Re: Beer-Lambert Law

[herbgottlieb@juno.com (Herbert H. Gottlieb)]

On Mon, 15 Jun 1998 23:03:36 -0400 "Bob Sciamanda" <trebor@velocity.net>
writes:
> Herb,
> Beer's law is routinely used in spectrophotometric analysis 
> to measure chemical concentrations.

Bob....    I am familiar with the Beers law that is related to 
chemical concentrations. It appears in almost all of my 
reference books.   However, I was concerned with the 
Beer-Lambert Law that is concerned  with the attenuation 
of a light beam by filters.   In the event that you did not 
find it anywhere, I am forwarding a copy of Hugh Logan's 
response to me (see below).

From: Hugh Logan <hlogan@ix.netcom.com>
>
Beer's law is frequently used in connection with the absorption and gain
in a laser medium. Let I(0) = intensity (power per unit area) of the
light beam entering the medium and I(z) the intensity after it has
traveled a distance z through the medium. Then Beer's law states

	I(z)=I(0)*exp[-alpha(nu)*z]

where alpha is the absorption coefficient, which is specified as a
function of the frequency nu. This equation can also include the case
of gain rather than attenuation, in which case the attenuation is
negative. In fact Beer's law can be written in terms of a gain
coefficient g(nu) = -alpha(nu), i.e.

	I(z)=I(0)*exp[g(nu)*z]   .

On the basis of a simplified, idealized two energy-level model of the
laser medium, the gain coefficient g may be written as

	g = [N(2) - N(1)]*h*[nu(21)*[B(21)]*(n/c)
where
        n = index of refraction of the medium,
        N(2) = population density of upper laser level,
        N(1) = population density of lower laser level,
        h = Planck's constant, 
        nu(21) = frequency of stimulated (or absorbed) photon
                         in electron transition between upper and lower
level
                 (for stimulated emission) or between lower and
                   upper  level (for absorption),
        c = speed of light in vacuum (so that c/n is the speed in the
             medium)
        B(21) is the Einstein coefficient having to do with
stimulated emission. The Einstein coefficient for
absorption, B(12),  doesn't appear, since it can be shown that
B(12) = B(21).

Amplification will occur if N(2) > N(1) so that g is positive; i.e. if
one has a population inversion. If N(2) < N(1), g becomes negative (or
alpha becomes positive). The above is based on the simplified treatment
in the very elementary text, _Lasers, Principles and Applications_ by 
J. Wilson and J.F.B. Hawkes, Prentice Hall (UK), 1987, pp. 13-18. The
possibility of degeneracy of the energy levels is neglected. Another
text, _Laser Fundamentals_ by William T. Silfvast, Cambridge University
Press, 1996 actually refers to the first equation of this posting as
Beer's law (p. 209). Silfvast takes degeneracy into account, so that
N(2)- N(1) is replaced by N(2) - {[g(2)]/[g(1)]}*N(1), where g(2) and
g(1) are the degeneracies of the upper and lower levels, respectively.
In this case, one would have a population inversion if

	N(2) > {[g(2)]/[g(1)]}*N(1)          .

I recall an experiment based on Beer's law in which the attenuation
of a beam of light is measured after passing through a stack of
increasing numbers of microscope slides (to vary z). One of my
colleagues used a dye solution in a tank of fixed length, but varied the
concentration of the dye (as discussed in Bob Sciamanda's posting in
this thread).

Hugh Logan

..


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