Coherent Bremsstrahlung and the Quantum Theory of Measurement E.H

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

HAL Id: jpa-00247055 https://hal.archives-ouvertes.fr/jpa-00247055 Submitted on 1 Jan 1995

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés.

(1995)

Phys. I OEYance

245-262 5 1995, 245 FEBRUARY PAGE

Classification

Physics Abstracts

78.70F 03.658

Bremsstrahlung Quantum Theory Coherent and the of

Measurement

Voorthuysen E-H- Marchie

du van

Gromngen, Physics Centre, Nuclear Solid and Materials Nijenborgh University State Science of

Groningen, Netherlands NL 4, AG The 9747

(Received accepted September 1994) 1993, October December revised 29 19g4, 21 18

(CB) brernsstrahlung Bragg Coherent Abstract. is the result inelastic of scattering of elec-

crystal. kev, hundreds elfect of few of collective

trie whole microscope

of In electron trons

a a an

theoretically possible inelastically electron of where the deterrnine trie is

it atoms to

row was

contradictory.

path

These

Optical scattered. calculations made kev for 160 staternents

are are

[Ill] by inelastically crystal scattered silicon compared with

electrons and the results

a are

application by rneasured CB The results understood be Kampen's

theory Van of spectra. can

Experimental CB of mechanical of the result be quantum spectra measurement. tum out to

summation coherent within and incoherent the of intensitie8 from of scattering atoms

row, same

words,

other the electrons CB locabsed

in within atomic dilferent that

rows, or, cause are rows.

Introduction l.

During of of of collisions electrons few hundreds

BREMSSTRAHLUNG. COHERENT 1-1. a

resulting

following shell

the ionisation inelastic inner in kev with events

atoms can occur:

for resulting charac- X-ray radiation, shell and ii) ionisation excitation in charactenstic outer

(EELS) iii) region, loss the eV and peaks the electron emission in 10 in teristic spectrum energy

bremsstrahlung changed by Coulomb the photon the electron the field of of when orbit is

nudeus. atomic

physics partide electromagnetic when is

that radiation dassical electron emits is In it

an a

/3(t)

velocity bremsstrahlung by electron

The of emitted with intensity the accelerated.

an =

v(t)/c is [1]

(l) d~I/duJdfl ((n

/3) /(1

e~ /(47r~c) e~~(~~~'~(~)/~)dt(~ /3)

n)~j x x =

r(t)

connecting

the of and emitted radiation

the direction the the where is in vector

vector unit n

without bremsstrahlung continuous, Normally the is detector. with spectrum the electron the

structure. any

interatomic

equidistant with

trajectory close of with consider electron atoms

We to

a row an a

We change

velocity Ad nucleus. each undergoes time L.

This

distance electron it

passes a a

IOIIRNAL PHYSIQUE11 5, OE T N° PEIIRUARY 1995

2, 4

Physique Q de Les Editions 1995

PHYSIQUE JOURNAL DE I N°2 246

(Ad(

straight (p(,

fl and

fine time between trajectory is the the successive

< a + so assume

L/(flc).

velocity change At is for The interactions is /3 radiative except

constant,

zero =

bremsstrahlung

frequency Fourier of with the the the is integer,

mât C spectrum

t +

so m =

equidistant sharp pulses the domain: of in time transform

j.27rflc/(L(1

f(uJ) /3))] (2) j à Const integer.

[uJ = =

perpendicular ground the of the photons for direction electrons that emitted the We observe to

velocity proportional proportional the frequency electron the and inverse interatomic is to to

distance.

spread spread the angular incoming

electrons of and the the beam the If in

energy are

regarded regarded but be sufliciently

small be entities, the localised electrons

must not may as

electron, plane

the which assumed of

manifestations wavefunction be of the is

wave, a as a

wavelength wavelength

Àf, the electron leaves with When

with The À;. À;. Àf atom

> we wave

crystal,

scattered from

of dealing extended the all

with like atoms,

structure

a waves an are

(the

added and number of electrons detected far intensity be the

atoms must away screen on a

time) squared.

equal of the modulus of

of Constructive the is unit

unit wave, per sum per area

scattering, Àf, interference effects like the of where elastic in and destructive

À;

occur, case =

(electron diffraction).

Bragg

scattering the well give effects As known

these rise matter to a

elastically

practically

leaving fact, by crystal electrons that scattered of the ail thin

are are a

interference place. inelastically where takes crystal directions For scattered in constructive

completely by

bremsstrahlung if electrons the the removed

true energy same excess one is is

Bragg photon. construction, which useful Ewald for visualising The is scattering,

very con

coherent used Figure therefore illustrate inelastic electron also scattering, Coherent be 1. to

(CB)

bremsstrahlung k; kf relation satisfied the when and

(kf(,

vector (k;(

+g >

occurs q is =

1/À,

k connecting lattice the electron k points is is

vector,

(k( in

two vector

g e wave a =

photon. reciprocal hq the of the CB and is momentum space,

photon of The the CB

is energy [2]

g2/2k;)/(1

pcos9) hep(g~ (3)

~ =

projection

of k; the the of with angle

direction and of the the between direction

g gz on

(3). k;. channeling and Effects

photon of the neglected Diffraction radiation effects in are

bremsstrahlung neglected, lattice itself with wavelength of the also the coherent of the are

bremsstrahlung generally large. is too

parallel

incoming electron

When beam the points

of reciprocal the all to

axis, is zone a

belonging

lattice have Figures

Laue the the value for They ld.

and to

same same zone gz,

peak bremsstrahlung the the

contribute k;, in because and

equations the spectrum to < same g

(2) (3) equivalent. and

are

between observed the CB be characteristic the in X-rays, continuum Figure The CB 2. can

peaks marked

equal

numbers the with of the

order Laue Vecchio and

Williams to

are [3] zone.

have study made expenmental of the extensive optimum producing conditions for peaks. CB an

(no These conditions

specimens thin X-rays elements impurities), of

intervening from are: pure

surfaces,

crystal dean (to orientation, low vibrations),

lattice axis reduce temperature and zone

detection with semi-conductor angle detector solid with

steradians

0.01

direction

> a a a m

perpendicular Carstanjen

beam. the from and observed CB

Sigle quasicrystal. to

[4] a

bremsstrahlung

considered effect be analytical side elec- Coherent is in

to annoymg an

X-ray

peaks

peaks from

impurities CB obscure

lead microscopy,

tron to wrong or can can

problem

interesting CB lead X-rays.

of But also impurity determinations

to intensity an can

fundamental of nature.

COHERENT N°2 BREMSSTRAHLUNG QUANTUM

AND THEORY

247

. ,

i~ .

o . ~

b) .

k ç

-3 ~

) q 2

-2

. ~ .

)

-1 -

-o ~

. . ,

a) b)

(kf(; reciprocal (k,( Ewald lattice: elastic Fig. constructions the inelastic scattering, l.

in =

hq photon; the CB inelastic of (kf(, c)

the emitted (k,(

(k,( scattering, scattering, momentum > > is

towards third

the order k, axis trie (kf(,

axis, Laue scattering point

zone; in zone a on zone a ii

k, (kf(,

in first

towards order Laue the scattering inelastic point scattering,

(k,(

> axis,

zone a a ii

zone.

intriguing physics One of the of the is PROBLEM.

DEFINITION 1.2. aspects most THE oF

particle

electron localised manifestation duality. question

the the particle-wave The is it

is or a

nicely experiment,

the double-slit for extended illustrated with instance of wave? be

see can an

detector,

function of measured

position The

flux of the is Feynman electron it

[5]. on as as a

factors, and

slits, diffraction factor of the

behind wall with the

consists two

two

a a screen

factor, diffraction. Fraunhofer interference observe

interference When it like is not

we m an

PHYSIQUE JOURNAL DE N°2 I 248

Cu sumoeak Si Si Fe Cu Fe

7 5

JO 6 8 4 2

Ioss (kev) energy

[loo] crystal Experimental

photon A

silicon bombarded Fig. temperature 2. spectrum. at

is room

[Ill] microscope.

parallel electrons axis JEOL JEM CX Photons by

kev 200 120 to

in a are an

Si(Li)

angle 90° detector eV. The of by with resolution of130 sobd

direction H detected the in

a a

10~~

by peaks X-rays

dominated of characteristic that

is The 1.8 srad. spectrum is acceptance

X are

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

indicating they by

bremsstrahlung marked numbers the from which Laue

peaks of coherent

zone are

origmate.

actually really

observed which electron through possible deduce slit the

went. to measure or

the

detect the slits disturbance electron of As of the

try

passage cause we we one a soon as

interference the phase

of the wavefunction the electron such that factor of result

the

as a m

terminology through

measuring The that the of the value.

usual into constant act

turns is a

collapsed. wavefunction electron has the of slit electron the the at

electron electron diffraction observed be for the of said the The it is in

can same case as

There which

passed microscope. by determine the

electron without atomic is

way no row was

destroying diffraction of the pattern spots.

leading of bremsstrahlung

Bragg the

In inelastic scattering

observe coherent to

case can we

photon photons the well the information about electrons. The

collec- spectrum contains

as as a

crystal effect interference ail of the between from scattering ail At tive 1.e. the atoms, atoms.

provides electrons information the of time, the location where scattering

the detection

sonne on

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

Heisenberg: by angle uncertainty relation of if scattering the given bounded within the is is

uncertainty

the of the electron the of small.

is transverse

momentum

too too cone, a narrow

the that assumption the where

make the For the scattered be electron moment

row we can

crystal

slightly electron The quite reduced, still determined. leaves the with accurately but a

high-resolution

possible known electron In be

good make

must to energy. microscope a a

crystal resolution,

of with two-dimensional the atomic position-sensitive

elec- mount to image a

image plane, the

electronic coincidence and deterrnme

for detector each at tron to

unit an m

bremsstrahlung. photon passed

which producing detected the the atomic while If electron row

bremsstrahlung

the coherent that interference factors contains

between spectrum

we assume

factors

atomic

these

different become of the wùl

constant

measurement

rows, as soon as a row

change the CB

made. made of So the is spectrum atomic is must

measurement

as soon as a

photon passed

by: trajec- the of caused electron that the the where the

measurement row

reduces intensity peaks

CB through crystal the of and the increases of the electron tory some

througu

collapse famous wavefunction trie of peaks. of Tuis tue otuer intensity act tue is

COHERENT N°2 QUANTUM BREMSSTRAHLUNG AND THEORY 249

fi Mfi- 2

9 j

3-&-

à4

A y

~ ~

X X Y Y B

llit

+Î

MEMORY

gate

yes =

Fig. paradox.

photons CB The Switches down:

atornic detected without of 3. the rneasurement are

photons Manipulation

CB with sirnultaneous switches detected

of the

rneasurernent

up: row, are row.

pulses delay and the switch annihilation

of creation of the that

the it units, count

rneans in or means

l) depends

measured setting incoming electron,

A the switch the future.

in in counter rate

as on

3) aligned parallel

crystal 2) incorning E. with

electron beam. electron the

atomic to energy rows

making lens, electron, outgoing

resolution. trie detector with

atomic

image energy on an =

voltage proportional 5) position pulses sensitive

z-coordinate. E electron detector. with the to e.

voltage proportional pulse electron, pulses y-coordinate. 8)

with the detection of and

means a an

known; photon with

the from is information stored CB this the

in atomic

rnemory. row carne is

Il) 10)

proportional pulse by voltage logic pulses the

with if within window

to set

energy E, e, e is as

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

belong sirnultaneously pulse delay

pulses containing DELAY: the

event. unit

to arriving many sarne

through voltage pulses photon speed and with DET: detector. it with conserved. constant running

puise voltage voltage

analyser; the

single

channel and within

if the SCA: output

gives rneasures is a

specified interval.

place collapse wuen last tue

Now tue let tuat takes is connection measurement.

us assume

photon-electron complicated needed for tue coincidence circuitry electronic that

made tue is m

Figure switcues uigu-resolution Tue last

electron 3. tue in sit-

measurement microscope, are

be stored. will

wuere tue information

detectors and tue far from tue uated to counter near

typically

but pulses tue from tue detectors Travelling 10 of electronic

times to memory ps, are

milliseconds, seconds uours delay be mcreased

tuese times by of units

or even can means

delay Cuanging the of tue switch would that loss setting without

in units

accuracy. any mean

PHYSIQUE JOURNAL N°2 DE I 250

pulses

pulses annihilated other electronic and If

immediately created. electronic

are some are

photon

pulses follows the this with that of the it

accept spectrum

not consequence, a we

dependent

already position of switch height pulse distribution that the the is present

on were

manipulated: equal

the A long switch the in before the is rate count to count rate counter was

dear-sighted

telescope

make

unique later. that could

hour with In B in

counter case a one we a

stock-exchange.

possibility rich the become to on

solving problem:

propositions for this this discuss will In

two paper we

manipulation of

electronic the of the with equipment, is atomic 1)

act not measurement a row

place

outgoing the that the electrons

focused but takes

with at moment

are on screen a

resolution; atomic

do interference 2) coming factors from CB between different contain

spectra not

waves rows.

SOME MECHANICAL MEASUREMENT. REMARKS QUANTUM the

In 1.3. ABOUT common

(the

interpretation) Copenhagen

of called interpretation mechanics of the quantum act sc-

physical physical

measuring quantity

has effect the world: wavefunction the

an on a enormous

drastically, changes quantities physical and be determined

other with

cannot any some more

knowledge "holy"

only suflicient Before the of of

the world

is measurement act

accuracy. our a

possibilities, superposition complex administrated

of numbers, coherent with the

many wave-

objectively

after probabilities, function.

the world of the existing But is

measurement sum a

real,

probability positive One

with numbers. read perform such administrated

out (1.e.

may a a

measurement)

knowledge world,

and of the dassical does make

get it not,

not more may or one

difference for the of there the world. So between difference essential

rest quantum any an

mechanical dassical and measurement measurement.

change wavefunction sudden The "collapse the the wavefunction". called of in the the is In

interpretation of collapse

mechanics of the of the

result the

quantum

common occurrence as a

imposed

has theory be postulate:

into the projection the put measurement to

separate as a

postulate.

According

Bour mecuanical

terminated wuen is quantum tue to measurement outcome a

macroscopically been bas recorded

dassical made in difference

Bour between apparatus.

a a

world, microscopic tue where mechanics

the rules of

be applied, and the quantum to are

world, physics dassical where

valid.

macroscopic is

usually macroscopic large

of A consists partides

number of resulting system in

very a a

high density of

In mechanical

microscopic

states. quantum in

measurement system

very a a a

(nearly)

brought is

the

into with As change state

result contact system.

macroscopic pure a a

brought

about the in consequently macroscopic

and microscopic the the system in system: is

collapse wavefunction of the microscopic of the Because large of the difference system. in

density of both change between the probability irreversible: is systems the states both that

their original

neglected. be systems to state return can

According Kampen Van huge density

the

the of feature of is main to states

[6] macroscopic a

rules of The apply the mechanics system. but the eigenvalues quantum of the system, to

bave

distance ôE tuat mucu smaller tuan operator

tue

best energy average an accuracy

obtained tuat

be in Manipulations

expenment. influence tue

system

can an macroscopic a on

of wavefunction tue but tue result system, mucu small is observable. be So of tue rules to to

good dassical valid

approximation. mechanics

to very are a

measuring defines mechanical Kampen macroscopic Van

system quantum apparatus

as a a

metastable for opinion To the condition metastable is

state not in

state.

necessary a my a

of exarnple

Kampen nice gives Van measuring mechanical

instrument. quantum [6] a a

metastable

excited,

consisting mechanical measuring

atom

apparatus quantum an an m

BREMSSTRAHLUNG COHERENT QUANTUM N°2 AND THEORY 251

electromagnetic surrounding and free the field the In

ii. photon emitted

finds state space an

modes,

density photon nearly of different the of large. the A

electron fast states

cc so very

able consequently trigger metastable the and neighbourhood is existence atom, its the in to

change electromagnetic of the form of

irreversible measured the is the

in field. atom

an m

Kampen descnbes Van the whole

mechanical

the measuring in quantum

process way, a so

Schrôdinger

has wavefunction

The equation works the whole apparatus and

too. system a on

this of

demonstrated that is triggering the it destruction of

atom consequence as a means

possibility

of interference the electron. In the with of this Section

will

2 such I

paper use an

analyse problem approach Section the mentioned in 1.2. to

of'a

example Another

measuring mechanical the of of instrument foi] is

quantum

use a

solid the fast

for measuring existence of electron. The foi] is

matter system

macroscopic a a

undergoes change irreversible the Ioss of that due the the electron foil. This in

to energy an

electronically

change by semiconductor,

if the

be be read foil observed is it out

can or can a

measuring foil of

if calorimeter. When the has

rise the electron is temperature part

a a m

deposited

This the foil this

measured last

be in

measurement not. energy energy may or a

wavefunction dassical because the foil describing macroscopic The

is system. measurement a

foil of changed by the multitude the

excitations this such in is that in

not measurement way a

change change observable. The effect of

original this wavefunction this of the the fast is on

negligible. completely only positioning deed the electron It is the of foil of trajectory the is in

nothing electron, influences the of the electron that wavefunction the else.

theory interpretation A the modal of of related mechanics Dieks is quantum [7].

approach practice, Another isolated of Zurek macroscopic be that In systems is

[8]. can never

Leakage

of completely outside world. leads loss of coherence from the between to energy

eleménts

off-diagonal

disappear words, density different the of the

other matrix states,

or, m

havmg of dassical meaning

positive and left with the elements: real numbers the trace

we are

probability.

approaches boundary

microscopic for between and In ail mentioned there need

macro- no a is

mechanics) (quantum always, valid principle, microscopic for the the world

rules In scopic. are

of The physics of dassical without

loss

the rules be used

but

accuracy, may any very soon

"collapse of the

physics, induding wavefunction"

mechanics dassical the

quantum step

is -

projection the

complexity trie of and growing of the time just

system

goes on consequence as a

of nothing deed

change wavefunction has do with postulate left The the

be in to

out. a can

being. measuring by human some

amplitude(~ (probability

probability mechanical calculation the In real must

quantum

a =

develop

has calculated the mechanical started be

quantum system

to to

macro- as a soon as

of development follow

different would if The would be the scopic stage. outcome not one

impossible although further, would become due the calculations wavefunction

the to soon any

the complexity independent So the is

macroscopic of the tremendous system.

outcome on a

early. In the probability, provided done arbitrary when calculate the it is choice not to to

pho- the developed

when "paradox" Figure of wavefunction has into stage the

3 macroscopic a

deposited

in photon the electron has dissipated the detector and has its in ton energy energy

Any manipulations

influence the further do electron detector. position trie sensitive not wave-

So measurable.

microscopic the

such that the

function in system

are consequences way on a

telescope creationlannihilation dear-sighted pulses does and cannot electronic of

not occur, a

built. be

PHYSIQUE JOURNAL DE I N°2 252

Model The 2.

simple validity Section

propositions order of the

In the in GENERAL. 1.2 2.1. test to a

Following applied Kampen be simulations. Van made the model that in computer is

can [GI

full

by wavefunction. mechanical We will make complete described quantum system

a is a

exduding induding detectors, the but Figure description of the the electronics. 3 in system

symbolic

complexity ourselves presentation

of of the have rather Because restrict to

to a we

subsystems

by of

Dieks wavefunction where A is

the components. treatment ngorous given [7]

subsystems

plays decomposition of important biorthonormal rote. the an

following product wavefunction of

total the of wavefunctions the The components:

is an

crystal,

the incoming containing outgoing electron, electron electron lens wherein

space a an

plane of

image be focal the lens moving, electron that the in the in put is

an screen can or

electromagnetic surrounding field, photon detector, plane,

the whole and wall the set-up

a a

crystal. apply completely. thin-crystal lens of of

point The focal the the the middle the We is m

complete

neglected. limit, wavefunction Bloch The is written

so waves are as

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

with

plane j there the of h with intersection 1: atom,

ahj row an is "

plane j intersection there the of h with is 0:

at atom.

ahj row no =

develops electron, which from: incoming time il in >

(il+

(il

packet

and with

vector

to >: >= energy mean a wave wave mean

(il (il~

incommg electron.

>: >= no

develops outgoing

electron, the which from time in (4l >:

electron, outgoing

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

h,j outgoing electron 4l kf, with k emitted and

atom

vector >= >:

(4l(~~ wave =

finally back to

Ill

(4l~ >. >=

(S develops the from which

time

> screen, m

(S~

the by electron, hit (S

not to >= >: screen, an

(S(~~

(S by the

position electron hit

from with

coming k

vector >= >:

screen, an wave x

h, j. atom

(E electromagnetic field, develops from the which time in >:

(E (E~

electromagnetic field,

empty to >: >=

(El (E

bremsstrahlung photon

with crystal. has left the

j h, Indices

>= >:

energy a E are

dropped large wavelength because of the photon: of the j h and cannot

(E be measured from Finally depend h,

j. does it

not >, > so on

develops back to

(E (E~

>. >=

detector, photon trie develops which from D >:

(D~

by photon, photon trie

detector hit is not to >: >= a

IDI

ID by

photon

the

photon with hit detector

>: >= energy is a E.

the crystal. W around tue watt >

Starting incoming witu electron tue bearn: in an

£ahJ(il+

(E~ (D~ (W~ (5) (S~

(4l~ > > > > >

x = hJ

BREMSSTRAHLUNG N°2 COHERENT QUANTUM AND THEORY 253

develops wavefunction influence tue tue under of equation. inelastically Dirac electron Tue is

bremsstraulung photon scattered j and

witu

emitted. Tue is atom interaction

a energy E

bremsstraulung electron

between tue fast and tue tuat tue

in of atom causes energy region

kev)

be located few assumed

region around in small is tue atomic nudeus. interest to a

used, drop approximation

Tue unperturbed tue Born incoming After

time

is we wave. some

crystal packet tue electron between tue photon and tue packet is and tue

screen, wave wave

crystal between the the detector: and

(El

ahj(il~ (D~

(S~ (W~

(6)

> >

> > > > (4l(~~ X "

hjke

photon the

position hits the The electron hits and the detector the wall

screen x or

IDI

(E~ (W~ ahj(4f~

(S(~~

(4l~

> >

> > > > x =

hjkxe

ahj(il~ (E~ (D~

(S(~~

(7)

(4l~ + > > > > > > (W~+

hjkxe

yield. photon for important The the measured second is term not

developed

wavefunction

excitations the electron The has stage:

into

macroscopic now a on

photon Therefore

probability trie detector. calculate the

and that in

screen we can now a

photon photon the of detected the detector that electron has the in and hit is

energy screen: E

Î(E) x(E)x*(E)

iÎ~ 4Î~ 4Î~ iÎ~

< >

tlhjtlh,j,

" "

jj' kk' hh' xx'

Dl Dl

(8)

E~ E~ S(~,~,~, >< < >< > S(~~~

(il

really

After know and the that electron time that is

empty, certain

>, >, > (4l a we are

crystal crystal front and and the for of the and between the the in

not not

screen, is sure

probabilities crystal and detector. Therefore the il~ il~ photon between the the is >, not <

equal

and E~ 4l~

4l~ E~

>, < < > one. are

coherence forms there between is Each of the

separate system, part

macroscopic no so screen a

only the wavefunctions different summation in

in the parts,

terms [7].

survive x

efliciency photon equal

of which make trie detector

trie detection

< > we

shape of trie CB becomes: The spectrum

~~j'k'x

Î(E) ~~kx

< ~

~hjah'j' ~~~ "

hh'jj'kk'x

depending

of electrons the and value, incoming

the the with

constant (k(

energy energy a on

loss e.

correspondence

plane lens there positioned focal of the

the one-tc-one If the in

is a screen is

Therefore

with position

the and k the

terms between trie

vector vector screen on x. wave

k' CB becomes

So trie # do k spectrum not [7]. survive

(1°)

~~j'k

Sljk Î(E)

ÎXk(E)Î~

< ~

tlhjtlh'j' " "

hh,jj,k k

S~k (11) xk(E)

tlhj # hj

PHYSIQUE JOURNAL DE I N°2 254

plane; if

focal the the lens removed the electron and electrons the in with electron

is screen

(10)

Equation

the result. chamber far

of the hit the expect wall

away we same means vacuum

probability amplitudes coming for from of the electrons ail for each k that value atoms must

intensity calculated. the The CB of intensity that the be is the added and

alter be must sum

for ail k intensities vectors.

positioned image plane crystal makes the of if image the the

and lens If the is in

an screen

resolution, from different different hit coming electrons atomic the with atomic

at rows screen

correspondence

Therefore the index and

between positions there is h the

one-tc-one

row x. a

index and the h do the So CB

position with summation. in spectrum

terms not

survive x

good during imagmg of the electrons is

£ ~ ~~~J

~~k ~~'k' ~~k Î(E)

~Î~

< ~ ~~~~ tlhjtlhj' "

hjj'kk' J~ ~

amplitudes (12) probability

for for from Equation that eacu electrons tue

coming means row

different be

going different wituin the the directions added after k in and that

must atoms row

CB of the for

intensity intensity calculated. ail be The is the intensities must

sum rows.

focusing

using the under the CB An alternative calculate condition spectrum is to way

calculating Huygens' principle image plane intensity for the distribution the of the in out

(11), amplitude plane, equation

focal and the for distribution the the whole

intensities

sum in

plane. image

(12) difference

(10) proposition If there between and Section If there 2 in is

1.2

true.

is no

difference, influence be able CB

proposition and the with is 1 spectrum

true,

to may are we a

manipulations

outgoing large crystal, example the of

the electrons from

distance at

on an a a

delayed-choice experiment [9]

(10), (11), (12) equations CALCULATIONS. From the that the coherent and learn 2.2. we

ahj(S(~ yield photons

bremsstrahlung depends

of prob- £~~ with

This

the is

>. energy E on

amplitude ability

by electron hit

that the

with Ebeam electron that is

screen energy an E

from plane scattered the of j

and h the direction k intersection

atom at

an was row (,

elegant h, j. perform

An analogy summed all

such calculation using the

atoms to

way over a is

analogy optics between

This and mechanics Feynman's example of integral path is

[Si. wave an

[loi. method

following photon

At number of energies

the

calculations each done. For and

atom a E were

phase calculated,

Figure scattering each direction 4,

p a is see

P=alk>1+blkfl> (13)

phase the shift which assumed that in is the it

melastic independent in scattering

process is

of outgoing k; direction of

the the kf k and the

of the and

incoming

vectors wave;

are wave =

respectively. electron, the outgoing

plane,

focal For the each corresponding

probability

k, the spot total

to vector

on x wave

amplitude calculated: is

Xx(E)

(14) e~~~~,

(11).

equivalent equation intensity photons which of The CB is with

the and with

to energy E

plane proportional

the focal is electron to at screen

I(E) lxx(E)l~ (15) =

COHERENT BREMSSTRAHLUNG

N°2 QUANTUM AND THEORY 255

h ç

lrow

atom l'

j plane

1 plane focal F'

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

angle values

extended with the all

scattering summation with 8 smaller than critical

over a x

angle 8~r.

[Il], (3.31) (1.12)) (Refs. Huygens' principle Eq. Eq. and used the is calculate [12], to

probability plane amplitude image intensity from distribution the distribution the in the in

(14). plane, equation focal

(15) (14) for directions equations kf calculations of the The and be made ail of where must

large. amplitude single-atom amplitude

The

scattering function: taken is is scattenng step

a as

angle 8~r 8~r. for 8 scattering for 8 and 0 1 > <

angular distribution lost

Sommerfeld of the for that electrons their kinetic

[13] gave some

bremsstrahlung

the non-relativistic The maximal

the for intensity is in

process energy case.

(kf()/(k;(.

((k;( Hi reduced 8 and half maximum For kev electrons is 160 0 at

to at

/2 " =

magnitude angle This of the of the the losing kev order deflection 3 0.5°.

is sonne as /2 "

Bragg peak, for the Sommerfeld below. made first diffracted descnbed order 0.85° situation

taking screening nudei, of Coulomb of field the

the the atomic without

mto account pure use

surrounding effect of the electrons.

bremsstrahlung taking compilation formulae, relativity of Motz

into

Koch and made

[14] ac- a

screening IBS, their the Thomas-Fermi the using approximation formula and Born From count.

calculated, Fig- angular inelastically of electrons be function, distribution scattered the

can

the by effect, screening Hi 8°, of which larger The much value caused the is restricts 5.

ure /2,

values. smaller impact of domain parameters to

il11]

bombarding crystals with electrons for kev silicon the

Calculations made 160 two were

crystal,

different

along containing

beam, 840 electron the thin 4 105 2 2 axis atoms

x nm, x on a

crystal, thick containing atomic 8400 2 40

atomic 105 2

atoms nm, rows, x x rows. or a on

PHYSIQUE JOURNAL DE I N°2 256

g 2

à Î

20 10 30 0

scattering angle (degrees) inefastfc

d~a/dedQphatonsin8d8

angle function the

Fig.

Dilferential of section scattering 5.

cross as a

inelastically producing scattered electrons kev silicon

8 for trie of electrons 160 atorn

a on a

perpendicular photon

ernitted in brernsstrahlung kev is direction the electron. bearn. of that 3 to a

Boundary Angle Spectra 8~r Calculation 2.5° of CB with 3.

angles

scattering assurned that inelastic than the this In small- it is section greater at

occur can

angles. reciprocal reflection

Bragg points lattice So for each Laue contribute several est zone

bremsstrahlung, Figure coherent ld. the to

CB SPECTRUM Figure

THE ELECTRON SCREEN

FOCAL PLANE. 3,1. 6 WITH THE AT THE

calculated gives equation crystal. according the for CB

(15) calculated thin The spectrum to a

Figure peaks and has features measured

the

and spectrum spectra, 2 6 2

[2]: as same are

neighbours. peaks absent, and than their

8 4 intense

are more

A from

just CB calculated atomic containing

has the 8 spectrum atoms,

one row, same

shape Figure

and factor of105 mtensity calculated

smaller. So 5 the its is spectrum,

as as a

interpreted

expenmental intensity

well the be of from summations spectra, spectra

must as as

wavefunction, mterference

from different the absent in

separate atomic terms

rows; rows are

angle boundary

8~r large proposition for Section of is 2

1.2 true. so

CB contributions from the of reciprocal peak number the result of A lattice in

spectrum

is a

the angle points loss from Laue kf k; The smaller between if

the and

energy same zone. is

(3),

second Figure finestructure also 1d. order in In increases,

calculate the term in to

see a

peak the thickness Laue of Figure crystal the be result for increased. shows the the 7 must

peak broadening kev. It complete that width of the this determines 3.3 is peak the at vector g

Ils]

CB SPECTRUM SCREEN IMAGE THE 3.2.

ELECTRON PLANE.

The AT wiTH THE THE

complex plane amplitude calculated from the

image intensity the distribution distribution in is

plane application Huygens' by principle. intensity of the calculated focal distribution The in

crystal.

the gives good image of the atomic in

rows a

of equal plane. the the the in image intensity of For intensity photons The CB

sum is

complete

absence difference

between of result the

all that calculated the

any was a were cases

(which

plane

the focal equivalent yield photons of CB the the electron is with at to screen

lens)

plane. image the the of and absence the at screen

COHERENT BREMSSTRAHLUNG N°2 QUANTUM AND THEORY 257

2 4 3

kev)

- 6. Fig.

CB The a loss

energy 4 [Ill] = thick. crystal energy.

Silicon

peaks

the are frorn

which 20ne

Large

arked

hey

by Laue

nurnbers

originate.

angular

electrons: of = scattered Hcr

elastically 2.5°.

distribution

3.0

foss

(kev) energy

- 7. Fig.

40 nm.

crystal

3.3.

DiscussioN.

The

result -

detected. way the from Whether

coherent

the the

ectrons

spectrum are

electrons not any relevance outgomg or is the

are

focused not of

1.2 when

1 the pecial

not

roposition

Section Nothing

outgoing electrons true. happens

effect is quite

The does

are reason that

no obvious. here

wavefunction The

rows.

terms interference

rows

between tomic

amplitudes contain

probability

from

different

well with

scattered but

be very can

imagèd

a electrons,

inelastically

would

difference makes the no for that result.

row, destroy the mterference,

of the position on any of photon the

effect absence The screen

PHYSIQUE JOURNAL DE N°2 I 258

plane of distribution

made intensity image of in the the electron calculation the In

use was

(12) applying calculational

of equation instead of because principle. This done Huygens' was

Huygens' principle the of

should the the be but the result conveniance,

same. conserves norm

already proposition be

for this 1 electron true.

cannot

wave, reason so

theory of mechanical this of the

interpret We situation quantum terms

measurement. may m

separately. of

from each It the

photon of atomic is

spectra The spectrurn consists

sum row a

photons

originate

where known find when the the that atomic expects

to spectrum rows are one

electron).

(= partide,

manipulation

made by the the We associated of measured

on means a

(transverse

explicitly

order the outgoing wavefunction of the electron the in to not

row measure

transfer) (momentum

placing plane. position) by the focal angle

in scattering but the

a screen

observables,

non-commuting that and Position expect measurement

momentum are, so we

obtaining position for that the the uncertainty in is

transverse momentum necessary assures

this interference photons different But nevertheless ab- from is

interference between rows.

of electron Apparently, position scattering the is contained the in the centre transverse sent.

wiped

that wavefunction such it in cannot out.

way a

flying

regard electron

theory of the frorn the

is it In

measurement terms away may as we

brought by

change inelastic

which

irreversible about the crystal

macroscopic in system

an is as a

higu density large of from The the scattering Section

1.3. states

necessary comes process, see

read

macroscopic of allowed. One this of the that number directions

wavevector out may are

by itself, putting image

order this done is the the know the in

system to at very screen row

photon

the plane; tue does For in

omit tuis matter, step.

measurement it not

may a or one

completed

of been mecuanical tue bas atomic

quantum sense row

bremsstrahlung

they normally is,

A result of the calculations that coherent spectra,

as are

observed, words, interference different of

do from other contain in

not terms rows, waves or

crystal

the localised within electrons the in separate atomic

are rows.

probability amplitudes

Feynman added, should for what

be critenon intensities:

[5] gave a or

probability they amplitudes Using only indistinguishable. add if

criterion, could this

are one

argued beginning amplitudes from have from the that and

intensities should separate not rows

added, possibility apriori doubt be because their image

the the is with

to to reason no rows

outgoing using with the electrons resolution. atomic microscope a

possible calculations explain that experimental the From it is CB it turns not out spectrum to

first assuming scattering

afterwards

elastic and scattering,

inelastic do make

not

we errors so

neglecting by

Spence Bloch also the ai.

et [16]. waves, see

Figure paradox clear that of already wavefunction does exist. It the 3 has The is not

"collapsed"

early crystal.

after the the electron has left stage,

very soon a m

Angle

Spectra Calculation CB with Boundary 8~r of 0.2° 4.

always

angle Now that the angle lowest smaller than the inelastic where is scattenng

we assume

Bragg place. rather

reflection take situation, This abnormal only be achieved is that

can a can

crystallographic crystallographic choosing by another with

least

direction index 1, at » one

8~r sake discussion reduce but for the of keep just the the and the electron bearn parameter we

[Ill]

parallel reciprocal direction.

only picture the the reciprocal that lattice In it to means

bremsstrahlung, contribute coherent Figure lattice that lie the axis the points lc. to

on zone

[Ill] only

points of In silicon eacu Laue bave turee

the

out

zones zone one case axis. on

disappearance reciprocal of

the the lattice An other interference points. Destructive causes

peak Figure corresponds 4, Peak example interference the 4

such of destructive is 8. to see

outgoing of kev with the incoming beat electron 156.7 kev. of the 160

pattern wave wave

high perfect peak find

This with beat intensity with is 4

pattern

row, resonance one so we m a

N°2 COHERENT

BREMSSTRAHLUNG QUANTUM AND THEORY 259

0.235 ~ nm

~0.314nm

il11] Fig.

The three of

silicon

beat types with corresponding peak

the CB in

4 to rows m waves

spectrum.

3 ~

0 3 2 4 6 8 5 9 7 ID

(kev) foss energy

angular inelastically Fig. Figure but electrons, 6, Sartre for srnall distribution of the 9. scattered

0.2°. 8cr

only

has

from which Figure shape calculated the that the in 6. spectrum

row, was one as same

Si[Ill]

of that shifted built of three with

each other. atomic is types respect to

eut rows are

from 120°, shift beat different of phase that The the such between is patterns types is rows

probability amplitudes phase

coming different from of the the outgoing between is types rows

only

interfere if the destructive outgoing result therefore and 120°

too, way waves a as m a

forward allowed. scattering is

Figure

SCREEN PLANE.

ELECTRON FOCAL 9

THE CB SPECTRUM 4,1. THE AT WITH THE

(15). Figure Compared

equation from calculated

the CB

to spectrum

it many gives was as

difference effects. there dear interference So

is peaks disappeared due destructive

have to a

calculated interference

before: and that CB the between this spectra terms spectrum were

peaks

result from different atomic show with the that photons coming

between

many up, rows

expected that

therefore

absent. It simultaneous be CB the measurement is in to spectrum are

peaks photon onginated the of will the where from trie of

trie atomic

reappearance cause row

7, 1, 4, 5, Laue etc. zones

PHYSIQUE JOURNAL DE I N°2 260

ELECTRON SCREEN PLANE. CB SPECTRUM

The IMAGE THE 4.2. WITH AT THE THE

plane electron the image gives exactly of with the the CB the spectrum calculation at screen

plane,

expected the focal last with the what result the in in at contrast to

screen as was same

photons from paragraph. interference The destructive between dilferent remains active. rows

large crystal boundary

8~r

angle image of the with there the In situation is to

contrast on no

plane

illumination the of image the without the is structure. screen,

boundary

angle 8~r calculated small CB DIscussIoN. The do with contain 4.3. spectra a

changes drastically that

expected interference the be the

it spectrum is terms, to

soon so as as

photon dealing where created But

measured. here the the CB is with atomic

was we are row

Abbe, imaging saying theory beams

of least dilfracted that make

two

to at necessary are an

inelastically regular angular distribution the In this the

of of scattered

structure. image case a

loss,

for given electrons for

that, assumed

given Laue be is

energy narrow, zone, so a so a

contributing only diflracted the formation

image essentially beam So is is it to at most. one

by perform

optical of the inelastically impossible for atomic

measurement to means row a

angles. restricted forward if scattered the electrons is scattenng to

explain why possible alternative determine An the coordinates it is not transverse to to way

scattering with suilicient uncertainty of the of Heisenberg. the relation is Be- centre accuracy

spread

angle, reduced scattering of the in

the the of the electron transverse momentum cause

boundaries fixed

within of the smaller the position and hence electron transverse cannot is

bremsstrahlung be determined that

known the be where such in photon the

way con a row

originated.

for Figure by The condition photons given

measured that 9 is

spectrum

measunng a as are

they only

electrons if with that angle associated

smaller scattered that is

are were an over

Bragg than angle. by the first This bombarding fulfilled condition be crystal

the in

con a

high-index

direction. just

the But could well (10) summation restrict in equation

to as we

angles artificial by done This be small tuning electrou Figure [eus the

means. con m

position-sensitive plane, detector adjusting that the such by focal the single- in is and the

analysers and channel SCA SCA photons

that

such X Y only counted if 8 in B is counter are

By smaller than

moving value. the swich certain Figure upwards, original the in 3 spectrum a

(countrate

function

of the surgie-chanuel B setting of the analyser in counter behind the

a as

detector),

Figure changes photon

shape Figure 6, the of by just 9,

intercepting of to number a

puises

electronic coming photon detector. from the Ail

classical this technology, measuring is

nothing has with and paradox do the it mentioned electrons The large in to section with 2.

angles scattering puises and Y caused X

that by accepted

single-chaunel analysers. the not were

These electrons

classical measured

the in But they surely

measured not in

were sense. were

they quantummechanical

dissipated detector,

in the

slits, the

collimator

in energy

sense: or m

chamber. of watt the In the scattermg discussion of argued Section

that 3 it the electrons was

flying already "measured"

while

from crystal. the

ail So electrons caused that were away

bremsstrahlung photons

quantummechanical measured

in and the calculation of in are sense,

(10) photon using the

total

equation the summation spectrum

k-values the of must

over run

inelastically ail electrons. scattered

following.

Section from The conclusion interference this the The is the CB in spectrum terms

manipulation optical destroyed by

electrons, because with the imaging be outgoing Carnot an

impossible. So,

8~r

that for the paradox small of also is Figure the of atomic the

case rows is

for exist, the that of imaging impossible. Also

does proposition 3

atomic

Dot reason rows is

of is Section 2.1 true. not

BREMSSTRAHLUNG COHERENT N°2 QUANTUM AND THEORY 261

0.2°

B~, =

2.2° 8~,

'éi

oss

- 10. Fig.

distribution Bottom:

angular of electrons. wide

63Cu2sFe12.

Top:

nelastically

scattered

angular

istribution.

5. CB

the evidence to order increase In

a quasicrystal see for ordered the

kind,

epeated

dilferent

structure

[17]

built

a structure not

aud

[17].

asicrystal

efereuces is

in out

periodic

are

in planes,

the

in and just ows evertheless toms

ganised

rdinary

crystals.

in m

is a [17]. perspective

view

atoms

quasicrystal given

Coherent

were lock

remsstrahlung pectra

size

rom

alculated

this is

axis. a to

side

parallel

two-fold

The of

eam electron

parallel

kev

of CB with distribution axis.

the a

angular

wide

scattered

pectrum

The

inelastically

from separate as

the CB has shape

8~r = alculated 2.2°,

pectra

electrons,

aine

see so scattered is

there

robability amplitudes electrons

10, Figure

interference

between

a small angle,

B~r

calculated with

the

dilferent

oundary from CB

rows.

spectrum

between

because a number have

disappeared

peaks electrous

interference

electrons detected dilference ou is between 10. No dilfereut

Figure

rom

found rows,

the plane plane.

at the

at or

mage focal

Conclusions 6.

loosmg few kev brem- of160 coherent

Optical path made kev electrons calculations

a are

angular

il11] assumptions: broad dilferent crystal under sstrahlung

silicon in emission

two a a

bremsstrahlung

photon

and the

The outgoing electrons distribution of the

one. or narrow a

expected entangled that

It therefore be

is inelastically scattered electron in state. to

are an

depends

the of of conditions electron. photon intensity the measured the measurement on

manipulations crystal the far from depend optical using does

The CB not spectrum away on

PHYSIQUE JOURNAL DE I N°2 262

paradox electrons, Figure inelastically

scattered

of outgoing the the does the exist. 3 In not so

angular of the broad distribution interference between electron coming

no case occurs waves

the angular distribution diflerent

from this atomic In interference

case rows. a narrow causes

peaks

possible the absence of several the CB in

It the interference is spectrum. not to remove

theory

imaging by microscope, electron Abbe's the imaging impossible. makes in

rows an

between of There correlation the absence interference the is CB in one-to-one spectrum terms

photons diflerent possibility from

between and the the

transverse to coming rows measure

optical

by of of the imaging using outgoing position

electrons. these The calculated

means rows

independence photon manipulation of from

they electrons the the after have spectrum

any on

crystal left conclusion that of position leads the the the the scattering the of measurement to

quautummechanical completed long already

before the electron is

reach in centre

sense con a

inelastically electron, spread

So scattered that moving available for lens. the is in it

to space an

Bohr,

macroscopic Kampen

forms the of and measuring

suitable for Van system out,

a sense m

position the of the transverse row.

bremsstrahlung,

experimental they of coherent observed

tilt inter- spectra

as now, are up

belonging

between from diflerent scattering

ference does the atomic atoms to

not occur; rows

only

localised within

regions electron that incoming is contain

row. one

Acknowledgments

Nieborg,

sample prepared by bremsstrahlung

The coherent silicon Mr. Il-B- the

spec- was

by Kuipers. Many

J-B- fruitful discussions measured made Dr. Ii. with B-J-

trum

was were

Hoenders.

References

iii (John Electrodynamics 1975). Sous, J-D-, York, Wiley & Classical Jackson New

(1984) Spence G-M-, J-C-H-, N., Reese Yamamoto Mag. Philos. A 697. 49 [2]

(1987) K-S-, D.B., Vecchio Williams J. Microsc. 147 15. [3]

Sigle Carstanjen W., (1992) Mag. H-D-, Philos. B 533. 66 [4]

Feynman R-P-, R-B-, Leighton Sands (Addison-Wesley, Feynman M., Physics

The Lectures [5] on

1965) chap. III, vol. 1.

(1988) Physica N-G-, Kampen

153 97. [6] van

(1994) D., Phys- Dieks Reu. A 49 2290. [7]

(oct.

1991) Physics W.H., Today Zurek

36. [8] p.

Quantum J-A-, W-H-, (Princeton Zurek Wheeler Theory and Uuiversity Press, Measurement [9]

1983). Princeton, NJ,

[loi

Feynman Quantum R-P-, A.R., Hibbs (McGraw-Hill Integrals Mechanics Path and Book Com-

1965). pany,

[Il]

L., (Springer-Verlag, Reimer Microscopy Transmission Electron 1984) chap. 5.2.

G-O-, Reynolds G-B-, J-B-, Thompson B-J-, Develis Parrent Physical The Optics [12] New Notebook:

Tutorials Optics, (Bellington, Fourier Optical Society SPIE-The in Intemational for engineering

1989). Washington,

(Friedr- Spectrallinien A., Sommerfeld Vieweg Sohn, 1939) Atombau Braunschweig, und & [13]

chap. II.Bond, 7.

Phys. Koch (1959) J-W-, Reu. Med. H-W- Motz and 31 920. [14]

(1986) J-C-H-, Spence Cryst G., Acta A Reese 577. 42 [15]

Spence (1983) J-C-H-, Mag. G-, N., G., Philos. Kurizki Reese Yamamoto L39. [16] B 48

(1993) Marchie du Voorthuysen E-H-,

S., P-J-M-, Smaalen Smulders Phys. [17] Reu. B 48

van van 9374.