Dynamics of collective parameters

Volume 94B, number 3

PHYSICS LETTERS

11 August 1980

DYNAMICS OF COLLECTIVE PARAMETERS A.S. JENSEN and S.M. LARSEN

Institute of Physics, University of Aarhus, DK-8000 Aarhus C, Denmark Received 9 July 1978 Revised manuscript received 8 May 1980

Dynamical equations for collective parameters are derived from a variational principle. They are essentially expressed only in terms of the density distribution function through which the parameters enter. The energy of the system has to be given as a functional of the density e.g. by the energy density approach or in the Thomas-Fermi and Hartree approximations. The method is used to study monopole vibrations in nuclei. Excellent agreement with available RPS calculations for the same force is found and the RPA state is explained in terms of a very simple density vibration.

Time dependent H a r t r e e - F o c k (TDHF) calculations have recently been carried out numerically in several cases [1]. The very encouraging results call for more investigations. Unfortunately, the computations are very elaborate for most realistic cases and it is at the moment extremely difficult to obtain any kind o f systematics. In an attempt to simplify this problem the TDHF determinant was divided into two classes, the "core" described by a few collective parameters and the remaining part, the "particles" which were treated explicitly [2]. In a simple model, it was shown that the extreme reduction to only a core could reproduce the full TDHF calculation to a very good approximation. As it stands this core treatment of the nucleus assumes knowledge o f the individual single-particle wavefunctions constituting the core. The saving obtained is in general probably not substantial, because the majority o f the computer time is spent on constructing the average potential from the single particle wavefunctions. The purpose of this letter is to reformulate the core treatment in terms of the density itself. When the average potential is a functional of the density, the computations simplify enormously. In fact, at present such a reformulation makes the difference between doing more than a few test cases and performing a large number of calculations. 280

Besides the computational gain, we obtain the advantage of relating the density o f the nucleus directly to the collective parameters without any intermediate steps. This is clearly more satisfying, since the density, rather than the single-particle wavefunctions, usually carries the collective characteristics. A very good example is the fission process which directly is related to a certain development of the density. The formulation also allows degrees of freedom otherwise difficult to parametrize and study, e.g. diffuseness vibrations. Let us now proceed as in ref. [2] and derive the equations of motion for one collective parameter q and the corresponding " m o m e n t u m " p. The total wavefunction xI, for an A nucleon system is a Slater determinant q~ = (A!) -1/2 det {~k}

(1) = (A !)-1/2 det {exp [ipa(x, q)] q~k (x, q ) } , where {4~k, k = 1. . . . . A } is an orthonormal set o f functions depending on q and the single-particle coordinates x. The spacial dependent phase factor contain p and the function Q. In the static case p = 0 the wavefunction is qb -- (A !)-1/2 det {dPk(X, q) }.

(2)

Through a variational formulation [3] we arrive at the dynamical equations

Volume 94B, number 3

{P,q)~l = 3H/3p ,

PHYSICS LETTERS

{q,p}~ = 3H/Oq ,

(3)

11 August 1980

(i) the energy density approximation [4] where

E=E [p] is a functional o f p . where (a, b } is the antisymmetric Lagrangc bracket [3], and H = H(q, p) is given as (4)

g = (,I, I/]rl ,I,), where/2/is the many-body Hamilton operator. To calculate {p, q} we need A

O~ i ~ 3P

k=l

(ii) Using a T h o m a s - F e r m i type of approximation for the kinetic energy part and omitting the exchange term in the potential energy or maybe approximating it in terms of the density. The equations of motion, eq. (3), are now

B~ = OH/~p = 2pK , (12)

(5)

Q(Xk, q)" qd

leading to

- B~ = 3H/Oq = - F +p23K/3q, where B and K are given by eqs. (6) and (9) and the driving force F in the q direction is

B ( q ) - {p,q}=fi

*

Q(Xk,q) 3q

F = -(3/3q)~l[q~) = -3E/Oq.

(13)

Eliminating p form eqs. (12) we find

3~* ~k Q(Xk,q) ~](d.r)A

(6)

B2 ..

~q

=Fi f ~ Q(Xk, q) ~q ('I'*'I') (dr)A

LoB+

B

(14)

Since F is the force we can identify the mass in the collective motion along q by

M = B2/2K.

= ~ f Q(x,q) 3p(x,q) dr,

3q

(15)

The important thing is that B, K and F are all expressed in terms o f q through the density and the function

where

Q.

A

p(x,q)=k~=l [qbk(X,q)[2 .

(7)

We now turn to the kinetic energy part of H, which may be expressed as (qz] Tiff') = (qblTlqb) + p2K(q),

(8)

where

K(q) =-(h2/2m)

.

- F = qZ~K-

f i VQl2p(x, q) d r .

(9)

When the potential energy can be expressed in terms of the density, we have (tI, I V I ~ ) = ( ~ 1 V I ~ ) .

(10)

The function H is H = (q'l[/I q~) + p2K(N) =-E(q) +pNK(q),

(11)

where the p dependence is collected in the last term. If eq. (10) is not valid it may be possible to find the p dependent correction explicitly. We want E expressed in terms of the density p and its derivatives. This can be achieved at least in two ways:

Let us now see how Q enter the description. The current is given by ]/ A ] (,Is) = - - ~ (¢~ I¢ ffk - ffk V ~,~) =/" (q~) +/ip P ~¢Q 2im k=l m (16) When the wavefunctions q5k are real,j(q~) = 0 and the function Q is the velocity potential describing the current arising from variations o f q . Since the only timedependence of p is through the collective coordinate the continuity equation is then ~ . Op_ - ~ p div(p VQ) + q 3qq - 0 .

(17)

As an example we consider two identical particles 1 1 of mass m at points i R and - ~ R moving with opposite velocities. The collective parameter is the distance R between the particles. The density distribution is a sum of two 6 functions at ½R and - ½ R . The current, which should be identified with (3Q/3x)O, is proportional to the difference between these two 6 functions. Both B and K are now easily found to be

281

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'

PHYSICS LETTERS

m

1 \_..x=Ri2 !

where

'

1

leading to the mass M = ~ m, which is the reduced mass of the system. Another example is the translation of a particle. Its density distribution is a 6 function at the point R. Again we easily find

OQj B= l~-~X x= R ,

h2 [OQ]

K,

~2 -~m ~,TXlx=R] '

(19)

leading to the mass M = m as it should. The formalism above can be generalized to more than one dimension. The 'I' wavefunction is now q~=(A!) 1/2 det [i~nPnQn(x,q)](~k(x,q)

,

(20)

where q stands for the set q = ( q l , q2 . . . . }. The equations of motion are

[(qn,qm}gim + {qn,pm)pm] =3H/oqn ,

m

(21a)

[{Pn,qm)gim + (Pn,Pm }Pm = 3H/OPn • (21b)

m

11 August 1980

r rLOanOqm

A£,m

fen Oqm oo dr__B n m , ,

(

LOqn3qm

a

3qn 3qn J

(q~[TI q~)= (q~lTl~)+ ~

n,m

h2

Kn, m = 2m

pnPmKn,m ,

f(v, Qn)- (•mQm)P

(23)

0k = ~

X

Ckxbx ,

(29)

13(oc;

Ik= X \ ~qn Oqm

dr.

{Pn,Pm) = 0 .

(24)

(25)

For the coordinates we find for normalized single particle wavefunctions

= Pfn,m, 1

282

3q n 3q m

ock oc; ) 3qn ~

(30) "

Thus I k = 0 when the expansion coefficients may be chosen as real numbers. The equations of motion are then

~ B n mgim - OH _ ~ P m ( K m n+Kn,m), m ' 3Pn m '

3q n 3qm_] dr (26)

(31a)

~m [ ~ P ] Z f n , m g i m - B m , n [ ) m 1 81-1_

aqn

_Fn + ~ p m p j

aK],m

,

m, j

Oqn where the driving force F n = -OE/Oqn.

(31b)

The current given in eq. (16) is now

j(tP)=f(~b)+~O ~n pnVQn ,

Two other types of Lagrange brackets are now needed. When the Q functions are real we obtain for the momenta

{qn, qm } = ifi

(28)

dr.

This can be shown by expansion on a q-independent orthonormal basis {bx}

_

(22)

(27)

In eq. (26) we have used that I k = O, where

We find in analogy with eqs. (6), (8) and (9)

{Pn,qm} =h

] d~.

~

(32)

and the continuity equation becomes

~Pn n

30=

div(PVQn) + ~ gin ~ n n

O.

(33)

Under the assumptions where this formulation is valid, it is identical to the Hartree-Fock treatment of the core [2] except for the treatment of the energy. Everything else has only been evaluation of the same quantities in more convenient ways. The different treatments of the energy lie either in the energy density approximation or in the T h o m a s Fermi treatment of the kinetic energy and the exclusion or density expressed approximation of the exchange term in the potential energy. The merits of the equations of motion can now be

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investigated numerically. This has essentially been done already for one-dimensional monopole vibrations. The density distribution was that of the lowest orbits of a harmonic oscillator. The q parameter was an overall scaling of the density keeping the normalization to a fixed number o f particles. The function Q was x 2 and the interaction a gaussian two-body force. The result was compared to the full time dependent Hartree solution. The amplitude and frequency o f the vibration only deviate by a few percent. It should be mentioned that in the above numerical example the kinetic energy was treated directly while the formulation in this letter require a T h o m a s - F e r m i type of approximation. Provided this is accurate, and investigations claiming this have been published [5], the comparison described here would only change insignificantly. As a test o f the method we apply it for spherical nuclei in the study o f two types o f monopole vibrational modes, i.e. overall density scaling and surface thickness vibration. The parametrization of the density is

p(r, b, d) =

Po(b, d) b)/bd] '

1 + exp[R(r-

r"

dr"

Q(r)=-2qfp(r,)(r,,)2 f 0

dr'(r') 2

/

oo

i i 50 I

_~.~

~

L

n JJJ°S:------'---'--"

b'///j..

.....

n.d ~ "

[J~,d I

._.

b,o

i

r

J 2

3

I,

AI13

5

6

7

Fig. 1. Monopole vibrations energies/~co as function of nucleon number A for two different forces. The filled circles correspond to the energy functional in ref. [4] with the parameter values of set II. The open circles are obtained with the Skyrme III force [6] without spin-orbit terms and vfith the kinetic energy replaced by the Thomas-Fermi expression of ref, [4]. The labels on the curves refer to the different modes parametrized by b and d. The energy of the normal mode n coincide within a few percent with the energy of the lowest lying individual mode.

(34)

where Po(b, d) is determined by particle number conservation. The equilibrium value R of b is obtained at the energy minimum point in the space spanned by d and b (=R). Assuming spherical symmetry for b o t h p and Q we can solve eq. (17) and find r

11 August 1980

ap(r') ~q '

(35)

0

where we have assumed Q(O) = 0 and ~ = pq. 2fi/m. For the scaling degree o f freedom we can integrate analytically and obtain Qb(r) = r 2. This is the reason for the choice o f proportionality constant between and p. For the surface thickness degree o f freedom we have relied on numerical integration. The energy functional used in number II o f ref. [4] and the Skyrme III force [6] without s p i n - o r b i t terms and with the kinetic energy replaced by the T h o m a s - F e r m i expression o f ref. [4]. For b o t h forces we assumed proportionality between neutron and proton density distributions. The Q functions are now found from eq. (35) and

the B, K and A coefficients from eqs. (6), (9) and (27). The equations of motion, eq. (31), are then solved for small amplitude vibrations around the energy minimum. First we consider only one degree of freedom at a time and extract the energies corresponding to the vibrational frequencies in the harmonic motion. Afterwards we include both modes simultaneously and extract the energies and amplitudes o f the normal modes. The energies are shown in fig. 1. The two forces both claiming to reproduce static properties lead to very different vibrational energies. We notice the expected [7] A-1/3 behaviour for large A for the b mode. The normal mode o f lowest energy coincide (within a few percent) for both forces with the lowest lying individual mode. The reason is that this normal mode is an almost pure b mode, except for the very light systems where it is an almost pure d mode. When the energies o f the b and d mode cross, the lowest normal mode is an even mixture of the same energy. The energy of the highest normal mode is on the other hand appreciably higher than the highest individual energy reflecting the fact that this mode contains 283

Volume 94B, number 3

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significant contributions from both individual modes. Thus the b and d modes are not orthogonal and in principle a parametrization, where this is the case, could be used instead of eq. (34). Fortunately RPA energies for the Skyrme III force are available [7] for a few nuclei. They are all within a few percent of the energy of our lowest normal mode. Thus our calculation show thow the RPA state can be explained in terms of a simple density vibration, e.g. it is an overall scaling vibrational state for all nuclei of A>16. Summing up we have derived a set of dynamical equations for collective parameters. We assume the energy of the system is given as a functional of the density e.g. by the energy density approach or in the T h o m a s - F e r m i and Hartree approximations. The necessary a priori choice of parameters for the process to be studied can then be made directly in the nuclear density distribution function. The equations of motion are approximations to the TDHF equations. An application to the monopole vibrational mode in nuclei has been made. Excellent agreement with available RPA energies are found and the RPA state is explained in terms of a very simple density vibration. The authors wish to acknowledge Dr. A. Miranda for discussions and careful reading of the manuscript.

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11 August 1980

References [1 ] P. Bonche, S.E. Koonin and J.W. Negele, Phys. Rev. C13 (1976) 1226; R.Y. Cusson, R.K. Smith and J. Maruhn, Phys. Rev. Lett. 36 (1976) 1166; J. Maruhn and R.Y. Cusson, Nucl. Phys. A270 (1976) 471; J. Blocki and H. Flocard, private communication; A.K. Dhar and B.S. Nilsson, Phys. Lett. 77B (1978) 50; H. Flocard, S.E. Koonin and M.S. Weiss, Phys. Rev. C17 (1978) 1682; P. Bonche, B. Grammaticos and S.E. Koonin, Phys. Rev. C17 (1978) 1700; [2] A.S. Jensen and S.E. Koonin, Phys. Lett. 73B (1978) 243. [3] A.K. Kerman and S.E. Koonin, Ann. Phys. 100 (1976) 332. [4] R.J. Lombard, Ann. Phys. 77 (1973) 380; M. Beiner and R.J. Lombard, Ann. Phys. 86 (1974) 262. [5] C.Y. Wong, Phys. Lett. 63B (1976) 395; M. Brack, B.K. Jennings and Y.H. Chu, Phys. Lett. 65B (1976) 1. [6] M. Beiner, H. Flocard, N. Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. [7] J.P. Blaizot, D. Gogny and B. Grammaticos, Nucl. Phys. A265 (1976) 315.