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Spectroscopy of Ions Using Fast Beams and Ion Traps 18.2 Spectroscopy Using Ion Traps 275

70 µm) as a light source without a further entrance

slit, a high-efﬁciency transmission grating spectrom-

eter, and a position-sensitive detector, lines from two

Tl isotopes were recorded simultaneously [18.55]. The

differential effects of the two nuclei yielded additional

information on the charge distribution in the nucleus,

on top of the magnetic moment distribution that makes

up the dominant effect. These results are required for

a better interpretation of parity nonconservation (PNC;

Chapt. 29) measurements of neutral Tl atoms, in which

the nuclear magnetic moments so far have been treated

only as a point dipole. Corresponding measurements

are also being done on Li-like ions, and again the

most precise data so far (on Bi) have been obtained

at SuperEBIT.

One of the pertinent problems in astrophysics is the

cataloguing of spectral lines in all spectral ranges, both

for identifying spectral features and thus learning about

the composition of a light source, and for modeling

of the light source in order to understand its “operat-

ing conditions”. Even data bases that claim “practical

completeness”, however, are found to be grossly incom-

plete in the EUV and soft-X-ray ranges that have been

opened to high-resolution observations by the grating

spectrometers on board the Chandra and XMM-Newton

spacecrafts. Here observations at electron beam ion traps

ﬁll in much of the needed data, at comparable quality.

Moreover, an EBIT as a kind of analog computer does

more than provide correct line positions: the spectra

show line ratios from a light source with known elec-

tron energy (and the option of known temperature by

simulating a Maxwellian energy distribution [18.56])

and particle density. This serves both as a check on

collisional-radiative models and as an immediate data

resource for astrophysics.

Most X-ray data from EBITs have been collected

using solid state detectors [Si(Li), Ge] that offer

high detection efﬁciency (large solid angle) and sig-

nal timing on the microsecond scale, but feature poor

spectral resolution. High-resolution instruments like

crystal spectrometers are necessary to analyze spectra

in any detail; equipped with position-sensitive detec-

tors, such instruments do much better than scanning

spectrometers in terms of data collection rate and cal-

ibration. However, they suffer from the low diffraction

efﬁciency of the crystals. Recently a new device, the

microcalorimeter, has started to show its interesting

properties. In these devices, small absorbers at mK-

temperatures show a measurable temperature increase

when absorbing an X-ray photon, and the signal is pro-

portional to the deposited photon energy [18.57]. The

linewidth of the best devices is below 10 eV (not as

good as a crystal spectrometer, but much better than

a traditional solid state diode), the sensor pixels can

be grouped in arrays to make for a larger area and for

cross references among pixels (which helps with cal-

ibration and with the rejection of cosmic ray events),

and the signal processing is fast enough to permit time

resolution on the millisecond range. A spaceﬂight engi-

neering spare has been used at the Livermore EBITs

to study, for example, soft-X-rays of light elements

as seen from CX near comets [18.58], or measure the

(10 ms) time constant of the M3 decay in a Ni-like ion

(Xe

26+

) [18.59].

Last, but not least, the well deﬁned and adjustable

electron beam energy permits detailed studies of the in-

teraction of fast electrons with highly charged ions. This

includes, for example, the spectroscopy of dielectronic

recombination (DR) resonances, or the exploration of ra-

diative recombination (RR)(Chapt.55). These processes

can be investigated up to the highest ion charges, where

relativistic and QED contributions (like the Generalized

Breit interaction) matter.

18.2.2 Heavy-Ion Storage Rings

With foil-excited ion beams and long level-lifetimes,

decay curves spread out along the beam, and the sig-

nal from a given width of the ﬁeld of view may drop

to the detector background level. The decay lengths of

microsecond lifetime levels are on the order of 10 m,

and those of millisecond levels are tens of kilometers.

In such cases it is advantageous to curve the beam line

around on itself, forming a storage ring, in which the

ions pass in front of the detector over and over again.

Heavy-ion storage rings need excellent vacuum condi-

tions (10

−11

mbar and better) to reach storage times of

seconds, minutes, or hours, depending on the electronic

structure of the stored ions and on the ion beam en-

ergy. The dominant loss processes are electron capture

and loss, small angle scattering, and large angle (in-

elastic) scattering as with any fast ion beams [18.60],

but these are aggravated here by the much longer path

lengths.

Storage rings (for example, TSR Heidelberg,

ASTRID Aarhus, CRYRING Stockholm, ESR Darm-

stadt), with magnetic dipoles and quadrupoles for beam

transport and focusing, sort the stored particles by their

momentum. The typical ion beam energies range from

a few dozen keV total to hundreds of MeV per nucleon

(ESR). Electrostatic storage rings (ELISA Aarhus and

more under construction) have only electrical ﬁelds and

Part B 18.2

276 Part B Atoms

select by particle energy, usually below about 100 keV

(for a review, see [18.61]). They are more suitable for

low-charge heavy ions and ion molecules, including

biomolecules, than the magnetic rings that can han-

dle very fast particles. Magnetic storage rings usually

have electron cooler sections in which a “cool” elec-

tron beam (with a low longitudinal velocity spread

from kinematic compression) of almost the same ve-

locity as the circulating ion beam is merged with the

latter for a path of a few meters (and is then de-

ﬂected out again). By scattering among electrons and

ions, the momentum spread of the electrons (small) and

ions (larger) equilibrates, leaving the ion beam with

a narrower momentum distribution and thus cooled.

Cooling, which typically takes a few seconds, im-

proves the storage behaviour of the ion beam and

the energy resolution of, for example, dielectronic re-

combination (DR) studies (Chapt. 55). For these, the

same electron cooler is now tuned to provide elec-

trons at a well deﬁned but different velocity. Thus the

electron cooler can serve as an electron target, with-

out the complications of a foil target in beam-foil

spectroscopy. The difference velocity can be chosen

from a wide range, including zero. Extremely low en-

ergy collisions are being investigated for the study

of DR and for the recombination of molecular rad-

icals. When a beam of molecular ions is injected,

storage is long enough to let some of the internal

degrees of freedom relax, and then a beam of mo-

lecular ions can be extracted that are closer to their

ground state.

A cooled ion beam, with its narrow velocity dis-

tribution, is also of interest for laser-ion interaction

studies, offering higher resolution and better signal.

Laser-assisted electron capture in the electron cooler,

as well as laser spectroscopy on high-lying levels pop-

ulated by DR, are possible. One of the problems with

precision wavelength measurements involving fast ions

is, as always the accurate determination of the veloc-

ity, as a step towards determining Doppler corrections.

At TSR Heidelberg, a beam of Li

ions was subjected

to a laser beam from ahead, tuned to one of the 2s–2p

transitions. A second laser beam from behind probed

the position of the Lamb dip in the velocity distri-

bution and thus assured that it would meet the same

velocity group of the multi-MeV stored ions. Accurate

off-line calibrations of the laser frequencies then permit-

ted a test of the Doppler formula to a relative precision

of 2.2×10

−7

[18.62].

The Doppler shift determination in any observation

of fast ions requires accurate angle measurements. These

are nontrivial, because the detection efﬁciency of any

ﬁnite size spectrometer or extended detector may be

non-uniform as a function of position or angle. One tech-

nique calls for segmented (“granular”) X-ray detectors,

the strips of which are calibrated individually [18.63].

Relativity changes the emission pattern seen in the lab-

oratory rest frame to one that favours forward emission.

This is beneﬁcial for zero-degree spectroscopy, that is, an

observation along the ion beam path. At ESR Darmstadt,

bare ions captured an electron in the electron cooler sec-

tion, and the resulting X-rays were detected from straight

ahead (behind the next dipole magnet that deﬂected the

ion beam). This geometry maximizes the Doppler shift,

but minimizes the uncertainty relating to geometry. Also,

after electron capture, the ion in the bending magnet

section follows a trajectory that differs from that of the

unperturbed ions. The ion can be detected and, in coin-

cidence with the X-ray detector, make for a very clean

and charge-speciﬁc spectrum. Similar coincidence mea-

surements make it possible to use a low-pressure gas jet

target in a high-energy ion storage ring, evaluating only

coincidences of X-ray photons and charge-changing

events [18.63]. Fast ions (energetic enough to achieve the

desired charge state) can also be decelerated in a storage

ring, which helps to do systematic checks of the Doppler

effect and to work at lower Doppler shift [18.63].

As mentioned in Sect. 18.2.1, laser-resonance tech-

niques have been used to ﬁnd the ground state hyperﬁne

transition in two H-like heavy isotopes. In one of them,

the lifetime was also measured, by recording the ﬂuo-

rescence decay from the ion beam after switching off

the laser [18.54]. For lower charge states, one can ex-

ploit the excitation that ions carry into the ring from their

production in the ion source, or from stripping processes

in the injector accelerator [18.64]. Lifetimes from half

a millisecond to several seconds have been measured this

way by passive observation [18.50], with an accuracy of

better than 0.2% in favourable cases. Other techniques

use excitation by DR in the ring [18.65], or laser prob-

ing of the remaining metastable level population so that

ﬂuorescence is emitted near a photomultiplier detec-

tor [18.66]. With a stored beam of negative ions, even

blackbody radiation may be sufﬁcient to photodetach the

weakly bound last electron; the ensuing neutral atom is

not deﬂected at the next bending section and leaves the

ring to be detected. All in all, lifetime measurements at

storage rings reach from 10 µs to about 1 min.

Part B 18.2

Spectroscopy of Ions Using Fast Beams and Ion Traps References 277

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Part B 18

279

Line Shapes an

19. Line Shapes and Radiation Transfer

The shapes of collisionally broadened atomic lines

is a topic almost as old as Fraunhofer’s discovery

of the existence of discrete lines. Lorentz pro-

vided the ﬁrst quantitative theory in 1906 [19.1],

and Weisskopf advanced this to the impact the-

ory by 1933 [19.2]. Holtsmark [19.3], Kuhn [19.4]

and Margenau [19.5] meanwhile developed the

quasistatic or statistical theory which describes

the line wing, and Jablonski put this on a quan-

tum mechanical footing in the context of free–free

molecular radiation [19.6,7]. By the 1940s, satellite

bands in the line wings, and a variety of high and

low pressure line shapes and broadening rates

had been measured. Initial confusion regarding

the validity of the contrasting impact versus static

approaches was largely resolved by uniﬁed treat-

ments of the Fourier integral theory [19.8–13].

Baranger then provided a quantum basis for the

impact theory, including level degeneracies [19.14].

Descriptions can be found in a variety of reviews

and references therein, including [19.2, 5, 15–20].

The broadening of molecular lines involves the

additional complication of rotationally nonadia-

batic collisions; this was initially addressed by

Anderson [19.12,13] and later with great thorough-

ness by van Kranendonk [19.21]. This chapter and

most of the above theories are concerned with

19.1 Collisional Line Shapes ......................... 279

19.1.1 Voigt Line Shape ....................... 279

19.1.2 Interaction Potentials ................ 280

19.1.3 Classical Oscillator Approximation 280

19.1.4 Impact Approximation ............... 281

19.1.5 Examples: Line Core................... 282

19.1.6 ∆ and γ

Characteristics............. 284

19.1.7 Quasistatic Approximation.......... 284

19.1.8 Satellites .................................. 285

19.1.9 Bound States

and Other Quantum Effects......... 286

19.1.10 Einstein A and B Coefﬁcients ...... 286

19.2 Radiation Trapping.............................. 287

19.2.1 Holstein–Biberman Theory ......... 287

19.2.2 Additional Factors ..................... 289

19.2.3 Measurements .......................... 290

References .................................................. 292

neutral atomic gases, which is sometimes

called pressure broadening. In plasmas, elec-

tron, ion, and neutral collisions all contribute

to the line shapes and strengths; thus the

emitted lines provide a powerful diagnostic of

plasma conditions. Neither molecular nor plasma

broadening will be covered here; the latter is

reviewed in [19.17–20, 22], and in Chapts. 59

and 48.

19.1 Collisional Line Shapes

The neutral-gas theories described above generally used

phenomenological or long range forms of the atomic

and molecular interactions, and most measurements

were not sufﬁciently detailed to test the validity of

these parameterizations or the theoretical approxima-

tions. A great deal of the work since mid century has

been directed towards obtaining more realistic and ac-

curate descriptions of these interactions and of the full

line shapes. In addition, many new types of observa-

tions have stimulated variations on the basic theories

and descriptions. This includes topics such as collision-

induced forbidden transitions, satellite shapes, spectral

and polarization redistribution, orientation and align-

ment effects, Doppler-free spectroscopy and very low

temperature collisions. Most of these topics are beyond

the scope of this brief and basic description of neutral

collisional line shapes.

19.1.1 Voigt Line Shape

An atomic (or molecular or ionic) line has an intrinsic

Lorentzian shape that reﬂects the Fourier transform of

the exponentially decaying spontaneous emission. For

a spontaneous decay rate Γ , the fullwidth at half max-

imum (FWHM)is∆ω = Γ . Due to the Maxwellian

distribution of atomic velocities v in a thermal va-

Part B 19

280 Part B Atoms

por, and the Doppler shift ∆ω

= v/λ, the resonance

frequencies of atoms in the laboratory frame have

a Gaussian distribution. The Doppler width of this

distribution is approximately ω

u/c,whereω

is the

resonant frequency and u is the mean thermal vel-

ocity. The full line shape is thus a convolution of

the natural Lorentzian with the thermal Gaussian;

a Voigt proﬁle. In the presence of collisions, the

line from each atom broadens, shifts and becomes

asymmetric, and this is normally convoluted with the

Gaussian velocity distribution to obtain the complete

line shape. Collisions may also cause weak or dipole-

forbidden transitions to become stronger as well as

broader.

19.1.2 Interaction Potentials

Theories of collisional line shapes consider an en-

semble average of collisional interactions. For atomic

gases, the description of each individual collision or

interaction generally starts with a molecular model for

a pair of interacting atoms, since the Born–Oppenheimer

approximation is appropriate for thermal atomic colli-

sions (electron velocities  nuclear velocities) [19.23].

The radiative transition then occurs between adiabatic

02 4 6810

Energy (eV)

R(Å)

–V

Fig. 19.1 Diagrammatic representation of a free–free molecular

radiative transition and the classical Franck–Condon principle. Adia-

batic potentials (V

and V

), the difference potential equal to the

classical transition frequency (dashed), nuclear kinetic energies and

wave functions Φ

(R) and Φ

(R) are indicated. The position R

is the classical radiation position for the initial (E

)andﬁnal(E

)

energies shown

electronic molecular states that separate to the atomic

states of the transition under consideration. The nota-

tion V

(R) denotes the electronic energy of the upper

state, and V

(R) the lower state, where R is the inter-

nuclear separation, and the total atomic energies are E

and E

with E

− E

= hω

(Fig. 19.1). The next sim-

pliﬁcation is to assume that the atom-pair statistically

branch into adiabatic motion along each of the molecu-

lar states associated with the initial atomic state, and

radiation to each of the ﬁnal states is summed inde-

pendently, assuming they are also completely adiabatic.

This ignores an inevitable nonadiabatic coupling be-

tween the states as their energy separation decreases to

zero at large R, but as discussed below this has been

shown to have a very minor effect on line broadening.

The single-collision problem then reduces to calculat-

ing the spectrum of a molecular transition between

each upper and lower pair of adiabatic states, for each

initial state of internuclear motion and all possible ﬁ-

nal motions, and summing these weighted by the rate

of initial collisional motions. However, such calcula-

tions are only necessary to elucidate particular quantum

features, because the classical oscillator approxima-

tion, impact approximation, quasistatic approximation,

and classical Franck–Condon principle provide major

conceptual and calculational simpliﬁcations. These are

presented in the context of free–free transitions in the

following sections.

19.1.3 Classical Oscillator Approximation

Consider a free–free molecular radiative transition of

energy

ω = E

− E

between upper (u) and lower (g)

states of total energy E

and E

, as shown in Fig. 19.1.

Each elastic scattering state is the product of an elec-

tronic adiabatic state φ(r, R) and a nuclear motion state

Φ(R) of the molecule. The electronic states u and g have

effective potentials

= V

+l(l +1)/R

, (19.1)

where q = uorg.ExamplesofsuchV(R) and Φ(R)

for the case of l = 0areshowninFig.19.1. The elec-

tric dipole radiation operator that couples these states is

normally a weak perturbation that does not alter these

potentials or wave functions. For deﬁniteness assume

the atom is initially in the upper state, and that an atom

pair approach with kinetic energy T

and separate with

, as indicated. The intensity I, or transition prob-

ability, is proportional to the squared matrix element of

the dipole operator er = eΣ

between initial and ﬁnal

Part B 19.1

Line Shapes and Radiation Transfer 19.1 Collisional Line Shapes 281

states:

I ∝





rΦ

(R)

∗

(r, R)

∗

erφ

(r, R)Φ

(R)



= µ





RΦ

(R)

∗

(R)



, (19.2)

where the last factor is the Franck–Condon factor, µ is

the dipole moment, here assumed to be the atomic

value independent of R,andd

R = 4πR

dR for di-

atomics. The Φ(R) can be represented, in the WKB

approximation, as

RΦ

= k

−1/2

exp





−i

∞



dRk

(R)





, (19.3)

where





−V

(R)





1/2

/ (19.4)

and M is the reduced mass of the atom pair (

/2M

is the kinetic energy in Fig. 19.1).

The integrand in the second line of (19.2) then con-

tains exp[i



R(k

−k

)]. If one multiplies k

−k

)/(k

), the exponent becomes

2M(ω −ω

)

and if one further deﬁnes a variable t(R) by

dt = dR/v(R) with v(R) =

)/2M, the expo-

nent becomes i





)dt



−iωt,where

(R) =[V

(R) −V

(R)]/ (19.5)

and R(t) is the classical trajectory in the average po-

tential. Thus, the Franck–Condon factor reduces to the

squared Fourier transform of exp[i





)dt



],where

(R) = V

(R) −V

(R) (Fig. 19.1). This is the classi-

cal oscillator approximation (COA), in which the atoms

are considered to be a classical oscillator at ω

(R) dur-

ing the collisional interaction, and R(t) is the classical

orbit.

The

√

k in the denominator of the WKB wave func-

tionleadsto1/v(R) in the nuclear density at R,again

corresponding to classical motion. This COA is used as

the starting point in many line shape calculations. It is

apparent that the COA breaks down when the initial- and

ﬁnal-state orbits are signiﬁcantly different, or when the

Born–Oppenheimer or WKB approximation is not valid.

However, the COA is normally a very good approxima-

tion for thermal atomic vapors when h(ω −ω

)  k

T ,

since most orbits are then straight lines within the R re-

gion that yields the observed radiation.

One further conceptual and mathematical simpliﬁca-

tion is to subtract ω

from ω



) and from ω by deﬁning

∆ω

= ω

−ω

and ∆ω = ω −ω

. The Franck–Condon

factor in (19.2) then becomes

I(ω) =



4µ







dt exp







∆ω



)dt



−i∆ωt







(19.6)

where the prefactors that yield the correct intensity are

explained in [19.12, 13] (Sect. 10.6). The average is, in

general, over a sequence of collisions leading to an

integral over collision velocity and impact parameter,

weighted by the rate of occurrence.

In the next section, the character of the broadened

line core is obtained by the traditional method of evalu-

ating (19.6) using the impact approximation.

19.1.4 Impact Approximation

Atomic collisions in thermal vapors typically occur in

a time interval τ

< 1 ps, whereas the time between sig-

niﬁcant collisions is much longer at vapor pressures

below a few atmospheres. Thus, one can consider the

atom as radiating its unperturbed frequency ω

most

of the time, but occasionally suffering a rapid, strong

perturbation. If the duration of a collision is τ

,then

φ(t) =



∆ω



)dt



in the exponent of (19.6) under-

goes a net phase shift ∆θ in a time τ

, and is constant

between collisions. If ∆ω  1/τ

, then one can ap-

proximate this phase shift as instantaneous; this is the

essential assumption of the impact approximation.When

the factor exp[i∆θ(b,v)], which then occurs in (19.6)is

averaged over collision rates with velocity v and im-

pact parameter b this leads to (γ

−iδ)t in place of φ(t),

where γ

and δ are given by [19.15]

∞



n(v)v dv

∞



2πb[1−cos ∆θ(b,v)]db ,

δ =

∞



n(v)v dv

∞



2πb sin ∆θ(b,v)db .

Including the spontaneous emission dipole decay rate

exp(−Γt/2), the Fourier integral produces the normal-

Part B 19.1

282 Part B Atoms

ized Lorentzian line shape

I(ω) =

(Γ/2+γ

)



(Γ/2+γ

)

+(∆ω −δ)



(19.7)

The full line shape is then a Voigt proﬁle with

a Lorentzian-component half width Γ +2γ

and a shift

of δ from ω

. The shift can be understood as the

perturbed oscillator frequency advancing relative to

by the average value of sin ∆θ, or alternately

as indicating the direction of the frequency shift of

−V

. The Lorentz approximation corresponds to tak-

ing 1 −cos ∆θ = 1 or 0, with the former representing

a collision. This gives γ = 1/∆t

,where∆t

is the

average time between strong collisions.

The b for which ∆θ(b, v) = 1iscalledtheWeiss-

kopf radius (R

). The collisional line width 2γ

can be

thought of as a collisional rate nvQ,whereQ =πR

The importance of R

is that γ

and δ result primarily

from atomic interactions in the region R ≈ R

.Since

the size of an electronic wave function increases with in-

creasing excitation energy, |∆V(R)|=|V(R) −V(∞)|is

normally larger for the upper state of a transition. (High

Rydberg states can be an exception.) Since ∆θ =1for

b = R

and ∆θ ≈(∆V/ )τ

,then∆V(R

) ≈ /τ

.For

a typical τ

= 1 ps this implies ∆V(R

)/hc ≈ 5cm

−1

which is a relatively weak, long range interaction. Thus,

reasonable approximations to the necessary V(R) can of-

ten be obtained from atomic perturbation theory. Simple

expressions for γ

and δ are obtained for V

−V

= A/R

with n > 2 [19.24]. The case of n = 3 corresponds to

the resonant interaction between identical atoms, while

n = 6 corresponds to the van der Waals interaction,

which holds at long range for foreign gas interactions.

This result for one pair of adiabatic states must be av-

eraged over all of the pairs that separate to each atomic

state.

Higher order approximations obtain an asymmetric

line, rather than a pure Lorentzian, due to the asymme-

try of ∆V(R) [19.24]. In the line core this asymmetry

appears as a multiplicative correction 1+∆ω/D,where

|D|≈1/τ

[19.25], and with increasing detuning it in-

creases until at |∆ω| > 1/τ

the static wing approaches

the quasistatic limit described in Sect. 19.1.7.If∆V(R)

changes monotonically with decreasing R, only one side

of the line has a static contribution and the other, anti-

static side falls off exponentially at |∆ω| >τ

[19.11].

However, this situation is seldom observed, as more

than one difference potential generally contributes and

there is usually a static contribution on both sides of the

line. Another factor that produces a small divergence

from the Voigt proﬁle is the velocity dependence of the

shift and width. When combined with the higher vel-

ocities of atoms emitting or absorbing in the Doppler

wing, this produces an asymmetry in the Doppler

wings [19.26,27].

19.1.5 Examples: Line Core

It is possible to deconvolve a Voigt line shape to separate

the Doppler and Lorentzian components, and thereby de-

duce broadenings of considerably less than the Doppler

width ([19.28] and references therein). However, the

broadening is most easily observed at perturber densities

where the collisional broadening exceeds the Doppler

broadening. Such a pressure-dependent line shape is

shown in Fig. 19.2, for a range of perturber density n

such that the broadening exceeds the Doppler width

and hyperﬁne structure, yet the line core is described

by the impact theory [19.29]. In Fig. 19.2 the normal-

ized line intensity has been divided by n

; as the line

wings are proportional to n

they are constant in such

a plot, while the line center broadens and shifts with

increasing n

. For this case of fairly heavy atoms,

(2πcτ

)

−1

= ∆k

corresponds to ≈ 0.5cm

−1

,andthe

line becomes asymmetric and non-Lorentzian beyond

≈ 1cm

−1

(Fig. 19.3a); the red wing intensity falls more

slowly and the blue wing more rapidly than (∆ω)

−2

This behavior is typical for most heavy perturbers, and is

–18

–19

–20

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

∆k (cm

–1

)

I(λ)/N兰 I(λ) dλ(cm

/Å)

Fig. 19.2 Normalized line shape of the Rb 5P

3/2

−5S

1/2

transition broadened by Kr, for Kr densities of 4.5, 9, 18,

and 27 × 10

−3

(top to bottom). Hyperﬁne structure and

instrumental resolution cause ≈ 0.3cm

−1

of the broadening

shown

Part B 19.1

Line Shapes and Radiation Transfer 19.1 Collisional Line Shapes 283

–20

–21

–22

–23

–24

–25

–1000 –100 –10 –1 1 10 100 1000

I(λ)/N兰 I(λ) dλ(m

/Å)

R(a

)

k–k

A–X

B–X

Fig. 19.3a,b Normalized intensity in the wings of the Rb

3/2

−5S

1/2

transition broadened by Kr, in frequency

units of k = 1/λ. The measured spectrum in

(a) is from

[19.29–31]. The solid line is at 310 K and the dashed line

at 540 K. The difference potentials corresponding to the A,

B and X states of Rb −Kr, taken from [19.32], are shown

(b)

attributed to a long range attractive V

which dominates

−V

For the lowest n

shown in Fig. 19.2, a convolution

with the Doppler, hyperﬁne and instrumental broaden-

ings showed that the line is essentially a symmetric

Lorentzian for |∆k| < ∆k

[19.29]. However, at the

highest density, the half-height point is beginning to fall

outside of ∆k

; the impact approximation is marginally

valid for describing the half width of the line at this

density. Most early experiments were done at more

than 10 times this density [19.15]; most of the line-

core was at |∆k| > ∆k

and describable by the static

theory (Sect. 19.1.7) rather than the impact approxi-

mation. The impact approximation was also not valid

under these conditions because collisions overlap in

time. These very broad lines are well represented by the

multiple-perturber, static theories that assume scalarly

additive perturber interactions [19.5, 8]. This transition

between an impact and quasistatic line core, and to mul-

tiple perturber interactions, occurs at lower pressures

for transitions to higher states, as the interactions have

a longer range. In addition, nearby intensity peaks or

satellites often occur, and strongly affect the line as

pressure increases. An example calculation, based on

an interpretation of measured spectra [19.16], is shown

in Fig. 19.4. This shows how a line with a satellite fea-

ture progressively broadens and ﬁnally blends with the

satellite as n

increases.

With the advent of saturated-absorption (Doppler

free) spectroscopy, collisional line broadening can be

measured at much lower densities, where 2γ

 ∆ω

In principle, this can allow measurement of line broaden-

ings and shifts, although with a complication that affects

the line shape; the same collisions that produce optical

phase shifts also change the atomic velocity. These vel-

ocity changes have a minor effect outside the Doppler

envelope where high pressure measurements are nor-

mally made, but they are quite important in saturated

absorption line shapes. This affects primarily the low in-

tensity wings of the line, so it does not preventmeasuring

the broadening and shift of the nearly Lorentzian core.

–46 –38 –30 –22 –14 –6 +2

Intensity

Frequency (cm

–1

)

1 × 10

2.5 × 10

5 × 10

1 × 10

1.5 × 10

2 × 10

Fig. 19.4 Calculated line shapes of the Cs(9P

1/2

−6S

1/2

)

line broadened by Xe at the densities indicated (from

[19.16]). The assumed interaction is based on measured line

shapes, but data corresponding to the calculated conditions

are not available

Part B 19.1

284 Part B Atoms

Details can be found in [19.33] and references therein.

Two-photon absorption yields Doppler free lines that

are not affected by velocity changing collisions. This

provides the most exacting test of line shapes. These

narrow lines are precisely Lorentzian, with a broaden-

ing that reﬂects the upper state interaction since this is

usually much stronger than that of the ground state. The

technique has been used to measure the broadening of

two photon transitions to many excited states [19.34,35].

19.1.6 ∆ and γ

Characteristics

Since 1970, neutral broadening has generally been

measured in the |∆ω| < 1/τ

region where the

impact approximation and (19.7) is valid. Measure-

ments through 1982 are tabulated in [19.16], and

through 1992 in [19.36]. More recent measurements

are tabulated in the NIST Reference Data bibliog-

raphy, which is accessible (free) at the web site

http://physics.nist.gov/PhysRefData. These involve pri-

marily metal vapor resonance lines broadened by noble

gases. For collisions with the heavier (more polariz-

able) gases, the sizes of these measured broadening

rate coefﬁcients generally fall within a factor of 10

range, and approximately ﬁt the prediction of (19.7)

with ∆V

(R) −∆V

(R) = C

−6

[19.37]andC

given

by a simple effective quantum number formula. This

occurs because the potentials are fairly close to van

der Waals for R ≥ R

and the broadening is insensi-

tive to details of the potentials at R < R

since cos ∆θ

in (19.7)averagesto≈ 0 for the closer (strong) colli-

sions. It also occurs because the full quantum solution

for broadening by a van der Waals interaction, with Zee-

man degeneracies, yields nearly the same result as the

above single-level theory with an average C

[19.38].

For the heavy, more polarizable perturbers, the ex-

cited state interactions are attractive and red shifts occur,

but the measured shifts have a very poor correlation

with the van der Waals prediction. As b decreases and

∆θ(b,v)increases, sin ∆θ oscillates and major cancel-

lations occur in the average of sin ∆θ in (19.7). The

shift is therefore only a fraction of the broadening and

is very sensitive to the interaction throughout the region

R ≈ R

. This often differs considerably from the van

der Waals form, at the typical ≈ 5cm

−1

interaction en-

ergy at R

. The shape of the red wing just beyond ∆ω

also frequently fails to ﬁt that expected for a van der

Waals interaction [19.29]. Thus, the often good agree-

ment of γ

with van der Waals numbers is not a reliable

indicator of the actual V(R) in the relevant R region,

even for heavy noble gas perturbers.

For He and sometimes Ne perturbers, a repulsive

interaction due to charge overlap normally dominates

at R ≈ R

, causing a blue shift as well as a larger

broadening than the van der Waals prediction.

19.1.7 Quasistatic Approximation

The impact approximation is valid for |∆ω|1/τ

where the 1/τ

is typically 1–10 cm

−1

.Forlarger|∆ω|

the line shape becomes asymmetric, with higher in-

tensity on the wing corresponding to the long-range

(R) −V

(R). At large detunings where ∆ω  1/τ

a major simpliﬁcation occurs. The COA describes the

interacting atom pair as an oscillator of frequency

(R) =



(R) −V

(R)



/ when at separation R.

Since R is time dependent during the classical orbit,

is as well and the Fourier spectrum is broadened

relative to the simple distribution of ω

(t) that occurs

during the orbit. But if the motion is sufﬁciently slow,

the intensity at ω reduces to the probability of ﬁnding

the atom pair at the appropriate R(ω = ω

). The spec-

trum then reduces, at low pressure, to the probability

distribution of pair separations R, subject to (19.5)be-

tween R and ω. This is the binary quasistatic, static,

or statistical spectrum, which accurately describes most

line wings for |∆ω| > 1/τ

. When the pressure is large

enough to yield a signiﬁcant probability of one perturber

at R ≤ R

, multiple-perturber interactions must also be

considered as in [19.5].

This intuitive deduction of the statistical spectrum

from the COA [19.4] can also be obtained more for-

mally from (19.6) by expanding the exponent about

the time during a collision when ω

(t) = ω. Alter-

natively, it follows directly from (19.2)usingWKB

wave functions to evaluate free–free molecular Franck–

Condon factors [19.6, 7]. This result is identical to the

classical Franck–Condon principle (CFCP), originally

established in the context of bound–bound molecular

radiation.

The CFCP yields important insights for all molecular

radiation. Again consider (19.2) with the substitution of

the WKB wave functions φ

, given below it. Examples

of φ

and φ

are given in Fig. 19.1. For large detun-

ings ω −ω

,asshowninFig.19.1, the integrand on the

right side oscillates rapidly everywhere except at the

stationary phase point R

,wherek

= k

. As a conse-

quence, the dominant contribution to the integral occurs

at R

and one can consider the transition to be localized

at R

.Sincek

) = k

), T

) = T

also holds, and as can be seen in Fig. 19.1 it then

follows that

ω = V

) −V

). Thus, radiation at

Part B 19.1