Theory of Auger ejection of electrons from metals by ions

11

THEORY OF AllER EJ&::TlON OF EL&::TRONS FROM MEI'ALS BY lONS

by

Eric P. Wenaas

and

A. J.

Howsmon

Manuscript received December,

1968.

March,

1969.

UTlAS Report No.

137

f

ACKNOWLEDG~

The authors wish to express their appreciation to the fo11owing for their assistance and support: to the Computing Center at the State University of New York at Buffa10 for the use of the C. D. C.

6400

Computer; to the United States Air Force Office of Scientific Researchunder Grant ·Number AF-AFOSR-68-1481 and to the Nationa1 Research Counci1 of Canada under Grant Number

A5555

for their partia1 financia1 support; to Miss Do~na Braun for typing the manuscript; and to Miss Syde1 Mi11er for preparing the manusc±ppt for public-ation.

SUMMARY

A theory of the ejection of electrons from atomically clean metals by slow ions of the noble gases arising from Auger transitions is presented. Scattering matrix elements are evaluated from first principles and the second-ary ejected electron energy distribution and total yield are found. The theory accounts specifically for ion kinetic energy and quantitatively describes the experimentally observed secondary electron energy distribption and total yield.

The approach used in the solutio~ of this problem is that of the quantum mechanical theory of scattering, rather than first order perturbation theory used previously by other authors. ~his approach eliminates the artificial concept of determining transition matrix elements per unit time as a function of the distance of the ion from the metal surface, and will allow the direct calculation of the secondary electron e~ergy distribution, total yield, spatial distribution, position of most probable neutralization, and depth of origina-tion of Auger electrons.

The theory results in an integral equation for the scattered state vector for which the Born approximation is not directly applicable because of the strong ion-surface interaction. By choosing a suitable form for the Hamilton-ian and by treating part of the problem exactly, the integral equation is rearranged in such a way that the distorted-wave Born approximation becomes valid. The state vector of the system and therefore the scattering matrix elements describing the emission process are thus determined. There are several possible forms for the scattering matrix elements because the Auger process can be treated as a re arrangement collision where an initial and re-arranged state description are both possible. The most suitable form for the matrix elements is chosen resulting in a solution which gives the probability of finding the system in a final state given that it was in some particular initial state. By averaging over initial states, the secondary electron ener-gy distribution is found; by summing over final states, t,he total secondary electron yield is found. By rearranging summations the spatial distribution of the secondaries mayalso be found.

The formal solution is applied to the situation of He+ ions of low energy incident on atomically clean tungsten. Normal incidence is treated although the theory is not restricted to this case. Typical theoretical curves are presented and compared with the experimental res~lts of others. A discussion of such effects as energy broadening, effective ionization energy, uncertainty principle effects, inelastic electron scattering, and density of electron states in metals is included.

1.0 2.0

3.0

4.0

6.0

TABLE OF CONTENTS NOTATION INTRODUCTION SCATTERING THEORY 2.1 Quantum Formalism 2.2 Scattering Formalism

2.3

The Scattering Matrix

2.4

Ordinary Scattering

2.5

Re~arrangement Scattering

THE HAMILTONIAN AND-. UNPERTURBED STATES THE SCATTERING MATRIX ELEMENTS

THE MCDEL AND CALCULATIONS

5.1

The Crystal Electrons

5.2

~he Incident Ion

5.3

The Excited Electron

5.4

The Reflected Atom

5.5

The Initial Distorted Wave

5.6

The Final DistDvted Wave

5.7

The Interaction Potential

5.8

The Matrix Elements

RESULTS AND DISCUSSIONS SUMMARY

REFERENCES ₎ APPENDIX A:

APPENDIX B:

APPENDIX C:

9UT~INE OF THE SOLUTION WITH THE LATTICE HAMILTONIAN

SOLUTION OF SCHRODINGER'S EQUATION WITH AN EXPONENTIAL POTENTIAL COMPUTER PROGRAM iv PAGE v 1

5

7

9

14

16

20

25

30

34 34 38 38

39

42

44

46

46

50

NCYrATION

+

A Metastable atom

akI ,bk' Crystal electron wave function constants

~",bk" Crystal electron wave function constants

~,bk Excited electron wave function constants

b Hydrogenic atom wave constant c Subscript referring to the crystal cill

v· Solid angle elemen~

e Electron

f Final state

f' Final rearranged state g Reciprocal lattice vector

G Green's function

H Hamiltonian operator

H

0 Unperturbed system Hamiltonian H'

0 Post-interaction unperturbed system Hamiltonian

Hfi Matrix elements

Hfi (~) Direct matrix elements Hfi(E) Exchange ma~rix element s

i Initial state

I Projectile ionization potential It Modified Bessel function

...

k Excited electron wave vector with components k, k , and k

y z

k'

Crystal electron wave vector with components k'

,

k' and k'

y' z

kil _{Crystal electron wave vector with components kil}

_,

kil _{and kil}

y' Z

...

K Incident ion wave vector with components K, K , and K _y z

...

KI m M M(n) S s(x) u,v

v

V o 0:,0:1 TJ_o ~ ep epi epf X~ x:( +) x·( -) '.l'

1.'( +)

'.l'( - )

n

(+ )

(-)

1 2 3

Reflected atom wave vector with components KI, KI, _y and KI

Z

Tangential component of wave vector Normal component of wave vector

Subscript referring to the crystal lattice points Electron and projectile masses

Crystal lattice point masses Metal with n electrons

Scattering matrix Unit step function Spin functions Potential

Perturbing potential depending upon particle-surface separ-ation

Perturbing potential depending upon internal coordinates of the projectile and target

Internal system variables

Total electron yield per incident ion

(k - kil)2 + (k - kl)2

Y Y z Z

(kl I + k l _ k )2 + (kl I + k l _ k )2 + b 2

Y

Y

Y

z z z

Density of electron states Phase angle

Fermi level

Unperturbed system wave function

Initial state unperturbed system wave function Final state unperturbed system wave function Distored wave function

but-going distortè.d", wave In-going distorted wave Total system wave function Out-going system wave function In-going system wave function Operator

Gut-going In-going

Crystal electron and excited electron identification nuIDber Crystal electron and atomic electron illdentification nuIDber , Incident and reflected projectile idèntification nuIDber

..

1.0 INTRODUCTION

This thesis is a study of secondary electron emission from a crystal caused by a beam of monoenergetic ions incident on the surface. The emission process that occurs some distance from the surface and involves the potential

energy of exci tation of the int::ident ion can be distinguished from processes ,IC , C

that occur much closer to the surface and involve the kinetic energy of the ion. The former process is usually referred to as potentialor Auger electron ejection and the latter as kinetic ejection. This work will be restricted to low ion energies (less than 600 eV) where potential ejection is predominant. Both processes rnay occur simultaneo'\lsly, however, and care must be taken to

disting~ish between the two, particularly when interpreting experimental

results.

Four possible transition processes have been identified: A~er ne'ltralization, Auger de-excitation, resonance ionization, and resonance

ne~tralization. They can be described as follows:

where

Auger neutralization Auger de-excitation Resonance ne»tralization

Resonance ionization

A: ground state atom

A+: excited atom (metastable) e: free electron

I: ion

M(n): metal with n electrons

I + M(n) ~ A + M(n-2) + e + M(n-l) A + M(n) ~A + + e I + M(n) ~ A + + M(n-l) + + M(n) ~ + M(n+l) A I

Two processes invlove an incident ion and secondary electron emission: Auger neutralization, and resonance neutralization followed by

Aug~r de-excitation; while two other processes invètve an incident metastable atom and secondary emission: resonance ionization followed by Auger

neutral-ization, and Auger de-excitation. Of the two possible processes involving an incident ion, Auger+neut~aliz~tion is ~n many cases the only process that can occur (e.g., for He ,

Ar ,

Kr , áhd Xe incident on tungsten). When the two processes can occur simultaneo~sly, both must be treated theoretically and their relative contributions to the emission phenomena determined.

The Auger neutralization process is schernatical~y depicted in

Fig-~e 1. The crystal is represented by a potential barrier of depth D which is filled with electrons up to the Fermi level~. Initially, the ion is incident on the crystal in which two of the many electrons are represented by dots in

Figure la. In Auger neutralization of the ion, one of the crystal electrons neutralizes the incident ion to the groundrstate which is then reflected as an atom, while the second electron receives energy from the neutralization process and may escape from the crystal as a secondary electron as shown in

Figure lb.

Secondary electron emission resulting from ion impact was first

by Thomson [2], who used alpha rays from polonium as a source of ions. The

phenomena was subsequently studied and interpreted by others including

Logeman

[3],

F~chtbauer

[4],

Bumbstead

[5],

Campbell

[6],

Pound

[7],

and

McLennan

[8].

It was during this period that the extreme sensitivity of the

electron emission process to gases adsorbed on the surface was discovered. A summary of this early work with alpha rays is presented in a review article by Geiger

[9].

Experimental work between

1915

and

1930

was performed with alkali

ions obtained from thermionic sources. Most of the data thus obtained by

Cheney

[10],

Holst and Oosterhuis

[11],

Klein

[22],

Jackson

[13],

and others

were conflicting due to the low ionization potentials of the alkali ions and

the presence of an uRdetermined amount of adsorbed gas on the target surface.

Significant results were obtained in

1930

by Gliphant

[14],

who

used a beam of helium ions which have a much higher ionization pbtential than

the alkali ions previously used. Because of the higher ionization potential

of the helium atom, he was able to experimentally detect and determine the

importance of the ionization energy in the emission process. Measurements obtained by Oliphant include the secondary electron energy distribution, total yield, and the effect of angle of incidence of the ion beam.

In the

20

years that fOllowed, scores of investigators measured the

secondary electron emission from various target materials arising from

incid-ent ion beams of noble gases, atmospheric gases, alkali metals, and others.

Results most frequently referenced include those of Pennig

[15],

Rostagni

[16],

Healea and Houtermans

[17],

D'Ans, DaRious, and Malaspina

[18],

Greene

[19],

Molnar

[20],

Lauer

[21],

and Varney

[22].

Most of these 'results must be discounted, however, b€cause they were obtained without specific

know-ledge of the condition of the surface or the incident beam.

More recent investigations, notably those of Hagstrum

[23-28],

and

Pro~st and Lûscher

[29],

were performed with much better control of

experi-mental conditions. Hagstr.um

[30],

for example, was able to show conclusively

that he was working with an atomically clean surface and an ion beam of known composition. He obtained data on electron energy distributions and total yields for noble gas ions incident on molybdenum, tantalum, tungsten, nickel, germanium, and silicon. Many others have obtained reliable results using

various ions and target materials. Klein

[31]

measured the spatial

distrib-ution of secondary electrons ejected from tungsten by noble gas ions with an

energy range of

300

to

4000

eV. Vance

[32]

measured the total secondary

electron yield as a function of incident ion beam angle for nOble gas ions

incident on tungsten. Very recently, French and Prince

[33]

measured the

electron energy distribution and total yield from tungsten caused by low energy diaûomic ionso

Because of the significant evidence of the validity of Hagstrum's and Probst and Lüscher's experimental data, which include measurements of secondary electron energy distribution and total yield, their experimental results will be compared with the theoretical results presented in this thes-is ..

Four fundamentally different theoretical approaches have been form-ulated to treat the Auger neutralization process. The first approach was the

2

'

.

treatment of Kapitza [34] who hypothesized that the incident ion lost all of its energy to a very small portion of the surface of the crystal causing local heating o~tthe order of several thousand degrees. This would result in therm-ionic emission which was, according to him, the observed secondary electron e.mission. Based on Richardson' s thearetical work [35] with emission of electrons from hot bodies, Kapitza made some calculations which seemed to agree with the limited experimental results of the day. Later, the experi~ ments of Compton and VanVorrhis [36] indicated that ions captured electrons from a metal before making thermal contact with the surface. , In light of these experiments, Oliphant and Moon [37] discussed two alternate processes by which ion neutralization and subsequent electron emission might occur.

The first supposition discussed by Qliphant and Moon was that neutral-ization is due to auto-electronic emission under the influence of the electro-static field of the approaching positive ion. This suggestion was actually first made by Holst and öosterhuis [38J. Oliphant and Moon used the theory of Fowler and Nordheim [39], and ~he Nordheim image-force formulas [40] in an attempt to calculate the probability of ion neutralization, but were only able to conclude that most of the slow ions would probably be neutralized before reaching the metal surface. ~he main objection to their theory, as they pointed out, was that it ignored the difference in the potential energy of ionization of differen~ kinds of atoms.

The other viewpoint, first suggested by Oliphant and Moon [37], form-ed the basis for the first detailform-ed treatment. They regardform-ed the process as the transition through the potential barrier of an electron of a given energy, from the metal to a state of equal energy associated with the ion. Oliphant and Moon did not make any calculations of secondary electron emission using this approach, but they did discuss theoretical upper and lower limits of secondary electron energy based on Sommerfield's wave theory of the state of electrons in metals.

Massey [41, :42] further developed the treatment of the Auger process based on the viewpoint of Oliphant and Moon using the first order perturbation theory of Oppenheimer [43] .. He calculated the probability that a system

initially in' a state with an ion at some finite distance in front of the metal and an electron inside the metal will make a transition to a state with an atom at the same distance in front of the metal and an electron outside the metal by considering the coulomb interaction potential between the two elect-rons as the perturba~ion. Massey viewed the process adiabatically, assuming that the time for the transition of the electron from intttal to final states was much less than the characteristic time of the incident ion. Thus, the transition matrix elements as calculated by Massey are a function of the

ion-surface separation. Massey applied this theory to the processes of the neutralization of positive hydrogen ions açd the de-excitation of metastable atoms incident on a metal surface.

Cobas and Lamb [44] applied Massey's formalism to the process by which a helium or a 23s metastable helium atom extracts an electnon from a metal. They also corrected an error in Massey's analysis of the metastable process. Varnerinf[45] examined the resonance transfer of an electron be-tween a metal and an ion in close proximity to the §urface, and conclu~ed that the results of Cobas and Lamb for the ejection of electrons by He from a molybdenum surface were seriously in error.

In both of the previously mentioned treatments, a free electron model

for the metal was assumed, but the exponential Ittailslt _{for the electron wave}

functions outside the metal were neglected. Shekhter

[46]

included the

ex-ponential tails in his analysis of the Auger"neutralization process and was

able to show that a sizable error results when these tails are neglected. He

was the first to show that the amount of energy released from the neutral-ization of an ion and subsequently radiat.ed by photoemission was small

com-pared to the energy transmitted to a secondary electron, although Oliphant

and Moon

[37]

were the first to suggest that this was the case. He was also

responsible for identifying both two-electron processes as examples of the

Auger effect. Sternberg

[47]

attempted to simplify Shekhter's results by

using an approximation valid only for large ion-surface separations. He treat-ed the four transition processes previously mentiontreat-ed and discusstreat-ed the

secondary electron emission process in some detail.

The primary emphasis in all the treatments discussed above centers around the calculation of matrix elements which describe an electron trans-ition from the metal to an ion located at a fixed distance from the metal

surface. In these ~reatments no serious attempt was made to calculate

measur-able quantities such as the secondary electron energy distribution, spatial distribution, and total yield per incident ion.

Hagstrum

[48]

developed an approach to calculate some of these

measur-able quantities but did not evaluate matrix elements from first principles. He assumed, apriori, a form for the matrix elements based on the work of

Shekhter

[46],

and Cobas and Lamb

[44],

and used experimental data from Auger

neutralization studies in order to evaluate certain parameters. He calculated the energy distribution of secondary electrons ejected by ions of the noble

gases incident on tungsten. However, he was not able to quantitatively

pre-dict the experimentally observed dependence of the total electron yield on the

incident ion kinetic energy. Later Hagstrum

[49]

extended his theory of Auger

neutralization to include semiconductors as well as metals. Other papers by

Hagstrum on the subject of ion neutralization at solid surfaces have been

published

[50-58].

An excellent sLunmary of the details of his theory is

con-tained in a book on atomic and ionic impact phenomena by Kaminsky

[59].

In the fourth approach, which has been treated in detail only by Probst [60], the electron transition is considered to be due to an internal

absorptîon of radiation. The radiation field set up by the first electron as

it neutralizes the ion is treated as the perturbation that causes the

second-ary electron emission. The problem can be conveniently divided into two

parts: the calculation of the probability that the first electron radiates

energy, and the calculation of the probability that a second electron absorbs the radiation. This approach accounts for two factors that have not been included in previous analyses, namely, the perturbation of the wave function of the metallic electron by the coulomb potentialof the ion, and the effect

of el*ctron-electron interactions. Probst applied his analysis to the case

of He on tungsten.

Burhop

[61]

has ShDwÁ that in the non-relativistic limit the approach

that treats the transitions as caused by the direct interaction of the metalic electrons leads to the same results as the approach that treats the trans-itions as caused by an internal absorbtion of radiation.

Recently Rydnik and Yavorskii

[62]

theoretically investigated the emission process with the semiclassical approach of Oliphant and Moon which used the Fowler-Nordheim theory. They claim to have arrived at the same

q~alitative, and to a certain extent, quantitative results as those given by the quantum approaches previously discussed. They applied their method to the process of the neutralization of a metastable atom incident on a metal surface, but no theoretical results were given for any of the measurable quantities.

Many other Soviet authors have contributed to the li~erature of electron excitation in solids including Abroyan, Arivof, Dorozhkin, Eremeev, Petrov, Rakhimov, and Tel'kovskii. The publications of these authors is too extensive to enumerate here, but Abroyan, Eremeev, and Petrov

[63J

have com-piled a survey on the subject listing nearly 200 recent references.

There are serious objections to all the previous treatments which used either radiation or perturbation theory to obtain transition matrix elements as a function of ion surface separation, including the following: it is necessary to fit to experimental data rather than to evaluate matrix elements from first principles; the total secondary electron yield as a function of incident ion energy cannot be predicted; the depth of origination of the Auger electrons cannot be determined; the spatial distribution of the secondary

electrons cannot be found; and the predicted yields in certain cases can differ from the experimen~al results by as much as

50%

.

Apart from the difficulties in obtaining certain results, there is a more fundament al problem i~ the theory arising from the use of a physically nonmeasurable transition matrix element which is a function of ion-surface separation. That approach requires the subsequent use of classical ideas to arrive at the physically measureable transition probability (i.e., secondary electron energy distribution and total yield). In order to enhance agreement of the theory with experi:rrent, it is necessary to add some quantum ideas such as energy level perturbations, effective ionization potentials, and uncertainty effects into the approach.

The theory presented here will attempt to overcome these difficulties by using a quaIftum scattering approach instead of the perturbation or ra.diation approach. A formal derivation of the scattering matrix elements will be given

(Chapter 2), and the Hamiltonian, the appropriate wave functions, and the forms of the matrix elements will be given (Chapter

3).

An approximate model useful for calculations will be presented (Chapter

4).

The expression for the second-ary electron energy distribution, total~yield, and spatial distribution will be presented in terms of the matrix elements of the previous section. The

indicated computations will be carried out to calculat~ the secondary electron energy distribution ~d total yield for the case of He on tungsten (Chapter

5).

The results and a discussion of the results (Chapter 6) will be given. Finally, future applications of ~he theory developed in this thesis will be discussed

(Chapter

7).

2.0 SCATTERING THEORY

Consider once again the system of Figure 1-1 comprised of an ion and two of the many conduction electrons that exist below the Fermi level in a crystal. The incident ion and the two electrons in the crystal will be de-signated the initial state of the system, while the reflected atom and second-ary ejected electron will be designated the final state of the system. ~he

problem is to find a method to determine the probability of a transition from the initial state to the final state.

The scattering and perturbation approaches both describe transitions occurring in the system and both crufbe formally derived from the same time-dependent quantum theory, but they convey different information about trans-itions. The choice of the approach depends upon the information desired. The perturbation approach describes the transition of the initial state with an

ion at a distance, s, in front of the crystal surface to the final state ~th

an atom also at a distance, s. This approach does not describe the total or cumulative transition probability as the ion moves toward the surface and the atom scatters away from the surface. The scattering approach, on the other hand, describes the total probability that a system in an initial state with an ion far from the crystal scatters into a final state with a secondary electron and an atom again far from the crystal.

Because the measurable quantities--including incident ion energy, secondary electron energy distribution, and total electron yield--are mea-sured at points far removed from the scattering center (i.e., the crystal surface), it is natural that the scattering approach, which describes this

sitriation, should be used. The following difficulties arise if the

pertur-bation approach is used:

(aj A classical calculation is necessary to find the total tran-sition probability from the trantran-sition probability as a function of ion-surface separation. This calculation becomes so complex that an approximation is necessary whereby most of the inform-ation originally contained in the transition probability function is lost.

(B) The initial and final states have not really existed for a long

time at each point, s, and therefore, have finite lifetimes. (c) The initial and final ion and surface states are mutually

per-turbed because they are separated by a finite distance, s. This causes a shift of the edergy levels in the crystal, the ion, and the atom, so tha~ initial and final wave functions are thus no

longer known. Because of the level shifts, the concept of an

effective ionization energy must be introduced.

The objections are removed by using scattering theory. The initial and final particle energies and wave functions need to be known only for large ion-surface separations where the mutual perturbation is absent and the

energies are well specified because they are measurable. The calculation is completely within the framework of quantum theory, and therefore, uncertainty

principle effects need not be considered separately. Finally, and most

important, the scattering matrix elements, which are not a function of ion-surface separation, describe the total probability of the transition so that the secondary electron energy distribution, total yield, spatial distribution, position of most probable neutralizatiorr, and depth of origination of Auger electrons can be directly calculated.

The formalism of quantum mechanics and quantum scattering theory will be briefly reviewed in order to provide a background for the development of

the present theory of Auger neutralization. The review is similar to the presentation in Merzba~~r

[64],

Goldberger and Watson

[65],

and Rodberg and ~haler

[66].

A discussion of these topics can also be found in the original

work of Dirac

[67],

~ippmann and Schwinger

[68],

Gell-Mann and Goldberger

[69]

and Lippmann [70]. The state vector approach has been chosen to simplify notation and make the discpssion as general as possible. Reference will be made to the more familiar coordinate representation wherever deemed necessary for the sake of clarity.

2~1 Quant~ Formalism

The state, or state vector ~, is a vector in an n-dimensional vector space such that each dynamical state of the sys~em can be represented by a vector in this space. In the bracket notation of Dirac, the state vector is denoted by ~ or equivalently by l~~>. The usual laws of vector algebra and

operational calculus are assumed to be valid, and one can form a scalar pro-duct by introducing a dual space. The vectors of the dual space are called bra vectors and are described in the Dirac notation by.(~ I. There is a one-to-one correspondence between the vectors of the two spaces, a correspondence which is assumed to be antilinear.

The scalar product <~I~»represents the overlap of the two states

and is interpreted as representing ~e probability amplitude for finding the

system simultaneously in a state ~ and~. The absolute square of the

prob-ability amplitude when properly normalized is the probability of finding the system simultaneously in states ~ and ~.

A giveR state may be transformed into another state by means of an

operator. The operator

n,

for example, transforms the state ~a into ~b:

cp =

n

~

b a

The expected value of an operator, which in quantum mechanics must be a

measurable quantity, or simply an observable, is given by

(2-1)

(2-

2

)

A set of eigenstates may be defined for the operator

n

by an eigen-value equation

n~

=

a~

a a

(2-3)

~he set of states ~ associated with an observable forms a complete set so that any arbitrary state can be represented by a superposition of such s~ates.

For example, the state ~ can be expanded in terms of the eigenfunction ~a

(2 ... 4)

The coefficients c can be found by using the orthogonal property of the set a

of ~I s

The quantity c therefore is interpreted as being the probability amplitude of finding theasystem simultaneously in state ~ and ~. By inserting

Equation (2-5) into Equation (2-4) a

I~>

=

I

II~)< ~

a

I~

>

(2-6)

a

and comparing with Equation (2-4), we obtain the closure relation

(2-7)

a

If a is a continuous variable, the closure relation becomes

~ I~(a)>-

da

<~(a)

I = 1 (2-8)

The state vector is postulated to satisfy Schrodingeris equation of motion gi ven by .».è~(t) h l

öt

=

H(t) ~(t) (2-9) where H(t) along with all time. explicitly

is the Hamiltonian operator. The specification of the Hamiltonian an initial condition is sufficient to describe the state ~(t) for If the system is conservative so that the Hamiltonian does not depend upon time, then the formal solution to Equation (2-9) is

~(t) = exp[-(i/~) H(t - t )]~(t )

o 0

or by setting t = 0, the solution becomes o

~(t) = exp[-(i/~) H(t~]~(O)

(2-10 )

where ~(O) is the initial state of the system which is assumed to be known. One type of solution having the form of Equation (2-10) is a stationary state given by

The time-independent state ~k is a function only of the coordinates in space. A wave packet may be fonmèd by the superposition of stationary

\

r.

dk

.a. ~(t)

=

J(2H)3

A(k)~k exp[-(i/n)Ekt] ( 2-12) state states (2-13) The substitution of Equation (2-13) into Equation (2-9) leads to the

time-independent equation

(2-14) The coordinate representation of Equation (2-14) can be found af ter defining the coordinate representation of the state vector and operator. The coordinate representation of ~ is

The coordinate representation of nl~) is found using Equation (2-8)

(?lnl'1?=

J<~lnl;~ ~,

(;,

I~)

The coordinate representation of the operator

n

is

For a loc al operator

and therefore, Equation (2-16) becomes

<rlnl~>=

f

n(r')5 (; -

;')~(r')dr'

n(;)~(r) . (2-16) (2-17) (2-18) (2-19) (2-20)

The equation of motion (2-14) can now be put into the coordinate representat~on

using Equation (2-20)

(2-21)

and finally

(2-22)

2.2 Scattering Formalism

In the scattering experiment a beam of partieles from a source is focused on a target, and the resultant produets emerging from the target area are subsequently observed at a detector. When the target dimensions are finite, the short range forces of the target do not affect the partieles as they are emitted from the souree or collected at the detector. In this case, the Hamiltonian describing the system can be divided into two parts - one des-cribing the unperturbed system when the partiele and target are far removed from each other, and the other describing the perturbing potential acting between the target and partiele when they are close together.

The initial unperturbed state of the system can be represented by a wave packet of the form

(2-23)

-where k refers to the incident partiele momentum and

a

refers to internal variables of the partiele and target. Ideally, the properties of the souree are such that

A(k)

is peaked ~bout some initial momen~m

k

=

k

.

with a very small spread in wave number Lk. If the detector is constructeà so that it measures a momentum spread IlUlch larger than the momentum spread of the initial packe.t, the initial state can be approximated by the stationary state

and the final state can be represented by the stationary state

(2-25)

The notations i

_...

=

_-

(k,a) and f

=

_-

(k',

a') refer to initial and fmnal states, k,

and k' refer to total initial and final particle momentum, and

a

and

a'

refer

to initial and final internal variables.

The central problem of scattering theory is to predict the probability

that a particle initially in a given state ~k (t) scatters into a final state

~~k'

,(tJ.

According to the theory of the

ri~st

section, the probability

ampli~ude

for this scattering is given by the overlap of the final stationary

state of interest with the total state vector ~(t) in the limit of very late

time (t ~oo). The state vector describing the scattering process is

post-ulated to be of the very general form

(2-26)

which satisfies the time-dependent equation (2-9) when

(2-27)

Before evaluating the overlap of ~k (t) with ~(t), we must find a

suitable expres sion for the

time-independentast~te ~k

satisfying Equation

(2-27) where H is the Hamiltonian describing the

entlt~

system consistmng of.

the incident particle and target. The Hamiltonian can be arranged in two fundamentally different ways depending on whether or not the system has been

nearranged. In ordinary scattering problems, the incident particle is the

same as the reflected particle, and therefore, the initial and final unper-turbed state of the system may be described by the same Hamiltonian, H. The potentialof interaction between the target and particle is denoted byOV, so that the total Hamiltonian becomes

H=H *'V

o

(2-28)

In a re arrangement scattering the final particle is different from the initial particle so that both the initial and final state of the system

cannot be described by the same Hamiltonian. In this case, the Hamiltonian

can be written in two ways

H = H' = H + V = H ' + V'

o 0 (2-29)

where the primed Hamiltonian refers to the final state description, and the unprimed Hamiltonian refers to the initial state description. The two cases must be treated differently, as will be discussed later.

In both cases the solution to the Schrodinger equation with the Hamiltonian H describing the unperturbed initial state of the system is given by

~k

_,

áOwhere

(2-30)

The time-independent state describing the entire system can be written 'f... = cp.... + 'f

k,a k,a sc

(2-31)

where

'Pk:'

arepresents the state of the system before any scattering has occurrea:~ and'f represents the state during the scattering. The state

~

...

a

is usuallySfeferred to as the incident wave or wave packet, and 'f as

t~è

scattered wave, but this is not strictly correct unless the targetS8an be represented by a potential only. If the target has any structure, the

inform-ation on the internal states of the target must be contained in

CPk

and 'f . The state 'f

_k

a

describes all possible eigenstates av~lable to the'~ystem sc

during the sèattering, but the particular eigenstate occupied by the system at any given instant is controlled by the time-dependent portion of 'f(t) which contains 'f

k

_,

a'

In scattering problems it is far more convenient to rearrange the differential equation

(2-14)

into the form of an integral equation. To do this, we begin by substituting Equation

(2-28)

into Equation

(2-14)

(Ho + V) /'fk> E /'f~

(2-32)

where the

a'

s have been suppressed. Svbstituting for 'f

_k

from Equat,ion

(2-31)

(E - H )/'f ~

=

V/'f-"'7

o sc! k

Formally' inverting the operator (E-H ) leads to o 'f sc 1 E i-oH _V'fk o

so that Equation

(2-31)

becomes

(2-33

(2-34)

(2-35)

(2-36)

The quantity (E-H )-1 is the Green's fumction for the unperturbed

o

system and is denoted by G. To avoid a singularity at

E=H ,

the quantity

o 0

~ iE is usually introduced and the expEEssion is later evaluated in the limit

as E approaches O. The Green's function then becomes

. )-1

lE

(2-37)

so that Equation

(2-36)

becomes the integral equation 'f ... (±)

~

=

:-

cP_

... 1

k h k E- H ± lE .

V'fkJ~)

(2-38)

The state

'Y~+)

with the (+) is an outgoing-wave solution representing the incident wave and the outgoing waves subsequently caused by the scattering of the initial wave from the target. When

'Y~+)

is used in Equation (2-13), 'Y(t) approaches

~k

for very early time and approaches

'Y~:)

for very late time. On the other hand, the state

'Y~-)

with the (-) is an ingoing-wave solution repres-enting the final outgoing wave and the incoming waves which WEU}d have to be scattered from the target to cause such a final wave. When 'Y

k- is used in Equation (2-13), 'Y(t) approaches

'Y(-)

for very early time and approaches

~k

for very late time. The ingoing

wavess~escribed

by 'Y&-) do not actually exist in the scattering experiment, but rather, are a result of the mathematical formal-ism.

An

interesting relationship exists between 'YL+) and

'Y~-)

when the

k k

scattering potential V is realo The integral equation for

'Y~+)

is

(2-39) and the complex conjugate is given by

f+)* * . )-1* * f+)*

'Y~ = ~- + (E - H + lE V 'Ik ....

k k 0 (2-40)

~.a.

+ (E - H - iE)-lwt+)*

-k 0 k (2-41)

The integral equation for

'Y~-)

is

'Yt-)

=

~

.. + (E - H -

iE)-lw~-)

k k o k (2-42)

Substituting

-k

for kin Equation (2-42),

'Y(=~

becomes

'Y

L-)

=~

--+ (E - H - iEflwL-)

k -k 0 -k (2-43)

Comparing Equation (2-41) arrd Equation (8-43), we find

'Y

_

~-) ='Y~+)*

(2-44)

and by taking the complex conjugate 'I

_~-)* =

'Yt+)

This relatiorrship has the following physical interpretation. Thef~

state

'Y_~-)

represents a set of ingoing waves scattered into a single outgoing wave

~+k'

The comples conjugate of 'I

~-),

whieh in quantum mechanios corres-ponds to time reversal, is equal to

'Y~)

which is a single ingoing wave

~+i"'

scattered into a set of outgngng.waves. Since'Y

L-)

=

'Y

L+)*, the state 'I

~-)

-k k -k

ean be described as a time reversed state.

The integral ~quation

(2-38)

is more easily recognized as an integral equation when it is expressed in the coordinate representatlion. This is accom-plished by forming the following sealar product with Equation

(2-38)

<;I'Y(±)

>

=<rl~~'\.

+(

.

;1

1 V

/'YkL±»

k

i/

E - H

±

iE (2-46)

Using the closure re1ation of Equation ~2:..a;tv U '1 (~,.:.J) t±) - - -( -

J

1-'

1

~k

(r) -

~k

r) + , r E - Ho ±i€ The quantities G(r,r') using Equations (2-17) and V(r) are and (2-18)

defined in the coordinate representation

<;'E _

Hl ± o

, .... >

....

, r' =G(rr') l.E 0 ' (2-48) (2-49) In coordinate space the integral equation becomes

(2-50) where the Green's function i s

*

(±) .... ,

_

-

3

f

A,

~k' C~) ~k'

(t')

G (r,r) - (2~) dk E _ E + 'E

o k k' - l. (2-51)

and ~k is the known wave function representing the unperturbed system des-cri bea: by H .

o

A formal solution to the integral equatjonnexpreSSed(~~ Equati~n

(2-38) can be found in the following manner. First, using G -0 for B-H +' ,

. ( 8) _l.E

Equatl.on 2-3 oecomes 0

or

(1 -

G(±)

V)~~±) = ~.

o k k

Formal1y inverting the operator leads to

Using the identity

~~±)*

(1 - G(±)

V)-l

~­

k o k ( AB

)

-~

=

B A -1 -1 (2-52) (2-53) (2-54) (2-55) (2-56) (2-57)

Equation

(2-56)

becomes 1'(±) k

cp-

k + cp~ + k 1 =

CPk

+ E - H

±

iE VCPk

Finally,

1'~),

exprèssed only in terms of the incident wave, becomes

where ~~±) k 1

( 2-58)

(2-59)

(2-60)

(2-61)

E - H

± iE (

2-62)

The function G(±) differs from the Green's function

G~±),

which was expressed in terms of the known state cp~r) as shown in Equation

(2-51).

The Green's function G(±) is expressed in terms of

1'~±)

which must be obtained by solving the differential equation H'l' gp. Thus, the evaluation of the formal solution becomes as difficult as solving the original differential equation.

2.3

The Scattering Matrix

With a formal expression for

1'~±),

given by Equation

(2-61),

we are now in a position to determine the probability that a particle in an initial

~t

ate CP~:~

scatters into a final state

cp&7~a.

As previously described, this probability amplitude is proportional to the overlap of a fin al stationary state CPk4 ,

,ei/h~,t

with the total state 1'(t) in the limit of very late time

,a

lim(Pk'

,a'

11'(t» exp

[(i/h)~,t]

t---+ 00

This expres sion can be easily evaluated by introducing the ingoing wave solution 1'L- ) - cp ...

R'

,

a' -

k'

,

a'

(2-64)

which, af ter rearranging becomes

CP... _k'

_,

_{a' -}-

1'~-)

_k'

_,

_a' - (Fe _-k' - H - iE)-l vcp ... _k'

_,a'

(2-65)

The overlap of the second term on the right side of Equation

(2-65)

with 1'(t)

in the limit of late time is zero, so that Equation

(2-63)

for the overlap becomes

(2-66)

lim Jdk

A(k)

<'l'~::a' I'l'~:~)

_{exp [-(i/.fl)(Ek - Ek' )t]}

(2-67)

t-t>óI) co

The quantity

('l'&:)a'

I'l'i+)is zero unless Ek ::: Ek' as will be sh0wn ;Later. Therefore, Equatlon

(2-gt~

is independent or time ànd has the limit

(2-68)

The scatte,ring matrix is defined.to be that part of Equation

(2-68)

in brackets

Sk .... '

ry'.

k- rv

:::~'l't-)

I'l'.'-+)

'7

(2-69)

,= ,,"'"

k'

,a'

k,a

and af ter replacing (k,~) with i and

(k'

,a')

with f, the scattering matrix elements become

In view of the meaning of 'l'(:) and'l' (-), the scalar productcrep-resents the probability amplitude forlfinding afsystem simultaneeusly in a state such that

a) a set of outgoing-scattered waves is formed at the scattering center as a result of an incident wave ~., and

b) a final wave ~f is formed at the scattering center as a result of a set of ingoing-scattered waves.

In other words, the S matrix represents the probability amplitudes for scattering from an initial state ~i to a final state ~f'

The expression for S given by Equation

(2-70)

is useful for several reasons .' Time is no longer a parameter in the resul t, so that the time-dependent wave packet_description is not necessary. The expression is symmetrie between initial and final states and does not depend upon any par-ticular separation of the Hamiltonian. The scatt~ring problem has been red-uced to solving the two time-independent integral equati0ns

(2-71)

(2-72)

with formal solutions

(2-73)

(2-74~

a form more suitable for explicit evaluation. The specific form is

deter-mined by whether the scattering is ordinary or rearranged. Each of these

cases will be considered in the next two sections.

2.4

Ordinary Scattering

In ordinary scattering the initial and final arrangement of the system is the same, so that both the initial and final states can be re-presented by the same unperturbed Hamiltonian H. The total Hamiltonian is

. b 0

g1.ven y

H 2 H + V

(2-VB1

7

5

)

o

where the initial and fif~t states corresponding to Ho are ~i and ~f' The

integral equations for ~ - are

~~±)

1 _~(±) ~.

...

,j, 1. 1. E. - H

±

i~ i 1. 0 ~(±) ~f + 1 V~(-) f E - H± iE f f 0

and the formal sol'utions are

~~±)

₊ 1 . V~. 1. ~i E. - H I 1.E 1.

(2-78)

1. ~(±) _~f + 1 iE V~f = - H ± f _Ef

The matrix elements are

8fi

=

(~~-) I~i+)

(2-eO)

which can be evaluated directly because both

~(;)

and

~(:)are

expressed as a

sum over eigenstates of the same operator, (E-Ho±E)-~

iglo

Equation

(2-76)

through Equation

(2-79).

A.

One Potential FormUla

When only orre potential is present in the system, the Hamiltonian of

Equation

(2-75)

may be used without modification. The solution to the

Schrodinger equation, ~., for the unperturbed Hamiltonian is assumed to be

1. knowno

Equation

(2-80)

can be simplified by using the identity

1 1

Ef - H - iE Ef - H + iE + _{2ni5 (Ef - H)}

(2-81)

in

Equa~ion

(2-79),

whereby

~~-)

becomes

~(-)

=

~(+)

+ 2ni5(E -

Hf

v~

f f f f

(2-82)

Equation

(2-80)

for the matrix elements becomes

uct

(2-83) scal ar

prod-(2-84 ) (2-85)

Using the integra1 equation for

'l'~+)

in the first term ontthe right hand

side of Equation (2-85)

<'l'i +)

I'l'~

+»

::::~i

ICPf'> +

<

_{CPi I E - H + lE} 1 .t- vl''l'(+) _f

f 0

+ E _

~

_ . <CP. Iv I'l'f( +»

i f lE l

Fina11y, the scattering matrix given by Equation (2-83) becomes

8_fi :::: 5_fi - 2ni5(E_f - Ei) <CPfIVI'l'i+»

(2-86)

(2-87)

(2-88)

The reciproca1 form of Equation (2-89) can be obtained by using

'l'~-)

::::

'l'~+)

+ 2ni5(E. - a)vcp. (2-90)

l l l l

rather than Equation (2-82). The fina1 resu1t is

(2-91) Usua11y the state 'l' cannot be determined exact1y. In these cases the expression for the scattering matrix, Equation (2-89) or Equation (2-91),

must be eva1uated by an approximate method. One method of ten app1icab1e

consists of retaining on1y the first term in the formal solution for the

state 'l' which is given be10w

\T{~. +~

:::: cp. + 1 vcp:::::; cp

Il l E. - H + iE i i (2-92)

Af ter ~ubstituting ~. for ~ in Equation

(2-89),

the scattering matrix becomes

~

(2-93)

This approximate method is known as the first Born approximation. B. Two Potentials

~he potential V usua1ly depends upon internal coordinates of the particle and target, and coordinates of their relative separation. Of ten the potentia1

V

can be split into two parts

V

and

V ,

where

V

depends only on the relative separation of the target andOpartic!e. In

th~s

case the Hamil-tonian H +

V

describes the internal parts of the partic1e and target and the

potentia~ of ~nteraction between them. This leads to an equation describing

the e1astic scattering of two complex particles, a prob1em which can be solv-ed exact1y in many cases. The advantage of this division of potentials is that it allows one to account for the stronger interactions exactly, so that approximate methods can be successfully used on the wekker interactions.

The total Hamiltonian is

H=H

+v +v

o 0 àl

(2-94)

where the solution for H + V is assumed to be availab1e. Substituting V

o + V1 for V in EquatioR (2-~~), the S matrix becomes

For convenience, the quant2ty in brackets is defined as H fi:

(2-96)

(2-97)

The quantity X(!)iS defined by the integral equation

x(±) =

~

+ 1 V x(±)

f f E - H

±

iE 0 f

o

(2-98)

which has the formal sol ut ion

X _f (± ) = m _{'t' f} +

=----=_l--=--..,._~u

_{l!: - H V ± iE·}v m

0 't' f

(2-99)

o 0

The state x(i)represents the elastic scattering of a

parti~t,

with a Hami1-tonian H from a potential V , where Equation

(2-98)

for X

f

is assumed to be solub1e.o Using ~f from Equa~ion

(2-99)

in the second term on the right side of Equation

(2-97),

Hfi becomes

-

<~flvo

E _ H _lV ± iE

v11~~+»

(2-100)

o 0

This expression ean be simp1ified by rep1aeing

~~+)

in the first term on the right side by an equivalent quantity found by ustng the identity

1

t: :

H + iE o 1

E - H - V

+ iE o 0 1 ~ 1 E - H - V + iE 0 E - H + iE o 0 0 (2-101)

By using Equation (2-101) in the integra1 equation for

~~+),

we obtain

1

~~+)

=

~.

+ 1 (V + V )

~~+)

1 1 E - H + iE 0 1 1 o

=

~

+ _ _ _

1-==-_~

(V + V

)~~+)

i E - H - V + iE 0 1 1 o 0 1 V 1

~V

+ V

~~~

+) E - H - V + iE 0 E - H + iE , 0 1 1 0 0 0 1 _{. (V +V}

_)~~+)

=

_~i + V E - H

-

+ lE 0 1 1 0 0

=

x(+) + 1 V

~(+)

i E - H - V + iE 1 i (2-102) o 0

S~bstituting Equation (2-102) in the first term of Equation (2-100), Hfi beeomes

(2-103)

The seattering matrix e1ements beeome

+

<x(-)

Iv

I':l:'~+»

] (2-105)

f 0 l

The reciproeal form ean be found by starting with Equation (2-91) for Sfi rather than Equation (2-89). The result is

Sf'

=

Bf' - 27TiB(E_f - E.)[<xf(-)Iv Icp.) + ('lf(-)Iv

Ix~+~J

(2-106)

l l l O l O l

2.5 Rearrangement Seattering

In rearrangement seattering, the initial and final arrangement of the system is not the same, so that the initial and final states eannot be deseribed by the same unperturbed Hamiltonian H. If the initial state Hamil-tonian is represented without primes and the figal state HamilHamil-tonian with primes, then the Mamiltoniarrbeeomes

H

=

H + V H' + V'

o 0 (2-107)

The initial and final sê~yes eor{~,POnding to Ho and H5 are CPi and CPf'. The ináegral equation for ':l:' i and ':l:' f' are

=----=1=--~~

V

':l:'~

+ ) CPi + E - H + iE l o (2-108) (-) ':l:'f'

=

CPf' + _ _ _{E - H ' - iE} 1=--_ _ V'':l:'(-) _f' (2-109) o

with formal solutions given by

':l:'~+) =

CP. + 1 . Vcp.

l l E - H + lE l (2-110 )

(-) 1

':l:'f' = CPf' + -E--";;';""H---'-i-E V'CPf' (2-111) The seattering matrix elements of Equation (2-70) are still valid beeause they do not depef~)upon(~y partie~làrrseparation of the Hamiltont

_i

y'

but t~~)sealar produet<':l:' f'

I

':l:' . )eannot be direetly evaluated beeause ':l:' . and ':l:' f' as expressed by Equations (2-10821- (2-109) are su~d over gige~­

st~yes of different operators, (E-H + iE) -V an~ )E-H' - iE)-~'. Either ':l:' i must be expanded in terms of H~oand V', or ':l:' ;, mu~t beeexpanded in terms

of H and V.

o

Dhoosing the first alternative, we begin with the identity

1 E - H + iE o

______ *1:.-___

+ 1 E - H ' + iE E - H + iE o 0 (V' - V) E _ Hl/ o + iE

1 1

=

~~~-~- + --~-~-E - H' + iE E - H' + iE o 0 1 (V I -V) -E---H~-+"""i-E o (2-112)

which is the same as the matrix identity

A- 1 _ B- 1 = A- 1 (B _ A)B-1 _(2-113)

By using the identity of Equation (2-112) in Equation (2-108),

~(~)becomes

1

1 ( , 1 (+) + E-H'+iE

V -V)

E-H +iE

VW

i

o 0 1 (+) =

~i

+ E-H'+iE

~i

1 + ---:-~ E-H'+iE o (V' - V)

(~~+)

-

~.)

1 1 o 1 1 I (+) = [1 - E H,+· ·(H -H')] ~. + E H'+' V~. - 0 l ' 0 0 1 - Ó lE 1

If there is no rearrangement,

V = V'

and H

o

H'

0'

~~+) =~.

+ _ _

1~--:-_ ~~+)

1 1 E - H + iE 1 o

which is identical to the original equation (2-108).

(2-114 )

(2-115 )

(2-116)

However, if there is a true rearrangement, ~i is an eigenfunction of H but !\ot of H'. Therefore, H cp.

=

~., but H'~.

f

Np. and

o 0 O l 1 O l 1 1 E-H'+iE o (H

-H

~

)~

.

=

(E-H'+iE)-l(H

-H'+iE)~.

0 0 1 0 0 0 1

Equation (2-114) for

~(~)

becomes

1

=

(E-H'+iE)-l(E-H

'

+iE)~.

o 0 1

=

~. 1 1 V'~(+) ~E:---~H~'~+--:-i-E i o (2-117) (2-118) (2-119)

The scattering matrix e1ements can now be eva1uated with the aid of Equation (2-119)

A. One potentia1

The Hami1tonian is expressib1e as

H = H + V = H ' + V'

o 0

(2-120) and the matrix e1ements from Equation (2-80) are

s

= <'1'( -) I'l'( +»

f'i f' i (2-121)

(± )

The state '1' f' can be exp~essed as

(±) (~) - . ,

'1'f' = '1'f' _{+ 27Tl5(Ef , -} H)V CPf' (2-122)

in a manner ana1ogous to Equation (2-82). With the substitution of

'l'(;~

from Equation (2-112) into Equation (2-121), the scattering matrix e1ements become

-/, (+) I (+» ./, I '5( - H) I\T/(.+»

Sf' i -,'l'f' 'l'i - 27Tl'\CPf' V Ef' I l (2-123)

The

qUantitY('l'(;~ I'l'~+»can

be eva1uated in a manner

used with Equation (2-84) l

similar to that

( 'l'(+)'l'(+» =

<cp

I'l'(+)

'>

+

(cp

Iv' 1 . Ur/-f2-)

f'

1

i f ' ~ l' ' E - - l E r I l f' (2-124) ( +)

Using Eqttation (2-119) for 'l' . , the first term in Equation (2-124) becomes l 1 (+» H

!

+ iE V' 1'1' i o 1 and therefore

Equation (2-123) for Sf'i becomes

Sf'i = _{-27Ti5(Ef , - Ei) <CPf' Iv' I'l'i+»}

(2-125)

(2-126)

,.

The reciproca1 form for Equation (2-127) is

(2-128)

B. Two potentials

As in the case of ordinary scattering, the potentia1 can of ten be divided into two parts so that the Hami1tonian can be expressed as

H=H +V +V =H'+V ' +V '

o 0 1 0 0 1 (2-129)

where it is assumed that the states x(:)andXX(;?, corresponding to Ho+V_o and

H'+V' , can be determined. The state X(±)iS defined by

o 0 f'

(±) 1 V ' (±)

Xf ' = ~f' + E - H ± iE 0 Xf '

o

and the formal solution to Equation (2-130) is

(± ) Xf ' 1 - m + V 'm - ~f' E - H ' - V ' ± iE 0 ~f' o 0 (2-130) ( 2-131)

The expression for the scattering matrix e1ements is found by sub-stituting V' + V' into Equation (2-127J

o 1

The quantity in brackets is again defined as Hf'i Hf''; =

/~

Iv' +V

'

I'f'~+)

....

" f '

0 1 ~ Rearranging Equation (2-131~ m - ( - ) 1 V ' lfI ~f' - Xf ' - E - H ' - V • - iE 0 'Yf' o 0'

and inserting into Equation (2-134), Hf'i becomes Hf'i

=<~f'

Ivo' I'f'i+)

>

+

<X~~)

_{Iv1 ' I'f'i+»}

Ar> Iv' 1 V ' + _{iE V1'} I'f't +»

-'~f' 0 E - H ' .... o 0 (2-132) (2-133) (2-134 ) (2-135) (2-136)

With V'

=

V' + Vi Equation (2-119) for"'f(-:)becomes

o l ' ~

1 I (+)

E - H '-V ,+ iE V1 'fi

o 0

which can then be substituted into the third term in Equation (2-136). thus becomes

The scattering matrix e1ements become

Sf'i

=

_{-27Ti5(Efl -} Ei)

<x~

~)

Iv₁'

l'f~+»

(2-137)

(2-138)

The reciproca1 form of Equation (2-139) is found by starting with Equation

(2-128) and is given by

(2-140)

Notice that the resu1ts, Equation (2-139) and Equation (2-140), do not exp1icit

-1y contain the potentia1 V or V' . The effect of this potentia1 is inc1uded in

+1,,~ 0 0

th~

wave x(:)or x(;), which is simp1y the initia1 wave

~i

or the fina1 wave

~f'

distorted by the presence of the potentia1 V or V' •

o 0

An approximate method of determining Sf"i consists of using on1y the first term in the formal solution for '1'(:):

~

(2-141) Substituting the first term of Equation (2-141) in Equation (2-139), Sfii becomes

(2-142)

The approximate method is known as the distorted Born approximation. Equation

(2-142) can be expressed in the coordinate representation by using the c10sure re1ation of Equation (2-8)

Sfli = -2m.5(E

(2-143)

3.0

THE HAMILTONIAN ANTI UNPERTURBED STATES

"

3.1 The Hamiltonian

In the Auger neutralization process the initial state of the system consists of a crystal with n valence electrons, and an ion incident on the crystal surface. The final state of the system consists of the crystal with n-2 valence electrons, a reflected atom, and an excited secondary electron. This process may be described as a re arrangement collision in which the tar-get and projectile have become rearranged--part of the tartar-get leaves with the reflected projectile. The theory for such a collision describing the prob-ability of scattering from the initial to final=_state was given in Section

;

2.5.

The total HamiltoRian consists of the kinetic energy and interaction potentials of all particles in the system:

H = \ '

p;

+

L2m

_ e e

(3-1)

In this expression me' m, and M~ are the valence electron, projectile, and crystal atomic core or lattice point masses; Pe' p, and p~ are the momenta of the valence electrons, projectile, and lattice points; and the remaining terms

...

-

...

are interaction potentials, where re' r, and r~ refer to the coordinates of the electron, ion, and latticeppoints.

The solution to a problem with such a Hamilto~ian would lead to formidable mathematical difficulties, and therefore many of the usual approx-imations in the description of the crystal must be made. These approximations are discussed in detail elsewhere [71, 72J and are only briefly stated here.

The solid target is assumed to be a perfect crystal lattice Which consists of periodic unit cells, each containing one or more a~oms. The motion of the crystal nuclei is as~umed to be independent of electron motion, and the harmonic approximation for the nuclear interaction potentials is assumed to be valid. The adiabatic approximation for the crystal electrons is as sumed , whereby the motion of the electrons is determined using the nuc-lear positions as a parameter. Electron-electron interactions are ignored

iple must be satisfied. These electron-electron interactions which must be considered in the Auger neutralization process are included as a perturbation. The incident ion is assumed to be a point particle, and the reflected atom is assumed to be a hydrogenic atom. A central potential is assumed to exist be-tween the projectile and the other particles in the crystal.

With these assumptions, the Hamiltonian becomes

2 { \ ' p2" \ \ "'" }

H =

~

+

L

éM'"

+

LL

cp

a~

u

a ("

),u~

(" ' )

t

"

o:~"'"

(3-2)

where the first term describes the unperturbed state of the projectile, the second group of terms describes the unperturbed state of the crystal nuclei, the third group of terms describes the unperturbed state of the electrons, and the last group of terms represent the perturbing potentials. By assuming that the initial and final states of all valence electrons except the two involved in the Auger process are unchanged, and by assuming that the effect of all other electrons on the incident projectile and two Auger electrons can be in-cluded in an effective lattice potential, then only that part of tbe Hamilton-ian involving the two electrons need be considered. The HamiltonHamilton-ian of Equa-tion

(3-2)

reduces to

2 2

H=:L+\P"

2m

L

2M"

"

or rewriting in a more compact form

(3-4)

The following notation applies:

V

= l-c

L

V(;l'

~)

t

L

(...

... V = V r 2,r

t

)

2-c

t

V 3-c

I

V(;3 -

~)

t

. SUBSCRIPI' P.A..RTICLE STATE DESCRIPrION

..

1 crystal electro n k i initial state

...

1 secondary electron k final state

...

2 crystal electron kY1 initial state

2 atomic electron b final state

...

3

incident ion K initial state

.,-3 reflected ion Ki _{final state}

c crystal c in,itial

&

final

The solution to the Auger neutralization problem using Equation (3-4) is ~ow possible although the calculations are quite tedious. A tremendous simplification is possible by assuming that ~he initial and final states ~e~the

~uclei in the crystal remain unchanged during the proceS5 so that the second

and thè±äaterms in Equation (3-4) for the Hamiltonian need not be considered. The Hamiltonian therefore reduces to

2 2

P3

Pl

H = + -2Ir] 2~ 2 P

+L+

2~

v

l-c + V 2-c +v +V +V +V 3-c 1-3 2-3 - 1-2

This assumption precludes the consideration of energy exchange be-tween the projectile and the crystal, and therefore is equivalent to assuming that the projectile is elastically scattered by the crystal.