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

Подождите немного. Документ загружается.

898 Part E Scattering Experiment

the ring. More recently, Ellman et al. [60.67]haveused

the CRYRING facility (at the Manne Siegbahn Labora-

tory in Stockholm) to measure the radiative lifetime of

a bound excited state of a negative ion. In this proof-of-

principle experiment, the lifetime of the 5p

1/2

level

of Te

−

was measured to be 0.42( 5) s. This value is in ex-

cellent agreement with the result of a multi-conﬁguration

Dirac Hartree-Fock (MCDHF) calculation. The J = 1/2

level radiatively decays to the J = 3/2 ground level, pri-

marily via M1 transitions. The idea of the experiment

was to monitor the population of the J = 1/2 level as

a function of time after injection of the Te

−

into the

ring. This was accomplished by selectively photodetach-

ing ions in the J = 1/2 level as the Te

−

ions repeatedly

passed through the ﬁeld of a laser beam situated along

one arm of the ring. The neutral Te atoms thus pro-

duced were used as the signal. Corrections were made

for collisionally-induced detachment and repopulation.

Data was taken at four different ring pressures. A lin-

ear ﬁt to this data yielded the zero-pressure radiative

lifetime of the excited J = 1/2level.

60.4.4 Multielectron Detachment

Synchrotron radiation has been used over the past few

years in order to study how negative ions respond to

the absorption of high-energy photons. Photons in the

VUV and X-ray regions will excite and/or detach inner

shell electrons. Multiple electron detachment appears to

be initiated by the detachment of a core electron. This

process triggers the ejection of one or more valence

electrons either by shake off or by interactions of the

detached core electron with the valence electrons as it

leaves the atom. The ALS (Advanced Light Source) has

been used to investigate multiple electron detachment.

Measurements of the absolute cross sections for the de-

tachment of two electrons from the closed shell ions

−

[60.68]andF

−

[60.69] have been reported.

References

60.1 H. S. W. Massey: Negatives Ions (Cambridge Univ.

Press, London 1976)

60.2 B. M. Smirnov: Negative Ions (Mcgraw-Hill, New

York 1972)

60.3 D. R. Bates: Adv. At. Mol. and Opt. Phys. 27, 1 (1991)

60.4 S. J. Buckman, C. W. Clark: Rev. Mod. Phys. 66,539

(1994)

60.5 C. Blondel: Physica Scripta T58, 31 (1995)

60.6 T. Andersen: Physica Scripta T43, 23 (1991)

60.7 T. Andersen, H. K. Haugen, H. Hotop: J. Phys. Chem.

Ref. Data 28, 1511 (1999)

60.8 R. C. Bilodeau, H. K. Haugen: Photonic, Electronic

and Atomic Collisions (Rinto Press, New York 2002)

60.9 U. Fano: Rep. Prog. Phys. 46, 96 (1983)

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

60.11 T. F. O’Malley: Phys. Rev. 137, 1668 (1965)

60.12 S. Watanabe, C. H. Greene: Phys. Rev. A 22,158

(1980)

60.13 J. Sandstroem, G. Haefﬂer, I. Kiyan, U. Berzinsh,

D. Hanstorp, D. J. Pegg, J C. Hunnell, S. J. Ward:

Phys. Rev. A 70, 052707 (2004)

60.14 P. Kristensen, U. V. Pedersen, V. V. Petrunin, T. An-

dersen, K. T. Chung: Phys. Rev. A 55, 978 (1997)

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

60.16 B. W. Shore: Phys. Rev. 171, 43 (1968)

60.17 M. Crance: Comments At. Mol. Phys. 2, 95 (1990)

60.18 M. D. Davidson, H. G. Muller, H. B. van Linden van

den Heuvell: Comments At. Mol. Phys. 29, 65 (1993)

60.19 R. Middleton: Nucl. Instrum. Methods 214,139

(1983)

60.20 D. J. Larson C. J. Edge, R. E. Elmquist, N. B. Mansour,

R. Trainham: Physica Scripta T22, 183 (1988)

60.21 D. Hanstorp, M. Gustafsson: J. Phys. B 25, 1773

(1992)

60.22 S. L. Kauffman: Opt. Comm. 17, 309 (1976)

60.23 M. Scheer, R. C. Bilodeau, H. K. Haugen: Phys. Rev.

Lett 80, 2562 (1998)

60.24 M. Scheer, R. Bilodeau, J. Thägersen, H. K. Haugen:

Phys. Rev. A 57, 1493 (1998)

60.25 M. Scheer, R. Bilodeau, C. A. Brodie, H. K. Haugen:

Phys. Rev. A 58, 2844 (1998)

60.26 D. J. Pegg, J. S. Thompson, R. N. Compton, G. D. Al-

ton: Phys. Rev. Lett. 59, 2267 (1987)

60.27 C. Froese Fischer, J. B. Lagowski, S. H. Vosko: Phys.

Rev. Lett. 59, 2263 (1987)

60.28 T. Andersen, H. H. Andersen, P. Balling, P. Kris-

tensen, V. V. Petrunin: J. Phys. B 30, 3317 (1997)

60.29 V. V. Petrunin, H. H. Andersen, P. Balling, T. Ander-

sen: Phys. Rev. Lett. 76, 744 (1996)

60.30 P. Kristensen, C. A. Brodie, U. V. Pedersen,

V. V. Petrunin, T. Andersen: Phys. Rev. Lett. 78,2329

(1997)

60.31 V. V. Petrunin, J. D. Voldstad, P. Balling, P. Kris-

tensen, T. Andersen, H. K. Haugen: Phys. Rev. Lett.

75, 3317 (1995)

60.32 S. Salmonson, H. Warston, I. Lindgren: Phys. Rev.

Lett. 76, 3092 (1996)

60.33 G. Haefﬂer, D. Hanstorp, I. Yu. Kiyan, A. E. Klink-

müller, U. Ljungblad, D. J. Pegg: Phys. Rev. A 53,

4127 (1996)

Part E 60

Photodetachment References 899

60.34 K. T. Andersen, J. Sandström, I. Yu. Kiyan,

D. Hanstorp, D. J. Pegg: Phys. Rev. A 62, 022503

(2000)

60.35 P. Juncar, C. R. Bingham, J. A. Bounds, D. J. Pegg,

H. K. Carter, R. L. Mlekodaj, J. D. Cole: Phys. Rev.

Lett. 54, 11 (1985)

60.36 J. Sandstroem P. Andersson, K. Fritioff, D. Hanstorp,

R. Thomas, D. J. Pegg, K. Wendt: Nucl. Instrum.

Methods B 217, 513 (2004)

60.37 H. C. Bryant, B. D. Dieterle, J. Donahue, H. Sar-

iﬁan, H. Tootoonchi, D. M. Wolfe, P. A. M. Gram,

M. A. Yates-Williams: Phys. Rev. Lett. 38, 228 (1977)

60.38 D. W. MacArthur, K. B. Butterﬁeld, D. A. Clark,

J. B. Donahue, P. A. M. Gram, H. C. Bryant, C. J. Har-

vey, W. W. Smith, G. Comtet: Phys. Rev. A 32,1921

(1985)

60.39 P. G. Harris, H. C. Bryant, A. H. Mohagheghi,

R. A. Reeder, C. Y. Tang, J. B. Donahue, C. R. Quick:

Phys. Rev. A 42, 6443 (1990)

60.40 P. Balling, P. Kristensen, H. H. Andersen, U. V. Ped-

ersen, V. V. Petrunin, L. Præstegaard, H. K. Haugen,

T. Andersen: Phys. Rev. Lett. 77, 2905 (1996)

60.41 H. H. Andersen, P. Balling, P. Kristensen, U. V. Ped-

ersen, S. A. Aseyev, V. V. Petrunin, T. Andersen:

Phys. Rev. Lett. 79, 4770 (1997)

60.42 M. Gailitis, R. Damburg: Sov. Phys. JETP 17, 1107

(1963)

60.43 E. Lindroth, A. Burgers, N. Brandefelt: Phys. Rev.

57, 685 (1998)

60.44 I. Yu. Kiyan, U. Berzinsh, D. Hanstorp, D. J. Pegg:

Phys. Rev. Lett. 81, 2874 (1998)

60.45 U. Berzinsh, G. Haefﬂer, D. Hanstorp, A. Klinkmül-

ler, E. Lindroth, U. Ljungblad, D. J. Pegg: Phys. Rev.

Lett. 74, 4795 (1995)

60.46 U. Ljungblad, D. Hanstorp, U. Berzinsh, D. J. Pegg:

Phys. Rev. Lett. 77, 3751 (1996)

60.47 G. Haefﬂer, I. Yu. Kiyan, U. Berzinsh, D. Hanstorp,

N. Brandefelt, E. Lindroth, D. J. Pegg: Phys. Rev. A

63, 053409 (2001)

60.48 G. Haefﬂer, I. Yu. Kiyan, D. Hanstorp, B. J. Davies,

D. J. Pegg: Phys Rev. A 59, 3655 (1999)

60.49 I. Yu Kiyan, U. Berzinsh, J. Sandström, D. Hanstorp,

D. J. Pegg: Phys. Rev. Lett. 84, 5979 (2000)

60.50 C. Pan, A. F. Starace, C. H. Greene: J Phys. B 27,137

(1994)

60.51 C. Pan, A. F. Starace, C. H. Greene: Phys. Rev. A 53,

840 (1996)

60.52 C-N. Liu, A. F. Starace: Phys Rev. A 58, 4997 (1998)

60.53 C-N. Liu, A. F. Starace: Phys Rev. A 59, 3643 (1999)

60.54 A. E. Klinkmüller, G. Haefﬂer, D. Hanstorp,

I. Yu. Kiyan, U. Berzinsh, C. W. Ingram, D. J. Pegg,

J. Peterson: Phys. Rev. A 56, 2788 (1997)

60.55 A. E. Klinkmüller, G. Haefﬂer, D. Hanstorp,

I. Yu. Kiyan, U. Berzinsh, D. J. Pegg: J. Phys. B 31,

2549 (1998)

60.56 H. K. Kjeldsen, P. Andersen, F. Folkmann, B. Kris-

tensen, T. Andersen: J. Phys. B 34, L353 (2001)

60.57 N. Berrah: Phys. Rev. Lett. 87, 253002 (2001)

60.58 N. Berrah, J. D. Bozek, G. Turi, G. Akerman, B. Rude,

H.-L. Zhou, S. T. Manson: Phys. Rev. Lett. 88,

093001 (2002)

60.59 N. D. Gibson et al.: Phys. Rev. A 67, 03070 (2003)

60.60 A. M. Covington, A. Aguilar, V. T. Davis, I. Alvarez,

H. C. Bryant, C. Cisneros, M. Halka, D. Hanstorp,

G. Hinojosa, A. S. Schlacter, J. S. Thompson,

D. J. Pegg: J. Phys. B 34, L735 (2001)

60.61 H.-L. Zhou, S. T. Manson, L. Voky, N. Feautrier,

A. Hibbert: Phys. Rev. Lett. 87, 02301 (2001)

60.62 H.-L. Zhou, S. T. Manson, L. Voky, A. Hibbert,

N. Feautrier: Phys, Rev. A 64, 012714 (2001)

60.63 O. Satsarinny, T. W. Gorczyca, C. Froese Fischer: J.

Phys. B 35, 4161 (2002)

60.64 H. K. Haugen, L. H. Andersen, T. Andersen,

P. Balling, N. Hertel, P. Hvelplund, S. D. M‘ller:

Phys. Rev. A 46, R1 (1992)

60.65 P. Balling, L. H. Andersen, T. Andersen, H. K. Hau-

gen, P. Hvelplund, K. Taulbjerg: Phys. Rev. Lett. 69,

1042 (1992)

60.66 T. Andersen, L. H. Andersen, P. Balling, H. K. Hau-

gen, P. Hvelplund, W. W. Smith, K. Taulbjerg: Phys.

Rev. A 47, 890 (1993)

60.67 A. Ellmann, P. Schef, P. Lundin, P. Royen, S. Man-

nervik, K. Fritioff, P. Andersson, D. Hanstorp,

C. Froese Fischer, F. Österdahl, D. J. Pegg, N. D. Gib-

son, H. Danared, A. Källberg: Phys. Rev. Lett. 92,

253002 (2004)

60.68 A. Aguillar, J. S. Thompson, D. Calabrese, A. M. Cov-

ington, C. Cisneros, V. T. Davis, M. S. Gul-

ley, M. Halka, D. Hanstorp, J. Sandström,

B. M. McLaughlin, D. J. Pegg: Phys. Rev.A 69, 022711

(2004)

60.69 V. T. Davis, A. Aguilar, J. S. Thompson, D. Calabrese,

A. M. Covington, C. Cisneros, M. S. Gulley, M. Halka,

D. Hanstorp, J. Sandström, B. M. Mclaughlin,

G. F. Gribakin, D. J. Pegg: J. Phys. B (to be pub-

lished)

Part E 60

901

Photon–Atom

61. Photon–Atom Interactions: Low Energy

Theoretical and experimental aspects of the atomic

photoelectric effect at photon energies up to

about 1 keV are presented. Relevant formulae

and interpretations are given for the various

excitation and decay processes. Techniques

and results of photoelectron spectrometry in

conjunction with synchrotron radiation are

emphasized.

61.1 Theoretical Concepts ............................ 901

61.1.1 Differential Analysis .................. 901

61.1.2 Electron Correlation Effects......... 904

61.2 Experimental Methods ......................... 907

61.2.1 Synchrotron Radiation Source ..... 907

61.2.2 Photoelectron Spectrometry ....... 908

61.2.3 Resolution and Natural Width..... 910

61.3 Additional Considerations .................... 911

References .................................................. 912

61.1 Theoretical Concepts

Scattering of low-energy photons proceeds predomi-

nantly through the photoelectric effect. In this process

a photon γ of energy hν, angular momentum j

= 1,

and parity π

=−1 interacts with a free atom or mol-

ecule A, having total energy E

, angular momentum J

and parity π

to produce an electron of energy ε,spin

s =1/2, orbital angular momentum , total angular mo-

mentum j, and parity π

= (−1)



and an ion A

with

ﬁnal total energy E

, angular momentum J

, and parity

. This process can be written as the reaction

γ(hν, j

=1,π

=−1) + A(E

, J

,π

)

→ A

, J

,π

) +e

−

[ε, sj,π

=(−1)



] .

(61.1)

Conservation laws require that

hν + E

= ε +E

+ j

= J

+s+,

·π

= π

·π

= (−1)



·π

. (61.2)

Since E

−E

becomes quite large for inner shells or

deep core levels, scattering of low-energy photons in-

volves the removal of an electron from a valence or

shallow core level. In the low-energy regime, from the

ﬁrst ionization threshold to hν ≈ 1 keV, the photoelec-

tric effect accounts for more than 99.6% of the photon

interactions in the elements, with elastic scattering con-

tributing the remainder [61.1, 2]. Ionization by inelastic

scattering, the Compton effect, assumes increasing im-

portance with the higher photon energies and the lower

Z elements. Above the ﬁrst ionization potential, the

total photoabsorption cross section and the photoioniza-

tion cross section are essentially equivalent at the lower

photon energies.

The cross section σ

for producing a given ﬁnal

ionic state in the photoionization process is given by

4π



| Ψ

T | Ψ

|

, (61.3)

where k is the photon momentum,

T is the transition op-

erator, Ψ

and Ψ

are the wave functions of the initial

and ﬁnal states, and the summation includes an average

over all initial states and a summation over all the unob-

served variables in the ﬁnal state. A detailed derivation

of this expression, including the different forms for

T ,

is given in the articles by Fano and Cooper [61.3]and

Starace [61.4] and in Chapt. 24. The total cross section

is given by the sum of all these different partial cross

sections, σ

61.1.1 Differential Analysis

Detailed information about the photoionization process

can be obtained most directly in emission measurements,

especially those involving the photoelectron. The re-

sulting photoelectron spectrum yields the energy and

intensity for a given interaction. Further differentiation

is obtained by varying the angle of observation and by

a spin analysis of the photoelectron. Hence, electron

emission analysis can reveal all energetically allowed

photoprocesses connecting an initial atomic state i to

a ﬁnal ionic state f and yield their dynamic prop-

Part E 61

902 Part E Scattering Experiment

erties. When averaged over the spin, the differential

cross section dσ

/dΩ is given in terms of the par-

tial cross section σ

and an expression involving an

expansion in Legendre polynomials of order n with the

coefﬁcients B

dσ

dΩ



4π





(cos θ) , (61.4)

where the angle θ is measured between the direction of

the emitted electron and the unpolarized incoming pho-

ton beam. In the dipole approximation, which describes

the dominant process at low energy, only the terms con-

taining P

and P

contribute. Then (61.4) reduces, for

a photon beam with linear polarization p,to

dσ

dΩ



4π





1 +

(1 +3 p cos 2θ)



(61.5)

where the angle θ lies in the plane perpendicular to

the direction of propagation and is measured with re-

spect to the major axis of the polarization ellipse [61.5].

Then, the differential cross section, or photoelectron an-

gular distribution, is characterized by the single angular

distribution or anisotropy parameter β

for a partic-

ular process i → f . For observation at the so-called

pseudomagic angle θ

,deﬁnedas

cos

−1



−1



(61.6)

the differential cross section dσ

/dΩ becomes propor-

tional to the angle-integrated, or partial, cross section

In the absence of correlation effects, the partial

cross section σ

for the production of an individual

ﬁnal state and the corresponding anisotropy parame-

ter β

are given by simple expressions derived from

a single-particle model [61.1, 4]. For the central ﬁeld

potential,

4π

n

hν





2 +1



−



 +1

2 +1





(61.7)

where N

n

is the occupation number of the subshell, and

( −1)R

−

+( +1)( +2)R

(2 +1)



R

−

+( +1)R



−

6( +1)R

−

cos ∆

(2 +1)



R

−

+( +1)R



(61.8)

The subscripts + and − refer to the ( +1) and ( −1)

channels respectively, and ∆ = δ

−δ

−

is the differ-

ence in phase shift between these two allowed outgoing

waves. The parameter R

is the radial dipole matrix el-

ement connecting the electron in the bound orbital with

orbital angular momentum  with the outgoing wave

having orbital angular momentum  ±1.

Effects of the electron correlation on the direct pho-

toionization process can result in values for β which are

not reproduced by the Cooper–Zare expression (61.8)

[61.4]. The contribution of the different partial waves to

the outgoing wave function can, however, be ascertained

through the angular momentum transfer formalism de-

veloped by Fano and Dill [61.6]. In this approach, one

deﬁnes the angular momentum transferred from the

photon to the unobserved variables j

= j

− = J

+s− J

, (61.9)

where the second portion of the equality results from the

conservation of angular momentum. For each allowed

value of j

the associated transfer can be deﬁned as either

parity favored or parity unfavored according to whether

the product π

is equal to +(−1)

or −(−1)

re-

spectively. (All symbols have the same deﬁnition as in

(61.1).) Calculation of the partial cross section for the

production of a given ﬁnal state characterized by the

values J

and s is then determined from the cross sec-

tion corresponding to each angular momentum transfer

according to



σ( j

). (61.10)

The associated anisotropy parameter β

is derived from

a similar sum:



=fav

σ( j

)

fav

β( j

)

fav

−



=unfav

σ( j

)

unfav

(61.11)

The second equation derives from the fact that β( j

)

for each parity-favored value must be calculated sepa-

rately, whereas for the parity-unfavored case β( j

) =−1

always.

The physical effect described by the angular mo-

mentum transfer approach is the interaction between

an electron and the anisotropic distribution of the other

electrons in the atom. Thus, it becomes most useful in

the case of ionization from an open-shell atom having

an extra electron or a hole in a shell with  = 0. An il-

lustrative example is the 3s ionization of chlorine. Here

= 2 identically in (61.8) because only the single

Part E 61.1

Photon–Atom Interactions: Low Energy 61.1 Theoretical Concepts 903

value  = 1 is allowed in the single-particle model. How-

ever, the three possible values, j

= 0, 1, 2 are allowed,

of which only the ﬁrst corresponds to the Cooper–Zare or

single-particle, central ﬁeld result [61.4]. That β = 2for

3s ionization of atomic chlorine has been demonstrated

experimentally [61.7].

It is generally the case for ionization of elements

which are found naturally in the atomic state that there

is an equal population in all the ﬁne-structure compo-

nents of the initial state. This is because of the relatively

small energies associated with the ﬁne-structure split-

ting. (This does not necessarily apply to atomic species

generated through a process of molecular dissociation or

high-temperature metal vaporization.) Thus, the deter-

mination of all cross sections and angular distributions

involves an average over these ﬁne-structure compo-

nents. However, it is possible to generate atoms in

which one of the ﬁne-structure components is pref-

erentially populated. In this case there can also be

a preferential ionization to a particular J-component

of the ﬁnal ionic state even in the limit in which the

electron correlation is neglected, i. e., the geometri-

cal limit. The partial intensities for the production of

a given ionic state characterized by the angular mo-

menta L

by removal of an electron from an

orbital  of a state characterized by L

are given



][L

][S

]

[1/2][]

g(, L

, S

, L

, S

)

j=+1/2



j=−1/2

[j]











 1/2 j











. (61.12)

Here the term in curly brackets is a 9–j symbol,

and the notation [J]=2J +1 is used. The quantities

g(, L

, S

, L

, S

) are weighting factors determined

solely by the initial-state wave function. For the case

in which  represents a closed shell, these factors are

equal to unity [61.8].

In situations in which the target atoms possess an

initial orientation, i. e., have an average value  J

 = 0,

or if the ionization is performed with circularly polar-

ized radiation, the electrons which are produced have

a net spin [61.9, 10]. It is also possible that unpolarized

atoms which are ionized by unpolarized photons can

have a net spin, provided that the detection is carried out

at a speciﬁc angle, and the ionization is from a given

ﬁne-structure component of the initial state to a given

ﬁne-structure component of the ﬁnal state. In the latter

case, the transverse spin polarization is given by

P =

−2ξ sin θ cos θ

1 +βP

(cos θ)

(61.13)

for linearly polarized radiation, and by

P =

2ξ sin θ cos θ

2 −βP

(cos θ)

(61.14)

for unpolarized radiation [61.11, 12]. The angle θ is the

same as in the angular distribution measurement; the

parameter ξ is the spin parameter analogous to β;and

(cos θ) is the Legendre polynomial of order 2.

Yet another parameter which describes the differ-

entiation inherent in the photoionization process is

the alignment A, which reﬂects an anisotropy in the

quadrupole distribution of the angular momentum J

the ion [61.12]. For the cylindrically symmetric coor-

dinate system appropriate to dipole photon excitation,

only one moment A

of the distribution is nonzero. This

is deﬁned by





− J

+1)



σ(m

)

+1)



σ(m

)

(61.15)

where σ(m

) is the partial cross section for produc-

tion of a given m

component of J

. A very useful

approach to the interpretation of the alignment can be

obtained through the angular-momentum-transfer for-

malism [61.13]. In this approach the angular momentum

transfer j

is deﬁned as

= j

− J

. (61.16)

In contrast to the case of the electron angular distribu-

tion, it is possible to derive an alignment for each value

of j

as a function of the angular momentum J

of the

ion. The net alignment is then the incoherent sum of the

contributions corresponding to each:



( j

)σ( j

)





σ( j

) (61.17)

If the photoionization produces an ion in an excited state

which decays by photoemission, the parameter A

reﬂected either in the angular distribution of the ﬂuores-

cence photons I(θ) or the linear polarization P measured

at one angle, typically 90

◦

, according to

I(θ) = I



1 −

(2)

(cos θ)

(2)

sin

θ cos(2χ) cos(2η)



(61.18)

Part E 61.1

904 Part E Scattering Experiment

or, for θ =π/2andχ =0,

P =

I(η = 0) −I(η = π/2)

I(η = 0) +I(η = π/2)

(2)

4 +h

(2)

(61.19)

respectively. The angle θ is the angle at which the ﬂu-

orescence is determined, and the angle χ is measured

between the axis of the polarization selected by the

detector and the quantization axis. The polarization of

the ﬂuorescence is given by ζ = (cos η, isinη, 0).The

quantity h

(2)

is a ratio of 6–j symbols depending on the

angular momenta J

of the intermediate ion and the ﬁnal

state J



(2)

= (−1)

−J





11J









11J



(61.20)

When it is energetically allowed, a hole in a shal-

low inner-shell will preferentially undergo Auger decay,

emitting an electron with an energy ε

determined

by the energy difference between the energy E

the ion and E



of the state of the doubly-charged

ion to which the decay occurs. Angular analysis of

the Auger electrons reﬂects the alignment of the in-

termediate ionic state, which is different from, and

does not bear a one-to-one relationship to, the angular

distribution parameter β of the photoelectrons. Nor-

mally, Auger decay is regarded as a two-step process

in which the ﬁrst step is the production of the hole

and the release of the primary photoelectron, followed

by the decay and the release of the second electron.

Within this approximation [61.14], the angular dis-

tribution of the Auger electrons takes on the simple

form

I(θ) =



4π



[1 +α

(cos θ)] . (61.21)

Here P

(cos θ) is the second-order Legendre polyno-

mial, and α

is the matrix element corresponding to the

Auger decay. For the speciﬁc case in which the Auger de-

cay is to a ﬁnal ionic state of

symmetry, α

is purely

geometric, and a measurement of the angular distribu-

tion leads directly to a determination of the alignment.

Correspondingly, if the alignment of a speciﬁc state can

be determined through such a decay, then analysis of the

angular distribution of the decay to other states provides

avalueforα

61.1.2 Electron Correlation Effects

The primary focus of advanced studies in photoion-

ization is to determine the role played by electron

correlation in the structure and dynamics of electron

motion above the lowest ionization threshold. Because

the form of the interaction potential for the Coulomb in-

teraction is very well known, theory can focus on the

many-body aspects of the process (Chapt. 23). Electron

correlation manifests itself in many ways. Most promi-

nent are the appearance of autoionization structure due

to the excitation of one or two electrons, the production

of correlation satellites due primarily to the ionization of

one electron accompanied by the excitation of another,

and the creation of two continuum electrons in a double

ionization process.

Autoionization resonances are perhaps the oldest

known features associated with electron correlation

(Chapt. 25). These features arise when the absorption

of a photon creates a localized state which lies in en-

ergy above at least one ionization limit. This state is

then degenerate in energy with a state of an electron in

the continuum, and the interaction between these states

results in the decay of the quasi-localized state into the

continuum. Such resonance states appear in an absorp-

tion spectrum in the form of strong, localized variations

over an energy range characteristic of the width Γ of the

feature, which is in turn related to the lifetime τ of the

state by

Γ =

/τ . (61.22)

In contrast to absorption features between bound states,

autoionization resonances are characterized by having

an asymmetric line shape. When only one localized state

and one continuum are involved, these line shapes can be

derived analytically, as ﬁrst shown by Fano [61.15]and

later by Shore [61.16], resulting in simple parametrized

forms which are suitable for numerical calculation of

overlap integrals for determining widths. For the Fano

proﬁle

σ() = σ

( +q)



+σ

, (61.23)

with

 =

E − E

(Γ/2)

(61.24)

the parameter q describes the asymmetry of the line, E

the resonance position, and Γ is the width of the line. The

parameters σ

and σ

reﬂect the relative contributions to

states in the continuum which do and do not interact with

Part E 61.1

Photon–Atom Interactions: Low Energy 61.1 Theoretical Concepts 905

the autoionizing state respectively. The energy E

does

not correspond to the peak energy E

of the resonance

feature but is related to the maximum through

= E

(61.25)

TheShoreproﬁle,

σ() = C() +

A + B



(61.26)

describes the same phenomenon except that the inter-

pretation of the parameters A and B is different. In this

case, they represent products of dipole and Coulomb

matrix elements. C() is the continuum contribution.

From an experimental point of view, the

parametrized forms for the Fano and Shore proﬁles are

very useful as a basis for ﬁtting autoionization spectra.

However, they both have the limitation that they only

describe the interaction of an isolated state with the con-

tinuum. While they can be extended to include several

continuua [61.4], they do not allow for an interaction

among two or more localized states [61.17, 18]. Never-

theless, it is possible to use these functions to achieve

often good ﬁts of states which do interact with each other,

as these functions are mathematical representations of

localized resonances in a continuous spectral distribu-

tion. If this is done, the parameters no longer have the

physical meaning which they have for the noninteracting

case.

Mixing of discrete ionization channels with com-

peting continuum channels adds complexity to the

photoionization process, not just in the classical au-

toioinization regime but also in the vicinity of inner

shells [61.19–21]. In a rigorous application of the

Mies formalism [61.17], feasible with modern com-

puter power, even complex experimental spectra can

now be satisfactorily interpreted and reproduced. A case

in point is the excitation spectrum from the 2p level of

the open-shell chlorine atom [61.19].

The process of autoionization is discussed in more

detail in Chapt. 25.InFig.61.1 an example is shown of

the set of 2s

(

S) → 2s2p

np, n ≥ 3 autoionization

resonances which decay into the

P ground state of the

ion [61.22]. The energies E

of these resonances are

related to the ionization limit E

∞

of the series by the

Rydberg formula

= E

∞

− R

∞

/(n −µ

)

(61.27)

where n is the principal quantum number and µ

the quantum defect characteristic of a given series and

reﬂecting the short-range electrostatic interactions of the

electron with the ion core. Values of µ

fors,p,d,and

f electrons have been calculated for atoms and ions up

to Z = 50 [61.23]. For high precision work, the reduced

Rydberg constant

∞

1 +5.485 799 × 10

−4

/(M

−m

)

(61.28)

should be used instead of the value R

∞

for inﬁnite nu-

clear mass (Chapt. 1). The atomic mass M

and the

electron mass m

are in a.u.

A process closely related to the autoionization

phenomenon is resonant Auger decay. This process

differs from the ordinary Auger process [61.24]in

that an electron from an inner shell is not ionized

but excited to either a partially ﬁlled or an empty

subshell. It may be viewed either as an Auger pro-

cess or as autoionization. Such an inner-shell excited

state of a neutral atom (molecule) lies above one

or more of the ionization limits of the singly ion-

ized species and consequently must decay by electron

emission unless the decay is forbidden by selection

rules. As a result, resonance structure will appear

superimposed on the continua of direct photoioniza-

tion from the various subshells. From a most general

point of view, the resonant Auger process can be

considered as resonances in the continua of single

photoionization, while the ordinary Auger process can

be regarded as resonances in the continua of double

photoionization. If excitation proceeds to a partially

ﬁlled subshell within a principal shell, as, for ex-

ample, Mn 3p → 3d [61.20], interference between the

direct photoionization channels and the indirect reso-

nance channel may be strong, and the lineshapes are

Photon energy (eV)

17.5 18.0 18.5 19.0 19.5 20.0 20.5

n =3 4 5 6 78

Cross section (Mb)

(

Fig. 61.1 Autoionization resonances 2s

(

S) →2s2p

in atomic nitrogen

Part E 61.1

906 Part E Scattering Experiment

givenby(61.23) with arbitrary q values and σ

/σ

ra-

tios. If, however, the excitation proceeds to an empty

shell, as, for example, Mg 2p → nsorn d, n ≥ 4, in-

terference with the direct channels is likely to be

negligible, and the resulting resonances are distin-

guished by essentially Lorentzian line shapes (as for

normal Auger lines) with q  1andσ

/σ

 1in

(61.23). For a given excitation state a number of

resonance peaks may arise because more than one

ionization channel is usually available and, in addi-

tion, the excited electron can change its orbital from

n to n



= n ± 1, 2, ... in a shakeup or shakedown pro-

cess [61.25].

As a consequence of the electron–electron interac-

tions which occur simultaneously with the electron–

photon interaction, ions are produced in states which

do not correspond to those which would be expected

based on an interpretation using an independent particle

model, which allows for only a single-electron transi-

tion (Chapt. 24). Evidence for these states appears as

correlation satellites in the photoelectron spectrum, the

Auger electron spectrum or the X-ray spectrum [61.26].

Figure 61.2 [61.27] presents as an example the pho-

toelectron spectrum of argon produced by photons

with hν =60.6 eV. In addition to the 3s main line

of single electron photoionization (and the 3p main

lines not shown) numerous satellite lines are seen as

the manifestation of two-electron transitions involving

ionization-with-excitation correlations. It is convenient

to categorize the satellites in a photoelectron spec-

trum according to various electron correlations, as,

for example, initial state interactions which mix dif-

28 31 34 37 40 43 46

counts/ channel

Binding energy (eV)

55°

×3

Fig. 61.2 Photoelectron spectrum (PES)of3s, 3p satellites

in argon at a photon energy of 60.6 eV. Note the reduced

intensity of the satellites compared to the 3s main line

ferent conﬁgurations into the initial state, and ﬁnal

state interactions, which include core relaxation and

electron–electron interactions in the ﬁnal ionic state,

electron–continuum, and continuum–continuum inter-

actions. While initial-state correlations are essentially

independent of the photon energy, ﬁnal-state correla-

tions depend on the energy of the photon through the

interactions with the continuum channels. However, the

heuristic value of placing correlation effects into a strict

classiﬁcation scheme is limited by the fact that their rela-

tive strengths depend on the basis set used in a particular

theoretical model and its expansion into a “fully corre-

lated” system within a given gauge [61.3,4, 28, 29](see

also Chapt. 24).

Another manifestation of double-electron processes

is the simultaneous excitation of two electrons to bound

states. These states may decay by electron or photon

emission and are seen as resonance structures above the

thresholds of inner-shell ionization or near autoionizing

members of Rydberg series. As single or double ion-

ization continua are usually strong in the spectral range

of the double excitations, interference occurs, and the

lineshapes can display dispersion forms. Typically, the

cross section for the sum of all correlation processes

is between 10 and 30% of that for single photoioniza-

tion, but may exceed this range considerably in special

cases.

Photon scattering near thresholds is complex be-

cause of the possibility of strong interactions between

the various particles created and the different modes

of deexcitation (Chapt. 62). In the case of ionization-

with-excitation processes, the threshold cross section is

ﬁnite, as it is for single electron photoionization, in ac-

cord with Wigner’s theorem [61.30] (Sect. 60.2.1). In

the case of double photoionization, the cross section is

zero at threshold and then rises according to Wannier’s

law [61.31](Chapt.52). For the motion of two electrons

with essentially zero kinetic energies in the ﬁeld of the

ionic core,

σ ∝ E

(2µ−1)/4

, (61.29)

where µ depends on the value of the nuclear charge Z

through

µ =



(100Z −9)

(4Z −1)



. (61.30)

For Z = 1 the Wannier exponent has the value 1.127.

In the case of Auger decay following ionization

at threshold, interaction between the two electrons

results in a shift in the energy of the Auger

Part E 61.1

Photon–Atom Interactions: Low Energy 61.2 Experimental Methods 907

electron and a corresponding shift in that of the

photoelectron (to conserve energy), as well as an

asymmetry in the shape of the Auger electron peak

and a corresponding asymmetry in the photoelectron

peak shape. In this so-called post-collision interaction

(Chapt. 62), the lineshape, averaged over angles, has the

form [61.32]

K(ε) =

(Γ/2π)

(ε −ε

)

+(Γ/2)

f(ε) (61.31)

with

f(ε) =

πψ

sinh(πψ)

exp



2ψ tan

−1



ε −ε

(Γ/2)



(61.32)

In the above equations, ε is the energy of the Auger line,

is the nominal Auger energy, Γ is the initial hole-

state width, and the parameter ψ = 1/

√

2ε

−1/

√

2ε

with ε

being the energy of the photoelectron, and

≥ ε

61.2 Experimental Methods

An overview of the experimental approaches to the

study of photon interactions at low energies is given

in Fig. 61.3. The sketch emphasizes the interaction of

a polarized photon beam with a small static or particle-

beam target of atoms, ions, molecules, or clusters, and

the detection of the reaction products at various angles

in a plane perpendicular to the direction of propaga-

tion of the photon beam, where the general equation

(61.5) is valid. Emission products, such as electrons,

ions, or photons, may be studied by way of the total

Dispersive element

and detector

Photon monitor

(flux, polarization)

Dispersive element and

detector (electron, fluor-

escence, or ion spectro-

meter, spin or polariza-

tion analyzer)

Target source

(oven, gas discharge, plasma,

static gas, particle beam)

Photon source

(laser, lamp, plasma,

synchrotron)

Total yield

detector (ions,

electrons,

photons)

zˆ

yˆ

xˆ

Fig. 61.3 Generic arrangement for detection of particles

in an emission measurement. The incoming radiation is

assumed to be linearly polarized along the z-axis

yields, which can be related to the total photoionization

cross section, or by differential analysis in a spectrome-

ter according to energy, intensity, emission angle, and

polarization. The various particles may be measured

independently, simultaneously, or in coincidence. The

photon monitor provides the information for normaliza-

tion of the data with regard to ﬂux and polarization. The

photon monitor can also be used for a measurement of

the total photoionization cross section, equivalent to the

photoabsorption or photoattenuation coefﬁcient at low

photon energies. For this purpose, the size of the tar-

get source is advantageously increased in the direction

of the photon beam. While experimental apparatus dif-

fers, sometimes drastically, for the photon sources as

well as for the spectrometry of electrons, ions, and ﬂu-

orescence photons, many features are common, and the

relationships of the measured quantities to basic prop-

erties of the atoms and the photon–atom interaction are

similar. Thus, the following will place emphasis only on

the roles of the synchrotron radiation source and pho-

toelectron spectrometry, whereas speciﬁc references to

other methods can be found elsewhere [61.5, 33–38].

61.2.1 Synchrotron Radiation Source

The primary source of photons over a broad energy

range for experiments in the VUV andsoftX-rayre-

gion of the spectrum is the synchrotron radiation source

[61.39, 40]. In a synchrotron or electron storage ring,

radiation is produced as the electrons are bent to main-

tain the closed orbit. Such bending magnet radiation is

emitted in a broad continuous spectrum which begins in

the infra-red and ends sharply at a critical photon en-

ergy given by hν

= κE

,whereE

is the energy of the

electrons in the ring and κ is a constant characteristic of

the ring. Synchrotron radiation can also be generated by

introducing additional magnetic ﬁeld structures [61.41]

Part E 61.2

908 Part E Scattering Experiment

into the ring, such as undulators or wigglers, which pro-

duce a deviation of the electron motion from a straight

path in a well-deﬁned manner. Wiggler radiation has

the same spectrum as a bending magnet, except that the

critical energy is generally much higher because the ef-

fective magnetic ﬁelds can be larger than those of the

bending magnet. Undulator radiation is very different in

that it consists of a sharp spiked proﬁle of about a 1%

bandwidth at energies determined by the magnetic ﬁeld

within the undulator and by the electron beam energy in

the storage ring.

Synchrotron radiation, no matter what the magnetic

ﬁeld structure of the source, requires monochromati-

zation before it can be used for experiments. For the

wavelengths of interest in low-energy photon scattering,

this can be achieved by using grating instruments with

a metallic coating on the grating surface. The highest res-

olution possible is obtained through the use of a normal

incidence monochromator (NIM) with a plane grating

set at normal incidence. However, because the reﬂectiv-

ity of the metallic coating at normal beam incidence

decreases drastically as the photon energy increases,

use of a NIM has an upper limit of about 40 eV. At

energies above this, up to about 1 keV, gratings can

still be used but must be mounted at grazing inci-

dence. There is a number of functional designs for these

grazing-incidence instruments which vary in the shape

of the grating – spherical, toroidal, or plane surfaces

(SGM, TGM, PGM) – and the associated optics. Above

1 keV, gratings are no longer suitable, and crystal diffrac-

tion must be used. While the radiation emerging from

a beamline which couples the monochromator to a bend-

ing magnet, wiggler, or undulator has a high degree

of linear polarization, varying from 80 to 99% in the

plane of the electron orbit, a useful ﬂux of circularly

polarized radiation can be derived from out-of-plane ra-

diation [61.33], by the use of multiple reﬂection optics,

or from a helical undulator.

61.2.2 Photoelectron Spectrometry

The primary particle emitted in photoionization is the

photoelectron. Hence, a photoelectron spectrum pro-

vides a detailed view of the photon interaction by (a)

specifying the individual processes from an initial state i

to a ﬁnal state f by way of the electron energy, (b)

determining their differential and partial cross sections

by recording the number of electrons as a function of

emission angle, and (c) measuring the polarization of

the electrons by a spin analysis (spin polarimetry). The

experimental approach is governed largely by the rela-

tions (61.3), (61.5), and (61.6). The number of electrons

(e) detected per unit time at an angle θ within an en-

ergy interval dε and within a solid angle dΩ is given

(e) = GN(hν)N( A) f(hν) f(ε)

dσ

dΩ

dε

(61.33)

where G is a geometry factor, which includes the source

dimensions, N(hν) the number of photons, N(A) the

number of atoms in the source, f(hν)and f(ε) efﬁciency

factors depending, respectively, on photon and elec-

tron kinetic energies, and dσ

/dΩ is the differential

cross section for a particular transition i → f . Equa-

tion (61.33) assumes that dΩ and dε are sufﬁciently

small that integration over the pertinent parameters is

not needed. Since N

∝ dσ

/dΩ, a measurement at

two angles, e.g., θ =0

◦

and 90

◦

, yields the electron

angular distribution parameter β

according to (61.5),

and a measurement at θ

(61.6) yields the partial cross

section σ

. In the case of closed-shell atoms, n j no-

tation is sufﬁcient to designate single ionization to an

ε continuum, e.g., 3p

1/2,3/2

→ εsorε d in argon, but

for open-shell atoms LSJ notation is required, e.g.,

(

3/2

) → 3p

(

2,1,0

)ε(

)

in chlorine. Similarly, for ionization-with-excitation

transitions, the ﬁnal state requires an open-shell des-

ignation.

The sum of the partial cross sections is equal to the

total photoionization or absorption cross section

tot



i, f

(61.34)

where the σ

encompass (a) single ionization events in

all energetically accessible subshells n j,ortheLSJ

multiplet components, (b) ionization-with-excitation

events (shakeup or shakedown), and (c) double ion-

ization events (shakeoff). All σ

can be determined

from a photoelectron spectrum from its discrete peaks

(cf. Fig. 61.2) and from the continuum distribution of

multiple ionization. However, the latter process is meas-

ured more readily by observing the multiply charged

ions in a mass spectrometer.

The differentiation afforded by measuring the var-

ious partial cross sections, and the associated β

parameters, can be augmented by differentiating the

continuum channels according to the spin using a spin

polarimeter [61.33]. In closed-shell atoms, this allows

for an experimental determination of the relevant matrix

elements and phase shifts, and hence for a direct com-

parison with theory at the most basic level [61.34, 35].

In a more global measurement, the cross section σ

tot

Part E 61.2