Coherent Bremsstrahlung and the Quantum Theory of Measurement

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Coherent Bremsstrahlung and the Quantum Theory of Measurement

E.H. Du Marchie van Voorthuysen

To cite this version:

E.H. Du Marchie van Voorthuysen. Coherent Bremsstrahlung and the Quantum Theory of Mea- surement. Journal de Physique I, EDP Sciences, 1995, 5 (2), pp.245-262. �10.1051/jp1:1995126�.

�jpa-00247055�

Classification Physics Abstracts

03.658 78.70F

Coherent Bremsstrahlung and the Quantum Theory of

Measurement

E-H- du Marchie _van

Voorthuysen

Nuclear Solid State Physics and Materials Science Centre, University of

Gromngen,

Nijenborgh 4, NL 9747 AG Groningen, The Netherlands

(Received

29 December 1993, revised 21 September _19g4, accepted 18 October

1994)

Abstract. Coherent brernsstrahlung

(CB)

is the result of inelastic Bragg scattering of elec- trons of _a few hundreds of kev, _a collective elfect of trie whole crystal. In

an electron microscope it is theoretically possible to deterrnine trie _row of atoms where the electron

was inelastically scattered. These staternents are contradictory. Optical path calculations _are made for 160 kev electrons inelastically scattered by _a silicon

[Ill]

crystal and the results _are compared with rneasured CB spectra. ^{The results} can be understood by application of Van Kampen's theory of quantum mechanical measurement. Experimental CB spectra tum out to be the result of

coherent scattering ^of atoms within the

same row, and incoherent summation of intensitie8 from dilferent rows, or, in other words, the electrons that _cause CB _are locabsed within atomic _rows.

l. Introduction

1-1. COHERENT BREMSSTRAHLUNG.

During

collisions of electrons of _a few hundreds of

kev with atoms the

following

inelastic events _{can occur:}

i)

inner shell ionisation

resulting

in

charactenstic

X-ray radiation, ii)

outer shell ionisation and

for

excitation

resulting

in charac- teristic

peaks

in the electron _energy loss spectrum

(EELS)

in the 10 eV

region,

and

iii)

emission of _a

bremsstrahlung photon

when the electron orbit is

changed by

the Coulomb field of the

atomic nudeus.

In dassical

physics

_an electron is _a

partide

that emits

electromagnetic

radiation when it is accelerated. The

intensity

of the

bremsstrahlung

emitted

by

_an electron with

velocity /3(t)

=

v(t)/c

is [1]

d~I/duJdfl

= e~

/(47r~c) /

in

^x

⁽⁽ⁿ

^{/3) x}

^{/3) /(1}

^/3

n)~j

e~~(~~~'~(~)/~)dt(~

(l)

where _n is the unit vector in the direction of the emitted radiation and

r(t)

the _vector

connecting

the electron with the detector.

Normally

the

bremsstrahlung

spectrum is continuous, without any structure.

We consider _an electron with _a

trajectory

close to _{a row} of

equidistant

atoms with interatomic distance L. This electron

undergoes

_a

velocity change Ad

each time it passes a nucleus. We

IOIIRNAL OEPHYSIQUE11 T5, N° 2, PEIIRUARY 1995 4

Q Les Editions de Physique 1995

assume

(Ad(

<

fl

+

(p(,

_so the

trajectory

is _a

straight

fine and the time between successive radiative interactions is constant, At

=

L/(flc).

The

velocity change

/3 is _zero except for t = C + mât with _m

integer,

_so the

frequency

spectrum of the

bremsstrahlung

is the Fourier

transform of

equidistant sharp pulses

in the time domain:

f(uJ)

= Const à_[uJ

j.27rflc/(L(1

_n

/3))]

,

j

₌ integer.

(2)

We observe that for

photons

emitted

perpendicular

to the direction of the electrons the

ground frequency

is

proportional

to the electron

velocity

and inverse

proportional

_to the interatomic

distance.

If the energy

spread

and the

angular ^spread

^of the electrons in the

incoming

beam _are

sufliciently

^{small the} electrons may not be

regarded

_as localised entities, but _must be

regarded

as manifestations of _a wave, the wavefunction of the

electron,

which is assumed _to be _a

plane

wave with

wavelength

À;. The electron leaves the atom with

wavelength

Àf,Àf > À;. ^When we are

dealing

with _an extended structure of atoms, like _a

crystal,

the scattered _waves from all

atoms must be added and the

intensity (the

number of electrons detected _{on a screen} far away

per unit of _{area per} unit of

time)

is

equal

to the modulus of the _sum wave,

squared.

Constructive and destructive interference effects _occur, like in the _case of elastic

scattering,

_À;

= Àf, where these effects

give

rise to the well known

Bragg scattering (electron diffraction).

As _a matter of

fact, practically

ail electrons that _are

elastically

scattered

by

_a thin

crystal

_are

leaving

the

crystal

in directions where constructive interference takes

place.

For

inelastically

scattered electrons the _{same is} true if the _excess energy is

completely

removed

by

_one

bremsstrahlung photon.

The Ewald

construction,

which is very useful for

visualising Bragg scattering,

_con therefore be used also to illustrate inelastic coherent electron

scattering, Figure

1. Coherent

bremsstrahlung (CB)

_occurs when the vector relation kf

=

k;

_{+g q is} satisfied and (k;( >

(kf(,

k is the electron _wave vector, (k( e k ₌

1/À,

_g is _a vector

connecting

two lattice

points

in

reciprocal

_space, and

hq

is the momentum of the CB

photon.

The _energy of the CB

photon

_{is [2]}

~ =

hep(g~ g2/2k;)/(1 pcos9) (3)

with gz the

projection

of _{g on} the direction of k; and the

angle

between the direction of the

photon

and k;. Effects of

channeling

radiation _are

neglected

in

(3).

Diffraction effects of the

bremsstrahlung

itself with the lattice _are also

neglected,

the

wavelength

of coherent

bremsstrahlung

is

generally

too

large.

When the

incoming

electron beam _is

parallel

_{to a zone axis,} all

points

of the

reciprocal

lattice

belonging

to the _same Laue _zone have the _same value for _gz,

Figures

lc and ld.

They

contribute to the _same

peak

in the

bremsstrahlung

spectrum because _g <

k;,

and the equations

(2)

and

(3)

_are equivalent.

CB _can be observed in the continuum between the characteristic

X-rays, Figure

2. The CB peaks are marked with numbers

equal

to the order of the Laue _zone. Vecchio and Williams [3]

have made _an extensive expenmental

study

of the optimum conditions for

producing

CB

peaks.

These conditions _are: thin

specimens

of _pure elements

(no intervening X-rays

from

impurities),

dean

crystal surfaces,

_zone axis orientation, low temperature

(to

reduce lattice

vibrations),

and detection with _a semi-conductor detector with _a solid

angle

> 0.01 steradians _{m a} direction

perpendicular

to the beam.

Sigle

^and

Carstanjen

_[4] observed CB from _a

quasicrystal.

Coherent

bremsstrahlung

is considered to be _an annoymg side effect in

analytical

elec- tron microscopy, CB

peaks

_can obscure

X-ray

peaks from impurities _{or can} lead _to wrong

intensity determinations of

impurity

X-rays. ^But CB can also lead to an

interesting problem

of fundamental nature.

, o .

,

k~

i~ _.

.

.i _1.

o ~ .

.

a) b)

i~

,

i~

k ç

~ 3 -3

)

^q . ~ 2

. -2

-

)

_-1

~ o

Î

-o

. .

c)

,

d)

Fig. l. Ewald constructions _in the reciprocal lattice: a) elastic scattering, (k,(

= (kf(; b) inelastic scattering, (k,( > (kf(, hq _is the momentum of the emitted CB photon; c) inelastic scattering, (k,( >

(kf(, k, _{ii a zone} axis, scattering towards

a point _on the

zone axis _in trie third order Laue _zone;

d)

inelastic scattering, (k,( > (kf(, ^k, _{ii a} zone axis, scattering towards _a point ^{in the first} order Laue

zone.

1.2. DEFINITION _{oF THE} PROBLEM. One of the most

intriguing

aspects of

physics

^{is the}

particle-wave duality.

The

question

is the electron _a localised

particle

_or is it the manifestation of _an extended wave? _can be illustrated

nicely

with the double-slit

experiment,

_see for instance

Feynman

_[5]. The electron flux _{as a} function of position of the detector, _as it is measured on

a screen behind the wall with the two slits, consists of two

factors,

_a diffraction factor and

an interference

factor,

like _m Fraunhofer diffraction. When _we observe interference it is not

Si Si sumoeak Fe Fe Cu Cu

5 7

2 4 6 8 JO

energy Ioss (kev)

Fig. 2. Experimental photon spectrum. A silicon [loo] crystal at _room temperature is bombarded

by 120 kev electrons parallel to _an

[Ill]

axis

in a JEOL JEM 200 CX microscope. Photons _are detected in the direction H

= 90° by _a

Si(Li)

detector with _a resolution of130 eV. The sobd angle of acceptance ^{is 1.8} X 10~~ _{srad. The spectrum} is dominated by peaks of characteristic X-rays that _are

indicated by their chemical symbol; the height of the Si-Ka peak at 1.8 kev is 120,ooo courts. The

peaks of coherent bremsstrahlung _are marked by numbers indicating the Laue _zone from which they origmate.

possible

_{to measure or} to deduce

through

^which slit the

actually

observed electron

really

went.

As _{soon as we} try to detect the _passage of the electron at _one of the slits _{we cause a} disturbance

m the

phase

of the wavefunction of the electron such that _{as a} result the interference factor

turns into _a constant value. The usual

terminology

_is that

through

the act of

measuring

the

electron at the slit the wavefunction of the electron has

collapsed.

The _{same can} be said for the _case of electron diffraction _as it is observed in the electron

microscope.

There is _no way ^to determine which atomic _{row was}

passed by

the electron without

destroying

the pattern ^of diffraction spots.

In the _case of inelastic

Bragg scattering leading

to coherent

bremsstrahlung

_{we can} observe the

photons

as well _as the electrons. The

photon

spectrum contains information about _a collec- tive effect of ail

crystal

_atoms, 1.e. the interference between

scattering

from ail atoms. At the

sonne

time,

detection of the electrons

provides

information _on the location where the

scattering

occurred. The maximum accuracy in the measurement of the location of the

scattering

centre is

given by

the

uncertainty

relation of

Heisenberg:

if the

scattering angle

is bounded within

a too narrow cone, the

uncertainty

of the transverse momentum of the electron is too small.

For the moment we make the

assumption

that the _row where the electron scattered _can be determined. The electron leaves the

crystal

with _a

slightly reduced,

but still

quite accurately

known _energy. In _a

high-resolution

electron _microscope it must be

possible

to make _a

good

image of the

crystal

with atomic

resolution,

to mount a two-dimensional position-sensitive elec- tron detector at the

image plane,

and to deterrnme _{m an} electronic coincidence _unit for each detected

photon

which atomic _row the electron

passed

while

producing

the

bremsstrahlung.

If

we assume that the coherent

bremsstrahlung

_spectrum contains interference factors between

different atomic _rows, these factors wùl become constant _{as soon as} a measurement of the _row is made. So the CB spectrum must

change

_{as soon as a} measurement is made of the atomic

row where the electron that caused the

photon passed by:

the measurement of the

trajec-

tory of the electron

through

the

crystal

reduces the

intensity

of _some CB peaks and increases tue

intensity

of otuer

peaks.

Tuis _is tue famous wavefunction

collapse througu

trie act of

2

fi

Mfi-

j ⁹

3-&- W

à4

y A

~ ~ II

X X Y Y B

8

MEMORY

+Î

llit

gate

= yes

Fig. 3. The paradox. Switches down: CB photons _are detected without rneasurement of the atornic row, switches up: CB photons _are detected with sirnultaneous rneasurernent of the _row. Manipulation

of the switch _rneans annihilation and creation of pulses _in the delay units, _or it _means that the count rate _as measured in counter A depends

on the switch setting in the future.

l)

incoming electron,

energy E. 2) crystal with atomic _rows aligned parallel to the incorning electron beam. 3) electron

lens, making _{an image on} trie detector with atomic resolution. 4) outgoing electron, _energy ^E'

=

E _e. 5) position sensitive electron detector. 6) pulses with voltage proportional to the z-coordinate.

7) pulses with voltage proportional to the y-coordinate. 8) _a pulse _means detection of _an electron, and

the atomic row it _carne from is known; this information _is stored in the _rnemory. 9) CB photon with

energy E, 10) pulses with voltage proportional to e,

Il)

logic pulse if _{e is} within the window _as set by

the SCA. ADC: analogue to digital converter. CNT: pulse counter. COIN: coincidence unit, logic and;

pulses _arriving sirnultaneously belong to the _sarne event. DELAY: pulse delay unit containing _many pulses _running through it with constant speed and with voltage conserved. DET: photon detector.

SCA: single channel analyser; _rneasures the voltage and _gives output puise if the voltage _is within _a

specified

interval.

measurement. Now let _{us assume} tuat tue

collapse

takes

place

wuen tue last connection is made _m tue

complicated

electronic

circuitry

that is needed for tue

photon-electron

coincidence

measurement in tue

uigu-resolution

electron microscope,

Figure

^3. Tue last switcues _are sit- uated far from tue detectors and _near to tue counter wuere tue information will be stored.

Travelling

times of electronic

pulses

from tue detectors to tue _{memory are}

typically

10 ps, but

by

_means of

delay

units tuese times _can be mcreased to

milliseconds,

seconds _{or even} uours

without _any loss of _accuracy.

Cuanging

tue switch

setting

^would mean that in the

delay

units

immediately

_some electronic

pulses

_are annihilated and other electronic

pulses

_are created. If

we do not accept ^this consequence, it follows that the

pulses

of the

photon

spectrum ^with a

pulse height

distribution that is

dependent

_on the

position

of the switch _were

already

present

long

before the switch _was

manipulated:

the count rate in counter A is

equal

to the count rate

in counter B _one hour later. In that _{case we} could make _a

dear-sighted telescope

with _a

unique

possibility

to become rich _on the

stock-exchange.

In this paper ^we will discuss two propositions for

solving

this

problem:

1) ^the act of measurement of the atomic _row is not _a

manipulation

with electronic

equipment,

but takes

place

at the moment that the

outgoing

electrons _are focused _{on a screen} with atomic

resolution;

2)

CB spectra ^do not contain interference factors between _waves

coming

from different _rows.

1.3. SOME REMARKS _ABOUT QUANTUM MECHANICAL MEASUREMENT. In the _common

interpretation ^of quantum mechanics

(the

_sc- called

Copenhagen interpretation)

the act of

measuring

_a

physical quantity

has _{an enormous} effect _on the

physical

world: the wavefunction

changes drastically,

and _some other

physical quantities

cannot be determined _{any more} with suflicient _accuracy. Before the

"holy"

act of measurement our

knowledge

of the world is

only

_a

coherent

superposition

of many

possibilities,

administrated with

complex numbers,

the _wave- function. But after the measurement the world is _{a sum} of

objectively existing probabilities,

administrated with

real, positive

numbers. One _may read _out such _a

probability

(1.e.

perform

_a

dassical

measurement)

and get more

knowledge

of the

world,

_{or one may} not, it does not make any difference for the rest of the world. So there is _an essential difference between quantum

mechanical measurement and dassical measurement.

The sudden

change

in the wavefunction is called the

"collapse

of the wavefunction". In the

common

interpretation

of quantum mechanics the _occurrence of the

collapse

_{as a} result of the

imposed

measurement has to be put into the

theory

_{as a separate}

postulate:

the

projection postulate.

According

to Bour _a quantum ^mecuanical measurement is terminated wuen tue _outcome bas been

macroscopically

recorded in _a dassical apparatus. Bour made _a difference between tue

microscopic world,

where the rules of quantum mechanics _are to be

applied,

and the

macroscopic

world,

where dassical

physics

_is valid.

A

macroscopic

system

usually

consists of _{a very}

large

number of

partides resulting

in _a very

high density

of states. In _a quantum mechanical measurement _a

microscopic

system in _a

(nearly)

_pure state is

brought

into contact with the macroscopic system. As _a result _a

change

is

brought

about in the

macroscopic

system ^and

consequently

in the

microscopic

_system: the

collapse

of the wavefunction of the

microscopic

system. Because of the

large

difference in

density

of states between both systems the

change

is irreversible: the

probability

that both systems ^{return to} their

original

state _can be

neglected.

According

to Van

Kampen

_{[6] the}

huge density

of states is the main feature of _a macroscopic system. ^The rules of quantum mechanics

apply

to the system, but the

eigenvalues

of the energy operator ^bave an average distance ôE tuat is mucu smaller tuan tue best _accuracy tuat _can be obtained in _an expenment.

Manipulations

_{on a} macroscopic system influence tue

wavefunction of tue system, but tue result is mucu to small to be observable. So tue rules of dassical mechanics _are valid _{to a very}

good approximation.

Van

Kampen

^defines _a quantum ^mechanical

measuring

_{apparatus as a} macroscopic system in a metastable state. To _my

opinion

the metastable state is not a necessary condition for

a quantum ^mechanical

measuring

instrument. Van Kampen _[6] gives a nice exarnple of _a quantum mechanical

measuring

apparatus consisting ^of an atom m an

excited,

metastable

state and the free _space

surrounding

ii. In the

electromagnetic

field _an emitted

photon

finds

nearly

_cc different

modes,

_so the

density

of states of the

photon

is very

large.

A fast electron is able to

trigger

the metastable atom, and

consequently

its existence in the

neighbourhood

of the atom is measured in the form of _an irreversible

change

_m the

electromagnetic

field.

Van

Kampen

descnbes the whole _process in _a quantum mechanical _{way, so} the

measuring

apparatus ^has a wavefunction _too. The

Schrôdinger

equation works _on the whole system ^and

as a consequence of this it is demonstrated that

triggering

the atom _means destruction of the

possibility

of interference with the electron. In Section 2 of this _paper I will _use such _an

approach

to

analyse

the

problem

mentioned in Section 1.2.

Another

example

^of'a quantum mechanical

measuring

instrument is the _use of _a foi] of solid matter for

measuring

the existence of _a fast electron. The foi] is _a macroscopic system

that

undergoes

_an irreversible

change

due to the _energy Ioss of the electron in the foil. This

change

_can be read out

electronically

if the foil is _a

semiconductor,

_or it _can be observed

by

measuring

_a rise _m temperature ^if the foil is part ^of a calorimeter. When the electron has

deposited

_energy in the foil this energy may be measured _or not. This last measurement is _a dassical measurement because the foil is _a

macroscopic

system. ^The wavefunction

describing

the multitude of excitations in the foil is not

changed by

this measurement in such _a way that this

change

is observable. The effect of this

change

_on the wavefunction of the

original

fast electron is

completely negligible.

It is

only

the deed of

positioning

the foil in the

trajectory

of the electron that influences the wavefunction of the

electron, nothing

else.

A related

theory

is the modal

interpretation

of quantum mechanics of Dieks [7].

Another

approach

is that of Zurek [8]. ^In

practice,

macroscopic systems can never be isolated

completely

from the outside world.

Leakage

of _energy leads to loss of coherence between different states, or, m other

words,

the

off-diagonal

eleménts of the

density

matrix

disappear

and _{we are} left with the trace elements: real

positive

numbers

havmg

the

meaning

of dassical

probability.

In ail mentioned

approaches

there _{is no} need for _a

boundary

between

microscopic

and _macro- scopic. ^In

principle,

the rules for the

microscopic

world

(quantum mechanics)

are valid

always,

but very soon the rules of dassical

physics

_may be used without _any loss of _accuracy, The step quantum mechanics _- dassical

physics, induding

the

"collapse

of the wavefunction" _is

just

a consequence of the

growing complexity

of trie system as time goes ^on and the

projection

postulate

_can be left out. The

change

in the wavefunction has

nothing

to do with _a deed of

measuring by

_some human

being.

In _a real quantum ^mechanical calculation the

probability

₌

(probability amplitude(~

must be calculated _{as soon as} the quantum mechanical system ^has started to

develop

to a macro-

scopic stage. ^The outcome would not be different if _one would follow the

development

of the wavefunction _any

further, although

the calculations would _soon become

impossible

due to the tremendous

complexity

of _a

macroscopic

system. ^{So the} outcome is

independent

_on the

arbitrary

choice when to calculate the

probability, provided

it is not done to

early.

In the

"paradox"

of

Figure

3 the wavefunction has

developed

into _a macroscopic stage ^{when the}

pho-

ton has

dissipated

its energy in the

photon

detector and the electron has

deposited

_energy ⁱⁿ

trie

position

sensitive electron detector.

Any

further

manipulations

do not influence the wave-

function in such _{a way} that the consequences on the microscopic system are measurable. So

creationlannihilation

of electronic

pulses

does not occur, and _a

dear-sighted telescope

cannot be built.

2. The Model

2.1. GENERAL. In order _{to test} the

validity

of the

propositions

in Section 1.2 _a

simple

model is made that _can be

applied

in computer simulations.

Following

Van

Kampen

_[GI the

complete

system _is described

by

_a wavefunction. We will make _a full quantum mechanical

description

of the system ⁱⁿ

Figure

3

induding

the

detectors,

but

exduding

the electronics.

Because of the

complexity

_we have to restrict ourselves _{to a} rather

symbolic presentation

of the wavefunction components. A ngorous ^treatment of

subsystems

is given

by

Dieks _[7] where the biorthonormal

decomposition

of

subsystems plays

_an

important

rote.

The total wavefunction _is the

product

of wavefunctions of the

following

components: an

incoming

electron, _a

crystal,

_space

containing

_an electron lens wherein the

outgoing

electron is

moving,

_an electron _screen that _can be put ^{in the} focal

plane

of the lens _or in the

image

plane,

the

electromagnetic field,

_a

photon detector,

and _a wall

surrounding

the whole set-up

completely.

The focal point of the lens is _m the middle of the

crystal.

We

apply

the

thin-crystal limit,

_so Bloch _{waves are}

neglected.

The

complete

wavefunction is written _as

x=alil>14l>lS>lE>lD>lW>. (4)

with

ahj " 1: at the intersection of _row h with

plane j

there _{is an} atom,

ahj = 0: at the intersection of _row h with

plane j

there is _no atom.

il _>

incoming electron,

which

develops

in time from:

(il

>=

(il+

_{>: a wave}

packet

with _{mean wave} vector k; and _{mean energy}

E,

to

(il

_>=

(il~

_{>: no}

incommg

electron.

(4l >: the

outgoing electron,

which

develops

in time from

(4l >= (4l~ >: no

outgoing electron,

to

4l _>= (4l(~~ ^>:

outgoing

electron emitted

by

atom

h,j

with _wave vector k

=

kf,

and

finally

back to

Ill

_{>= (4l~} >.

(S

_> the _screen, which

develops

_m time from

(S

>=

(S~

_>: the _screen, not hit

by

an

electron,

to

(S

>=

(S(~~

>: the _screen, hit at position _x

by

_an electron with _wave vector k

coming

from

atom

h, j.

(E

_>: the

electromagnetic field,

which

develops

in time from

(E

>=

(E~

>: empty

electromagnetic field,

to

(E

>=

(El

_{>: a}

bremsstrahlung photon

with _{energy E} has left the

crystal.

Indices h,

j

_are

dropped

because of the

large wavelength

of the

photon:

h and

j

cannot be measured from

El

_{>, so} it does not

depend

_on

h, j. Finally (E

>

develops

back _to

(E

_>=

(E~

>.

D _>: trie

photon detector,

which

develops

from

ID

>=

(D~

>: trie

photon

detector is not hit

by

a

photon,

to

ID

_>=

IDI

_>: the photon detector _is hit

by

_a photon with energy ^E.

W > the _watt around tue

crystal.

Starting

witu _an electron in tue incoming bearn:

x =

£ahJ(il+

_{> (4l~ >}

_(S~

_>

_(E~

_>

(D~

_>

(W~

_>

(5)

hJ

tue wavefunction

develops

under influence of tue Dirac

equation.

Tue electron is

inelastically

scattered

by

atom

h, j

and _a

bremsstraulung photon

witu energy E is emitted. Tue interaction between tue fast electron and tue atom tuat _causes

bremsstraulung

in tue energy region of

interest

(a

few

kev)

is assumed to be located in _a small

region

around tue atomic nudeus.

Tue Born

approximation

_is

used,

_we

drop

tue

unperturbed incoming

wave. After _some time tue electron _wave

packet

is between tue

crystal

and tue _screen, and tue

photon

_wave

packet

is

between the

crystal

and the detector:

X "

£ _ahj(il~

_>

(4l(~~ ^>

(S~

>

(El

_>

(D~

_>

(W~

>

(6)

hjke

The electron hits the _screen at position x and the

photon

hits the detector _or the wall

x =

£ _ahj(4f~

_{> (4l~ >}

_(S(~~

_>

_(E~

_>

IDI

_>

_(W~

_>

hjkxe

+

£ _ahj(il~

_{> (4l~ >}

_(S(~~

_>

_(E~

_>

_(D~

_{> (W~+ >}

₍₇₎

hjkxe

The second term is not important for the measured

photon yield.

The wavefunction has

developed

_now into a macroscopic stage: excitations _on the electron

screen and in trie

photon

detector. Therefore _{we can} calculate _now the

probability

that _a

photon

of _{energy E} is detected in the

photon

detector and that the electron has hit the _screen:

Î(E)

"

x(E)x*(E)

"

£

tlhjtlh,j, ^{< iÎ~ iÎ~ >< 4Î~ 4Î~ >}

hh' jj' kk' xx'

<

S(~~~ S(~,~,~,

^{>< E~ E~ ><}

Dl Dl

_>

(8)

After _a certain time _we know that

(il

_{>, (4l >,} and

(E

_{> are}

really

empty, that is the electron

is for _sure not in front of the

crystal

and not between the

crystal

and the screen, and the photon is not between the

crystal

and the detector. Therefore the

probabilities

< il~ il~ >,

< 4l~ 4l~ >, and _< E~ E~ _{> are}

equal

to one.

Each part ^{of the} screen forms _a separate macroscopic system, so there is _no coherence between the wavefunctions in different parts,

only

terms x = x' _survive in the summation [7].

<

Dl Dl

_{> is} trie detection

efliciency

of trie

photon

detector which _we make

equal

to 1.

The

shape

of trie CB spectrum becomes:

Î(E)

_"

~

_{~hjah'j' <}

_{~~kx ~~j'k'x}

_{~ ~~~}

hh'jj'kk'x

with (k( a constant

value, depending

_on the energy of the

incoming

electrons and the _energy loss _e.

If the _{screen is}

positioned

in the focal

plane

of the lens there is a one-tc-one

correspondence

between trie _wave vector k and the

position

vector on the _{screen x.} Therefore terms with

k

#

k' do not _survive [7]. ^{So trie} CB spectrum ^becomes

Î(E)

_"

~

_{tlhjtlh'j' <}

_{Sljk ~~j'k}

_~

"

~

_ÎXk(E)Î~

(1°)

hh,jj,k k

xk(E)

_#

£

_tlhj

_S~k

_>,

(11)

hj

with the electron _screen in the focal

plane;

if the electron lens _is removed and the electrons hit the wall of the _vacuum chamber far _{away we} expect the _same result.

Equation (10)

_means

that for each value of k the

probability amplitudes

for electrons

coming

from ail _{atoms must} be added and alter that the

intensity

must be calculated. The CB

intensity

is the _sum of the

intensities for ail k vectors.

If the _screen is

positioned

in the

image plane

^{and if} the lens makes _an

image

of the

crystal

with atomic resolution, electrons

coming

from different atomic _rows hit the _{screen at} different

positions

_x. Therefore there is _a one-tc-one

correspondence

between the _row index h and the position index _x and terms with h

#

h' do not _survive in the summation. So the CB spectrum

during good imagmg

of the electrons is

Î(E)

"

£

tlhjtlhj' ^<

~~k ~~'k'

~

~ ~~~J _{~~k ~Î~}

_~~~~

hjj'kk' ~ J~

Equation (12)

_means that for eacu _row tue

probability amplitudes

for electrons _coming from different atoms wituin the _row

going

in different directions k must be added and after that the

intensity

must be calculated. The CB intensity is the _sum of the intensities for ail _rows.

An alternative _way to calculate the CB spectrum under the

focusing

condition is

using Huygens' principle

for

calculating

the intensity distribution in the

image plane

out of the

amplitude

distribution _in the focal

plane, equation (11),

and _sum the intensities for the whole

image plane.

If there _{is no} difference between

(10)

and

(12)

,

proposition

2 in Section 1.2 is true. If there is _a

difference, proposition

1 may be true, and _{we are} able to influence the CB spectrum with

manipulations

_on the

outgoing

electrons at _a

large

distance from the

crystal,

_an

example

of _a

delayed-choice experiment

_[9]

2.2. CALCULATIONS. From the

equations (10), (11),

and

(12)

_we learn that the coherent

bremsstrahlung yield

of

photons

with energy E

depends

_on

£~~ ahj(S(~

>. This is the

prob- ability amplitude

that the electron _screen is hit

by

_an electron with _energy Ebeam _E that

was scattered from _{an atom at} the intersection of _row h and

plane j

in the direction

k/

k _(, summed _over all _atoms h, j. An

elegant

_way to perform such _a calculation _is using the

analogy

between

optics

and _wave mechanics [Si. This

analogy

is _an

example

of

Feynman's path integral

method

[loi.

At _a number of

photon energies

E the

following

calculations _were done. For each atom and each

scattering

direction _a

phase

_{p is}

calculated,

_see

Figure

4,

P=alk>1+blkfl> (13)

in which it is assumed that the

phase

shift in the melastic scattering _{process is}

independent

of the direction of the

outgoing

_wave; k; and kf ₌ k _are the _{wave vectors} of the incoming and

the

outgoing electron, respectively.

For each spot x on the focal

plane, corresponding

to _wave vector k, ^the total

probability amplitude

is calculated:

Xx(E)

_"

£

_e~~~~,

₍₁₄₎

hj

which is

equivalent

to

equation (11).

The

intensity

of CB

photons

with energy E and with the electron screen at the focal

plane

is

proportional

to

I(E)

=

L _{lxx(E)l~ (15)}

ç lrow

^h

F

a l' atom

plane j

b

1 _F' focal plane

1,

Fig. 4. Definition of optical paths, not to scale.

with the summation extended _over all _x values with

scattering angle

8 smaller than _a critical

angle 8~r.

Huygens' principle (Refs. [Il], Eq. (3.31)

and

[12], Eq. (1.12))

is used to calculate the

intensity

distribution in the

image plane

from the

probability amplitude

distribution in the focal

plane, equation (14).

The calculations of

equations (14)

and

(15)

must be made for ail directions of kf where the

single-atom scattering amplitude

is

large.

The scattenng

amplitude

is taken _{as a} step ^function:

1 for

scattering angle

8 < 8~r and 0 for 8 > 8~r.

Sommerfeld [13] gave the

angular

distribution for electrons that lost _some of their kinetic energy in the

bremsstrahlung

_process for the non-relativistic _case. The intensity is maximal at 8

= 0 and is reduced to half maximum at Hi_{/2 "}

((k;( (kf()/(k;(.

For 160 kev electrons

losing

3 kev

Ri

_/2

" 0.5°. This _is the _sonne order of

magnitude

as the deflection

angle

of the

first order

Bragg

diffracted

peak,

0.85° for the situation descnbed below. Sommerfeld made

use of the pure Coulomb field of the atomic nudei, without

taking

mto account the

screening

effect of the

surrounding

electrons.

Koch and Motz [14] ^made a

compilation

of

bremsstrahlung formulae, taking relativity

into ac-

count. From their formula

IBS, using

^the Born

approximation

and the Thomas-Fermi

screening function,

the

angular

distribution of

inelastically

scattered electrons _can be

calculated, Fig-

ure 5. The much

larger

value of

Hi

_/2, 8°, is caused

by

the

screening effect,

which restricts the domain of

impact

parameters to smaller values.

Calculations _were made for 160 kev electrons

bombarding

two silicon

crystals

with the

il11]

axis

along

the electron beam, a thin

crystal,

2 x 2 _x 4 nm,

containing

840 atoms _on 105 different atomic _{rows, or a} thick

crystal,

2 x 2 _x 40 nm,

containing

8400 atoms _on 105 atomic _rows.

3

g

g

~ 2 ai

_oc

à Î

w

Ô

0 10 20 30

inefastfc scattering angle (degrees)

Fig. 5. Dilferential _cross section

d~a/dedQphatonsin8d8

_{as a} function of the scattering angle

8 of trie electrons for inelastically scattered electrons of 160 kev _{on a} silicon atorn producing _a

brernsstrahlung photon of 3 kev that is ernitted in _a direction perpendicular to the electron. bearn.

3. Calculation of CB

Spectra

with

Boundary Angle

8~r ₌ 2.5°

In this section it is assurned that inelastic

scattering

_{can occur} at

angles

greater than the small- est

Bragg

^reflection

angles.

So for each Laue _zone several

reciprocal

lattice

points

contribute to the coherent

bremsstrahlung, Figure

ld.

3,1. THE CB SPECTRUM _{WITH THE} ELECTRON SCREEN _{AT THE} FOCAL PLANE.

Figure

6

gives

the calculated CB spectrum

according

to

equation (15)

^for _a thin

crystal.

The calculated spectrum has the _same features _as measured spectra,

Figure

2 and [2]:

peaks

2 and 6 _are

absent, peaks

4 and 8 _{are more intense} than their

neighbours.

A CB spectrum calculated from

just

_one atomic _row,

containing

⁸ atoms, has the _same

shape

_as

Figure

^{5 and its}

mtensity

is _a factor of105 smaller. So the calculated spectrum, as

well _as the

expenmental

spectra, must be

interpreted

_as

intensity

summations of spectra from separate atomic rows; mterference _terms from different _{rows are} absent in the

wavefunction,

so for

large boundary angle

8~r

proposition

2 of Section 1.2 is true.

A peak in the CB spectrum is the result of contributions from _a number of

reciprocal

lattice

points

from the _same Laue _zone. The _energy loss _is smaller if the

angle

between k; and

kf

increases, second term in

(3),

_see also

Figure

1d. In order to calculate the finestructure in _a Laue

peak

the thickness of the

crystal

must be increased.

Figure

7 shows the result for the

peak

at 3.3 kev. It is this g ^vector

broadening

that determines the width of the

complete peak Ils]

3.2. THE CB SPECTRUM _{wiTH THE} ELECTRON SCREEN _{AT THE} IMAGE PLANE. The

intensity

distribution in the

image plane

is calculated from the

complex amplitude

distribution in the focal

plane by application

of

Huygens' principle.

The calculated

intensity

distribution

gives

a

good image

of the atomic _rows in the

crystal.

The

intensity

of CB

photons

is

equal

to the _sum of the

intensity

in the

image plane.

For

all _cases that _were calculated the result _{was a}

complete

absence of any difference between the

yield

of CB

photons

with the electron _screen at the focal

plane (which

is

equivalent

to the absence of the

lens)

and the _{screen at} the

image plane.

g

~ g

é

é 1

0 2 3 4

kev)

Fig. ^6. -

a energy loss = CB energy. _Silicon [Ill] crystal 4 nm thick.

The

peaks

are ^arked by nurnbers the _Laue 20ne frorn which hey

originate. Large

angular

distribution

of elastically scattered electrons: Hcr = _2.5°.

4

3.0

energy foss (kev)

Fig. 7. -

crystal 40 nm.

3.3. DiscussioN.

- The result

the coherent _spectrum from the way the ectrons

are

detected.

Whether

the outgomg

electrons

are focused or not is not _of any relevance

roposition 1 of

Section 1.2 _is not

true. Nothing ^pecial happens when the _outgoing

electrons

are

_The reason that here is no

effect is quite

obvious. The wavefunction does

contain interference terms _between

probability amplitudes _from

different

rows.

tomic rows

can be imagèd very

well with

inelastically scattered _electrons, but _a

row, that

would

destroy the

mterference,

makes

no

difference

for

the result.

The ^absence of

any

_effect

of

the

screen position on the photon