Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
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Collisional Broadening of Spectral Lines 59.2 Isolated Lines 877
If Ψ
(
r, R
)
is expanded in the form
Ψ(r, R) =
j
a
ji
(t)ψ
j
(r) exp(−iE
j
t/ ), (59.14)
where initially at time t =−∞
a
ji
(
−∞
)
= δ
ji
, (59.15)
and (59.12)–(59.14)give
i
da
ji
dt
=
k
a
ki
V
jk
exp
iω
jk
t
,
i, j, k = 0, 1, 2,... ,
(59.16)
where
V
jk
(
R
)
=
ψ
∗
j
(
r
)
V
(
r, R
)
ψ
k
(
r
)
dr
(59.17)
in which V
(
r, R
)
is the emitter-perturber interaction and
ω
jk
= E
j
−E
k
. (59.18)
Integration of equations (59.16)for−∞ ≤ t ≤∞gives
the unitary scattering matrix S, with elements
S
ji
(
b, v
)
≡ a
ji
(
∞
)
, i, j = 0, 1, 2,... .
(59.19)
Then
w +id = 2πN
∞
0
v f
(
v
)
dv
×
∞
0
δ
i
i
δ
f
f
−S
i
i
(
b, v
)
S
∗
f
f
(
b, v
)
av
bdb ,
(59.20)
where J
i
= J
i
and J
f
= J
f
, N is the perturber
density and [···]
av
denotes an average over all orien-
tations of the collision and over the magnetic sublevels
[see Eq. (59.9)]. In (59.20), f
(
v
)
is the Maxwell veloc-
ity distribution at temperature T for an emitter-perturber
system of reduced mass µ:
f
(
v
)
= 4πv
2
µ
2πk
B
T
3/2
exp
−
µv
2
2k
B
T
,
∞
0
f
(
v
)
dv = 1 . (59.21)
59.2.2 Simple Formulae
These are useful for making quick estimates, but in indi-
vidual cases may give results in error by a factor of two
ormore.Ifitisassumedin(59.17)that
V
ij
(
R
)
= V
jj
(
R
)
δ
ij
, (59.22)
where V
jj
(
R
)
is a simple central potential
V
jj
(
R
)
=
C
j
R
−p
, j = i, f , (59.23)
and C
j
depends only on the state j of the emitter, and if
the relative motion is along the straight line
R= b+vt ,
(59.24)
then
S
i
i
(
b, v
)
S
∗
f
f
(
b, v
)
av
= exp
2i
η
i
−η
f
,
(59.25)
where the phase shifts are
η
j
(
b,v
)
=−
1
2
∞
−∞
V
jj
(
R
)
dt , j = i, f .
(59.26)
Equations (59.20)–(59.26)give
w +id = πN
¯
v
β
p
C
p
¯
v
2/
(
p−1
)
× Γ
p −3
p −1
exp
±
iπ
p −1
α
p
, (59.27)
where
α
p
= Γ
2p −3
p −1
π
4
−1/
(
p−1
)
,
β
p
=
√
π
Γ
1
2
(
p −1
)
Γ
1
2
p
,
C
p
= C
i
−C
f
,
¯
v =
∞
0
v f
(
v
)
dv =
8k
B
T
πµ
1/2
. (59.28)
In (59.27)the± sign indicates the sign of C
p
, α
p
1
for p ≥ 3, and Γ
(
···
)
is the gamma function.
The cases p = 3, 4, and 6 correspond to reso-
nance, quadratic Stark and van der Waals broadening,
respectively. This approximation is invalid for the
dipole case ( p = 2) for which (59.27) is not finite.
The dipole–dipole interaction ( p = 3) occurs when
emitter and perturber are identical atoms (apart from
isotopic differences). If the level i is connected to
the ground state by a strong allowed transition with
Part D 59.2
878 Part D Scattering Theory
absorption oscillator strength f
gi
, and the perturba-
tion of the level f can be neglected by comparison,
then
C
3
= c
d
q
2
f
gi
2m
e
ω
gi
,
c
d
= 1 +
1
2
√
3
ln
2 +
√
3
= 1.380 173 . (59.29)
Also, if g
j
is the statistical weight of level j, the constant
c
d
may be replaced by an empirical value
c
d
=
4
π
g
g
g
i
1/2
, (59.30)
and this gives a width correct to about 10% [59.14].
Equation (59.27) does not predict a finite shift. Quadratic
Stark broadening occurs when a nonhydrogenic emitter
is polarized by electron perturbers. Then
C
4
=−
q
2
2
α
i
−α
f
,
(59.31)
where α
i
and α
f
are the dipole polarizabilities of states
i and f , respectively. Van der Waals broadening occurs
when the emitter and perturber are nonidentical neutral
atoms. If energy level separations of importance in the
perturbing atom are much greater than those of the emit-
ter (e.g., alkali spectra broadened by noble gases), C
6
is
given by
C
6
=−
q
2
α
d
r
2
i
−r
2
f
,
(59.32)
where α
d
is the dipole polarizability of the perturber.
The mean square radii can be calculated from the nor-
malized radial wave functions
1
r
P
n
∗
j
l
j
(
r
)
or estimated
from
r
2
j
=
∞
0
P
2
n
∗
j
l
j
(
r
)
r
2
dr
n
∗2
j
a
2
0
2z
2
5n
∗2
j
+1 −3l
j
l
j
+1
,
j = i, f
(59.33)
in which the effective principal quantum numbers n
∗
j
are
given by
E
j
≡−
z
2
n
∗2
j
I
h
, z = Z
e
+1 , (59.34)
where I
h
= hcR
∞
is the Rydberg energy, Z
e
is the
charge on the emitter and z = 1 in this case.
59.2.3 Perturbation Theory
An approximate solution of (59.16) is given by [59.8–
10, 15, 16]
S
ji
(
b, v
)
= δ
ji
−
i
∞
−∞
V
ji
(
t
)
exp
iω
ji
t
dt
−
1
2
k
∞
−∞
V
jk
(
t
)
exp
iω
jk
t
dt
×
t
−∞
V
ki
t
exp
iω
ki
t
dt
.
(59.35)
This gives a cross section for the collisional transition
i → j:
σ
ij
(
v
)
= 2π
∞
0
P
ij
(
b, v
)
av
b db ,
i, j = 0, 1, 2,... ,
(59.36)
where
P
ij
(
b, v
)
=
δ
ji
−S
ji
(
b, v
)
2
, (59.37)
2Re
[
1 −S
ii
(
b, v
)
]
=
j
P
ij
(
b, v
)
,
(59.38)
correct to second-order in V
(
r, R
)
on both sides. Using
(59.10), (59.20)and(59.36)–(59.38), the full width is
γ = N
∞
0
v f
(
v
)
dv
×
j=i
σ
ij
(
v
)
+
j= f
σ
fj
(
v
)
+σ
if
(
v
)
,
(59.39)
where the sums are taken over all energy-changing
transitions, the tilde indicates an interference term, and
σ
if
(
v
)
= 2π
∞
0
P
if
(
b, v
)
av
b db , (59.40)
in which
P
if
(
b, v
)
=
1
∞
−∞
V
ii
(
t
)
−V
ff
(
t
)
dt
2
.
(59.41)
Part D 59.2
Collisional Broadening of Spectral Lines 59.2 Isolated Lines 879
59.2.4 Broadening by Charged Particles
The total emitter-perturber interaction is
zZ
p
q
2
R
−
Z
p
q
2
|
R−r
|
V
0
(
R
)
+V
(
r, R
)
,
(59.42)
where Z
p
is the charge on the perturber and
V
0
(
R
)
=
Z
e
Z
p
q
2
R
, V
(
r, R
)
=−Z
p
q
2
r · R
R
3
.
(59.43)
If Z
e
= 0, the relative motion is described by (59.24),
but if Z
e
= 0, the trajectory is hyperbolic and is given
by
µ
d
2
R
dt
2
=−∇V
0
=
Z
e
Z
p
q
2
R
3
R, (59.44)
with the resulting hyperbola characterized by a semi-
major axis a and an eccentricity ,where
b
2
= a
2
2
−1
, a =
Z
e
Z
p
q
2
µv
2
. (59.45)
On using (59.17), (59.20), (59.36)–(59.41)and(59.43),
V
ii
(
t
)
= 0 ,
P
if
(
b, v
)
= 0 , σ
if
(
v
)
= 0 ,
(59.46)
and
w +id = 2πN
∞
0
v f
(
v
)
dv
×
∞
0
j=i
Q
ij
(
b,v
)
+
j= f
Q
fj
(
b,v
)
b db ,
(59.47)
where
Q
ij
(
v
)
=
4Z
2
p
I
2
h
a
2
0
m
e
ω
ij
f
ij
b
2
v
2
[
A
(
β, ξ
)
+iB
(
β, ξ
)
]
,
2Re[Q
ij
(
v
)
]=
P
ij
(
b, v
)
av
. (59.48)
If
ξ ≡
a
ω
ij
v
,β≡ ξ , δ ≡
2
−1
2
, (59.49)
the functions A
(
β, ξ
)
and B
(
β, ξ
)
in (59.48)aregiven
by
A
(
β, ξ
)
= δ exp
(
∓πξ
)
β
2
×
K
iξ
(
β
)
2
+δ
K
iξ
(
β
)
2
(59.50)
B
(
β, ξ
)
=
2β
π
℘
∞
0
A
β
,ξ
dβ
β
2
−β
2
,
(59.51)
where K
iξ
(
β
)
is a modified Bessel function. In (59.50),
the ∓ sign corresponds to Z
e
Z
p
=±
Z
e
Z
p
, and in
(59.51), ℘ indicates the Cauchy principal value. If Z
e
=
0, then
ξ = 0 ,β= b
ω
ij
v
,δ=1
(59.52)
in (59.50)and(59.51).
Approximation (59.48) breaks down at small values
of b because of assumption (59.43) and the lack of uni-
tarity of Sas given by (59.35). This problem is discussed
elsewhere [59.8, 15, 16], and all methods used involve
choosing a cutoff at b = b
0
, where b
2
0
r
2
f
, and using
(59.48) only for b > b
0
. For b ≤b
0
, an effective con-
stant probability is introduced and the method works
well as long as the contribution from b ≤ b
0
is small.
For b > b
0
(or β>β
0
), the contribution to σ
ij
(
v
)
in
(59.39) is evaluated using (59.47)–(59.50)and(59.52),
where
∞
b
0
A
(
β, ξ
)
db
b
=−e
∓πξ
β
0
K
iξ
(
β
0
)
K
iξ
(
β
0
)
.
(59.53)
A similar treatment exists for the quadrupole contribu-
tion to V
(
r, R
)
in (59.43) [59.8, 15, 16].
59.2.5 Empirical Formulae
An empirical formula based on the theory of Sect. 59.2.4
for the width of an atomic line Stark broadened by elec-
trons has been developed [59.17]. Konjevi
´
c [59.18]has
reviewed the data available for nonhydrogenic lines and
has provided simple analytical representations of the
experimental results for widths and shifts.
The full half-width is given by (59.39), where σ
if
=
0and
∞
0
v f
e
(v)
k=j
σ
jk
(v) dv =
8π
2
3
√
3
m
e
2
v
−1
×
T
j
g(x
j
) +
l
j
=l
j
±1
T
jj
g(x
jj
)
, j = i, f ,
(59.54)
Part D 59.2
880 Part D Scattering Theory
where
v
−1
=
∞
0
v
−1
f
e
(
v
)
dv =
2m
e
πk
B
T
1/2
, (59.55)
and f
e
(
v
)
= f
(
v
)
with µ = m
e
.In(59.54),
T
j
=
3n
∗
j
2z
2
1
9
n
∗2
j
+3l
j
l
j
+1
+11
, (59.56)
T
jj
=
l
>
2l
j
+1
R
2
jj
n
∗
l
>
, n
∗
l
<
, l
>
,
(59.57)
l
<
= min
l
j
, l
j
, l
>
= max
l
j
, l
j
,
with
R
jj
n
∗
l
>
, n
∗
l
<
, l
>
≡ a
−1
0
∞
0
P
n
∗
l
>
l
>
(
r
)
rP
n
∗
l
<
l
<
(
r
)
dr .
(59.58)
The radial matrix element (59.58) can be written as
R
jj
n
∗
l
>
, n
∗
l
<
, l
>
=
R
jj
n
∗
l
>
, l
>
φ
n
∗
l
>
, n
∗
l
<
, l
>
≡
3n
∗
l
>
2z
n
∗2
l
>
−l
2
>
1/2
φ
n
∗
l
>
, n
∗
l
<
, l
>
(59.59)
and φ
n
∗
l
>
, n
∗
l
<
, l
>
is tabulated elsewhere [59.19]. The
effective principal quantum numbers n
∗
l
>
and n
∗
l
<
of
the states j and j
in (59.57)–(59.59) both corre-
spond to principal quantum number n
j
and φ 1for
n
∗
l
>
−n
∗
l
<
1. The effective Gaunt factors g
(
x
)
and
g
(
x
)
are given by
g
(
x
)
= 0.7 −1.1/z +g
(
x
)
, x =
3k
B
T
2∆E
,
(59.60)
where
x ≤ 2351030100
g
(
x
)
0.20 0.24 0.33 0.56 0.98 1.33
is used for x < 50, and for x > 50
g
(
x
)
= g
(
x
)
=
√
3
π
1
2
+ln
4
3z
|
E
|
I
h
x
, (59.61)
with (59.60)and(59.61) joined smoothly near x = 50.
The energy E = E
j
is given by (59.34)andx
j
and x
jj
in (59.54) are evaluated using
∆E
j
=
2z
2
n
∗3
j
I
h
,∆E
jj
=
E
j
−E
j
.
(59.62)
For Z
e
= 2 and 3, (59.54) is generally accurate to within
±30% and ±50%. F or Z
e
≥ 4, (59.54) is less accu-
rate, as relativistic effects and resonances become more
important. Accuracy increases for transitions to higher
Rydberg levels as long as the line remains isolated.
Tables in appendices IV and V of [59.8] give widths
for atoms with Z
e
= 0, 1 and other semi-empirical
formulas based on detailed calculations have been de-
veloped by Seaton [59.20, 21] for use in the Opacity
Project where simple estimates of many thousands of
line widths are required.
59.3 Overlapping Lines
59.3.1 Transitions in Hydrogen
and Hydrogenic Ions
The most important case is that of lines of hydrogenic
systems emitted by a plasma with overall electrical neu-
trality, broadened by perturbing electrons and atomic
ions. The line profile is given by (59.4)–(59.9)inwhich
(59.20) is generalized to give
i
f
∗
w +id
if
∗
= 2πN
∞
0
v f
(
v
)
dv
×
∞
0
δ
i
i
δ
f
f
−S
i
i
(
b, v
)
S
∗
f
f
(
b, v
)
av
b db .
(59.63)
The superscripts and suffices e and i will be used to de-
note electron and ion quantities, and em indicates that
averaging over magnetic quantum numbers has not been
carried out. In the impact approximation, electron and
ion contributions evaluated using (59.63) are additive
and the matrix to be inverted is of order n
i
n
f
.How-
ever, under typical conditions in a laboratory plasma,
e.g., a hydrogen plasma with N
e
= N
i
= 10
22
m
−3
,per-
turbing atomic ions cannot be treated using the impact
approximation. The ions collectively generate a static
field at the emitter which produces first-order Stark
splitting of the upper and lower levels. The ions are
randomly distributed around the emitter and the field
distribution W
(
F
)
used assumes that each ion is De-
bye screened by electrons; allow ance is made for these
heavy composite perturbers interacting with each other
as well as with the emitter. If the ion field has a slow
Part D 59.3
Collisional Broadening of Spectral Lines 59.3 Overlapping Lines 881
time variation, ion dynamic effects on the line are pro-
duced [59.8, 9, 22].
The shift produced by electron perturbers is very
small and the usual model adopted is to assume that
the ions split the line into its Stark components, and
that each component is broadened by electron impact.
In both cases, only the dipole interactions in (59.43)are
included and the profile is symmetric. Then (59.4) takes
the form
I
(
ω
)
=
1
π
Re
W
(
F
)
dF
ii
ff
if
∗
|
∆
|
i
f
∗
×
i
f
∗
w +id−i
ω −ω
if
I
−1
if
∗
,
(59.64)
and the destruction of the degeneracy by the ion field
means that the matrix to be inverted in (59.64)isofor-
der
n
i
n
f
2
. Inclusion of higher multipoles in V(r, R) in
(59.43) introduces small asymmetries. The Stark repre-
sentation for the hydrogenic wave functions is often used
because it diagonalizes the shift matrix in (59.64). The
transformation is given by (see Sect. 9.1.2 and 13.4.2)
n
j
K
j
m
j
=
n
j
−1
l
j
=
|
m
j
|
(
−1
)
K
2l
j
+1
1/2
×
NN l
j
M
1
M
2
−m
j
n
j
l
j
m
j
,
j = i, i
, f, f
, (59.65)
where quantum number K
j
replaces l
j
and
n
j
= K
j
+K
j
+
m
j
+1 , N =
1
2
n
j
−1
,
K =
1
2
2K
j
+
m
j
+m
j
+1 ,
0 ≤ K
j
≤
n
j
−1
,
M
1
=
1
2
m
j
+K
j
−K
j
,
M
2
=
1
2
m
j
+K
j
−K
j
.
(59.66)
For the electron impact broadening, it is convenient
to separate the energy-changing and the zero energy-
change transitions, so that (59.39) is generalized to
give
γ
em
= γ
0
em
+
˜
γ
em
≡
i
f
∗
|
2w
e
|
if
∗
,
(59.67)
where
γ
0
em
≡
i
f
∗
2w
0
e
if
∗
= N
e
∞
0
v f
e
(
v
)
dv
×
j=i
σ
em
ij
(
v
)
+
j= f
σ
em
fj
(
v
)
δ
ii
δ
ff
,
(59.68)
˜
γ
em
≡
i
f
∗
|
2w
e
|
if
∗
= N
e
∞
0
v f
e
(
v
)
dv
˜
σ
em
i
f
if
(
v
)
.
(59.69)
In (59.68),
n
i
−n
j
= 0and
n
f
−n
j
= 0in
the first and second terms, respectively, and in
(59.69)
(
n
i
−n
i
)
=
n
f
−n
f
= 0 . The matrix elem-
ent (59.68) can be evaluated using (59.36), (59.48),
(59.50), and (59.52) as before. In (59.68),
σ
em
ij
(
v
)
= 2π
∞
0
P
e
ij
(
b, v
)
av
0
bdb ,
i, j = 0, 1, 2,... ,
(59.70)
and in (59.69)
˜
σ
em
i
f
if
(
v
)
= 2π
∞
0
˜
P
e
i
f
if
(
b, v
)
av
0
b db (59.71)
by analogy with (59.36)and(59.40), where a v
0
indicates
an average over all orientations of the collision only. On
using (59.48)–(59.50), (59.59)and(59.8) with j
i
=l
i
,
j
i
=l
i
, j
f
=l
f
, and j
f
=l
f
,
˜
P
e
i
f
if
(
b, v
)
av
0
=
8I
h
a
2
0
3m
e
b
2
v
2
D
ifi
f
˜
R
i
f
if
A(0, 0),
(59.72)
where
˜
R
i
f
if
≡
l
j
=l
i
±1
˜
R
2
ij
(
n
i
, l
i>
)
+
l
j
=l
f
±1
˜
R
2
fj
n
f
, l
f>
× δ
l
i
l
i
δ
l
f
l
f
−2
˜
R
i
i
(
n
i
, l
i>
)
˜
R
f
f
n
f
, l
f>
δ
l
i
l
i
±1
δ
l
f
l
f
±1
.
(59.73)
From (59.45)and(59.49)–(59.53)
A
(
0, 0
)
= δ,
Part D 59.3
882 Part D Scattering Theory
b
1
b
0
A
(
0, 0
)
db
b
=
ln
(
1
/
0
)
, Z
e
= 0 ,
ln
(
b
1
/b
0
)
, Z
e
= 0 .
(59.74)
(59.53). The impact approximation neglects electron–
electron correlations and the finite duration of collisions,
so the long-range dipole interaction leads to a logarith-
mic divergence at large impact parameters in (59.74).
Therefore, a second cutoff parameter is introduced
which is chosen to be the smaller of the Debye length
b
D
and vτ:
b
1
= min
b
D
≡
k
B
T
4πq
2
N
e
1/2
,vτ
, (59.75)
but estimating τ in this case is not straightforward; it
depends on the splitting of the Stark components [59.8].
59.3.2 Infrared and Radio Lines
If the density N
i
is low enough, the impact approxima-
tion becomes valid for the perturbing atomic ions, and
since impact shifts are unimportant, (59.6)gives
I
(
ω
)
=
1
π
Re
ii
ff
if
∗
∆
i
f
∗
×
i
f
∗
w
e
+w
i
−i
ω −ω
if
I
−1
if
∗
(59.76)
where in (59.76)
γ
e,i
= γ
0
e,i
+
˜
γ
e,i
≡
i
f
∗
2w
e,i
if
∗
,
(59.77)
γ
0
e,i
≡
i
f
∗
2w
0
e,i
if
∗
= N
e,i
∞
0
v f
e,i
(
v
)
dv
×
j=i
σ
e,i
ij
(v) +
j= f
σ
e,i
fj
(
v
)
δ
i
i
δ
f
f
, (59.78)
˜
γ
e,i
≡
i
f
∗
2
˜
w
e,i
if
∗
= N
e,i
∞
0
v f
e,i
(
v
)
dv
˜
σ
e,i
i
f
if
(
v
)
.
(59.79)
In general, cross sections for electron and heavy-particle
impact are roughly comparable for the same velocity
and hence different impact energies. Therefore, using
(59.78)and(59.79),
γ
0
e
γ
0
i
,
˜
γ
e
˜
γ
i
, (59.80)
and this result is consistent with approximation (59.64)
for high density plasmas. If
n
i
−n
f
= 1, 2, say, as n
f
increases, the relative contributions from (59.78)and
(59.79) decrease because there is increasing coherence,
and hence cancellation in
˜
σ
e
i
f
if
(
v
)
and
˜
σ
i
i
f
if
(
v
)
be-
tween the effects of levels i, i
and f, f
.
Radio lines of hydrogen are observed in galactic HII
regions where principal quantum numbers are of the or-
der of n
f
100, temperatures are T
e
= T
i
10
4
K, and
densities are N
e
= N
i
10
9
m
−3
.Ifγ is the full-half
width and
˜
γ is the full-half width when only contribu-
tions (59.79) are retained, the effect of cancellation is
illustrated by
n
i
− n
f
= 1 electrons protons + electrons
n
f
˜
γ/γ
˜
γ/γ
5 0.81 0.99
10 0.44 0.95
15 0.21 0.87
20 0.11 0.75
25 0.06 0.61
50 0.00 0.32
100 0.00 0.16
59.4 Quantum-Mechanical Theory
59.4.1 Impact Approximation
The scattering amplitude for a collisional transition
i → j is given in terms of elements of the transition
matrix T = 1−S by
f
k
j
, k
i
≡ f
χ
j
M
j
k
j
,χ
i
M
i
k
i
=
2πi
k
i
k
j
1/2
lml
m
i
l−l
Y
∗
lm
ˆ
k
i
Y
l
m
ˆ
k
j
× T
χ
j
M
j
l
m
;χ
i
M
i
lm
, (59.81)
where the quantities k
i
lm and k
j
l
m
refer to the
motion of the perturber relative to the emitter be-
fore and after the collision, and χ
i
and χ
j
represent
all nonmagnetic quantum numbers associated with
the unperturbed states i and j of the emitter.
The total energy of the emitter-perturber system is
given by
E
J
= E
j
+ε
j
,ε
j
=
2
2µ
k
2
j
,
(
J, j
)
=
(
I, i
)
,
(
F, f
)
,
(59.82)
Part D 59.4
Collisional Broadening of Spectral Lines 59.4 Quantum-Mechanical Theory 883
and for an isolated line, γ isgivenby(59.39), where
σ
ij
(
v
)
=
k
j
k
i
1
4πg
i
×
M
i
M
j
f
χ
j
M
j
k
j
,χ
i
M
i
k
i
2
d
ˆ
k
i
d
ˆ
k
j
,
k
i
= µv/ , (59.83)
and the interference term
˜
σ
if
(
v
)
≡
˜
σ
ifif
(
v
)
is given by
˜
σ
if
(v) =
1
4π
M
i
M
i
M
f
M
f
D
ifi
f
f(χ
i
M
i
k
,χ
i
M
i
k)
− f(χ
f
M
f
k
,χ
f
M
f
k)
2
d
ˆ
k d
ˆ
k
, (59.84)
with k = k
= µv/ .From(59.8)and(59.20),
w +id = π
µ
2
N
∞
0
1
v
f(v) dv
×
∞
l=0
(2l +1)
1 −S
ii
(l,v)S
∗
ff
(l,v)
,
(59.85)
where
(
µvb
)
2
⇒
2
l
(
l +1
)
(59.86)
and the integral over b has been replaced by a summation
over l . In (59.85)), S
ii
(
l,v
)
S
∗
ff
(
l,v
)
is given by
S
ii
(
l,v
)
S
∗
ff
(
l,v
)
=
1
(
2l +1
)
M
i
M
i
M
f
M
f
mm
D
ifi
f
S
I
χ
i
M
i
lm
;χ
i
M
i
lm
× S
∗
F
χ
f
M
f
lm
;χ
f
M
f
lm
(59.87)
and subscripts I and F are introduced to empha-
size that the S-matrix elements correspond to different
total energies E
I
and E
F
defined by (59.82). If
scattering by the emitter in a state j is treated
using a central potential, the amplitude for elastic
scattering is
f
k
, k
=
i
2k
∞
l=0
(
2l +1
)
T
jj
(
l,v
)
P
l
ˆ
k
·
ˆ
k
,
(59.88)
where
1 −T
jj
(
l,v
)
= S
jj
(
l,v
)
= exp
2iη
j
(
l, k
)
,
j = i, f ,
(59.89)
[cf. (59.23), (59.25), and (59.26)]. For the case of over-
lapping lines, (59.63) becomes
i
f
∗
w +id
if
∗
= π
µ
2
N
∞
0
1
v
f
(
v
)
dv
×
∞
l=0
(
2l +1
)
δ
i
i
δ
f
f
−S
i
i
(
l,v
)
S
∗
f
f
(
l,v
)
(59.90)
on generalizing (59.85) and using (59.87). Formulae
(59.85)and(59.90) have been obtained by assuming
that a collision produces no change in the angular mo-
mentum of the relative emitter-perturber motion. This
corresponds to the assumption of a common trajectory
in semiclassical theory, and means that the total angular
momentum of the emitter-perturber system is not con-
served. This assumption is removed in the derivation
of the more general expressions given in the following
sections.
59.4.2 Broadening by Electrons
Different coupling schemes can be used to describe the
emitter-perturber collision. For LS coupling,
χ
j
M
j
⇒ χ
0
j
L
j
M
j
SM
S
, j = i, i
, f, f
, (59.91)
in (59.81), where χ
0
j
denotes all other quantum numbers
required to describe state j that do not change during
the collision. Then
L
j
M
j
SM
S
lm
1
2
m
s
=
L
T
j
M
T
j
S
T
M
S
C
L
j
lL
T
j
M
j
mM
T
S
C
S
1
2
S
T
M
S
m
s
M
T
S
×
L
j
Sl
1
2
L
T
j
M
T
j
S
T
M
T
S
,
(59.92)
where C
j
1
j
2
j
3
m
1
m
2
m
3
is a vector coupling coefficient, the
superscript T denotes quantum numbers of the emitter-
perturber system, and
1
2
, m
s
are the spin quantum
numbers of the scattered electron. On using (59.92),
Part D 59.4
884 Part D Scattering Theory
(59.90) is replaced by
i
f
∗
w +id
if
∗
= π
(
/m
e
)
2
N
×
L
T
i
L
T
f
S
T
ll
(
−1
)
L
i
+L
i
+l+l
2L
T
i
+1
2L
T
f
+1
×
2S
T
+1
2
(
2S +1
)
L
T
f
L
T
i
1
L
i
L
f
l
L
T
f
L
T
i
1
L
i
L
f
l
×
∞
0
1
v
f
e
(v) dv
δ
l
l
δ
L
i
L
i
δ
L
f
L
f
− S
I
L
i
Sl
1
2
L
T
i
S
T
; L
i
Sl
1
2
L
T
i
S
T
× S
∗
F
L
f
Sl
1
2
L
T
f
S
T
; L
f
Sl
1
2
L
T
f
S
T
,
(59.93)
where, for an isolated line, the width and shift are given
by (59.93) with L
i
= L
i
and L
f
= L
f
. For hydrogenic
systems, where states i, i
and f, f
with different an-
gular momenta are degenerate, a logarithmic divergence
occurs for large values of l and l
[cf. Eq. (59.74)], and
must be removed by using (59.75)and(59.86).
If a jj coupling scheme is used χ
j
M
j
⇒ χ
0
j
J
j
M
j
,
j = i, i
, f, f
in (59.81), and
J
j
M
j
lm
1
2
m
s
=
J
T
j
M
T
j
jm
C
J
j
jJ
T
j
M
j
m
M
T
j
C
l
1
2
j
mm
s
m
J
j
ljJ
T
j
M
T
j
,
(59.94)
then (59.93) becomes
i
f
∗
w +id
if
∗
= π
(
/m
e
)
2
N
×
J
T
i
J
T
f
jj
ll
(−1)
J
i
+J
i
+2J
T
f
+j+j
1
2
(2J
T
i
+1)(2J
T
f
+1)
×
J
T
f
J
T
i
1
J
i
J
f
j
J
T
f
J
T
i
1
J
i
J
f
j
×
∞
0
1
v
f
e
(v) dv
δ
l
l
δ
j
j
δ
J
i
J
i
δ
J
f
J
f
− S
I
(J
i
l
j
J
T
i
; J
i
ljJ
T
i
) S
∗
F
(J
f
l
j
J
T
f
; J
f
ljJ
T
f
)
,
(59.95)
where J
i
= J
i
and J
f
= J
f
for an isolated line. If the
spectrum of the emitter is classified using LS coupling,
it is often sufficient to use energies defined by
E
L
j
S
=
J
j
2J
j
+1
2L
j
+1
2S
j
+1
E
L
j
SJ
j
, (59.96)
and obtain the S-matrix elements in an LS coupling
scheme. They are then transformed to jj coupling by
using the algebraic transformation
S
J
j
l
j
J
T
j
; J
j
ljJ
T
j
=
2J
j
+1
2J
j
+1
(
2 j +1
)
2 j
+1
1/2
×
L
T
j
S
T
(2L
T
j
+1)(2S
T
+1)
×
L
j
lL
T
j
S
1
2
S
T
J
j
jJ
T
j
L
j
l
L
T
j
S
1
2
S
T
J
j
j
J
T
j
× S
L
j
Sl
1
2
L
T
j
S
T
; L
j
Sl
1
2
L
T
j
S
T
,
(59.97)
and introducing the splitting of the fine structure com-
ponents in (59.4)or(59.6). If in LS coupling the line
is isolated, but nevertheless the broadened fine structure
components overlap significantly, then the interference
terms in (59.6) must be included.
59.4.3 Broadening by Atoms
The formal result is very similar to (59.95), but in this
case, the relative motion only gives rise to orbital angular
momentum. Thus
i
f
∗
w +id
if
∗
= π
(
/µ
)
2
N
×
J
T
i
J
T
f
ll
(−1)
J
i
+J
i
+2J
T
f
+l+l
(2J
T
i
+1)(2J
T
f
+1)
×
J
T
f
J
T
i
1
J
i
J
f
l
J
T
f
J
T
i
1
J
i
J
f
l
×
∞
0
1
v
f(v) dv
δ
l
l
δ
J
i
J
i
δ
J
f
J
f
− S
I
J
i
l
J
T
i
; J
i
lJ
T
i
S
∗
F
J
f
l
J
T
f
; J
f
lJ
T
f
.
(59.98)
For many cases of practical interest, transitions of type
J
i
→ J
f
are isolated and so have line profiles given by
(59.3), (59.10), and (59.98), where J
i
= J
i
and J
f
= J
f
[59.14]. In order to obtain the S-matrix elements in
Part D 59.4
Collisional Broadening of Spectral Lines 59.5 One-Perturber Approximation 885
(59.98), it is usually sufficient to use adiabatic poten-
tials for the emitter-perturber system that have been
calculated neglecting fine structure. Since T is typi-
cally a few hundred degrees, only coupling between
adiabatic states that tend to the appropriate separated-
atom limit are retained in the scattering problem. The
coupled scattering equations are then solved with fine
structure introduced by applying an algebraic trans-
formation to the adiabatic potentials, and using the
observed splittings of the energy levels. The Born–
Oppenheimer approximation is valid, and details are
given in [59.23].
59.5 One-Perturber Approximation
59.5.1 General Approach and Utility
If only one perturber is effective in producing broad-
ening, I
(
ω
)
can be obtained by considering a dipole
transition between initial and final states I and F of the
emitter-perturber system. Then P
(
ω
)
isgivenby(59.2),
where
I
(
ω
)
=
δ
(
ω −ω
IF
)
IF
∗
|
∆
|
IF
∗
av
,
ω
IF
= E
I
−E
F
, (59.99)
and av denotes an average over states I and a sum over
states F [59.9]. Wave functions Ψ
J
are given by
Ψ
J
(
r, R
)
= O
j
ψ
j
(
r
)
φ
k
j
, k
j
0
; R
, (59.100)
where J = I, F,andO is an operator that takes account
of any symmetry properties of the emitter-perturber sys-
tem. The energies E
I
and E
F
are given by (59.82). The
perturber wave functions for initial state j
0
and final
state j are expanded in the form
φ
k
j
, k
j
0
; R
= 2πi
l
j
0
m
j
0
l
j
m
j
i
l
j
0
k
−1/2
j
0
Y
∗
l
j
0
m
j
0
ˆ
k
j
0
× Y
l
j
m
j
ˆ
R
1
R
F
Γ
j
,Γ
j
0
; R
(59.101)
where Γ
j
denotes a channel characterized by
Γ
j
= χ
j
M
j
l
j
m
j
, j = 0, 1, 2,... , (59.102)
[see Eq. (59.81)]. In (59.101), the radial perturber wave
function has the limiting forms
F
Γ
j
,Γ
j
0
; R
R→0
∼
R
l
j
+1
R→∞
∼
k
−1/2
j
δ
Γ
j
Γ
j
0
exp
−iθ
j
−S
J
Γ
j
;Γ
j
0
exp
iθ
j
,
(59.103)
with
θ
j
= k
j
R −
1
2
l
j
π −
z
k
i
ln
2k
j
R
+arg Γ
l
j
+1 +i
z
k
j
,
z =
µq
2
2
Z
e
Z
p
. (59.104)
The coupled equations obtained by using (59.12),
(59.13), (59.100), and (59.101), where
Ψ
(
r, R, t
)
= Ψ
J
(
r, R
)
exp
(
−iE
J
t/
)
,
(59.105)
are integrated to give functions F
Γ
j
,Γ
j
0
; R
. Using
(59.100)and(59.101), (59.99) becomes
I
(
ω
)
=
1
2
N
Γ
i
0
Γ
i
Γ
i
Γ
f
0
Γ
f
Γ
f
u
i
0
Γ
i
Γ
f
∗
|
∆
|
Γ
i
Γ
f
∗
×
∞
0
1
v
f(v) dv F
(
Γ, v
)
,
(59.106)
where
F
(
Γ, v
)
=
∞
0
F
∗
Γ
i
,Γ
i
0
; R
F
Γ
f
,Γ
f
0
; R
dR
×
∞
0
F
Γ
i
,Γ
i
0
; R
F
∗
Γ
f
,Γ
f
0
; R
dR ,
(59.107)
u
i
0
= g
i
0
(
i
0
g
i
0
,v=
k
i
0
µ
.
(59.108)
The one-perturber approximation is valid when
∆ω ≡|ω −ω
if
|w ; V
¯
V , (59.109)
where V is the effective interaction potential required to
produce a shift ∆ω . In the center of the line, many-body
effects are always important and the one-perturber ap-
proximation diverges as ∆ω →0 . In many cases, there
Part D 59.5
886 Part D Scattering Theory
is a region of overlap where criteria (59.1)and(59.109)
are all valid, but when ∆ω τ 1, (59.99) is a static ap-
proximation, since the average time between collisions
is ∆ω
−1
.
59.5.2 Broadening by Electrons
If LS coupling is used, definition of channel Γ
j
in
(59.104) is replaced by
Γ
j
= L
j
Sl
j
1
2
L
T
j
S
T
, j = i
0
, i, i
, f
0
, f, f
,
(59.110)
[cf. (59.91)and(59.92)]. Then assuming that the weights
u
i
0
of all the levels i
0
that effectively contribute to the
line are the same, (59.106) becomes
I
(
ω
)
=
1
2
N
e
Γ
i
0
Γ
i
Γ
i
Γ
f
0
Γ
f
Γ
f
L
i
S
L
f
S
∗
∆
L
i
S
L
f
S
∗
× δ
l
i
l
f
δ
l
i
l
f
δ
L
T
i
0
L
T
i
δ
L
T
f
0
L
T
f
δ
L
T
i
0
L
T
i
δ
L
T
f
0
L
T
f
×
(
−1
)
L
i
+L
i
+l
i
+l
i
2S
T
+1
2
(
2S +1
)
×
2L
T
i
+1
2L
T
f
+1
×
L
T
f
L
T
i
1
L
i
L
f
l
L
T
f
L
T
i
1
L
i
L
f
l
i
×
∞
0
1
v
f
e
(v) dv F (Γ, v) , (59.111)
where F
(
Γ, v
)
is defined by (59.107)and(59.111). If
the functions F
Γ
j
,Γ
j
0
; R
in (59.107) are replaced by
their asymptotic forms (59.103), then
F
(
Γ, v
)
∆ω
−2
(
/m
e
)
2
×
δ
Γ
i
0
Γ
i
δ
Γ
f
0
Γ
f
−S
I
Γ
i
;Γ
i
0
S
∗
F
Γ
f
;Γ
f
0
×
δ
Γ
i
0
Γ
i
δ
Γ
f
0
Γ
f
−S
∗
I
Γ
i
;Γ
i
0
S
F
Γ
f
;Γ
f
0
.
(59.112)
On substituting (59.112)into(59.111), summing over
Γ
i
0
and Γ
f
0
and using the unitary property of the
S-matrix,
I
(
ω
)
=
1
π∆ω
2
L
T
i
L
T
f
S
T
ll
L
i
S
L
f
S
∗
∆
L
i
S
L
f
S
∗
×
i
f
∗
w
if
∗
,
(59.113)
where
i
f
∗
w
if
∗
isgivenby(59.93). Line shape
(59.113) is identical to that obtained from (59.6)when
∆ω w.If the jj coupling scheme specified by (59.94)
is used, and channel Γ
j
is defined by
Γ
j
= J
j
l
j
j
j
J
T
j
, j = i
0
, i, i
, f
0
, f, f
, (59.114)
Equation (59.106) becomes [cf. (59.95)]
I(ω)
=
1
2
N
e
Γ
i
0
Γ
i
Γ
i
Γ
f
0
Γ
f
Γ
f
J
i
J
f
∗
∆
J
i
J
f
∗
× δ
l
i
l
f
δ
l
i
l
f
δ
j
i
j
f
δ
j
i
j
f
δ
J
T
i
0
J
T
i
δ
J
T
f
0
J
T
f
δ
J
T
i
0
J
T
i
δ
J
T
f
0
J
T
f
× (−1)
J
i
+J
i
+2J
T
f
+j
i
+j
i
×
1
2
2J
T
i
+1
2J
T
f
+1
×
J
T
f
J
T
i
1
J
i
J
f
j
i
J
T
f
J
T
i
1
J
i
J
f
j
i
×
∞
0
1
v
f
e
(v) dv F (Γ, v) , (59.115)
where F
(
Γ, v
)
is given by (59.107)and(59.114).
59.5.3 Broadening by Atoms
In the wings of a line where
|
∆ω
|
E
J
i
−E
J
i
, j =
i
0
, i, i
, f
0
, f, f
, coupling between the fine structure
levels is important. If channel Γ
j
is defined by
Γ
j
= J
j
l
j
J
T
j
, j = i
0
, i, i
, f
0
, f, f
, (59.116)
Equation (59.106) becomes [cf. (59.115)]
I
(
ω
)
=
1
2
N
Γ
i
0
Γ
i
Γ
i
Γ
f
0
Γ
f
Γ
f
J
i
J
f
∗
∆
J
i
J
f
∗
× δ
l
i
l
f
δ
l
i
l
f
δ
J
T
i
0
J
T
i
δ
J
T
f
0
J
T
f
δ
J
T
i
0
J
T
i
δ
J
T
f
0
J
T
f
× (−1)
J
i
+J
i
+2J
T
f
+l
i
+l
i
2J
T
i
+1
2J
T
f
+1
×
J
T
f
J
T
i
1
J
i
J
f
l
i
J
T
f
J
T
i
1
J
i
J
f
l
i
×
∞
0
1
v
f
(
v
)
dv F (Γ, v) ,
(59.117)
where F
(
Γ, v
)
is given by (59.107)and(59.116).
In the far wings, where
|
∆ω
|
E
J
i
−E
J
i
, j =
i
0
, i, i
, f
0
, f, f
, an adiabatic approximation is valid.
Part D 59.5