COLLISION BROADENING

Let us write Eq. (3.7.24) as

f₁₂¼ 1 g1

2mv₂₁ 3h

X

m1m2

jxm1m2j² (3:7:25)

for the transition between levels 1 and 2, wherev21¼(E2E1)=h (.0). Interchanging 1 and 2, we define

f21¼ 1 g2

2mv12

3h X

m1m2

jxm1m2j²¼ g1

g2

f12, (3:7:26)

which is negative;f12andf21are called oscillator strengths for absorption and emission, respect- ively. One motivation for introducing a separate oscillator strength for emission, which is not really necessary for our purposes, may be found at the end of Section 3.14. † The degenerate states belonging to a given atomic energy level are those correspond- ing to different magnetic quantum numbersm. The application of a relatively weak mag- netic fieldBestablishes a preferredzdirection and removes the degeneracy: Each of the states is shifted in energy by an amount proportional tomjBj. This is theZeeman effect discussed in many textbooks on quantum mechanics. The different values of m are defined with respect to some “z” direction, such as the direction of the magnetic field in the Zeeman effect. If the atom is exposed to isotropic (e.g., thermal) radiation, then by spherical symmetry the different magnetic substates must have equal populations, so that (3.7.19) is an exact consequence of the more general (3.7.12). In the case of atoms in unidirectional narrowband light, simplified rate equations such as (3.7.19) are often a good approximation if collisions between atoms or between atoms and con- tainer walls are effective in maintaining a nearly equal population distribution among degenerate magnetic substates, or if the intensity of the light is not large enough to pro- duce a significant change in an initial thermal distribution of level populations. In the latter case the generalization of Eq. (3.6.7),

N₂

N₁¼g₂N2(m₂) g₁N1(m₁)¼g₂

g₁e^(E2E1)=kBT, (3:7:27) ensures the validity of (3.7.19). As discussed in Section 14.3, however, atoms can be

“optically pumped” or “aligned” preferentially in certain magnetic substates.

approach the question of absorption and lineshape from a more fundamental viewpoint, focusing our attention on “line-broadening” mechanisms in gases to answer the question of the origin of the frictional coefficientb.

It is a well-known result of experiment that, for sufficiently large pressures, the width of an absorption line in a gas increases as the pressure increases. This broadening is due to collisions of the atoms and is therefore calledcollision broadening or sometimes pressure broadening. Collision broadening is the most important line-broadening mech- anism in gases at atmospheric pressures and is often dominant at much lower pressures as well. We will begin our study by considering the details of collision broadening.

Our treatment will follow the original approach of Lorentz. We will find, for instance, that a kind of frictional force arises naturally as a result of collisions, and that the damp- ing ratebcan be interpreted as simply the collision rate. We start with (3.4.1)withoutthe frictional term and introduce the oscillator momentumpby writing

dx dt ; p

m: (3:8:1)

Then (3.4.1) can be rewritten in simple complex form as d

dt xþi p mv₀

þiv₀ xþi p mv₀

¼i^1 e

mv₀^{E(R,t), (3:8:2)} where one can easily check that the real part simply repeats the defining relation (3.8.1), and with the use of (3.8.1) the imaginary part is nothing other than (3.4.1) with the fric- tional force omitted. It will be convenient to have a shorthand form of this equation, so we define

xþi p

mv₀^{;S (3:8:3)}

and will examine the solution of theSequation, d

dtSþiv₀S¼i^1 e

mv₀^{E(R,t), (3:8:4)}

under the following interpretation of the effect of collisions.

We imagine collisions to occur in billiard-ball fashion, each collision lasting for a time that is very short compared to the time between collisions. We suppose that, immediately prior to a collision, the active electrons in an atom are oscillating along the axis defined by the field polarization, as indicated by (3.4.8). During a collision, the interaction between the two atoms causes a reorientation of the axes of oscillation.

Since each atom in a gas may be bombarded by other atoms from any direction, we can assume that on the average all orientations of the displacements and momenta of the atomic electrons are equally probable following a collision, so after a collision the displacement and momentum both vanish on average. This is the assumption made by Lorentz. It is an assumption about the statistics of a large number of collisions rather than about the details of a single collision.

We will examine the consequences of this assumption as follows. We consider the evolution of the electron displacements of atoms that underwent their most recent col- lision at the representative earlier timet1. The time of the last collision is known only in a statistical sense, and we will obtain a picture valid for the electrons in all the atoms by averaging overt1. The simplest statistical model for the frequency of collisions is “Markovian,” meaning memoryless. The fraction of atoms without a collision sincet1

must decrease during the waiting interval betweentandtþdt, and in the memoryless model this change is directly proportional to the length of the intervaldtbut does not depend ont.

Specifically, iff(t;t1) is the fraction of dipoles at timetnot having suffered a collision since t1, then the fraction collision free at a time dt later is smaller by an amount proportional to the fraction “available” for collision, namelyf itself, and to the time intervaldt:

f(tþdt;t₁)¼f(t;t₁)g_cf(t;t₁)dt, (3:8:5) where the proportionality constantgcis the rate at which collisions are occurring. In the limit of very smalldt, this recipe for the surviving fraction becomes the simple equation

df

dt ¼ g_cf, (3:8:6)

and so long asg_cis not dependent ontitself (the collisions are Markovian) the solution is f(t;t₁)¼e^gc(tt1): (3:8:7) Since the probability that a collision occurs in the timedt1is g_cf dt1, we can take account of all collisions by integrating overt1. We will now indicate the time of latest collision explicitly and write the complex displacement asS(t,t1), where we will enforce S(t1,t1)¼0, corresponding to the fact that the starting displacement and momentum were zero following the collision. The collision-averaged complex displacement S, which we will denoteS(t), is thus given by

S(t); ð_t

1S(t;t₁)g_cf(t;t₁)dt₁: (3:8:8) By differentiatingSwith respect totwe find that it satisfies a simple equation, one with an obvious physical interpretation. Using the evolution equation (3.8.4), and remember- ing thatS(t₁,t₁)¼0, we obtain

d

dtSþi(v₀ig_c)S¼i^1 e

mv₀^{E(R,t), (3:8:9)}

which we see to be exactly the same as Eq. (3.8.4) for the collisionless displacement, except for the appearance of the newg_c term. Such an extra term is exactly what is needed to reproduce the b coefficient in the electron oscillator equation: writing S¼xþ(i=mv₀)p in (3.8.9) and then taking the real and imaginary parts of that

3.8 COLLISION BROADENING 101

equation, we obtain d²x

dt²þ2g_c^dx

dt þ(v²₀þg_c²)x¼1^e

mE(R,t): (3:8:10) While collisions occur frequently on a “normal” time scale, they are rare on the scale of an optical period (roughly 1 fs), so

v²₀g_c², (3:8:11) and one safely ignores the contribution ofg_c²to the last term on the left side of (3.8.10).

Then (3.8.10) is exactly the same as (3.4.4) when we equatebin (3.4.4) to the collision rategc. In other words, an averaged treatment of collisions leads directly to “frictional”

drag in the electron oscillation. The linewidth of the collision-broadened lineshape is [Eq. (3.4.23)]

dn0¼ g_c 2p^¼

1

2p(collision rate): (3:8:12) Collision broadening is often described in terms of a “dephasing” of the electron oscillators, as follows. Immediately after a collision the phase of the electron’s oscil- lation has no correlation with the precollision phase. Collisions have the effect of “inter- rupting” the phase of oscillation, leading to an overall decay of the average electron displacement from equilibrium (Fig. 3.14). The damping rategcis sometimes called a dephasing rate in order to distinguish it from an “energy decay” rate. The latter would appear as a frictional term in the equation of motion of each electron oscillator as well as in the average equation. In the absence of any inelastic collisions to decrease the energy of the electron oscillators, each oscillator would satisfy the Newton equation (3.2.18b) with no damping term. Due to elastic collisions, that is, collisions that only

Collision Collision Collision t

t

t

Atom 1

Atom 2

Atom 3

Figure 3.14 Electron oscillations in three different atoms in a gas. Collisions completely interrupt the phase of the oscillation. The average electron displacement associated with all the atoms in the gas therefore decays to zero at a rate given byg_c, the inverse of the mean time between collisions.

interrupt the phase of the oscillation but do not produce any change in energy, the averageelectron displacement follows Eq. (3.4.4), which includes damping.

Collision Cross Sections

The collision rateg_cmay be expressed in terms of the number densityNof atoms, the collision cross sectionsbetween atoms, and the average relative velocityvof the atoms.

Imagine some particular atom to be at rest and bombarded by a stream of identical atoms of velocityv. If the number of atoms per unit volume in the stream isN, then the number of collisions per unit time undergone by the atom at rest isNs_v, where the areasis the collision cross sectionbetween the atom at rest and the atoms in the stream. The number of collisions per second is the same as if all the stream atoms within a cross-sectional area scollide with the stationary atom. The idea here is the same as that used to define the absorption cross section for incident light.

According to the kinetic theory of gases, an atom of massmXhas a mean velocity vrms¼ 8k_BT

pmX

₁₌₂

(3:8:13) in a gas in thermal equilibrium at temperatureT. To obtain the average relative velocity vrelof colliding atoms of massesmXandmYin the gas, we replacemXin (3.8.13) by the reduced mass

m_X,Y¼ m_Xm_Y

m_Xþm_Y¼ 1 m_Xþ 1

m_Y

1

: (3:8:14)

Thus,

vrel ¼ 8kBT p

1 m_Xþ 1

m_Y

₁₌₂

: (3:8:15)

It is convenient to express this in terms of the atomic (or molecular) weightsMXandMY:

vrel ¼ 8RT p

1 M_Xþ 1

M_Y

₁₌₂

, (3:8:16)

whereR, the universal gas constant, is Boltzmann’s constant times Avogadro’s number.

The collision rate for molecules of type X is therefore g_c¼X

Y

N(Y)s(X, Y)vrel(X, Y)

¼X

Y

N(Y)s(X, Y) 8RT p

1 M_Xþ 1

M_Y

₁₌₂

, (3:8:17) where the sum is over all species Y, including X.

3.8 COLLISION BROADENING 103

The important “unknowns” in expression (3.8.17) are the collision cross sections s(X, Y), which often are not known very accurately. The simplest approximation to the cross section is the “hard-sphere” approximation. We write

s(X, Y)¼p

4(d_Xþd_Y)², (3:8:18) wheredX anddY are the hard-sphere molecular diameters, estimates of which can be made from measurements of various transport quantities such as thermal conductivities or diffusion constants;s(X,Y) is the area of a circle of diameterdXþdY, just what we would expect if the molecules acted like spheres of diametersdXanddY. For CO2, for example, the hard-sphere diameter is about 0.4 nm. From (3.8.18), therefore, the hard- sphere cross section for two CO2molecules iss(CO2, CO2)¼5.0310²¹⁹m². For a gas of pure CO2atT¼300K we find the average relative velocity of two colliding CO2

molecules to bevrel¼5:3710² m=s. The collision rate (3.8.17) in the hard-sphere approximation is therefore

g_c¼N(5:0310¹⁹m²)(5:3710²m=s)¼2:7010¹⁶N=s, (3:8:19) whereNis the number of CO2molecules per cubic meter. For an ideal gas we calculate (Problem 3.10)

N ¼9:6510²⁴P(Torr)

T , (3:8:20)

whereP(Torr) is the pressure in Torr (1 atm¼760 Torr) andTis the temperature (K).

From (3.8.17), finally, the collision rate for a gas of CO2at 300K is

g_c¼8:6910⁶P(Torr) s¹: (3:8:21) Thus, at a pressure of 1 atm we calculate the collision rate

g_c¼6:6010⁹ s¹, (3:8:22) and from (3.8.12) the collision-broadened linewidth

dn₀¼1:0510⁹Hz: (3:8:23) The actual collision-broadened linewidths can be larger, by as much as an order of magnitude or more, than those calculated in the hard-sphere approximation. The value calculated above, however, is reasonable, and it allows us to point out some general features of collision-broadened linewidths. First, we note that the collision rate (3.8.22) is very much smaller than an optical frequency, as assumed in (3.8.11). The linewidthdn0 is thus also orders of magnitude less than an optical frequency. This explains why we can speak of absorption “lines” in a gas, even though the absorption occurs over a band of frequencies: the band has a width (2dn₀) that is very small compared to the resonance frequencyn0.

From (3.8.21) we note that the linewidth is linearly proportional to the pressure. For this reason, experimental results for collision-broadened linewidths are often reported in

units such as MHz-Torr²¹. The linewidth calculated above, for instance, may be expressed as 1.38 MHz-Torr²¹at 300K.

Our treatment of collision broadening only highlights some general features of a complex subject. In actual calculations we prefer always to use measured values of the collision-broadened linewidths. We note parenthetically that, for the 10.6-mm CO2laser line, the linewidth (1.38 MHz-Torr²¹) computed above is about three times smaller than the experimentally determined value. It is possible to calculate these widths more accurately, but this will not concern us. See also Problem 3.11.

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