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An Analytical Formula for the Continuous Absorption Coefficient and the Scattering Amplitude of the Electron Detachment for the Negative Hydrogen Ion by Electron Impact.

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ANNALEN D E R PHYSIK
7.FOLGE
BAND23,
*
HEFT7/8
*
1969
An Analytical Formula for the Continuous Absorption
Coefficient and the Scattering Amplitude of the Electron
Detachment for the Negative Hydrogen Ion
by Electron Impact
By T.TIETZ
Abstract
I n this paper we derive an approximate analytical formula for the continuous absorption
coefficient of the hydrogen negative ion. The numerical results of this formula are compared with the results of CHANDRASEKH~R
and ELBERT
who used many-parameter groundstate function with the dipole-velocity matrix element. Here we derive also an analytical
formula for the scattering amplitude for the detachment of electrons by electron impact
from hydrogen negative ion in its ground '8,state. This paper contains numerical tables for
the continuous absorption coefficient and other results.
1. Introduction
It is well known that the continuous absorption coefficient of H- is of great
astrophysical interest. This coefficient has been calculated by several authors
[ l -131. The accurate numerical calculations being those of GIANDRASEEHAR
and of JOHN.
The previous theoretical calculations have been carried out with
many-parameter ground state wave functions e. g. CHANDRASEEHAR and
ELBERT
[14] used a HARTand HERZBERG
20-parameter ground-state wave
function and a HARTREE
approximation for the continuum wave function. The
simplest calculations concerning the continuous absorption coefficient for H are given by GELTMANand the author. GELTMAN'S
considerations do not give an
analytical formula for the continuous absorption coefficient for the hydrogen
negative ion.
The purpose of this note is to derive without much trouble, by a simple
assumption an approximate analytical formula for the continuous absorption
coefficient of H - and an analytical formula for the scattering amplitude of the
electron detachment for H - by electron impact.
2. The approximate ground-state function for HI n order to obtain an analytical function for qo(r)we must solve the following
SCHR~DINGER
equation written in atomic units
[A - 2V(r)
+ 2E01 %(r) = 0
(2.1)
where V ( r )is the exact spherical potential and 214, I = k;; E, is the bound state
having an energy equal i n magnitude to the accepted affinity of the hydrogen
20
Ann. Phye. 'I.
Folge, Bd. 23
306
Annalen der Phyaik
*
7.Folge & Band 23, Heft 718
*
1969
atom. The exact potential appearing in the last equation is given by:
As known the SCHRODINCER
equation for this potential cannot be solved exactly
therefore we have approximated V ( r )given by eq. (2.2)as follows:
where a, b, c and d are constants. Substituting the approximate potential V(r)
given in the last formula in the SCHR~DINOER
equation we see that po(r)is
I n this case the constants b and d appearing in eq. (2.3)are given by the independent constants a, c as follows
+ a)
U
b =(1 - c) (2k0
2
a=
+ a)
(1 - 4 (2kO + 4
w
(2.5)
o
No appearing in vo(r)given by (2.4)is the radail normalization constant
kO(2kO
+ a ) PIC, +
(ko
+ 2a)
1.(2.6)
The eigenfunction To(?+)for the ground-state corresponds t o an eigenvalue Eo
where
I n the physical literature there are known in atomic units some numerical values
for k:. The HEINRICH'S
value of kg is kg = 0.05466. The JOHN'S
values of k; is
kg = 0.055289 and the PEEERIS
values of k% is kg = 0.0555027. Eq.(2.5)
shows that if the constants a and c are given numerically then the approximate
potential V ( r )given by eq. (2.3) and vo(r)by eq. (2.4)are determined. The constants a and c will be determined by help of variational method.
3. The variational method for hydrogen negative ion
I n order t o determine the constants a and c appearing in the previous
chapter we write the SCHRODINGER
equation for H - in atomic units as follows:
[- +dl - -12A 2
1
1
1
-TI - r2 + , - E l @ ( , , r
r 2) = O
(34
where E is the energy of the hydrogen negative ion in its groundstate and
is the corresponding normalized wave function. The indices appearing
in the last equation reffer t o the first and second electron. Since @(rl, r2) is
normalized so that
1
E = JJ@*(r,, r,)
sd2 - 1- 1 L ] @ ( r l , r 2 )dt, dt,.
(3.2)
ri
ra
Putting
(3.3)
@;P(rvr2) = vo(rd vo(r2)
where cpo(r)is given by (2.4)and substituting the last equation in eq. (3.2) we
see that after some simple but long calciilations we obtain for E the following
@ ( r l , r,)
[-
-
+
307
T. TIETZ:
Scattering Amplitude of the Electron Detachment
formula :
+ 2(2k0 + a) (ko + 2a) c + (2ki -I- 3k0a + 2a2)C21
+ +4
8(24 +
(5+ w
C2
+--(In
(4+ 2a)
+
WkO 3 4 (k,
(4ko 5a)'
+ 5a)+(41C,3 +~ 7 )4 ~
2c In (4k0
>I
+
+ c21n
+
4(2k0
(4k0 7 4 2
where N2 appearing here is given by (2.6). The numerical values of the constants
a and c we determine from the minimum value of E. It has been found that E
has a minimum in the neighbourhood of a = 3k0 and c = 1.31. If we accept the
PEKERIS'
values of ko we get E = -0.4856. I n case if ko takes the JOHN'S value
we obtain for E the following value E = -0.4855. We see that the JOHN'S and
PEKERIS'
value of ko give nearly the same values for E. It has been found too
Table 1
A comparison of t h e a p p r o x i m a t e p o t e n t i a l g i v e n b y eq. (2.3) with t h e e x a c t
p o t e n t i a l given b y eq. (2.2)
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.2
0.3
0.4
0.7
1.0
2.0
-m
20*
I
I
+m
99.00
49.00
32.33
24.00
19.00
15.67
13.29
11.50
10.12
9.01
4.02
2.38
1.57
0.60
0.27
0.028
0.000
+m
I
90.89
45.44
30.28
22.70
18.14
15.09
12.92
11.28
10.01
8.98
4.33
2.73
1.92
0.84
0.053
0.33
0.000
308
Annden der Physik
*
7.Folge
*
Band 23, Heft 718
*
1969
that E given by the last formula has another minimum in the neighbourhood
of a = 8ko and c = -0.125. For the PERERIS’
and JOHN’Svalues of k, the
numerical values of E are respectively E = -0.4853 and E = -0.4852.
Thus we have determined the numerical values of the constants a and c appearing
in all above given formulas. I n Table 1 we have compared our approximate
potential V ( r )given by eq. (2.3) with the exact potential V ( r )given by eq. (2.2).
The comparison has been carried out for the numerical values a = 8k, and
c = -0.125. We have used here the PEKERIS’
value of k,. The constants b and d
appearing in the approximate potential V ( r ) are calculated from eq. [2.5).
Table 1 shows that for small values of r the approximate potential - V ( r )
given by eq. (2.3) fits well the exact potential - V ( r ) given by eq. (2.2).
4. The continuous absorption coefficient of HThe purpose of this chapter is to derive an analytical formula for the continuous absorption coefficient of H-. It is known that the total continuous
absorption coefficient x, for H - is given by
radius, [x is the fine structure constant andv,is the normalwhere a, is the BOHR
ized ground-state function which in our case is given by eq. (2.4), k2 = 2kk where
Ekis the continuum state energy. Between the frequency v, E,, and Ek as it is
known there exists the relation E, - Ek = hv. The continuum wave function
qkas we know may be expended in term of LEQENDRE
polynomials as
The partial wave xlin eq. (4.1) contributes to the dipole matrix element in the
pwave. We do not make a great error if we adopt for x1 the solution of the
SCHR~DINQER
equation for V = 0. Which is known and can be written in the
following form :
sin kr
%=-cos k r .
(4.3)
kr
Substituting 9, given by eq. (2.4) ans xl given by eq. (4.3) in the formula for
the total continuous absorption coefficient given by eq. (4.1) we obtain after
some calculations the following simple analytical formula for x, in our case.
In Table 2. We have compared the numerical values of x, given by eq. (4.4)
for the HEINRICH’S
value of k, with the corresponding numerical values given by
CHANDRASERHAR
and ELBERT
for the constants a and c we have accepted the
numerical values a = 8ko and c = -0.125 discussed in the previous chapter.
Table 2 shows that our results for x, calculated from eq. (4.4) agree well with
the corresponding numerical values of CHANDRASEEHAR
and ELBERT.
I n Table 2
we have some numerical values given by eq. (4.4)in case of a = 8ko and
309
T. TIETZ:Scattering Amplitude of the Electron Detachment
Table 2
A comparison of o u r c o n t i n u o u s a b s o r p t i o n coefficient of H eq. (4.4) w i t h t h e c o r r e s p o n d i n g n u m e r i c a l r e s u l t s of CHANDRASEKAAR and ELBERT
x, (1O-I’ cma)
k2
Our results eq. (4I4)
0
0.010
0.020
0.030
0.035
0.045
0.050
0.055
0.060
0.070
0.080
0.090
0.100
0.125
0.175
0.250
0.500
8.000
16533
13994
12131
10706
10112
9 102
8 669
8275
7 916
7 283
6 744
6279
5 875
5 059
3960
2 987
1642
1066
CHANDRASEKHAR
and
results
ELBERT’S
0
1.70
3.12
3.93
4.17
4.42
4.47
4.48
4.47
4.38
4.24
4.09
3.91
3.49
2.76
2.00
0.93
0.50
0
2.83
4.03
4.37
4.45
4.49
4.41
4.09
3.90
3.41
2.30
1.83
0.75
0.34
c = -0.125 for the PEKERIS’ and JOHN’Svalues for ko. I n Table 2 we have also
calculated some numerical values of x, from eq. (4.4)for the PERERIS’
value
of ko if a = 3k0 and c = 1.31.
Table 3
Some n u m e r i c a l v a l u e s f o r x, given b y eq. (4.4) f o r a = 3k0, c = 1.31 i f
k, is given b y PEKERIS’
a n d a = 8 4 , c = -0.125 i f k, is given b y PEEERIS
a n d JOHN
cm2)
a = 8kO c = -0,125
for PEKERIS’
value of k,
a = 8ko c = -0,125
for JOHN’S
0
1.64
3.04
3.84
4.08
4.34
4.40
4.42
4.41
4.33
4.20
4.04
3.88
3.46
2.75
2.00
0.93
0.50
0
1.66
3.06
3.86
4.10
4.36
4.41
4.43
4.42
4.34
4.21
4.05
3.88
3.46
2.75
2.00
0.93
0.50
x, (1O-l’
ka
0
0.010
0.020
0.030
0.035
0.045
0.050
0.055
0.060
0.070
0.080
0.090
0.100
0.125
0.175
0.250
0.500
0.800
a
34, c = 1.31
for PEKERIS’
value of k,
A
0
1.61
2.97
3.81
4.00
4.27
4.32
4.34
4.33
4.25
4.13
3.99
3.83
3.43
2.73
2.01
0.95
0.54
value of k,
310
Annalen der Physik
*
7.Folge
*
Band 23, Heft 7/8
*
1969
Table 3 shows that the PERERIS’
value of ko for the case a = 8k0 and
= -0.125 gives for x, a little lower values than the values obtained for this
case for the JOHN’S
values of ko. The case a = 3k0 and c = 1.31 as visible from
Table 3 gives lower values for x, than the case a = 8ko and c = -0.125. If we
G
compare the numerical values for x,, given in Table 2 and 3 with the numerical of
x, given by SMITH
and BURCH
[15] we see that our numerical values of x, are in
a good accordance with the experimental values.
6. An analyticaI formuIa for the scattering amplitude of the electron detachment
for €
by
Ielectron impact
As known the total cross-section [16] for the detachment of an electron by
an electron impact in its ground ISo state in the BORN-OPPENHEIMER
approximation depends on the following scattering amplitudes
where k,, k, and x are the incident, scattered and ejected wave vectors respectively, K = k, - k, - x and
oi,
A = j dz yOqH, J ( K ) = drrqo sin Kr.
(5.2)
0
In the last formula yo(r)is given by eq. (2.4) and rpH(r)= 1
e-T. Substituting
G
qo(r)given by eq. (2.4) in the formula for d and J ( K ) we obtain the following
formulas :
and
values for ko we obtain for A the value d = 0.8949. I n
Using the PERERIS’
Table 4 we have some numerical values for J ( K ) / K calculated from eq. (5.4)
for the PEIIERIS’
value of k;.
Table 4
Some numerical values of J ( K ) / K given by eq. (5.4)
K
10.1 1 0 . 2 10.3 10.4 10.5 10.6 10.7 10.8 / 0 . 9 11.0
J ( K ) / K 1 04.12 3.51 2.37 1.54 1.04 0.706 0.506 0.374 0.284 0.220 0.174
1
Since J ( E ) / Kis given analytically so that &l),
= gkl) are also given
analytically then we can derive an approximate formula for the total crosssection for the detachment of an electron by electron impact in its ground ISo
state in the BORN-OPPENHEIMER
approximation., These calculations are in
progress.
T. !I!IETZ: Scattering Amplitude of the Electron Detachment
311
6. Consequences and discussion
The importance of stellar atmospheres of the continuous absorption of radiation by free-free transitions of electrons in the field of hydrogen atoms has long
been known. All physical problems connected with H - cannot be done exactly
since this would require a knowledge of the exact wave function of the hydrogen
negative ion, therefore these problems can be solved only approximately. The
numerical values of x, and J ( K ) / K show that the approximate simple formula
for Q ) ~ ( T )given by eq. (2.4) allows us to calculate integrals appearing in different
problems e.g. the scattering amplitudes of the electron detachment for H - by
electron impact, the total cross-section for detachment of an electron by an
electron impact in its ground 'So state in the BORN-OPPENHEIMER
approximation,
detachment from the negative hydrogen ion by electron or positron impact
[17] and other problems.
References
[l] JEN,C.K., Phys. Rev. 48 (1933) 640.
[2] lliZ4ssEY, H.S.W., and R.A.SYITH,Proc. Roy. SOC.,Lond. A. 17,166 (1936) 47.
, W., and D.R.BATEs, Astrophys. J. 91 (1940) 202.
[3] M ~ S S E YH.S.
[4] WILLIAMS,
R.E., Aetrophys. J. 96 (1942) 438.
[5] HEINRICH,
L. S., Astrophys. J. 99 (1943) 59.
[6] CHANDRASEEHAR,
S., Revs. Modern Ph 8.16 (1944) 301; Astrophys. J. 100 (1944) 176.
[7] CHANDRASEKHAR,
S., Astrophys. J. lOi(1945 )223; 102 (1945) 395.
[S] GELTMAN,
S., Phys. Rev. 104 (1956) 346.
[9] CEANDRASERHAR,
S., Astrophys. J. 128 (1958) 114.
[lo] CHANDRASEKHAR,
S., and D. D. ELBERT,
Astrophys. J. 128 (1958) 633.
[ll] SMITH,S.J., and D.S.BURCH,Phys. Rev. 116 (1959) 1125.
[12] JOHN,T.L., Monthly Notices Roy. Astron. SOC.121 (1960) 41.
[13] TIETZ,T., Phys. Rev. 124 (1961) 493.
[14] see ref. [lo].
r151 see ref. r111.
[16j see ref. r8].[17] RUDQE,M.R.N., Proc. phys. SOC.88 (1964) 419.
L6di (Poland), Department of Theoretical Physics, University of U d i .
Bei der Redaktion eingegangen am 3.Dezember 1968.
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