Macroscopic quantum theory of transition radiation : Application to oblique incidence and small precipitates

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Macroscopic quantum theory of transition radiation : Application to oblique incidence and small precipitates

M. Natta

To cite this version:

M. Natta. Macroscopic quantum theory of transition radiation : Application to oblique incidence

and small precipitates. Journal de Physique, 1971, 32 (8-9), pp.639-649. �10.1051/jphys:01971003208-

9063900�. �jpa-00207120�

MACROSCOPIC QUANTUM THEORY OF TRANSITION RADIATION :

APPLICATION TO OBLIQUE ^{INCIDENCE AND SMALL} PRECIPITATES

M. NATTA

Centre d’Etudes

Nucléaires,

^{CEDEX n°}

85, 38, Grenoble-Gare,

^France

(Reçu

^{le 30 novembre}

1970)

Résumé. ²⁰¹⁴ Nous développons ^{une théorie} quantique macroscopique ^{de la} radiation émise par les particules ultra-relativistes au passage d’une interface entre deux diélectriques.

Le résultat de Garibian établi dans le cas de l’incidence normale (~ ⁼ 0) ^{est étendu à} l’incidence oblique ~ ~ 0 ; l’énergie ^{totale de} la radiation de transition est :

(0394E

^{= (e2/3c) 03A9p y cos}

~),

^pour

une interface traversée. Nous discutons ensuite le cas de précipités

sphériques ;

^{il y a deux} régimes principaux :

1)

^Les grandes valeurs du _rayon

a((a03A9p/c) ~ 1)

^pour

lesquelles

l’énergie

perdue

par la particule

incidente par radiation de transition est proportionnelle à 03B3, ^sa valeur étant en fait la valeur moyenne du résultat obtenu dans le cas plan.

2) ^Les petites valeurs du rayon

((a03C9p/c)

^~

1)

^pour lesquelles ^{dans le cas des} métaux les plasmons

de surface sont excités avec une probabilité proportionnelle ^à log ^{03B3. Enfin nous discutons de la} possibilité d’utiliser la radiation dans la région

optique

(03C9 ~

03C9p)

pour l’identification de particules

isolées : cette solution ne paraît pas très intéressante.

Abstract. ²⁰¹⁴ A macroscopic quantum theory of transition radiation is developped for relativistic particles. Garibian’s result established for normal incidence (~ ⁼

0)

is extended to

oblique

^inci-

dence (~ ~ 0) ; the energy of the transition radiation par interface is

(0394E

⁼

^(e2/3c)

^{03A9p 03B3 cos}

~).

The case of small spherical precipitates ^is investigated ; ^the particle ^{size is} important ^{in determi-} ning ^the exact nature of the transition radiation.

1) ^For large ^{values of the radius}

(a(03A9p a/c) ~ 1)

^{the energy loss of the} particle by radiation is

proportional

to 03B3, ^{its value} being ^the average of the result given by ^the plane ^surface.

2) For small values of the radius, in metallic spheres,

((a03C9p/c) ~ 1)

^surface plasmons ^{are excited}

with a log y dependence. Finally, ^the possibility ^of using ^the optical region (03C9 ~ 03C9p) ^{of the tran-}

sition radiation is discussed : it is found not to be _very appropriate ^{for the} identification of a single particle.

,* Classification Physics Abstracts :

16.95, ^17.10

Introduction. ^-

Recently

renewed interest has

ibeen

shown in transition

radiation,

^first

predicted by Ghinsburg

^{and Frank}

[1],

^{because it}

might permit

^a

determination of the

speed v (and consequently

^the

nature)

of ultrarelativistic

particle

^{of known} energy : indeed the total _{energy AE} radiated

by

the passage of

one

particle through

^an interface between a dielectric and the vacuum is

proportional,

^{as shown}

by

^Gari-

bian

[3],

^to

(y

^{= 1 -}

(v2Ic2)-’h.) according

^{to the}

formula :

Dp

^{is a}

^{plasma frequency}

^which

only depends

^{on the}

total electron

density N, by

^{the usual formula}

The order of

hQ,

^{is 20} electron volts.

This

phenomenon

^is

usually [3]

calculated

by

^the

Landau and Lifshitz

[4]

classical method in which :

1)

^The encountered media are characterized

by

their dielectric and

magnetic

^constants.

2)

The Maxwell

equations

^{of the} system ^are expres- sed

taking

^{account of the}

incoming charge density

Ze

b(r - vt).

The so-obtained result includes all

possible photon

excitations from the

optical

^{to the}

X-rays region

^{and in}

particular

^{in the case of metals}

the so-called surface

plasmon

excitations. However this automatic

proceedure

^{turns out to be rather}

difficult when the symmetry ^{of the}

problem

^{is too}

low,

which is the case of non central incidence on dielectric

spheres.

Another

approach

^{consists of}

treating

^the

problem

as

scattering

of the incident

particle by

^the

elementary

excitations of the isolated system. ^{Such a}

proceedure

described

by

^Mott

[5]

^{has been used}

mainly by

Ferrell

[6]

^{for the case} of volume and surface

plasmon

excited

by

non-relativistic

particles.

^An

important point

about such a quantum ^{method is} that it is

possible

^{to make}

approximations

^{which are valid in a}

given

range, and then

by physical

^{arguments make}

LE JOURNAL DE PHYSIQUE. ^- T. 32, ^{N °} 8-9, AOUT-SEPTEMBRE 1971

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01971003208-9063900

further

corrections ;

^{this has been}

nicely

illustrated

by

^Ferrell

[6].

^{In this article we shall} extend such an

approach

^to the whole

photon

^{domain where the}

electric field can no

longer

^be

approximated by

^the

gradient

^{of a scalar}

potential

^{but also must be consi-}

dered to be

dependent

^{on the vector}

potential

^A.

In fact this

extension,

^{carried out in} part

I,

^is

especially

useful for relativistic

particles

^{so that we}

only

^deal

with this case. Two

applications

^are made. Part II treats the

oblique

^{incidence on a}

plane

^surface

Part III is the relativistic extension of the

scattering

^of

particles

^on

spherical precipitates.

I.

Quantum theory

^of transition radiation. - The _energy of a

particle

^{of mass m and}

charge

^{e, in a}

field

(E, H)

which derives from the

potential ^{(ç, A)}

^{is :}

It is

always possible

^to

choose qJ

^z

0,

^{so that to} first order in A the hamiltonian can be written :

with

For

highly

relativistic

particles (cp > mc’)

^this

reduces to

In the first Born

approximation

^the

scattering

pro-

bability

^{of the}

particle

^from

state | K

> to

state K’

^>

with the

production

^{of a}

photon

^of

frequency

^{(J) is :}

with :

In

general

^{the VE term can be}

neglected.

We now consider the

photon

^{field. We} shall still describe the dielectrics

by

^the

macroscopic isotropic

constants

e(m)

^and

p(co) :

^{this is valid at}

high

^fre-

quency ; the determination of the

electromagnetic

field is then a classical

boundary

^value

problem [7], [8]

which has

already

been solved for

special geometries.

The next step ^{is to}

quantize

^{the field} energy which is

given by :

This can be done

by enclosing

^{the field in a}

large

^box

of infinite

conductivity

^{and of volume L’}

(Fig. ^la).

FIG. 1. ^- Quantization ^{of the} electromagnetic ^field

la. ^- Stationnary conditions.

As we look for radiative

modes,

^{that is to} say, solutions

propagating

outside volume 0, ^the volume

L 3

can be taken as

large

^{as we want, then}

in

calculating

^{the total energy}

(8)

^{we can omit}

volume Q. We can then

replace statiônary boundary

conditions

by

Born-Von Kermann conditions and calculate the _energy outside the dielectric 0

(Fig. 2b).

The

problem

looks then more like its

experimental

realisation which is to detect

photons

^{in a}

given

direction. In short the

quantization

^{can be done in}

various _ways, and must be done in the most convenient

one or in the one which

corresponds

^{to the}

experiment

in

question.

This will become apparent ^{in what follows.}

1 b. ^- Bom - Von Kermann conditions.

II.

Oblique

^{incidence on a}

plane

^{surface. -}

II. 1 FIELD SOLUTION AND QUANTIZATION. ^- It iS

appropriate

^to choose the so-called

[8]

^transverse

magnetic (TM),

^and transverse electric fields

(TE).

They

^can be described

by

^{the function}

with two

possible

^choices for q in

region

⁰

(qll’

^±

^{q, _L)}

and one in

region

^Q2

(qll, q21) if it

is intended to detect the radiation in the

region

Q. This is shown schemati-

cally

ⁱⁿ

figure

²

together

^{with the notations}

adopted.

FIG. 2. ^- The eleotromagnetic ^{field in the case of a} plane surface.

The

(TM)

^and

(TE)

^{solutions are}

given

ⁱⁿ

appendix

A with values of coefficients

(A 1, Bl, A2).

^We

only

quote ^here the results for the _energy in the volume SL

(S :

surface of the

slab,

^{and L : a}

large depth

^{in the oz}

direction in the second

medium).

For the TM mode we find :

For the TE mode we find :

In the

following

^we shall take ,u - u2 - 1 .

By

^{use of the} transformation

with

and

the TM energy

(8)

^can be written as :

which makes P+ _appear as a

photon

^creation operator.

II.2 SCATTERING PROBABILITY AND ENERGY LOSS. ^-

The

scattering probability

^{of the}

particle

^{with the}

production

^{of one}

single photon

^is

(with

^{K’ = K +}

Q) :

Let the

particle

be incident in the oxz

plane

^with

the

angle cp

^to the normal oz

(Fig. 3),

g2 ⁼

(q2.Ll qil

^cos

§, _qn

^sin

§)

^{and let 0 be the}

angle

of observa- tion

of the light :

FIG. 3. ^- Geometry ^and notations for the spherical precipitate

case.

Straightforward,

^but

lengthy

calculations

give :

Where TM and TE

correspond respectively

^{to the}

TM and the TE modes.

Integration

^over

^{d 2} Q

_Il gets ^{rid of the two delta} functions :

with now

The

expression (15)

^includes

only

the radiative modes. Volume and surface

plasmon

excitations which

are non radiative for

semi-infinite

^media

[6]

^{do not}

appear in this

scattering probability.

^The

quantity P,

sometimes

[10], [14] called ’1,

^{is the} total number of

photons,

^{that are} emited with various kinds of

frequencies.

When ₉ ⁼ 0 this

expression for ’1

^{is similar to that}

of Garibian

[3].

^In

principle

^{it is}

only

^{valid for semi-}

infinite media. For a finite

slab,

^of

thickness a,

^inter- ference effects will be

important

^{if the}

integration

over dz in the slab cannot be

approximated by

^its

value _up to + oo, that is to say if the condition

(Ql

⁺

qii) a » 1

^{its not} fulfilled. This

gives

^the

lower limit

for a, occuring when §

⁼

0,

^{0 =} cp, with the

high frequency

limit for the constant si :

with

Let us now

take,

^{as does}

Garibian,

82 ⁼

1,

E1 ⁼

E(w).

Then formula

(15)

^{can be} written with for short

t ⁼

clv

⁼

fi-’,

^{u = sin 0 sin ç cos}

tk, v

^{= cos 8 cos} (p :

To get ^the energy radiated

by

the incident

particle through

^the

interface,

^the

integrant

^{over dw must be}

multiplied by

^hm.

Formula

(18)

^{includes Cherenkov} radiation when

8 > 1 in the

denominator ;

^this contribution

diverges

since the medium Q is

supposed

^to be infinite. If we

restrict ourselves ^to transition radiation which is finite we see that it occurs

mainly

^{when e is close to}

unity ;

the main contribution comes from the

high frequency region

^{where a is of the form}

(16).

^Further

examination of the denominator shows that the mean

upper limit of the spectrum ^is

At that

frequency

^{the formula}

(17)

^{reduces to}

with

Integration

^{over dO)}

gives :

This

expression

^can be evaluated

by noticing

^that

the smallest value of the denominator occurs when

§

⁼

0,

⁰ = (p, which

corresponds

^{to a}

light

^emission

in the

incoming particle

direction. Here the for- mula

(19)

^{reduces to} roM ⁼

Sp

^y.

^Making

^limited

expansions

^{near this}

point

^we

finally

^{find :}

which for _lp ⁼ 0 is Garibian’s result.

The formula

(21)

^{shows the}

advantage

^{of normal}

incidence.

At

present

transition radiation is

mainly

^investi-

gated

for the detection of relativistic

particles ^{[2], [9],}

^its

main defect comes from its _poor statistical _accuracy, due to the

relatively

^{low number of}

photons produced.

This number

only depends [3]

^on

log

y and his of the order of

e 2lhC@

^{which means that with a} system ^of 1 000 foils the total number of

photons

^{would be}

n ⁼ 10 which

gives

^{an error of}

^n-l-

^{= 30}

%.

^The

number of foils cannot be

augmented indefinitly

because of

absorption

effects which are

important

^as

the radiation is in the forward direction.

III. Radiation

from spherical particles.

^{- The} geo- metry ^{is defined in}

figure

^{3. The two} dielectrics Q and 02 ^are

separated by

^a

sphere

^{of radius a. The situa-}

tion

usually

studied is the ^«

precipitate

^{case » where}

the medium

CD ..is

^a metal and medium Q is a vacuum or an ideal dielectric with constant 82 ⁼ 8rn. The initial

study

^{of this case is due to Jensen}

[10]

^who

predicted

that the ratio of the surface to the volume

plasma

oscillations should be

1 /Î3.

The

complete

calculation of the _energy loss and radiation of non-relativistic electrons on such

precipi-

tates has been carried out

by Fujimoto

and Komaki

[11]

with the

hydrodynamic

^Bloch

equations [10], [12], [13].

Hovever as this method does not include the

magnetic

field

retardation,

^it turns out to be valid

only

^for

small radii

(a

¹⁰⁰

Á) (1).

^{In fact the same result}

can be obtained

by

the classical method if one defines the electric field

by

^the

potential

lp, and if one takes

the dielectric constant in the

vicinity

of the volume

plasmon

excitation as :

where

û)p

⁼

(4ne2Im)

n,,,

(n,,

⁼ conduction electron

density) and g

represents ^{a small}

damping

^term.

In real

metals g N

^{0.1 m. In}

principle

this Maxwell

equations

^{method can be}

applied

^{in its}

complete

correct form but it turns out to be rather difficult when the

particle

^{does not} pass

through

^{the center 0 of the}

sphere

^{in which case the}

cylindrical

symmetry ^is

destroyed.

^On the other hand it is

straighforward

to define the surface

plasmon

^{modes of the} system ⁱⁿ the absence of the external field and to use a quantum method in which the

impact point

^{is no}

longer

^defined.

This has the

advantage

^of

including

^the retardation

effect ;

ⁱⁿ

particular

^{the common}

formula,

which

gives

^the

frequencies

^of the surface modes in metallic

precipitates

^is

only

^{valid for}

[14], [15]

^small

values of their radius a :

(m/c) a «

^1.

The

complementary

^« bubble situation » where medium Q is a

cavity

^{has been}

investigated by Hénoc, Henry

^{and the} present ^author

[14], [18].

Here we shall _{carry out} the quantum ^{extension to} relativistic

particles

^{in the two limits :}

1)

Transition radiation in the U. V. X.

region,

extension which is valid for any

material ; 2)

^Surface

plasmon

excitations in the case of metals.

The use of the quantum method rather than the Landau and Lifshitz one is

justified by

the fact that the classical method turns out to be very difficult to

apply

^{in the case} where the electric field is

correctly

defined

by

^{the vector}

potential A ;

^{in fact to} get ^both

light intensity

^{and its}

angular

distribution would

require

^{the same amount of work in both}

methods,

but in the quantum ^{one it is}

possible

^to calculate the

intensity

^{in a}

simple

way and _guess at its

angular

distribution.

III.1 FIELD SOLUTION AND QUANTIZATION. ^- The solution for the field is found

[8]

from the functions

where q2 - (C02/C2)

ep

(here

^{we shall}

consider y

^=-

1).

[Y,’,,(O, lp)]

^{is the}

^complete

^set of normalized

sphe-

rical harmonics with

polar angles

^{0 and} lp for r.

zl is a linear combination of the two

spherical

^Bessel

functions

(q

^{can be}

imaginary).

The solutions for E and H are

recapitulated

ⁱⁿ

Appendix

^{B. The nature of the}

function zi changes

^at

the surface r = a and must

satisfy

^{the usual}

matching

conditions for E and H.

In

region

^{1 :}

In

region

^{2 :}

The

quantization of q2

^{is found}

by

^the

stationnary

condition that the transverse electric field is zero on the upper

sphere

^{of radius R}

(Fig. 3).

^This

gives by making

^{use of the}

asymptotic

^{form of}

(25),

for the TM mode

for the TE mode

The

phase

^shift

ô ,

is found from the

matching

^{on the}

interface of radius _a, its

general

^{behaviour on w is}

shown on

figure

^{4. The total}

electromagnetic

energy which can be written in a

symbolic

^{manner for the TM}

mode :

Fie. 4. ^- The ^« Phase Shift »,ôl ^{as a} function fo the frequency ^M.

as R can be taken as

big

^we

like,

^this then reduces to :

For the TE field ₁₁₂ would

replace

82. In the

following

we shall take ₈₂ ⁼

1,

81 ⁼

8(ro)

^and ,ui = 112 ⁼ 1.

(1) ^In fact Bloch equations describe quite ^{well the} high ^fre-

quency behaviour of _any material if we take as electron density the total electron density. ^{The reason} why they turn out to be unvalid is that they ^{do not} take into account the magnetic field.

III.2 SCATTERING PROBABILITY. ^-

Up

^{to now We}

have not caracterized the reference axis oz of the

quantization.

^{The reason} is that in order to

simplify

the calculation we shall take a

moving

^{reference axis}

along Q

^{= K’ - K}

(Fig. 3).

^{The matrix elements are}

then of two types ^which

depend

^{on the} quantum number ^m. The calculation is outlined in

Appendix

^B.

The total

scattering probability

^{can be written in a}

way similar to

(12) :

The contribution from the transverse

magnetic

mode reduces to three terms

(m

⁼

0,

±

1) :

The contribution from the transverse electric mode reduces to two terms

(m

⁼ ±

1).

The

expression (28)

^is

general

but rather

compli- cated ;

it includes the entire spectrum. ^It is convenient to

separate

^{it in two}

parts.

^{The first can} be called the continuum which

spreads

^{over the whole} range of

frequencies

^{and whose}

integral

^over dom is the transi- tion radiation. The other one is the low _energy or

optical

part which shows

peaks ;

^those

peaks,

^{in the}

case of metals are the surface

plasmon

excitations.

This

separation corresponds

ⁱⁿ the mathematics to two different

approximations

^{to the}

spherical

^Bessel

functions.

III.3 TRANSITION RADIATION. ^- It is out of the

question

^to

integrate

^{such an}

expression

^as

(28).

To get the characteristic of the transition radiation we

shall

proceed

^{in a} somewhat devious manner. First

we note that as in the case of the

plane

^{the main}

contribution will come from

points

^{where the deno-}

minator of

expression (30)

^and

(31)

^is

nearly vanishing.

The

integration

^over

dQ gives Q

^{= -}

(wlvu)

^where

u ⁼ cos

OQ,

Then the denominator is :

As the form

(16)

^{is valid for c at}

high frequency

^this

means that the

interesting point

^{is M - 2013 1}

(as Q

^is

positive) ;

^{and co --* + oo. Hovever the}

important

limit for ^ro is rather

Now this

divergency

^{of the}

integrant

^{over du and dru}

will be effective if it is not

damped by

^{a zero of the}

numerator at this

point.

^Since

only high

^{values of ce}

are

important,

the condition

(w/c) a >

^{1 is}

fulfilled ;

this

give

^{for instance in}

(31) :

when u --> - 1 and W ⁼ WM ⁼

Qp

^y.

Since b ;

(s

^-

1) ma/2

^{c = -}

Qp ^al2

^{cy, the condi-}

tion

(34)

is fulfilled if :

This condition is the same as

(17).

^{With the condi-}

tion

(35)

^{it is also}

possible

^to take averages of the square of the

asymptotic

values of the Bessel func- tions. The _square of the left hand, side

of (34)

averages to

The last

point

^{is to} estimate the summation over 1 which would be

divergent

^{if the}

asymptotic

^{form of the}

Bessel function was valid for _every 1. In fact the limit of the

validity

^{is around}

lM

⁼

(ro/e)

^{a, which we shall}

take as an upper limit of summation. With all those

rough

estimates the total

scattering probability

^{can be}

written as :

where t

= p-l.

^To get ^the energy from this number of

photons

^{we must}

multiply

^the

integrant by

^{hco. The}

first term is a constant. The second is _y

dependent

^and

is :

In fact this _very

rough

estimation is

just

^{twice what}

we would get

by taking

^the average of the formula

(21)

over the

sphere.

^{The correct} formula should then be :

For

practical

^{use the}

important point

about transi- tion radiation is its

spatial

distribution. In view of the here taken

quantization

^one could think that it is distributed as the

spherical

harmonic functions.

Hovever because of the

degeneracy

^{this is not true :}

if it is correct to take any

complete

^{set of functions to}

apply

the Fermi

golden-rule,

^{it is not correct when}

we want to calculate the

angular

distribution. In fact what is is then observed is a

plane

^{wave, that is to} say,

a coherent

superposition

^{of solutions of the} type

(23)

with different 1. The

right

way ^to solve the

problem

^is

then to take a

plane

^{wave in}

region

^{Q and to match}

it at the surface r ⁼ a. But we then have the same

difficult

problem

^{to solve as for the} classical method.

However we can say that as

long

^as the radiation transition

regime

^is excited then the condition

(35) holds,

^{which means} that the radius of the surface is very

big

^with respect ^{to the wave}

lenght

of the emitted

light

and makes the

sphere

^{to behave as a}

plane

^at

each

impact point :

the emitted

light

is in the

incoming particle

^direction.

The result

(38)

holds for the inverse « bubble case »

where medium Q ^is the vacuum

(or

^a gas of low

density)

^and

medium (t

^a dielectric. This can be of interest for the detection of

photons.

111.4 SURFACE PLASMON EXCITATION. ^-

a) Light produced by

^a

single particle hitting

^the

sphere. -

In the case of metals well defined

modes,

^whose

frequencies

^are

given by

^the

expression (21),

^are

excited at least with low

speed particles.

^{In fact those}

modes are also present in formula

(28)

^but

they

^will

correspond

to resonant excitations

only

^when

This is the other

regime

^{for the}

spherical

^{Bessel func-}

tions,

which allows us to write :

with _{q2 a} ⁼

(colc) a «

^{1. In} that range ni »

ji,

which means that the main contribution to the

integral

over do) comes from the

points

^{where sin2}

bl

^is

maximum

(Fig. 4),

^{that is to} say 1 + 1 + la ⁼ 0 which is formula

(21)

^for ro,. In fact this surface

plas-

mon contribution is the main part of the radiation

as the condition

(35)

^{is not} satisfied. The number of

photons

^obtained per

particle hitting

^the

sphere

^with

unit

probability

^{is :}

where :

is the function

(*)

introduced

by Fujimoto

^{and Ko-}

maki

[11].

^{The first term which comes} from the m ⁼ 0 modes is in fact the

asymptotic

limit of the formula of

Fujimoto

^{and Komaki}

[11 ].

^{The other is}

log

y

depen-

dent and is of relativistic

origin.

^The

particularity

^of

the relativistic correction in the small

sphere

^{case is}

that the

intensity

^shows

peaks corresponding

^to

surface

plasmons.

^{In fact in all cases the}

background

behaves as

log

y, but in the case of thin

plane

^foils

the

log

y term shows no

peaks.

^{This can be verified in}

the formula

given

^{for that case}

by

Ritchie and Eld-

ridge [19] ;

^{the reason} for this behaviour is

given

ⁱⁿ

Ferrell’s article

[6] :

^{the rate} of radiation

responsible

for the

log

y term is slower than its

damping.

The

expression (40)

^{is a} function of

which can be written as :

where :

The function

il(x, y)

which is the number of

photons

obtained if the

particle

^{hits the}

^sphere

^is

^plotted

^on

figure

^{5 for}

coplg

^{= 20.}

(*) ^{In fact the} exact upper limit of the integral ^{is 2} Ka ; ^for the radii a considered here 2 Ka » 1, ^{so that it can be} replaced by + oo. To apply ^{the result} to very small systems (for ^instance nucleus, a « ^10-3 À) ^{must be} replaced by :

FiG. 5. ^- Number 1’/ ^of photons ^{as a} function of the quantity

x ⁼ QrofJ/c, (hère coplg ⁼ 20) ^for aluminium ; x ^{== 1} corresponds

to ^a = 132 A.

1. ^- Total number of photons ^for y = 100 : 1’/ ; ^{2. -} Non-rela- tivistic term constant with _{y : n2} 3. ^- Coefficient of (log y - 0.5) : Fy3 ; 4. ^- Total constant term : 14 ⁼ n2 - 0.5 n3

n/(y) ⁼ 1’/4 + 1’/3 log y.

Note that the scale changes for curves 1, 2, 3 and also that _1’/4 is multiplied by 100.

b) Angular

distribution

of

the radiation. ^- The

following

table shows the contribution of the different quantum ^{numbers to} the formula

r¡(x, y)

Contribution of dif ferent l number.

MM

- The first ^row

is the constant non relativistic term. The second row

is the coefficients of

(log y - 1)

in the relativistic term.

We see that _up to the values of x which are of interest the 1 ⁼ 1 term

gives

the main contribution.

It should be noted

that,

^{as there is no} energy

degeneracy

between the different 1

numbers,

^{the diffe-}

rent surface

plasmon

^emission processes ^{are not}

mixed,

^{so that the}

angular

distribution of the

intensity

is

given by

the distribution of each

(1, m)

^{mode which}

is

proportional

^{to :}

But in this formula the

angular

co-ordinates

(0, (p)

refer to the axis oz’ which is taken

along Q.

^However

the main contribution to the

intensity

^{for the}

(m

⁼

1)

modes comes from the

point

^where

Q

^is

along

^oz

which means that in formula

(43) (0, cp)

^{can be taken}

as

polar angles

in the oxyz reference system. ^The

intensity

then varies as 1 +

cos’ 0

for the relativistic

1 m 1

^{= 1 terms}

(Fig. 6).

FiG. 6. ^- Angular distribution of the relativistic contribution to the radiation. The distribution is of revolution around oz.

For the non relativistic term the situation is more

complicated

^as the function

fl(x), given by

^for-

mula

(40),

^{does not have a} strong variation ^{for small x.}

On the average the distribution is much more

isotropic

than sin’

0,

^{as shown}

by

^the

Fujimoto

^{and Komaki}

calculation

[11].

c) Interference effect

^{in a} system

of

many small metallic

spheres.

^{- The mean} number of

photons produced by

^one

particle hitting

^one

sphere

^{is of the}

order of one

tenth,

^{so that a}

practical

system ^must contain _many

spheres,

^or many incident

particles.

In this second

situation,

^the

phenomenon

^{can be}

studied

accurately ;

^{let us look at the first one. If}

the volume

density

_np ^of

precipitates

^{is too}

large,

interference effects arise. Those effects cannot be calculated

exactly

^{as in the case of}

parallel planes [3], [19] ;

^however

they

^{can be estimated in the}

following

way.

First we note that the method we have taken is correct if there are no other

precipitate

^{in the}

sphere r

⁼ R, however this

quantity

^{which is in}

principle

^as

large

as we want, ^{must be a least}

equal

^{to :}

if we want to preserve the

log y dependence.

^{This is}

shown is

Appendix

^B. This condition

(44)

^{is not too}

drastic in the X

region

because there it reduces to the usual condition R >

À,

^y, but in the

optical region

^it

gives

^{a rather}

big radius ;

for instance if _y ⁼ 100 :

R,,, -

^{0.1 mm.}

In

practice

^the real condition is not so severe. To get ^a y

dependence,

^the

phase

^{between the}

light

^and

the electron must be conserved _up to the radius

Rm ;

in fact we can say that if the

photon

^encounters

another

sphere

^its

phase

^is

complety destroyed

^because

it is in resonance with the

sphere

^as

, ro..I - n/2 (see Fig. 4) ;

^{otherwise we shall} say that the

phase

^shift

is

complety destroyed.

^{This means} that the calculation

will be correct if an

appreciable

part ^{of the}

photon

field is left

unscattered,

^{when r =}

Rm.

^With

rough

estimates this

gives

^{the condition}

and leads to the

limiting

^{value of the}

precipitates density

The maximum total number

NT

of surface

plasmons given by

^one

particle through

^a

path

^{d of such a} system ^{is :}

or :

where

Rm

^is

given by expression (44).

This formula can

be

interpretated

^{in the}

following

way : the system behaves as if the

spheres

^were

gathered

ⁱⁿ compact

planes

distant of

Rm

^{from each other :}

figure

^7.

FIG. 7. ^- Interprétation ^{of the} discussion of interference effccts.

This

interpretation

shows that the discussion of a

similar effect on

plane

surfaces with

roughness [20]

of the order of

c/wp

^{can be made from the}

^preceeding

considerations.

Let us finish this discussion

by

^an

example.

^For

y ⁼

100, Rm N

^{0.1 mm.} Then with one meter of

such a system

(d

^{= 1}

m)

^the

log

y

dependent

^{term is}

of the order of

NT

^{= 1000}

photons.

^{This means that}

the statistical error is at least of the order

of NÎ V’

^{= 3}

%.

d)

Conclusion. ^- As there seems to be as yet ^no available

experiments

^{of the}

required

type, ^for- mula

(41)

^cannot be checked.

Now,

^{if we} consider the other

practical problems,

^to

identify

^one

particle

^{with such a} system ^{seems to be}

hopeless.

^For instance in the

exemple of §

^{c to make}

the difference between a

proton

^{and an}

hyperon

^E+

would

require

^{for the}

photon

^{detection a}

precision

of

(0.05 - 0.03)

^{= 2}

%,

^{which is}

quite

^a

good performance.

General conclusion. ^- In this article we have not

compared thouroughly

^{our result} with what is obtained in the case of

plane surfaces ;

^{in that last case there}

are several similar situation which are described

by

the formula

given by

^{Ritchie and}

Eldridge [19].

^In

fact the main

advantage

^{of small metallic}

spheres

^is

the presence of the

log

y term in the surface

plasmons peaks.

^{On the whole we can then} say that the use of the transition radiation _up to the

X-ray region

^{as it is}

presently investigated [3]

^{is the best mean}

of identifiying single particles (2).

On the quantum ^{method we have shown that it is}

sufficiently adaptable

^{to treat}

problems

^{where the}

classical calculation

fails ;

^{but we must once}

again emphasize

^{that this}

approach

^{is correct}

only

^{when the}

impact point

^{can be taken as random.}

Acknowledgments. -

^{It is a}

pleasure

^{to thank}

Dr. C.

Zadje

^{and Dr. R. H. Wade for} discussions and

helpful

^{comments. ,}

(2) ^The problem ^of measuring ^the energy of an intense beam of particules ^{is a different one ; this can be done} by collecting transition radiation at very low angle « y-’) ⁱⁿ the forward direction. The emission then shows a y4 dependence : ^{on that} subject ^{see reference} [2].

APPENDIX A

The field with a

plane

interface. ^- From the function U ⁼ A exp ⁱ

(qr - rot)

^{we can construct the}

electromagnetic

^field.

The TM field

(Hz

⁼

0)

^{is :}

The

matching

^{at the surface z = 0}

gives

^with

straighforward

^notations

where q2 = «02/C2)

8/1 :

The

electromagnetic

energy in the volume SL is

The TE field

(E,,

⁼

0)

^{is : with the} coefficients

APPENDIX B

Calculation of matrix élément in

spherical

^{case. -}

We now outline the calculation of the matrix elements which is

simple

but involves rather cumbursome

expressions. Figure

^{3 shows} the notations :

iz,

^and

ix,

are unit vector

along

^{oz’ and ox’. We can write :}

K =

K(iz,

^cos

OQ

⁺

^ix,

^sin

OQ) . ^{(B. .1)}

Now

taking ox’yz’

^{as reference} system ^{we know how}

to

develop

^the two terms. The

developments

^are

given

in the book of Stratton

[7] (p. 419-420)

^{to which}

we refer for the notations

1,

m, ^{n :}

With those

expressions

^the

integrations

^over

(0, lp)

are

straightforward.

^{We are} left with the

integration

over dr in the two domains of definition of the func- tion U whose coefficients A and B are found

by matching (E, H)

^{on the} boundaries. The

integration

over R is done

by

^{use of}

Zieldjes integrals (B.6),

and relation between A and B

This

procedure

^is illustrated ^on the ^{m =} + 1 ^trans-

verse

magnetic

^{mode. After}

integration

^over

angles

we find :

In those

expressions

^{the vector}

1,

m, ^{n are}

projected

on to the

spherical

coordinate system

(ir, ie, ilp).

In that system ^{we can} write the solution from the function U

(24) (see ^{Angot [8]}

^p.

463).

The

magnetic

^{field H need not be writen as it matches}

automatically

^{with E.}

The transverse electric field is :

where

can be written in the form I ⁼

Ia

⁺

IR

These

expressions

^are

unchanged if ji and y;

^{are both}

replaced, respectively ^by (Ilr) (rj)’

^and

(1 /r) (ry)’.

B and A are related

by :

The

expression (B. 9)

^{can be} réduced and

gives

^the

expression (30)

for the matrix element.

This is

possible

^{if the term}

IR

which is due to the

sphere r

^{= R does not} interfere in a destructive manner

with first one. The

quantization of q2

^{is found for}

the relation

(ryl)’

^{= 0 which means that :} q2

Ryi

^{= ± 1.}

When

Q

^is very different

from q2

the value of

IR

^is

random with respect

to la :

^{there is no} interference

effet and the

expression (30)

is the contribution of the r ⁼ a surface. Now when

Q -->

q2 ^{we can} write if a « R :

Then the result

(30)

^{is correct if}

(Q - q2) R

>

1,

which

gives

^the rather drastic condition :

otherwise there are interference effects and the

depen-

dence of the relativist correction _{upon y}

disappears.

References

[1 ] ^GHINZBURG (V. L.) ^{and FRANK} (I. M.), ^J. Exptl.

Theoret. Phys., USSR, 1946, 16, 15.

[2] ALIKHANIAN (A. I.), ^AVAKINA (K. M.), ^GARIBIAN (G. M.), ^LORIKIAN (M.

P.)

^and SHIKHLIAROV (K.

K.),

Phys. ^Rev. Letters, 1970, 25, ⁶³⁵ (this

paper gives ^{the main} references on the

subject).

[3] ^GARIBIAN (G. M.), ^Sov. Phys.

(JETP),

1960, 37, ^372.

[4] ^LANDAU (L. D.) and LIFSHITZ (E. M.), Electrodynamics of continuous media, Pergamon ^Press (1960).

[5] ^See for instance: MOTT (N. F.) and MASSEY (H. ^S. W.),

The theory of ^Atomic Collisions, Oxford (1965).

[6] ^FERRELL (R. A.), Phys. Rev., 1958, 111, ^1214.

[7] ^STRATTON (J. A.), Electromagnetic theory, ^{M. C.}

Graw Hill Book Comp. (1941).

[8] ^ANGOT (A.),

Compléments

^de Mathématiques, ^{C. N.-}

E. T. (1961).

[9] ^YUAN (L. ^C. L.), ^WANG (C. L.), ^UTO (H.) ^{and PRUNS-}

TER

(S.),

Phys. Letters, 1970, 31B, ^603.

[10] ^JENSEN

(H.),

^Z. Phys., 1937, 106, ^363.

[11] ^FUJIMOTO (F.) and KOMAKI (K.), ^J. Phys. ^Soc. Japan, 1968, 25, ^1679.

[12] ^BLOCH

(F.),

^Z. Phys., 1933, 81, ^363.

[13] ^RITCHIE (R. H.), Phys. Rev., 1957, 106, ^874.

[14] ^NATTA

(M.),

Solid State Com., 1969, 7, ^823.

[15] ^NATTA (M.), Thesis, ^Paris

(1969) (unpublished).

[16] ^NATTA (M.), ^J. Physique, Colloque ^C1

supplément

au n° 4, 1970, 31, ^C1-53.

[17] ^HENOC

(P.)

^{and HENRY} (L.), ^J. Physique, Colloque ^C1 supplément ^{au n°} 4, 1970, 31, ^C1-55.

[18] ^HENOC (P.), ^NATTA (M). ^{and HENRY} (L.), Microscopie Electronique 1970, Grenoble, 1970, ^Tome II, p.123.

[19] ^RITCHIE (R. H.), and ELDRIDGE (H. B.), Phys. Rev., 1962,126,1935.

[20] ^See for instance: RAETHER, ^J. Physique, Colloque ^C1

supplément

^{au n°} 4, 1970, 31, ^C1-59.