Complex representation of the quantumstatistical second virial coefficient

Volume 29A, number 8

COMPLEX

PHYSICS

LETTERS

30 June 1969

REPRESENTATION OF THE QUANTUMSTATISTICAL VIRIAL COEFFICIENT

SECOND

W. D. KRAEFT, W. EBELING and D. KREMP Universittit Restock,

Sektion Physik der

Restock,

DDR

Received 20 May 1969

Using the resolvent representation the second virial coefficient of quantum systems is expressed by contour integrals over complex scattering quantities (e.g. Jost functions).

We consider a dilute quantum gas of species a, b, c with densities na,n b, . . . and spins s,, sb, . . . in a volume V. The free energy of the system can be represented by the cluster expansion F = Fid-kTV(pb%q,Baj,(T)+...)

0)

Using the well known resolvent representation of the exponential operators we get for the second virial coefficient [l, 21

&(I+

kfi

Bab

bab)!

(T) = (2 sa+ i)(2Sb + 1)27ri cJexp(-Pz)F(z)

dz;

(2)

where xab = ti (2 mab kT)-+ and H,b the Hamiltonian of relative motion (trace over relative motion and spin). The contour C encircles all singularities of the resolvent. The trace of the resolvent F(Z) can be calculated using the spectral representation of the resolvent. We get F(z) = .$ g

P(E) = Tr (eE - Pgo)

+ n-

Because eq. (3) is a Cauchy type [3] integral, the function F(z) has the following analytic properties : F(z) is an analytic function with a branch cut along the positive real axis; the jump is F(E + k) - F(E - ic) = 2nip(E). The poles of F(z) are located at the bound states z = En,

T&.

p(E) may be determined in the case of a general potential by the t-matrix

the residues are

(T-matrix of relative mo-

tion) p(E) =

c

spin I

$a(p

a2011 Ret (E+ic)lalu2P)*

+

From eq. (2) we may derive the known form Bat(T)

= 47&X3

(1+6ab)!

ab (2sa+1)(2sb+l)

]c

exp(-PE,) + ra

exp(-@%(E)

1

(5)

For systems without spin we may deBy inserting (4) into (5) we get a t-matrix representation of B .“hb’ rive from (4) and (5) after a partial integration the result wluc was given by Baumgartl [Z]. For spherically symmetric potentials we have

466

PHYSICS

Volume ZSA, number 8

LETTERS

(-If ab 2sa+l

30 June 1969

1

1 d$(E) 1~ dE

(6)

In this case F(z) may be given explicitely using the fact, that an analytic function is determined by its singularities. As well known from scattering theory the Jost function Dl (z) [4]‘has zeroes at the bound states and a branch cut along the positive real axis. It can be proved easily that the following function F(z) = (sa+I)(ab+I) (1+6ab)!

5 (%+l)[I f bab &](-I) 1=0

$ln Dl(Z)

has the same analytic properties as the trace of the resolvent (3), i.e., -(d/dz) lnD&z) has poles at the bound states with the corresponding residues 1, and (7) has a branch cut along the positive real axis with the discontinuity (6). Eqs. (2) and (7) yield a new useful representation of the second virial coefficient if the Jost function is known explicitely. For the special case of Coulomb interaction a corresponding representation was given in an earlier paper [5].

References 1. K.M.Watson, Phys. Rev. 103 (1956) 489. 2. 3. 4. 5.

B. J. Baumgartl, 2. Physik 198 (1967) 148. N. J. Muschelischwili, Sing&ire Integralgleichungen,(Akademie- Verlag, Berlin 1965) M. L. Goldberg and K. M. Watson, Collision theory, (Wiley and Sons, New York 1964) W. Ebeling, Ann. Physik 22 (1968) 33. *****

CALCULATION

OF

THE lD-l IN THE

Po TRANSITION Be SEQUENCE

PROBABILITY

J. LINDERBERG Department

of Chemistry,

Aarhus University, Denmark

Received 13 May 1969

Transition moment and energy has been calculated by a perturbation method for the transition (3p2)lD to (~~2p)lpO in beryllium like ions. Comparison with experimental values shows better agreement than earlier calculations.

Experiments on the life-times of the (2p2)lD state of the ions in the/beryllium sequence have been performed by several groups [l-4]. The results have been in considerable disaccord with the calculations by Weiss [5]. It was then considered to be relevant to try an alternative calculational scheme and to extend some previous investigations .r61 on transition enerties to obtain also the transition moments. We have employed the Hylleraas perturbation method [?] with the inverse atomic number as the expansion parameter. The general technique has 2

been discussed at length by Cohen and Dalgarno [8]. The two quantities of importance for the evaluation of the transition probability are the energy and the dipole matrix element. An approximate linear function has been calculated earlier for the transition energy [Sj and is

E(lD) - E(IPO) = 0.0645 2 - 0.1896+, . a.u.

(1)

= 0.0645 (Z- 2.94) a.u. The formula is reasonably accurate over the range of atomic numbers considered here as can be seen in table 1. The calculation of the dipole

467