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arXiv:nucl-th/9808073v2 28 Aug 1998

Nonlinear Enhancement of the Multiphonon Coulomb Excitation

in Relativistic Heavy Ion Collisions

M.S. Hussein, A.F.R. de Toledo Piza, O.K. Vorov

Instituto de Fisica, Universidade de Sao Paulo

Caixa Postal 66318, 05315-970,

Sao Paulo, SP, Brasil

(13 August 1998)

Abstract

We propose a soluble model to incorporate the nonlinear effects in the tran-

sition probabilities of the multiphonon Giant Dipole Resonances based on

the SU(1,1) algebra. Analytical expressions for the multi-phonon transition

probabilities are derived. Enhancement of the Double Giant Resonance exci-

tation probabilities in relativistic ion collisions scales as (2k+1)(2k)−1for the

degree of nonlinearity (2k)−1and is able to reach values 1.5 − 2 compatible

with experimental data. The enhancement factor is found to decrease with

increasing bombarding energy.

[KEYWORDS: Relativistic Heavy Ion Collisions,Double Giant Resonance]

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Coulomb Excitation in collisions of relativistic ions is one of the most promising methods

in modern nuclear physics [1–5]. One of the most interesting applications of this method

to studies of nuclear structure is the possibility to observe and study the multi phonon

Giant Resonances [1]. In particular, the double Dipole Giant Resonances (DGDR) have

been observed in a number of nuclei [6–8]. The “bulk properties” of the one- and two-

phonon GDR are now partly understood [1] and they are in a reasonable agreement with

the theoretical picture based on the concept of GDR-phonons as almost harmonic quantized

vibrations.

Despite that, there is a persisting discrepancy between the theory and the data, observed

in various experiments [6–10] that still remains to be understood: the double GDR excitation

cross sections are found enhanced by factor 1.3 − 2 with respect to the predictions of the

harmonic phonon picture [1], [3], [11], [12]. This discrepancy, which almost disappears at high

bombarding energy, has attracted much attention in current literature [4], [13–17], [22–24];

among the approaches to resolve the problem are the higher order perturbation theory

treatment [17], and studies of anharmonic/nonlinear aspects of GDR dynamics [4], [22,23],

[24]. Recently, the concept of hot phonons [15], [16] within Brink-Axel mechanism was

proposed that provides microscopic explanation of the effect. These seemingly orthogonal

explanations deserve clarification which we try to supply here.

The purpose of this work is to examine, within a a soluble model the role of the nonlinear

effects on the transition amplitudes that connect the multiphonon states in a heavy-ion

Coulomb excitation process. Most studies of anharmonic corrections [22–24] concentrated

on their effect in the spectrum [25,26]. Within our model, the nonlinear effects are described

by a single parameter, and the model contains the harmonic model as its limiting case when

the nonlinearity goes to zero.We obtain analytical expressions for the probabilities of

excitation of multiphonon states which substitutes the Poisson formula of the harmonic

phonon theory. For the reasonable values of the nonlinearity, the present model is able to

describe the observed enhancement of the double GDR cross sections quoted above.

Having in mind to show how analytical results follows from the nonlinear model, and

to explain how the model works, we restrict ourselves here to its simplest version (trans-

verse approximation, or SU(1,1) dynamics) and keep numerics up to minimum level. We

postpone till further publications detailed numerical analysis and comparison with the data.

Microscopic origins of the nonlinear effects (which are considered here phenomenologically)

are also beyond the scope of this presentation.

We work in a semiclassical approach [11] to the coupled-channels problem, i.e., the

projectile-target relative motion is approximated by a classical trajectory and the excitation

of the Giant Resonances is treated quantum mechanically [13]. The use of this method

is justified due to the small wavelenghts associated with the relative motion in relativistic

heavy ion collisions. The separation coordinate is treated as a classical time dependent

variable, and the projectile motion is assumed to be a straight line [2].

The intrinsic dynamics of excited nucleus is governed by a time dependent quantum

Hamiltonian (see Refs. [11], [12]). The intrinsic state |ψ(t) > of excited nucleus is the

solution of the time dependent Schr¨ odinger equation

i∂|ψ(t) >

∂t

= [H0+ V (t))] |ψ(t) >(1)

where H0is the intrinsic Hamiltonian and V is the channel-coupling interaction. We use

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the system of units where ¯ h = 1, c = 1. The standard coupled-channel problem for the

amplitudes an(t) reads

|ψ(t) >=

?

n=0

an(t) |n > exp

?

− iEnt

?

,(2)

where En is the energy of the state |n > in the wave packet |ψ?. In our treatment, the

nuclear states are specified by the numbers of excited GDR phonons, N or n. Taking scalar

product with the states < N|, we get the set of coupled equations for the amplitudes anas

functions of impact parameter b

i˙ aN(t) =

?

n=0

< N|V |n > ei(EN−En)tan(t). (3)

We assume the colliding nuclei to be in their ground states before the collision. The am-

plitudes obey the initial condition an(t → −∞) = δ(n,0) and they tend to constant values

as t → ±∞ (the interaction V (t) dies out at t → ±∞). The excitation probability of an

intrinsic state |N > in a collision with impact parameter b is given as

WN(b) = |aN(∞)|2.

and the total cross section for excitation of the state |N > is given by the integral over the

imact parameter

(4)

σN= 2π

∞

?

bgr

bWN(b)db(5)

with the grazing value bgr= 1.2(A1/3

(sp) refer to the excited (spectator) partner in a colliding projectile-target pair. We neglect

the here nuclear contribution [21] to the excitation process.

In the following, it is convenient to treat the coupled channel equations (1),(3) in terms

of the unitary operator UI(in the interaction representation) that acts in the reference basis

of multiphonon states (including the ground state |0?):

id

dtUI(t) = VI(t)UI(t),

exc+ A1/3

sp) as the lower limit. Hereafter, the labels exc

VI(t) = eiH0tV (t)e−iH0t,UI(t = −∞) = I,(6)

and the time-dependent Hamiltonian H(t) = H0+V (t) that acts in the intrinsic multi-GDR

states is given by

H0= ωNd,Nd≡

?

m

d+

mdm,

V (t) = v1(t)[(E1−1)†− (E1+1)†] + v0(t)(E10)†+ Herm.Conj.(7)

where E1†

GDR states. In the harmonic approximation, they are given by the GDR phonon creation

and destruction operators of corresponding angular momentum projection m, E1†

The function v1(t) is given [2] by

mand E1mare the dimensionless operators acting in the internal space of the multi-

m= d†

m.

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v1(t) =

w

bt)2]3/2,

[1 + (γv

w = ρZspe2γ

2b2

?

?

?

?

NexcZexc

A2/3

excmN· 80MeV

, (8)

(the corresponding expression for v0(t) can be found in Ref. [2]). Here, mN and e are the

proton mass and charge, Z, N and A denote the nuclear charge, the neutron number and

the mass number of the colliding partners, γ = (1 − v2)−1/2is relativistic factor, v is the

velocity and the parameter ρ is the deal of the strength absorbed by the collective motion

(usually assumed to be close to unity) [1].

The harmonic approximation (ideal bosons) yields the transition probabilities between

the states with numbers of GDR phonons differing by unity to grow linearly ∝ N. This

model of ideal bosons has well known exact solution (see, e.g. [12]) that is given by the

Poisson formula for the excitation probabilities

WN= e−|αharm|2|αharm|2N

N!

,

|αharm|2=

?

m=0,±1

|αharm

m

|2= 2|αharm

1

|2+ |αharm

0

|2

(9)

where the amplitudes αharm

the colliding energies sufficiently high, the longitudinal contribution (∝ |αharm

pressed by a factor proportional to γ−2[3]. In the following, we will work in the “transverse

approximation” dropping the longitudinal term for the sake of simplicity. This is a good

approximation at high energies and the results are still qualitatively valid at lower energies.

Our idea is to keep the spectrum of GDR system harmonic with the Hamiltonian H0=

ωN. That is supported by the systematics of the observed DGDR energies, E2, which yields

E2≃ (1.75 − 2)ω [1]. The conclusion on the weak anharmonicity in the spectrum follows

also from theoretical considerations [25], [26].

The transition operators E1†,E1 can however include nonlinear effects. (In particular,

this could be a result of nonlinearities in the phonon Hamiltonian obtained in higher orders

of perturbation theory). A reasonable accounting for these nonlinear effects that we adopt

in this work consists of generalization of the transition operators

m

are expressed in terms of the modified Bessel functions. At

0

|2) is sup-

E1†

m= d†

m

?

1 +

1

2kNd,E1m=

?

1 +

1

2kNd

d†

m, (10)

where the parameter k ?= 0 determines the strength of the nonlinear effects, i.e., the problem

reduces to the harmonic oscillator with linear coupling when

1

2k→ 0.(11)

If k < 0, the transition probabilities are suppressed as N grows. Positive values of k that

we will consider here correspond to the enhancement of the matrix elements.

It is convenient to introduce the three following operators D+, D−and D0

D+=

1

21/2(d+

+1− d+

−1)

D0=1

?

2k + Nd,D−=

1

21/2

?

2k + Nd(d+1− d−1),

4

?

(d+

+1− d+

−1)(d+1− d−1) + 2(2k + Nd)

?

,(12)

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with Nd≡ d+

relation for the noncompact SU(1,1) algebra

+1d+1+ d+

−1d−1. It is easy to check that they obey the standard commutation

?

D−,D0?

= D−,

?

D+,D0?

= −D+,

?

D−,D+?

= 2D0.(13)

The dynamics of the our system, in transverse approximation, can be written in terms of

the operators D+, D−and D0(12) only. In the interaction representation, the evolution

equation (6) takes the form

id

dtUI(t) =

?v1(t)

√k

eiωtD†+v1(t)

√k

e−iωtD−

?

UI(t),(14)

for UI(t) (the last equation follows from (6), (7) after using the commutation relations (13)

and [Nd,D±] = ±D±.

From purely mathematical viewpoint, the problem described by the last equation drops

into the universality class of the systems with SU(1,1) dynamics that can be analyzed by

means of generalized coherent states [29]for the SU(1,1) algebra. In particular, the problem

of the parametric excitation of a one-dimensional harmonic oscillator ( [29], [30]) belongs to

the same class. The physical meaning of the algebra generators (12) in our case is of course

quite different from that of [29,30]. (For other algeabraic approaches to scattering problems,

see Refs. [19], [20]).

The formal solution of our problem is given by the expression for the unitary operator

UI(t) as a time-ordered exponential (see, e.g., [31])

UI(t) = Texp

−i

t

?

−∞

dt′VI(t′)

(15)

Due to closure of the commutation relations between the operators D+,D−and D0that

enter the exponential in Eq.(15), the time-ordered exponential can be represented in an-

other equivalent form that involve ordinary operator exponentials only (see, e.g., [18]). The

operator UI(t) can be expressed as

UI(t) = exp

?α(t)

√kD+

?

exp

??

log

?

1 −|α(t)|2

k

?

− iφ(t)

?

D0

?

exp

?

−α∗(t)

√k

D−

?

(16)

where one has the product of ordinary exponentials [29] which contain some time-dependent

complex number α(t) (star means complex conjugation) and real number φ(t) (phase). The

functions α(t) and φ(t) can be found from simple differential equations which relate the

unknown α(t) and φ(t) with the function v1(t) in the Hamiltonian H(t). These equations

(see below) can be restored after substituting the right hand side of Eq.(16) into the left

hand side of the Schr¨ odinger equation for the operator UI(t) (14) and collecting the terms

which have the same operator structure.

Proceeding this way, we obtain, after some algebraic manipulations, from (14) with using

the commutation relations (13), the formula

d

dteA=

1

?

0

dτeτA

?d

dtA

?

e(1−τ)A

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