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

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690 Part D Scattering Theory

45.5 G. F. Gribakin, V. V. Flambaum: Phys. Rev. A 48,546

(1993)

45.6 Z. L. Petovi c, T. F. O’Malley, R. W. Crompton: J. Phys.

B 28, 3309 (1995)

45.7 K. W. Ford, J. A. Wheeler: Ann. Phys. (NY) 7,259

(1959)

45.8 K. W. Ford, J. A. Wheeler: Ann. Phys. (NY) 7, 287

(1959)

45.9 M. V. Berry, K. E. Mount: Rep. Prog. Phys. 35,315

(1972)

45.10 H. S. W. Massey, E. H. S. Burhop, H. B. Gilbody

(Eds.): Electronic and Ionic Impact Phenomena,

Vol. 1–5 (Clarendon, Oxford 1969–1974)

45.11 E. W. McDaniel, J. B. A. Mitchell, M. E. Rudd: Atomic

Collisions: Heavy Particle Projectiles (Wiley, New

York 1993) Chap. 1, p. 47

(A) Textbooks on Scattering Theory

G.1 N. F. Mott, H. S. W. Massey: The Theory of Atom-

ic Collisions, 3rd edn., (Clarendon, Oxford 1965)

Chaps. 2–5

G.2 R. G. Newton: Scattering Theory of Waves and Par-

ticles (McGraw-Hill, New York 1966)

G.3 L. S. Rodberg, R. M. Thaler: Introduction to the

Quantum Theory of Scattering (Academic, New York

1967)

G.4 H. S. W. Massey, E. H. S. Burhop, H. B. Gilbody (Eds.):

Electronic and Ionic Impact Phenomena (Clarendon,

Oxford 1969 –1974) Vols. 1–5

G.5 M. R. C. McDowell, J. P. Coleman: Introduction to

the Theory of Ion-Atom Collisions (North-Holland,

Amsterdam 1970)

G.6 M. S. Child: Molecular Collision Theory (Academic,

New York 1974)

G.7 C. J. Joachain: Quantum Collision Theory (North-

Holland, Amsterdam 1975)

G.8 L. D. Landau, E. M. Lifshitz: Course of Theoretical

Physics, 3rd edn. (Pergamon, Oxford 1976) Vol. 1,

Chap.4,p.41

G.9 H. Eyring, S. H. Lin, S. M. Lin: Basic Chem-

ical Kinetics (Wiley, New York 1980) Chap. 3,

p. 81

G.10 B. H. Bransden: Atomic Collision Theory,

2nd edn. (Benjamin-Cummings, Menlo Park

1983)

G.11 B. H. Bransden, C. J. Joachain: Physics of Atoms and

Molecules (Wiley, New York 1983)

G.12 E. E. Nikitin, S. Ya. Umanski

ı: Theory of Slow

Atomic Collisions (Springer, Berlin, Heidelberg

1984)

G.13 E. W. McDaniel: Atomic Collisions: Electron and Pho-

ton Projectiles (Wiley, New York 1989) Chaps. 3 and

4, p. 71

G.14 M. S. Child: Semiclassical Mechanics with Molecular

Applications (Clarendon, Oxford 1991)

G.15 A. G. Sitenko: Scattering Theory (Springer, Berlin,

Heidelberg 1991)

G.16 E. W. McDaniel, J. B. A. Mitchell, M. E. Rudd: Atomic

Collisions: Heavy Particle Projectiles (Wiley, New

York 1993) Chap. 1, p. 1

(B) Chapters in Edited Books

G.17 E. H. Burhop: Quantum Theory I. Elements,ed.by

D. R. Bates (Academic, New York 1961), Chap. 9, p.

300

G.18 E. A. Mason, J. T. Vanderslice: High-Energy Elas-

tic Scattering of Atoms, Molecules and Ions,

in Atomic and Molecular Processes,ed.by

D. R. Bates (Academic, New York 1962) Chap. 17,

p. 663

G.19 B. L. Moiseiwitsch: Atomic and Molecular Processes,

edited by D. R. Bates (Academic, New York 1962),

Chap.9,p.281

G.20 A. Dalgarno: Ion-Molecule Reactions,ed.by

E. W. McDaniel, V.

Cerm

ak, A. Dalgarno, E. E. Fer-

guson, L. Friedman (Wiley, New York 1970) Chap. 3,

p. 159

G.21 K. W. Ford, J. A. Wheeler: Semiclassical Description

of Scattering, Ann. Phys. (NY) 7, 259 (1959)

G.22 K. W. Ford, J. A. Wheeler: Applications of Semiclas-

sical Scattering Analysis, ibid. 287 (1959)

G.23 M. V. Berry, K. E. Mount: Semiclassical Approxima-

tions in Wave Mechanics, Rep. Prog. Phys. 35,315

(1972)

G.24 W. H. Miller: Adv. Chem. Phys. 25, 69 (1974)

G.25 W. H. Miller: Adv. Chem. Phys. 30, 77 (1975)

G.26 M. S. Child: Dynamics of Molecular Collisions: Part

B, edited by W. H. Miller (Plenum Press, New York

1976) Chap. 4, p. 171

G.27 H. Pauly: Atom-Molecule Collision Theory: A Guide

for the Experimentalist ed. by R. B. Bernstein

(Plenum Press, New York 1979) Chap. 4, p. 111

Part D G

Elastic Scattering: Classical, Quantal, and Semiclassical General References 691

G.28 S. Stolte, J. Reuss: Atom-Molecule Collision Theory:

A Guide for the Experime?ntalist ed.byR.B.Bern-

stein (Plenum Press, New York 1979) Chap. 5, p.

231

G.29 J. N. L. Connor: Semiclassical Methods in Molecu-

lar Scattering and Spectroscopy, ed. by M. S. Child

(D. Reidel, Boston 1980) p. 45

G.30 R. E. Johnson: Encylopedia of Physical Science and

Technology (Academic, New York 1987) Vol. 2, p. 224

G.31 J. J. H. van Biesen: Vol. I, ed. by G. Scoles (Oxford

Univ. Press, New York 1988) Chap. 19, p. 472

G.32 U. Buck: Atomic and Molecular Beam Methods,ed.

by G. Scoles (Oxford University Press, New York,

1988) Chap. 20, p. 499

Part D G

693

Orientation an

46. Orientation and Alignment in Atomic

and Molecular Collisions

This chapter deals with the concepts of orientation

and alignment in atomic and molecular physics.

The terms refer to parameters related to the shape

and dynamics of an excited atomic or molecu-

lar level, as it is manifested in a nonstatistical

population of the magnetic sublevels. To take full

advantages of the possibilities of this approach,

one utilizes “third generation experiments”, i. e.,

scattering experiments which exploit the planar

scattering symmetry, contrary to an angular dif-

ferential cross section determination (a “second

generation experiment”) having cylindrical sym-

metry, or a total cross section measurement (a “ﬁrst

generation experiment”) integrating over all scat-

tering angles. In this way one is often able to probe

atomic collision theories at a more fundamental

level, and in favorable cases approach a “per-

fect scattering experiment” in which the complex

quantal scattering amplitudes are completely de-

termined. This term was coined by Bederson [46.1]

and has since served as an ideal towards which

scattering experiments attempt to strive.

46.1 Collisions Involving Unpolarized Beams . 694

46.1.1 The Fully Coherent Case.............. 694

46.1.2 The Incoherent Case

with Conservation of Atomic

Reﬂection Symmetry.................. 697

46.1.3 The Incoherent Case

without Conservation of Atomic

Reﬂection Symmetry.................. 697

46.2 Collisions Involving

Spin-Polarized Beams .......................... 699

46.2.1 The Fully Coherent Case.............. 699

46.2.2 The Incoherent Case

with Conservation of Atomic

Reﬂection Symmetry.................. 699

46.2.3 The Incoherent Case

without Conservation of Atomic

Reﬂection Symmetry.................. 700

46.3 Example ............................................. 702

46.3.1 The First Born Approximation ..... 702

46.4 Recent Developments........................... 703

46.4.1 S → D Excitation ....................... 703

46.4.2 P → P Excitation ....................... 703

46.4.3 Relativistic Effects

in S → P Excitation.................... 703

46.5 Summary ............................................ 703

References .................................................. 703

The study of anisotropies has a long history in atomic

physics, with light polarization, or Stokes parame-

ter analysis, as a prominent example. A pioneering

review by Fano and Macek [46.2] layed the mathe-

matical and conceptual foundation for most of the later

work. Advances in coincidence techniques, laser prepa-

ration methods, and development of efﬁcient sources

for polarized electrons [46.3] have boosted the ﬁeld fur-

ther. Parallel developments of powerful computational

codes have enabled a matching theoretical effort. In

this way, detailed insights into the collision dynam-

ics have been obtained, such as the locking radius

model for low energy atomic collisions, propensity

rules for orientation in fast electronic and atomic colli-

sions, and spin effects in polarized electron-heavy atom

scattering. Presentations of these developments are con-

tained in [46.4]andChapts.37, 50, 51, 63, 64,and

66. A comprehensive and critical review of the litera-

ture and the mathematical formalism was initiated by

NIST [46.5–7].

Space limitations allow only presentation of the

formalism for the simplest case of excitation. Recent

developments in the description of processes involving

polarized beams are included. The presentation is re-

stricted to atomic outer-shell excitation. Other reviews

describe excitation of inner shells [46.8] and molecu-

lar levels [46.9]. Related material on density matrices is

contained in Chapt. 7 [46.10].

Part D 46

694 Part D Scattering Theory

46.1 Collisions Involving Unpolarized Beams

46.1.1 The Fully Coherent Case

Consider ﬁrst the simplest nontrivial case of S →Pexci-

tation. A general property of an atomic collision is that

the total reﬂection symmetry with respect to the scat-

tering plane of the total wave function describing the

system is conserved. In simple cases, such as electron

impact excitation of He singlet states, or atom excita-

tion by fast, heavy particle impact, the projectile acts

as a structureless particle, with only the target atom

changing its quantum state. In this case, the reﬂection

symmetry of the target wave function alone is conserved.

Figure 46.1 shows the angular parts of the simplest

|lm states with l = 0, 1, 2. The arrows indicate how the

two families of states with positive or negative reﬂection

symmetry may couple internally. Thus, in an S → P

transition, an atom initially in an S-state may be excited

to the (p

, p

−1

) subspace, while the p

state is not

accessible. A characteristic feature of the corresponding

electron charge cloud is that it has “zero height”, i. e.,

zero electron density in the direction perpendicular to

–2

–1

––

Reflection

symmetry

Fig. 46.1 Illustration of the reﬂection symmetry of the

simplest spherical harmonics, corresponding to S, P, and

D states

the scattering plane. Furthermore, the expectation value

of the orbital angular momentum has a nonvanishing

component in this direction only.

The resulting electron charge cloud may thus have

a shape as shown on Fig. 46.2a. We shall now discuss

the parametrization of the wave function of this state

and analyze the connection between the wave function

and the corresponding photon radiation pattern emitted

when the state decays back to the initial S-state, using

the properties of electric dipole radiation.

Basic Deﬁnitions and Coordinate Frames

The coordinate frames of particular use in describing

the wave function are as follows. The collision frame,

, y

, z

), is deﬁned by

 k

and

 k

× k

out

The natural frame,(x

, y

, z

), is deﬁned by

 k

and

 k

× k

out

. Finally, the atomic frame is iden-

tical to the natural frame, except that the frame is

rotated by an angle γ around z

= z

such that x

is parallel to the major symmetry axis of the P-state

charge cloud, as shown in Fig. 46.2. Most scattering

a) b)

c) d)

–

, z

, y

coll

, x

out

, y

, x

Fig. 46.2 (a) Shows the shape, or the angular part, of

a P-state with positive reﬂection symmetry with respect to

the scattering plane. Some relevant coordinate frames are

also indicated;

(b) shows a cut of this shape in the scatter-

ing plane.

in the scattering plane;

(d) is the polarization ellipse for

light emitted in the direction perpendicular to the plane, as

observed from above

Part D 46.1

Orientation and Alignment in Atomic and Molecular Collisions 46.1 Collisions Involving Unpolarized Beams 695

calculations use the collision coordinate system as ref-

erence frame, while mathematical analysis is often most

conveniently performed in the natural frame, where the

algebra is particularly simple. Many expressions are

even simpler in the atomic frame, but it has the disadvan-

tage that the angle γ varies with collision parameters,

such as impact velocity, impact parameter, etc.

The expansion of the P-state wave function in the

three coordinate systems is

|Ψ =a









−1



−1



(46.1)

= a





−1



−1



(46.2)

= a





−1



−1



(46.3)

Conservation of reﬂection symmetry in the scattering

plane implies

=−a

−1

, (46.4)

= 0 , (46.5)

= 0 . (46.6)

The normalization condition implies



−1



−1



= 1 . (46.7)

Finally, the a

coefﬁcients are related to the scattering

amplitudes f

and the differential cross section σ(θ) by

= f

out

)

σ(θ)

−

(46.8)

σ(θ) = (k

out

)







, etc.

(46.9)

Except for an arbitrary phase factor, the pure state (46.1)

may thus be characterized by two dimensionless param-

eters. Traditionally, they have been chosen as (λ, χ)

deﬁned as [46.11]

λ =



, (46.10)

χ = arg



∗



(46.11)

An alternative parametrization (L

⊥

,γ) is given by [46.5]

⊥



−



−1



−1



, (46.12)

γ =

arg



−1

∗



=−

(δ ±π) ,

(46.13)

–1

Fig. 46.3 Fully coherent S → P excitation may be

described by two independent scattering amplitudes, char-

acterized by their relative size and phase

with the notation of Fig. 46.3. Referring to the natural

coordinate frame, the expectation value of the electronic

orbital angular momentum is thus

Ψ |L|Ψ =(0, 0, L

⊥

). (46.14)

Coherence Analysis: Stokes Parameters

We now discuss the information obtainable from a po-

larization analysis of the emitted light. In the notation

of classical optics (see, e.g., Born and Wolf [46.12])

the components of the Stokes vector (P

, P

) meas-

ured in the direction +y

(+z

) perpendicular to the

scattering plane and deﬁned by

= I(0

◦

) −I(90

◦

), (46.15)

= I(45

◦

) −I(135

◦

), (46.16)

= I(RHC) − I(LHC), (46.17)

with

I = I(0

◦

) +I(90

◦

)

= I(45

◦

) +I(135

◦

)

= I(RHC) + I(LHC),

(46.18)

are given by

= 2λ −1 , (46.19)

=−2



λ(1 −λ) cos χ, (46.20)

= 2



λ(1 −λ) sin χ, (46.21)

or, alternatively,

= P

cos 2γ, (46.22)

= P

sin 2γ, (46.23)

=−L

⊥

, (46.24)

with



+ P

. (46.25)

Part D 46.1

696 Part D Scattering Theory

Here, I(θ) is the intensity transmitted through an ideal

linear polarizer with transmission direction tilted by an

angle θ with respect to the z

or x

direction. Similarly,

RHC (LHC) refers to photons with negative (posi-

tive) helicity. Inspection of (46.19)to(46.25)shows

that determination of the Stokes vector in the direc-

tion perpendicular to the collision plane determines the

wave function (46.1) completely. A determination of

the Stokes vector thus constitutes a “perfect scattering

experiment”.

The Stokes vector (P

, P

) is a unit vector char-

acterizing the state |Ψ , and it may be represented by

a point on the Poincaré sphere, see Fig. 46.4.

Correlation Analysis

Another experimental approach is to map the angular

distribution of the photons emitted in the subsequent

S → P decay. For this purpose we ﬁrst note that the

angular part Υ(θ, φ) of the electron probability density

Ψ |Ψ  corresponding to the wave function (46.1)may

be written as

Υ(θ, φ) =

[1 + P

cos 2(φ −γ)]sin

θ. (46.26)

Figure 46.2(b) shows a cut of this charge cloud in the

scattering plane,



π, φ



[1 + P

cos 2(φ −γ)] . (46.27)

)

2γ

Fig. 46.4 The Poincaré sphere. The Stokes vector for a pure

state |Ψ corresponds to a characteristic polarization ellipse,

represented by a point on the unit sphere

The relative length, l, and width, w of the charge cloud

are given by

l =

(1 + P

), (46.28)

w =

(1 − P

). (46.29)

The degree of linear polarization P

is thus a measure of

the shape of the charge cloud, since

l−w

l+w

(46.30)

According to the properties of electric dipole radia-

tion, the pattern I(φ) in the scattering plane is

I(φ) =

[1 − P

cos 2(φ −γ)] . (46.31)

A mapping of the radiation pattern in the collision plane

determines γ and P

, and thereby the absolute value of

the angular momentum by

⊥



1 − P

. (46.32)

However, the sign of L

⊥

cannot be determined by cor-

relation analysis, and this approach accordingly does

not qualify for classiﬁcation as “a perfect scattering

experiment”.

Density Matrix Representation

Recalling that ρ

= a

∗

, the density matices in the

various basis sets introduced above are

√







√

(1−P

) −P

+iP

−

√

(1−P

)

−P

−iP

√

2(1+P

) P

+iP

−

√

(1−P

) P

−iP

√

(1−P

)







(46.33)







1 +L

⊥

0 −P

−2iγ

00 0

−P

2iγ

01−L

⊥







(46.34)







1 +L

⊥

0 −P

000

−P

01− L

⊥







(46.35)

The density matrices illustrate the algebraic simpli-

ﬁcations obtained using the natural coordinate frame.

In the following, this frame will be used unless other-

wise stated, and we therefore suppress the superscript n

below.

Part D 46.1

Orientation and Alignment in Atomic and Molecular Collisions 46.1 Collisions Involving Unpolarized Beams 697

Postcollisional Depolarization due to Fine

Structure and Hyperﬁne Structure Effects

After the collision, the isolated atom evolves under

the inﬂuence of internal forces, such as ﬁne structure

and hyperﬁne structure, until the optical decay. While

typical collision times are of the order 10

−15

s, the

light emission occurs after a time interval of 10

−9

This is long compared with the Larmor precession time

of the electron spin (∼10

−12

s) and one can thus assume

that the atom has completely relaxed into, e.g., its

1/2

and

3/2

states before the photon emission. For the sim-

plest nontrivial case of electron spin S =

, the Stokes

vector P(S) of the subsequent

P →

S transition is

modiﬁed to become

1,2





1,2

(0), (46.36)





(0), (46.37)

and the Stokes vector is thus no longer a unit vector. In

general, the Stokes vector components are reduced by

depolarization factors c

(a table of c

coefﬁcients for

the most common values of electron and nuclear spin is

given in Appendix B of [46.5])

(S) = c

(0). (46.38)

This suggests introduction of reduced polarizations

= P

(S)/c

. (46.39)

The reduced Stokes vector (P

, P

) is again a unit

vector, and the formalism developed above for the spin-

less case can then be applied. We shall assume that

this correction has been made, and the “bar” will be

dropped.

46.1.2 The Incoherent Case

with Conservation of Atomic

Reﬂection Symmetry

The picture outlined above has to be modiﬁed in

cases where the experiment sums over several, in prin-

ciple distinguishable, channels, each of which, however,

conserves reﬂection symmetry. Prototype examples are

electron impact excitation of hydrogen or light alkali

atoms. Here, the excitation process is described by

singlet and triplet scattering amplitudes respectively,

and we have the possibility of direct and exchange

scattering. This doubles the number of scattering am-

plitudes from two to four. The experimental results are

thus an incoherent sum over these channels (singlet

and triplet), the unraveling of which would require ap-

plication of spin-polarized beams, see Sect. 46.2.The

atomic P-state can no longer be described as a sin-

gle pure state, (46.1), but as a mixed state (Chapt. 7).

The expressions (46.33)to(46.35) for the density ma-

trix are unchanged, but the matrix elements are now

sums of the contributions from the individual channels

(We shall discuss its decomposition below). P

and L

⊥

are now independent quantities, and the (reduced)

Stokes vector P is generally no longer a unit vector.

The degree of polarization P may thus be less than

one, i. e.,

= P

+ P

= P

⊥

≤ 1 . (46.40)

For electron impact excitation, deviation of the param-

eter P from unity may thus serve as a measure of the

effect of electron exchange. We have accordingly three

independent observables, e.g., (L

⊥

,γ,P

). This set of

variables is frame-independent. A photon correlation ex-

periment in the scattering plane may extract the (γ, P

)

pair, while coherence analysis in the z direction provides

the complete set. An alternative set of (frame-dependent)

parameters used in particular for hydrogen is (λ, R, I),

with

λ =

(1 + P

), (46.41)

R =−

√

, (46.42)

I =

√

. (46.43)

46.1.3 The Incoherent Case

without Conservation of Atomic

Reﬂection Symmetry

In the general case, the assumption of positive reﬂec-

tion symmetry with respect to the scattering plane for

the wave function of the excited atom cannot be main-

tained. For example, for electron impact excitation of

a P-state of a heavy atom, spin-orbit effects may be

so strong that they ﬂip the electron spin, thereby al-

lowing population of the |p

 state. Thus, the total

number of scattering amplitudes is now six. Typical ex-

amples are mercury, or the heavy noble gases. The total

angular momentum of the excited state is J = 1, the

ﬁne structure being completely resolved in these cases.

Strictly speaking, the density matrix elements no longer

describe the electronic charge cloud, but rather the ex-

cited state (J = 1) distribution. Similarly, L

⊥

should be

Part D 46.1

698 Part D Scattering Theory

replaced by J

⊥

, but for simplicity we keep the nota-

tion. This state radiates as a set of classical oscillators,

completely analogous to the

state described above,

so we shall maintain our previous notation, simply re-

placing the term charge cloud density by oscillator

density. The main difference from the cases consid-

ered above is that the density now displays a height,

see Fig. 46.5.

Blum, da Paix

ao and collaborators, [46.13,14], were

the ﬁrst to formulate a parametrization of the general

case. Here, we shall use as starting point the expression

for the density matrix in the natural frame and decom-

pose it into the two terms with positive and negative

reﬂection symmetry, respectively,

= (1 −h)







1 − P

0 −P

−2iγ

00 0

−P

2iγ

01+ P













000

010

000







(46.44)

a) b)

Fig. 46.5 The top row shows classical oscillator densities

for height parameters h = 0andh = 1/3, respectively. The

alignment angle is γ = 35

◦

and P

= 0.6 in both cases.

Below are shown cuts along the symmetry axes

where the linear polarization P

is labeled with a “+”

referring to the positive reﬂection symmetry. Similarly,

we deﬁne

⊥

≡−P

, (46.45)

≡ P

+ P

≤ 1 . (46.46)

The shape of the density is now characterized by the

three parameters

l = (1 −h)

(1 + P

), (46.47)

w = (1 −h)

(1 − P

), (46.48)

h = ρ

, (46.49)

with l +w +h = 1. There are thus four independent ob-

servables, chosen as (L

⊥

,γ,P

, h). Again, this set of

variables is frame-independent. Determination of the

height parameter evidently requires observation from

a direction other than the z

-direction. Traditionally,

the Stokes parameter (P

, 0, 0) is measured from the

-direction to obtain

h =

(1 + P

)(1 − P

)

4 −(1 − P

)(1 − P

)

(46.50)

Thus, all four parameters may be obtained from analysis

of the light coherence, but two directions of observa-

tion are necessary. Similarly, photon correlation analysis

in two planes are required to extract (γ, P

, h), see,

e.g., [46.5] for a discussion.

The various cases with unpolarized beams discussed

in this Section are summarized in Table 46.1.

Table 46.1 Summary of cases of increasing complexity,

and the orientation and alignment parameters necessary

for unpolarized beams. N

is the number of independent

parameters, and N

is the number of observation directions

required

Variable Sect.46.1.1 Sect.46.1.2 Sect.46.1.3

Forces Coulomb +exchange +spin–orbit

Representation wave func. ρ

Reﬂ. symmetry + + +, −

Ang. mom. L

⊥

Align. angle γ γ γ

Linear pol. P

Degree of pol. P =1 P ≤ 1 P

≤ 1

Height h = 0 h = 0 h ≥ 0

2 3 4

1 1 2

Part D 46.1

Orientation and Alignment in Atomic and Molecular Collisions 46.2 Collisions Involving Spin-Polarized Beams 699

46.2 Collisions Involving Spin-Polarized Beams

In this section we discuss the additional informa-

tion that can be gained by application of a polarized

beam compared with an unpolarized beam. Colli-

sions with polarized electron beams are discussed in

particular, but most of the ideas presented are eas-

ily generalized to beams of atoms in polarized, or

otherwise prepared, states. We shall keep the order

of increasing complexity introduced in the previous

section.

46.2.1 The Fully Coherent Case

As seen in Sect. 46.1.2, a simple example of a fully co-

herent excitation process is He 1

S → n

P excitation,

for which complete information can be obtained from

experiments using unpolarized beams. Consequently, in

this case application of polarized electrons will add noth-

ing new. Also, the polarization of the scattered electron

can be trivially predicted, since no change is possible in

the scattering process.

46.2.2 The Incoherent Case

with Conservation of Atomic

Reﬂection Symmetry

The targets of interest here are hydrogen or light al-

kali atoms with an electron spin S =

.Sincewe

now have the possibility of triplet (t) and singlet

(s) scattering, this doubles the number of scatter-

–1

∆

–1

Fig. 46.6 For

S →

P electron impact excitation of hy-

drogen or light alkali atoms, four scattering amplitudes

come into play

ing amplitudes from two to four (recall that f

= 0)

see Fig. 46.6.

The amplitudes of interest are

= α

iφ

, (46.51)

−1

= α

−

iφ

−

, (46.52)

= β

iψ

, (46.53)

−1

= β

−

iψ

−

. (46.54)

Neglecting an overall phase, seven independent par-

ameters are needed to characterize the amplitudes

completely. Traditionally, one is chosen as the differ-

ential cross section σ

corresponding to unpolarized

particles. Six additional dimensionless parameters may

be deﬁned: three to characterize the relative lengths of

the four vectors, and three to deﬁne their relative phase

angles.

The density matrix is parametrized, in analogy to the

unpolarized beam case, according to [46.15]

= σ







1 +L

⊥

0 −P

−2iγ

00 0

−P

2iγ

01−L

⊥







(46.55)

= σ







1 +L

⊥

0 −P

−2iγ

00 0

−P

2iγ

01−L

⊥







(46.56)

where

= α

+α

−

, (46.57)

= β

+β

−

, (46.58)

⊥



−α

−



(46.59)

⊥



−β

−



(46.60)

2iγ

= P

+i P

=−

2α

−

−iδ

, (46.61)

2iγ

= P

+i P

=−

2β

−

−iδ

. (46.62)

In the case of an unpolarized beam, the total density

matrix becomes the weighted sum of the two matrices

Part D 46.2

700 Part D Scattering Theory

and ρ

,i.e.,

= σ







1 +L

⊥

0 −P

−2iγ

00 0

−P

2iγ

01−L

⊥







= 3w







1 +L

⊥

0 −P

−2iγ

00 0

−P

2iγ

01−L

⊥













1 +L

⊥

0 −P

−2iγ

00 0

−P

2iγ

01−L

⊥







(46.63)

where

+3σ

4σ

, (46.64)

+3σ

4σ

= 1 −3w

, (46.65)

= (3w

)σ

. (46.66)

The six parameters σ

, L

⊥

, L

⊥

,γ

,andγ

have now

been introduced, leaving one parameter still to be cho-

sen. Inspection of Fig. 46.6 suggests, for example, the

angle ∆

. The fourth angle, ∆

−

, is then ﬁxed through

the relation

∆

−∆

−

= δ

−δ

= 2



−γ



(46.67)

The following set of six dimensioness parameters is thus

complete:

, L

⊥

, L

⊥

,γ

,∆

. (46.68)

Detailed recipes for extraction of the parameters from

coherence experiments are somewhat complicated, and

we refer to discussions in the literature [46.7, 15, 16].

Using spin-polarized electrons and spin-polarized tar-

gets, all parameters may be determined, except for

information about singlet-triplet phase differences, such

as ∆

The reduced Stokes vector P of the unpolarized

beam experiment is given by the singlet and triplet (unit)

Stokes vectors P

s,t

P = 3 w

, (46.69)

from which the set of parameters (L

⊥

,γ,P

) for the

unpolarized beam experiment may be evaluated from

⊥

= 3 w

⊥

, (46.70)

2iγ

= 3w

2iγ

. (46.71)

Since in general, L

⊥

= L

⊥

and γ

= γ

, this causes the

(reduced) degree of polarization P to be smaller than

unity.

To summarize this section, Stokes parameter anal-

ysis may provide ﬁve dimensionless parameters, the

relative phase between the two f

and f

−1

ampli-

tudes and the relative phase between the two f

and f

−1

amplitudes, as well as the relative sizes of

all four amplitudes. However, none of the relative

phases between any triplet and singlet amplitude can

be determined, and coherence analysis alone is thus

not able to provide a “perfect scattering experiment”.

The missing phase may be extracted from the STU

parameters of the scattered electron, ( [46.16, 17]and

Chapt. 7).

46.2.3 The Incoherent Case

without Conservation of Atomic

Reﬂection Symmetry

The results of coherence analysis of a J = 0

→ J = 1

transition will now be discussed. We only analyze

the photon polarization in the exit channel, not the

electron spin parameters. There are six independent

scattering amplitudes for a J = 0 → J = 1 transition

(Fig. 46.7), thereby requiring the determination of one

absolute differential cross section, ﬁve relative magni-

tudes, and another ﬁve relative phases of the scattering

amplitudes.

This large number of independent parameters

leads to considerable complications. Nevertheless, the

∆

–1

∆

–

Fig. 46.7 For J = 0 → J = 1 electron impact excitation of

heavy atoms six scattering amplitudes come into play since

spin-ﬂip may occur

Part D 46.2