Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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8.7 Rotational energy levels of polyatomic molecules 325
the direction of the applied field since J carries out a precessional motion. Thus
the maximum component of the angular momentum vector cannot be as large
as the vector itself. The reaction of the molecule to an applied field is called space
quantization.
Since the molecule has an electric dipole moment, the different orientations of
J relative to the applied field will correspond to different energies. In a magnetic
field, all states with different M will have different energies. This observation is
called Zeeman effect. In case that J = 2, energy levels corresponding to M =
0, ±1, ±2 would be observed. In an electric field only those states have different
energy which differ in |M|. Thus, for J = 2 only three different energy levels M =
0, |±1|, |±2| appear. The splitting of the energy levels in the electric field is known
as Stark effect which often permits precise measurements of the electric dipole
moment.
Without going into any details but to demonstrate that the quantum theory
describing rotating molecules is very complex, we wish to point out in which
way further complications may occur. In addition to the molecular rotation angular
momentum, some molecules have angular momentum resulting from the nuclear
spin of one or more of their nuclei. A complete description of the rotational
states of the molecule must take into account both contributions to the total angu-
lar momentum. There exists a nuclear spin quantum number usually denoted by
I = 0, 1/2, 1, 3/2,...The spin angular momentum is given by
√
I (I + 1)h/2π.If
there is no interaction (coupling) between the orientation of the nucleus and that of
the molecule, the molecule will rotate and leave the spinning nuclei unchanged
in orientation. In this case the energy of the given molecular rotational state
described in terms of J will not be influenced by the nuclear spin. However,
if there is a molecular rotation–nuclear spin coupling the energy of the system
will depend on the orientation of the nuclear spin relative to that of the molec-
ular rotation. This coupling will cause a hyperfine splitting of rotational energy
values.
8.7.2 Symmetric top molecules
In case of symmetric top molecules, two quantum numbers are needed to specify
the rotational states. These are usually denoted by J and K . The axis along which
I
A
lies is usually called the figure axis of the molecule.IfI
A
< I
B
we speak of
a prolate symmetric top, and if I
A
> I
B
an oblate symmetric top. The rotational
energy levels for such a molecule are given by
E
J,K
=
h
2
8π
2
J (J + 1)
I
B
+ K
2
1
I
A
−
1
I
B
(8.222)
326 Absorption by gases
The derivation of this equation can be found, for example, in Pauling and Wilson
(1935). The quantum number K has the physical significance that Kh/2π
is the component along the figure axis of the resultant angular momentum
√
J (J + 1)h/2π. K takes values between −J and +J .
Equation (8.222) verifies that the energy does not depend on the magnetic quan-
tum number M = 0, ±1, ±2,...,±J and is independent of the sign of K . There-
fore, the total degeneracy of the energy level is 2J + 1ifK = 0 and 4J + 2ifK
differs from zero. Most symmetric top molecules that have a dipole moment have
this dipole moment in the direction of the figure axis. In such cases the selection
rules for the absorption and emission of electromagnetic energy are
J = 0, ±1, K = 0, K = 0
J =±1, K = 0, K = 0
(8.223)
For absorption experiments, that part of the selection rules applies which increases
the energy of the system, that is
J =+1, K = 0 (8.224)
8.7.3 Spherical top molecules
The energy levels are given by the same simple formula stated in (8.181). For
each value of J there are again 2J + 1 values of K . Due to the high degree of
symmetry, spherical top molecules do not have a permanent dipole moment so that
pure rotational spectra do not occur.
8.7.4 Asymmetric top molecules
Theoretical considerations are much more involved than those for linear and sym-
metric molecules. If we ignore nuclear-spin angular momentum, the total angu-
lar momentum is quantized according to the relation (8.221). Unlike the sym-
metric top molecule, no component of this angular momentum is quantized. No
closed general expression is available for any but the lowest few energy values
for an asymmetric top molecule. Nevertheless, it is possible to obtain approximate
energy values by interpolation involving oblate and prolate symmetric top energy
diagrams.
A wealth of information about the approximate formulas of the partition function
and the matrix elements can be found in Herzberg’s books Spectra of Diatomic
Molecules
(1964a) and Infrared and Raman Spectra (1964b) as well as in Penner’s
book Quantitative Molecular Spectroscopy and Gas Emissivities (1959). These
8.8 Appendix 327
books, however, are not introductions to the field of spectroscopy and require much
additional reading. A brief treatment for important atmospheric cases is given by
Goody (1964a) who assumes some knowledge of spectroscopy. A condensed and
quite readable treatment of most of the topics we have introduced in this chapter is
given in Infrared Physics by Houghton and Smith (1966). Many newer books, too
numerous to be mentioned, may serve as guides to get a deeper understanding of the
interesting but difficult topics treated in spectroscopy, e.g. Molecular Spectroscopy
by Barrow (1962), Molecular Spectroscopy by Levine (1975), Molecular Rotation
Spectra by Kroto (1975), Physikalische Chemie by Atkins (1993), Molek
¨
ulphysik
by Demtr¨oder (2003), and Lehrbuch der Physikalischen Chemie by Wedler (2004).
Many of the newer books give only very brief outlines of the quantum mechanical
derivations or simply state results. For details they often refer to the older textbooks
by Pauling and Wilson (1935) and Eyring et al. (1965).
8.8 Appendix
8.8.1 The Hamilton function
The Lagrangian function L is defined by
L = K (q
k
,
˙
q
k
) − V (q
k
) (8.225)
where K is the kinetic energy and V the potential function. The symbol
q
k
(k = 1, 2,...) represents the generalized coordinates and
˙
q
k
their time change.
Lagrange’s equation of motion is defined by
d
dt
∂ L
∂
˙
q
k
=
∂ L
∂q
k
(8.226)
The generalized momenta are
p
k
=
∂ L
∂
˙
q
k
=
∂ K
∂
˙
q
k
(8.227)
Now consider the function
H =
k
˙
q
k
p
k
− L (8.228)
which is a function of the generalized coordinates. Euler’s theorem for a homoge-
neous function f of degree n in the variables x
1
, x
2
,...,x
r
states that
r
i=1
x
i
∂ f
∂x
i
= nf (8.229)
328 Absorption by gases
Application of this theorem to the first term on the right-hand side of (8.228) yields
k
˙
q
k
p
k
=
k
˙
q
k
∂ K
∂
˙
q
k
= 2K (8.230)
which leads to
H = 2K − K + V = K + V
(8.231)
The function H is known as the classical Hamilton function. Detailed treatments
of L and H can be found in textbooks on theoretical mechanics, for example in
Fowles (1966).
8.8.2 Macroscopic fields and Maxwell’s equations
In order to describe the absorption and emission of infrared radiation, a num-
ber of relations from classical electromagnetic theory will be used which are
summarized in this appendix. As before, we will use the mks rationalized units.
The electromagnetic state of matter at a given point is described by the following
four quantities:
(1) volume density of electric charge ρ;
(2) volume density of electric dipoles, called the polarization P;
(3) volume density of magnetic dipoles, called the magnetization M;
(4) electric current per unit area, called the current density J.
These quantities are considered macroscopically averaged in order to smooth
out microscopic variations due to the atomic structure of matter. They are related to
the macroscopically averaged electric and magnetic fields E and H by Maxwell’s
equations
∇×E =−µ
0
∂H
∂t
− µ
0
∂M
∂t
∇×H = ε
0
∂E
∂t
+
∂P
∂t
+ J
∇·E =−
1
ε
0
(∇·P − ρ)
∇·H =−∇·M
(8.232)
The constants ε
0
and µ
0
are called the permittivity and the permeability of the
vacuum, respectively. Introducing the abbreviations D for electric displacement
and magnetic induction B
D = ε
0
E + P, B = µ
0
(H + M) (8.233)
8.8 Appendix 329
into (8.232) then Maxwell’s equations assume the compact form
(a) ∇×E =−
∂B
∂t
(b) ∇×H =
∂D
∂t
+ J
(c) ∇·D = ρ
(d) ∇·B = 0
(8.234)
The response of the conduction electrons to the electric field is given by
J = σ E (8.235)
where σ is the electrical conductivity. Additionally, we introduce the constitutive
relation
D = εE (8.236)
which describes the aggregate response of the bound charges to the electric field
and the corresponding magnetic relation
B = µH (8.237)
Using (8.233) and (8.236) the polarization P may also be expressed by
P = (ε − ε
0
)E = χε
0
E (8.238)
where χ is known as the electric susceptibility.
χ =
ε
ε
0
− 1 (8.239)
This equation is another way of expressing the response of the bound charges to
the impressed electric field. For isotropic media such as glass, the susceptibility
is a scalar. It is also customary to introduce the so-called relative permittivity or
dielectric constant ε
r
and the relative permeability µ
r
as stated in
ε
r
=
ε
ε
0
, µ
r
=
µ
µ
0
(8.240)
For non-magnetic media the relative permeability is equal to 1.
Next we are going to separate the electric and magnetic fields E and H from the
curl relations (8.234a,b). First we take the curl of (8.234a) and the partial derivative
of (8.234b) with respect to time. In a region where the volume density of electric
charges is zero, we obtain after a few elementary steps the wave equation
∇
2
E − εµ
∂
2
E
∂t
2
− µσ
∂E
∂t
= 0
(8.241)
330 Absorption by gases
Analogously we obtain the corresponding wave equation for H
∇
2
H − εµ
∂
2
H
∂t
2
− µσ
∂H
∂t
= 0
(8.242)
In order to establish two useful operator identities in connection with harmonic
waves described by the wave vector k and the angular frequency ω, we perform the
following operations
∂
∂t
exp[i(k · r − ωt)] =−iω exp[i(k · r − ωt)]
∇ exp[i(k · r − ωt )] = ik exp[i(k · r − ωt)]
or
∂
∂t
→−iω, ∇→ik
(8.243)
Momentarily we consider a nonconducting medium σ = 0 and the case ρ = 0so
that (8.234) reduces to
∇×E =−µ
∂H
∂t
∇×H = ε
∂E
∂t
∇·E = 0
∇·H = 0
(8.244)
Application of (8.243) to (8.244) leads to Maxwell’s equation in the form
k × E = µωH
k × H =−εωE
k · E = 0
k · H = 0
(8.245)
From the latter set of equations we recognize that the three vectors E, H and k
form a mutually orthogonal triad. The electric and the magnetic field vectors are
perpendicular to each other, and they are both perpendicular to the wave vector k
pointing in the direction of propagation. Furthermore, it may be easily seen that the
magnitudes of the fields are related by
H =
E
µc
= εcE since c =
ω
k
, k = ke
k
(8.246)
Here, k = 2π/λ is the magnitude of the wave vector, λ is the wavelength, c is the
speed of light in the medium and e
k
is the unit vector in the direction of k.
8.8 Appendix 331
The time rate of flow of electromagnetic energy per unit area is given by the
Poynting vector which is defined as the cross product of the field vectors
N = E × H (8.247)
By considering E and H as real plane harmonic waves, i.e.
E = E
0
cos(k · r − ωt), H = H
0
cos(k · r − ωt)
(8.248)
the Poynting vector N is given by
N = E
0
× H
0
cos
2
(k · r − ωt) (8.249)
In the mks-system of units N is expressed in Watts per square meter.
Since the average value of the cosine squared is 1/2, we find that the average
value of the Poynting vector is given by
< N > =
1
2
E
0
× H
0
=
E
2
0
2µω
k =
E
2
0
2µc
e
k
= I e
k
(8.250)
Hence, it is seen that in isotropic media the direction of N and k coincide. For
nonisotropic media such as crystals this is not always the case. The magnitude of the
averaged Poynting vector is the intensity I , also called irradiance. The intensity is
proportional to the square of E
0
. In a conducting medium there is a phase difference
between E and H. Nevertheless, the intensity I is still proportional to E
2
0
.
Now we return to the wave equation (8.241). It is easy to verify that the function
E = E
0
exp[i(kz − ωt)] (8.251)
is a solution of (8.241) provided that the following relation is satisfied
k
2
= ω
2
µ
ε +
iσ
ω
(8.252)
This solution refers to propagation along the z-axis so that z is the distance from a
fixed origin.
Finally, we wish to relate (8.252) to the index of refraction. For a nonabsorbing
medium the refractive index is defined as N = c/v, where c is the speed of light
in vacuum. Since for an absorbing medium k is a complex expression, the index of
refraction is also a complex quantity and is given by
N = n + iκ =
ck
ω
where c =
1
√
ε
0
µ
0
(8.253)
332 Absorption by gases
By separating the real and imaginary parts of the complex refractive index we find
from (8.252) and (8.253) the important relations
n
2
− κ
2
= ε
r
µ
r
,2nκ =
σµ
r
ε
0
ω
(8.254)
Expressing the wave number k in the solution (8.251) in terms of the complex
index of refraction we obtain
E = E
0
exp
−
ωκz
c
exp
iω
nz
c
− t
(8.255)
The first exponential factor describes the attenuation along the z-axis from an
arbitrary origin, while the second exponential defines the shape of the wave.
By introducing in (8.250) the attenuation along the z-axis we find
I =
1
2µc
E
2
0
exp
−
2ωκz
c
(8.256)
Denoting the initial value of the intensity by I (z = 0) = I
0
we obtain Beer’s law
I = I
0
exp
−
2ωκz
c
= I
0
exp
(
−αz
)
(8.257)
where the absorption coefficient α = 4πκ/λis expressed as the reciprocal of length.
More complete treatments of the equations of electromagnetic theory are given
in standard textbooks on electrodynamics and in related books. Our reference goes
to Fowles (1966) and to Houghton and Smith (1966).
8.9 Problems
8.1: In connection with Figure 8.5, show that y
1
= y
3
.
8.2: Verify equations (8.28) and (8.29).
8.3: Prove equation (8.32).
8.4: Show the validity of equations (8.33).
8.5: Verify equations (8.37), (8.38) and (8.41).
8.6: Show by direct computations that the wave functions for the simple harmonic
oscillator are normalized. Use the special cases n = 1 and n = 2. Show also
by direct computations that the wave functions ψ
1
and ψ
2
are orthogonal.
8.7: Show that the selection rules for the harmonic oscillator are given by (8.200).
8.8: Verify that the squares of the matrix elements for the harmonic oscillator are
given by (8.201).
8.9: Show the validity of equations (8.210) and (8.211).
9
Light scattering theory for spheres
9.1 Introduction
The evaluation of the radiative transfer equation requires a detailed knowledge of the
extinction and scattering properties of atmospheric particles. In most cases we rely
on theoretical calculations assuming that the scattering particles have a spherical
shape. In order to carry out the calculations we must specify the particle size and
the wavelength-dependent complex index of refraction. The required calculations
are known as Mie calculations while the entire theory is called the Mie theory. This
goes back to Gustav Mie who published the whole theory in 1908. Nowadays a
large number of reliable computer programs are available to provide the required
information without understanding the theory behind it. This might be sufficient
in some cases, but it is more valuable to comprehend the theoretical background
which will be presented in the following sections.
The Mie theory of light scattering by homogeneous spheres is based on the for-
mal solution of Maxwell’s equations using proper boundary conditions. In this text
we will follow Stratton’s discussion of the Electromagnetic Theory (1941). In van
De Hulst’s (1957) treatise Light Scattering by Small Particles many mathematical
details of the Mie theory are omitted, but numerous helpful physical explanations
are given. In particular he gives a number of approximate formulas for special
cases. Due to modern computers the approximate formulas are rarely used since
the Mie series solutions can now be evaluated efficiently for practically all situa-
tions of interest. A newer more complete treatment of the Mie theory is given, for
example, by Bohren and Huffman (1983). We should not overlook the important ref-
erence Principles of Optics by Born and Wolf (1965) where a detailed treatment of
the theory can also be found. While the theory is named after Mie (1908), some of the
notations which we will use were introduced by Debye (1909). Special cases
of the scattering theory are discussed in various textbooks on electromagnetic
theory.
333
334 Light scattering theory for spheres
We assume that the student has some familiarity with Bessel functions since
they are introduced in most introductory courses on differential equations. The
Mie theory makes use of the elements of the so-called field theory. In the present
case this means that if the scalar function ρ and the vector function J denoting the
charge density and the distribution of currents are known, we may seek a solution
for the vectors E and H of the electric and the magnetic field by using Maxwell’s
equations. Finally, the entire problem may be reduced to the solution of a standard
partial differential equation. A brief and simple but very illuminating mathematical
introduction to this type of problem can be found, for example, in Chapter 8 of
Buck’s Advanced Calculus (1965).
A full treatment of the entire Mie theory will now be presented giving all required
mathematical details.
9.2 Maxwell’s equations
As stated above, the Mie formulas are an exact mathematical solution of Maxwell’s
equations which contain the basic physics of the problem. From Appendix 8.8.2 of
the previous chapter we restate the curl equations (8.234a,b)
∇×E +
∂B
∂t
= 0, ∇×H −
∂D
∂t
− J = 0
(9.1)
The relations involving the vectors B, D, E, H, J are repeated as
∇·B = 0, ∇·D = ρ, J = σ E, D = εE, B = µH
(9.2)
According to (8.241) and (8.242) the wave equations for E and H are given by
∇
2
E − εµ
∂
2
E
∂t
2
− µσ
∂E
∂t
= 0
∇
2
H − εµ
∂
2
H
∂t
2
− µσ
∂H
∂t
= 0
(9.3)
As usual, we assume that the time dependency of the field vectors is harmonic.
Thus we may write for E and H
E =
˜
E exp
(
−iωt
)
, H =
˜
H exp
(
−iωt
)
(9.4)
where
˜
E and
˜
H denote the time-independent parts. Substituting (9.4) into (9.3) we
find
∇
2
E + k
2
E = 0, ∇
2
H + k
2
H = 0 (9.5)