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

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Photon–Atom Interactions: Low Energy 61.2 Experimental Methods 909

obtained by ion or mass spectrometry from

tot

= σ(A

) +σ( A

) +···. (61.35)

Generally, the charge states can be correlated with the

various initial photoionization processes if allowance is

made for Auger transitions and the ﬂuorescence yield

upon exceeding the binding energies of core levels.

If the charge states are not distinguished, as in a total

ion yield measurement, σ

tot

is obtained directly. Simi-

larly, a direct measurement of the global quantity σ

tot

is obtained by the total electron yield, although care

must be exercised to avoid discrimination by angular

distribution effects. At photon energies below about

1 keV, ionization, absorption, and attenuation are vir-

tually equivalent, and σ

tot

can also be determined in an

ion chamber setup [61.36] or in a photoabsorption mea-

surement in which the number of photons ∆ N absorbed

in a source of length d and having an atom density n is

given by

∆N = N(ph)[1 −exp(−σ

tot

nd)] (61.36)

with N(ph) being the ﬂux of incident photons. As a rule,

in all experiments employing the relation (61.5), the

total, partial, or differential cross sections are determined

on a relative rather than an absolute scale because it is

very difﬁcult to know accurately such factors as the

Photon energy, hv

Electron kinetic energy

CIS (E

= hv– = const.)

CKE (

= hv– E

= const.)

PES (

hv = E

+ = const.)

Fig. 61.4 Energy relationship among the three different

operational modes of the technique of photoelectron spec-

trometry. E

is the binding energy of the level

geometry of the source volume and the number density.

However, once a single absolute value of σ

tot

or any σ

is available, all relative values of the other quantities can

be converted to absolute values.

An electron spectrometric experiment can be car-

ried out in three different operational modes, as deﬁned

in Fig. 61.4. In the most conventional mode, PES,the

photon energy is ﬁxed, and a scan of the electron ki-

netic energy reveals all the electron-emission processes

possible and yields their properties. The CIS (constant

ionic state) mode is especially suited to follow con-

tinuously a selected process as a function of photon

energy by locking onto a given state E

−E

(denoted

by E

) which requires a strict synchronization of the

photon energy (hν) and electron kinetic energy (ε) dur-

ing a scan. This mode is particularly advantageous to

elucidate resonance features, such as autoionization res-

onances. Finally, a CKE (constant kinetic energy) scan

allows one to access various processes sequentially or,

most importantly, follow a process of ﬁxed energy, such

as an Auger transition, as a function of photon energy.

This description also includes the technique of zero-

(

)

3s3p

np(

)

n = 100

n = 6

n = 5

PES

⁄

n = 4

CIS– σ

tot

3s np

26.62 eV

Fig. 61.5 Connection between the PES and CIS techniques

as illustrated by the 3s → np autoionization resonances in

argon

Part E 61.2

910 Part E Scattering Experiment

kinetic-energy measurements. Most frequently, the PES

and CIS modes are employed, and Fig. 61.5 gives a self-

explanatory example of an actual experiment directed at

the characterization of the argon 3s → np autoionizing

resonances. It should be stressed that the cross section

tot

can be partitioned into its components by CIS scans

that differentiate between the 3p

1/2

and 3p

3/2

doublet

states.

Energy analysis can be performed either by electro-

static energy analyzers or by time-of-ﬂight techniques.

The latter is well-suited to those electrons which have

very low kinetic energy, including threshold electrons

with ε ≈0 eV. Of the electrostatic energy analyzers now

in use, two designs are prevalent, the cylindrical mir-

ror analyzer (CMA) and the hemispherical analyzer,

where the latter readily lends itself to the application

of multichannel detectors.

61.2.3 Resolution and Natural Width

The details that can be gleaned from an experiment us-

ing photons depend on the resolution achievable with

the particular photon source and spectrometry used, the

particular excitation or analysis modes and the target

conditions chosen, and, ultimately, on the natural width

of either the levels or transitions examined as well as

any ﬁne structure present. Generally, the instrumental

and operational resolution should approach, but need

not exceed by much, the natural width inherent in the

photoprocess under scrutiny. The demands are most se-

vere for processes involving outer levels because of their

typically very narrow widths, and are relatively mild for

processes involving inner levels [61.5,35]. It is desirable

that in the former case the resolving power (the inverse

of the resolution) of the instrument exceed 10

, while in

the latter case 10

may sufﬁce. If the target atoms move

randomly, a resolution limit is set by the thermal motion,

namely

∆ε = 0.723 (εT/M)

1/2

(meV), (61.37)

where, in an experiment involving photoelectron spec-

trometry, ε is the kinetic energy of the photoelectron in

eV , T the temperature in K, and M the mass in a.u.

of the target atom. This contribution can be limited

in ﬁrst order by employing a suitably directed atomic

beam.

The experimental peak-width and shape generally

contain the natural width; the extent to which instru-

mental factors enter depends on the speciﬁc experiment.

In a measurement of the total or partial photoionization

cross section in which either the ﬂuorescence photons,

the electrons, or the ions are monitored, the resolution of

the photon source (often called the bandpass) is the only

instrumental contributor. This applies speciﬁcally to the

CIS mode of electron spectrometry, in which features

are scanned in photon energy and the electron serves

solely as a monitor. However, in such a CIS study, or

a corresponding ﬂuorescence study, the resolution of

the electron or ﬂuorescence spectrometer must be ade-

quate to be able to distinguish adjacent processes. For

the example of Fig. 61.5, the 3p

3/2

and 3p

1/2

levels of Ar

need to be separated if the partial photoionization cross

sections are to be determined across the resonances. In

such a case of more than one open ionization chan-

nel, the natural widths of the features will be identical

in all channels [61.4], but the shapes may be different.

In emission processes subsequent to initial photoioniza-

tion, namely electron (Auger) decay or photon (X-ray)

emission, the resolution of only the spectrometer per-

forming the detection counts on the instrumental side.

In the PES mode (Fig. 61.4) the observed lines con-

tain contributions from all sources, the photon source or

photon monochromator, the electron analyzer, thermal

broadening, and the natural level width. In the special

case of photoprocesses near inner thresholds, the post-

collision interaction inﬂuences the position and shape of

photoelectron and Auger lines (Chapt. 62).

Excluding threshold regions and the resonant Ra-

man effect, the line proﬁle observed in the various

experiments is given by the Voigt function

V(ω, ω

) =



L(ω



−ω)G(ω



−ω

)dω



. (61.38)

In this the Lorentzian function L represents the natural

level or transition proﬁle, and the Gaussian function G

is representative of the window functions of the dis-

persive apparatus. Although the integral representing

the Voigt proﬁle has no analytic form, it can be repre-

sented for practical purposes by the analytic Pearson-7

function [61.42]

(ε) = A



1 +

(ε −ε

)



−C

, (61.39)

where A is the peak height, ε

the peak position, B the

nominal half-width-half-maximum of the peak, and C

the shape of the peak. In the limit in which C = 1, this

function is identically a Lorentzian; in the limit C →∞,

the function is essentially Gaussian. Use of this function

allows one to ﬁt the resulting photoelectron spectrum

using standard numerical techniques. If the width of

Part E 61.2

Photon–Atom Interactions: Low Energy 61.3 Additional Considerations 911

the feature is the only quantity of interest, the simple

approximate expression

= 1 −





(61.40)

which relates the Voigt width Γ

with the Lorentzian

width Γ

and the Gaussian width Γ

can be used

to determine either Γ

or Γ

from the measured

[61.5]. Often the observed feature exhibits a dis-

persive shape given by the Fano or Shore proﬁle. In

this instance the instrumental function must be con-

voluted with the resonance proﬁles given by (61.23)

or (61.26) in order to ﬁt the data and to extract the

parameters [61.43].

For the special case of resonant Auger decay in

which the bandpass of the exciting radiation is very

narrow compared with the natural width of the excited

state, the experimental linewidth is governed by the

width of the exciting radiation, and will be more narrow

than the natural width of the line. The resulting line-

shape in this resonant Raman effect is then the simple

product [61.24]



(ω, ω

) = L(ω)G(ω −ω

), (61.41)

where L(ω) is the line proﬁle as determined by

the natural width, typically Lorentzian for resonant

Auger decay, and G(ω −ω

) is the, usually, Gaus-

sian function representing the bandpass of the exciting

radiation.

61.3 Additional Considerations

Although low-energy photon interactions are well de-

scribed nonrelativistically in the dipole approximation,

relativistic and higher multipole effects which become

increasingly important at higher energies cannot be

ignored even below 1 keV. Spin-orbit effects [61.44]

and relativistic effects [61.45] are of special signiﬁ-

cance even at low energies in Cooper minima, where

one of the transition matrix elements becomes zero.

Moreover, the use of intermediate coupling, which

includes both the spin-orbit and electrostatic interac-

tions [61.46], is required in open-shell systems, as

exempliﬁed for the halogen atoms and atomic oxy-

gen [61.47]. Level energies of heavy elements also

require a relativistic treatment [61.48](Chapt.22), and

it is natural to employ relativistic formulations for

calculating the spin parameters appearing in photoion-

ization [61.49].

Although low energy photon scattering is domi-

nated by the dipole contribution, experiments and theory

have shown higher multipole effects to be present

at hν<1 keV [61.50–57]. As a result, measurements

that take the dipole formulations as a basis (61.5)

and (61.6) can incur a discernible error in both the

differential cross sections dσ

/dΩ and the partial

cross sections σ

. A more accurate determination of

dσ

/dΩ can be made [61.51,55, 56] on the basis of the

equation

dσ

dΩ



4π



1 +βP

(cos θ)

+(δ +γ cos

θ) sin θ cos φ



, (61.42)

using linearly polarized radiation, where P

(cos θ) is

the second-order Legendre polynomial, θ is the angle

between the electron emission direction and the elec-

tric vector, φ is the angle between the electron and

photon directions, β is the angular distribution param-

eter related to B

, and the parameters δ and γ are

related to B

and B

in (61.4). Figure 61.6 shows

the geometry for the relationship between the pho-

toelectron momentum vector, the polarization vector,

and the photon propagation vector as used in (61.42).

It serves as the template for the arrangement and

motion of the electron detector, or, for added efﬁ-

ciency and accuracy, several detectors [61.51–53, 56].

The parameters β, δ,andγ have been calculated

for most subshells of the noble gases for photo-

electron energies between 100 eV and 5 keV [61.55]

Polarization (E)

Photon (k)

Electron (p)

Fig. 61.6 Geometry of the relationship between the photo-

electron momentum p, the polarization vector E,andthe

photon momentum k. (Courtesy of O. Hemmers)

Part E 61.3

912 Part E Scattering Experiment

and 20 eV to 5 keV [61.57]. Generally, the electric

quadrupole (E2) and magnetic dipole (M1) photoion-

ization channels are the most important beyond the

dipole (E1) photoionization channel. Similar to the

dipole angular distribution parameter β, the parame-

ter γ may also be subject to interchannel coupling

and relativistic effects, as demonstrated for xenon 5s

photoionization [61.51].

References

61.1 W. J. Veigele: At. Data Nucl. Data Tables 5, 51 (1973)

61.2 J. H. Hubbell, M. J. Berger: Photon cross section,

attenuation coefﬁcients, and energy absorption

coefﬁcients. In: Engineering Compendium on Ra-

diation Shielding, Vol. 1, ed. by R. G. Jaeger

(Springer, Berlin 1968) p. 167

61.3 U.Fano,J.W.Cooper:Rev.Mod.Phys.40, 441 (1963)

61.4 A. F. Starace: Theory of photoionization. In: Hand-

buch der Physik, Vol. XXI, ed. by S. Flügge,

W. Mehlhorn (Springer, Berlin 1982) p. 1

61.5 M. O. Krause: Electron spectrometry of atoms and

molecules. In: Synchrotron Radiation Research,ed.

by H. Winick, S. Doniach (Plenum, New York 1980)

Chap. 5

61.6 U. Fano, D. Dill: Phys. Rev. A 6, 185 (1972)

61.7 S. B. Whitﬁeld, K. Kehoe, M. O. Krause, C. D. Cald-

well: Phys. Rev. Lett. 84, 4818 (2000)

61.8 J. Schirmer, L. S. Cederbaum, J. Kiessling: Phys. Rev.

A 22, 2696 (1990)

61.9 V. W. Hughes, R. L. Long Jr., M. S. Lubell, M. Posner,

W. Raith: Phys. Rev. A 5, 195 (1972)

61.10 U. Fano: Phys. Rev. 178, 131 (1969)

61.11 N. A. Cherepkov: Adv. At. Mol. Phys. 19, 395 (1983)

61.12 U. Fano, J. H. Macek: Rev. Mod. Phys. 45, 553 (1973)

61.13 C. H. Greene, R. N. Zare: Phys. Rev. A 25, 2031 (1982)

61.14 W. Mehlhorn: Auger-electron spectrometry of core

levels in atoms. In: Atomic Inner-Shell Physics,ed.

by B. Crasemann (Plenum, New York 1985) Chap. 4

61.15 U. Fano: Phys. Rev. 124, 1866 (1961)

61.16 B. Shore: J. Opt. Soc. Am. 57, 881 (1967)

61.17 F. H. Mies: Phys. Rev. 175,164(1968)

61.18 J. P. Connerade, A. M. Lane: Rep. Prog. Phys. 51,

1439 (1988)

61.19 M. Martins: J. Phys. B 34, 1321 (2001)

61.20 S. B. Whitﬁeld, M. O. Krause, P. van der Meulen,

C. D. Caldwell: Phys. Rev. A 50, 1269 (1994)

61.21 T. W. Gorczyca, B.M. McLaughlin: J. Phys. B 33,859

(2000)

61.22 S. J. Schaphorst, S. B. Whitﬁeld, H. P. Saha,

C. D. Caldwell, Y. Azuma: Phys. Rev. A 47, 3007 (1993)

61.23 C. E. Theodosiou, M. Inokuti, S. T. Manson: At. Data

Nucl. Data Tables 35,473(1986)

61.24 T. Åberg, G. Howat: Theory of the Auger effect. In:

Handbuch der Physik, Vol. XXI, ed. by S. Flügge,

W. Mehlhorn (Springer, Berlin 1982) p. 469

61.25 S. B. Whitﬁeld, J. Tulkki, T. Åberg: Phys. Rev. A 44,

R6985 (1994)

61.26 M. O. Krause: J. Phys. (Paris) 32, C4–67 (1971)

61.27 M. O. Krause, S. B. Whitﬁeld, C. D. Caldwell,

J. -Z. Wu, P. van der Meulen, C. A. de Lange,

R. W. C. Hansen: J. Elec. Spectros. Rel. Phen. 58,

79 (1992)

61.28 R. L. Martin, D. A. Shirley: Phys. Rev. A 13,1475

(1976)

61.29 T. Åberg: Nucl. Instrum. Methods B87, 5 (1994)

61.30 E. P. Wigner: Phys. Rev. 73, 1002 (1948)

61.31 G. H. Wannier: Phys. Rev. 90,817(1953)

61.32 M. Y. Kuchiev, S. A. Sheinermann: Sov. Phys. JETP

63,986(1986)

61.33 U. Heinzmann: J. Phys. B 13, 4353 (1980)

61.34 V. Schmidt: Rep. Prog. Phys. 55, 1483 (1992)

61.35 U. Becker, D. A. Shirley (Eds.): VUV and Soft X-ray

Radiation Studies (Plenum, New York 1996)

61.36 J. A. R. Samson: Atomic photoionization. In: Hand-

buch der Physik, Vol. XXI, ed. by S. Flügge,

W. Mehlhorn (Springer, Berlin 1982) p. 123

61.37 B. Sonntag, P. Zimmermann: Rep. Prog. Phys. 55,

911 (1992)

61.38 C. A. de Lange: Adv. Chem. Phys. 117, 1 (2001)

61.39 H. Winick, G. Brown: Nucl. Instrum. Methods 195,1

(1987)

61.40 E. E. Koch: Handbook of Synchrotron Radiation

(North-Holland, Amsterdam 1983)

61.41 J. A. Clarke: The Science, Technology of Undulators,

Wigglers (Oxford Univ. Press, Oxford 2004)

61.42 S. B. Whitﬁeld, C. D. Caldwell, D. X. Huang,

M. O. Krause: J. Phys. B 25, 4755 (1992)

61.43 J. Jiménez-Mier: J. Quant. Spectros. Radiat. Trans-

fer 51, 741 (1994)

61.44 U. Fano: Phys. Rev. 178, 131 (1969)

61.45 S. T. Manson, Z. J. Lee, R. H. Pratt, I. B. Goldberg,

B. R. Tambe, A. Ron: Phys. Rev. A 28, 2885 (1983)

61.46 R. D. Cowan: The Theory of Atomic Structure and

Spectra (Univ. of California Press, Berkeley 1981)

61.47 C. D. Caldwell, M. O. Krause: Rad. Phys. Chem. 70,

43 (2004)

61.48 I. P. Grant: Aust. J. Phys. 39,649(1986)

61.49 K. N. Huang, W. R. Johnson, K. T. Cheng: At. Data

Nucl. Data Tables 26, 33 (1981)

61.50 H. K. Tseng, R. H. Pratt, S. Yu, A. Ron: Phys. Rev. A

17, 1060 (1978)

61.51 O. Hemmers, R. Guillemin, D. W. Lindle: Rad. Phys.

Chem. 70, 123 (2004)

61.52 N. L. S. Martin, D. B. Thompson, R. P. Bauman,

C. D. Caldwell, M. O. Krause, S. P. Frigo, M. Wilson:

Rev. Lett. 81, 1199 (1998)

Part E 61

Photon–Atom Interactions: Low Energy References 913

61.53 R. W. Dunford, E. P. Kanter, B. Krässig, S.H. South-

worth, L. Young: Rad. Phys. Chem. 70, 149 (2004)

61.54 M. O. Krause: Phys. Rev. 177, 151 (1969)

61.55 J. W. Cooper: Phys. Rev. A 47, 1841 (1993)

61.56 P. S. Shaw, U. Arp, S. H. Southworth: Phys. Rev. A

54, 1463 (1996)

61.57 A. Derevianko, W. R. Johnson, K. T. Cheng: At. Data

Nucl. Data Tables 73, 153 (1999)

Part E 61

915

Photon–Atom

62. Photon–Atom Interactions:

Intermediate Energies

The main photon-atom interaction processes in the

energy range from ≈ 1keV to ≈ 1MeV are outlined.

The atomic response to inelastic photon scattering

is discussed; essential aspects of radiative and

radiationless transitions are described, including

hole-state widths and ﬂuorescence yields, in

the two-step approximation which applies well

above threshold where atomic de-excitation is

nearly independent of the excitation process.

Multi-electron photoexcitation that transcends

the independent-electron model is brieﬂy

considered.

Threshold phenomena that arise when the

two-step model breaks down, so that excitation

and de-excitation must be treated as a single

second-order quantum process are described.

The time-independent resonant scattering

approach that leads to a uniﬁed description of

photoexcitation, ionization, and Auger electron or

X-ray emission is outlined. In terms of this theory,

Raman processes and post-collision interaction

are described.

62.1 Overview............................................. 915

62.1.1 Photon-Atom Processes ............. 915

62.2 Elastic Photon-Atom Scattering ............ 916

62.2.1 Rayleigh Scattering.................... 916

62.2.2 Nuclear Scattering ..................... 917

62.3 Inelastic Photon-Atom Interactions....... 918

62.3.1 Photoionization ........................ 918

62.3.2 Compton Scattering ................... 919

62.4 Atomic Response to Inelastic

Photon-Atom Interactions.................... 919

62.4.1 Auger Transitions ...................... 919

62.4.2 X-Ray Emission ......................... 921

62.4.3 Widths and Fluorescence Yields .. 921

62.4.4 Multi-Electron Excitations .......... 921

62.4.5 Momentum Spectroscopy ........... 922

62.4.6 Ultrashort Light Pulses ............... 922

62.4.7 Nondipolar Interactions ............. 923

62.5 Threshold Phenomena ......................... 923

62.5.1 Raman Processes....................... 924

62.5.2 Post-Collision Interaction........... 925

References .................................................. 925

62.1 Overview

62.1.1 Photon-Atom Processes

It is convenient to divide photon-atom interactions into

elastic and inelastic processes, depending on whether or

not the photon energy changes in the c.m. frame. This

separation, while useful, is somewhat arbitrary [62.1]:

the radiative corrections of quantum electrodynamics

(Chapt. 27) and the possibility of emitting very soft pho-

tons, as well as target recoil, make all processes in fact

inelastic, while experimental comparison of incident and

scattered photon energies is limited by source bandwidth

and detector resolution.

Elastic photon-atom interactions (Chapt. 45) include

scattering from bound electrons (Rayleigh scattering)

with transition amplitude A

and nuclear Thomson

scattering, Delbrück scattering, and nuclear resonance

scattering, with amplitudes A

, A

,andA

,re-

spectively. These amplitudes are coherent, i. e., not

physically distinguishable: they add, retaining relative

phase information, in the total atom elastic scattering

amplitude A [62.1]. In the commonly used approxima-

tion, that is however “neither unique nor exact” [62.2],

we have

A = A

+ A

. (62.1)

The total atom elastic cross section is proportional to A

Inelastic photon-atom interactions are photoex-

citation (including ionization) (Chapt. 24), Compton

scattering, and pair production. The pair produc-

tion threshold at 2m

lies above the energy region

considered here (Chapt. 27). Inelastic scattering pro-

cesses are generally incoherent, e.g., ejected Compton

electrons originating from different orbitals can be dis-

tinguished [62.1]. However, coherence can enter even

between photoionization and Compton scattering in the

infrared-divergence limit [62.1, 3–5].

Part E 62

916 Part E Scattering Experiment

Fig. 62.1 Contributions of the photoeffect, elastic, and in-

elastic scattering to the total photon interaction cross section

of copper. (After [62.6], by permission, University of Cal-

ifornia, Lawrence Livermore National Laboratory; work

performed under the auspices of the U.S. Department of

Energy)

The cross sections for the main elastic and inelas-

tic photon-atom processes comprising the photoeffect

are illustrated in Fig. 62.1 for Cu [62.6]inwhich,how-

ever, resonant pieces of the inelastic (Compton) cross

section are omitted. Elastic exceeds inelastic scatter-

ing up to well above the K edge; this is true for all

atomic numbers. The total cross section is dominated

by the photoeffect up to photon energies that increase

with atomic number, from ≈ 6 keV for He [62.5]to

≈ 25 keV for C and ≈ 700 keV for U [62.6–8]. (See also

Sect. 24.2.2 and Sect. 92.3.2). For a summary of some

current developments in the ﬁeld see [62.9]

–1

–2

–1

11010

Cross section (b)

Photon energy (keV)

Total

Elastic

Photoelectric

Inelastic

62.2 Elastic Photon-Atom Scattering

62.2.1 Rayleigh Scattering

The term Rayleigh scattering is applied to elastic photon

scattering from bound atomic electrons (Sect. 68.6.1).

The Rayleigh scattering amplitude for neutral atoms

dominates the total atom elastic scattering amplitude,

at all angles, for photon energies below ≈ 1MeV and

dominates forward scattering at higher energies [62.1].

At angles other than 0°, Rayleigh scattering occurs

increasingly from inner atomic shells as the photon en-

ergy is raised. Depending on the energy, the process

can thus sensitively probe atomic structure in distinct

regions [62.1].

Rayleigh scattering has been reviewed by

Henke [62.10], Gavrila [62.11], Kissel and Pratt [62.1],

and Kane et al. [62.12]. The process has commonly

been estimated in a simple form-factor approxima-

tion, which represents scattering from a free charge

distribution; this underlies comprehensive tabulations

of elastic scattering cross sections and attenuation

coefﬁcients [62.7, 8]. The form factor relates cross sec-

tions to classical theory, and has been derived from

both nonrelativistic and relativistic quantum mechan-

ics. It is attractive because of calculational simpli-

city [62.2, 13].

The form factor f(q) for a spherically symmetric

charge number density ρ(r) is the Fourier transform of

the charge density:

f(q) ≡



ρ( r) e

iq·r

dr =

∞



ρ(r)

sin(qr)

dr ,

(62.2)

where

q = (2 ω/c) sin

(

θ/2

)

(62.3)

is the momentum transfer, and θ the scattering angle.

In elastic photon scattering, f(q) corrects the Thom-

son point charge scattering cross section for scattering

from an extended charge distribution. In the form fac-

tor approximation, the differential Rayleigh scattering

cross section (neglecting other coherent processes) for

unpolarized photons, averaged over ﬁnal polarization, is

dσ

dΩ



1 + cos



| f(q)|

, (62.4)

where r

= e





is the classical electron radius

and the terms multiplying | f(q)|

are the Thomson cross

section.

Even with modiﬁcations and corrections [62.1, 2],

the range of validity for the form factor approxima-

tion for elastic photon scattering is limited [62.13].

Predictions based on this approach are often wrong

in quantitative detail, although they lend insight into

Part E 62.2

Photon–Atom Interactions: Intermediate Energies 62.2 Elastic Photon-Atom Scattering 917

qualitative features of the Rayleigh amplitude [62.2]

Fig. 62.2.

The shortcomings of the form factor approximation

arise from the absence of binding effects in the Thomson

cross section, which pertains to photon scattering from

a free charged particle. Speciﬁcally, accurate Rayleigh

amplitudes must include the interaction of the electron

with the atomic potential in the initial, intermediate, and

ﬁnal states [62.1]. The form factor approximation fails,

especially for photons of low energy compared with the

electron binding energy, and for large scattering angles

(high momentum transfer), but O



Zα



discrepancies

remain at all energies and angles [62.2].

Substantial progress in the understanding of elas-

tic photon-atom scattering was made with the advent

of numerical calculations of the amplitudes based on

the second-order S-matrix of quantum electrodynamics

and relativistic wave functions [62.2,14]. These numer-

ical partial wave calculations include binding effects in

the intermediate state exactly to all orders. They consti-

tute an approach that leads to amplitudes with errors of

O(1%). However, some limitations do exist: (1) great

computer time requirements make it necessary to em-

ploy simpler (e.g., modiﬁed form factor) methods for

outer shell contributions, (2) only lowest-order terms

of the S-matrix expansion in powers of e are included,

and (3) most electron-electron correlation effects are not

taken into account [62.1].

Measurements of Rayleigh scattering in the en-

ergy range here considered are rather scarce [62.12].

Gamma rays from radioactive sources have been used

in most experiments, but discrimination from inelastic

contributions has been a problem [62.15]. Comparison

of measurements on medium- and high-Z targets with

theory, albeit incomplete, has however been encourag-

ing [62.1, 12, 15].

62.2.2 Nuclear Scattering

The nucleus contributes to the total atom elastic scatter-

ing amplitude through nuclear Thomson, Delbrück, and

nuclear resonance scattering. Only the ﬁrst two of these

processes are important in the energy range considered

here.

Nuclear Thomson Scattering

Photon scattering by the nuclear charge can be ap-

proximated by treating the nucleus as a rigid, spinless

sphere of charge Ze,massM, and radius r

.Inthesim-

plest point-charge form factor approximation, nuclear

Thomson scattering is included by replacing | f(q)| in

30°

60°

90°

120°

60°

90°

120°

30°0

150° 180° 150°

10 keV

100 200 300

3 keV

1 keV

30 keV

300 keV

100 keV

Fig. 62.2 Differential elastic photon-atom scattering cross

section (in b/atom) for neutral lead in form factor approx-

imation. The contribution of nuclear Thomson scattering

(see Sect. 62.2.2) to the amplitude is included (After [62.1])

(62.4)by



f(q) + Z

/M)



. If the ﬁnite nuclear ra-

dius r

is taken into account, the nuclear form factor is

approximately [62.1]

≈ Z



1 −

r





. (62.5)

The nuclear Thomson scattering cross section is

dσ

dΩ



(1 + cos

θ) (62.6)

if other coherent processes can be neglected. For some

energies, scattering angles, and target atoms, the nuclear

Thomson scattering contribution to elastic scattering can

dominate. A few measurements exist that conﬁrm the

preceding predictions with f

∼

[62.1].

Delbrück Scattering

Scattering of photons by virtual electron-positron pairs

created in the screened nuclear Coulomb potential is

called Delbrück scattering [62.1, 16]. The imaginary

part of the amplitude describes photon absorption due

Part E 62.2

918 Part E Scattering Experiment

to pair production; it vanishes for ω ≤ 2m

.The

real part of the amplitude, which adds coherently to

the Rayleigh and nuclear Thomson amplitudes, arises

from vacuum polarization. Together with photon-photon

scattering, photon splitting and coalescence in an exter-

nal ﬁeld, Delbrück scattering belongs to the nonlinear

effects of quantum electrodynamics that have no clas-

sical analog (Sect. 27.2) [62.17]. In the photon energy

range 0.1–1 MeV, the Delbrück amplitude can be sig-

niﬁcant at intermediate scattering angles for high-Z

targets [62.1]. Theory and experiment have been re-

viewed by Papatzacos and Mork [62.16]andmore

recently by Kissel and Pratt [62.1] and by Milstein

and Schumacher [62.18]. Lowest-order Born approxi-

mation calculations of the Delbrück amplitude scale as



Zα



. Experiments to verify the process have been

performed since 1933, but only recently has the neces-

sary accuracy been attained [62.18]; a major obstacle

in the interpretation was the lack of accurate predic-

tions of the Rayleigh amplitude, a problem that is now

being solved to a considerable extent (Sect. 62.2.1).

Recent measurements agree with the Born approxi-

mation predictions to a few percent below Z = 60;

but for heavier targets, Coulomb corections appear to

play a signiﬁcant role that have only recently been

calculated [62.19].

62.3 Inelastic Photon-Atom Interactions

62.3.1 Photoionization

Photoexcitation and ionization are important processes

for the investigation of atomic structure and dynamics

(Chapt. 61). Early reviews of the theory of the atomic

photoelectric effect are by Fano and Cooper [62.20]

(low photon energy) and by Pratt et al. [62.21]

(

ω>10 keV). See also recent comprehensive mono-

graphs by Starace [62.22]andAmusia [62.23]. In the

present volume, the theory of atomic photoionization

for photon energies below 1 keV is summarized in

Chapt. 24. Experimental aspects have been reviewed,

e.g., by Samson [62.22, 24] and by Siegbahn and Karls-

son [62.25].

An extensive compilation of measured X-ray atten-

uation coefﬁcients or total absorption cross sections, as

well as calculated photoelectric cross sections (com-

puted with relativistic Hartree-Slater wave functions),

has been published by Saloman et al. [62.26]. Based on

the National Bureau of Standards (now the National In-

stitute of Standards and Technology) data base, this work

covers the photon energy range from 0.1–100 keV and

includes all elements with atomic numbers 1 ≤ Z ≤ 92;

an extensive bibliography is included. Very useful tables

of theoretical subshell photoionization cross sections

covering all elements and the photon energy range

1–1500 keV have been computed by Scoﬁeld [62.27];

the atomic electrons are treated relativistically as mov-

ing in a Hartree-Slater central potential, and all relevant

multipoles as well as retardation effects are included.

Accuracy of the results is borne out by systematic com-

parisons with experiment and detailed tests [62.28]. A

useful energy range for applications in electron spec-

trometry is covered by a tabulation of subshell cross

sections calculated by Yeh and Lindau [62.29]inthe

dipole approximation with nonrelativistic Hartree-Fock-

Slater wave functions. Because they were obtained in

a frozen-core model, these cross sections like those

of Scoﬁeld [62.27] automatically include all multi-

ple excitations [62.30]. Such multi-electron processes

(Sect. 62.4.4) can make substantial contributions, e.g.,

≈ 40% of the 1s photoionization in the threshold region

of Kr [62.31]. A very useful family of codes being devel-

oped by I Grant’s Oxford group is “GRASP – A General

Purpose Relativistic Atomic Structure Program”, sev-

eral versions of which are being distributed through the

Computer Physics Communications Program Library

(Elsevier Science).

The usefulness of photoexcitation and ionization

for experimental studies of atomic structure and pro-

cesses has been greatly enhanced by the advent

of synchrotron radiation sources beginning in the

1960’s [62.1, 32, 33]. At ﬁrst, experiments were per-

formed mostly “parasitically”. Later, they were done

with dedicated sources, taking advantage of the high

brightness, narrow bandwidth and wide tunability of

the radiation (with suitable monochromators, from the

visible to the hard X-ray regime), its high degree of

polarization and sharp time structure [62.34]. In the

1990’s, “third-generation” sources are being commis-

sioned in which radiation is primarily derived from

insertion devices, wigglers and undulators [62.35],

entailing further increase of the power of this experi-

mental tool. There is a growing literature on research

applications of synchrotron radiation in atomic and

molecular physics, including handbooks [62.36–38],

proceedings of the series of conferences on X-ray

and atomic inner-shell physics [62.39], and mono-

Part E 62.3

Photon–Atom Interactions: Intermediate Energies 62.4 Atomic Response to Inelastic Photon-Atom Interactions 919

graphs [62.40–43]. Much of the physics described in

the remainder of this chapter is being explored with

synchrotron radiation.

62.3.2 Compton Scattering

Compton scattering denotes the scattering of photons

from free electrons. The term is also used for in-

elastic photon scattering from bound electrons, which

approaches the free electron case when the photon en-

ergy greatly exceeds the electron binding energy [62.1].

The theory of the process is discussed in standard text-

books [62.17, 44, 45]. Photon scattering from a free

electron at rest is expressed, within lowest-order rela-

tivistic quantum electrodynamics, by the Klein–Nishina

formula. A low energy approximation to the Klein–

Nishina cross section is

= σ



1 +

+···



, ω  m

(62.7)

where σ

is the classical total Thomson cross section,

= (8π/3)r

. (62.8)

The energy ω



of a photon scattered through an angle

θ is related to the incident photon energy

ω by the

Compton relation



1 +



ω/m



1 − cos θ



. (62.9)

The differential cross section for Compton scattering

from free electrons, averaged over initial and summed

over ﬁnal photon and electron polarizations, is

dσ

dΩ











− sin



. (62.10)

In the limit ω



→ ω, this reduces to the Thomson differ-

ential cross section,

dσ

dΩ



1 + cos



(62.11)

contained in (62.4).

Compton scattering from bound electrons has

extensive applications in the study of electron mo-

mentum distributions in atoms and solids [62.46].

An exact second-order S-matrix code for the rel-

ativistic numerical calculation of cross sections for

Compton scattering of photons by bound electrons

within the independent particle approximation has been

developed [62.47].

A systematic treatment of elastic and inelastic pho-

ton scattering, starting from many-body formalism, has

been described by Pratt, Kissel,andBergstrom [62.48].

62.4 Atomic Response to Inelastic Photon-Atom Interactions

62.4.1 Auger Transitions

Atomic inner shell vacancy states tend to decay pre-

dominantly through radiationless or Auger transitions,

which are autoionization processes (see also Chapt. 25

and Chapt. 61) that arise from the Coulomb interac-

tion between electrons [62.1,1,22,49–55]. Radiationless

transitions dominate over radiative ones, except for 1s

vacancies in atoms with atomic numbers Z < 30, pri-

marily because of the magnitude of the matrix elements

and because far fewer channels are allowed by elec-

tric dipole X-ray emission selection rules (Sect. 62.4.2,

62.4.3) than by the selection rules for Auger transitions,

whichinpureLS coupling are

∆L = ∆S = ∆M

= ∆M

= 0 ,

∆ J = ∆M = 0 ,Π

= Π

. (62.12)

Thus, for example, 2784 electron-electron interaction

matrix elements can contribute to the radiationless decay

of a 2p

3/2

hole state in Hg.

In the simplest approach, ionization and subsequent

Auger decay are treated as two distinct steps. This ap-

proximation is valid if the electron which is ejected in

the ionization process is sufﬁciently energetic so that it

does not interact signiﬁcantly with the Auger electron,

and the interaction of the core hole state with the Auger

continuum is weak. Then the hole state can be consid-

ered to be quasi-stationary and the decay rate can be

expressed according to Wentzel’s ansatz, formulated in

1927 and later known as Fermi’s Golden Rule No. 2

of time dependent perturbation theory [62.49]. In the

independent-electron central-ﬁeld approximation, this

leads to the following nonrelativistic matrix element for

a direct Auger transition:

D =



∗







(1)ψ

∗

∞

(2)



− r



× ψ

n j

(1)ψ



(2)dτ

dτ

, (62.13)

where the quantum numbers n, , j characterize elec-

trons that are identiﬁed schematically in Fig. 62.3.The

Part E 62.4