2.3 Voigt Profile
29
broadening and the Lorentzian due to natural broadening. The Voigt functions are
given by
U(x, t) =
1
√
4πt
∞
−∞
exp
−
(x−y) 2
4t
1 + y 2
dy
(2.14a)
V (x, t) =
1
√
4πt
∞
−∞
y exp
−
(x−y) 2
4t
1 + y 2
dy, x ∈ R, t > 0,
(2.14b)
and the line broadening function [5]
H (a, u) =
a
π
∞
−∞
exp(−t 2 )
(u − t) 2 + a 2 dt,
(2.14c)
with
a =
√
ln2δν L
δν D
, u = 2
√
ln2
ν − ν 0
δν D
.
(2.14d)
ν is the frequency, ν 0 the central frequency of the Gaussian, δν D the full width
half maximum (FWHM) due to the Doppler broadening, and δν L the FWHM of the
Lorentzian.
The Voigt functions are connected [4] as follows:
U(x, t) + iV (x, t) =
π
4t
exp(z
2 ) erfc(z), with z =
1 − ix
2
√
t
.
(2.15)
Some results are plotted in Fig. 2.3. The line broadening function is given by
H (a, u) =
1
a
√
π
U
u
a
,
1
4a 2
.
(2.16)
0
5
10
15
x
0.2
0.4
0.6
0.8
U
0
5
10
15
x
0
0.1
0.2
0.3
0.4
V
Fig. 2.3 On the left-hand side, the Voigt function U(x, t), and on the right-hand side, V (x, t),
with t equal to 0.1, 1, 2.5, 5, and 10 from top to bottom