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10.10: APPENDIX B- Radiation Damping as Functions of Angular Frequency, Frequency and Wavelength

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It occurred to me while preparing this Chapter as well as the preceding and following ones, that sometimes I have been using angular frequency as argument, sometimes frequency, and sometimes wavelength. In this Appendix, I bring together the salient formulas for radiation damping in terms of ω=ωω0,ν=νν0 and λ=λλ0. I reproduce equation 10.2.11 for the absorption coefficient for a set of forced, damped oscillators, except that I replace n, the number per unit volume of oscillators with n1f12, the effective number of atoms per unit volume in the lower level of a line, and I replace the classical damping constant γ with Γ, which may include a pressure broadening component.

α=n1f12Γe2ω2mε0c[(ω2ω20)2+Γ2ω2]m1.

You should check that the dimensions of this expression are L1, which is appropriate for linear absorption coefficient. You may note that [e2/ε0]ML3T2 and [Γ]T1. Indeed check the dimensions of all expressions that follow, at each stage.

We can write ω2ω20=(ωω0)(ω+ω0)=ω(2ω0+ω), and the equation becomes

α=n1f12Γe2(ω0+ω)2mε0c[(ω)2(2ω0+ω)2+Γ2(ω0+ω)2]m1.

Now I think it will be owned that the width of a spectrum line is very, very much smaller than its actual wavelength, except perhaps for extremely Stark-broadened hydrogen lines, so that, in the immediate vicinity of a line, ω can be neglected compared with ω0; and a very long way from the line, where this might not be so, the expression is close to zero anyway. (Note that you can neglect ω only with respect to ω; you cannot just put ω=0 where it lies alone in the denominator!) In any case, I have no compunction at all in making the approximation

α(ω)=n1f12Γe24mε0c[(ω)2+(12Γ)2]m1.

The maximum of the α(ω) curve is

α(0)=e2n1f12mε0cΓm1.

The optical thickness at the line centre (whether or not the line is optically thin) is

τ(0)=e2N1f12mε0cΓ.

N1 is the number of atoms in level 1 per unit area in the line of sight, whereas n1 is the number per unit volume.

The HWHM of α(ω) curve is

HWHM=12Γrad s1.

The area under the α(ω) curve is

Area=πe2n1f122mε0cm1rad s1.

As expected, the area does not depend upon Γ.

To express the absorption coefficient as a function of frequency, we note that ω=2πν, and we obtain

α(ν)=n1f12Γe216π2mε0c[(ν)2+(Γ4π)2]m1.

The maximum of this is (of course) the same as equation 10.B.4.

The HWHM of the α(ν) curve is

HWHM=Γ/(4π)s1.

The area under the α(ν) curve is

Area=e2n1f124mε0cm1s1.

α=n1f12Γe2mε0c(λ2λ404π2c2(λ20λ2)2+λ2λ40Γ2)m1.

In a manner similar to our procedure following equation 10.B.12, we write λ20λ2=(λ0λ)(λ0+λ), and , λ=λ0+λ, and neglect λ with respect to λ0, and we obtain:

α(λ)=n1f12Γe216π2mε0c3(λ40(λ)2+λ40Γ216π2c2)m1.

The maximum of this is (of course) the same as equation 10.B.4. (Verifying this will serve as a check on the algebra.)

The HWHM of the α(λ) curve is

HWHM=λ20Γ4πcm.

The area under the α(λ) curve is

Area=λ20e2n1f124mε0c2.

Did I forget to write down the units after this equation?

These results for α might be useful in tabular form. For τ, replace n1 by N1.

ωνλΓe2n1f124mε0c[(ω)2+(12Γ)2]Γe2n1f1216π2mε0c[(ν)2+(Γ4π)2]Γe2λ40n1f1216π2mε0c3[(λ)2+λ40Γ216π2c2]Heighte2n1f12mε0cΓe2n1f12mε0cΓe2n1f12mε0cΓAreaπe2n1f122mε0ce2n1f124mε0cλ20e2n1f124mε0c2HWMH12ΓΓ/(4π)λ20Γ4πc

It is to be noted that if the radiation damping profile is thermally broadened, the height of the absorption coefficient curve diminishes, while the area is unaltered provided that the line is optically thin. The optically thick situation is dealt with in the following chapter. It might also be useful to note that a gaussian profile of the form

α(λ)=α(0)exp(c2(λ)V2mλ20)

has an area of λ20e2n1f124mε0c2 if

α(0)=λ0e2n1f124πmε0cVm.


This page titled 10.10: APPENDIX B- Radiation Damping as Functions of Angular Frequency, Frequency and Wavelength is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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