1945ApJ...102..223C

ON THE CONTINUOUS ABSORPTION COEFFICIENT OF THE NEGATIVE HYDROGEN ION
S. CHANDRASEKHAR
Yerkes Observatory Received June 25, 1945

ABSTRACT
In this paper it is shown that the continuous absorption coefficient of the negative hydrogen ion is most reliably determined by a formula for the absorption cross-section which involves the matrix element of the momentum operator. A new absorption curve for H- has been determined which places the maximum at X8500 A; at this wave length the atomic absorption coefficient has the value 4.37 X 10-17 cm2•
I. Introduction.-In earlier discussions1 by the writer attention has been drawn to the fact that the continuous absorption coefficient of the negative hydrogen ion, evaluated in terms of the matrix element
(1)

(where '1i'a denotes the wave function of the ground state of the ion and '¥c the wave function belonging to a continuous state normalized to correspond to an outgoing electron of unit density), depends very much on '¥a in regions of the configuration space which are relatively far from the hydrogenic core. This has the consequence that the absorption cro~s-sections are not trustworthily determined if wave functions derived by applications of the Ritz principle are used in the calculation of the matrix elements according to equation (1). This is evident, for example, from Figure 1, in which we have plotted the absorption coefficients as determined by Williamson2 and Henrich,3 using wave functions of the forms
( 2)
and

+ + + + + t + + o '¥a = <Dl e-o._•/2 ( 1 {3u "(t 2

s 1: s2

u 2 X6t4 x1t6

} (3)

+ + + xst4U 2 x9t2u2 x10t2u4) ,

respectively. (In eqs. [2] and [3] <Dl is the normalizing factor; and a, /3, 'Y, etc., are con-
stants determined by the Ritz condition of minimum ener_gy,

s=r2+r1, t=r2-r1, and u=r12,

(4)

where r1, r2, and r12 are the distances of the two electrons from the nucleus and from each other, respectively.) The wide divergence between the two curves in Figure 1 is too large to be explained in terms of only the improvement in energy effected by the wave function (3): it must arise principally from the fact that in the evaluation of the matrix elements according to equation (1) parts of the wave function are used which do not contribute appreciably to the energy integral-and are therefore poorly determined. Indeed, this sen-

1 Ap. J., 100, 176, 1944; also Rev.Mod. Phys., 16,301, 1944. 2 Ap. J., 96, 438, 1942. a Ap. J., 99, 59, 1944.
223

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1945ApJ...102..223C

224

S. CHANDRASEKHAR

sitiveness of the derived absorption coefficients to wave functions effecting only relatively slight improvements in the energy makes it difficult to assess the reliability of the computed absorption coefficients. However, in this paper we shall show how these difficulties can be avoided by using a somewhat different formula for the absorption cross-section.
2. Alternative formulae for evaluating the absorption coe.fficient.-It is well known that in the classical theory the radiative characteristics of an oscillating dipole can be expressed in terms of either its dipole moment, its momentum, or its acceleration. There are, of course, analogous formulations in the quantum theory, the matrix element

(a I zi I b) = Sfa* z 11hd r

(5)

4

l.

n

8000 10000 12000
A.-
FIG. 1.-A comparison of the continuous absorption coefficient of H- computed according to formula
(I) and with wave functions of forms (2) (curve I) and (3) (curve II). The ordinates denote the absorption coefficients in units of 10-17 cm2; the abscissae, the wave length in angstroms.

for the co-ordinate z; of the jth electron in an atom being simply related to the corre-

sponding matrix element of the momentum operator or the acceleration. Thus, we have

the relations

•

•

J J (a I Zjl

b)

=

1 (Ea -Eb)

af: OZj

Vlbdr

=

-

1 (Ea-&)

* afb Via OZj dr

(6)

and

(7)

if all the quantities are measured in Hartree's atomic units and where Ea and Eb denote the energies of the states indicated by the letters d and band where V denotes the potential energy arising from Coulomb interactions between the particles. More particularly for an atom (or ion) with two electrons, we have

µz = S'1rd (z1 + Z2)'¥cdr,

(8)

f * ( a a ) · µz=-(Ed-E1 c) 'Yd OZ1+0Z2 'Ycdr,

(9)

and

f µ.= (Ed2Ec) 2 '1rd (;;+;i)'1rcdr.

(10)

While the foregoing formulae are entirely equivalent to each other if 'Yd abd 'Ye are exact solutions of the wave equation, they are of different merits for the evaluation of µz if approximate wave functions are used. Thus, it is evident that formula (8) uses parts of the

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1945ApJ...102..223C

NEGATIVE HYDROGEN ION

225

:on:figuration space, which are more distant than relevant, for example, in the evaluation )f the energy; similarly,'formula (10) uses the wave functions in regions much nearer the xigin. It would appear that formula (9) is the most suitable one for the evaluation of u,, particularly when wave functions derived by applications of the Ritz principle are '1sed. The calculations which we shall present in the following sections confirm this anticipation; but before we proceed to such calcul::ttions, it is useful to have the explicit formulae for the absorption cross-sections on the basis of equations (8), (9), and (10) ..
In ordinary (c.g.s.) units the standard formula for the atomic absorption coefficient Kv for radiation of frequency v, in which an electron with a velocity vis ejected, is

K,=

321r4 m2 e2 3 h3c

vv\S':ItJ(z1+z2)'¥edr[

2

,
,

(11)

where m, e, h, and c have their usual meanings. (In writing eq. [11] it has been assumed

that the electron is ejected in the z-direction; see eq. [15] below.) By inserting the numeri-

cal values for the various atomic constants equation (11) can be expressed in the form

K, = 8. 5 61 X 10-19 (Patk Iµz [2) cm2 ,

(12)

where k denotes the momentum of the ejected electron and Pat the frequency of the

radiation absorbed, both measured in atomic units, and where, moreover, the matrix

element µz has also to be evaluated in atomic units.

If I denotes the electron affinity (also expressed in atomic units)

47!"Pat = k 2 + 2I ,

(13)

and depending on which of the formulae (8), (9), and (10) we use for evaluating Kv, we

. have

Kv=6.812X 10-20 k(k 2 +2I) I S':Itd (z1+z2)'¥cdr\ 2 •

(I)

I f + J2, Kv= 2. 725 X 10-19 (k2~ 2[) 'Yd (a~l a~JjtcdT

(II)

and

f Kv = 1.090 X 10-is (k 2: 2[) 3 l

2
'Yd(:~+ :i)'YcdT 1

(III)

Finally, we may note that if X denotes the wave length of the radiation measured in

angstroms, then

911. 3
X= k2+ 2I A.

(14)

3. The continuous absorption coefficient of H- evaluated according to formula (III).-

As we have already indicated, in the customary evaluations of Kv accord1ng to formula

(I) the relatively more distant parts of the wave function are used. It is evident that we

shall be going to the opposite extreme in using the wave function principally only near

the origin if we evaluate Kv according to formula (III). For this reason it is of interest to

consider first the absorption coefficient as determined by this formula.

•

In evaluating K, according to formula (III), we shall use for 'Yd a wave function of

form (3) and for 'Ye a plane wave representation of the outgoing electron:

(15)

(In § 5 we refer to an improvement in 'Ye which can be incorporated without much diffi-
culty at this stage.) For 'Yd and 'Ye of forms (3) and (15) the evaluation of the matrix element
(16)

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1945ApJ...102..223C

226

S. CHANDRASEKHAR

is straightforward, though it is somewhat involved. We find

f :i :p ! 1 + '¥d (

'¥cd r = - (20481r3) 1/2 (1 a) 3 2 [t//£}a)

~ x ~ +

.!£<.1+2a) _
~11

-lo/3 ( 1 +

a) 4 { ~ a s<.1+2aJ + ~11

~

b
1

.c<.1+2aJ
1

1t] ,

i=-2

i=-1

i=-1

)

where we have used the following abbreviations:

!f}P) = lco e-PY ( k cos ky - sin/y) yidy

(j = - 2, -1. .... ) '

= (j - 1) !pi {j pk cos [ (j + 1) ~] - sin j 0 (j '?; 1) .

= pk cos~ - ~

(j = 0) '

os:

=P~ -k

(j = - 1).

_J_) =..!2. (pk p2

(j = - 2).

S(P) = f 00 e-Pvyi sin ky dy
1 Jo
+ = j!pi+1 sin [ (j 1) ~]

1 Ci= - 1. o ..... ) .
~ ~j=0,1, .... ).

(19

=~

(j = - 1) .

j

and

j + C(P) = 00 e-Pvyi cos ky dy = j!pi+l cos [ (j 1) ~]

1

0

(j = o. 1••••• ) '} (20

(-P- \ ~I) = [ 00 e-Pv (eav ..:_cos ky) dy = log

sec

o

Y

p-a

( j = - 1).

where

1
+ P = (k2 p2) 1;2 and

(21

and

l-2 = 4q2{3 ' l-1 = l+q (t/3+ 8) + 12q2 ( 'Y+e+r) +360q4 (x6+X9) + 20,160q6(x1+xs+x10) ,

lo= (o+/3) -6q ("f-e) -120q3 (2x6+X9) -5040q5 (3x1+2xs+x10),

l1

=

-

{3

6.

+ q

('Y+e+r) +

24q2 (3x6+X9) +

120q4 (45x1+ 2 lxs+ 13x10) . ,

r l2 = - 3 q-4q (3x6+2x9) -40q3 (30x1+13xs+12x10)

(

la= (x6+fxg)+4q2 (45x1+29xs+21x10).

l4 = _ 3xq9 - 2q (9x1+ 12xs+ 7x10.) .

l5 = x1+ li- Cxs+x10) .

.

1

l6 = -3q Cxs+ 2x10) .

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1945ApJ...102..223C

NEGATIVE HYDROGEN ION X-2=-4{3q2 ;

227 ( 23) •

where

and w·here

a1

=

677 4

[ 477k 2

(5a4

-

10a2 k2 +

k4 )

+

(a 4 -

+ a O =671 3 a[1671k 2 (a2 -k2 )

(a2 -3k2 )],

+ a 1 = 377 2 [ 471k 2 (3a 2 - k2) (a 2 - k2)] •

+ a2 = '17 a ( 817 k2 1) .

, aa = 77 k2 .

6a2 k2 +

k4 )] ,

= + + b1

2471 4 ak [77 (a4 - 10a2k 2

5k4

)

-

(a 2 -

k2 )].

b0 = -671 3k [471 (a 4 -6a 2k2 +k4 ) - (3a 2 -k 2)],

bi= - 671 2 ak [ 277 (a 2 - 3k 2) - 1].

b2 = - 11 k [411 ( a2 - k2 ) - 1] .

ba = - 71 ak .

(24) ( 25) ( 26) ( 2 7)

Putting X6 = x1 = .... = x10 = 0 in the foregoing equations, we shall obtain the for-
mulae which can be used with a wave function of form (2). By using for the constants of wave functions (2) and (3) the values determined by
Williamson and Henrich, the atomic absorption coefficient K, has been computed according to the foregoing formulae for various wave lengths. The results of the calculations are given in Table 1 and are further illustrated in Figure 2. It is seen that, in contrast to what happened when formula (I) was used (cf. Fig. 1), wave function (2) now predicts systematically larger values for Kv than does wave function (3). The divergence between the two curves must now be attributed to the overweighting of the wave function near the origin, where it is again poorly determined by the Ritz method.
4. The continuous absorption coefficient of H- evaluated according to formula (1I).Finally, returning to formula (II), which would appear to have the best chances for determining Kv most reliably, the calculations were again carried through for wave1unctions '¥d of forms (2) and (3) and for '¥c of form (15). Before we give the results of the calculations, we may note that for '¥d of form (3) and for '¥c of form (15)

f + _i_ * (__j_+~) d '¥d dZ1 OZ2 '¥c 7 -

-

(2048 3)1/2
7r

(1

<ol
a)

•
3

k2

r ~[ ,D(a)
L ~ l i;;c_,i

k2 S(a) 6 + ~ +1 A.9?,(1+2a) +

{ ~7 s

+ ~ +1 S(l+2a) }]

~11
i=-1

~11

~11

'

i=O

i=O

(28)

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1945ApJ...102..223C

TABLE 1
THE CONTINUOUS .ABSORPTION COEFFICIENT OF H- COMPUTED
ACCORDING TO FORMULA ffi 'AND ·WITH WAVE FuNCTIONS
OF FORMS (2) AND {3)

X K). 1017 C.M2

X K"),.

1017 CM2

).. (AJ

' With Wave With Wave Function (3) Function (2)

).. (A)

With Wave With Wave Function (3) Function (2)

1000 ....... 2000 ....... 2500 ....... 3000 ....... 3500 ....... 4000 ....... 4500 ....... 5000 ....... 5500 ....... 6000 . . . . . . . 6500 .......

0.225 0.955 • 1.459 2.010 2.580 3.139 3.657 4.118 4.505 4.812 5.036

0.241 1.010 1.538 2.125 2.752 3.400 4.046 4.676 5.271 5.820 6.310

7000 ...... 7500 ...... 8000 ...... 8500 ......
9000 ...... 9500 ...... 10000 ...... 12000 ...... 14000 ...... 16000 ......

5.173 5.225 5.204 5.106 4.946 4.724 4.453 3.031 1.407 0.149

6.732 7.070 7.333 7.496 7.567 7.536 7 .411 5.952 3.355 0.401

8 7 6 5

3 2

0

2000 4000 6000 aooo A~ooo 12000

FIG. 2.-A comparison of the continuous absorption coefficient of H- computed according to formula (III) and with wave functions of forms (2) (curve I) and (3) (curve II). The ordinates denote the absorption coefficients in units of 10-17 cm2; the abscissae, the wave length in angstroms.

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1945ApJ...102..223C

NEGATIVE HYDROGEN ION

where

l-1 = 4 /3 q3 ; lo = 0 ; 11 = - {3 q .

l

12 = - 2qt -40q3x9- 1680q5 (x8 + 2x1o). [3 = 16q2x9 + 960q4 Cxs + x10), ,

j

[4 = - 2qx9 - 80q3 (3xs + 2x10) .

ls= 3 2q2 (xs + x10) ; 1 ls= - 2q (xs + 2x10) ,

Ao= - 4{3 q2 ; So= 4{3 q2 ,
S1 = 1+3qo+12q2 (,y+e+r) +360q4 (x6+X9) +20,160q6 (x1+xs+X10),

S2= Co+B)-6q c,,-E)-120q2 (2xa+x9)-5040q5 (3x1+2xs+x10)'
= S3 ('}'+e+r) +24q2 (3x6+X9) +120q4 (45x1+21xs+13x10), s4 = -6 q ( 2xa+xg) -80 q3 (15x1+6xs+Sx10),
ss = Cxa+x9) +4q2 (45x1+21xs+13x10),
sa = -6q (3x1+2xs+x10), s1 = x1+xs+x10

and

229 ( 29) (30) (31) (3 2)

Further, in equation (28) the quantities 1:_;Pl, sJvl, and q have the same meanings as in equations (18), (19), (21), and (24).
TABLE 2
THE CONTINUOUS ABSORPTION COEFFICIENT OF H- COMPUTED
ACCORDING TO FORMULA II AND WITH WAVE FUNCTIONS
OF FORMS (2) AND (3)

K'l\ X 1017 CM2

K'l\ X 1017 CM2

A (AJ

With Wave With Wave Function (3) Function (2)

'A (A)

With Wave With Wave Function (3) Function (2)

1000 ....... 2000 . . . . . . . 2500 . . . . . . . 3000 ....... 3500 ....... 4000 ....... 4500 ....... 5000 ....... 5500 .......
6000 .......
6500 .......

0.271 0.945 1.335 1.730 2.119 2.498 2.860 3.197 3.504 3. 773 3.998

0.270 0.991 1.461
1.955 2.437 2.880 3.265 3.581 3.822 3.989 4.084

7000 ...... 7500 ...... 8000 ...... 8500 ...... 9000 ...... 9500 ...... 10000 ...... 12000 ...... 14000 ...... 16000 ......

4.174 4.296 , 4.363 4.372 4.324 4.221 4.065 2.995 1.502 0.167

4.113 4.080 3.993 3.858 3.682 3.471
3.233 2.108 0.954 0.097

The absorption cross-sections, as calculated according to formula (II), and the fore-

going equations are given in Table 2 and further illustrated in Figure 3. It is seen that, as

anticipated, the two curves now do not diverge more than can be reasonably attributed

to the betterment of the wave function in consequence of the increased number of param-

eters used in the Ritz method.

•

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1945ApJ...102..223C

230

S. CHANDRASEKHAR

5. Concluding remarks.-A comparison of Figures 1, 2, and 3 clearly illustrates th superiority of formula (II) for the purposes of evaluating the continuous absorption co efficient of the negative hydrogen ion. The general reliability of the absorption cross
5

4

3

2

2000

4000

6000

8000

10000

).,-

12000

14000

FrG. 3.-A comparison of the continuous absorption coefficient of H- computed according to formul (II) and with wave functions of forms (2) (curve I) and (3) (curve II). The ordinates denote the ab sorption coefficients in units of 10-17 cm2; the abscissae, the wave length in angstroms.

5 4 3 2

2000

4000

6000

8000

10000

i\.-

12000

14000

16000

FIG. 4.-A comparison of the continuous absorption coefficient of H- computed according to formul (I) (curve I), (II) (curve II), and (III) (curve III) with a wave function of form (3). The ordinates ci note the absorption coefficients in units of 10-17 cm2; the abscissae, the wave length in angstroms. •

sections derived on the basis of formula (II) and wave function (3) can be seen in a other way. In Figure 4 we have plotted K, as given by the three formulae and as obtain, in each case with wave function (3). It is seen that, while the cross-sections given by fc mula (II) agree with those given by formula (I) in the visual and the violet part of t

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1945ApJ...102..223C

NEGATIVE HYDROGEN ION

231

•spectrum (X < 6000 A), they agree with those given by formula (III) in the infrared (X > 12,000 A). This is readily understood when it is remembered that on all the three
. formulae the absorption cross-sections in the infrared are relatively more dependent on the wave function at large distances than they are in the visual and the violet parts of the spectrum. Accordingly, it is to be expected that, as we approach the absorption limit : of H- at 16,550 A, formula (III) must give less unreliable values than it does at shorter . wave lengths; formula (I), of course, ceases to be valid in the infrared. It is also clear that, as we go toward the violet, we have the converse situation.
Summarizing our conclusions so far, it may be said that in the framework of the approximation .in which a plane-wave representation of the outgoing electron is used, formula (II), together with wave function (3), gives sufficiently reliable values for the absorption coefficient over the entire range of the spectrum. Attention may be particularly drawn to the fact that the maximum of the absorption-curve is now placed at X8500 A, where K}.. = 4.37 X 10-17 cm2•
The question still remains as to the improvements which can be effected in the choice of 'Ye. As shown in an earlier paper,4 it may be sufficient to use for 'Ye the wave functions in the Hartree field of a hydrogen atom. On this approximation we should use (op. cit., eq. [15] )

'Ye=

.

1
12

{

e-

r

00
,L

_i ik

(2l+l)Pz(cos13-2)xz(r2;k)

V 7r

Z=O r!I

(33)

where xz is the solution of the equation

d2xz d r 2

+ISk2_

l

(l+l) r2 .

+2(1+.!) r

e-2ri )

=0 Xz

'

(34)

which tends to a pure sinusoidal wave of unit amplitude at infinity. We shall return to these further improvements in a later paper.

It is a pleasure to acknowledge my indebtedness to Professor E. P. Wigner for many helpful discussions and much valuable advice. My thanks are also due to Mrs. Frances Herman Breen for assistance with the numerical work.
4 Ap. J., 100, 176, 1944.

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