 115 I IS 117 117 117 118 123 123 124 126 126 128 128 129 134) 134) 131 131 131 133 134 134

6. Adiabatic approximation for broken pairs 6.1. At the start 6.2. At a crossing: The harmless part 6.3. Healing of the crossing 6.4. Terms in the total energy 6.5. Mass parameter 7. Others adiabatic approximations 7.1. Adiabatic cranking model 7.2. Adiabatic time dependent Hartree—Fock 8. An example: Fissioning ~U 9. Extension of the time dependent Flartrce—Fock 9.1. The problem 9.2. Pair breaking by the residual interaction 9.3. Lifetime of static pair excitations Discussion Appendix A. Two levels Al. The problem A2. Asymptotic behaviour A3. Different initial conditions A4. Analytic structure References

134 135 135 137 138 139 144) 1411 141 142 145 145 145 146 149 l54) 151) 152 153 154 155

Abstract: The excitation of a nucleus during deformation at low energies is exemplified by the fission process. The basic formulas are derived from the TDHF-approach. In the framework of the cranking model the fundamental role of crossings and pseudo-crossings for the intrinsic excitation of the nucleus is demonstrated. A clear distinction between intrinsic and collective kinetic energy is obtained. The theory is applied to a fissioning uranium nucleus.

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G. Schune. Quan7um mechanical descnpnon ofthe inrn.nsic excitatum offissioning nuclei

ItS

1. Introduction 1.1. Physical situation A nucleus undergoing spontaneous or induced fission deforms from ellipsoidal shapes of the compound nucleus to the scission point where the fragments do no longer interact except for the Coulomb force. The driving force for this deformation comes from the Coulomb potential which is almost but not quite compensated by the surface energy [1~.These macroscopic potential energies can be calculated either in the liquid drop model [2. 31 or by folding an effective interaction with two single particle densities [4,5]. In addition microscopic forces arise from the variation of the single particle level density at the Fermi energy as a function of deformation [6—121.These shell corrections can be calculated by a constrained Hartree—Fock approach [13, 141 which confirms the procedure proposed by Strutinsky [15]. The sum of the macroscopic and the microscopic potentials is called the collective potential. It has been calculated as a function of two or three deformation parameters [16. 171. Generally, it exhibits two minima and two saddle points for the lighter actinide nuclei. The first minimum gives the ground state deformation while the second minimum at larger deformations gives rise to the shape isomers [181.The two minima are separated by the “inner” or “first” saddle. The “outer” or “second” saddle leads from the second minimum to the scission point where the fragments separate. Experimentally, excitation functions of neutron induced fission confirm the calculated collective potential [19]. Small discrepancies are due to the additional and simplifying assumption that the collective potential can be represented by a series of smoothly joined parabolas. This article is concerned with a microscopic picture for the excitation of the nucleus and finally of the fragments during the deformation from the outer saddle to the scission point. The total energy is assumed to be a little larger than the collective potential at the start of this deformation interval. 1.2. Theoretical approaches Imagine now a nucleus at the outer saddle or a point just beyond. It is then driven to the scission point by the collective potential. All that happens during this deformation is determined by the time dependent Schrödinger equation for the A-particle nucleus. This has been approximated by the time dependent Hartree—Fock (TDHF) equation [20—251.The direct solution [26] of this equation for a fissioning actinide nucleus implies a huge numerical effort. Therefore, additional approximations are desirable which also should give physical insight into the excitation of a nucleus during deformation. Two lines have been followed. One is the adiabatic approximation (ATDHF) [27—331 reconsidered in section 7. It is assumed that at each deformation the nucleus is mainly in its lowest static state and that the deformation velocity is small. These two assumptions, however, are contradictory: Either the nucleus does not get excited strongly and the entire available collective potential energy is converted into collective kinetic energy yielding a fast deformation process or the major part of the available energy must be converted into intrinsic energy yielding a small deformation velocity for a highly excited nucleus. Leaving the question of justification aside, the ATDHF approach retains the selfconsistent definition of the single particle potential. It, thereby, allows to determine the sequence of deformations followed by the fissioning nucleus the deformation path selfconsistently. In the second approximation to TDHF the time dependence of the single particle potential is parametrized by time dependent deformation parameters. The deformation path is guessed. The deformation velocity along this path can then be determined by energy conservation. The time dependent Schrödinger equation with this time —

—

116

G. Sc/tulle. Quantum mechanical de.wriptürn of the intrinsic excitation of fissioning nuclei

dependent single particle potential is solved without assumption on the deformation velocity or the intrinsic excitation energy. This approach is chosen in this article. The basic excitation mechanism will turn out to come from Landau—Zener transitions at crossings of single particle levels with the Fermi energy. In section 2 it is shown that the TDHF equations including the pairing force can be written in such a form as to exhibit this excitation mechanism in a fully selfconsistent theory. In an independent particle model the sudden approximation exemplifies in a simple way the excitation mechanism for large deformations [34]: Single particle levels are occupied with certain probabilities which are kept fixed while the single particle states are deformed keeping their nodal structure. Due to their dependence o~ the deformation occupied levels leave the Fermi sea and unoccupied levels dive into it. Thus, multi-particle—multi-hole excitations are created relative to the Slater determinant which has the lowest energy at each deformation. This simple picture is changed by the pairing force, which inhibits due to the pairing gap the excitation across the Fermi energy. The transition to excited states will be shown in section 4 to be of the Landau—Zener type [35]. In appendix A these transitions between two levels are studied, especially their an~ilyticdependence on the deformation and the deformation velocity. From that it follows that the transitions over the pairing gap do not give any contribution to the collective kinetic deformation energy and, hence, not to the mass parameter. In a slight generalization this study shows that crossings or pseudo-crossings do not contribute to the mass parameter. The spikes and strong fluctuations in the mass parameter obtained in [6.36—38] at close pseudo-crossings are the result of an erroneous application of the adiabatic approximation. In the picture just given only the coupling of states at the Fermi energy was included. There is, of course, also coupling due to the time dependence of the mean field between static single particle states which are separated by several MeV, i.e.: by essentially 2hw. This coupling can he treated in the adiabatic approximation as shown in section 5. Thereby a term linear in the actual deformation velocity is obtained in the wave function and hence a term quadratic in the actual velocity is found in the energy. iiis term is interpreted as collective kinetic energy since every classical kinetic energy is a quadratic form in the actual velocities. The corresponding mass parameter is given in subsection 6.5. In summary the total energy as discussed in subsection 6.4 contains a part which depends on the deformation only, a part which is quadratic in the actual deformation velocity and a term which depends non analytically on deformation velocities which occurred in the past i.e.: at the crossings of single particle levels with the Fermi energy. Due to this dependence the terms are identified as collective potential, collective kinetic energy and intrinsic energy respectively at each deformation. The picture of independent particles or quasi-particles has been extended to incorporate collisions due to an effective residual interaction between two nucleons which move in the average field. First order perturbation theory in this residual interaction leads to the Boltzmann equation 1391. In the fission process it is important to incorporate the energy transfer between the deforming potential and the colliding two nucleon subsystem giving rise to otT-shell collisions between the two particles. This effect has been completely neglected in an attempt to derive a master equation for the transitions between different orthogonal TDHF solutions [40].A critique of this work which is also wrong in other aspects is given in [41]. For high excitation energies of ten or more MeV above the outer saddle the level density becomes high enough and the mixing of any simple basis states becomes strong enough that a description of the deformation process by a statistical transport theory becomes relevant [42,43). For the low energies considered here this is not possible. An extension of the cranking model for independent quasi-particles is given in section 9. The pair excitations in which two quasi-particles occupy time reversed states and which are strongly excited during the fission process are coupled to

G. &hüae. Quantum mechanical description of the intrinsic excitation of fissioning nuclei

117

more complicated states by residual interactions. Statistical arguments are used to estimate odd-even effects in fragment yields. 1.3. Experimental information at the scission point

The physical interesting part of the deformation where the dynamical excitation occurs lies prior to the scission point. This is, however, not a region accessible experimentally. Only asymptotic quantities can be measured like the distribution of the kinetic energy and the excitation energy of the fragments, the yields in neutron and proton numbers, the angular distribution of alpha-particles emitted during fission, and the probability with which a muon is attached to the light or heavy fragment in a muon induced fission process. Neutron and y-emission from the fragments, the deflection of alpha-particles in the Coulomb field of the fragments, or the transition of a muon between molecular levels enter into the calculations of such quantities. The configuration at the scission point comes into play as the initial conditions for these calculations fixing the relative velocity, the deformation and the intrinsic excitations of the nascent fragments. It appears that these “initial conditions” are not uniquely determined by looking at only one class of experimental results. An example is the fit of the total kinetic energy of the fragments by different hydrodynamical calculations which yield different deformations and intrinsic excitations of the fragments at the scission point [44, 451. It is not possible to discuss here all the possible experimental information on the scission point configuration. The attitude of the work reviewed here is different. The first aim is to find a reasonable and yet simple quantum mechanical picture for the deformation process which allows to calculate the scission point configuration. Subsequently one may control whether this is in accord with the experimental result.

2. Selfconsistent approach 2.!. Time dependent Hariree—Fock method Although there are many presentations of the time dependent Hartree—Fock (TDHF) approximation

[491 the basic formulas and the extension to time dependent Hartree—Fock—Bogoliubov ~TDHFB)[471 to include the pairing interaction are repeated in this section. The aim is to exhibit the excitation of the nucleus via Landau—Zener transitions over the pairing gap. This basic mechanism will be studied in later sections in the cranking model. The TDHF approximation can be formulated in terms of a determinental wave function 4 determined from the variational principle

oJ~oIiat.-HI~d:’=o.

(2.1)

The single particle wave functions q’,, contained in ~ fulfill the Schrödinger equation

ii.. = hço,,

(2.2)

G. &hüne. Quantum mechanical description ofthe intrinsic excitation offissioning nuclei

118

with the s.p. Hamiltonian h. Alternatively, the equation of motion for the single particle (s.p.) density p is i1i=[h,p].

(2.3)

The s.p. Hamiltonian is (2.4) The s.p. potential U is defined as tr(pV) in terms of an effective interaction V. It must be borne in mind that although ~t is an A-particle wave function it should not be used to calculate the expectation values of many particleoperators. because not many particle correlations are contained in it [23).With this precaution in mind the two formulations of the TDHF approach are completely equivalent. In the following sections the formulation in terms of the wave function will be used, because it gives the closest connection to the physical picture for the nuclear excitation presented in sections 3 and 4. A vital part in this picture is played by the pairing force, so that the more general TDI-IFB method will he described in the next subsections. 2.2. 11w time dependent Hariree —Fock Bogoliuhov method —

The aim of the TDHFB is to include the attractive residual interaction between two nucleons in time reversed states in the time dependent Schrodinger equation. This is accomplished by the transformation of particle creation and annihilation operators a and a~to quasi-particle operators U~a,. V~a,

=

—

a~= U*a~+ V,.a.

The time reversed state of v is denoted by U~=U~,

V0=—V,

~.

(2.5)

The convention (2.6)

will be used for the amplitudes. The more general transformation superposing different s.p. states reduces to the form (2.5) by a suitable choice of the s.p. states as proved in ref. [47).These special states turn out to be the TDHF s.p. states. The state ~=fl(U~—V~aa~O) I.

(2.7)

>0

with 10) being the bare vacuum is then the vacuum with respect to the quasi-particle operators (2.5). Since the transformation (2.5) does not conserve the particle number the constraint NE (~X~)= (~~ aa,jofr) = const.

(2.8)

is introduced fixing the average number of particles to its initial value. The time dependent Schrödinger

G. Schüne, Quantum mechanical descnptwtt of the intrinsic excitation offissioning nuclei

119

equation is solved in the space of all quasi-particle vacua with the constraint (2.8). The solution can be determined from the variational principle a

J

(~H AX ia,~~c) dt’ = 0 —

(2.9)

—

in which the integral is varied with respect to U1., V, and the sp. wave functions. The Lagrange multiplier A is the chemical potential. The Hamiltonian consists of the kinetic energy and a two body

force H = ~(vItl,L)aaM

~~~PIV’v’)a~aa1..aM.

+

(2.10)

with antisymmetrized matrix elements

(j.tvIVl,&’i”)= —(~u’~V]v’~’).

(2.11)

When the Hamiltonian (2.10) is transformed [48,49) to quasi-particle (qp) operators according to (2.5) terms with no, two, or four qp-operators occur. The last terms coupling the vacuum directly to

4qp-excitations are neglected. The resulting approximate Hamiltonian is denoted by H. In the expression (~—AX)çb=(E—AN+H~,)t/i

(2.12)

the total energy is E=(~pIHI.fr) 2 ~ —

=

~ h1..I V1.~

VMI2I VM12

(JLvI V1~v)f

—

~ 4MUMV

(2.13)

and the particle number is N=2~.,IV i>0

2.

(2.14)

1.l

The last term in eq. (2.12) is =

~ {[(vfrIii)-. A8

1.~+

~

(J.L~vIVI,.L?,L)tVM.l2]ZJvV,~ ~4VMUVUM + Li ~MVVVM}aa~.Js. —

(2.15)

The Hermitean conjugate to H~oapplied an .fr vanishes, because ~~‘ is the vacuum for a1.. The pairing matrix is defined by ~

(v,iIV]~f,i’)UZ.V~..

(2.16)

G. &hüue. Quantum mechanical description of the intrinsic excitation offissioning nuclei

The Schrödinger equation or the variational principle (2.9) with !~is solved, if the projection of (H AX i3,)~con any element of the Hubert space vanishes. Since neither H nor 8, (see subsection —

—

3.3) couple to four or higher quasi-particle excitations, only the projection on ~c and on states of the form aati/i need to be considered. By a proper choice of the phase of ~frone can ensure

(2.17) The projection on the 2qp-excitations yield a diagonal and an off-diagonal set of equations [46) (2.18) (hkl i(kI!)+ hç~—u(cIr))U,Vk + 4k,UkU: —4 ~ —

= 0,

k 1.

(2.19)

The last two terms in (2.19) are neglected, because the sum in (2.16) is restricted to terms around the Fermi energy so that 4~M is large only for v, ~ close to the Fermi energy according to the product of U:. and VM.. These terms, however, cancel to a large extent in (2.19). The time reversal invariance of the Hamiltonian ensures the symmetry of the equations for time reversed states. Eq. (2.18) is split into two parts demanding that the first term and the rest vanish separately. This leads together with (2.19) to the s.p. equations

(2.20)

hp~

I’Pk =

with the s.p. Hamiltonian given by h,,,

=

(k 1:11)

—

Aôk, +

~ (~kI V~!)I v~

12.

(2.21)

The solutions of (2.20) give the special form for the time dependent vacuum (2.7). The self consistent potential given by the last term of eq. (2.21) depends on time via the s.p. states tp~and the occupation probabilities I VIJ42. Various parametrizations for this time dependence by time dependent deformation parameters have been proposed [50, 511. In this work the folded Yukawa potential [52, 53]

=

J

exp(—~Ir—r’I) d3r

(2.22)

will be used. Here p is a uniform density with a sharp surface with some given deformation so that the integral extends over the volume inside that surface. The range of the Yukawa interaction is = 1.25 fm~.Fig. I shows a single particle spectrum calculated with this potential as a function of deformation. The prominent feature of this spectrum is also found in other s.p. potentials [7,54, 551: the violent dependence on the deformation. Most of the pseudo-crossings are close so that an estimate with the Landau—Zener formula yields a jump probability close to unity for the deformation velocities obtained in the fission process. Therefore, the solutions of (2.20) resemble closely the diabatic states during the first stages of the deformation process, if they started out as eigen states of h. For these times a solution of (2.20) has essentially the form

G. Schüae, Quaniwn mechanical description of the intrinsic excitation offissioning nuclei

121

236U

Protons

o~

‘,

—.

-

-2

..4~. \

~

~ /

~-6[

~ -~-~~::.. ~

-8~

/

~

-loP ~23~

2.7

R/R 0 Fig. 1. Proton Icvcls in the folded Yuk~wapotential as a function

= q,~exp{_i

f

of

R along the path of deformation used in

e~di’)

section

8.

(2.23)

where ço~ are the static diabatic states and e°& the diabatic crossing energies. Later, when many pseudo-crossings have been passed and when the coupling due to the 8,-operator mixed eigen states of h which correspond to well separated eigenvalues, this simple picture will eventually break down. The second part of eq. (2.18)

(2.24) is, of course, the time dependent version of the usual BCS-equation. Writing the single particle phases

explicitly: i’&exp{ife& dr}= Vk,

ákk exp{2i

J

4 dr} =

Ukexp{—tfekdr)

Uk,

(2.25)

G. Schüae. Quantum mechanicaldescription

122

of the intrinsic excitation offissioning nuclei

leads to

—4kkUk+~ikkVk+2ekUkVk=i(Ck~—UkVk). The eigenvalues of h are denoted by

ek.

(2.26)

Define static BCS-amplitudes

e’—A 2k’Vfrk —A)2+(ReJkk)2 v~=I—u~. 2_’f

ttk

(2.27)

Inserting the transformation Uk =

ukdk + VkC,,

= vkdk

—

144

exp{_2i

J

e~dr}

exp{—2i

J

4 dr)

(2.28)

-

into eq. (2.26) yields {2e~u~v~ Re l1kk(u~ v~)}(d~c~)—2dk4ek —

—

—

i[d~(ukl.~g,

—

UkVk)

—

~kdk+

(U&V

—

140k)Ck +

4(1k)

—

i(d~+ E~)Im.i~= 0.

(2.29)

The abbreviation

4 = c~exp(_2i

J

e~dr)

(2.30)

V’(e,._A)2+(Rejkkf

(2.31)

and the definition ~

have been used. The curly bracket vanishes identically because of the definition (2.27). The imaginary part of the gap parameter may be expected to be small, because it is a sum over terms with varying phases Im

hi&k

=

—

~ (k°IZ°j VI,u°,i’) (e:dM

—

ë~d).

(2.32)

It is therefore, of the order of one term. Furthermore, if the nucleus starts in the BCS-groundstate the cM are zero (see subsection 5.3). During the deformation process the c. which get excited attain values

G. Schüae. Quantum mechanical descnpsson of the intrinsic excitation offissioning nuclei

123

of about 0.3 yielding an imaginary part of about 50 keV. In contrast, the real part Re Skk =

~

—

(k°k°~ Vtu°ii°){vMu$(1dM 12_ 1cM 12) + (v~ u~XdMë + dy,. )} —

(2.33)

M>O

has a coherent sum as the bulk part. Therefore the last term in eq. (2.29) is neglected. The second part in the curly bracket of (2.33) is small by the same arguments as used for the imaginary part. The first part exhibits a reduction of the pairing gap due to excitation of about 20% for the fission process. The second line in (2.29) vanishes, if the pair of differential equations 4

=

(~~t3~ — ukvk)expt2i

=

—(Ukl3k

—

UkVk)

J

4

exp(_2i

dr} dk

J

~

dr}

Ck

(2.34)

is fulfilled. This pair of equations is a key to the excitation mechanism of a nucleus during deformation studied in the following sections. For an essentially constant chemical potential, gap, and slope of ek the system (2.34) is identical to the Landau—Zener problem studied in appendix A. In eq. (2.34) the jump occurs over the gap 2 Re 4. -

3. Cranking model

3.1. The single particle potential In this and the following sections the selfconsistent TDHF approach is replaced by a cranking or squeezing model. This consists of a deforming s.p. potential e.g. a Saxon—Woods [6) or a folded Yukawa potential [521 in which nucleons move about interacting via a pairing force. Figs. I and 2 show the s.p.

spectrum of such a folded Yukawa potential. The time dependent Schrodinger equation will then determine the wave function of that system. No special approximation as e.g. the adiabatic approximation for the solution of this equation is here implied in the name “cranking model”. It is natural to think of this s.p. potential as an approximation to the selfconsistent potential of the preceding section. One could in principle choose a monotonically increasing expectation value of the s.c. solution a(t), invert this function and insert it into the selfconsistent potential U(t(a)). This is certainly possible for any initial condition. If in a cranking model calculation the nucleus starts in these initial conditions the wave function at any time t will be just the same as that obtained in a selfconsistent calculation. Nevertheless, the physical interpretation of terms in the total energy is different in the two approaches. So, as shown in section 6 part of the wave function describes the collective deformation of the nucleus with a certain velocity a. If. therefore, the selfconsistent potential is calculated with this wave function it evidently depends on the deformation velocity. This dependence has submerged into a pure deformation dependence by the above mentioned parametrization. The conclusions of the following study of the cranking model are independent of the exact definition of the s.p. potential so that one could also start with “the best” s.p. potential not depending on the actual collective deformation velocity. Using this potential in the time dependent Schrödinger equation

124

0. Schüne. Quantum mechanical descsiptson of the intnr,sic excitation offissioning nuclei

236U

Neutrons

4

-6~~’~]~

~

-8’

;~I

~

—4 -io~ 1.5

/

—.-..--.‘--.

1.9

2.3

R/R 0 Fig. 2. Neutron levels in analogy

to 11g.

I.

yields a wave function depending on the actual deformation velocity as shown in section 6 and which will yield a velocity dependent s.p. potential. With it the time dependent Schrödinger equation must in principle be solved again etc. until selfconsistency is achieved. In the following sections it will become clear that even the first step of such an iteration the cranking model yields valuable insight into the excitation mechanism during collective deformation. —

—

3.2. Independent quasi-particles

In such a cranking approach the time dependent Sthrodingerequatioñ (X—ia,).fr=0

(3.1)

must be solved with a Hamiltonian for independent quasi-particles

~W E0+ ~ e1(aa1.+aa,).

(3.2)

The BcS-energies e1.=V(e1._A)2+z12

(3.3)

0. &hüae. Quantum mechanical description of the intrinsic excitation of fissioning nuclei

125

depend on the deformation via the chemical potential A. the gap parameter 4 and the s.p. energies e~. These are eigenvalues of an approximate selfconsistent s.p. Hamiltonian h as discussed above. E0 is the lowest energy of a nuclear state compatible with the actual deformation. The qp-operators are defined in terms of static s.p. operators creating or annihilating the static eigenstates of h:

a

u~a

=

—

v,a0,

a = u~a + ~

(3.4)

The inverse transformation is given by =

u,.a

+

v~a0.

(3.5)

The coefficients are given by 2/~ (36) tv~J 2” V(e~_A)2+4 The occupation probability v~tends to one for states far below A and to zero for states well outside the Fermi sea. The opposite trend is shown by u~.In this simplest BCS-version [47]the gap parameter 4 is assumed to be independent of the state v. In the calculation in section 8 the static deformation dependent 4 is used. A sclfconsistcntly defined gap parameter is probably about 20% smaller (see eq. (2.33)). The real BCS-ground state at each deformation is given by

fl (ii,.

IBCS> =

—

v,,a ~a

) 0)

(3.7)

V>0

with the bare vacuum 0). The state (3.7) defines the vacuum of the qp-operators a~.At each deformation the Hamiltonian (3.2) defines a complete basis set. It can be created out of the BCS-ground state by applying any even number of qp-operators:

lv”

v 2k)a’

..a~IBCS).

The special states I~v>or Ii~vt

...

(3.8)

i~v~) are called single and multiple pair excitations respectively.

Numbering the states (3.8) by In) and denoting the corresponding eigenvalues (which are sums of e.,) by E~the wave function ifr can be expanded at each deformation a in the form =

a~exp[—i

J

E~d’r}In).

(3.9)

If this expansion is inserted into the Schrödinger equation (3.1) the system of equations =

—~

J

i(nI8,Im) exp{i (E~

—

E~n)dr}am

(3.10)

is obtained, lithe Hamiltonian in the Schrödinger equation differs from the one defining the basis by a

12h

0. &häue. Quantum mechanical description ofthe intrinsic excitation of fi.ssioning nuclei

residual interaction V eqs. (3.10) must be replaced by the system

= ~

[—i(n~ê,~m) + (n( VrI,n)1 exp{i

J

(En

—

Em) d~r}am.

(3.11)

In this section independent quasi-particles described by (3.10) are studied. Residual interactions will be incorporated in section 9. 3.3. Coupling matrix elements

The operator 8, can create or annihilate two quasi-particles out of any excited state (3.8) or it can change one quasi-particle to another level. The corresponding matrix elements of 8, can be calculated with the help of the identity =

(á.,v~—u,.i3,)a

+ VVvM)

0— ~(uvuM

(,.~It9,Iv)aM~—~(uVvM vvun)(~Id,Iv)aa. —

(3.12)

The s.p. matrix elements are real, because 8, is diagonal in the magnetic quantum number. Because the states depend on time via a time dependent,,parameter a, the 8, can be substituted by á8,,. These elements are [57):

(1v’1c9,,I0) = —(vi’Ia,,IO) = (uVvV. UV.VV)(v18,,Iv) —

,

(vvI8~,l0)a(UVVV— ~

,

for v v’

(er—)

(/~l/22I8d,I3’I1~2) = 5MIs.,(U$2Uv~+

(v~I8~,I/i,L) = (v~tl8..lir’).

(3.l3a)

2e~

(3.13b)

v~ivv~)(/.L2I 8aI ~2)+8$,~(uMIuVt+ v~,v,.~)(~iI8,,Ivi)

(3.13c)

(3.13d)

The elements (3.13a) create two arbitrary quasi-particles corresponding essentially to particle—hole excitations as seen from the combination of u,. and V~in (3.13a). The second type (3.l3b) creates one pair and in (3.13c) one particle or one hole is changed. In the special case v2 = i~this last group breaks a pair 11V5 which may have been excited earlier during the deformation. Matrix elements containing more than two quasi-particles reduce to those given in (3.13) due to the gap parameter being state independent, e.g.

(nmiv’I8alnm)= (iv’I8~40).

(3.14)

3.4. Harmonic oscillator It is convenient and illuminating to have a simple model, in which s.p. energies and matrix elements can be calculated. As such the deformed axially symmetric harmonic oscillator is used which is characterized by the oscillator constants w~and w~.Due to volume conservation (3.15)

0. Scleuue. Quantum mechanicaldescription of the intrinsic excitation offissioning nuclei

127

the deformation can be characterized by one parameter (3.16) An axis ratio w~/w2of two characteristics of the deformation of fission isomers corresponds to a = 1.6. The outer saddle of a uranium nucleus corresponds to a = 1.8. The single particle wave functions can be characterized by the number of nodes n. and n,, in the z-direction and2-term in theisperpendicular direction included and therefore respectively and the magnetic quantum number. No spin-orbit force or l the spin degree of freedom is dropped. The s.p. energies are given by eV=(nZ+

2)+wOVa(2n9+ImI+1).

(3.17)

In this spectrum there are no pseudo-crossings between s.p. levels. Hence, the eigenstates of the harmonic oscillator are identical with the diabatic basis. For more general s.p. potentials there are pseudo-crossings and the diabatic basis must be defined such that these pseudo-crossings are changed to crossings. For this basis the deformed harmonic oscillator may be a reasonable approximation to estimate the orders of magnitude. In section 8 it is shown how to treat a realistic case. The s.p. matrix elements entering (3.13) are (nn,m’I8,,ln~n0m)=

Smm(ö

+

;[on;.n,+2V’n~(n: 1)

c~n.,i:~2Vn:(Pz:—

I))

S .,,[S~,~,_1Vn9(n0+ Imt) Sn,.,i;_t’V’n~(n,~+ Im I)1}. —

(3.18)

These matrix elements depend very weakly on the deformation, in the deformation internal correspondingto the passage from saddle to scission they change by about 20%. At saddle point deformations the Fermi energy is crossed from above by s.p. levels with n~about 10. Such a level is coupled e.g. to a level with n5 = 8. The matrix element (3.18) is about 1.2. The slope of the s.p. energy e~for a state p = (n. = 8, n. = 0, m = I) at a = 2is —9(MeV/unita). In fig. 3 the matrix element (3. 13a) for the 2qp-excitation with the above ii and with v’ = (n~= 10, n~= 0, in = 1) is shown as a function of a. A constant gap parameter 4 = 0.76 MeV is used. It is assumed that the constant chemical potential A is crossed by e~at a = 2. The matrix element attains its maximum value at the deformation with e~= —e~~ it changes more rapidly at the deformations where eV and e~cross the Fermi energy. This small andsmooth matrix element is compared to the pair creating matrix element (3.13b). This has a pronounced maximum of e’/24 6 at a = 2 where eV crosses the Fermi energy. It drops to half its value over an interval a=4/e~~ss0.08.

(3.19)

The energy differences between the ground state and the 2qp-state I~v’)and between the ground state and the pair excitation I~v)are shown in fig. 4. The smaller energy difference to the pair excitation also favours the excitation of these states. It is the strong time dependence of the matrix element and the pair excitation energy which causes the strong excitation of the nucleus and the break down of the adiabatic approximation for these pair excitations. It will be realized in subsection 4.3 that the analytic properties of these transitions give a proof of this statement independent of any estimates by purely analytical arguments.

128

0. Schütte. Quantum mechanical description of the intrinsic excitation offissioning nuclei

i:

~: ~

1.8

2.0

1.9

2.2 ~

2.1

2.3

Fig, 3. Matns elements for pair excitation and for pair breaking.

:

1.8

II

1.9

2.0

2.1

2.2 ~

2.3

Fig. 4. Energy difference of the ground state to a pair excitation and a broken pair state as a function of deformation.

4. Zeroth order solution 4.1. HiTher: space and form of the solution Due to the estimates of subsection 3.4 it is natural to incorporate in the lowest order solution of the Schrödinger eq. (3.1) only pair excitations. The projection onto that Hilbert space spanned by the states

IBCS>,

ii’),

Iiiv,~2v2),

(4.1)

. . .

is denoted by P0. In other words, the Schrödinger equation —

ia)Poip

=

0

(4.2)

is solved in this section. The solution can be written in terms of particle creation operators [581: =

fl>1) (U~

—

V1,aa~)I0)

(4.3)

i’

or in terms of qp-operators =

fl (d~+ c~exp(_2i 0

V>

J 0

e~dT)aa)IBCS).

(44)

The superscript h standing for homogenous will become meaningful in the following section. The amplitudes CV and d~in (4.4) are connected to 1.J~and V~by

=

VV

=

d~u~ + vVc~exp(_2i

dVVV

—

uVcV exp(_2i

I I

EV

dr) (4.5)

EV

dr).

0. Schüae. Quantum mechanical description of the intrinsic excitation offissioning nuclei

129

The wave function (4.4) has components of the BCS-ground state and single and multiple pair excitations. The total probability that a pair ëc~~ is present in any of these excitations is Ic~I. In an independent particle picture (4.4) corresponds to a wave function which contains 2p.-2h. 4p—4h etc. components relative to the static “ground state” determinant at each deformation. Whenever a s.p. energy e~crosses the Fermi energy from below two particles are added to the nucleus with amplitude c~, and whenever a s.p. energy dives into the Fermi sea two holes are created. On the average as many s.p. levels should leave the Fermi sea as enter it, because of the volume conservation of the total nucleus. 4.2. Prooffor the solution In order to prove that (4.4) is a solution of eq. (4.2) one has to insert the various parts of P0 according to (4.1) on the far left side of this equation. E.g., the projection onto the BCS ground state leads to the condition —

~ d;’[d~ + (018,I,5p) c,, exp(_2i

I

~

di.)]

0.

(4.6)

Projection onto an arbitrary pair excitation (ëo’l gives

c~,,exp(_2i

+

I

~

c,, exp(_2i

di.) + d,,(t*rja,1O)

J

e~di.)

~

d~

+ c~cxp(_2i

J

di.) (oIaiI~5p)]=0.

(4.7)

Analogous conditions are obtained by projecting on multiple pair excitations. All these conditions are fulfilled, if the amplitudes c,, and 4, obey the set of pairs of differential equations

ci,,

J

=

—(óirJt94O) exp{2i

=

—(0f8,jôrr) exp{_2i

e,, dr} 4,

j~ e~,dr} ce,.

(4.8)

Multiplying c, by a constant phase

ë~r=c~rexp{_2t Je.rdr}

(49)

shifts the lower bound of the time-integrals in (4.8) to 1,, when e0. crosses A. Inserting the matrix element (3 13b) in eqs (4 8) shows the correspondence to the eq (2 34) in the self-consistent approach

130

0. &hsiue. Quantum mechanical description ofthe intrinsic excitation offissioning nuclei

4.3. Correspondence to the Landau—Zener problem The shift in phase (4.9) brings (4.8) into the form of the Landau—Zener problem discussed in appendix A with the following correspondence e2— e1 4

=

=

2(e0,

—

A)

V.

(4.10)

Therefore, the amplitudes c0. and d,2has arenoone-valued analyticoffunctions of thevelocity two variables positive powers the collective a, if it ise~/e,~á writtenand as Hence, the probability Ic~I a4/e,’,á. function of deformation a and deformation velocity a. If a, e,~.or 4 are not strictly constant, jc,,-J ~S determined by the values these quantities had at time 1,, because the coupling in (4.8) is localized around that time. If c,, is zero long before :,,. the jump probability Ic,12 tends to the asymptotic value 2

Icc

7l

/

D•42

exP~(A)?.)

(4.11)

where the velocity at t,, is denoted by a,,. Accordingly, the excitation described by the wave function (4.4) proceeds as follows. The nucleus starts in the BCS-ground state just beyond theluter saddle, i.e. all the c’s are zero. At the first crossing a1 of a s.p. energy with the chemical potential the corresponding 1-pair excitation gets occupied with amplitude c1. At the crossing of a second s.p. energy with A another 1-pair state is excited with amplitude c2d1 at the same time the 2-pair state (1122) is occupied with amplitude c1 c2. This process Continues to the scission point or somewhat before where the s.p. energies become constant as a function of the relative distance between the two nascent fragments. At this time the excitation mechanism described above is no longer effective. 4.4. The energy expression In the spirit of the cranking model [591the expectation value of the Hamiltonian (3.2) is taken as the total energy E

=

(*IXJ~c)

=

2 ~ EVIcVI

+

E0.

(4.12)

The “ground state energy” E0 is the expectation value of the Hamiltonian with the state of lowest energy at each deformation. The probabilities Ic~I contain only negative powers of a. When the scission point is approached most of the times tV2.lie in shows the farexplicitly past andthe the non-analytic asymptotic expression be This dependence(4.11) on a.can Since inserted for most of the probabilities IcVI the qp-energies e,, do not depend on the velocity a, the energy (4.12) does not contain any term which could be interpreted as collective kinetic energy. In a self consistent theory the total energy is given by expression (2.13). The self consistent s.p. Hamiltonian depends on the amplitudes C~and 4,. Therefore, no analytic dependence on the actual velocity is induced in the qp-energies in the approximation where all the s.p.-matrix elements of 8, have been neglected. Even if s.p. couplings at s.p.-pseudo crossings

0. &hüne. Quantum mechanical description of the intnnsic e.xcitatzon offissioning nuclei

131

were taken into account, some more Landau—Zener transitions would occur giving rise to a corresponding non-analytic dependence on the velocity with which this pseudo-crossing was passed. On the other hand, not to find a term in the total energy which can be interpreted as a collective kinetic energy is disturbing, not because it contradicts the adiabatic approximation but because it is in conflict with basic physical intuition. First, for larger systems which ultimately could reach macroscopic dimensions such a term must emerge due to the correspondence principle. Secondly, for a fissioning nucleus the collective kinetic energy at the scission point is known: one half the reduced mass times the square of the relative velocity. It is highly unplausible that such a term should not be found before the scission point even though it might be negligibly small. These arguments apply for any hypothetical nuclei with almost arbitrary interaction. Therefore, missing the collective kinetic energy cannot be explained by the neglect of residual interactions. Hence, the collective kinetic energy must be described by the 8,-coupling between s.p. states which have eigenvalues separated by about 2hw.

S. Broken pairs In this section the a,-coupling between diahatic single particle states is included in solving the time dependent Schrödinger equation with the Hamiltonian (3.2). As argued above, only this coupling can finally yield a term in the energy which can be identified with the collective kinetic energy. For the qp-picture this means including arbitrary qp-excitations in which the quasi-particles do not occupy time reversed static states. These states will be called broken pairs. In a TDHF calculation this coupling is included in the TDHF s.p. wave functions. This observation also shows that “broken pairs” in the static basis do not imply “broken pairs” in every dynamically defined basis, because it was shown in section 2 that a TDI-IFB calculation always yields a completely paired state in the s.p. basis defined by (2.20). 5.1. Extended Hubert space The neglect of the 8,-coupling between s.p. states led to the restriction of the Hilbert space to the Po space. In addition, there are, of course, states of the form (5.1)

IfllflVIVI...VkVk)

where n ill and k may be zero. For arbitrary pairs i~...v~the projection operator onto the space spanned by the states (5.1) is denoted by Pnm. The states (5.1) have the same structure as the states (4.1) expect that the BCS-ground state is substituted by the broken pair state mm). Analogous projection operators ~ are defined. All these projection operators commute with each other and with the Hamiltonian. For an arbitrary wave function ~s the components in the subspaces are denoted by *o=P0tP

~

etc.

(5.2)

52. Projection formalism

Projecting the time dependent Schrödinger equation (W—i8,)ifr=0

(5.3)

0. Schütte. Quantum mechanical description of the intrinsic eicitaiion offissioning nuclei

132

onto the various subspaces gives P0(X i81).4’o = iP0 ~

(5.4a)

—

P,, (X

—

= ~Pnmlfio +

~

Pm,,ç1~i+ i ~ P,,,,,II’,,,,k,.

~

(5.4b)

,~n.m

For the low energy fission considered here it is always assumed that the nucleus starts on top of the outer barrier or just beyond essentially in the lowest possible state, i.e. in the state IBCS). In this case two pair breaking matrix elements (3.13a) are needed to produce a component ~4’nmkt.Since this consideration is limited to the first order these terms are neglected in (5.4b). Similarly in the second term on the rhs of this equation. first the two quasi-particles ki must be created by a matrix element (3.13a) and secondly either n, m must be produced by the 8,-operator (the dot on *kt) or k, I must be changed to n, m by an element (3.13c). Hence this term is of second order, too, and is neglected. The remaining equations —

i8,)*o = i!’0 ~ ifr,,,,,

(5.5a)

i8,).Ji~.~ = IPnntIJ(p

(5.5b)

Pnm(~(—

arc read as inhomogeneous equations for ~su and ‘j’~,,,respectively. The homogeneous part of (5.5a) is identical to (4.2) and has been solved in section 4. The homogeneous equation P.,,,, (Z i8,)*,,,,, —

=

0

(5.6)

has the very same structure as (4.2) except that the ground state is replaced by mm). The solution is consequently given by

=

exp{_i

J

cr.,

jj

+ ~m)di.}

0

V

n.m

(ci;’~+ c~exP{_2i

J

e~di.) a~tcr;)inm>.

(5.7)

0

m and dm must all fulfill the same The superscript stands homogeneous. The specialized amplitudes toc the case ii’ = v. However, the initial differential eqs. h(4.8) due for to the relation (3.14) conditions could in principle result in amplitudes depending on n and m. (For the special initial conditions mentioned above, however, the amplitudes will be independent of n and in.) For each 2qp state eq. (5.6) has a complete set of solutions in the sense P.,.,,

=

~

Icb~m(j)Xcfr~m(j)L

(5.8)

They can, e.g., be generated in the form (5.7) by starting with any sample of c~,,mequal to unity at time zero the other c’s being zero. The corresponding pair excitation in which ~m(~) starts is abbreviated by i. The case with no pairs excited is denoted by i = 0. The solutions are orthogonal for different i. For a proof one has to notice that the overlap ( m(j)I~it~m(j))is constant due to eq. (5.6) and that it vanishes at the start by construction. Analogously, P 0 can be written in the form (5.8) in terms of i~c~(i).

0. &hüae. Quantum med,anical description of the intrinsic excilasion offlrnoning nuclei

133

Any solution of the inhomogeneous eq. (5.5b) can be written in the form

=

~

—

I

(P~m(~)k&o) dt’} *~im(~)

(5.9)

as can be verified by insertion. The constants A’.,,,, have finally to be determined by the initial conditions. For the above mentioned initial conditions they will be chosen to be of first order or zero. An expression similar to (5.9) holds for ~‘~:

=~

{A’O_J

~ (ul1~(J)k&m)df’}~(/).

(5.10)

Eqs. (5.9) and (5.10) can be iterated by inserting ~, A~i/s~(j) on the rhs of (5.9) for *ii inserting the resulting ~~nmback into (5.10). This yields a better approximation for ç(i~to be put into (5.9) etc. The first correction term in i/,,~thus obtained is of second order. It is therefore sufficient to insert the lowest order ~ for tI’~in (5.9).

5.3. Initial conditions

It is assumed that the nucleus starts its fatal deformation process at the outer saddle without any intrinsic excitation neither single particle nor collective vibrations. Since the pair excitations were shown to contain intrinsic energy only, coefficients A’ do not enter into the description of the kinetic energy, if the nucleus starts in the state IBCS) at the saddle. Therefore the initial conditions give A~A~,m0

forj~’0,

A~to be of zero order, and ~

(5.11) of first order. As stated above the amplitudes ~ and d in ~O) and

11th,,,,, (0) have initial values c~=0 d~=1 c~m=0 d~m=1.

(5.12)

Because the differential eqs. (4.8) for these amplitudes are independent of n and m, the amplitudes themselves are independent of the superscript. The initial normalization

IA?12 +

~ IA.imI2 = 1

(5.13)

shows that A~deviates from 1 by second order terms. As the initial conditions are chosen the functions ./i~(i) and 11’~,,(i)for i 0 do not come into play. For this to be true the overlap (~m(i)I11t~(0)) and its analogue in (5.10) must vanish. For a proof one has to realize that 11’~m(1)for i 0 and i~~(0) differ by the 2qp-excitation nm and by at least one pair. The s.p. operator ö~can consequently not couple the two states.

134

0. Schüae. Quantum mechanical description of the intrinsic excitation offissioning nuclei

5.4. Solution The wave function finally attains the form =

~‘J

{A0

(11i~I~nm) dt’}

+

~

fA~m

~‘

-

J

(~mI~)dt’}11,~m.

(5.14)

The sum over n and m does not include any pair states. The index 0 characterizing the initial conditions has been dropped. The norm of this wave function is constant. It solves the time dependent Schrödinger eq. (3.1) for independent quasi-particles. The excitation of the nucleus due to the crossings at the Fermi energy has been treated exactly while the coupling between diabatic states separated by larger energy differences is incorporated in first order. 5.5. Total energy

The expectation value of the Hamiltonian is through second order

(11’1XI11) +

~‘

n.m

=

E,, + A,,,,,

—

2

~

J

2r,,Ic,.1

I’~) di’f

(11’~...

2

[r,,(I

—

21c,, j2

+

r,,, (I

—

21Cm 12)1

(5.15)

.

0

Any 2qp-excitation must he counted only once. From section 4 it is evident that any term proportional to the square of the deformation velocity must be contained in the absolute square in the second line; all other terms have a completely different dependence on the velocity. The overlap integral can be calculated explicitly: = [ci..ci.~exp{i

J

(e,,

+ Em)

J

di.) + C~Cm eXP{_i (e,,

+ Em)

di.)] (nmj8,IBCS)

+ [~mdn exp{iJfr~_E~.)di.}_C~dmexp{iJ(en _em)dr)](nmI8:Ithnl).

(5.16)

The magnetic quantum numbers in n and in must be reversed, otherwise the overlap vanishes. The symmetry relation (3.l3d) has been used. 6. Adiabatic approximation for broken pairs It has been shown in subsection 3.4 that s.p. levels underlying the 2qp-excitations occunng in (5.16) arc separated by 2Iiw~or 2hw~in the case of a harmonic oscillator. It is assumed that these orders of magnitude also hold for a “realistic diabatic’ basis. It is obvious how to change from two adiabatic

0. Schüne. Quantum mechanical description of the intrinsic excitation of fissioning nuclei

135

states of the TDHF-Hamiltonian which show a narrow pseudo-crossing to two diabatic states which cross and which do not change their nodal character at the pseudo-crossing. It is, however, not obvious and in fact unknown how to define the “best” diabatic basis in the general case. Therefore, for the following considerations the deforming harmonic oscillator is taken as a guide-line. Consequently, the first line in (5.16) has rapidly oscillating phases and partial integration yields a good approximation. In the second line the phase is stationary where e,, equals e,,, which occurs for e,, = —em. From that deformation the energy difference E., ,,, increases to several MeV. Straightforward partial integration would lead to a pole term which shows that this adiabatic approximation fails. It will, however, be shown in subsection 6.3 that the matrix element in the second line of (5.16) can be renormalized in such a way that partial integration of the phase does not lead to a pole term. —

6.1. At the start

First, the region close to the start is considered when no crossing at the Fermi-surface has been passed and consequently all c’s are equal to zero and all d’s equal to one. Only the first term in (5.16) contributes:

J(ti1~mlc~)dTa

71!~o~Iexp{iJ(e,, +~m)dT}_~

~

(6.1)

Since no confusion will be possible the state IBCS) is denoted by 10) for brevity. The initial conditions are chosen such that the coefficients Anm

~~

(6.2)

cancel the last term of (6.1) in the expression (5.15). The conditions (6.2) are identical to the initial conditions usually used in the Inglis cranking model [591.They are interpreted as describing the finite deformation velocity for the nucleus at the start. In accord, the total energy (5.15) has the form

(11,I5ti~i’)= E() +

a2.

(6.3)

The second term looks like a collective kinetic energy. It differs from the Inglis formula [e.g. 6~,because the pair excitations do not contribute, indicated by the prime on the summation sign. The reason for this is the form of the non analytic dependence on a of the amplitudes of pair excitations discussed in section 4 and appendix A. 6.2. At a crossing: The harmless par:

When : and the deformation increase one crossing after another of s.p. levels with the Fermi energy is passed. The corresponding amplitudes become different from zero and the other terms in (5.16) come into play. In this subsection the effect of having passed a crossing n is considered. The diabatic states m

0. Schürte. Quantum mechanical description of the intrinsic excitation offissioning nuclei

136

which are coupled by a, to this state n must have a similar nodal structure and hence their energies have a similar slope. These energies lie about 2hw apart. That is why the deformations a, and a,,, where those energies cross the Fermi energy lie far apart. The estimate for the deformed harmonic oscillator shows that the difference between a,, and am is larger than the entire interval from saddle to scission. Therefore, cm and thus the second term in (5.16) always vanish and in the second line only the first term contributes. As a consequence, it may be assumed that c., and d, have attained their asymptotic values while c,, vanishes and d,,, is equal to unity. The integral over (5.16)

/

~ I11~)dt’ = +

J

c,,(nm

J

d,,(nm I8,.I0) exp(i

Ia,.Irnm

exp{_iJ (e,,

—

J

(e,,

Em)

+ Em)

di.]

di.) dt’

(6.4)

then consists of two parts. In this subsection the first term in (6.4) is discussed. At a,, the BCS-amplitudes u,,, v,, and the amplitude d,, change abruptly. Therefore, a partial integration of the phase over the entire interval is questionable (see eq. (6.5) for the form of the correction term). But the interval (0, t) can be~Jividedinto a part (0, r2) containing the deformation a,, with the more rapid variation of the amplitudes and a part (r2, t) in which these amplitudes are smooth (see fig. 3). The integrand over the first subinterval depends, of course, only on deformations and velocities which occurred before ~2. It does not depend on the actual velocity. Therefore, it does not contribute to the kinetic energy. The integral over the later subinterval is

J

d,,d (nmla~J0)exp{iJ(e9+ e~)di.]dt’

d,

—

‘~°~

J~

[i,,

~

exp{i

/

(e,,

+ Em)

exp{i

J

di.) d., ~ —

(e,,

+ Em)

di.) dt’.

I

exp{i

J

(e.,

+ Em)

di.)

(6.5)

The first term is retained in the usual adiabatic approximation and gives rise to a dependence on the actual velocity a while the second term depends on the velocity at ~2. The last term is the correction term. It must be small compared to the lhs of (6.5). This can be seen using the numbers of subsection 3.4. If the deformation a,, comes sufficiently late after the start, the first interval (0, ‘r2) can be subdivided by r~such that all the amplitudes are smooth in the interval (0, r1) and the crossing at a, is contained in (r,, r2). Partial integration is then justified in the first interval yielding a form similar to (6.5). The term corresponding to the second term on the rhs of (6.5) is in this case cancelled by the choice (6.2) of the initial conditions A,,,,,.

0. Schüae. Quantum mechanical description of the intrinsic excitation of fissioning nuclei

137

6.3. Healing of the crossing Performing the partial integration in the second term of (6.4) in analogy to the preceding subsection leads to a calamity at the deformation where e, and Em cross. This happens, if the s.p. energies lie below and above the Fermi energy by the same amount: e, A = —(em A). Only after the renormalization described below the partial integration and hence the adiabatic approximation can be justified. The matrix element at this deformation is denoted by —

Mnm

=

—

(nmlö,lñim)L,...f,,.

(6.6)

With a deformation velocity of about 0.1 MeV eqs. (3.13c) and (3.18) give M,,,, ~0.02 MeV, if the states n and m differ by the number of nodes in the z-direction. For the states differing in the quantum number n,, the matrix element is about 0.01 MeV. If this small and constant coupling between the two crossing states is treated in all orders in the time dependent Schrodinger equation the result is a Landau—Zener transition which depends on the local velocity. The nucleon follows the static diabatic qp-level with the “jump probability” ,

=

~

~

(r.,

M ,im —

Em)a

)

(6.7)

nsO.99

using the numbers of a deforming harmonic oscillator. Therefore, one can view the diabatic basis as a very good approximation to a time dependent basis in which the coupling M,,,,, is included. What remains to be treated in first order time dependent perturbation theory is then a renormalized matrix element (nmla,lthm) M,,,,,. Proceeding now exactly as in the preceding subsection the integral over the last subinterval (72. t) of the second term in (6.4) can be rewritten as —

/

c,((nm Ia,lthm) Mm~)exp{_i —

=

—c,a (nm Ia,,Ithrn)— M,,,, x exp{—i

J

(e,

—

J

(C,

—

/

Em)

exp~—i (e,

Em)di.} +

J

di.) di.’

—

e,,,

+ c,,a (nm Io,I,nm)

~ [c.. (~~m1~mm~M,m] exp(_i

M,,m

J

(e,

—

e~)di.) (it’. (6.8)

The pole in the first term has disappeared. Using de l’Hospital’s rule and the typical numbers of subsection 3.4 one obtains (nm Ia,I,nnz) M,,m —

i(e.,

—

Em)

—

0.004.

(6.9)

13$

0. Schüne. Quantum mechanical description of the intrinsic excitation offissioning nuclei

Summarizing, the integral is

(J+J+J)(~mki4)dt’_A~m

=

(nmla,l0) Em)

+

—

+

d, ~

d,,

1. 1 exp~Ij

‘1 (i,, + C,,,)

I exp(i J fr,, I, J exp{i

~

(e,,

dr

J+ Rnm(tn)

+ C,,,)

(6.10)

di.) + c~(nmf9(thm)—M,,m

+ Em) dr}

—

Cn (nmlt3tltñrn)—M,,m

exp(_i

exp{_i

J

fr,,

J —

(e,,

—

C,,,)

Em)

di.)

di.).

The integral over the interval (i.1, 72) is denoted by R,,.,,. It depends on the deformation and the deformation velocity at a,,. 6.4. Terms in the total energy In the total energy (5.15) the absolute square of (6.10) enters. All terms are quadratic in velocities but one has to distinguish the actual velocity and the velocities that occurred in the past especially at the level-crossings at the Fermi energy. Hence the total energy has the form E = E~+2

~

2+O(~~~)+ O(a,,â~).

E~Ic~t~+ ~Ba

(6.11)

Only the third part can be interpreted as collective kineticenergy. All other contributions depending on velocities a~must be intrinsic energy. The terms of order aa~and a,.a~will be seen to oscillate fast leading to a small amount of energy oscillating back and forth between collective and intrinsic degrees of freedom. If the energy is averaged over a time interval long compared to the period of this oscillation but small compared to the total time of the deformation process, these terms can be ignored. The total energy (6.11) is a sum of a potential energy, an intrinsic excitation energy and a collective kinetic energy in accord with physical intuitation. In the example in section 8 it is demonstrated that the intrinsic energy is large compared to the kinetic energy until the scission point is approached and the fragments slip apart without further mutual excitation except for Coulomb effects. The fundamental difference of (6.11) to the adiabatic approximation is the explicit appearance of intrinsic excitation energy. This difference is not a result of considering a more or less violent deformation process but comes from applying the adiabatic approximation to the Landau—Zener transitions. In the case of nuclear fission it could even be argued physically that the deformation velocity is either slow implying a small kinetic energy and, hence, by energyconservation a large intrinsic excitation energy or the velocity is large implying a violent deformation process. Both possibilities contradict the adiabatic assumptions of a nucleus deforming slowly essentially in the lowest state at each deformation.

0. &hütxe. Quantum mechanical description of the üunnsic excitation offissioning nuclei

139

6.5. Mass parameter

in the energy expression (6.11) is given by

The mass parameter B

B=

2

~‘

d,, (~~aI0) exp{i

/

(C,, + Em)

di.)

—

~,,

(nm

Iaalthrn)— Mm,, expl,I—i

/

—

Xfr,,(1—21c,,12)+gm).

E,,,)

dr}I 2 (6.12)

According to the estimates given in subsection 3.4 at least one s.p. energy e,, or em crosses A outside the interval (0, t). The terms in the last sum in (5.15) are numbered such that this is em. That is why formula (6.12) looks asymmetric in m and n. If a, lies outside the interval (0, t), too, c,, vanishes and expression (6.12) reduces to the last term in (6.3). The sum B

2 1=

2 n.m ~

(6.13)

C,, + C,,, (nml~,,I0)

over all unpaired 2qp-excitations is the mass parameter for a nucleus without excitation at any given deformation. It differs from the formula obtained in the adiabatic Inglis approximation by the exclusion of the pair excitations. Deviations from this value due to the excitation of the nucleus during the deformation follow from (6.12). One such correction term is

21 d,

~

X (C,, + Em

exp{i

—

/

.,

(C,, + Em) d:)

2E,,Ic,,11

_______ /

2

~,

—

(nn: Ia.~ñirn)—M,,m exp(_i

/

(C,,

—

Cm)

dr}l z

~

C,,

C,,,

.

(6.14)

For fixed n there are four values of m in the case of the harmonic oscillator contributing to (6.14). If reasonable numbers are inserted, the absolute squares of the individual terms in (6.14) for the four rn-values contribute about —0.0 15 MeV~to the mass parameter. In the total interval from the saddle point to the scission point about 20 neutron levels and 10 proton levels cross the Fermi energy. Hence, the decrease of the mass parameter due to these terms is about 0.5 MeV1. In addition, there are the cross products of the two terms in the absolute square in (6.14). They have an amplitude of about 0.1 MeV and varying sign. Therefore, they do not add up coherently and their total sum will be less than one MeV. As a result, the mass parameter changes only little as the deformation process proceeds. Since the mass parameter (6.13) has not yet been calculated I give here as a reference the value of 80 MeV’ for the irrotational incompressible flow for a uranium nucleus at saddle point deformations (see section 8).

144)

0. Schuue. Quantum mechanical description of the int,insic excitation of fissioning nuclei

7. Others’ adiabatic approximations In the light of the picture obtained in the previous section a comment on the adiabatic approximations to be found in the literature is in place. Two versions have been employed phrased in terms of the wave function and in terms of the s.p. density. These are the adiabatic cranking model [60] and the ATDHF [29—33]. 7.1. Adiabatic cranking model

Any self-consistent time dependent theory determines a time dependent average s.p. potential. In the cranking model this time dependence is parametrized by time dependent deformation parameters. This leads to a model Hamiltonian like in section 3 which depends on the deformation. The eigen states of ~k°define the adiabatic basis ~‘

TIn)

=

E,,In)

(7.1)

.

Expanding the wave function

~‘

amplitudes a,,

a,, =

—~

in this basis like in eq. (3.9) yields the Schrödinger equation for the

J

th(nIa~lm)exp{i (E,, E~~)d;) a,,, —

(7.2)

.

Iwo assumptions arc made to obtain the adiabatic approximation to the solution of (7.2). First, the amplitude of all excited states arc assumed to he so small that only the term containing the ground state amplitude a~in (7.2) must he retained which is put equal to unity. It is demonstrated in the example of section 8 that the ground state probability decreases to a few per cent during the fission process and that many states become occupied so that many terms in the sum (7.2) must be kept. Secondly, it is assumed that the velocity 6, the matrix elements and the energies E,, E~are essentially constant. Integrating the remaining term from (7.2) —

a,, = —a(nla,,I0) exp{i

J

(E,,

—

E0) di.)

(7.3)

then yields a,, (t)

=

~

exp{i

J

(E.,

—

E(,) di.)

—

c,,.

(7.4)

The initial conditions arc chosen such that the numbers c,, all vanish. It is noticed that the term (7.4) comes from a partial integration of the rhs of (7.3). The “correction term” is —

J

~

[i~hl~j~0)]

exp{i

J

(E,,

—

E,,) di.) dt’.

(7.5)

0. Schãne. Quantum mechanical description of the inmnsic excuanon offissioning nuclei

141

The energy in the cranking model is given by E = (~I4Xl4t)= E,)+ ~ (E,

—

E0)la,,12.

(7.6)

If (7.4) is inserted the total energy appears as the sum of a potential energy E 0 and a kinetic energy (7.7)

2 E=E(,+~Bâ

with the Inglis mass parameter (7.8) It has been argued that in the derivation given above, the ground state may be replaced by any excited state by starting the nucleus in the appropriate state. It may then happen that at some deformation E,, crosses or almost crosses an energy E 1. At that deformation the mass parameter (7.8) exhibits a pole or. if the crossing is avoided by a residual interaction, it becomes very large. To see that this behaviour is unphysical one can start with crossing levels E0 and E1 assuming that the matrix element (lIa,,IO) vanishes. This is the usual situation in a deforming harmonic oscillator. Of course, there is no pole in this case in B. Now a tiny interaction V. between the two states ~I) and 10) is switched on. This interaction can be diagonalized in the basis changing the states 10), ~l) into states Ilow) lup) and the energies (see appendix A) into E1.,... and ~ In this new adiabatic basis the energy denominator (E~~E1,,~)becomes very small, equal to 2 V at the deformation where E0 and E1 crosses. The matrix element (see eq. (A6)) at that deformation is (E~ E~)f4V where the prime denotes differentiation with respect to the deformation. The consequence is a spike in the collective mass parameter due to an arbitrarily small interaction between two nucleons. This strange feature is not caused by any physics but by the adiabatic approximation. The integral over the rhs of (7.3) is well behaved at a crossing or a close pseudo-crossing of E,, and E3. What is wrong is the assumption that the matrix element and the energy difference are constant and the main time dependence comes from the phase. The “correction term” (7.5) has the same pole or spike as the part which is kept, but this is thrown away in the adiabatic approximation. The result of the preceding sections and of the appendix is that the nucleus described by the time dependent Schrödinger equation of the cranking model will make a Landau—Zener transition over a close pseudo-crossing thereby acquiring some intrinsic excitation energy. From the analytic properties of this transition it is obvious that this energy does not contribute to the kinetic energy or to the mass parameter. The arguments given in the preceding section are, however, more general. They assure that no Landau—Zener transition contributes to the kinetic energy whether it occurs at a narrow or a wide pseudo-crossing. It was shown that the transitions over the pairing gap were of that nature. If the pairing gap decreases and finally vanishes due to the excitation as in a temperature dependent BCS-theory [47] the Inglis mass parameter becomes singular [61,621by exactly the reason given above: at pseudo-crossings the adiabatic approximation breaks down. —

—

7.2. Adiabatic time dependent Hariree—Fock Using the language of the s.p. density instead of wave functions two slightly different versions of the

142

0. Schuae. Quantum mechanical description ofthe intrinsic excitation of fissioning nuclei

adiabatic approximation to the TDHF approach have been formulated in refs. [281and [29]. In both, the s.p. density p is expanded around a static density pa: (7.9)

4~PI4P2~”.

P=PO

There are two expansion parameters which are treated on equal footing. One is a(d,,)/~iEwhich becomes small for small deformation velocities. The other parameter is the 49,-operator. (It should be noted that 49, is not a linear operator, because it also acts on time dependent coefficients of a linear combination of states.) It is believed that each application of 49, increases the order by one. If, however, a pseudo-crossing occurs increasing powers of 49, are applied for example on the energy denominator

(V(e,._~)2+j2)

=

(7.10)

At the pseudo-crossing this sequence diverges showing that for large amplitude deformations the 49,-operator does not add powers in a smallness parameter. The complete expansion in powers of the two expansion parameters is given in ref. [63]. The s.p. Hamiltonian W = t + tr Vp is expanded in analogy to (7.9) as W=W,,+W 1+W,

-.

(7.11)

by inserting (7.9) into tr~pV). In one version (ref. [281)the zeroth order is defined by IHhpj=0

(7.12)

where H~1differs from W’~by a constraining force. In the other version no constraining force is introduced and no zero order equation is considered. In both versions the commutator IWij,p~Iis treated as a second order term. Consequently, in a basis which diagonalizes Wu, the elements of pa differ from zero and unity by second order terms. In contrast, the zero order solution in section 4 leads to a s.p. density which contains the Landau—Zener transitions at the Fermi surface. These are not of second nor of any positive order. The difference to the ATDHF is (apart from the self-consistency) that in section 4 the zero-order solution is defined by a time dependent Schrödinger equation into which no assumption on the smallness of the excitation enters and not by an essentially static equation like (7.12).

8. An example: Fissioning ~‘U All the quantities in the energy expression (5.15) have a well defined quantum mechanical meaning: A constrained HF yields E,1, the4. mass givenapproximations in section 6 and the occupation 2arecalculation determined in section With parameter the help of issome to these quantities Ic~I calculation can be performed once the static s.p. energies are known. aprobabilities simple dynamical The collective potential can be calculated as the sum of a macroscopic energy, e.g.: a liquid drop energy or a folded “Yukawa plus exponential potential”, and shell corrections calculated for the s.p.

G. Schüue. Quantum mechanical de~cripri,rnof the intrinsic excitation offissioning nude,

143

spectra shown in figs. 1 and 2. The mass parameter given in section 6 does not have the violent fluctuations at pseudo-crossings but is a smooth function of the deformation. Therefore, an effective constant value is taken. It is chosen according to the mass of an incompressible irrotational flow [641 B

=~t ~

17

~

/ 128/R

3\\

(8.1)

The reduced mass ~ for a symmetrically fissioning nucleus is one quarter of the total mass. The distance R between the two halves of the nucleus is measured in units of the radius R,, of the united spherical nucleus. For a ~“U-nucleus B” decreases from 87 MeV~to 77 MeV~during the deformation from the saddle to the “scission point”. The s.p. potential is calculated by folding a Yukawa interaction with a constant s.p. density with a sharp surface. The deformation of that surface is described by three smoothly joined quadratic forms [SI). The path taken by the deforming nucleus is determined from a classical calculation with one body dissipation that fitted the mean kinetic energies of fission fragments of a ~“U-nucleus. The spectrum of this s.p. potential along this path is shown in figs. 1 and 2. According to section 4 the slope of these s.p. energies at the crossing with the Fermi energy determines the excitation of the nucleus. The intrinsic excitation energy should be calculated in the following way. For all s.p. levels crossing the Fermi energy between saddle and scission the pairs of differential eqs. (4.8) must be solved. The additional condition of energy conservation determines the velocity at each time step. Although the energy conservation couples all the pairs of differential equations the problem is much simplified compared to a full coupled channel calculation. The calculation can, however, he still further simplified because of three reasons: (I) For the s.p. levels with large slopes which obtain most of the excitation ener~ythe amplitude c,. rises from zero to its asymptotic value in a narrow interval &r. Therefore, the probabilities can be approximated by a step function. For s.p. levels less steep this approximation is less good but they are also less important. (II) The asymptotic value of lc~l2is determined by local values of the gap, the slope of the s.p. energyand the velocity a~at the pseudo-crossing of the qp-state with the Fermi energy. This is obvious from eq. (A6) because the coupling becomes weak when going away from a~,see fig. 5. (III) The s.p. spectrums of figs. 1 and 2 show that the s.p. levels cross the Fermi energy in groups. These features suggest the following scheme for a dynamical calculation. First, the total interval of the deformation (R 1, R,) is divided into sub-intervals by points R,, such that these points are separated from the steep crossings. Secondly, an interpolation formula for the velocity R between R,, and R,, is chosen, in this example a constant acceleration leading to +

=

+ RR.:~~k(I~+~ — ,~)]if2.

(8.2)

Hence, knowing R,, the velocity at R,, and guessing R,,÷,the velocity is fixed in the interval (R,,, R,,+1). Third, for any s.p. level crossing the Fermi energy in (R,,, R,,+1) the asymptotic occupation probability (4.1!) is calculated using the velocity at that crossing fixed by (8.2). The occupation probabilities determined in previous subintervals are kept constant. If the results are inserted into the expression (6.11) for the total energy at R,,+, it will in general differ from the energy at R,. Therefore, step three must be iterated until R,,+, is found so as to yield the original total energy at R,,+~.

144

0. Schuue, Quantum mechanical descnpnon of the intrinsic excitation of fissioning nuclei

-0.2

0

I

I

0.2

I

0.6

0).

I

I

~ I

..1OM.V

Microscopic Colculotion

l.a

____

L~.

0

1.6

I

-0.2

0

0.2

0.4

0.6

a

Fig, 5. Jump probabilities (upper purl) for a level crossing ots.p. levels the slopes of which differ by e 10MeV and e’ 5MeV. The interaction between the two i.p. states is 0.7MeV. The velocity a (lower part) is determined (rum energy conservation. The asymptotic Landau—Zener jump probability using the velocity at the crossing is indicated by the horizontal lines in the upper part.

8

20

22

24

26

28

R/R~ Fig. 6. Collective potential and kinetic energy and intrinsic eScitation energy for a flssioning ~U.nucIeus.

In this way the intrinsic energy of a fissioning 2~Unucleus was calculated in ref. [65]. Once the s.p. spectrum is known the dynamic calculation as described above can be performed on a desk calculator. The s.p. levels exhibit pseudo-crossings most of which have extreme jump probabilities for the velocity

obtained above. Only a few with jump probabilities between 0.2 and 0.8 had to be treated explicitly in the Landau—Zener approximation, i.e. splitting occupation probabilities between pair excitations. Fig. 6 shows the various terms of the total energy (6.11): The collective potential Et), the collective kinetic energy counted from E 0 upwards, and the intrinsic excitation energy. As long as there are crossings of s.p. levels with the Fermi energy the excitation is sufficiently large to keep the velocity and thereby the collective energy small. After the deformation of about R 2.2 is passed the s.p. levels flatten out as a function of the relative distance R between the nascent fragments. Therefore, the strong excitation seizes and only residual interactions could cause additional excitation. In the example given here there is one exceptional level for the neutrons and one for the protons. They cross the Fermi energy steeply shortly before the scission point. Due to the large velocity already attained by the nucleus they obtain a large occupation probability, about 0.5, and give rise to a last considerable increase of the intrinsic excitation energy. At the scission point 9.3 MeV has been dissipated into intrinsic neutron energy and 4.3 MeV into intrinsic proton energy. The average occupation probability of a pair excitation of neutrons is 0.!!, of protons 0.05. At the scission point the ground state is occupied with the probability 0.05. The probability for any k neutron

G. Schüne. Quantum mechanical description of the intrinsic excitation of fissioning nuclei

pairs being excited out of the total of n pairs which have an average occupation probability = (~)ici~ id~’~.

145

2

id

is (8.3)

The average number of excited neutron pairs is ~kpk =nici2

17~0.11=1.9

(8.4)

and for protons 9 0.05 = 0.45. .

Up to now no residual interaction was incorporated. This will certainly contribute to the intrinsic excitation energy by coupling the pair excitations to unpaired states. Transport theories [66, 671 show that the excitation due to residual forces is due to the increase of the density of states with increasing energy. In contrast, the excitation of a deforming nucleus at low energies is dominated by the direct transport of pairs of quasi-particles out of the Fermi sea into regions of higher excitation energy.

9. Extension of TDHF 9.!. The problem In the preceding sections a non-perturhative solution of the cranking model has been studied. It can be viewed as an approximate TDHFB calculation. No two-body correlations in the wave function were admitted. These can be incorporated, e.g. in the equations of motion for Green’s functions [39). The one-body Green’s function is coupled to the two-body Green’s function etc. This hierarchy can be terminated by a suitable factorization of the two-body or three-body Green’s function depending on the desired order of approximation. The lowest order is the TDHF solution. Approximating the two-body Green’s function by first order perturbation theory yields a collision term which gives a slight change in the s.p. potential. This time dependent potential defines a complete basis set of solutions of the time dependent single particle Schrödinger equation. The collision term couples these basis states. With the help of statistical assumptions on the matrix elements of the effective residual interaction a master equation for the occupation numbers of these basis states may be obtained. In ref. [40) it has been tried to follow this line but unfortunately the transition matrix of the master equation has been calculated for a completely time independent problem. The master equation has even been applied to an isolated level crossing which is certainly wrong [411. 9.2. Pair breaking by the residual interaction

In this section the residual interaction to more complicated unpaired states is treated in second order to estimate the lifetime of the simple “TDHFB-solution” obtained in the previous sections with the initial conditions given in subsection 5.3. This solution determines the self-consistent qp-Hamiltonian which is then brought into diagonal form and approximated by the cranking model Hamiltonian (3.2). This defines a basis set ii’,. by the equation ~‘

(X—i49,)~=0.

(9.1)

146

0. Schüne. Quantum mechanical descnption ofthe intrinsic excitation of fissioning nuclei

The pairing character, i.e. the difference of the s.p. densities with positive and negative quantum numbers, of ~ is constant, i.e.: it is fixed by the initial conditions. Especially. if a was in a paired state at time zero, it will remain paired at any time. (‘This statement does not contradict to sections 5 and 6, see below!) For a proof the particle number operators N~= aa~and N; = a~a~ for positive and negative magnetic quantum numbers of any s.p. state v are defined. Simple algebra shows

~

(9.2)

[~,N—N;1=0 which proves the above statement. The solution of (Y(+ V—i3,)t~r=0

(9.3)

with the residual interaction V can be expanded in the form * = ~ a,*1.

(9.4)

The amplitudes a, satisfy the Schrödinger equation ia5 = ~(kiVIl)ai.

(9.5)

The system (9.5) can be split into two coupled subsystems, one for the amplitudes of paired basis states ~ç of which the s.p. densities with positive and with negative magnetic quantum numbers are equal and one for the amplitudes of the states *~ for which these two densities are unequal: it’..

~

(*..l ~

)aM +

~

VItft~)a,,

(~i~,(

(9.6) ~Clfl=

~.:(4!1,,~VI*M)a5. + ~

VI*,n)(l,u.

(lflfl~

No matrix elements of e9, occur, as these are taken care of in the basis. Since the coupling to unpaired states comes from the residual interaction only, the a,-coupling does not cause any odd fragmentation in

the mass or charge distribution of a fissioning even—even nucleus. 9.3. Lifetime of static pair excitations

In sections 3 to 6 the eq. (9.1) has been solved for special initial conditions *~(0).The preceding subsection shows that the “broken pair states” introduced in section 5 do not imply a physically unpaired wave function: There exists a linear transformation, namely to the TDHF-s.p. functions, in which the paired character of the solution fr(t) is exhibited. The bulk part of s/i(t) was contained in the static single and multiple pair excitations. Only during the last moments before the scission point is passed the D,-coupling to static broken pair states became larger due to the large deformation velocities attained by the nucleus. Therefore, this coupling of 0, will be neglected in the following estimate of the lifetime of the paired component of the total wave function which is approximately equal to the lifetime due to residual interactions of the static pair excitations of section 4. In the following the static pair excitations are denoted by Greek letters I”) and broken pair excitations by Roman letters In). The time dependent Schrödinger eq. (3.11) can again be split into two

0. Schüae. Quantum mechanical description of the intrinsic excitation of fissioning nuclei

147

parts

iá~=

—

+ ~

i(v~0,~) exp(i

J —

(E~ EM) dr)a~.

[—i(v~0,~n) + (vI V)n)) exp(i

iá.~ ~ [—i(nIa,~)+ (ni v~)1exp(i + ~

J

(E, E~)dr)an

J — J — (E,,

[—i(n~0,~,n) + (nl Vim)) exp(i

(9.7a)

—

E.)dr) aM

(E,,

Em) d’r) a,~.

(9.7b)

It is assumed that there is no first order residual interaction between the states Iv). As long as the paired states play the dominant role the amplitudes am are small. This, however, is not sufficient to neglect the last sum in (9.7b), because there are many more arbitrary qp-excitations than pair excitations. In addition, the sum must not ~omprise any coherent parts as function of n or in. The following argument can be made in favor of this assumption. Imagine the residual interaction V is diagonalized in the 0-space spanned by the states I~z).The resulting new state ~n) would be coupled by the 0,-operator. They are complicated linear combinations of simple quasiparticle excitations. Consequently, any matrix element of 0, is a complicated bilinear form in many amplitudes. For fixed ñ one can then expect that this form changes rapidly its sign as the state in is changed, because in should be a completely different linear combination of simple states due to the mixing by V. Still, this argument does not give a guarantee and it is a physical assumption that the last sum in (9.7b) is incoherent. On the ground of the above argument this part in (9.7b) is neglected. Because the 01-operator does not change the pairing character, the matrix elements (vIO,I,:) arc omitted in the following discussion. For the elements (vI V]n) a random sign is assumed for fixed v and varying n and vice versa in accord with the plausibility argument given above. The remaining equation is now integrated over time

a~(t)=—i

~J(ni Vliz)exp[i J

(E~ EM)dr} aM(t’)dt’.

(9.8)

—

Inserting into (9.7a) yields 4,,(t) =

—~

(vi0,I~)exp(i

J

— iJ ~ (vlVln), (nI

(E~ EM) dr} aM —

VIIL)1.exp(i[J (E~ E~)dr—J(En —

-

}

EM) dr] aM(t)dt.

(9.9)

14$

0. Schlitz,’. Quantum mechanical description of the intrinsic excUation of fissioning nuclei

According to the assumption of random sign of the elements (vI Vln) the sum over n is negligibly small, if v and jz are different. The diagonal term is large for t’ = t, because absolute squares are summed. For times t’ different from t the sum over n decreases, because of two possible reasons. First, the propagator gives a phase to each term. These phases lead to a decrease only, if the energies E~are randomly distributed. Otherwise, a periodicity will occur. Secondly, the matrix elements depend on time as indicated by the indices t and t’. because the states In) change their character rapidly due to the many crossings and pseudo-crossings of the qp-states. Again, the arguments are plausible. but one must be on guard that no collective states are formed by the residual interaction between the F- and the 0-space. These are just the physical manifestation of coherence effects in the matrix elements and the propagators. Especially, all the collective deformation modes of the fissioning nucleus must be contained in the P-space. Applying the statistical arguments given above to eq. (9.9). especially that the sum over n is a peaked function of t I’ so that the amplitude aM(l’) can he taken out of the integral over time at t’ = t, leads to: —

a,. = —~(:‘IaiI,.t)ex~{iJ(E1.— E~)dr}a~—~r,.a,.(t)

(9.10)

with

= ~

J

(a’~Vk), (ui Viv),. exp{_i

J -

(fi~, E,.) dr} dt’.

(9.11)

The width l~.can be estimated by suitably parametrizing the ingredients. The energies E~are sums of BCS-cnergies. After the state ~v) got occupied all these BCS-energies are rising functions of the deformation and hence of time. They are approximated by linear Functions (9.12) In contrast, the energies E,, contain increasing and decreasing BCS-energies so that they do not depend on time as strongly; they are essentially constants with some wiggles and, hence, are taken to he constant. A few MeV above the lowest state at each deformation the density p(E) of states In) coupled to a state Iv) becomes large enough to replace the sum over ii by an integral over E’. The statistical arguments can then be condensed to the parametrization

p(E’)(vIVIE’), (El VIv),’ 2p exp(_ V

~

(EI_E.;(l~fl2)exp(_ (t _e)2)

.

(9.13)

The first Gaussian shows the overall decrease of the residual coupling with increasing energy difference. D is the range of the interaction and is a few MeV. The strength V is about 0.1 MeV for low-lying

G. Schüne. Quantum mechanical description of the intrinsic excitation off&ssioning nuclei

149

states. This corresponds to the strength of the matrix element G in a static BCS-calculation in the actinide region [46). The expression V2p is proportional to the spreading width and depends only

weakly on energy. Estimating at low energies gives a value of about 0.1 MeV. The Gauss function of time in expression (9.13) accounts for the loss of coherence of the phases of the matrix elements for an increasing difference between t and t’. If (9.13) is inserted into the width (9.11) the integration over E’ leads to a Gaussian of (1— t’) with the width T defined by (9.14)

T2=~+~_+~,.

The first term is negligible, because f3 is 1 or 2 MeV according to the example of section S and D is a few MeV. The coherence time o- can be estimated to be 0.4 MeV~’from twice the distance between neighbouring crossings of one level with any other level obtained from figs. 1 or 2 divided by the typical deformation velocity 0.1 MeV. The resulting T’0.25 MeV’ justifies the Markov approximation taking aM(t) out of the time integral. The time integration finally yields r,.=iN2pDT.

(9.15)

Due to the neglect of the first term in (9.14) the decay rate 1”~is independent of v. Its numerical value for the neutrons is about 0.3 MeV. The lifetime of a pair excitation is therefore 2.2 X 1O~2lsec compared to the total time for the passage from saddle to scission of 7 X lO_21 sec. Using the average excitation probabilities Id2 from section 8 some combinotorics lead to the probability for finding the neutrons in any static paired state I dj2” l’(s—*i) 1 W= IcI2c~’~’~ IdI2e’~—

—

+IdI2N.

(9.16)

In this Formula it has been assumed that the times t~when the s.p. energies e~cross the Fermi energy are separated by the constant time interval i’. For the example presented in section 8 W is about 10% at the scission point and the “ground state” occupation probability is 5%.

Discussion It has been demonstrated that the excitation of a fissioning nucleus proceeds mainly via the Landau—Zener transitions over the pairing gap. The resulting pair excitations do not contain any part of the collective kinetic energy in the fission degree of freedom. The consequence is a decrease of the mass parameter compared to the “Inglis form”. For the rotation for which this moment of inertia was originally derived the intrinsic structure of the nucleus does not depend on the angle. Therefore, no pairs are excited in contrast to the case of deformations. It has been proved that for the nuclear deformation neither the pair excitation nor any pseudo-crossing contribute to the mass parameter. As a result the mass parameter is a smooth function of the deformation. The analytic dependence on the collective deformation velocity allowed for the first time the identification of terms in the expectation value of the Hamiltonian according to the classical picture: At

0. &hüne. Quantum mechanicaldescription of the intrinsic exciwAun of fissioning nuclei

150

each deformation the total energy is the sum of a potential energy, a collective kinetic energy, and intrinsic energy. Such a picture has already been used intuitively for the scattering of two oxygen nuclei [68].The numerical calculations performed were not able to identify the terms in the energy [69,701. It was shown that in principle these results can be carried over to a self-consistent formulation as in section 2. A more detailed analysis of the finite lifetime of the pair excitations and its consequences on the odd—even effects in nuclear fission [71—731 are currently under way. A preliminary estimate was given in [74]. In the time dependent theory applied to the fission process in this article initial conditions come into play. In subsection 5.3 the special example of a nucleus starting cold at the outer saddle point has been formulated. In this case the nucleus starts in the local BCS-ground state and any excitation compared to the initial state is inhibited by the pairing gap. If, in contrast, an initial state had been defined in subsection 5.3 containing quasi-particle excitations these parts of the wave function will rush to high excitation energies where they are statistically dissipated into the more complicated many quasi-particle excitations. This faster dissipation for excited nuclei containing pair excitations at the saddle will

certainly have a strong influence on the odd-even effects. It must be born in mind that in principle the “initial conditions” at the saddle are never experimental initial conditions, which lie in the far past not even mentioned in this review article. Therefore, a more complicated choice for the c’s and d’s in subsection 5.3 might be compulsory.

Acknowledgements The s.p. spectra used in this work have been calculated by P. Möller with the code of the Los Alamos group, and the author gratefully acknowledges his work. Also the discussions with R. Nix and H.A. Weidenmüller are appreciated. The stimulating collaboration with L. WHets on many aspects of the presented work was always a great joy for the author.

Appendix A. Two levels In the calculation of the excitation of a deforming nucleus and the interpretation of the various terms in the excitation energy extensive use was made of the Landau—Zener problem. Since more detailed features of this problem were used than are known among physicists a summary is given in this appendix. Especially, the analytic properties are studied and formulae for various initial conditions are

presented. Al. The problem Two levels e, and e2 with constant slopes as a function of deformation cross at time zero. The two corresponding states ~ and ~2 are coupled by a constant interaction V as shown in fig. 7. The deformation velocity a is constant. In this diabatic basis the time dependent Schrodinger equation is

=

VexP{iJ(ea_e2)dT}c2

i~2=Vexp{_iJ(ei_e2)dr}ci.

(Al)

0. Schüne. Quantum mechanical description of the intrinsic excitation offls.sioning nuclei

Fig. 7. Diahatic

(~)and

adiabatic

fr,)

151

levels lot thc Landau-Zener problem.

Diagonalization of the residual interaction V leads to the adiabatic basis consisting of an upper and

lower state p

cut,U~’i+Vcz,

1,,~vç1—tu~.

(A2)

2 + V2

(A3)

The eigenenergies are .

~(e1+ e2) ±V~(e1 e2) The coefficients =

—

v[~(i_

u=sign

e~—e

V(e

112

2

—.

2)]

2 e,) +4V

~)] e1) +4V 112

~ \/(e2

(A4)

~‘

have the asymptotic behaviour v—I—

2(e2—e1),,

v e IVI —

,

1—e2

a

U—

I I i

\

—

e2—e1 2 V

fort—’—~ \.

~ )

sign V

for t—. +~.

(A5)

2(e2—e,)j

The Schrödinger eq. (Al) transforms to =

C1

a ~exp(i

-a

~

I exp(_i

~

f

— (t~

ei) dr)

-

c1

Ei) dr) Cu~

(A6)

0. Schüne. Quantum mechanical description of the intrinsic excitation offissioning nuclei

152

for the amplitudes

= —exp(i ~

= exp(i

di-) (uc~exp(—i

J e~

dr)(vc~ exp(—i

J e1 di-)

+

cc2 exp(_i

J e2 di-))

J e1 di-) — uc2 exp(_i f e2 dr)).

(A7)

From eqs. (Al) a differential equation for c~is obtained which is transformed to Webers equation [74] (AS) by the ansatz 2)f(\/iêi). (A9) c~= exp(~iêt The time derivative é = é chosen positive. Eq. (AS) is solved by the parabolic cylinder function 2/ê and z 2= is—exp(.~iir)Vêi. D~(z)with n = —iV A2. Asymptotic behaviour

—

The asymptotic behaviour of D~(z)appears to apply for Vet large compared to V2/E. This is seen when the next order is included =

c 2=

(_VCtY’(l

—

I))

~

for 1-.

—i(—Ve:r exp(_~ief2)Vnz~-’(I~

~).

—~ (AlO)

2/2z2 is, however, imaginary and can be put into a phase. The remaining term is small if The term n V ~ e. The adiabatic amplitude C’,,,, has for negative times the behaviour

J

~e’ V .~exp{i (e,,,,— e~)dr}.

(E,,,, —

E

(All)

1)

This expression is relevant 2when thetotal energy difference is large to V.in Itmind wouldthat yield in the energy. However, it compared must be born thisa isterm an which is proportional to a approximate form of a non-analytic function in a and that in principle an asymptotic expansion is not suited for studying the analytic behaviour of a function. In the region of the crossing the asymptotic expansion becomes meaningless and all the excitation energy becomes according to its dependence on a intrinsic energy.

G. Schüue. Quantum mechanical description ofthe intrinsic excitation of fissioning nuclei

153

For large positive times the form c1

—

exp{—irV2/ê} (VEt)~

—

Vf’() exp{— ~iir}exp{— ~irV2/ê}(V~t)”

1-~+X

(Al2)

which is non-analytic in a when written as a function of a and a leads to the usual jump probability

2

IciI

—

—

IC,,,,1

(A13)

exp{—2irV2Iê}.

A3. Different initial conditions For different initial conditions the second linearly independent solution D_~_,(iz)of eq. (A8) comes into play. Two examples are relevant for a calculation as performed in section 8: (1) The crossing occurred in the far past and at the starting time t 0 level 1 is unoccupied; (II) the deformation process starts at the crossing and the upper level is empty. In the first case the amplitude is written as 2}N(D~ c, = cxp{~iét 1(iz0)D,,(z)— D~(z~) D~_,(iz)). (A14) 2 = I at time :0. The differential The normalization N is determined from the condition Ic:1 equation (Al) leads constant to =

NVE V’ID..~_t(izo)D~(z,

1)—D~(z0)D~.~_,(iz0)I

(Al5)

.

The prime in (A15) denotes differentiation with respect to

z.

Evaluating the Wronskian for large Zo

yields V I ITV1 N=—~= —Iexpf———-~’ ye ‘. 2e’ .

(A16)

.

The occupation probability is for large times I

2 2

IciI

V2i

V

e 2 expj~—2ir-~—~j. 2) This formula shows the reduction of the jump probability due to the late start. The second case C,,,, = 0 implies (e,

—

(A17)

c,(0) = —c 2(0).

(A18)

Positive V is taken for simplicity. The ansatz 2}(A D~(z)—BD_~_ =

exp{~iê:

1(iz))

(A19)

154

0. Schüne. Quantum mechanical description ofthe intrinsic excitation offissioning nuclei

gives ci(0) and (A20) c2(0) = (An D,,.,(0)+ Bi(l + n) D~_2(0))(—i Vé) V1 exp{.~iir} using a recursion formula for the derivatives of the parabolic cylinder function. The power series expansion (A24) for D~(:)exhibits the value at the crossing D~(0) r(~—n/2y

(A21)

Condition (A 18) and the normalization condition Ici(O)12 = ~ determine A and B. For the asymptotic behaviour of c 1 only A is important:

2=

1A1

~

exp{~ninI} ~(exp{~irInI} + exp{~irlnl})(i —

,~J ~= ~

~

+

(A22)

This leads to the expression

lcd

— ~+

(I +

~

,.~2 ~(l — !,)

-

~ ir

.

—

in!

This formula is suited A4. Analytic

. .

C~nmfI){(l +

for lizI of

~)

I~~Ilog2)

~InI~ + in + .tInI •I (~ + m )2 ((~+ n~ )~+ In/2I1)f~

( ~

the order of 1. The limit of very small velocities can he shown to be

~.

structure

For completeness the power series expansion is given D,,(Z) =

V ‘~

f(~_~n)e (I + I!(l

I)2Z

+

I~—~2)~(~ + ~—n/2I2~ ~

n/2)(~—P1/2)i4~

.

~

(A24)

This power series expansion converges for fixed z uniformly in any finite region of a, hence the series is an analytic function of n. Since also the inverse r-functions are analytic. c and c are analytic functions 2/é and e2/é. Therefore, if lcd2 is written as a function of deformation and 2velocity, it contains no of V positive powers of a. Up to now the residual interaction V was independent of the velocity a. If crossing diabatic states are coupled by an essentially constant matrix element a(lIa~I2)most of the preceding discussion can be carried through replacing V by a(1la~f2).Then positive powers of a appear in the amplitude c, and a contribution to the kinetic energy is obtained. Such a transition is usually not called a Landau—Zener transition, because there the a,-coupling between diabatic states is assumed to be zero.

0. schatre. Quantum mechanical description of the intrinsic excitation offissioning nuclei

155

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