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Application of curve of growth to absorption spectral lines
1. Quanr. Spectmsc. Radial. Transfer Vol. 21. pp. 19-22 0 Pergamon Press Ltd., 1979. Printed in Great Brilain
APPLICATION OF CURVE OF GROWTH TO ABSORPTION SPECTRAL LINES R. M. HILL Department of Atomic Physics, University of Newcastle upon Tyne, NEI 7RU. England (Received 19 May 1978)
AbstracbThe definition and use of the curve of growth is explained and a simple original approximation for it is given which is good enough for most experimental purposes. The derivation of the approximate expression is given in the Appendix. INTRODUCTION ABSORPTION spectra
are often used to deduce broadening parameters and associated interatomic potentials. The number of absorbing atoms is difficult to estimate by non-spectroscopic methods so one usually relies on the fact that the spectral integral of the extinction coefficient (WT) is proportional to the product nf of the line oscillator strength and the number of absorbing atoms per unit area seen through the cell. This integral may be represented by the equation 2
dv = e 47zomc nf = WT,
T(V)
where T(V) is the extinction coerlicient. The intensity Z, per unit frequency interval, of the transmitted light is related to the incident intensity IOby Z = lo{1- exp [--T(V)]}. A typical absorption profile is shown in Fig. 1. The equivalent width of the line W (defined as the shaded area divided by lo) may readily be measured and is related to T(V)by W=
I
b;[1- exp (--7(v))] dv.
(2)
Clearly, if T(V) is small (i.e. small peak absorption), one may use what is known as the transparent approximation w=w, and hence obtain nf. The relation between W and WT is known as the curve of growth. I=Io~l-expl-~)l A
Fig. I. 19
R. M. HILL
20
of curve of growth Unfortunately, at very small absorption, when the transparent approximation is valid, the signal to noise level is small and results in an inaccurate measurement of W and, even at peak absorptions of only 40%, WT may be up to 40% more than W. If T(V) can be assumed to be a Voigt profile, the relation between W and WT (the curve of growth) may be computed either by double integration (since the Voigt profile is itself a convolution integral) or by integrating certain series approximations [PENNER,“’ VANDER HELD(*)].In either case, the computations are not a trivial matter and have only been carried out over a limited range of the parameters. The author has deduced a simple approximate relation by expanding the exponential to second order and integrating term by term. The approximation is asymptotic for small absorption and comparison with computed values [LAPP’~‘] shows maximum deviations of less than 3% even up to 75% peak absorption. This covers the most useful range for absorption spectroscopy and the experimental errors are usually more than 3% anyway. The relation is simple enough to be easily incorporated into a processing computer programme and may be used for parameter values not covered by computed curves of growth. Use
Approximate
curve of growth relation
The Voigt profile is characterised by a Gaussian parameter g and a Lorentz damping constant y and has the form
where Av is the frequency deviation from the line centre. Here the lower limit of integration is chosen as --(x instead of -v. since the profile is effectively zero there anyway. The Lorentz half, half-width is y and the Gaussian half, half-width is g (In 2)“’ . It is convenient to express W and WT in units of g, thus giving the dimensionless quantities 0
=
w/g,
UT
=
wT/g.
Clearly, it does not matter what units are used as abscissae when the profile is plotted (frequency, wavenumber, wavelength, etc.) as long as both W and g are measured in the same units. The approximate relation, giving WTin terms of 0 is, (4)
where a = r/g is the Voigt parameter and q(x) = (exp x2)( 1 - erfx). V(x)t is familiar to spectroscopists involved in calculating the peak height of a Voigt profile and fast concise computer programmes are available [HILL(~)]which are accurate to better than two parts in 10”. Care should be taken if subroutines for erfx are used since the exponential can easily magnify rounding errors to significant proportions. The relation above also agrees with the Lorentz approximation at large a (> 10) to within 3% for up to 75% absorption. The Lorentz approximation is T& = 2
exp (-W*/27Fa)[Jo(iWT/27Taa)- iJj(ioT/27ra)J.
tFor 0
Wx)=;$;;(l-&+$-$3 >
Application of curve of growth to absorption spectral lines
21
Comparison between different authors’ curves of growth VAN DER HELD(*) and PENNER(‘) plot o/2 as ordinate on a logarithmic scale against log,, (10.6~~“*UT) with a as parameter. The values of a of van der Held differ by a factor of two from those of Penner whose values of a correspond to those used here. LAPP’~)plots curves of log,,, (10.6&‘* oT) against a on a logarithmic scale with o/2 as parameter. REFERENCES I. S. S. PENNER,Quanritatioe Molecular Spectroscopy and Gas Emissiuities, pp. 108-109.Addison-Wesley, Reading, Mass. (1959). 2. E. F. M. VANDERHELD,Z. f. Phys. 70, 508 (1931). 3. M. LAPP,private communication. 4. R. M. HILL,I. P. Sharp Associates Ltd., Workspace 2952801ERC (1977)and private communication.
APPENDIX Derivation of the expression for the curue of growth Let V(x) be a normalized Voigt profile with parameter a [i.e. J-“I V(x) dx = I]. Then
V(x) = a1r-3’2 The extinction coefficient r may be represented by V in the following way. If T is a Voigt profile with Gaussian parameter g (i.e. Gaussian half, half width = gvln 2) and Lorentz half, half-width y, then
where Av = Y- VO;from Eq. (l),
and, therefore,
Writing x = Av/g,
or 7 =
F
V(x) = orV(x).
Then, from Eq. (2)
or o=-=
w g
=
I_I
(1 - e -wrvcxl)dx.
The first four terms of the exponential give
0=
1 ~~V(X)-~!W:V~(X)+~O:V’(~)
V3(x)
1
dx.
The second integral can be carried out (most easily by a Fourier transformation or by writing it as a convolution) with the result x I -x
V’(x) dx = -&r)
q(av2)
where
T(x) = ex2erjc x.
22
R. M. HILL The series must be convergent for small oT or large a (since this makes V small). Hence, the first two terms may be used as long as oT is small or a large, i.e.
or, expressing this as series in w for OT to the third order, oT
= 0 +
&
Y(aV2)
+$P(ad2).
The third-order term in the exponential gives rise to jZ V-’dx, which cannot be evaluated in terms of known functions but with the coefficient 0:/3! it must contribute to the third term in Eq. (6). The limiting values of this integral for small and large values of the Voigt parameter a can be obtained as follows: As a -0. let V(x)+ G(x) = aY3”
em’*
I
-
dx,
-x m
=t -,,Z emXz
As a+m. let
It is then not difficult to evaluate the following integrals: I
1.. G”(x)
dx
=
n-l,+,,n-l),?,
(7)
[2(n - I)]!
L”(x) dx = ,(n _ l)l]~ Plnal-‘“-”
For large a, the third-order term in the exponential (0?/3!) JFXV’(x) dx becomes (0:/3!) I-“_L’(x) dx = oT3/(4na)* from Eq. (8). Since Y(a)+ (a’n)-“’ at large a. we see that this integral contributes a negative term half the size of the last term in Eq. (6). suggesting we should take only half this term, viz.
However, from Eq. (7). we see that (ors/3!) _fZ G3(x) dx contributes less to the last term in Eq. (6) so that Eq. (9) is not the best third order approximation for small a. Since the fourth-order term contributes positively, it is better for the third-order term of the truncated series to be overestimated so it is not surprising to find that Eq. (6) is better than Eq. (9) for small a. Equation (4). which is the mean of Eqs. (6) and (9). is a better compromise. In the Lorentz approximation, it amounts to the inclusion of half the third-order term in the truncated series for the integrand in the integral expression for o. This is usual in the truncation of series with alternating signs.
APPLICATION OF CURVE OF GROWTH TO ABSORPTION SPECTRAL LINES R. M. HILL Department of Atomic Physics, University of Newcastle upon Tyne, NEI 7RU. England (Received 19 May 1978)
AbstracbThe definition and use of the curve of growth is explained and a simple original approximation for it is given which is good enough for most experimental purposes. The derivation of the approximate expression is given in the Appendix. INTRODUCTION ABSORPTION spectra
are often used to deduce broadening parameters and associated interatomic potentials. The number of absorbing atoms is difficult to estimate by non-spectroscopic methods so one usually relies on the fact that the spectral integral of the extinction coefficient (WT) is proportional to the product nf of the line oscillator strength and the number of absorbing atoms per unit area seen through the cell. This integral may be represented by the equation 2
dv = e 47zomc nf = WT,
T(V)
where T(V) is the extinction coerlicient. The intensity Z, per unit frequency interval, of the transmitted light is related to the incident intensity IOby Z = lo{1- exp [--T(V)]}. A typical absorption profile is shown in Fig. 1. The equivalent width of the line W (defined as the shaded area divided by lo) may readily be measured and is related to T(V)by W=
I
b;[1- exp (--7(v))] dv.
(2)
Clearly, if T(V) is small (i.e. small peak absorption), one may use what is known as the transparent approximation w=w, and hence obtain nf. The relation between W and WT is known as the curve of growth. I=Io~l-expl-~)l A
Fig. I. 19
R. M. HILL
20
of curve of growth Unfortunately, at very small absorption, when the transparent approximation is valid, the signal to noise level is small and results in an inaccurate measurement of W and, even at peak absorptions of only 40%, WT may be up to 40% more than W. If T(V) can be assumed to be a Voigt profile, the relation between W and WT (the curve of growth) may be computed either by double integration (since the Voigt profile is itself a convolution integral) or by integrating certain series approximations [PENNER,“’ VANDER HELD(*)].In either case, the computations are not a trivial matter and have only been carried out over a limited range of the parameters. The author has deduced a simple approximate relation by expanding the exponential to second order and integrating term by term. The approximation is asymptotic for small absorption and comparison with computed values [LAPP’~‘] shows maximum deviations of less than 3% even up to 75% peak absorption. This covers the most useful range for absorption spectroscopy and the experimental errors are usually more than 3% anyway. The relation is simple enough to be easily incorporated into a processing computer programme and may be used for parameter values not covered by computed curves of growth. Use
Approximate
curve of growth relation
The Voigt profile is characterised by a Gaussian parameter g and a Lorentz damping constant y and has the form
where Av is the frequency deviation from the line centre. Here the lower limit of integration is chosen as --(x instead of -v. since the profile is effectively zero there anyway. The Lorentz half, half-width is y and the Gaussian half, half-width is g (In 2)“’ . It is convenient to express W and WT in units of g, thus giving the dimensionless quantities 0
=
w/g,
UT
=
wT/g.
Clearly, it does not matter what units are used as abscissae when the profile is plotted (frequency, wavenumber, wavelength, etc.) as long as both W and g are measured in the same units. The approximate relation, giving WTin terms of 0 is, (4)
where a = r/g is the Voigt parameter and q(x) = (exp x2)( 1 - erfx). V(x)t is familiar to spectroscopists involved in calculating the peak height of a Voigt profile and fast concise computer programmes are available [HILL(~)]which are accurate to better than two parts in 10”. Care should be taken if subroutines for erfx are used since the exponential can easily magnify rounding errors to significant proportions. The relation above also agrees with the Lorentz approximation at large a (> 10) to within 3% for up to 75% absorption. The Lorentz approximation is T& = 2
exp (-W*/27Fa)[Jo(iWT/27Taa)- iJj(ioT/27ra)J.
tFor 0
Wx)=;$;;(l-&+$-$3 >
Application of curve of growth to absorption spectral lines
21
Comparison between different authors’ curves of growth VAN DER HELD(*) and PENNER(‘) plot o/2 as ordinate on a logarithmic scale against log,, (10.6~~“*UT) with a as parameter. The values of a of van der Held differ by a factor of two from those of Penner whose values of a correspond to those used here. LAPP’~)plots curves of log,,, (10.6&‘* oT) against a on a logarithmic scale with o/2 as parameter. REFERENCES I. S. S. PENNER,Quanritatioe Molecular Spectroscopy and Gas Emissiuities, pp. 108-109.Addison-Wesley, Reading, Mass. (1959). 2. E. F. M. VANDERHELD,Z. f. Phys. 70, 508 (1931). 3. M. LAPP,private communication. 4. R. M. HILL,I. P. Sharp Associates Ltd., Workspace 2952801ERC (1977)and private communication.
APPENDIX Derivation of the expression for the curue of growth Let V(x) be a normalized Voigt profile with parameter a [i.e. J-“I V(x) dx = I]. Then
V(x) = a1r-3’2 The extinction coefficient r may be represented by V in the following way. If T is a Voigt profile with Gaussian parameter g (i.e. Gaussian half, half width = gvln 2) and Lorentz half, half-width y, then
where Av = Y- VO;from Eq. (l),
and, therefore,
Writing x = Av/g,
or 7 =
F
V(x) = orV(x).
Then, from Eq. (2)
or o=-=
w g
=
I_I
(1 - e -wrvcxl)dx.
The first four terms of the exponential give
0=
1 ~~V(X)-~!W:V~(X)+~O:V’(~)
V3(x)
1
dx.
The second integral can be carried out (most easily by a Fourier transformation or by writing it as a convolution) with the result x I -x
V’(x) dx = -&r)
q(av2)
where
T(x) = ex2erjc x.
22
R. M. HILL The series must be convergent for small oT or large a (since this makes V small). Hence, the first two terms may be used as long as oT is small or a large, i.e.
or, expressing this as series in w for OT to the third order, oT
= 0 +
&
Y(aV2)
+$P(ad2).
The third-order term in the exponential gives rise to jZ V-’dx, which cannot be evaluated in terms of known functions but with the coefficient 0:/3! it must contribute to the third term in Eq. (6). The limiting values of this integral for small and large values of the Voigt parameter a can be obtained as follows: As a -0. let V(x)+ G(x) = aY3”
em’*
I
-
dx,
-x m
=t -,,Z emXz
As a+m. let
It is then not difficult to evaluate the following integrals: I
1.. G”(x)
dx
=
n-l,+,,n-l),?,
(7)
[2(n - I)]!
L”(x) dx = ,(n _ l)l]~ Plnal-‘“-”
For large a, the third-order term in the exponential (0?/3!) JFXV’(x) dx becomes (0:/3!) I-“_L’(x) dx = oT3/(4na)* from Eq. (8). Since Y(a)+ (a’n)-“’ at large a. we see that this integral contributes a negative term half the size of the last term in Eq. (6). suggesting we should take only half this term, viz.
However, from Eq. (7). we see that (ors/3!) _fZ G3(x) dx contributes less to the last term in Eq. (6) so that Eq. (9) is not the best third order approximation for small a. Since the fourth-order term contributes positively, it is better for the third-order term of the truncated series to be overestimated so it is not surprising to find that Eq. (6) is better than Eq. (9) for small a. Equation (4). which is the mean of Eqs. (6) and (9). is a better compromise. In the Lorentz approximation, it amounts to the inclusion of half the third-order term in the truncated series for the integrand in the integral expression for o. This is usual in the truncation of series with alternating signs.