научная статья по теме THE PROCESS OF COULOMB DISSOCIATION OF WEAKLY BOUND RELATIVISTIC NUCLEI AND HYPERNUCLEI WITHIN THE TWO-CLUSTER MODEL Физика

Текст научной статьи на тему «THE PROCESS OF COULOMB DISSOCIATION OF WEAKLY BOUND RELATIVISTIC NUCLEI AND HYPERNUCLEI WITHIN THE TWO-CLUSTER MODEL»

ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 9, с. 1663-1667

ЯДРА. Теория

THE PROCESS OF COULOMB DISSOCIATION OF WEAKLY BOUND RELATIVISTIC NUCLEI AND HYPERNUCLEI WITHIN THE TWO-CLUSTER MODEL

© 2007 V. L. Lyuboshitz1), V. V. Lyuboshitz2)*

Received October 31, 2006

Using the analogy with the problem of ionization and excitation of atoms at the propagation of relativistic charged particles through a bulk of matter, the process of Coulomb dissociation of weakly bound relativistic nuclei and hypernuclei is theoretically investigated in the framework of the two-cluster deuteron-like model. Explicit expressions for the total cross section of Coulomb disintegration of weakly bound systems are derived taking into account the corrections due to a finite size of a target nucleus. Numerical estimations for the Coulomb dissociation of relativistic hypernuclei 3Ha and 6HeA are performed. It is shown that due to a sharp dependence of the Coulomb dissociation cross section on the binding energy, experimental measurements of the cross section allow one to determine the values of binding energy for these systems.

PACS:25.20.-x, 21.60.Gx, 21.80.+a

1. INTRODUCTION

The process of dissociation of nuclei in the Coulomb field of fast charged particles has been discussed repeatedly (see, e.g., [1—3]). In the recent years the interest to the process was revived in connection with the constructing of beams of relativistic nuclei and the problem of identification of relativistic hypernuclei [4, 5]. In [6], excitation and disintegration of relativistic nuclei and hypernuclei were studied using a direct analogy with the problem of ionization and excitation of atoms at the propagation of relativistic charged particles through a bulk of matter [7]. In this work (see also [8]), we will discuss application of the results of [6] to weakly bound deuteron-like systems consisting of two compact clusters (charged and neutral), the distance between which essentially exceeds sizes of clusters themselves as well as a target nucleus radius R. A sharp increase of the Coulomb dissociation cross section with the decrease of binding energies allows one to determine experimentally the binding energies of weakly bound nuclei and hypernuclei studying their disintegration at ultrarelativistic velocities. We will investigate also the role of a finite size of target nucleus.

"Veksle r—Baldin Laboratory of High Energies, Joint Institute

for Nuclear Research, Dubna, Russia.

2)Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Dubna, Russia. E-mail: Valery.Lyuboshitz@jinr.ru

2. EXCITATION AND DISINTEGRATION OF RELATIVISTIC NUCLEI IN THE COULOMB FIELD OF A POINT-LIKE CHARGE

It was shown in [6] that the total cross section of excitation and dissociation of a relativistic nucleus in the field of an immovable Coulomb center with the charge Ze can be presented in the following form:

a =

n = 0

4n(Za)2

qi +

£n0

Y2v2

-2

(1)

Ш0)

£n0

v

Y2v2

d(qi ).

Here, h = c = 1; a = e2 = 1/137 is the electromagnetic constant; q^ is the transverse momentum transferred to the nucleus; v = |v| is the velocity of the projectile nucleus in the rest frame of the Coulomb center (i.e., in the laboratory frame); (n|j|0) is the vector of the current of transition from the ground state |0) of the projectile nucleus to the excited state of the continuous or discrete spectrum |n); en0 is the excitation energy. The summation in Eq. (1) is performed over all quantum numbers of final states including spin and angular variables, and the upper bar denotes the averaging over polarizations of the

1663

x

2

v

2

1664

V.L. LYUBOSHITZ, VV LYUBOSHITZ

initial ground state of the projectile nucleus, which is assumed to be unpolarized3).

At small transverse and longitudinal momenta transferred to a projectile nucleus (|q±| < 1/Rpr, q\\ = tno/v < 1/Rpr, where Rpr is the radius of the projectile) the transition current is expressed directly through the matrix element of the dipole moment

(n|j|0) = -Σn0

n

E>

0

(2)

In Eq. (2), the summation is performed over the coordinates of all the protons in the projectile nucleus. In accordance with the rule of multiplication of matrices, taking into account the equality (0| Ep rp|0) = 0, that arises due to the space parity conservation, the following relation holds:

E

n = 0

/ \ 2 / ( \2

/ n » = » Erp

\ p / \ V p )

0 • (3)

Finally, dividing the integration range into two intervals corresponding to very small and larger q2, the following formula for the Coulomb dissociation cross section emerges [6]

3v2

E>

0

(4)

ln

Y2v2

4n<0|(£ p rp )2|0)

- 2A + B - v2

Here, ebin is the binding energy of a projectile nucleus, the constant

A = In — = (5)

ebin

En = 0 ME p rP10)12 ln(e™0/£bin)

<»ke p rP)2 |o)

involves a dependence of a minimal momentum transfer at the transition to excited states In) upon the excitation energy eno > ebin (e = ebineA < 1/Rpr).

The constant B describes a contribution of comparatively large transfers of transverse momentum. Calculations lead to the expressions

B = -3 ln y

d (G(y)\

dy\ y )

dy,

(6)

y = q2 0

E]

Taking into account the completeness condition one gets the following expression for the function G(y) :

G(y) = E

n = 0

n

^2exp(-iq± rp)

0

= (7)

^exp(-iq±rp) 0\ -

^exp(-iq± rP)

At y < 1 the function G(y) k> y/3; at y » 1 the function G(y) ^ z, where z is the number of protons in a projectile. For a projectile nucleus with the charge of unity we have: G(y) = 1 — F2(y), where

F (y) = F (q2± ) = <0| exp(—¿q±rp)|0) is the electromagnetic form factor of the ground state.

3. CONTRIBUTION OF FINITE SIZES OF THE TARGET NUCLEUS

Taking into account that a target nucleus is not point-like, one should subtract the correction term AB from the constant B in Eq. (4):

AB = 3

0

G(y)(1 - H (y))

y2

dy.

(8)

Here, G(y) is determined by Eq. (7) as before, and

\ 2

3) At the first glance, Eq. (1) for the cross section of Coulomb dissociation corresponds to the one-photon-exchange approximation, i.e., to the approximation Za C 1. However, the analysis shows that even at large values of Z, when Za ~ 1, the corrections to this formula still remain small. This is connected with the fact that the considered result can be justified in the framework of the impulse approach with the amplitude of the Coulomb scattering. In doing so, the exact amplitude of the Coulomb scattering at small transferred momenta, where the main contribution into a is provided, differs from the amplitude obtained within the Born approximation only by phase.

h (y) = h

0

^exp(-iq± rP)

0'

(9)

is the square of the electromagnetic form factor of the ground state of a target nucleus |0'). In doing so, rp is the coordinate of a proton in a target nucleus. For the uniform distribution of protons over a target nucleus

H (y) = 9

sin x

cos x

2

0

p

p

2

2

0

2

2

p

x

2

THE PROCESS OF COULOMB DISSOCIATION

1665

where

x — Vv

R

tag

y/№Y, P rP )2|0)

(ii:

Rtag is the radius of a target nucleus. It is clear that H(0) = 1.

Equation (8) describes the influence of a finite size of a target nucleus on the Coulomb dissociation of a projectile without excitation and breakup of a target nucleus. If a quantum state of a target nucleus is not fixed and transitions into all excited states are taken into account, the function H(y) should be replaced by the following expression (see [6]):

H (y) —

1

Z2

0'

0'

(9a)

It is easy to demonstrate that at a uniform distribution of the charge Z the functions H(y) and H(y) are connected by the relation

1 - H (y) — [1 - H (y)]

Z - 1 Z '

(9b)

a — a ln y + b,

(12)

a = o

MO =

where r = |r| is the distance between the clusters,

/ mm \ -1/2 p = \2~m~ ebinj

(14)

Here, m1 and m2 are the masses of charged and neutral clusters, respectively; M = m1 + m2 is the mass of the deuteron-like nucleus; e0 = (—ebin) is the energy of its bound state. In the given case the quantity (0|(£p rp)2|0) can be explicitly determined:

E'

° - ^ (£)' x

J 4 (r)r4dr = Mff P

(15)

—_z _

2 4 m1Mebil

where z is a number of protons in the charged cluster.

Using Eqs. (7) and (13) we obtain the analytical expression for the function G(y):

G(y)= (16)

— z2 1 -

Thus, in case of a heavy target nucleus the corrections AB and KB = [1 - (1/Z)]AB practically coincide.

It is clear from Eq. (4) that in the case of relativistic nuclei with small binding energies the principal contribution into the Coulomb dissociation cross section is conditioned by the logarithmic term being proportional to M72v2/ebin(0KSpr)2|0)]. At ultrarelativistic energies of projectile nucleus (v ^ 1, Y > 1) the Coulomb dissociation cross section of this nucleus, irrespective of the relation between radii of projectile and target nuclei, has a structure

J 42(r) exp(-¿q±rM) ^

— Z2 1 - 2Z2 y

arctan ( ^ 12y2J

Here, according to Eq. (15), 1

m2

= _ 2 2_

V = 4q±Z miMebh

(17)

According to Eq. (14), at very small binding energies the effective radius p of the projectile nucleus considerably exceeds the target radius R (p » R). Then, in the first approximation, we may take H(y) = = 1, considering the target nucleus as a point-like Coulomb center. Substituting (16) into Eq. (6), we obtain the following value for the constant B:

4. WEAKLY BOUND SYSTEMS.

TWO-CLUSTER MODEL

Now we consider the Coulomb disintegration of the "friable", deuteron-like nuclei consisting of two clusters, charged and neutral, the average distance between which is significantly larger than a radius of the force action as well as the cluster sizes. In this case excited bound states are absent, and the normalized wave function of the ground state corresponding to the zero orbital momentum has the following form:

1 exp(—r/p)

B — -3z2y ln y

0

d (1 2z2 dy\y y2

(18)

arctan I ^ 122Z2 J

dy — ln(2z2 ) + C,

where (see [6])

^ „ , , ,1 arctan u C — 6 ln u ^ +

u3 u4(1 + u2)

(19)

—(arctan u)2j ~ 0.316.

Meantime, taking into account that

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