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

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368 Part B Atoms

(23.43) resembles the ordinary Schrödinger equation,

because of the energy dependence of the self-energy

part

Σ(r, r



,ε), it is not the same. Consequently, the

wave function ψ

(r), often called a Dyson orbital, must

be normalized according to the condition

(ψ



|ψ

) = F

δ(ε



− ε) , (23.44)

where



1 −





∂

Σ(r, r



, E)

∂E







E=ε



−1

. (23.45)

This is different from that for ordinary wave functions

by the factor F

< 1. In F

, the matrix element of the

operator ∂

Σ(r, r



, E)/∂E is calculated between states

(r). It is seen that F

is the same spectroscopic factor

as determined by (23.37), but for a continuous spectrum

or an excited electron state.

The object described by the wave function ψ

(r)

differs from an individual electron because it can be

unstable, and its state is mixed with those of more com-

plicated conﬁgurations, such as “two electrons” k





–

one vacancy j



. This object is called a quasi-electron.

Equation (23.43) also determines the energies of

electrons in discrete levels which are shifted from their

HF values. Contrary to the case of deep vacancies, it is

impossible to predict the sign of the energy shift without

detailed calculations.

For incoming electron energies ε higher than the

target ionization threshold, the operator

Σ(r, r



,ε) ac-

quires an imaginary part, thus becoming an optical

potential for the projectile. The additional elastic scatter-

ing phase shifts from their HF values can be expressed

via matrix elements of

Σ(r, r



,ε) between the wave

functions φ

∗

(r) and ψ

(r); but to ﬁnd numbers for these

phase shifts, the self-energy part or polarization inter-

action

Σ(r, r



,ε)must be calculated. It appears that the

second-order projectile–target interactions

a) b)

εε

k''

exchange

terms

(23.46)

provide a reasonably good approximation [23.10, 12].

The expression for

Σ(r, r



,ε) simpliﬁes at dis-

tances far from the target. Only (23.46a) contributes

in this region, while other terms are exponentially

small. Expanding the Coulomb interelectron interac-

tions in (23.46a) in powers of r



 1, r



 1

gives

Σ(r, r



,ε) =−δ(r−r



)

(0)

, r, r



→∞

(23.47)

where α

(0) is the static (ω = 0) dipole polarizability

determined by (23.33).

Accounting for each additional interaction between

the projectile and target increases the power of r in the

denominator of (23.47). Most important is the inter-

action between target electrons. By including this, the

asymptotic expression (23.47) is also modiﬁed, where

instead of α

(0),theRPAE polarizability α

RPAE

(0) ap-

pears. If the target–projectile interelectron interaction is

taken into account to second order as in (23.46) while all

the rest is included exactly, the expression for

Σ(r, r



,ε)

for r, r



→∞is still given by (23.46), but with the exact

static polarizability.

Many experimental results for low energy electron

scattering by noble gases, alkalis, and alkaline earths

agree well with calculations of elastic scattering phase

shifts obtained by solving equation (23.42)inwhich

Σ(r, r



,ε) is given by (23.46) with RPAE corrections

taken into account. Total cross sections are reproduced

with an accuracy as high as several percent, including

the Ramsauer minimum region. This is illustrated in

Fig. 23.1 for the e

−

+ Xe case [23.18]. This approxima-

tion is also reasonably good in describing the angular

distributions. As the projectile energy ε increases, the

contribution of (23.46) decreases rapidly.

The approach presented here applies to other in-

coming particles, such as for instance positrons e

0246810

150

100

E (eV)

σ (E) (arb. units)

Fig. 23.1 Electron–Xe atom elastic scattering cross sec-

tion [23.12]. Solid line: including polarization interaction;

dashed line: HF; dash-dotted line: experiment

Part B 23.2

Many-Body Theory of Atomic Structure and Processes 23.2 Calculation of Atomic Properties 369

The exchange between e

and target electrons must be

omitted by discarding (23.6) and all but (23.46a) terms

in the polarization interaction

Σ(r, r



,ε).Theincom-

ing positron in its intermediate state k



[see (23.46a)]

interacts strongly with the virtually excited electron



, forming a positronium-like object. Corresponding

diagrams are obtained by inserting elements (23.4d)

for the e

–e

−

interaction into (23.46a). Summation

of the inﬁnite sequence of such diagrams corresponds

to substituting the product φ



)φ



) in the in-

termediate state of (23.46a) by the exact e

−

wave

function in the ﬁeld of a target with a vacancy j.Such

a program is very complicated, and a simpliﬁcation

has proved to be satisfactory. Only the positronium

(Ps) binding is taken into account by subtracting its

binding energy in the denominator of (23.13). This is

equivalent to adding the Ps ionization potential I

ε in (23.46) [23.19]. It enhances the polarization in-

teraction and leads to an interesting qualitative feature:

the possibility of alteration of the sign of the interac-

tion (23.46). Indeed, instead of α(0), the corresponding

expression for e

-atom collision includes α(I

).For

alkalis, the binding energy is less than I

,andfor

the energy region

ω>I

the polarizability (23.33)

is negative, while α(0)>0. This leads to a repulsive

polarization interaction, rather than the usual attractive

one. This difference affects the cross section qualita-

tively [23.20].

Another, more complicated, approximation substi-

tutes φ



)φ



) by the product of the precise Ps

wave function ψ

(|r

−r

|) and the wave function of

the free motion of the Ps center of gravity [23.21]

According to the diagrams (23.23), (23.43) describes

the target with an additional electron. If the target is

a neutral atom, solution of (23.43) with discrete energy

values describes negative ion states; both ground and

excited.

Again, as in Sect. 23.2.2,diagrams(23.46) with

RPAE corrections form a reasonably good starting point

for calculating the negative ion binding energies, even

in cases when this binding is comparatively small, as

in alkaline earth negative ions [23.22]. The inclusion

of only the outer shell polarizability ( j is a vacancy in

the outer shell) leads to overbinding of the additional

electron forming the negative ion. Only the inclusion

of screening due to inner shell excitations yields good

agreement with experiment. For instance, recent meas-

urements of Ca

−

afﬁnity [23.23] give about 20 meV,

while the calculations without inner shell excitations

give about 50 meV [23.22]. Their inclusion must con-

siderably reduce the theoretical value.

23.2.4 Two-Electron

and Two-Vacancy States

One can construct a diagrammatic equation for the wave

function of a two-electron or a two-vacancy state by

separating all diagrams describing two-electron (two-

vacancy) scattering which do not include these states

as intermediate, and denoting their total contribution by

a circle. Then the exact two-electron (vacancy) state is

determined by the inﬁnite sequence of diagrams

=+ + + …

+ …

kk'

(23.48)

The analytic equation for two electrons in an atom

interacting with each other can be written in the form



− ε



× ψ

· r

) = 0 . (23.49)

Here

1(2)

is the HF part of the one-particle Hamiltonian

in (23.24),

is the effective interelectron interaction

and

Q is the projection operator

Q = 1 − n

− n

, (23.50)

with n

1(2)

being the Fermi step function, n

1(2)

= 1for

1(2) ≤ F and n

1(2)

= 0for1(2)>F. The function n

1(2)

thus eliminates contributions of vacant states. The op-

erator

Q takes into account the fact that propagation of

two electrons (or, more precisely, quasi-electrons due to

the presence of

Σ) takes place in a system of other par-

ticles which occupy all levels up to the Fermi level.

The presence of

Q makes (23.49) essentially differ-

ent from a simple two-electron Schrödinger equation:

Q requires that after each act of interaction described

, both participants either remain electrons or

become vacancies.

Diagrams (23.48) describe the states of two electrons

outside the closed shell core, or electron scattering by

an atom with one electron outside the closed shells.

In general,

is nonlocal, dependent upon ε, and can

Part B 23.2

370 Part B Atoms

have an imaginary part. The lowest-order approximation

is V =|r

− r

−1

. Equation (23.49)mustbe

solved in order to obtain, for example, the excitation

spectrum of two electrons in atoms with two electrons

outside closed subshells, such as Ca. Instead of V,the

interaction Γ from (23.31) can be used, which would

account for screening due to virtual excitation of inner

shell electrons. For low level two-electron excitations,

the energy dependence of

can be neglected.

The same type of equation can be obtained for two-

vacancy states, describing their energies, decay widths,

and structure due to conﬁguration mixing with more

complex states. For inner and intermediate shell va-

cancies, however, the corresponding corrections can be

taken into account perturbatively, and the screening by

outer shells is not essential. The admixture of outer shell

excitation with inner vacancies is important, leading to

satellites of the main spectral lines.

The interaction between vacancies leads to correla-

tion two-vacancy decay processes in which the energy

is carried away by a single electron or photon. Some

diagrams exemplifying these processes in the lowest

possible order of interaction are

a) b)

(23.51)

Even in this order, there are several diagrams giving

together the amplitude of the correlation Auger (23.51a)

and radiative (23.51b) decay. For inner shells, a speciﬁc

feature of such processes is that the released energy is

about twice that for a single vacancy decay. Of course,

in these cases the decay probability is relatively small.

It is not, however, necessarily much smaller than that of

individual vacancies if the energies of the intermediate

and initial states are close to each other.

The presence of residual two-body forces leads to ef-

fective multiparticle interactions. The simplest diagram

presenting a three-electron interaction is given by

(23.52)

The role of multi-particle interactions in atomic struc-

ture is far from being clear. Diagrams similar to (23.52)

are important if it is of interest to calculate the energy

levels of atoms with three or more electrons (or vacan-

cies) outside of closed shells. This is a very complicated

calculation because even two electrons in the Coulomb

ﬁeld of the nucleus is a difﬁcult three-body problem.

23.2.5 Electron–Vacancy States

The one-electron–one-vacancy state is the simplest ex-

citation of a closed shell system under the action of an

external time-dependent ﬁeld, represented by (23.2a).

Beginning with electrons and vacancies described in the

HF approximation, the result of residual interactions

leads to excitations of more complex states, including

those with two or more electron–vacancy pairs. The in-

teraction can also lead to a single electron–vacancy pair.

Let us concentrate on the latter case and separate all

diagrams describing electron–vacancy interaction which

do not include these states as intermediate, and denote

them by a circle. Then the exact electron–vacancy state

is determined by the inﬁnite sequence of diagrams

+ …

i' i'' i'

+ …

k'' k'

(23.53)

Contrary to the electron–electron case in Sect. 23.2.4,

the analytical expression corresponding to (23.53) can-

not be represented as a Schrödinger-type equation.

Indeed, being symmetric under time reversal, (23.53)

leads to an equation depending, unlike (23.49), upon

the second power of the electron–vacancy energy

ω. Note that R in (23.53)and

 in (23.46)are

different.

It is necessary to solve (23.53) when calculating the

photoabsorption amplitude, which can then be repre-

sented as

(23.54)

The amplitude for elastic photon scattering is also ex-

pressed via the exact electron–vacancy state, determined

Part B 23.2

Many-Body Theory of Atomic Structure and Processes 23.2 Calculation of Atomic Properties 371

by (23.53)tobe

ωω

(23.55)

The other case where it is necessary to solve (23.53)is

the scattering of electrons, both elastic and inelastic, by

atoms with a vacancy in their outer shell, such as the

halogens.

The simplest approximation to R is given by the

Coulomb interaction to lowest order (23.4e) and (23.4f).

With such R, the sequence of diagrams (23.53)is

thesameas(23.31) and thus forms the RPAE,which

is often used to describe photoionization and other

atomic processes. Both terms (23.4e) and (23.4f) con-

tribute to the electron–vacancy in (23.53) only if

the external ﬁeld is spin independent, such as an

ordinary photon. For a magnetic interaction, which

is proportional to spin, the term (23.4e) does not

contribute.

23.2.6 Photoionization in RPAE and Beyond

In RPAE, the photoionization amplitude k|D(ω)|i is

determined by solving an integral equation obtained

from (23.53)and(23.54) using the correspondence

rule (23.14) [23.12]:

k|D(ω)|i

=k|d|i





>F;i



≤F



k



|D(ω)|i



ki



|V|ik



− k



i

ω − ε



+ ε



+ iη

−

i



|D(ω)|k



kk



|V|ii



− i



i

ω − ε



− ε



− iη



(23.56)

where d is the dipole operator describing the photon–

electron interaction and V =|r

−r

−1

. To obtain the

RPAE photoionization cross section, the usual expres-

sion (24.19) must be multiplied by the square modulus

of the ratio k|D(ω)|i/k|d|i, with ω = ε

+ I

.The

RPAE corrections described by the term in square brack-

ets in (23.56) are very large for outer and intermediate

electron shells. Through the sum over i



, if only terms

with the same energies ε



= ε

are included, (23.56) ac-

counts for intra-shell correlations. By adding terms with



= ε

, the effect of inter-shell correlations is taken into

account.

For atoms, (23.56) has to be solved numerically,

but can be presented in a symbolical operator form

that creates the possibility of qualitative analyzes of its

solutions:

D(ω) = d+ D(ω)χ(ω)U ,

(23.57)

where U is a combination of the direct V

and exchange V

Coulomb interelectron potentials,

U = V

− V

, χ(ω) = χ

(ω) + χ

(ω), χ

(ω) = 1/(ω −



+ iη) and χ

(ω) = 1/(ω + ω



), with ω



being the ex-

citation energy of the virtual electron–vacancy state.

Using Γ from (23.31), one can present D(ω) as

D(ω) = d+ dχ(ω)Γ(ω) .

(23.58)

Equation (23.57) allows a rather simple, also symbolic,

solution

D(ω) = d/[1 − χ(ω)U] .

(23.59)

If the denominator in (23.59) has a solution Ω deter-

mined by the equation

1 − χ(Ω)U = 0 ,

(23.60)

at Ω>I,whereI is the atomic ionization potential, then

the cross section has a powerful maximum called a giant

resonance with energy Ω.

A giant resonance is of a collective nature, in the

sense that it appears to be due to coherent virtual

excitation of all electrons of at least one considered

multi-electron subshell.

These intra-shell correlations are most important

for multielectron shells with large photoionization cross

sections. Their inclusion leads to a quantitative descrip-

tion of the above mentioned giant resonances – huge

maxima in the photoionization cross sections. An exam-

ple is the 4d

photoionization cross section of Xe shown

in Fig. 23.2 [23.12], where satisfactory agreement with

experiment is demonstrated.

It appears that all RPAE intra-shell time-forward dia-

grams, such as that on the ﬁrst line in (23.53), and the

ﬁrst term in brackets in (23.56) with ε



= ε

,maybe

taken into account by the matrix element 

k |d|i.This

is the one-electron approximation, but with the function

(r) calculated in the term-dependent HF approxima-

tion [23.12]. Term-dependency means that only the total

angular momentum and spin and their projections for the

electron–vacancy pair are conserved, being equal to that

of the incoming photon. The individual values for the

electron and vacancy angular momentum and spin are

not considered to be good quantum numbers. Thus the

term-dependent HF includes a large fraction of RPAE

correlations.

Part B 23.2

372 Part B Atoms

4567891011

(Mb)

ω (Ry)

Fig. 23.2 Photoabsorption in the vicinity of the 4d

sub-

shell threshold in Xe [23.7]. Solid line: RPAE; dashed line:

experiment

RPAE permitted to the prediction of interference res-

onances. To describe them, let us consider a situation in

which the direct HF amplitude d

is small, while there

are other electrons with large photoionization amplitude

, D

(ω)  d

. Then, from (23.57) one has

(ω) ≈ d

+ D

(ω)χ(ω)

≈ D

(ω)χ(ω)U

 d

(23.61)

if the inter-transition interaction U

is not too small.

The enhancement of the photoionization amplitude de-

scribed by (23.61) manifests itself as a resonance in

the partial cross section of s electrons photoioniza-

tion. Very often the term D

(ω)χ(ω)U

and d

are

of opposite sign, so that the total amplitude acquires

two minima, along with an extra maximum, thus form-

ing a rather complicated structure in the partial cross

section that was named interference or correlation

resonance.

Usually, these resonances are manifestations of

inter-shell correlations. These are taken into account if

the sum over i



in (23.56) includes terms with ε



= ε

An example is the 5s

subshell in Xe, which is strongly

affected by the outer 5p

and inner 4d

neighboring

electrons. Due to this interaction, the 5s

cross section

is completely altered, as illustrated in Fig. 23.3 [23.12].

The RPAE results predict a qualitative feature of the ex-

perimental data, namely the formation of a maximum

and two minima in the cross section. The second mini-

mum is not seen in Fig. 23.3, since it lies at considerably

higher ω.

RPAE is able to describe a number of other ef-

fects, such as giant autoionizational resonance (decay

of a powerful discrete excitation into a continuum,

with which the excitation interacts strongly), continu-

ous spectrum autoionization (modiﬁcation of a broad

continuous spectrum excitation due to its strong in-

teraction with a narrow continuum that happens in

negative ions [23.22]) and quadrupole giant reso-

nances [23.24].

Above we concentrated on dipole Giant resonances.

Quadrupole amplitudes in RPAE are determined by an

equation similar to (23.57):

Q(ω) = q + Q(ω)χ(ω)U ,

(23.62)

where q is the quadrupole amplitude in HF approxima-

tion.

Giant quadrupole resonance was found in excitations

of 4d

electrons in Xe [23.25]. Its direct observation in

photoabsorption is almost impossible, since the corres-

ponding cross section is very small due to the inclusion

of the extra factor α

= 1/c

≈ 10

−4

as compared to the

dipole cross section. However, the quadrupole amplitude

leads to noticeable corrections to the angular distribu-

tions of photoelectrons where their relative contribution

is considerably bigger.

Note that the amplitude of electron elastic scattering

on an atom with a vacancy is expressed in RPAE via Γ

given by (23.31) [23.26].

For the inner or deep intermediate shells, RPAE

proves to be insufﬁcient. First, screening of the Coulomb

interaction between the outgoing or virtually excited

electron and the vacancy [see (23.4f )] must be taken

into account. This can be done by replacing V by Γ

1.0

0.5

24 6 810

σ (Mb)

ω (Ry)

Fig. 23.3 Photoionization of 5s

electrons in Xe [23.7].

Solid line: RPAE with effects of 5p

and 4d

included;

dashedline:5s

electrons only; dash-dottedline: with effect

of 5p

electrons; dash-double-dotted line: with effect of

electrons; dotted line: experiment

Part B 23.2

Many-Body Theory of Atomic Structure and Processes 23.2 Calculation of Atomic Properties 373

from (23.31). The ionization potential (or the energy of

the vacancy i) must also be corrected, which requires

inclusion of at least the contribution from the ﬁrst term

of (23.38). It has been demonstrated [23.27] that the

screening of the electron–vacancy interaction can be

taken into account by calculating the wave function of

the virtually excited or outgoing electron in the self-

consistent HF ﬁeld of an ion instead of that of a neutral

atom. A method which uses only these one-particle wave

functions in (23.54)(23.56) is called the generalized

RPAE or GRPAE. The use of this approximation con-

siderably improves the agreement with experiment near

the intermediate shell thresholds, decreasing there the

cross section value and shifting its maximum to higher

energies. GRPAE permitted to disclose intra-doublet

resonances that results from interaction of electrons

belonging to two components of the spin-orbit dou-

blet, e.g., 3d

3/2

and 3d

5/2

inXe,CsandBaatoms

[23.24, 28].

RPAE and GRPAE corrections affect not only the

cross sections but also characteristics of the photoelec-

tron angular distribution, i. e., dipole and non-dipole

angular anisotropy parameters [23.12, 16, 24]. As an

example, Fig. 23.4a presents the partial cross sections

[23.28] while Fig. 23.4b depicts the dipole anisotropy

parameter β [23.24] for 3d

5/2

and 3d

3/2

electrons in

Cs. The effect of intra-doublet resonance – an additional

maximum in the 3d

5/2

cross section under the action of

3/2

electrons – is clearly seen.

Figure 23.5 depicts the non-dipole angular aniso-

tropy parameter γ

for 5s electrons in Xe [23.29]. The

parameter γ

(in Fig. 23.5, n = 5) is given by the simple

formula [23.16]

(ω) = 6[|Q

(ω)|/|D

(ω)|] cos(∆

− ∆

(23.63)

where Q

) are the RPAE (GRPAE) quadrupole

(dipole) photoionization amplitudes and ∆

(∆

) are

their phases. Thus, γ

(ω) is sensitive to the pres-

ence of interference, dipole, and quadrupole res-

onances. The latter is presented by a small but

noticeable maximum on the high energy slope of

the huge maximum, caused by the presence of

the giant dipole resonance. The variation of γ

near the 5s threshold is determined by the reso-

nant behavior of cos(∆

− ∆

), called phase reso-

nance [23.24].

Close to inner shell thresholds, the Auger decay

of a deep vacancy must be taken into account. Due

to decay, the photoelectron instantly ﬁnds itself in the

ﬁeld of at least two vacancies instead of one, leading

720 730 740 750 760 770

–1

720 730 740 750 760 770

Cross section, Mb

Photon energy, eV

Cs3d

Fig. 23.4a,b Intra-doublet resonance in 3d

Cs. (a) Par-

tial photoionization cross sections σ

5/2

and σ

3/2

[23.28],

(b) Dipole angular anisotropy parameters β

5/2

and β

3/2

Solid line:datafor5/2 with account of 3/2; Dash-dotted

line:datafor5/2 without account of 3/2; Dotted line:data

for 3/2 with account of 5/2; Dashed line:datafor3/2

without account of 5/2

to considerable growth of the threshold cross section.

Diagrammatically, the effect of decay may be described

(23.64)

Here, the double line emphasizes that starting from the

instant of decay, the photoelectron moves in the ﬁeld

of double instead of a single i vacancy. For inner

vacancies, this is a strong effect which can lead even to

recapture of the photoelectron into some of the discrete

levels in the ﬁeld of the double vacancy j

Part B 23.2

374 Part B Atoms

0.5

–0.5

–1

50 100 150 200

Photon energy, eV

RPAE

RRPA

Fig. 23.5 Nondipole anisotropy parameter γ

(ω) of 5s

electrons in Xe [23.29]

So-called Post-collision interactions in photoioniza-

tion can be taken into account by this diagram when

the Auger electron is much faster than the photoelectron

[23.30]. If their speed is of the same order, their mutual

Coulomb repulsion must be accounted for, leading to ad-

ditional alteration of energy and angular redistributions.

A photoelectron can excite or knock out another

atomic electron. To lowest order in the residual in-

terelectron interaction, this process can be represented

(23.65)

While the formation of an initial electron–vacancy ki

pair requires RPAE or GRPAE for its description, the

second step of (23.65) can be reasonably well repro-

duced by the lowest-order term in V. It appears that

process (23.65) has a high probability for not too fast

photoelectrons, changing considerably the cross sec-

tion [23.31] for ionization by creating a vacancy i

A comparatively simple diagram

(23.66)

describes the photoionization process in which a more

complicated state is created in the ion than a single

vacancy, e.g. a state with two vacancies and one electron.

Here the doubled arrow indicates that the electron is in

a discrete level n.

Diagrams (23.64–23.66) present some corrections

which mix electron–vacancy and two-electron–two-

vacancy conﬁgurations. Each additional interaction line

increases the number of possible physical processes

considerably. With growth of the number of particles

actively participating in a process, the calculational difﬁ-

culties increase enormously. However, this is not a short-

coming of the diagrammatic approach, but a speciﬁc

feature of more and more complex physical processes.

23.2.7 Photon Emission

and Bremsstrahlung

The amplitude of photon emission in lowest order is

given by the time-reverse of (23.2c):

kk'

(23.67)

This diagram represents ordinary Bremsstrahlung; i. e.

a process of projectile deceleration in the ﬁeld of the

target. If the target has internal structure as atoms do,

it can be really or virtually excited during the collision

process. The simplest excitation means creation of an

Part B 23.2

Many-Body Theory of Atomic Structure and Processes 23.3 Concluding Remarks 375

electron–vacancy pair. The annihilation of this pair re-

sults via the time-reverse of (23.2a) in photon emission.

The process thus looks like

kk'

k''

(23.68)

To obtain the total Bremsstrahlung amplitude, the terms

(23.67)and(23.68) must be summed. The polarization

radiation (PR) created by the mechanism (23.68)has

a number of features which are different from the ordi-

nary Bremsstrahlung (OB) represented by (23.67). The

intensity is proportional to 1/M

for OB,whereM

the projectile mass, and the spectrum, at least for high

, is proportional to 1/ω. On the other hand, the PR

intensity is almost completely independent of M

and

its frequency dependence is quite complex, being de-

termined by the target polarizability α(ω) [23.32]. PR

is most important for frequencies ω of the order of and

higher than the target’s ionization potential. At sufﬁ-

ciently large distances and for neutral targets, PR starts to

predominate over OB. Close to discrete excitations of the



electron, the contribution (23.68) becomes resonantly

enhanced.

Higher-order corrections are important in the PR

amplitude. First, the Coulomb interaction V in (23.68)

must be replaced by Γ from (23.31).

The analytical expression for the total Bremsstrahl-

ung amplitude, including RPAE corrections to (23.68),

is given by the expression

k| A(ω)|k



=k |d|k



+







>F;i≤F

ki|V|k







× lim

η→+0

2(ε



− ε

)

− (ε



− ε

)

+ iωη

i|D(ω)|k



 .

(23.69)

To derive the Bremsstrahlung spectrum, the usual

general expression must be multiplied by the square

modulus of the ratio k|A(ω)|k



/k|d|k



.Iftheincom-

ing electron is slow, corrections (23.46) also become

important. The intermediate state in (23.68) includes

two electrons k



and k, and a vacancy i. The extent of

interaction between them could be considerable.

An important feature of PR is that it is nonzero even if

the projectile is neutral, but is able to polarize the target.

For example, it leads to emission of continuous spectrum

radiation in atom–atom collisions, whose intensity for

frequencies of the order of the ionization potentials is

close to that in electron–atom collisions.

The second and higher orders in the residual interac-

tion involve processes more complicated than (23.68),

for instance those which include simultaneous photon

emission and target excitation (ionization) [23.33].

23.3 Concluding Remarks

It is most convenient to apply diagrammatic techniques

to closed shell atoms whose ground state is non-

degenerate. Degeneracy means that some of the energy

denominators (23.13) become zero with nonzero statis-

tical weight. All such contributions must be summed

to eliminate this degeneracy. This leads to strong mix-

ing of some states. For example, the energy required

for electron–vacancy transitions jn [see (23.38)] within

an open shell is zero, thus leading to strong mixing of

i and ijn states. If a pair with zero excitation energy

has nonzero angular momentum, taking into account the

mixing within such a pair destroys angular momentum

as a characteristic of a one-vacancy state. This makes all

calculations much more complicated, reﬂecting a spe-

ciﬁc feature of the degenerate physical system.

In using the diagrams and formulas presented above,

it is essential that the interelectron and electron–

nucleus interactions be purely potential. Inclusion of

retardation and spin-dependence in the interparticle

interaction makes the calculations much more com-

plicated. These parts of the interaction appear as

relativistic corrections. They are comparatively small

in all but the heaviest atoms, and can be taken into

account perturbatively. Beyond lowest order, these

additional interactions are strongly altered when vir-

tual excitations of electron–vacancy pairs and the

Coulomb interaction between them is taken into ac-

count. An example is given by the sequence of

diagrams

…

(23.70)

Part B 23.3

376 Part B Atoms

where the heavy dashed line stands for the spin-

dependent interelectron interaction. Note that here

there are no electron–vacancy loops as in (23.28)

because the Coulomb interaction is unable to af-

fect the electron spin and thus to transfer spin

excitations.

The same kind of diagram describes the one-particle

ﬁeld acting upon an electron or vacancy due to the

presence of spin-orbit interaction or weak interaction

between electrons and the nucleus. For instance, the ef-

fective weak potential includes contributions from the

sequence

…

wi wi

(23.71)

This is another example demonstrating how ﬂexible and

convenient the many-body approach is for considering

different processes and interactions.

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Part B 23