The absorption coefficient for the free-bound transition H (ls) + e(-)+ h omega yields H(-)(2 sq p,(3)P(e)) is calculated (together with the differential emission rate for the inverse process) using ls - 2s - 2p close coupling continuum wave functions and a Hylleraas bound state wave function. A maximum in the absorption and emission spectra is found to occur at a photon wavelength of 1219.5 A, which is 2 A closer to the Lyman alpha line than predicted by the calculations of Drake, and is in closer agreement with the stellar absorption feature identified by Heap and Stecher. The free-bound absorption process appears to be a significant source of continuous ultraviolet opacity

Bhatia, A. K.

Jacobs, V. L.

Temkin, A.

English

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'-RADIATIVE TRANSITIONS
INVOLVING THE (2p Pe METASTABLE
AUTODETACHING OF H 7
7-I
1(NASA-TM-X-70570) RADIATIVE TRANSITIONS V74-154 12
INVOLVING THE (2p2) (3 Pe) METASTABLE
AUTODETACHING OF H.(-) (NASA) 18 p HC
$3.00 CSCL 20H Unclas
_G3/24 27232
V. L. JACOBS
A. BHATIA 7 'BHA
. A. TEMKIN'
JANUARY 1974
GODDARD SPA LIGHT CENTER
" GREENBET, MARYLAND
https://ntrs.nasa.gov/search.jsp?R=19740007299 2020-03-23T13:16:59+00:00Z
RADIATIVE TRANSITIONS INVOLVING THE (2p 2 ) 3 pe
METASTABLE AUTODETACHING STATE OF H
V. L. JACOBS*, A. K. BHATIA, AND A. TEMKIN
THEORETICAL STUDIES GROUP
GODDARD SPACE FLIGHT CENTER
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
GREENBELT, MARYLAND 20771
*NATIONAL ACADEMY OF SCIENCES
NATIONAL RESEARCH COUNCIL
RESIDENT RESEARCH ASSOCIATE
ABSTRACT
The absorption coefficient for the free-bound transition H (is) +
e- +4~K + H- (2p2 , 3pe) is calculated (together with the differential
emission rate for the inverse process) using is - 2s - 2p close coupling
continuum wave functions and a Hylleraas bound state wave function. A
maximum in the absorption and emission spectra is found to occur at a
photon wavelength of 1219.5 AO, which is 2 Ao closer to the Lyman a line
than predicted by the calculations of Drake and is in closer agreement
with the stellar absorption feature identified by Heap and Stechar. The
free-bound absorption process appears to be a significant source of
continuous ultraviolet opacity.
2I. INTRODUCTION
The continuous photoabsorption processes
H- (llS) + -1 H (is) + e- (1)
H (Is) + e- + -r H (1s) + e- (2)
play an important role in determining the visible and infrared opacities
of stellar atmospheres. Elaborate variational calculations (Doughty,
Fraser, and McEachran 1966; Doughty and Fraser 1966) indicate that the
absorption coefficients are negligible at shorter wavelengths, but the
contributions from the autodetaching resonances of H- which can be
excited by ultraviolet photons were not considered. For photodetach-
ment from the H ground state, the autodetaching resonances are not
expected to significantly enhance the absorption coefficient because
they are the result of double-electron excitations. In the free-free
transitions of H-, however, the contributions from the final state
resonances could be more significant and should be evaluated.
Within the LS-coupling approximation, the doubly excited (2p2 )3pe
state of H' is bound by 0.0095 ev below the H (2p) + e- continuum
threshold (Aashamar 1969, Drake 1970, Bhatia 1970). Because autodetach-
ment into the H (is) + e- continuum would violate conservation of either
total orbital angular momentum or parity, the (2p2) 3pe state may be
described as a metastable autodetaching state. The close proximity of
the (2p2)3pe state to the excitation threshold has led Heap and Stecher
(1974) to attribute the broad stellar absorption feature which they have
detected on the long-wavelength wing of the Lyman a line to the continuous
(free-bound) absorption process.
H (is) + e- + 4w. - H- (2p2,3pe) (3)
Realistic variational calculations have been performed for the free-
bound absorption coefficient (Drake 1974) as well as for the differential
emission rate corresponding to the inverse (radiative decay) process
(Drake 1973). However, the predicted spectra rise rapidly with increas-
ing wavelength to a maximum value at 1221.5 Ao, which is about 3 Ao
further from the Lyman a line than the observed absorption feature. In
this paper, we report an independent calculation for the absorption and
emission processes which indicates that the maximum occurs at 1219.5 AO,
in closer agreement with the position of the observed spectral profile.
II. THEORY
The differential spontaneous emission rate for the (2p2 )3pe +
(ls kp)3po transition, which is the only possible electric-dipole decay
mode for the (2p2)3pe state of H-, is given in atomic units by
A = 3 33 I(ls kp, 3P rI  + r2 112 p 2 , 3 pe)I 2 (4)
where w is the photon energy and c is the fine structure constant. The
(is kp) 3po scattering continuum wave function is normalized per unit
electron energy interval. The power radiated per unit volume per unit
solid angle and angular frequency intervals is given by the emission
coefficient:
£ (s) = A (w) NH -  (5)
where NH- denotes the density of H- ions in the (2p2)3pe state.
The effective absorption coefficient is defined in terms of the
difference between absorption and stimulated emission and is given by
k () = e - (E-) (6)
under the conditions of local thermodynamic equilibrium(Griem 1964).
With use of the Saha equation
NH Ne- = 2 m KT 3 /2 AE (7)
NH- 9g- 2- -  KT ,
where AE is the H- (2p2,3pe) energy measured from the hydrogen ground
state, the expression for the absorption coefficient per unit hydrogen
atom density per unit electron density (in units of a0 5) may be written
in the form
k () 18 7 7/2 3/2 1 e _Ee
a () = =  - 1 - e e A (0) (8)
NH Ne- (rw) KT KT
EH denotes the ionization energy of ground state atomic hydrogen, and the
electron and photon energies are related by the energy conservation
requirement.
Ee
-
+ = AE = 0.37465 a.u. (9)
5equation (8) can also be derived from the expression for the free-free
absorption coefficient corresponding to the transition to the final state
continuum channel H(2p) + e-(k 2=1),3Pe. Except for our inclusion of
stimulated emission, equation (8) is equivalent to the expression for the
free-bound absorption coefficient derived independently by Drake (1974).
III. RESULTS AND DISCUSSION
In our evaluation of the differential spontaneous emission rate A (w),
the (2p2)3pe state of H- was represented by an 84 - term Hylleraas bound
state wave function (Bhatia 1970), and the (Is kp) 3po scattering states
were approximated by ls-2s-2 p close coupling continuum wave functions
(Burke and Schey 1962). The length and the velocity forms of the dipole
matrix element in equation (4) were evaluated over the allowed electron
energy range using the procedures developed for photoionization calcula-
tions (Burke and McVicar 1965, Jacobs 1971). In contrast, Drake (1973)
has evaluated A (w) using a 50 - term Hylleraas function together with a
discrete variational representation of and a free-electron approximation
for the continuum.
The position and shape of the low electron energy spectral feature
are found to be quite insensitive to the continuum approximation but are
found to be surprisingly sensitive to the number N of Hylleraas terms
included in the bound state wave function. The convergence of the dif-
ferential emission rates obtained with three different expansions is
nevertheless satisfactory when as many as 84 terms are included (figure 1).
The length and velocity results presented over a more extended energy
6range in Table 1 are in excellent agreement at low energies. For higher
energies there is good agreement with the calculation of Drake (1973),
but significant discrepancies are found near the maximum which are
described below in connection with absorption.
The insensitivity to the continuum approximation is attributable to
the fact that for very low electron energies the dominant contribution
to the transition integral comes from large values of the ejected
electron separation r where the continuum wave function may be repre-
sented by its asymptotic form. In addition, since the triplet P-wave
phase shift tends to zero at threshold, it was expected and found that the
Born approximation, i.e. replacement of the ejected electron continuum
function by j(kr), gave essentially the same results as the more
elaborate continuum functions. The same insensitivity to the continuum
function was found by Drake (1973). The low electron energy variation of
A (w) is analogous to but much more rapid than the behavior of the H-
photodetachment absorption coefficient (Doughty, Fraser, and McEachran
1966), which also rises from zero at threshold to a maximum value.
In figure 2 our 84 - term results for the free-bound absorption
coefficient per unit hydrogen atom density per unit electron density
are compared with the absorption coefficient calculated by Drake (1974)
for a temperature of 5000 Ko. We find a maximum at 1219.5 Ao which is
slightly larger and 2 A closer to the Lyman a line than predicted by
the calculation of Drake (1974). The variation of a (w) with temper-
ature which is illustrated in Table 2 was an important consideration in
7the identification of the (2p2)3pe state in the stellar absorption
spectrum (Heap and Stecher 1974).
Drake (1974) has pointed out that for Ne- / NH > 0.1 the free-
bound absorption coefficient per unit length is greater than that for
Lyman a resonance broadening (Sando, Doyle, and Dalgarno 1969) in the
wavelength interval 1220 Ao - 1300 Ao. We wish to emphasize the
dominance of the free-bound absorption process over the nonresonant
free-free absorption process (Doughty and Fraser 1966) in the far
ultraviolet, as illustrated by the comparison of the respective absorp-
tion coefficients given in figure 3.
A second maximum associated with the transition H- (2p2,3pe)
H- (2s 2p, 3PO) + hw is found to occur at a photon wavelength of 29,
138 Ao, corresponding to radiative decay via the (2s 2p) 3Po truly auto-
detaching state of H-. This far infrared emission feature is shown in
figure 4. The peak value of 4.42 x 109 sec- 1 a.u.- I is about 1/10 the
maximum value in the far ultraviolet. Using a Hylleraas representation
for the (2s 2p) 3po resonance, a procedure which does not predict the
resonant line shape, Drake (1973) has estimated that the inclusion of
this transition increases the integrated decay rate by only about 1%.
This autodetaching feature would in any event be suppressed in absorption
because of the dominance of the nonresonant free-free transitions in
the far infrared, but it might be observable in emission under non - LTE
excitation conditions.
Additional sources of Lyman a resonance broadening absorption are
expected from transitions to autodetaching resonances of H-, several of
8which are in close proximity to the n = 2 excitation threshold (Temkin
and Walker 1965). Such contributions to the H- free-free absorption
coefficient.could be as significant as that found for the (2p2 )3pe
metastable autodetaching state, and calculations for all important
resonances are in progress.
We are grateful to T. P. Stecher and S. Heap for bringing to our
attention their astrophysical observation and for their encouragement
in this investigation. We also thank Professor H. Griem for a helpful
discussion. The use of the close coupling program written by Professor
P. G. Burke is acknowledged. Finally, we would like to thank Mr. E.
Sullivan for his programming assistance.
REFERENCES
Aashamar, K. 1969, Institute for Theoretical Physics, University of Oslo,
Norway, Institute Report No. 35 (unpublished).
Bhatia, A. K. 1970, Phys. Rev. A, 2, 1667.
Burke, P. G., and McVicar, D. D. 1965, Proc. Phys. Soc. 86, 989.
Burke, P. G., and Schey, M. H. 1962, Phys. Rev., 126, 147.
Drake, G. W. F. 1970, Phys. Rev. Letters, 24,127.
Drake, G. W. F. 1973, Ap. J., 184, 145.
Drake, G. W. F. 1974, Ap. J. (in press).
Doughty, N. A., and Fraser, P. A. 1966, Mon. Nat. Roy. Astronom. Soc.,
132, 267.
Doughty, N. A., and Fraser, P. A., and McEachran, R. P,, 1966, Mon. Nat,
Roy. Astronom. Soc., 132, 255.
Griem, H. R. 1964, Plasma Spectroscopy (Mc Graw - Hill,,New York).
Heap, S. R., and Stecher, T. P. 1974, Ap. J. (in press).
Jacobs, V. L. 1971, Phys. Rev. A, 3, 289.
Sando, K. M., Doyle, R. 0., and Dalgarno, A. 1969, Ap. J. 157, L 143.
Temkin, A., and Walker, J. F., 1965, Phys. Rev. A, 140, 1520.
10
TABLE 1
The differential spontaneous emission rate A (W) (in units of 1010
sec- 1 a.u.- 1) for the transition H- (2p2,3pe) 4 H (is) + e- + 1t. The
length and velocity results are denoted by L and V, respectively. The
numbers in parenthesis are the powers of ten by which the entries are
to be multiplied.
Electron energy (a.u.) AL (w) AV (w)
0.001 6.576 6.594
0.002 5.370 5.395
0.003 4.158 4.185
0.004 3.450 3.481
0.005 2.981 3.015
0.010 1.582 1.617
0.015 9.533 (-1) 1.005
0.020 6.400 (-1) 6.610 (-1)
0.025 4.332 (-1) 4.806 (-1)
0.030 3.122 (-1) 3.740 (-1)
0.040 1.976 (-1) 2.088 (-1)
0.050 1.212 (-1) 1.279 (-1)
0.100 1.727 (-2) 2.035 (-2)
0.150 4.153 (-3) 5.453 (-3)
0.200 1.311 (-3) 1.876 (-3)
TABLE 2
The free-bound absorption coefficient a (w) (in units of cm5). The
numbers in parenthesis are the powers of ten by which the entries are to
be multiplied.
X (Ao) T = 5000 Ko T = 9000 Ko T = 1,5000 Ko
1216.9 6.49 (-38) 2.70 (-38) 1.26 (-38)
1217.7 1.17 (-37) 4.92 (-38) 2.30 (-38)
1218.6 1.41 (-37) 5.97 (-38) 2.80 (-38)
1219.4 1.46 (-37) 6.24 (-38) 2.94 (-38)
1220.2 1.42 (-37) 6.09 (-38) 2.88 (-38)
1221.0 1.33 (-37) 5.75 (-38) 2.73 (-38)
1221.8 1.23 (-37) 5.35 (-38) 2.55 (-38)
1222.7 1.13 (-37) 4.95 (-38) 2.36 (-38)
1224.3 9.60 (-38) 4.26 (-38) 2.05 (-38)
1225.9 8.28 (-38) 3.73 (-38) 1.80 (-38)
1229.2 6.50 (-38) 3.01 (-38) 1.48 (-38)
1232.6 5.32 (-38) 2.53 (-38) 1.26 (-38)
1249.5 2.13 (-38) 1.17 (-38) 6.26 (-39)
1266.8 9.62 (-39) 6.07 (-39) 3.48 (-39)
1284.7 4.83 (-39) 3.51 (-39) 2.15 (-39)
1303.1 2.57 (-39) 2.14 (-39) 1.41 (-39)
1322.0 1.41 (-39) 1.35 (-39) 9.61 (-40)
1361.5 4.56 (-40) 5.80 (-40) 4.72 (-40)
12
FIGURE CAPTIONS
Figure 1
The differential spontaneous emission rate A (w) for the transition
H- (2p2,3pe) + H (is) + e- +-bw plotted as a function of the ejected
electron energy and the emitted photon wavelength. N is the number
of Hylleraas terms in the bound state wave function.
Figure 2
The free-bound absorption coefficient a (w) per unit hydrogen atom
density per unit electron density for a temperature of 5000 Ko. The
results of Drake (1974) are included for comparison.
Figure 3
The absorption coefficients for free-bound and free-free (Doughty
and Fraser 1966) transitions of H- for a temperature of 5000 KO.
Figure 4
The resonant emission feature associated with the transition H-
(2p2,3pe) + H- (2s 2p3,po) + M to the (2s 2p) 3po autodetaching
resonance of H
PHOTON WAVELENGTH (A)
1216.2 1220 1224 1228
8-
7- N=56
0
r- N=84 N=35
'6 -
-
z
O
t: 3
z
UJ<0
2-
z
w
u1
L
0 0.001 0.002 0.003 0.004 0.005
ELECTRON ENERGY (a.u.)
k'igure 1
15 
-N=84
14-
13-
E 12-
DRAKE
11 /
10-
lOU-
. 9-
Z 8-o
7
6-
5-
4-
1216 1218 1220 1222 1224 1226 1228 1230 1232 1234
O
PHOTON WAVELENGTH (A)
Fiqure 2
10-38
10-39
FREE-FREE
Lu 10-40
C)
U-LU
0
z
0
p 10-41
0 FREE-BOUND
m
10-42
10 -4 3
1000 1500 2000 4000 6000 8000
PHOTON WAVELENGTH (A)
Figure 3
i10 9
-6
UJ
LU
10 6
10,000 20,000 30,000 40,000
-7
106
PHOTON WAVELENGTH (A)
Figure 4 NASA-GSFC


Radiative transitions involving the (2p2)(3 Pe) metastable autodetaching of H(-)

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