Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

Подождите немного. Документ загружается.

Line Shapes and Radiation Transfer 19.1 Collisional Line Shapes 285

frequency ω “occurs” when the atoms are near R

,where

the electronic state energies differ by

ω. Note that

this holds for all initial kinetic energies and angular

momenta, as long as the conditions for validity of the

Born–Oppenheimer and WKB approximations hold for

the initial and ﬁnal nuclear motions. This is the CFCP,

which is equivalent to the classical-oscillator model for

radiation at large detunings from the atomic transition.

Another insight evident from Fig. 19.1 is that the

photon energy associated with the frequency differ-

ence ω −ω

is supplied by nuclear kinetic energy

(ω −ω

) = T

−T

. This transformation of nuclear into

electronic energy takes place as the nuclei move from

large R to R

on one V(R) and back to large R on the

other.

If an absorbing or emitting atom interacts as V

(R)

with a density n

of perturbers in a vapor of tem-

perature T , the probability of a perturber at separation

R → R + dR is n

4πR

exp[−V

(R)/k

T ]dR if the in-

teratomic motion is in equilibrium. Inverting (19.5)for

R(ω) yields dR = dω/( dω/ dR), and this pair of re-

lations yields the (single perturber) quasistatic (QS)

spectrum

I(ω) = Nn

Γ 4πR(ω)

×exp[−V

(ω)/k

T ]/[dω(R)/ dR] , (19.8)

where N is the radiator density and I(ω) the radiation

per unit volume and frequency interval.

Figure 19.3a gives an example of far wing emission

line shapes versus photon energy in units of cm

−1

,for

the Kr broadened Rb D2 line for which ∆k

≈0.7cm

−1

These data are normalized by dividing by perturber den-

sity, so they are independent of perturber density for the

density region of the experiment. The excited state pro-

duces two V

(R), called the A and B states, while the

ground state produces one V

(R), called the X state.

Each of these potentials has a single minimum at long

range and is strongly repulsive at close range, but the

well depths and positions are very different [19.32].

This causes the complex forms of V

(R) −V

(R) that

are shown in Fig. 19.3b. There I have plotted ln(R)

vertically and ln(∆V(R)/hc) horizontally, where ∆V

refers to V

−V

and V

−V

. The right side of (19.8)

can also be written as the exponential and constant fac-

tors times dR

(ω)/dω. Since ln[R

(ω)]∝ln[R(ω)],the

static spectrum at ∆k = ∆ω/2πc is proportional to the

slopes of the curves in Fig. 19.3b, divided by |∆k| due

to the ln(∆k ) horizontal axis. One can see qualitatively

that the overall spectrum follows such a relation to the

lines in Fig. 19.3b; in fact in most spectral regions this

relation is quantitatively accurate.

The temperature dependence in Fig. 19.3a corre-

sponds to the exponential factors in (19.8) [19.30]. At

large R, both ∆V(R) are attractive,and this causes a large

intensity on the negative ∆k (red) wing. However, once

|∆k| exceeds ≈ 20 cm

−1

,whereV

−V

reverses direc-

tion, the red wing intensity drops rapidly. This extremum

in ∆V(R) causes a satellite at ≈−20 cm

−1

, although it

is spread out by the ﬁnite collision speed and does not

cause a distinct peak in the spectrum. Satellite features

are discussed in more detail in the next paragraph. The

antistatic blue wing drops rapidly for several decades,

then suddenly ﬂattens beyond ≈ 10 cm

−1

due to the pos-

itive portion of V

−V

at small R. The remaining blue

wing is the B–X band, and has a satellite at ≈ 350 cm

−1

as V

−V

passes through another extremum. The the-

ory predicts this at 800 cm

−1

, but clearly represents all

the basic aspects correctly. This satellite is also spread

out by ﬁnite collision speed, but a deﬁnite intensity peak

remains. The red wing beyond ≈ 50 cm

−1

is the A–X

band. The feature near −1000 cm

−1

is due to the expo-

nential factor in (19.8), not an extremum in ∆V(R);the

feature essentially disappears if the normalized intensity

is extrapolated to inﬁnite temperature.

19.1.8 Satellites

In regions of the wing where the intensity falls slowly

with increasing frequency, motional broadening of the

static spectrum is not noticeable and the static spec-

trum is a good approximation. However, if ∆V(R),or

equivalently ω

(R), has an extremum at some R

,the

denominator of (19.8) is zero at ω(R

) = ω

.Thispro-

duces a local maximum, or satellite, in the far wing

intensity, as seen in Fig. 19.3a at 350 cm

−1

. If one

expands ω(R) in a Taylor series about R = R

this

produces in (19.8) a square root divergence of ﬁnite

area, with no intensity beyond ω

. The area under this

feature is meaningful, but not its shape; the quasistatic

assumption is clearly not valid for such sharp features.

The more accurate satellite shape is obtained by re-

turning to (19.2) and expanding V

(R), V

(R) and the

WKB wave functions about R

,orusing(19.6) with

(t) expanded about t(R

). Sando and Wormhoudt

used the former method to obtain a universal satellite

shape [19.39]. Szudy and Baylis improved the expansion

to yield a smooth transition to the quasistatic spectrum

at smaller detunings [19.37]. This result is nearly the

same as Sando et al. in the spectral region of the satel-

lite, but it more accurately connects to the adjacent static

line wing. Intensity undulations between the satellite and

the line occur in this calculation; these arise from alter-

Part B 19.1

286 Part B Atoms

nating constructive and destructive interference between

two contributions to the same frequency from R > R

and R < R

. This can not be seen in the low resolu-

tion of Fig. 19.3a, but such undulations are seen near

the 350 cm

−1

satellite [19.31]. At antistatic detunings

beyond ω

, which are not quasistatically allowed, the

calculated intensities decay exponentially. This is also

observed experimentally [19.31] and is the same behav-

ior predicted for the antistatic wing of a line [19.11].

At higher perturber densities and closer to the line

core, corresponding to larger R interactions, the multi-

ple perturber probability distribution must be included.

If the interactions are additive, this leads to a sec-

ondary satellite at twice the detuning of the low pressure

satellite, as seen in Fig. 19.4.

The wings of a collisionally broadened atomic line

are molecular radiation. In the context of molecular

bound state spectroscopy, a satellite is a “head of band

heads,” corresponding to a frequency region where

bound–bound band heads congregate. This occurs, of

course, at the classical satellite frequency and when

(R) −V

(R) has an extremum [19.40].

An extremum in ∆V(R) is the most common cause

of satellites, but similar looking features can occur for

other reasons. Forbidden bands often appear in the wings

of forbidden atomic transitions, due to an increase in

the transition dipole moment µ(R) resulting from the

collisional interaction. These are described, in the QS

approximation, by (19.8) with Γ replaced by Γ(R).If

Γ(R) increases rapidly with decreasing R, the intensity

increases as ω moves into the far wing until the dR

and

exponential factors cause a net decrease at small R.This

leads to forbidden bands far from the atomic frequency,

such as those in [19.41]. In some cases, a collision-

induced feature also appears at the frequency of the

forbidden transition. The shapes of such features, which

also include radiative collisions, in which both atoms

change state, are calculated and reviewed in [19.42].

A variety of related line shape phenomena has been

investigated, including the relation between absorbed

and emitted wavelengths (spectral redistribution), the

dependence of ﬂuorescence polarization on absorbed

wavelength (polarization redistribution), and high power

effects. Some references regarding these phenomena

are [19.43–47].

19.1.9 Bound States

and Other Quantum Effects

The validity of the QS spectrum requires the validity

of the WKB approximation in the initial and ﬁnal state,

but it is not restricted to free–free molecular transitions.

In fact, the equilibrium probability distribution in (19.8)

must include bound states in an attractive V

(R).The

QS spectrum describes the average behavior of bound–

free and bound–bound molecular bands, as well as the

free–free radiation implied by the above method of

derivation. The quantum character is expressed in the

discrete bound–bound lines that make up this average,

and in Condon oscillations, where the intensity oscillates

about the average I

(ω). The latter occur as oscil-

lations in Franck–Condon factors in the bound–bound

case, and as smooth oscillations in bound–free spectra

and low resolution bound–bound spectra. An additional

quantum feature occurs in regions of the spectrum dom-

inated by classical turning points, usually at the far edge

of a line wing where the intensity is dropping rapidly.

There, quantum tunneling past the edge of the classically

allowed region spreads the spectrum. Yet another is the

energy hω

/2 of the ground vibrational state, which ef-

fectively adds to k

T in (19.8) for attractive V

.Allof

these quantum features become more pronounced as the

reduced mass decreases; examples and details can be

found in [19.40,48–50].

19.1.10 Einstein A and B Coefﬁcients

The relationship between absorption coefﬁcient B

(ω),

stimulated emission coefﬁcient B

(ω) and spontaneous

emission coefﬁcient A

(ω) are given by the Einstein

relations; A

= 8πhλ

−3

and B

= g

These relations are most familiar for atomic lines, but

if they are referred to the density of absorbers dN

/dω

and emitters dN

/dω that emit or absorb at ω, then they

also apply to the wings of lines, i. e.,

k(ω) = B

(ω)



2πh





dω

(ω)





dω

(19.9)

g(ω) =

(ω)

dω

(19.10)

I(ω) = ωA

(ω)

dω

(19.11)

Here k(ω) is the absorption coefﬁcient due to lower

state atoms, g(ω) is the stimulated emission coefﬁcient

and I(ω) the spontaneous emission due to excited state

population, and g

and g

are the statistical weights of

the atomic states. For absorbing atoms of density N

and perturber density n

,theQS approximation with

Part B 19.1

Line Shapes and Radiation Transfer 19.2 Radiation Trapping 287

equilibrated internuclear motion sets

dω

= N

4πR

dω

exp



−



(19.12)

and equivalently for a radiating atom density N

with

perturber interaction V

. Normally most of the radia-

tion, and dN

/dω, is concentrated at the atomic line, so

integrating over dω near the line leads to the relations

∞



k(ω) dω =





(19.13)

∞



I(ω) dω = A



hω

2π



, etc.

(19.14)

Note that

g(ω)/N

k(ω)/N

∝ exp



−



;

(19.15)

if N

is also in equilibrium at T , this yields the

correct equilibrium relation between k(ω), g(ω), I(ω),

and a black body spectrum. While these relations are

much more general than the QS theory, the latter pro-

vides a helpful conceptual basis. The above expressions

in terms of spontaneous emission thus cover all cases.

19.2 Radiation Trapping

Atoms and ions efﬁciently absorb their own resonance

radiation, and their emission can be reabsorbed be-

fore escaping a vapor. Molecules are less efﬁcient

absorbers, since each electronic transition branches into

multiple-line bands, but interesting effects result if such

reabsorption occurs. This emission and reabsorption

process is fundamental to the formation of stellar lines,

where it is called radiation transfer, and to conﬁned

vapors and plasmas where it is also called radiation

diffusion or trapping. Fraunhofer’s observation of dark

lines in the stellar spectrum result from this radiation

transfer process. Highly sophisticated treatments of line

formation in inhomogeneous and nonequilibrium plas-

mas containing many species [19.19, 20] also apply to

laboratory plasmas, but the simpliﬁcations inherent in

a one- or two-element, conﬁned plasma with cylindrical

or planar symmetry leads to easier treatments. This sec-

tions discusses only a uniform density and temperature,

conﬁned atomic vapor.

The ﬂourescent lamp in which 254 nm mercury ra-

diation diffuses to the walls and excites a phosphor,

provides a prime example of radiation trapping. Its im-

provement motivated the seminal Biberman [19.51, 52]

and Holstein [19.53, 54] theories, continuing through

modern theory and experiment that is particularly rele-

vant to electrodeless and compact lamps. Dense clouds

of cold, trapped atoms are also inﬂuenced by radia-

tion trapping. Reference [19.55] provides and excellent

overview of this topic, which we will not discuss here.

The effect of radiation trapping on the polarization of

ﬂourescent radiation played a major role in develop-

ing a correct understanding of the coherent response of

atoms to radiation. This is reviewed in [19.44], and will

not be covered here. Molisch and Oehry [19.56]have

provided a detailed discussion of research on radiation

transport up to 1998.

19.2.1 Holstein–Biberman Theory

An atom in a dense vapor may be excited by exter-

nally applied radiation plus the ﬂuorescence from other

excited atoms within the vapor, and it will decay by

spontaneous emission (neglecting stimulated emission).

This is expressed by the Holstein–Biberman equation

dn(r, t)/ dt = S(r, t) +γ



vol

K(r −r



)n(r



, t) d



−γn(r, t), (19.16)

where n(r, t) is the excited state density at position r,

S(r, t) is the excitation rate due to externally applied ra-

diation, γ is the spontaneous emission rate, the kernel

K(r −r



) is the probability of a reabsorption at r due to

ﬂuorescence by an atom at r



, and the integral is over

the vapor ﬁlled volume [19.51–54]. Since K(r, r



) is as-

sumed the same for all excited atoms, this contains an

implicit assumption that all atoms emit the same fully

redistributed spectrum. The solution of this linear inte-

gral equation, subject to boundary values at the vapor

boundary, can be expressed as a sum over an orthog-

onal set of solutions n(r, t)

= n(r)

exp(−g

γt) of the

homogeneous equation

n(r, t) =

∞



i=1

a(t)

n(r)

, (19.17)

Part B 19.2

288 Part B Atoms

where, if S(r, t) = S(r) f(t),thena(t)



−∞

f (t



)

×exp[−g

γ(t −t



)]dt



and



S(r)n(r)

r.Here

n(r, t)

is the ith normal mode and g

γ is the decay

rate of this mode, as it would decay without change in

its shape n(r)

from a pulse of excitation.

Two shapes of vapor regions have been studied in

detail: an inﬁnitely long cylinder of radius R and the

region between two inﬁnite parallel plates with sepa-

ration L. The ﬁrst three symmetric modes of the latter

slab geometry are shown with unit height in Fig. 19.5.

A spatial integration over the normalized i = 1 or fun-

damental mode yields 1 and all others integrate to zero,

so a(t)

equals the total excited state population. g

the escape probability; i. e., the probability of photon es-

cape averaged over the fundamental mode distribution

of emitters n(r)

.Sincen(r, t) must be everywhere posi-

tive, the negative contributions of the higher order modes

only reduce the density in some regions. The g

can vary

from 0 to 1 and increase with increasing i, so that higher

order modes die out faster after pulsed excitation. The

ratios of decay rates is g

: g

= 1 : 3.7 : 6.4forthe

symmetric slab modes shown in Fig. 19.5. For steady

state excitation, (19.17) yields a(t)

γ ,sothe

lower order modes are more heavily weighted because

they decay more slowly. The fundamental mode decay

rate g

γ is of primary interest in most situations, and we

will now discuss its properties.

1.0

0.5

–0.5

–1.0

–1.0 –0.5 0 0.5 1.0

Fig. 19.5 The ﬁrst three symmetric eigenfunctions ( j =

0.2, 4) of radiation trapping between slab windows, for

a Doppler line proﬁle, from [19.57–59]. The windows are

at ±1

The kernel K(x) is the probability of ﬂuorescence

transport over a distance x followed by reabsorption,

averaged over the emitted frequency distribution. It is

conceptually useful to express it in terms of the spec-

trally averaged transmission T (x)

K(x) =

4πx

dT (x)

(19.18)

T (x) =

∞



L(ω) exp[−k(ω)x]dω,

where L(ω) is the emission line shape normalized to

unit area, and x =|r −r



|. If one assumes that the ﬂuo-

rescence frequency of an atom does not depend on the

frequency it absorbed (i. e., complete spectral redistri-

bution), this leads to k(ω) = κL(ω),whereκ =(λ

/8π)

)nΓ and g

and g

are statistical weights. This

simpliﬁcation applies under most conditions and will

be used here; its range of validity and more accurate

treatments are discussed below.

The transmission factor L(ω) and the integrand

of (19.18)areshowninFig.19.6, for a Gaussian line

shape and several values of k

x,wherek

is the line

center absorption coefﬁcient. At small k

x,thetrans-

mitted spectrum is similar to L(ω); for these conditions

T (x)  exp(−k

x ),wherek

 0.7k

is the average

attenuation. For k

x > 5, the transmission is small ex-

cept at the edges of the line. The transmitted radiation

is then predominately in the ω region near ω

,deﬁned

by k

xL(ω

) =1. Since the integrand is sharply peaked

near ω

, this leads to simple analytic forms for T (x).In

–3 –2 –1 0 1 2

(ω – ω

)/0.6 ∆ω

×5

Fig. 19.6 Gaussian emission spectrum L(ω) (short-dash

line), transmissions T(ω) (long-dash lines), and transmitted

intensities (solid lines)fork

x = 2, 10, and 50

Part B 19.2

Line Shapes and Radiation Transfer 19.2 Radiation Trapping 289

the large k

x limit, T (x) [k

x (π ln k

x )

1/2

]

−1

in the

Gaussian case, and T (x )  (πk

x )

−1/2

for a Lorentzian

line shape. These asymptotic forms of T (x) are com-

pared with the exact T (x) in Fig. 19.7; T (x) follows

the asymptotic formulas for k

x > 5 and 10, respec-

tively. T (x) for several Voigt line shapes is also shown

in Fig. 19.7; these follow the Gaussian T (x) at smaller

x, then rise above as ω

moves into the Lorentzian

wing.

For radiative escape from a cell, transmission over

distances near the cell dimension (R or L) is most impor-

tant, since transport over this distance often escapes the

vapor and transport over much smaller distances does

not have much effect. The escape probability g

,aver-

aged over the fundamental mode distribution, is close to

T (R) or T (L/2), while the higher order modes are re-

lated to the same asymptotic forms of T (x) at smaller

distances. Thus, in the large k

L slab case,

(19.19a)











ln k

1/2

Gaussian line



1/2

Lorentzian line,

(19.19b)

with G

= 1.03 and G



 0.65. For an inﬁnite cylinder,

the same equations hold with L/2 → R and slightly

larger G

values. Exact G

and G



values can be found

in [19.57–59].

19.2.2 Additional Factors

As noted above, the line shape of a two-level atom

in a thermal vapor is a Voigt shape; a convolution of

a Lorentzian of width Γ +2γ

with a Gaussian of width

∆ω

. In most cases, ∆ω

 Γ , so in the absence of

a buffer gas the line shape is nearly Gaussian at low

density (n). As a result, ω

is in the Gaussian region of

the line at low density and g

behaves similarly to the

Gaussian transmission in Fig. 19.7 with x replaced by the

conﬁnement dimension. k

is proportional to n, so from

(19.19a) g

is approximately inversely proportional to n

for k

L > 5. As n increases, ω

moves further into the

wing of the line, and when ω

reaches the Lorentzian tail

of the Voigt line proﬁle a transition to (19.19b) occurs,

where k

corresponds to a purely Lorentzian line. (That

the core of the line does not have a Lorentzian shape

does not matter, since the fraction of emission well into

the Lorentzian wing is nearly the same as that of a pure

Lorentzian line.)

In the absence of a collision, a two level atom rera-

diates in its rest frame the same frequency it absorbed.

–1

–2

–3

log (k

–1 0 1 2 3

log transmission

Gaussian

Holstein Gauss approximation

Lorentzian

Holstein Lorentzian

approximation

a = 0.1

a =

0.01

a =

0.001

a = 0

Fig. 19.7 Transmission T(x) versus distance in units of

x, for Voigt line shapes with the a parameters indi-

cated, where a = (ln 2)

1/2

∆ω

Lor

/∆ω

Gauss

. The Gaussian

limit corresponds to a = 0 and the Lorentzian limit to

a = 1. The Holstein, large k

x, approximations are also

indicated

Thermal motion redistributes this coherent scattering

frequency within the Doppler envelope when the emis-

sion and absorption are in different directions, but it

does not transfer it into the natural Lorentzian wing

outside the Doppler envelope. This leads to the prop-

erty that an atomic vapor will scatter frequencies in

the natural wing, but will not emit in this wing un-

less it absorbed there or is excited by or during

a collision. With line broadening collisions, a frac-

tion Γ/(Γ +2γ

) of optical attenuation is coherently

scattered and a fraction 2γ

/(Γ +2γ

) is redistributed

into “incoherent” emission with a Lorentzian spectrum

of width Γ +2γ

centered at ω

+δ in the reference

frame of the moving atom. This redistributed emission

can escape in the Lorentzian wing of the Voigt line.

In this radiation transport problem, the consequence is

that (19.19b) with k

= n(λ

/2π)(Γ/γ

)(g

), corre-

sponding to a Lorentzian with Γ

Total

= 2γ

not Γ +2γ

provides the best approximation to g

in the density

region where ω

is in the Lorentzian wing of the

line. Since k

∝ n/γ

and in the absence of a buffer

gas γ

= k

n,wherek

is the rate coefﬁcient for self

broadening collisions, g

becomes independent of n.

In fact, k

∝ Γ as well, so g

is also independent of

Γ . For the case of a J = 0 ground state and a J = 1

excited state, g

= 0.21(λ/L)

1/2

; the broadening coef-

ﬁcient for other cases can be found in [19.60]. If the

Part B 19.2

290 Part B Atoms

broadening is due to a buffer gas, γ

= k

in (19.19b)

yields

∝





1/2

; (19.20)

this has been studied in [19.64].

Post et al. have numerically evaluated g

for all

values of k

L for slab and cylinder geometries, by in-

tegrating the radiative escape probability g(z) over the

fundamental mode distribution N(z),wherez is the po-

sition between the windows [19.65]. To obtain g(z) they

integrate over the angular distribution of the emission,

using T (x) from the exact line shape. Thus all features

of the calculation correspond to the Holstein–Biberman

theory for an isolated line without approximation. As

will now be discussed real atomic vapors are generally

not that simple.

Many atomic “lines” have multiple components due

to hyperﬁne structure and isotope shifts; some compo-

nents are isolated while others are separated by less than

a Doppler line width and overlap. The absorption line

shape then becomes a weighted sum over components,

each with an equivalent Voigt shape. In a high dens-

ity vapor or a plasma, collisions will usually distribute

–1

–2

–3

–4

–1

Optical depth (k

L/2)

eff

Sodium density (cm

–3

)

R = Γ

eff

=2γc

H(3P

3/2

)

H(3P

1/2

)

P(3P

3/2

)

Fig. 19.8 Radiative escape probability g

for Na vapor excited to

the 3P

3/2

state, for a slab geometry. The Holstein approximation

for the 3P

3/2

−3S

1/2

(D2) line and the 3P

1/2

−3S

1/2

(D1) line

are indicated as dashed lines. The Post-type calculation of [19.61]

for the D2 line is indicated as a solid line. Solid squares are data

from [19.62], and open circles are data from [19.63]. The effective

escape probability corresponds to the D2 line rate at low densities

but a combination of D1 and D2 at high densities

the excited state population between the isotopes and

hyperﬁne states in proportion to their isotopic fraction

and statistical weight. The emission line shape L(ω) is

then a similarly weighted distribution over components.

Since radiation only escapes in the wings of a line com-

ponent at high k

L, overlapping components act almost

as a single component. If the line has M isolated com-

ponents, the right-hand side of (19.19a)and(19.19b)

become sums over the fraction f

of the intensity in

the j component times the escape probability for that

component. The latter is obtained, for large k

L,byre-

placing k

with k

in (19.19a)and(19.19b). The net

result, after summing over components, is an increase in

by a factor of ≈ M in the Gaussian case and ≈ M

1/2

in the Lorentzian. This approximation was obtained by

Holstein in the context of the Hg 254 nm radiation under

conditions appropriate to the ﬂuorescent lamp [19.66].

Walsh made a more detailed study of these overlapping

components [19.67], and the dependence of g

on the

ratio of line separation to Doppler width is also given

in [19.63].

19.2.3 Measurements

The overall behavior of g

versus n is shown in Fig. 19.8

for the Na(3P

3/2

) or D2 resonance line in pure Na

vapor [19.62, 63]. In this type of experiment the funda-

mental mode decay rate is established by a combination

of optimally exciting that spatial mode and of wait-

ing until the ﬂuorescence decay is exponential in time

after termination of the excitation. A transition to ap-

proximately 1/n behavior, corresponding to (19.19a),

is seen to occur at k

L/2 ≈ 5. At k

L/2 ≈ 100 the

transition to n

behavior, corresponding to a self-

broadened Lorentzian line in (19.19b), can be seen.

The behavior at k

L < 5 ﬁts the Milne diffusion the-

ory [19.68] as well as the Post et al. theory shown

as a solid line; this is also similar to T (L/2),as

seen in Fig. 19.7.For5< k

L/2 < 100, the behav-

ior is similar to (19.19a) (dashed line), but the Post

et al. theory (solid line) is ≈ 20% higher due to the

inclusion of the Na hyperﬁne structure (hfs splitting

 Doppler width). For k

L/2 > 1000, the Post theory

converges to the Holstein–Lorentzian-line result with

Total

= 2γ

The experiment is complicated in the 50 < k

L/2 <

500 region by ﬁne structure mixing [19.62]. The 3P

3/2

state was excited, but at high densities, collisions popu-

late the 3P

1/2

state, which has a smaller g

than the 3P

3/2

state (Fig. 19.8). At low densities, g

eff

= g

(3P

3/2

),and

at high densities these states are statistically populated

Part B 19.2

Line Shapes and Radiation Transfer 19.2 Radiation Trapping 291

and g

eff

(3P

1/2

) +

(3P

3/2

). The transition

density where the ﬁne structure mixing rate R equals

eff

is indicated in Fig. 19.8. The theory is also compli-

cated in this intermediate k

L region by the necessity of

including incomplete frequency redistribution [19.65];

this leads to the dip in g

near k

L ≈ 500. While the

overall behavior of the data in Fig. 19.8 is consistent

with the Post et al. theory, there is ≈ 30% systematic

discrepancy at k

L/2 = 10−100 and the dip near 500

is not seen. Part of this difference probably results from

the experimental geometry, which was between a slab

and a cylinder of radius R = L/2; g

for the cylinder is

17% larger than the slab value used in Fig. 19.8.

The fundamental mode decay rate has also been

measured for the Hg 254 nm [19.69] and 185 nm [19.65]

lines, for the Ne resonance line [19.70]andfortheAr

resonance line [19.71]. The Hg measurements are com-

plicated by multiple isotopes and hyperﬁne structure,

producing a mixture of partially overlapping and isolated

lines combined with density-dependent uncertainties in

excited state populations of the various isotopes. Se-

rious efforts to model and measure these effects have

been made [19.65, 67, 69, 72].TheNeandArmeas-

urements have similar complications, as will now be

discussed.

In essence, g

behaves like the Gaussian T (x = L/2)

in Fig. 19.7 until n is large enough for ω

to ap-

proach the collision induced Lorentzian wing of the

Voigt line. g

then decreases more slowly since the

line wing does not fall off as rapidly as a Gaussian.

With continued increase in n, ω

moves further into

the Lorentzian wing, a broader spectral region escapes

and g

reaches a minimum. Finally, when the en-

tire escaping spectral region is Lorentzian, g

reaches

the constant value described above. Independent and

detailed treatments of this density region, including in-

complete frequency redistribution, predict a dip in g

seen in Fig. 19.8 [19.65, 71, 73, 74]. However, this has

not been clearly conﬁrmed experimentally. In Fig. 19.8

this dip occurs where ﬁne structure mixing also occurs,

and in addition the data are higher than the calcula-

tions throughout this n region. Post et al. [19.65]did

observe such a dip for the Hg (149 nm) resonance line,

but the data do not ﬁt the calculation at other densities;

hyperﬁne and isotopic structure within the line cause

major complications. This long-standing issue has ﬁ-

nally been clariﬁed by Menningen and Lawler [19.75],

who measured the decay of the Hg (185 nm) resonance

line following laser excitation. They observed a clear dip

in g

due to incomplete redistribution. They also carried

out sophisticated Monte Carlo simulations, obtaining

values that compared favorably with the measure-

ments. By extending the simulations over a largerange of

a parameter space, they constructed an analytic formula

for g

of a single-component line in cylindrical geome-

try [19.76]. This formula includes effects of incomplete

frequency redistribution and varying ratios of Doppler

broadening, radiative broadening and collisional broad-

ening, so that it can be applied to any resonance line.

Payne et al. [19.71] did not observe the predicted dip for

the Ar resonance line; again a minor isotope with an iso-

lated line occurs and could be very important at these

high optical depths. Phelps [19.70] reported such a dip

for the Ne 74.3 nm resonance line, but with rather large

uncertainties due to the necessity of correcting for other

collisional effects. Again there are isotopes with isolated

lines that may have effected the data. Thus, experiments

have veriﬁed the essential aspects of the above theories,

but quantitative agreement in all aspects has not yet been

achieved.

The fact that the escaping radiation is concentrated

in the wings of the line, near the unity optical depth

point ω

, is reﬂected in the emitted spectrum. Calculated

examples are shown in [19.74]; the Gaussian case looks

somewhat like the transmitted spectra in Fig. 19.6 for

x ≈ L/4. These spectra, and all results described so far,

are calculated assuming no motion of the atoms. This is

appropriate in the central region of the vapor, because

the distance moved in an excited state lifetime (L

v/Γ ) is much smaller than L. In fact, resonant collisions

between excited and ground state atoms further limits the

distance an excited atom moves in one direction before

transfer of excitation. However, near the window or wall

of the container, atomic motion will cause wall collisions

of excited atoms and loss of radiation. This loss will be

primarily within the Doppler core of the line, since these

frequencies can only escape if emitted near the vapor

edge. This loss depends on the excited state density in

the neighborhood of the wall, and can be signiﬁcant if

> 1/k

. The excited atom density near the wall must

be self consistent with the radiation transport and wall

quenching. This situation has been modeled and studied

experimentally ([19.77] and references therein).

Additional aspects of radiation trapping, such as

higher-order spatial modes and non-uniform absorber

distributions, can be signiﬁcant in lighting plasmas (and

trapped atom clouds). Propagator function techniques

have been developed for modeling radiation transport

when the excitation has unusual temporal or spatial

character [19.78,79]. Non-uniform absorber spatial dis-

tributions can be particularly important at high power

densities, and have been considered in [19.80].

Part B 19.2

292 Part B Atoms

References

19.1 H. A. Lorentz: Proc. Akad. Wet. (Amsterdam) 8,591

(1906)

19.2 V. Weisskopf: Phys. Z. 34, 1 (1933)

19.3 J. Holtsmark: Ann. Phys. (Leipzig) 58, 577 (1919)

19.4 H. G. Kuhn: Philos. Mag. 18,987(1934)

19.5 H. Margenau, W. W. Watson: Rev. Mod. Phys. 8,22

(1936)

19.6 A. Jablonski: Z. Physik 70,723(1931)

19.7 A. Jablonski: Acta Phys. Polon. 27, 49 (1965)

19.8 E. Lindholm: Ark. Fys. A 32,1(1945)

19.9 H. M. Foley: Phys. Rev. 69,616(1946)

19.10 H. M. Foley: Phys. Rev. 73,259(1948)

19.11 T. Holstein: Phys. Rev. 79,744(1950)

19.12 P. W. Anderson: Phys. Rev. 76, 647 (1949)

19.13 P. W. Anderson: Phys. Rev. 86,809(1952)

19.14 M. Baranger: Phys. Rev. 112,855(1958)

19.15 S-Y. Chen, M. Takeo: Rev. Mod. Phys. 29, 20 (1957)

19.16 N. Allard, J. Kielkopf: Rev. Mod. Phys. 54, 1103 (1982)

19.17 H. Griem: Plasma Spectroscopy (McGraw Hill, New

York 1964)

19.18 H. Griem: Spectral Line Broadening by Plasmas

(Academic, New York 1974)

19.19 J. Jeffries: Spectral Line Formation (Blaisdell,

Waltham 1968)

19.20 D. Mihalas: Stellar Atmospheres (Freeman, San

Francisco 1970)

19.21 J. van Kranendonk: Cnd. J. Phys. 46, 1173 (1968)

19.22 J. Cooper: Rep. Prog. Phys. 29, 35 (1966)

19.23 J. Simons: Energetic Principles of Chemical Reac-

tions (Jones Bartlett, Boston 1983)

19.24 J. Szudy, W. E. Baylis: J. Quant. Spectrosc. Radiat.

Trans. 17, 681 (1977)

19.25 R. E. Walkup, A. Spielﬁedel, D. E. Pritchard: Phys.

Rev. Lett. 45,986(1980)

19.26 J. Ward, J. Cooper, E. W. Smith: J. Quant. Spectrosc.

Radiat. Trans. 14, 555 (1974)

19.27 D. G. McCarten, N. Lwin: J. Phys. B 10, 17 (1977)

19.28 D. N. Stacey, R. C. Thompson: Acta Phys. Polon. A

54,833(1978)

19.29 Ch. Ottinger, R. Scheps, G. W. York, A. Gallagher:

Phys. Rev. A 11, 1815 (1975)

19.30 D. Drummond, A. Gallagher: J. Chem. Phys. 60,

3426 (1974)

19.31 C. G. Carrington, A. Gallagher: Phys. Rev. A 10,1464

(1974)

19.32 J. Pascale, J. Vandeplanque: J. Chem. Phys. 60, 2278

(1974)

19.33 M. J. O’Callaghan, J. Cooper: Phys. Rev. A 39,6206

(1989)

19.34 B. P. Stoicheff, E. Weinberger: Phys. Rev. Lett. 44,

733 (1980)

19.35 K. H. Weber, K. Niemax: Z. Phys. A 307, 13 (1982)

19.36 J. R. Fuhr, A. Lesage: Bibliography of Atomic Line

Shapes and Shifts, NIST Special Publication 366

(U. S. Gov’t. Printing Ofﬁce, Washington 1993),

Suppl. 4

19.37 J. Szudy, W. E. Baylis: J. Quant. Spectrosc. Radiat.

Trans. 15,641(1975)

19.38 D. N. Stacey, J. Cooper: J. Quant. Spectrosc. Radiat.

Trans. 11, 1271 (1971)

19.39 K. M. Sando, J. Wormhoudt: Phys. Rev. A 7, 1889

(1973)

19.40 L. K. Lam, M. M. Hessel, A. Gallagher: J. Chem. Phys.

66, 3550 (1977)

19.41 A. Tam, G. Moe, W. Park, W. Happer: Phys. Rev.

Lett. 35, 85 (1975)

19.42 T. Holstein, A. Gallagher: Phys. Rev. A 16, 2413 (1977)

19.43 J. L. Carlston, A. Szoke: J. Phys. B 9, L231 (1976)

19.44 A. Omont: Prog. Quant. Electr. 5, 69 (1977)

19.45 J. Light, A. Szoke: Phys. Rev. A 15, 1029 (1977)

19.46 K. Burnett, J. Cooper, R. J. Ballagh, E. W. Smith:

Phys. Rev. A 22, 2005 (1980)

19.47 A. Streater, J. Cooper, W. J. Sandle: J. Quant. Spec-

trosc. Radiat. Trans. 37,151(1987)

19.48 J. Tellinghausen: J. Mol. Spectrosc. 103,455(1984)

19.49 F. H. Mies: J. Chem. Phys. 48, 482 (1968)

19.50 C. G. Carrington, D. Drummond, A. V. Phelps, A. Gal-

lagher: Chem. Phys. Lett. 22, 511 (1973)

19.51 L. M. Biberman: J. Exp. Theor. Phys. U. S. S. R. 17,

416 (1947)

19.52 L. M. Biberman: J. Exp. Theor. Phys. U. S. S. R. 59,

659 (1948)

19.53 T. Holstein: Phys. Rev. 72, 1212 (1947)

19.54 T. Holstein: Phys. Rev. 83, 1159 (1951)

19.55 A. Fioretti, A. F. Molisch, J. H. Müller, P. Verkerk,

M. Allegrini: Opt. Commun. 149, 415 (1998)

19.56 A. F. Molisch, B. P. Oehry: Radiation Trapping in

Atomic Vapours (Clarendon, Oxford 1998)

19.57 C. van Trigt: Phys. Rev. 181, 97 (1969)

19.58 C. van Trigt: Phys. Rev. A 4, 1303 (1971)

19.59 C. van Trigt: Phys. Rev. 13,726(1976)

19.60 C. G. Carrington, D. N. Stacey, J. Cooper: J. Phys. B

6, 417 (1973)

19.61 J. Huennekins, T. Colbert: J. Quant. Spectrosc. Ra-

diat. Trans. 41, 439 (1989)

19.62 J. Huennekins, A. Gallagher: Phys. Rev. A 28,238

(1983)

19.63 T. Colbert, J. Huennekens: Phys. Rev. 41, 6145 (1990)

19.64 J. Huennekins, H. J. Park, T. Colbert, S. C. McClain:

Phys. Rev. 35, 2892 (1987)

19.65 H. A. Post, P. van de Weijer, R. M. M. Cremers: Phys.

Rev. A 33, 2003 (1986)

19.66 T. Holstein: Phys. Rev. 83, 1159 (1951)

19.67 P. J. Walsh: Phys. Rev. 116, 511 (1959)

19.68 E. Milne: J. Math. Soc. (London) 1, 40 (1926)

19.69 T. Holstein, D. Alpert, A. O. McCoubrey: Phys. Rev.

85, 985 (1952)

19.70 A. V. Phelps: Phys. Rev. 114, 1011 (1959)

Part B 19

Line Shapes and Radiation Transfer References 293

19.71 M. G. Payne, J. E. Talmage, G. S. Hurst, E. B. Wagner:

Phys. Rev. A 9, 1050 (1974)

19.72 J. B. Anderson, J. Maya, M. W. Grossman, R. La-

gushenko, J. F. Weymouth: Phys. Rev. A 3,2986

(1985)

19.73 G. J. Parker, W. N. G. Hitchon, J. E. Lawler: J. Phys.

B 26, 4643 (1993)

19.74 C. van Trigt: Phys. Rev. A 1, 1298 (1970)

19.75 K. L. Menningen, J. E. Lawler: J. Appl. Phys. 88, 3190

(2000)

19.76 J. E. Lawler, J. J. Curry, G. G. Lister: J. Phys. D 33,252

(2000)

19.77 A. Zajonc, A. V. Phelps: Phys. Rev. A 23,2479

(1981)

19.78 J. E. Lawler, G. J. Parker, W. N. G. Hitchon: J. Quant.

Spectrosc. Radiat. Trans. 49, 627 (1993)

19.79 G. J. Parker, W. N. G. Hitchon, J. E. Lawler: J. Phys.

B 26, 4643 (1993)

19.80 J. J. Curry, J. E. Lawler, G. G. Lister: J. Appl. Phys.

86, 731 (1999)

Part B 19

295

Thomas–Ferm

20. Thomas–Fermi

and Other Density-Functional

Theories

The key idea in Thomas–Fermi theory and its

generalizations is the replacement of complicated

terms in the kinetic energy and electron–electron

repulsion energy contributions to the total energy

by relatively simple functionals of the electron

density ρ. This chapter ﬁrst describes Thomas–

Fermi theory, and then its various generalizations

which attempt to correct, with varying success,

some of its deﬁciencies. It concludes with an

overview of the Hohenberg-Kohn and Kohn-Sham

density functional theories.

20.1 Thomas–Fermi Theory

and Its Extensions ............................... 296

20.1.1 Thomas–Fermi Theory................ 296

20.1.2 Thomas–Fermi–von Weizsäcker

Theory ..................................... 298

20.1.3 Thomas–Fermi–Dirac Theory....... 299

20.1.4 Thomas–Fermi–von Weizsäcker–

Dirac Theory ............................. 299

20.1.5 Thomas–Fermi Theory

with Different Spin Densities ...... 300

20.2 Nonrelativistic Energies of Heavy Atoms 300

20.3 General Density Functional Theory ........ 301

20.3.1 The Hohenberg–Kohn Theorem

for the One-Electron Density ...... 301

20.3.2 The Kohn–Sham Method

for Including Exchange

and Correlation Corrections ........ 302

20.3.3 Density Functional Theory

for Excited States ...................... 303

20.3.4 Relativistic and Quantum Field

Theoretic Density Functional

Theory ..................................... 303

20.4 Recent Developments........................... 303

References .................................................. 304

In the early years of quantum physics, Thomas [20.1]

and Fermi [20.2–5] independently invented a sim-

pliﬁed theory, subsequently known as Thomas–Fermi

theory, to describe nonrelativistically an atom or

atomic ion with a large nuclear charge Z and

a large number of electrons N. Many qualitative

features of this model can be studied analytically,

and the precise solution can be found by solv-

ing numerically a nonlinear ordinary differential

equation. Lenz [20.6] demonstrated that this equa-

tion for the electrostatic potential could be derived

from a variational expression for the energy as

a functional of the electron density. Reﬁnements to

Thomas–Fermi theory include a term in the energy

functional to account for electron exchange effects

introduced by Dirac [20.7], and nonlocal gradient

corrections to the kinetic energy introduced by von

Weizsäcker [20.8].

Although the Hartree–Fock method or other more

elaborate techniques for calculating electronic structure

now provide much more accurate results (Chapts. 21,

22,and23), Thomas–Fermi theory provides quick esti-

mates and global insight into the total energy and other

properties of a heavy atom or ion. A rigorous analysis

of Thomas–Fermi theory by Lieb and Simon [20.9, 10]

showed that it is asymptotically exact in that it yields

the correct leading asymptotic behavior, for both the

total nonrelativistic energy and the electronic density,

in the limit as both Z and N tend to inﬁnity, with the

ratio Z/N ﬁxed. (In a real atom, of course, relativis-

tic and other effects become increasingly important as

Z increases.) However, Thomas–Fermi theory has the

property that molecules do not bind, as ﬁrst noted by

Sheldon [20.11] and proved by Teller [20.12]. That the

interatomic potential energy curve for a homonuclear di-

atomic molecule is purely repulsive was demonstrated

by Balàzs [20.13]. This ‘no binding’ property of clus-

ters of atoms was used by Lieb and Thirring [20.14]

to prove the stability of matter, in the sense that as the

number of particles increases, the total nonrelativistic

energy decreases only linearly rather than as a higher

power of the number of particles, as it would if elec-

trons were bosons rather than fermions. Lieb went on

to explore the mathematical structure of the modiﬁca-

tions of the Thomas–Fermi model when gradient terms

(von Weizsäcker) and/or exchange (Dirac) terms are in-

cluded [20.15, 16]. A review article by Spruch [20.17]

explicates the linkage between long-developed physi-

Part B 20