Theory of nuclear friction

Theory of nuclear friction

Nuclear Physics A240 (1975) 412484; @ North-HolIand Publishing Co., Amsterh Not to be reproduced by photoprint or miao6lm without written permissio...

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Nuclear Physics A240 (1975) 412484;

@

North-HolIand Publishing Co., Amsterh

Not to be reproduced by photoprint or miao6lm without written permission from the publisher

THEORY OF NUCLEAR FRICTION D. H. E. GROSS Hahn-Meitner-Institut ftir Kernforschung Berlin GmbH, Bereich Kern- und Strahlenphysik, Berlin- West, Germany

Received 12 September 1974 (Revised 8 November 1974) Abstract: From the general many-body

Schrijdinger equation for colliding heavy ions a classical equation of motion including frictional forces is deduced. It is argued that one of the main origins of friction between two colliding heavy ions is the reflection of nucleons off the moving edges of the single-particle potentials. A simple classical analogue is given which illustrates how friction appears.

1. Introduction Recent experiments on multinucleon transfer reactions induced by heavy-ion collisions show that there is a high probability for excitations of more than 100 MeV to occur. Yet, in spite of these violent interactions, the individuality of the ions is preserved ‘). In such situations the level density of the final fragments at these high excitation energies is too large to allow resolution of individual intrinsic states. Here a microscopic analysis like the usual distorted wave Born approximation does not make sense. Moreover the cross sections for the relevant partial waves are so large - approaching the classical geometric values - that a Born approximation is invalid. While conceivable formally, a quanta1 coupled channel calculation is impractical. Here we will show how a workable classical theory can be derived from the quanta1 many-body equation of motion for two colliding heavy nuclei. In this theory the two nuclei move on classical paths subjected to a frictional force. The friction controls the energy dissipation from the relative motion into the internal degrees of freedom. Moreover it determines the effect of the excitation back on the relative motion of the two ions. In particular it sets conditions for fusion to occur. At present the friction model first proposed by Gross, Kalinowski and Beck ‘) is the only one that synthesizes highly inelastic collisions and fusion reactions in a quantitative way. With a simple parametrization of the friction tensor Gross and Kalinowski 3, were able to obtain an overall agreement with nearly all existing fusion data. More recent calculations 4, confirmed the basic simple approach of ref. 3, by studying more complicated parametrizations and finding these unnecessary. However the model is still in the stage of development. An unambiguous picture of these reactions has not yet emerged. 472

NUCLEAR FRICTION

473

Recently Beck and Gross ‘) sketched a microscopic theory of nuclear friction. Here the arguments are given in more detail, the logical deduction of the final formula is improved and the theory is both illustrated and checked for consistency by a simple classical analogue. The general theory in sect. 3 wili show that possibly the main mechanism leading to friction between colliding heavy ions is the scattering of nucleons of one ion by the moving edge of the single-particle potential of the other ion. To prepare for this we discuss in sect. 2 a simple classical model illustrating how the motion of a body through a low density gas is damped by frictional forces. In sect. 3 the quanta1 many-body scattering theory of two colliding heavy ions is investigated. The collision process is described in terms of the density matrix, which is the solution of Liouville’s equation. A corresponding classical equation of motion is obtained with the help of Ehrenfest’s theorem. We will show how a frictional force enters that equation. A second-order ~rturbation expression for the friction force is given. In sect. 4 the general formalism as developed in sect. 3 is applied for the classical model of sect. 2. The relation of the friction to the more familiar imaginary part of the optical model potential is established in sect. 5. We will see that both are strongly interrelated by the same underlying physics. Finally a short discussion of our results is given in sect. 6.

2. Classical piston model for friction Consider a large box filled with a Fermi gas of nucleons. In this box a small piston with mass M moves with velocity u in the x-direction. The velocity u is assumed to be constant during many scattering events of nucleons hitting the piston. We further assume that the probability of a nucleon hitting the piston twice is negligible. Therefore the effects on the motion of the piston of possible changes in the momentum distribution of the gas are neglected. In an individual reflection the momentum transferred onto the piston is:

where p, is the relative momentum of a nucleon moving against the piston, and l&Q2 is the reflection probability of a nucleon on the piston. The number of nucleans impinging on the piston from the left per unit time is given by their flux dj,(p) in the x-direction:

(2.2)

414

D. H. E. GROSS

where f(p) = [exp ((sp-- p)/kT } + l] - ’ is the occupation probability of the state p in the lab. frame, and A is the area of the piston orthogonal to the x-direction. Thus the force acting on the piston from the left is: F,(lzft) = ;l$ =

dp,dj,(p) s

2A

*

= ~

(27&)3m

s*

dp,&W12

+a0 dp, +SOdplf(p+mu)(l--(p’+mo)}. s -m s -@I

(2.3)

Here the additional factor 1 - f(p' -I- mu) takes account of the fact that the nucleon after reflection has the momentum p’ + mu = ( - p, + mu, p,,, p,> and can only go to an unoccupied state. A similar consideration applies for the reflection of nucleons impinging from the right onto the piston: +m +CCJ -2A ’ F,(right) = (2,rh)3m _ ~d~x~~I~(~,)12 _ adPY _ oD dtd-(p+mw-f(~‘+~e(2.4) s s s Thus the net force F, by which the nucleons of the gas act on the piston is: F, = ~~(right) + F,(left) 2A Z-----------

m

(27di)3m s o

+CC

~~,~~l~(~,)i2

+m

dp, dp,(f(p+mu)-f(p-mu)). f -CC s -tC

(2.5)

Here we made use of the relation f(p) = f( - p), Up to first order in u we can write for eq. (2.5) +m dp, & f(p). (2.6) -CC

5

Thus we have finally obtained a force, proportional to the velocity u, acting on the piston in the direction opposite to u (remember that df(p)/dp, c 0). This is an ordinary friction force. It appears because of the interaction of the piston with the internal degrees of freedom of the gas, by which energy dE = - Fvdt dissipates from the motion of the piston into the gas. This interaction is due to the scattering of single nucleons by the potential wall of the piston moving through the gas, thereby leading to succesive particle-hole excitations. 3. Microscopic theory of nuclear friction The previous simple case serves to clearly show the microscopic physical origin of a many-body friction phenomenon, namely the reflection of the individual nucleans by a moving potential wall. Here we shall treat the heavy-ion case more realistically.

415

NUCLEAR FRICTION

The motion of the two ions is determined by the Hamiltonian H = H,,,+&,,+V=&+K

(3.1)

with Zfre, = - (h2/2A4)AR+ U(R), R being the distance between the centers of mass of both ions and U(R)their first-order (Hartree-Fock like) interaction potential in the elastic channel. The intrinsic Hamiltonian of the two ions is called Hintr. The residual interaction V governs the mutual excitations and particle transfers and has no diagonal matrix elements between the intrinsic ground states. To describe the motion of the two ions we choose the time-de+ndent density matrix p(t) instead of a wave function. This simplifies the calculation somewhat and is more flexible for a later introduction of statistical excitations. The density matrix p(t) is the solution of the Liouville equation corresponding to Schriidinger’s equation : ih$

It has to satisfy the boundary

p(t)-

[H, p(t)] = 0.

(3.2)

condition: lim IMr)-~,(r)ll *+-CC

= 0,

(3.3)

where P&) describes the motion (wave packet) of the two ions without the interaction I/ far before they interact and [IA11is the norm of the operator A. The free particle density matrix is given by: P&) = l$Jt))($,(t)(

= e-‘i”)Ho’ e(i’fi)Hotj?O,

(3.4)

where I+&)) is the unperturbed initial wave packet. Following Zubarev 6, we construct the formal solution of (3.2) and (3.3) as: f & ee(r-t) e(i/fi)Wf-t) p,(z)e-(i/fi)H(T-O.

p”(t) = &

(3.5)

s -CD This satisfies Liousville’s equation up to a source term proportional ifi;

p”(t) = ihep&)+ [H, p”(t)].

to E: (3.6)

Therefore, in the limit E + 0 the expectation value of any operator A : tr (A@(r)) satisfies the same equation of motion as the true solution tr C&(t)}. This way of formulating the scattering states l+‘“(t)) is the normal way in scattering theory:

fs

dZe”(r-‘)e(‘/n)H(‘-t)l~~(~)). 11/P(t)) = lim E e-0 -m

(3.7)

To show that (3.5) satisfies the boundary condition (3.3) we perform a partial inte-

476

D. H. E. GROSS

gration of the right hand side of eq. (3.5) using (3.4): p”(r) = &At)+ $

1 dZe~(r-f)e(i/R)H(r-t,[l! p,(r)]e-(Ufi)H(~-O, s Q,

(3.8)

Therefore we have IIPY+-P&II

5 ;

s

1

ddlK ~&Ill.

(3.9)

a,

If lim,+_, ~_~dzl(~o(~)~V2~~o(~))~3 = 0, the left hand side of (3.9) goes to zero in the limit t --+ - co. Having constructed p(t) we can calculate the expectation value of the relative distance R between both ions as a function of time: <&A = tr (&)R,). Moreover i

(It,), =

M

Atr(R,[K p(t)]) = A (V,>, = f

-$@A + (V, WW, +

= 0.

(3.10)

Following Ehrenfest’s theorem, eq. (3.10) gives the connection to the classical equation-of-motion analogue to the quantum mechanical problem (3.2). We see that we get a correction - VvV to the usual force - V, U(R) from the coupling to the intrinsic degrees of freedom. We now will show how this term which we abbreviate by F, = V, F/ contains an ordinary friction force. For this purpose we restrict ourselves to a ~rturbation treatment only. We take the interaction V into account only in the lowest order for which (F,), does not vanish. To first order in perturbation theory (3.8) is

m = &do+ ;

s

dzee(r-I)e(i/fi)aO(r-r)[I!

;

Po(Z)]e-(i/“)HO(t-t),

(3.11)

m

and
1 t & &r-t) tr F e(i/h)fh(~ - ‘)[~7 po(z)]e - (i/fi)&t? - t)}3 =1--L 3 ih f -m

(3.12)

which can be simplified to (3.13)

NUCLEAR

FRICTION

417

with p(r) = e(i/~)HorV e - (Vh)Hor.

(3.14)

Here we have used the fact that the diagonal matrix element of V between the intrinsic ground state is zero and consequently tr (F,p,(t)) = 0. Writing (3.15)

PO(t) = IPitt) O)
where lpi(t)) describes the initial wave packet of relative motion propagating with If,, and IO) is the product of the intrinsic ground states, we can express (3.13) as ; dre”‘
@,A = $

Now we proceed to calculate (F,(t)). To this end we write the commutator V,),

= ;

(3.16)

in (3.16) explicitly:

0). ; dre”‘{(Pi(t), 0~Fye(i~*‘(Ho’-Eo’)TP,(7)~pi(t), s m - (pi(t), O(&(z)e-(i”‘)(Hol -Eol)‘Fylpi(t), 0)},

(3.17)

with the notation: H,,

= HP’,

f2(7) =

H,, = H;,’

(3.18)

e(ill)H02rye-(i/l)H02r.

By partial integration of the two terms in (3.17) and using

we find

@‘A = (0; -

+ (0:

1

(Pi(t)vOIFvH,, _i,,

+ (Pitt)7 Olv

_ihE VlPi(t)3O>

1

H,, -E,,

+ihE

+Aih ” d7e”‘
FylPi(t),0)

ewWfWoI -W4 Ho,-E,,-ih

(3.19)

C&u 9 9,(7)llP,(G

exp i - WWm -&1)7) If,,-E,,+ihE

P,b)l

The first two terms in (3.19) can (because Q(H,, -E,,)Q be taken together:

0)

F ,p,ttI ” ’ ’

.

2 E > 0, Q = 1 - lO)(Ol)

D. H. E. GROSS

478


‘_Ev

vH

-
01

IPi@)l 0).

(3.20)

01

This is simply a second-order correction to the real potential value is called P. In the last two terms in (3.19) we use:

U(r). The principal

(3.21) In the resulting expression we neglect the commutator [p, F], which is of the order of the momentum transfer Ap and thus small compared to the terms we keep. We then assume that the wave packet lpi(t)) behaves so classically that (p,/M)lpi(t)) = (p&$-z)) z lp~t)>e-(“*)E02’$. We find with = Mr)><~,A and e -(i~*)HozSlpi(t)>
(3.22) with E, = E,, +Eo2, H, = H,,+H02. We may split this into terms symmetric and anti-symmetric under exchange p c+ v:
(3.23)

= ${
’ =

%
- <&OF,)).

We thus obtain t (3.24)

P with e,,(r) = 2nfi
H ‘, 01

01

FplPi(th 0)‘.

(3.25)

We should mention that ~o(~o~ -E,,)Q, # 0 because the intrinsic spectrum is discrete near the ground state E, 1, and the ground state 10) does not occur in the sum over intermediate intrinsic states. We see that (3.24) is just an ordinary friction force proportional to the velocity with a (normally anisotropic) friction tensor c,,(t). t In formula (25) of the work of Beck and Gross s) the wave packets Ip,(t)) were missing. This result is erreonous and should be corrected by (3.25). * At this point the assumption is made that the correlation function ( ) in (3.22) is of a sufficient short range in T that in the time z the external force V,U(r) does not change the velocity u(f) much.

479

NUCLEAR FRICTION

The other term (F,)j’”

adds together with (F,): (eq. (3.20)) to give (3.26)

the full second-order correction to the real conservative potential. Using the Wigner transform of the friction operator, eq. (3.25) one may define a local position- and momentum-dependent friction tensor:

= 27rh d3p(r+&, s

F,

‘,

01 F, cS(H,-E,)~ 01

‘10, r -$)

exp

01

(3.27) From the structure of (3.25) we see that the friction is just due to the dissipation of energy by the system, when it makes real transition @(Ho -E,)!) into excited states. The formulation of the problem has shown how we get an irreversible dissipation term starting from the quanta1 many-body equation of motion (3.2). The violation of time-reversial invariance entered into the problem by definition (3.3) of the scattering-state boundary condition. Our result, (3.25) or (3.27), shows a close similarity to the force-force correlation function by which friction coefficients are often expressed in statistical mechanics ‘). The structure of eq. (3.25) shows that the friction between two colliding heavy ions is mainly a surface effect. The value of F = VV is largest at the surface. These surface effects are not taken into account in calculations of the friction between heavy ions, which start from the viscosity of nuclear matter a). By a simple estimate we can understand what is the main physical mechanism leading to friction in heavy-ion collisions. To do this we roughly estimate the probability W,(E) to excite a lp-lh state with energy E in one of the colliding nuclei by the single-particle potential U of the other. Then we compare this to the probability W,(E) to excite a lp-lh state in both nuclei simultaneously with summed energy E by an inelastic nucleon-nucleon collision. We assume both nuclei are equal. Let the matrix element J&an of the effective nucleon-nucleon interaction between particle-hole states a/? in nucleus 1 and particlehole states KAin nucleus 2 be nearly constant and equal to r Then we have (p is the Fermi surface) :

W,(E)x

;

IUIZp\l&l,(E) w ;

AZV2g2E,

E

W,(E) x ;

[VI2

s0

dep’,‘d_~(E-&)P12~~h(&)= ;

4E3

lJ’12gF.

480

D. H. E. GROSS

Following Ericson 9, we have taken the lp-lh level density to be g2E. We may take g = (6/a2)&4 (MeV-r) and find W,(E),‘W,(E) ei gZE2,‘A23! x 10-3E2.

(3.29)

By this simple estimate we see that one of the most important mechanisms leading to friction (at least not for too light nuclei) is the reflection of single nucleons in the tail of one of the ions off the moving edge of the single-particle potential generated by the other ion and vice uersa. It is not due to the individual inelastic collisions of nucleons belonging to the different ions while these are moving through one another. This picture of friction in heavy-ion collisions is close to the simple classical model discussed in sect. 2. 4. Application of the general theory of the piston model of sect. 2

In order to clarify our result (3.25) for the friction tensor we will explicitly calculate it for the piston model of sect. 2. The interaction I/ is taken as I/ = &6(x- <), where x is the position of the piston and 5 the position of a nucleon in the gas. We will show that (3.25) gives exactly the same result as eq. (2.5) in lowest-order perturbation theory. Let the piston have the length L (later on taken in the limit L --P a) in the y- and z-directions. Its motion in the x-direction may be described by a Gaussian wave packet lpi(t)). The Wigner transform of its density matrix pPi = lpi(t))(pi(t)l is explicitly written with the help of formula (4) ch. 4.4 in the book of Leighton lo):

pp,fpx, x, 0 =

2 exp[ - ~(p,-pi)“-

h (~-~O-

~ t>‘] .

(4.1)

Here pi, x0 are the moments and position of the wave packet at t = 0 and a is the width in x of the wave packet at t = 0. With p&,, x, t) we can write the friction coefficient c,,(t) given by eq. (3.25) as:

c,,(t) =

s

dxdp,

y-$-

Pp,(Px9 x9 +dx,

P,).

We therefore have to calculate in our model cXX(x,p,), which is defined as in (3.27). Because the system is translational invariant in x, C&Z, p) does not depend on x at all: (p,, Olt,,lj,,

0) =

s s

in:

e-ci~a)cp~-px)xc,.

( ~ 1= x,,

p,+P, 2

&J,-- lod?4.

(4.3)

The Fermi gas is described by plane waves:

e+(t) =

with [a:, a,,]+ = S3(q -q’),

(4.4)

481

NUCLEAR FRICTION

Ho,a~,u,~O)

= s

d3k;

a~a,u~~a,~O) =

!!2m

” G

+Eo,

>

af*a,lO),

(4.5)

where uf,u,lO) are the intrinsic particle-hole excited states of the gas and IO) is its ground state. We have to calculate the matrix element of the interaction force F, = (a/ax) x V,6(x- <) connecting the initial state (p,, 01 with an intermediate plane wave of relative motion lpi) and the excited state lq'q)of the gas:

(4.6) where we have written: d(k) = gk

iii/‘(k)

sin(g),

= 6(k).

(4.7)

Inserting (4.6) into our general expression for (p, , Olt,,lp,, 0) (c.f. eqs. (3.25), (3.27), (4.3), (4.7)) we have for very large L:

(4.8)

with n(q)= 1 if q2/2m 5 ,u, the Fermi energy of the gas, and n(q)= 0 otherwise. In order to perform the integrations we write: qo = %&+qx),

and approximate

k = ~~:-~x)>

(M % m, lp,l % lkl): s-k-

After integration cXX(PX)= x n(m+k,

2c (2nh)3mu

(4.9)

E)}+k

($-9}.(4.10)

over pk and q. we obtain with the help of eq. (4.3): +w _ m dk

m2V2 k2

s&n

(k)

d’q,(l

f&i

s - 2L? ql) = (2nnh)3mv oadkk21R(k)12 s

--n(mu+k

ss

d2q1{n(k+m~,

qJ-n(k-mu,

qJ>, (4.11)

D. H. E. GROSS

482

with Mv = p,. To first order in perturbation theory the quantity IR(k)l = Iml/,/hkl is the quanta1 reflection coefficient for a plane wave with momentum k impinging on the potential I/= V,6(x - 5). Noting further that the area of the piston in the y- and z-directions is A = I_?we see that this result is exactly the same as (2.5). By this analysis we show that our general microscopic and quantum mechanical expressions (3.25) and (3.27) for the friction tensor give exactly the same result in our simple model as did the direct classical calculation. 5. Connection of friction to the imaginary part of the optical model potential For the heavy-ion case it is quite interesting to relate the friction force to the imaginary potential. Both friction and the imaginary part of the optical potential, W(R, E), describe the coupling of the intrinsic excitations to the relative motion. Therefore it is quite natural that both are strongly interrelated. The inelastic transition rate per time is given by: dN X-h

2 - - W(R E),

(5.1)

as can be seen, e.g., from the continuity equation of the current under the presence of absorption. The corresponding amount of energy dissipating per unit time from the relative motion into the intrinsic degrees of freedom is: g

= (AE)

; W(R, E),

(5.2)

where (AE) is the mean energy transfer in a single collision. On the other hand the friction force leads to an energy dissipation of (5.3) Assuming for the moment the friction tensor to be isotropic, c,JR, p) = c(R, p)&,, we find

dE dt=

-c(R

~1;

{E-

(5.4)

W)).

Consequently we have the relation between the friction coeffkient and the imaginary potential : c(R, p) = - ; M(AE)

WR -3 E-

U(R) =

_

Mw

WR>El E-U(R)’

(5.5)

where we assumed (A E) x ho, the gap energy of the intrinsic single-particle states. This relation between c(R) and W(R, E) is expected to be quite helpful for order of magnitude comparisons of both quantities. Especially, because they are measured

NUCLEAR

FRICTION

483

in a quite different way: We probe W(R, E) in elastic collisions only in the tails of the strong interaction region, whereas in a frictional analysis of deep inelastic collisions we test much closer features of c(R). Relation (5.5) can easily be gained from our formal result (3.24) and (3.25): (5.6) Using eq. (3.21) and the approximations obtain :

made there now in the inverse direction, we

dE dt=

- g
O[V

E. _ j I

+ ihE UP,(t),

0)

0)

z (A@

f ( W(E))*

(5.7)

0

This is just (5.2), where we identified (AE) with the mean intrinsic excitation energy {(Ho1 -Eol)

term). 6. Conclusion

The insight we have achieved by the present discussion can be summarized as follows: The coupling of the internal degrees of freedom to the relative motion leads in a natural way to an ordinary friction force acting on the classical relative motion of the two ions. One of the main origins of friction are particle-hole excitations made by the edges of the single-particle potential of one ion moving through the density tails of the other ion. Because of this frictional process, energy dissipates from the relative motion into the intrinsic degrees of freedom. In the quanta1 description of relative motion by an effective Schrijdinger equation for elastic scattering the same coupling to the internal degrees of freedom leads to the imaginary partof the optical model potential, which describes the flux dissipation from the elastic channel into other states. As both types of dissipation have the same physical origin, they are closely related by eq. (5.5).

The author gratefully acknowledges illuminating discussions with R. Beck and St. Landowne.

References 1) A. G. Artukh, G. F. Gridnev, V. L. Mikheev, V. V. Volkov and J. Wilczynski, Nucl. Phys. A215 (1973) 91 2) D. H. E. Gross, H. Kalinowski and R. Beck, Proc. Int. Conf. on nuclear physics, Munich, August 27September 1, 1973. contribution 5.65

484

D. H. E. GROSS

3) D. H. E. Gross and H. Kalinowski, Phys. Lett. 4SB (1974) 302 4) J. P. Bondorf, Conf. on reactions between complex nuclei, Nashville, Tenn., USA, June 10-14, 1974 invited talk 5) R. Beck and D. H. E. Gross, Phys. Lett. 47B (1973) 143 6) N. Zubarev, Fortschr. Phys. 20 (1972) 471, 485 7) R. Becker, Theorie der Warme (Springer, Berlin, 1966); R. Kubo, Many body theory, 1965 Tokyo Summer School Lectures (Tokyo, 1966) 8) G. Wegmann, Phys. Lett. 50B (1974) 327 9) T. Ericson, Adv. in Phys. 9 (1960) 425 10) R. Leighton, Principles of modern physics (McGraw-Hill, New York, 1959)