α D ¼
ϑ 0
c
ffiffiffiffiffiffiffiffiffiffi ffi
2R g T
M M
r
¼
ϑ 0
c
ffiffiffiffiffiffiffiffi
2kT
m g
r
ð3:24Þ
where R g is the ideal gas constant and M M is the molar mass in [kg mol
À1
]. The
Doppler width is often expressed slightly differently as the full width (or half width)
at half maximum (FWHM) which is
Δϑ D ¼ 2
ϑ 0
c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2kT log e 2
m g
r
ð3:25Þ
These equations are sometimes seen in an alternative but analogous form with the
absorption cross-section written as
σ abs ¼ S s φ ϑ, ϑ 0
ð
Þ
ð3:26Þ
where φ(ϑ, ϑ 0 ) is the line-profile function that takes into account the Doppler
broadening. Here, ϑ 0 is the rest frequency of the emission line. The function can
be written as
φ ϑ, ϑ 0
ð
Þ¼
c
v th ϑ 0
ffiffiffi
π
p e
Àc 2 ϑÀϑ 0
ð
Þ
2
ϑ 0
2 v th
2
ð3:27Þ
where c is the speed of light and v th is the most probable velocity given by
v th ¼
ffiffiffiffiffiffiffiffi
2kT
m g
r
ð3:28Þ
A variable similar to α D can be defined for pressure broadening so that (LopezPuertas and Taylor 2001)
α L STP
ð
Þ ¼
1
2πct STP
ð3:29Þ
where t STP is the mean time between collisions at standard temperature and pressure
(STP). Then
k ϑ ϑ
ð Þ ¼
S s
π
α L
ϑ À ϑ 0
ð
Þ
2 þ α 2
L
ð3:30Þ
where
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3 Gas Emissions Near the Nucleus
ϑ 0
c
ffiffiffiffiffiffiffiffiffiffi ffi
2R g T
M M
r
¼
ϑ 0
c
ffiffiffiffiffiffiffiffi
2kT
m g
r
ð3:24Þ
where R g is the ideal gas constant and M M is the molar mass in [kg mol
À1
]. The
Doppler width is often expressed slightly differently as the full width (or half width)
at half maximum (FWHM) which is
Δϑ D ¼ 2
ϑ 0
c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2kT log e 2
m g
r
ð3:25Þ
These equations are sometimes seen in an alternative but analogous form with the
absorption cross-section written as
σ abs ¼ S s φ ϑ, ϑ 0
ð
Þ
ð3:26Þ
where φ(ϑ, ϑ 0 ) is the line-profile function that takes into account the Doppler
broadening. Here, ϑ 0 is the rest frequency of the emission line. The function can
be written as
φ ϑ, ϑ 0
ð
Þ¼
c
v th ϑ 0
ffiffiffi
π
p e
Àc 2 ϑÀϑ 0
ð
Þ
2
ϑ 0
2 v th
2
ð3:27Þ
where c is the speed of light and v th is the most probable velocity given by
v th ¼
ffiffiffiffiffiffiffiffi
2kT
m g
r
ð3:28Þ
A variable similar to α D can be defined for pressure broadening so that (LopezPuertas and Taylor 2001)
α L STP
ð
Þ ¼
1
2πct STP
ð3:29Þ
where t STP is the mean time between collisions at standard temperature and pressure
(STP). Then
k ϑ ϑ
ð Þ ¼
S s
π
α L
ϑ À ϑ 0
ð
Þ
2 þ α 2
L
ð3:30Þ
where
188
3 Gas Emissions Near the Nucleus