Bound-free continuum of O-

J. Quant. Specfrosc. Radiat. Transfer. Vol. 5, pp. 449-452.

Pergamon Press Ltd., 1965. Printed in Great Britain

NOTE

BOUND-FREE

CONTINUUM

OF 0-*

R. G. BREENE, JR.t Space Sciences Laboratory, Missiles and Space Vehicles Division General Electric Company, Valley Forge, Pennsylvania (Received 6 July 1964) Abstract-An approximate treatment is described for estimating cross sections in the bound-free tinuum of O-. Only the contribution of the free s-electron is included.

THE ABSORPTION cross

0. =

section

for an ionization

process

~~a~v+2Lr+ 1)(2&s+ 1)X MLML

may

c

2 I

con-

be written:

~~$(O-)M~(Oe)dr~ZJ (la)

MS

In equation (la) Y is the frequency of the light quantum absorbed; v is the velocity of the ejected electron while a9 is the atomic unit of length. Lc and St are respectively the total initial orbital and spin angular momentum; ML and MQ are respectively the orbital angular momentum projections for the initial and final states; MS is the final spin angular momentum projection. The O- bound-free continuum may be represented as: o-(2P) Only the free s-electron

+ hv + O(3P) +e(s,d)

is considered,

and for this case equation 1[@P)

Q = 2.0436 x lo-19k(k2+21)

C zi7,L(3~;s)d~j

IJ

wherein the symbol k represents electron affinity of O-. Equation in atomic units. The equivalent

(la) becomes:

I

the linear momentum of the free electron while I is the (lb) contains the dipole length matrix element expressed dipole velocity form(l) of the operator yields:

fs = 8.175 x 10-19

k

UC)

(k2+21)

When single determinant functions are used for the two system states, the matrix elements in equations (1) reduce to integrals over the wave functions of the detaching electron; these integrals are multiplied by an overlap factor which is equated to unity. * This work was supported by the General Electric Company Research Program. t Consultant: 4761 Mad River Road, Kettering 29, Ohio. 449

under its Contractors

Independent

450

R. G.

BREENE,

JR.

The wave functions for the free s-electron were obtained from the G.E.C. programsc2); these are, of course, tabular functions having the asymptotic form (kr)-l sin (kr + Se)

(2) where 60 is the s-wave phase shift. These phase shifts are displayed as a function of k in Fig. 1. The asymptotic form for the free function is used in all cases and the result supporting this approximation is discussed briefly below. 7-

5-

z 0) $ z a

3-

2-

I-

I

0

01

I

I

I

I

I

I

I

0.2

03

0.4

05

0.6

0.7

0.8

Linear

momentum-k

,

atomic

0.9

units

FIG. 1. The s-wave phase shift for the free electron in the presence of neutral oxygen.

When equation (2) is used for the free function and a single-exponent, function, the dipole length matrix element takes the following form: M

6(c4 -6Pkz+k4)

= 4(1577)1’2 245k(t2 -k2) L

3k

(52+@)4

'OS *'+

(t2+k2)4

5, for the bound

sin SO I

The evaluation of equations (3) and (lb) for our bound functions(s) produces results differing from experimental results@) (Curve I of Fig. 2) by some two orders of magnitude. The result of using the CLEMENTI-MCLEAN@) bound function in the same evaluation is plotted as Curve II of Fig. 2. The dipole velocity matrix element for a single exponential bound function and equation (2) may be expressed as: 6k25 cm So 2k(352 -k2) + 52 -k2 1M _ 4(52+‘2 sin So v---3 (P+k2)3 - [ (52+k2)3 k(12 + k2)2 I I 1

(4)

The results of this evaluation of equations (4) and (lb) for our single-exponential bound function and for the Clementi-McLean multiple-exponential function yield Curves III and IV, respectively, in Fig. 2. It is to be seen that the ab initio calculation represented by Curve IV of Fig. 2 differs from experiment by about 30 per cent and is of correct shape.

Bound-free

continuum

of O-

451

FIG. 2. The cross section for the photoionization of the negative oxygen ion. (I) The experimental curve; (11) Single-exponential bound functions and the dipole length operator; (III) Multiple-exponential bound functions and the dipole velocity operator; (IV) Multipleexponential bound functions and the dipole velocity operator; (V) Results using the tabular free function; (VI) The Klein and Brueckner calculation.

The dipole length form of the operator weights most heavily those regions of space far removed from the nucleus, and thus we might expect that the asymptotic form of the free function, equation (2), would introduce some error into the evaluation of equation (1~). Therefore, the evaluation of equation (lc) was carried out by using the tabular form of the free function and our single-exponential bound function. The resulting continuum, which is displayed as Curve V of Fig. 2, is identical with the result obtained by using the asymptotic form for the free function. This result agrees with the conclusion reached by KLEIN and BRUECKNER.(@ These authors have obtained good experimental agreement by means of a calculation in which the polarization potential parameter was adjusted so that the experimental binding energy of 1.45 V was obtained for 0-. Klein and Brueckner’s result is presented as Curve VI of Fig. 2. It should be noted that they considered their polarization potential to contain the effects of exchange, which may to some extent account for their result being superior to ours. We compare their results with our scattering length, ro, and effective range, a. In order to determine these quantities, KLEIN and BRUECKNERused the first two terms of the expansion(7): k cots0 = - ++

;kz +O(k4)

and obtained YO= 0.860 and a = 1.613. The two-term expansion does not fit the k cot curve accurately; consequently, we found it necessary to include the term in k4. Under

80

452

R. G. BREENE,JR.

these conditions we obtained YO= 1.951 and a = l-227. The second node in our s-wave function is somewhat closer to the nucleus than is that of Klein and Brueckner so that a smaller scattering length would be anticipated. Our polarized potential is such that an estimate of the effective range, taken from its graph, approximates to the value calculated here. The effective range computed by Klein and Brueckner is only about one-half of our result; again this is at least partially attributable to the fact that their method automatically includes an exchange contribution. REFERENCES 1. S. CHANDRASEKHAR, Astrophys. J. 102,223 (1945). 2. R. G. BREENE,JR., Phys. Rev. 115,93 (1959); ibid. 123, 1718 (1961). 3. R. G. BREENE,JR., Phys. Rev. 111, 1111 (1958); ibid. 113,809 (1959). 4. L. M. BRANSCOMBE, D. S. BIJRCH,S. J. SMITHand S. GELTMAN,Phys. Rev. 111,504 (1958). 5. E. CLEMENTIand A. D. MCLEAN, Phys. Rev. 133A, 419 (1964). 6. M. M. KLEIN and K. A. BRUECKNER, Phys. Rev. 111, 1115 (1958). 7. TA-YOU Wu and T. OHMURA. Quantum Theory of Scattering, p. 71. Prentice-Hall, Englewood Cliffs (1962).