Chapter 8

Radiation by Moving Charges

8.1 Introduction

The problem of radiation of electromagnetic waves by a single charged particle moving at an

arbitrary velocity had correctly been formulated independently by Lienard and Wiechert before

the advent of the special relativity theory. This is because once emitted from a charged particle,

electromagnetic waves propagate at the speed c irrespective of the velocity of the charged particle,

just as sound waves propagate at a speed independent of the source velocity. (The major �nding

made by Einstein was that electromagnetic waves still propagate at the speed c regardless of the

observer�s velocity in contrast to the case of sound waves.)

The scalar and vector potentials due to a moving charge can be found rigorously using the

Green�s function for the wave equation. Then radiation electromagnetic �elds can readily be cal-

culated. In nonrelativistic regime, the radiation power only depends on the acceleration of charged

particles. As the velocity approaches c; however, signi�cant increase in the radiation power occurs.

Furthermore, in highly relativistic limit, radiation occurs primarily along the direction of velocity

within an angular spread of order � ' 1= about the velocity irrespective of the direction of ac-

celeration. Here = 1=p1� �2 is the relativity factor with � = v=c: Hence radiation frequency

is subject to strong Doppler shift. For example, in synchrotron radiation due to highly relativistic

electron beam bent or undulated by a magnetic �eld, radiation even in hard x-ray regime can be

created.

In material medium, radiation processes without acceleration on charged particles are possible.

If the velocity of a charged particle exceeds the velocity of electromagnetic waves in the medium

v >1

p"�0

;

where " is the permittivity, Cherenkov radiation occurs. Furthermore, if a charged particle crosses a

boundary of two dielectric media, the transition radiation occurs even if the condition for Cherenkov

radiation is not met. Transition radiation is due to sudden change in the normalized velocity from

�1 = vp"1�0 to �2 = v

p"2�0 which may be regarded as an e¤ective acceleration even though the

1

particle velicty v remains constant.

8.2 Lienard-Wiechert Potentials

The charge and current densities of a moving point charge are singular and described by

�(r; t) = e�[r� rp(t)]; (8.1)

J(r; t) = ev(t)�[r� rp(t)]; (8.2)

where e is the charge, rp(t) is the instantaneous location of the charge and v(t) = drp(t)=dt is the

instantaneous velocity of the charge which may be changing with time. Exploiting the Green�s

function for the wave equation,

G(r� r0; t� t0) = 1

4� jr� r0j��t� t0 � jr� r

0jc

�; (8.3)

we can write down solutions for the inhomogeneous wave equations,�r2 � 1

c2@2

@t2

�� = � �

"0; (8.4)

�r2 � 1

c2@2

@t2

�A = ��0J; (8.5)

in the form

�(r; t) =e

4�"0

ZdV 0

Zdt0

1

jr� r0j�[r0�rp(t0)]�[f(t0)]; (8.6)

A(r; t) =�0e

4�

ZdV 0

Zdt0

v(t0)

jr� r0j�[r0�rp(t0)]�[f(t0)];

where

f(t0) = t0 � t+ jr� r0j

c: (8.7)

The volume integrations can be carried out immediately with the results

�(r; t) =e

4�"0

Z1

jr� rp(t0)j�[f(t0)]dt0; (8.8)

A(r; t) =�0e

4�

Zv(t0)

jr� rp(t0)j�[f(t0)]dt0; (8.9)

where f(t0) is now

f(t0) = t0 � t+ jr� rp(t0)j

c: (8.10)

2

The integral involving the delta function can be simpli�ed asZg(t0)�[f(t0)]dt0 =

Zg(t0)�[f(t0)]

1����df(t0)dt0

����df=

g(t0)

jdf=dt0j

����f=0

; (8.11)

where t0 is now understood as a solution for t0 satisfying f(t0) = 0 or

t0 � t+ jr� rp(t0)j

c= 0: (8.12)

This is in general an implicit equation for t0: The time derivative of f(t0) is

df

dt0= 1� n(t

0) � vp(t0)c

= 1� n(t0) � �(t0); (8.13)

where n(t0) is the unit vector along the relative distance r� rp(t0) and � = v=c: The observing

time t and time t0 are related through

[1� n(t0) � �(t0)]dt0 = dt;

ordt

dt0= 1� n(t0) � �(t0):

After performing time integration, we �nally obtain

�(r; t) =e

4�"0

1

1� n(t0) � �(t0)1

jr� rp(t0)j=

e

4�"0

1

�(t0) jr� rp(t0)j; (8.14)

A(r; t) =�0e

4�

1

1� n(t0) � �(t0)v(t0)

jr� rp(t0)j=�0e

4�

v(t0)

�(t0) jr� rp(t0)j; (8.15)

where

�(t0) = 1� n(t0) � �(t0): (8.16)

These retarded potentials, called Lienard-Wiechert potentials, had been formulated in 1898. They

are applicable to arbitrary velocity of the charged particle. Retarded nature of the potentials clearly

appears in the condition that all time varying quantities, rp(t0);v(t0);n(t0); must be evaluated at

t0; not at the observing time t because of �nite propagation speed of electromagnetic disturbance.

Having found the retarded potentials, we are now ready to calculate the electromagnetic �elds

due to a moving point charge. The electric �eld is to be found from

E(r; t) = �r�� @A@t; (8.17)

where spatial and time derivatives pertain to r (the coordinates of the observing location) and

3

t (observing time). Since the potentials � and A are implicit functions of r and t; it is more

convenient to use the original integral representations, Eqs. (8.8) and (8.9), respectively, for proper

di¤erentiation with respect to r and t: For example, the spatial derivative of the scalar potential

can be performed as follows. Letting R(t0) = jr� rp(t0)j ; and introducing a unit vector in thedirection r� rp(t0);

n(t0) =r� rp(t0)jr� rp(t0)j

; (8.18)

we �nd

r� =e

4�"0

Zr�

1

jr� rp(t0)j�[f(t0)]

�dt0

=e

4�"0

Zn@

@R

�1

R�[f(t0)]

�dt0

= � e

4�"0

Z �n

R2�[f(t0)]� n

cR

d

df�[f(t0)]

�dt0

= � e

4�"0

�n

�R2+1

c�

d

dt0

� n�R

��; (8.19)

where use is made of the integration by parts,Zg(t0)

d

df�[f(t0)]dt0

=

Zg(t0)

1

df

dt0

d

dt0�[f(t0)]dt0 (8.20)

= ��

1

df=dt0d

dt0

�g (t0)

df=dt0

��f(t0)=0

(8.21)

= ��1

� (t0)

d

dt0

�g (t0)

� (t0)

��f(t0)=0

: (8.22)

Similarly,@A

@t=

e

4�"0

1

c�

d

dt0

��

�R

�; (8.23)

and the electric �eld becomes

E(r; t) =e

4�"0

�n

�R2+1

c�

d

dt0

�n� ��R

��f(t0)=0

: (8.24)

To proceed further, we need concrete expressions for the derivatives,

dn

dt0and

d

dt0

�1

�R

�:

The unit vector n along the distance vector R = r� rp(t0) changes its direction only through the

4

velocity component perpendicular to R;

dn = �v?Rdt0; (8.25)

as can be seen in Fig. 8-1. This yields

Figure 8-1: The change in the unit vector n is caused by the perpendicular velocity v?:

dn

dt0= �v?

R: (8.26)

Also,

d

dt0

�1

�R

�= � 1

(�R)2d

dt0[(1� n � �)R]

= � 1

(�R)2

hv?�� � (n � _�)R� c(1� n � �)n � �

i= � c

(�R)2

��2 � n � � � 1

cR(n � _�)

�: (8.27)

Substitution of Eqs. (8.26) and (8.27) to Eq. (8.24) gives

E(r; t) =e

4�"0

"n� �(�R)2

� n� ��(�R)2

��2 � n � � � 1

cR(n � _�)

��

_�

c�2R

#

=e

4�"0

�1� �2

�3R2(n� �) + 1

c�3Rn� [(n� �)� _�]

�f(t0)=0

: (8.28)

The �rst term in the RHS,

ECoulomb =e

4�"0

�1� �2

�3R2(n� �)

�f(t0)=0

; (8.29)

5

is the Coulomb �eld corrected for relativistic e¤ects. It is proportional to 1=R2 and thus does not

contribute to radiation of energy. The second term,

Erad =e

4�"0c

�1

�3Rn� [(n� �)� _�]

�f(t0)=0

; (8.30)

contains acceleration _� and is proportional to 1=R: This is the desired radiation electric �eld due

to a moving charged particle.

The magnetic �eld can be calculated in a similar manner from

B = r�A=

1

c[n�E]f(t0)=0 : (8.31)

Derivation of this result is left for an exercise.

8.3 Radiation from a Charge under Linear Acceleration

If the acceleration is parallel (or anti-parallel) to the velocity, _� � � = 0; the radiation electric �eldreduces to

Erad =e

4�"0

"n� (n� _�)

c�3R

#f(t0)=0

: (8.32)

The angular distribution of radiation power at the observing time t is

dP (t)

d= c"0 jEradj2R2

=1

4�"0

e2

4�c

"n� (n� _�)

c�3

#2f(t0)=0

: (8.33)

Denoting the angle between _� and n by �; we have [n� (n� _�)]2 = _�2sin2 �; and thus

dP (t)

d=

1

4�"0

e2

4�c

"_�2sin2 �

(1� � cos �)6

#f(t0)=0

: (8.34)

However, the power P (t) in the above formulation is the rate of energy radiation at t; the observing

time, which is not necessarily equal to the energy loss rate of the charge at the retarded time t0

determined from f(t0) = 0: To �nd the radiation power at the retarded time P (t0) ; let us consider

the amount of di¤erential energy dE=d radiated during the time interval between t0 and t0+dt0: Byde�nition dE = P (t0)dt0: The radiation energy dE is sandwitched between two eccentric sphericalsurfaces with a volume

dV = R2cdt0(1� � cos �):

6

Therefore, the di¤erential radiation energy is

dEd

=1

2"0 jEradj2R2cdt0(1� � cos �);

and

dP (t0)

d=

1

4�"0

e2

4�c

"_�2sin2 �

(1� � cos �)6 � (1� � cos �)#f(t0)=0

=1

4�"0

e2

4�c

"_�2sin2 �

(1� � cos �)5

#f(t0)=0

: (8.35)

In nonrelativistic limit j�j � 1; the radiation occurs predominantly in the direction perpendicular

to the acceleration � = �=2. The total radiation power in this case is

P ' 1

4�"0

e2 _�2

4�c

Zsin2 �d

=1

4�"0

2e2 _�2

3c; � � 1: (8.36)

This is the well known Larmor�s formula for radiation power due to a nonrelativistic charge. Since

in nonrelativistic limit,

n� [(n� �)� _�] ' n� (n� _�); (8.37)

Larmor�s formula is applicable for acceleration in arbitrary direction relative to the velocity.

For arbitrary magnitude of the velocity �; the radiation power can be found from

P (t0) =1

4�"0

e2 _�2

4�c

Zsin2 �

(1� � cos �)5d

=1

4�"0

e2 _�2

4�c2�

Z �

0

sin3 �

(1� � cos �)5d�

=1

4�"0

2e2 _�2

3c 6; (8.38)

where

=1p1� �2

; (8.39)

is the relativity factor and the following integral is used,Z �

0

sin3 �

(1� � cos �)5d� =Z 1

�1

1� x2(1� �x)5dx =

4

3

1

(1� �2)3: (8.40)

7

The angular dependence of the radiation intensity,

sin2 �

(1� � cos �)6 ; (8.41)

peaks at angle �0 where

cos �0 =

p1 + 24�2 � 1

4�: (8.42)

In highly relativistic limit � 1; the angle �0 becomes of order

�0 '1

� 1; (8.43)

which indicates a very sharp pencil or beam of radiation along the direction of the velocity �: (This

is also the case for acceleration perpendicular to the velocity as shown in the following section.)

Angular distribution of radiation intensity I (�)) for � = 0; 0:2; 0:9 and 0:999 is shown below for a

common acceleration. Note that the radiation intensity rapidly increases with = 1=p1� �2:

­1­0.8­0.6­0.4­0.2

0

0.20.40.60.8

1

­1 ­0.8 ­0.4 0.2 0.4 0.6 0.8 1

� = 0 ( = 1:0):

­1.0 ­0.5 0.5

­1.0

­0.5

0.5

1.0

x

y

� = 0:2 ( = 1:02):

2000 4000 6000 8000 10000 12000 14000

­2000

0

2000

x

y

� = 0:9 ( = 2:3):

2e+8 4e+8 6e+8 8e+8 1e+9 1.2e+9­9e+7

9e+7

x

y

� = 0:99 ( = 7:09):

8

Polar plot of I(�) for several � factors but with common parallel acceleration.

For linear acceleration, the momentum change of the charged particle is

dp

dt=d

dt

mvp1� �2

!= m 3 _v: (8.44)

Therefore, the radiation power can be rewritten as

P (t0) =1

4�"0

2e2

3c3m2

�dp

dt0

�2=

1

4�"0

2e4

3c3m2E2ac

=1

4�"0

2e2

3c3m2

�dEdx0

�2; (8.45)

where Eac is an external acceleration electric �eld and

dEdx0;

is the energy gradient of a linear accelerator which is at most of the order of 100MeV/m in practice.

The radiation loss in linear accelerators is negligibly small compared with energy gain. This is one

of the advantages of high energy linear accelerators. Note that the radiation power due to linear

acceleration is independent of the particle energy or the relativity factor :

8.4 Radiation from a Charge in Circular Motion

In circular motion, the acceleration is perpendicular to the velocity. Here we consider highly

relativistic motion of a charged particle with � ' 1, for nonrelativistic case has already been

discussed in Chapter 4. A geometry convenient for analysis to follow is shown in Fig.8-2. A

particle undergoes circular motion with an orbit radius � in the x � z plane and it passes theorigin at t0 = 0: At that instant, the acceleration is in the x direction while the velocity is in the z

direction,_� =

��� _���� ex; � = �ez:

The radiation electric �eld can then be written down in terms of cartesian components,

E(r; t) =e

4�"0

1

c�3Rn� [(n� �)� _�]

=e

4�"0

��� _����c�3R

[(� � cos �) cos�e� + (1� � cos �) sin�e�] ; (8.46)

9

Figure 8-2: Particle undergoing circular motion in the x� z plane with radius � and frequency !0:At t = 0; the particle passes the origin.

and the angular distribution of radiation power is given by

dP (t0)

d=

1

4�"0

e2��� _����24�c

1

(1� � cos �)5

�(1� � cos �)2 � 1

2sin2 � cos2 �

�: (8.47)

The total radiation power is

P (t0) =1

4�"0

e2��� _����24�c

Z1

(1� � cos �)5

�(1� � cos �)2 � 1

2sin2 � cos2 �

�d

=1

4�"0

e2��� _����24�c

Z �

0sin �d�

Z 2�

0d�

1

(1� � cos �)5

�(1� � cos �)2 � 1

2sin2 � cos2 �

�=

1

4�"0

2e2j _vj2

3c3 4: (8.48)

Relevant integrals are: Z 1

�1

dx

(1� �x)3 =2

(1� �2)2= 2 4;

Z 1

�1

1� x2(1� �x)5dx =

4

3

1

(1� �2)3=4

3 6:

In highly relativistic case, the acceleration may be approximated by

j _vj = v2

�' c2

�: (8.49)

10

Then, the radiation power in terms of the orbit radius � is

P (t0) ' 1

4�"0

2e2c 4

3�2: (8.50)

To maintain the radiation loss in a circular accelerator at a tolerable level, the orbit radius must

be increased as the particle energy mc2 increases. Note that the radiation power is a sensitive

function of the particle energy in contrast to the case of linear acceleration.

As an example, let us consider the Betatron, the well known inductive electron accelerator

invented by Kerst. In the Betatron, the electron cyclotron orbit is maintained constant,

mc2

�= ecB (t) = ecB0 sin!t ' ecB0!t;

where only the initial phase of the sinusoidal magnetic �eld is useful for acceleration, !t� 1. Then

the rate of electron energy gain is

d

dt

� mc2

�= ec�B0! = const.

Equating this to the radiation energy loss,

P�t0�=

1

4�"0

2

3c3

�c2

�

�2 4;

we �nd

max '�3!eB0�

3

2mc2re

�1=4; (8.51)

where

re =e2

4�"0mc2= 2:8� 10�15 m; (8.52)

is the classical radius of electron. If � = 50 cm, ! = 2� � 60 rad/s, and B0 = 0:5 T (5 kG), the

upper limit of is about 400 and the maximum electron energy attainable is approximately 200

MeV.

8.5 Fourier Spectrum of Radiation Fields

The formulae such as Eqs. (8.45) and (8.50) only tell us the total radiation power integrated

over the frequency. Radiation �eld emitted by highly relativistic particle is hardly monochromatic

but cnsists of broad frequency spectrum. Knowing such frequency spectrum of radiation is of

practical importance for identifying radiation source. A typical example is synchrotron radiation

due to highly relativistic electrons bent by, or trapped in, a magnetic �eld. In nonrelativistic limit,

the radiation �elds all have a single frequency component corresponding to the classical electron

cyclotron frequency !c = eB=m: However, as the relativity factor increases, the radiation �elds

11

consist of harmonics of the fundamental frequency !c = eB= m. In highly relativistic case � 1;

the frequency spectrum becomes almost continuous peaking at the frequency ! ' 3!c = 2eB=m:Frequency spectrum of the radiation electric �eld can be formulated by directly applying Fourier

transformation on the �eld in Eq. (8.24),

E(r; !) =

Z 1

�1E(r; t)ei!tdt

=1

4�"0

e

c

Z 1

�1

"n� [(n� �)� _�]

�3R

#f(t0)=0

ei!tdt: (8.53)

Changing the variable from t to t0 and assuming r � rp; we obtain

E(r; !) =1

4�"0

e

c

ei!r=c

r

Z 1

�1

n� [(n� �)� _�]

�2exp

�i!�t0 � 1

cn � rp(t0)��dt0: (8.54)

The unit vector n(t0) may be regarded constant since r � rp and thus approximated by n ' r=r:Then,

n� [(n� �)� _�]

�2' d

dt0

�n� (n� �)

�

�; (8.55)

and Eq. (8.53) can be integrated by parts,

E(r; !) = � i!

4�"0

e

c

ei!r=c

r

Z 1

�1n� (n� �) exp

�i!�t0 � 1

cn � rp(t0)��dt0: (8.56)

Any time derivatives contained in the physical electric �eld are merely multiplied by �i! in theFourier space and the disappearance of the acceleration _� is not surprising. Amazing fact about

Eq. (8.56) is that it is applicable to radiation �elds which do not require particle acceleration

such as Cherenkov and transition radiation provided a proper velocity of electromagnetic waves in

dielectrics is substituted for c:

The radiation energy (not power) associated with the electric �eld is

c"0r2

Z ZjE(r; t)j2 dtd; (J). (8.57)

However, since the electric �eld E(r; t) and its Fourier transform E(r; !) are related through

Parseval�s theorem:ZjE(r; t)j2 dt = 1

2�

ZjE(r; !)j2 d!; (8.58)

the radiation energy can be written in terms of the Fourier transform E(r; !) as

1

2�c"0r

2

Z ZjE(r; !)j2 dd!: (8.59)

12

The quantitydI(!)

d� 1

2�c"0r

2 jE(r; !)j2 ; (8.60)

can therefore be identi�ed as the radiation energy per unit solid angle per unit frequency. Substi-

tuting Eq. (8.56), we �nd

dI(!)

d=

1

4�"0

e2

8�2c!2����Z n� (n� �) exp

�i!�t0 � 1

cn � rp(t0)��dt0����2 ; �1 < ! <1; (8.61)

or

dI(!)

d=

1

4�"0

e2

4�2c!2����Z n� (n� �) exp

�i!�t0 � 1

cn � rp(t0)��dt0����2 ; 0 < ! <1: (8.62)

Let us work on a few examples.

8.6 Synchrotron Radiation I

Synchrotron radiation is due to highly relativistic electrons trapped in a magnetic �eld. The

radiation beam rotates together with an electron and is directed along the direction of the velocity

with an angular spread of order �� ' 1= � 1: If the orbiting frequency is !0 = eB= me; the

radiation beam shines a detector for a duration

�t0 ' ��

!0=

1

!0; (8.63)

as seen by the electron at the retarded time t0. Since

�t

�t0= 1� n � � = 1� � ' 1

2 2; (8.64)

which is entirely due to Doppler e¤ect, the pulse width detected is of order

�t ' �t0

2 2=

1

2 3!0: (8.65)

Therefore, synchrotron radiation is dominated by frequency components in the range

! ' 3!0 = 2eB

me: (8.66)

In ultrarelativistic case, the frequency spectrum of synchrotron radiation can extend to very high

frequencies even for a modest magnetic �eld.

We calculate the amount of energy radiated in one period of cyclotron motion T = 2�=!0: Since

the radiation power is constant, the radiated energy is

E = 1

T

Z 1

0I (!) d! =

!02�

Z 1

0I (!) d!: (8.67)

13

The time integration in the energy spectrum,

dI (!)

d=

1

4�"0

e2

4�2c!2

�����Z T=2

�T=2n� (n� �) exp

�i!�t0 � 1

cn � rp(t0)��dt0

�����2

; 0 < ! <1; (8.68)

can be extended from �1 to 1 since the characteristic frequency of the radiation �eld is much

higher than the fundamental frequency ! � !0;

dI (!)

d=

1

4�"0

e2

4�2c!2����Z 1

�1n� (n� �) exp

�i!�t0 � 1

cn � rp(t0)��dt0����2 ; 0 < ! <1: (8.69)

To perform the integration, we assume the trajectory shown in Fig. (8-2) in which an electron

passes the origin at t = 0: The vector n is assumed to be in the y � z plane since the radiationpro�le is essentially symmetric about the z axis. The trajectory is described by

rp�t0�= �

��1� cos!0t0

�ex + sin!0tez

�; (8.70)

and the velocity is

� =!0�

c(sin!0tex + cos!0tez) : (8.71)

Then,1

cn � rp(t0) =

�

csin (!0t) cos �: (8.72)

Since

n� (n� �) = ��? = � sin!0tex + cos!0t sin �e?; (8.73)

where

e?= n� ex; (8.74)

is a unit vector perpendicular to both n and x axis, and the radiation lasts for a very short time

and is limited within a small angle �; Eq. (8.73) reduces to

��? ' �!0tex + �e?: (8.75)

Within the same order of accuracy, the phase function !(t� n � rp=c) can be approximated by

!�t� n � rp

c

�= !

�t� � sin!0t cos �

c

�' !

2

��1

2+ �2

�t+

c2

3�2t3�; (8.76)

where v has been approximated by c but 1� � by

1� � ' 1

2 2:

14

Then, Z 1

�1�? exp

hi!�t� n � rp

c

�idt

=

Z 1

�1(!0tex � �e?) exp

�i!

2

��1

2+ �2

�t+

c2

3�2t3��dt

= 2i!0

Z 1

0t sin

�!

2

��1

2+ �2

�t+

c2

3�2t3��dt ex

�2�Z 1

0cos

�!

2

��1

2+ �2

�t+

c2

3�2t3��dt e?: (8.77)

The integrals reduce to the modi�ed Bessel functions of fractional orders (or the Airy�s functions),Z 1

0cos

�3

2x

�� +

1

3�3��d� =

1p3K1=3 (x) ; (8.78)

Z 1

0� sin

�3

2x

�� +

1

3�3��d� =

1p3K2=3 (x) ; (8.79)

where

� = !0tp1 + 2�2

; x =!

3 3!0

�1 + 2�2

�3=2: (8.80)

Then

2i!0

Z 1

0t sin

�!

2

��1

2+ �2

�t+

c2

3�2t3��dt = 2i

1 2+ �2

!0

1p3K2=3 (x) ;

2�

Z 1

0cos

�!

2

��1

2+ �2

�t+

c2

3�2t3��dt = 2�

q1 2+ �2

!0

1p3K2=3 (x) ;

and for I (!) =d; we obtain

dI (!)

d=

1

4�"0

e2

3�2c

!2

!20

�1 2+ �2

� h�1 2+ �2

�K22=3 (x) + �

2K21=3 (x)

i=

1

4�"0

3e2 2

�2c!̂2�1 + �2

�n�1 + �2

�K22=3[!̂

�1 + �2

�3=2] + �2K2

1=3[!̂�1 + �2

�3=2]o;

where � = � and !̂ is the normalized frequency,

!̂ =!

3 3!0: (8.81)

The modi�ed Bessel functions K2=3 (x) and K1=3 (x) both diverge at x ! 0: However, xK2=3 (x)

and xK1=3 (x) are well behaving and vanish at small x: Fig. (8-3) shows 2x2K22=3 (x) and x

2K21=3 (x)

which represent radiation intensities associated with electric �eld polarization along ex (that is, in

the particle orbit plane) and e?; respectively.

The energy spectrum I (!) emitted during one revolution (T = 2�=!0) can be found by inte-

15

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

Figure 8-3: 2x2K22=3 (x) (solid line) and x

2K21=3 (x) (dotted line). The factor of 2 in 2x

2K22=3 (x)

assumes � ' 1= :

grating dI (!) =d over the solid angle,

I (!)

=1

4�"0

3e2 2

�2c!̂2Z �

0sin �0d�0

Z 2�

0d�0�1 + �2

��n�1 + �2

�K22=3[!̂

�1 + �2

�3=2] + �2K2

1=3[!̂�1 + �2

�3=2]o

=1

4�"0

6e2 2

�c!̂2Z �=2

��=2cos �d�

�1 + �2

�n�1 + �2

�K22=3[!̂

�1 + �2

�3=2] + �2K2

1=3[!̂�1 + �2

�3=2]o

' 1

4�"0

6e2

�c!̂2Z 1

�1

�1 + �2

�n�1 + �2

�K22=3[!̂

�1 + �2

�3=2] + �2K2

1=3[!̂�1 + �2

�3=2]od�

=1

4�"0

6e2

�cf (!̂) ; (8.82)

where �0 = �2 � � is the polar angle from the axis of electron revolution (y axis), �0 is the azimuthal

angle about the y axis, and the function f (!̂) is de�ned by

f (!̂) = 2!̂2Z 1

0

�1 + �2

�n�1 + �2

�K22=3[!̂

�1 + �2

�3=2] + �2K2

1=3[!̂�1 + �2

�3=2]od�; (8.83)

and shown in Fig. (8-4) (linear scale) and Fig. (8-5) (log-log scale). In Fig. (8-5), the straight line

in the low frequency regime�! � 3!0

�has a slope of 1=3; and indicates I (!) _ !1=3: f (!̂) peaks

at !̂ ' 0:14; or ! ' 0:42 3!0; and its peak value is about 0.83. In high frequency regime ! � 3!0;

the spectrum decays exponentially. The energy radiated per revolution can be calculated as

16

0

0.2

0.4

0.6

0.8

1 2 3 4 5

Figure 8-4: The function f (x) = f�!=3 3!0

�plotted in linear scale.

E =Z 1

0I (!) d! ' 1

4�"0

6e2

�c3 3!0

Z 1

0f (!̂) d!̂; (8.84)

where the integral numerically evaluated is approximatelyZ 1

0f (!̂) d!̂ ' 0:713:

Then

E =Z 1

0I (!) d! =

1

4�"0

4:08e2 4!0c

; (J)

and the radiation power is

P = E !02�=

1

4�"0

0:65e2c 4

�2; (8.85)

which agrees reasonably well with Eq. (8.50),

P =1

4�"0

2e2c 4

3�2' 1

4�"0

0:667e2c 4

�2; (W).

The discrepancy may be attributed to the various approximations made in the analysis.

8.7 Synchrotron Radiation II

An alternative approach to �nding the frequency spectrum of synchrotron radiation is to apply

discrete Fourier analysis in terms of harmonics of the fundamental frequency !0 directly to radiation

17

.1e­2

.1e­1

.1

.1e­3 .1e­2 .1e­1 .1 1.

Figure 8-5: Log-log plot of the function f (x) = f�!=3 3!0

�:

�elds. If a charge e is in circular motion with a constant angular frequency !0, the radiation �eld

contains higher harmonics of !0 and can be Fourier decomposed as follows. Let us recall the

Lienard-Wiechert vector potential,

A(r; t) =�04�

ev(t0)

(1� n � �)R(t0)

����f(t0)=0

; (8.86)

where R = jr � rp(t0)j and the subscript f(t0) = 0 indicates that all time dependent quantities

should be evaluated at the retarded time t0 determined from the implicit equation for t0,

f(t0) = t0 � t+ jr� rp(t0)j

c= 0: (8.87)

The vector potential can be Fourier decomposed as

A(r; t) =Xl

Al(r)e�il!0t; (8.88)

where

Al(r) =�0e

4�

1

T

Z T

0

ev(t0)

(1� n � �)Reil!0tdt; (8.89)

with T = 2�=!0 being the period of the circular motion. Changing the integration variable from t

to t0 by notingdt

dt0= 1� n � �; (8.90)

18

leads to

Al(r) '�0e

4�r

eiklr

T

Z T

0v(t0)ei(l!0t

0�kln�rp)dt0; (8.91)

where kl = l!0=c. Note that the period T remains unchanged through the transformation. Let the

particle trajectory be

rp(t0) = �(cos!0t

0ex + sin!0t0ey); (8.92)

v(t0) = �!0(� sin!0t0ex + cos!0t0ey): (8.93)

Since all radiation �elds rotate with the charge, the observing point can be chosen at arbitrary

azimuthal angle � and we choose � = �=2; so that n = (1; �; � = �=2): Then

n � rp(t0) = � sin � sin!0t0: (8.94)

The velocity in the spherical coordinates is

v(t0) = �!0(sin � cos!0t0er + cos � cos!0t

0e� + sin!0t0e�): (8.95)

Thus, the � component of Al(r) is given by

Al� =�0e

4�reiklr

�!0Tcos �

Z T

0cos!0t

0eil!0(t0�� sin � sin!0t0=c)dt0: (8.96)

Letting x = !0t0; and noting � = �!0=c; we can rewrite this as

Al� = eiklr�0e�!04�r

cos �

2�

Z 2�

0cosxeil(x�� sin � sinx)dx: (8.97)

The integral reduces to Z 2�

0cosxeil(x�� sin � sinx)dx

= �[Jl+1(l� sin �) + Jl�1(l� sin �)]

=2�

� sin �Jl(l� sin �);

and thus �nally,

Al� =�0ec

4�reiklr cot �Jl(l� sin �): (8.98)

Similarly,

Al� = i�0e�!04�r

eiklrJ 0l (l� sin �); (8.99)

where use has been made of the recurrence formula of the Bessel functions,

Jl�1(x)� Jl+1(x) = 2J 0l (x): (8.100)

19

The far-�eld radiation magnetic �eld can be found from

Hl 'i

�0kl �Al; (8.101)

which yields

Hl� =e�!0kl4�r

eiklrJ 0l (l� sin �); (8.102)

Hl� = ieckl4�r

eiklr cot �Jl(l� sin �): (8.103)

The radiation power associated with the l-th harmonic is

Pl = c�0r2

ZjHlj2d

=1

8��0

(el!0)2

c

Z �

0

��2J 02l (l� sin �) + cot

2 �J2l (l� sin �)�sin �d�: (8.104)

Since Pl = P�l, the total power is

P =

1Xl=1

Pl; (8.105)

where Pl is now

Pl =1

4��0

(el!0)2

c

Z �

0

��2J 02l (l� sin �) + cot

2 �J2l (l� sin �)�sin �d�; l � 1: (8.106)

In nonrelativistic limit � � 1; the l = 1 term is dominant. For x� 1;

J1(x) '1

2x; J 01(x) '

1

2: (8.107)

Then the lowest order radiation power agrees with the Larmor�s formula,

P1 ' 1

4��0

(e!0)2

c

�2

4

Z �

0(1 + cos2 �) sin �d�

=1

4��0

2

3

e2a2

c3; (8.108)

where a = v2=� is the acceleration.

The integral in Eq. (8.105) cannot be reduced to elementary functions. However, the total

power given in Eq. (8.105) should reduce to Eq. (8.50),

P =1

4��0

2

3

e2a2

c3 4 =

1

4��0

2

3

e2a2

c31

(1� �2)2: (8.109)

To show this, we modify the integral by noting

J2l (x) =1

�

Z �

0J0(2x sin �) cos(2l)d�; (8.110)

20

J 02l (x) =1

�

Z �

0J0(2x sin �)

�cos 2� � l2

x2

�cos(2l)d�: (8.111)

Then

�2J 02l (�l sin �) + cot2 �J2l (�l sin �) =

1

�

Z �

0J0(2�l sin � sinx)(�

2 cos 2x� 1) cos(2lx)dx: (8.112)

Furthermore, Z �

0J0(2�l sinx sin �) sin �d� =

sin(2�l sinx)

�l sinx; (8.113)

and the power Pl reduces to

Pl =1

4��0

(e!0l)2

�c

Z �

0

sin(2�l sinx)

�l sinx(�2 cos 2x� 1) cos(2lx)dx: (8.114)

Noting Z �

0sinx sin(2�l sinx) cos(2lx)dx

=1

2

Z �

0[sin(2l + 1)x+ sin(1� 2l)x] sin(2�l sinx)dx

=�

2[J2l+1(2�lx)� J2l�1(2�l)]

= ��J 02l(2�l) (8.115)

and Z �

0

sin(2�l sinx) cos(2lx)

sinxdx

= 2l

Z �

0

Z �

0cos(2l� sinx) cos(2lx)dxd�

= 2�l

Z �

0J2l(2l�)d�; (8.116)

the power Pl can be rewritten as

Pl =1

4��0

2(e!0l)2

c�l

��2J 02l(2l�)� l(1� �2)

Z �

0J2l(2l�)d�

�; (8.117)

and the total power is

P =

1Xl=1

Pl:

Relevant sum formula of the Bessel functions is

1Xl=1

J2l(2lx) =x2

2(1� x2) : (8.118)

21

Di¤erentiating by x; Xl

2lJ 02l(2lx) =x

(1� x2)2 : (8.119)

Also, Xl

l2Z �

0J2l(2lx)dx =

�3

6(1� �2)3: (8.120)

This is one of Kapteyn series formulae. (See for example, Mathematical Formulae (in Japanese),

(Iwanami, Tokyo, 1960), vol. 3, p. 212.) Then, �nally, the total radiation power becomes

P =1

4��0

2e2a2

3c3 4; a =

v2

�(8.121)

which is consistent with the known radiation power from a charge undergoing circular motion.

Analytic expression for the radiation power Pl can be found by exploiting following approxima-

tion,

J2l(2�l) '1p�l1=3

Ai

l2=3

2

!; (8.122)

where Ai(x) is the Airy function de�ned by

Ai(x) =1p�

Z 1

0cos

�1

3t3 + xt

�dt: (8.123)

f(x) =1p�

Z 20

0cos

�1

3t3 + xt

�dt

For large x� 1; the function takes the form

Ai(x) ' 1

2x1=4exp

��23x3=2

�; x� 1: (8.124)

Also, Ai0(0) = �0:4587: Using these approximations, we �nd the following approximate formulae,

Pl '1

4�"0� 0:5175e

2!20cl1=3; 1� l� 3; (8.125)

Pl '1

4�"0

e2!202p�c

sl

exp

��23

l

3

�; l� 3: (8.126)

Note that the radiation power increases with l in the manner Pl _ l1=3 up to l ' 3 beyond whichPl decays exponentially. This is consistent with the analysis in the preceding section.

8.8 Free Electron Laser

In a synchrotron radiation source, many beamlines can be installed by bending an electron beam.

In straight sections of race track, no radiation occurs. However, by inserting a device called wiggler,

22

Figure 8-6: In a wiggler, an electron beam is modulated by a periodic magnetic �eld. Electronsacquire spatially oscillating perpendicular displacement x(z) and velocity vx(z) which together withthe radiation magnetic �eld BRy produces a ponderomotive force vx(z)�BRy(z) directed in the zdirection. The force acts to cause electron bunching required for ampli�cation of coherent radiation.

high intensity radiation can be extracted. A wiggler consists of periodically alternating magnets

and gives an electron beam periodic kick perpendicular to both the beam velocity and magnetic

�eld. Electrons receive kicks at an interval

�t0 =�wc; (8.127)

where �w is the �wavelength�of the periodic wiggler structure. Because of Doppler shift, this time

interval is shortened by a factor1

1� � ' 2 2 for a stationary detector in front of the beam,

�t ' �w2 2c

: (8.128)

Therefore, the wavelength of resultant radiation is approximately given by

� ' �w2 2

� �w; (8.129)

and the frequency by

! ' 4� 2c

�w: (8.130)

The intensity of free electron laser can be orders of magnitude higher than that of synchrotron

radiation because of coherent ampli�cation through the periodic structure. Electrons tend to be

bunched in the wiggler as the electron beam travels through the periodic structure. In contrast, no

collective interaction between electrons and electromagnetic waves exists in synchrotron radiation.

In electron bunching, the magnetic ponderomotive force plays a major role. Let us assume a

23

periodic wiggler magnetic �eld in y direction,

By = B0 cos

�2�

�wz

�= B0 cos kwz; kw =

2�

�w: (8.131)

The Lorentz force is

F = ev �B;

or

Fx = �evzB0 cos kwz:

Then electron acquires a velocity vx in x direction,

vx =eB0m kw

sin kwz; (8.132)

and a resultant ponderomotive force is

�evx �BR; (8.133)

where BR is the radiation magnetic �eld propagating in the form ei(kz�!t) along the beam. Since

the acceleration due to the wiggler magnetic �eld is in x direction, the radiation electric �eld is

predominantly in x direction and radiation magnetic �eld BR is in y direction. Then the pondero-

motive force directed in z direction is proportional to ei[(k+kw)z�!t] and propagates at a velocity

!

k + kw:

When this propagation velocity matches the electron beam velocity �c, strong interaction between

the radiation �eld and electron motion takes place and electrons tend to be bunched. This results

in positive feedback for wave ampli�cation. From the condition

!

k + kw' �c; or k

k + kw= �;

we readily recover

k =�

1� �kw ' 2 2kw:

8.9 Radiation Accompanying � Decay

Equation (8.61) for the angular distribution of radiation energy can be applied to cases in which

particle acceleration is not involved explicitly. In � decay, an energetic electron (or positron) is

suddenly released from a nucleus together with neutrino. The situation is equivalent to sudden

acceleration of an electron. The duration of acceleration �t is limited by the uncertainty principle

24

�t mc2 & ~: Therefore, the upper limit of the frequency spectrum should be of the order of

!max ' mc2

~: (8.134)

In � decay, the maximum value of is of order of 30.

Let us assume that an electron suddenly acquires a velocity � = v=c and then travels at a

constant velocity. The integration in Eq. (8.61) is limited from t = 0 to 1;

dI(!)

d=

1

4�"0

e2

4�2c!2�2 sin2 �

����Z 1

0ei!(1�� cos �)tdt

����2=

1

4�"0

e2

4�2c

�2 sin2 �

(1� � cos �)2 ; (8.135)

where � is the angle between the velocity � and the unit vector n: Integration over the solid angle

yields

I(!) =1

4�"0

e2

�c

�1

�ln

�1 + �

1� �

�� 2�= const.; ! . !max =

mc2

~: (8.136)

The frequency spectrum is �at up to !max: Therefore, the total energy radiated through � decay

is approximately given by

E ' 1

4�"0

e2

�c

�1

�ln

�1 + �

1� �

�� 2� mc2

~' 1

�

�ln(4 2)� 2

�� mc2; (8.137)

where the dimensionless quantity �,

� =1

4�"0

e2

c~' 1

137; (8.138)

is the �ne structure constant. The energy emitted as radiation through � decay is a small fraction

of the electron energy.

8.10 Cherenkov Radiation

Cherenkov radiation occurs when a charged particle travels faster than electromagnetic waves in

a material medium. It does not require acceleration of charges and the basic mechanism is very

similar to that of sound shock waves in gases. As in the case of � decay, we assume a charge

travelling along a straight line at a velocity � = v=c(!); where

c(!) =1p"(!)�0

;

25

is the velocity of electromagnetic waves in a dielectric having a permittivity "(!): Eq. (8.61) should

be modi�ed as follows after taking into account the proper de�nition of c(!);

dI(!)

d=�04�

e2

4�2c(!)!2v2 sin2 �

����Z 1

�1ei!(1�� cos �)tdt

����2 : (8.139)

Note that the integration limits are from �1 to 1: The time integral is singular,Z 1

�1ei!(1�� cos �)tdt = 2��[!(1� � cos �)];

and the condition for radiation is

cos � =1

�< 1;

or

�(!) =v

c(!)> 1: (8.140)

Then,dI(!)

d=�04�

e2

(2�)2c(!)!2v2 sin2 ��2[!(1� � cos �)]: (8.141)

Square of a delta function is not integrable and the radiation energy simply diverges. This is merely

due to the assumption that the charge is radiating forever from t = �1 to 1 which is of course

unphysical. It is more appropriate to consider a radiation power rather than energy. For this

purpose, we consider a thin slab of the dielectric of thickness dz. The transit time over the distance

dz is �T = dz=v and we calculate energy radiated during that time,

dI(!)

d=�04�

e2

4�2c(!)!2v2 sin2 �

�����Z �T=2

��T=2ei!(1�� cos �)tdt

�����2

: (8.142)

The integral can be carried out easily,Z �T=2

��T=2ei!(1�� cos �)tdt =

2

!(1� � cos �) sin�(1� � cos �)�T!

2

�:

ThusdI(!)

d=�04�

e2!2

4�2c(!)sin2 �

�sin�

�

�2(dz)2; (8.143)

where

� =1

2(1� � cos �)!�T: (8.144)

26

In high frequency regime !�T � 1; the function (sin�=�)2 may be approximated by a delta

function ��(�): Integration over the solid angle yields

dI(!)

dz=

1

4�"0

�e

c0

�2!

�1� 1

�2

�=

1

4�"0

�e

c0

�2!

�1� 1

"(!)�0v2

�; (8.145)

where c0 = 1=p"0�0 is the speed of light in vacuum. The rate of energy loss due to Cherenkov

emission is given by

�dEdz=

1

4�"0

�e

c0

�2 Z 1

0!

�1� 1

"(!)�0v2

�d!; 0 < ! <1: (8.146)

The result obtained is meaningful only if

� > 1; or v > c(!) =1p"(!)�0

; (8.147)

which is the condition for Cherenkov radiation. The instantaneous radiation power can be estimated

from

P =d

dt

ZI(!)d!

=1

4�"0

�e

c0

�2v

Z!

�1� 1

"(!)�0v2

�d!: (8.148)

In order to �nd the �eld pro�les emitted through Cherenkov radiation, we start from the wave

equations for the potentials in a material medium,�r2 � �0~"

@2

@t2

�~"�(r; t) = ��free; (8.149)

�r2 � �0~"

@2

@t2

�A(r; t) = ��0J; (8.150)

where ~" is the dielectric operator containing time derivative,

~" = ~"

�@

@t

�: (8.151)

For a charged particle e travelling at a constant velocity v; the charge density and current density

are described by

� = e�(r� vt); (8.152)

J = ev�(r� vt): (8.153)

27

Then, after Fourier-Laplace transformation, the Fourier potentials can readily be found,

�(k; !) =2�e

"(!)

�k2 � !2

c2(!)

��(! � k � v); (8.154)

A(k; !) =2�e�0v

k2 � !2

c2(!)

�(! � k � v); (8.155)

where, as before,

c2(!) =1

"(!)�0; (8.156)

and the transformation Z Z�(r� vt)ei(!t�k�r)dV dt = 2��(! � k � v); (8.157)

is substituted. Since the physical electric �eld is

E(r; t) = �r�(r; t)� @

@tA(r; t); (8.158)

the Fourier component of the electric �eld is given by

E(k; !) = �ik�(k; !) + i!A(k; !)

= �2�iek� !

c2(!)v

"(!)

�k2 � !2

c2(!)

��(! � k � v): (8.159)

Similarly, the Fourier-Laplace component of the magnetic �eld is

B(k; !) = ik�A(k; !)

= 2�ie�0k� v

k2 � !2

c2(!)

�(! � k � v): (8.160)

The physical electromagnetic �elds can then be found through inverse transformations,

E(r; t) =1

(2�)4

Zd3k

Zd!E(k; !)ei(k�r�!t); (8.161)

B(r; t) =1

(2�)4

Zd3k

Zd!B(k; !)ei(k�r�!t): (8.162)

To proceed further, we assume that the charged particle is travelling along the z axis at a

constant velocity v: The system is symmetric about the axis and we may assume an observing

point in the x � z plane without loss of generality. We denote the cylindrical coordinates of the

28

Figure 8-7: Geometry for Fourier inverse transform.

observing point by (�; � = 0; z) and spherical Fourier coordinates by k =(k; �; �): Then,

k � r = k(z cos � + � sin � cos�):

Because of the cylindrical symmetry, radiation of energy is expected in the radial direction �; and

the relevant Poynting vector is

S� = (E�H�)�

= �Ez(r; t)H��(r; t): (8.163)

The energy radiated per unit length along the particle trajectory (z axis) can be calculated from

�dEdz

= �2��ZEz(r; t)H

��(r; t)dt

= ��ZEz(r; !)H

��(r; !)d!; (8.164)

where Ez(r; !) is the Laplace transform of the electric �eld,

Ez(r; !) =1

(2�)3

Zd3kEz(k; !)e

ik�r; (8.165)

and H�(r; !) is the Laplace transform of the magnetic �eld,

H�(r; !) =1

(2�)3

Zd3kH�(k; !)e

ik�r: (8.166)

29

The Laplace transform of the axial electric �eld Ez(r; !) can be calculated as follows:

Ez(r; !) =1

(2�)3

Zd3kEz(k; !)e

ik�r

= � ie

(2�)2

Z 1

0k2dk

Z �

0sin �d�

Z 2�

0d�

k cos � � !vc2(!)

"(!)�k2 � !2

c2(!)

��(! � kv cos �)eik(z cos �+� sin � cos�)= � i�0e!

(2�)2

Z 1

1=��d�

1�2� 1

�2 � 1

Z 2�

0exp

"i!

c(!)

(z

�+ ��

s1� 1

(��)2cos�

)#d�

= � i�0e!2�

exp

�i!z

�c(!)

�Z 1

1=��d�

1�2� 1

�2 � 1J0�!�

c(!)

q�2 � 1

�2

�; (8.167)

where

� =kc(!)

!; � =

v

c(!): (8.168)

Letting

�2 = �2 � 1

�2; (8.169)

we �nally obtain

Ez(r; !) =i�0e!

2�exp

�i!z

�c(!)

��1� 1

�2

�Z 1

0

�J0

�!��c(!)

��2 � 1 + 1

�2

d�

=ie�02�

! exp

�i!z

�c(!)

��1� 1

�2

�K0(��); (8.170)

where

� = � !

c (!)

s1

�2� 1; (8.171)

and use is made of the integral representation of the modi�ed Bessel function K0(ax);

K0(ax) =

Z 1

0

tJ0(at)

t2 + x2dt: (8.172)

In the asymptotic regime j�j �� 1; K0(��) approachesr�

2��e���: (8.173)

For this to be propagating radially outward in the form eik��; we must choose

� = � !

c (!)

s1

�2� 1

= �i !

c (!)

s1� 1

�2; � > 1: (8.174)

30

With this choice for �; the Laplace transform of the axial electric �eld becomes proportional to

exp

"i!

c(!)

s1� 1

�2�+

z

�

!#; (8.175)

and the radial and axial wavenumbers can be identi�ed as

k� =!

c(!)

s1� 1

�2; kz =

!

v; (8.176)

respectively. Cherenkov radiation is con�ned in a cone characterized by an angle �;

sin � =c(!)

v; (8.177)

as shown in Fig. 8-8.

Figure 8-8: Cherenkov cone. Radiation �elds are con�ned in the cone.

The Laplace transform of the azimuthal magnetic �eld is given by

H�(r; !) =1

(2�)3

Zd3kH�(k; !)e

ik�r

= � ie

(2�)2

Zk2dk

Z �

0sin �d�

Z 2�

0d�

kv sin �

k2 � !2

c2(!)

�(! � kv cos �)

�eik(z cos �+� sin � cos�)

=e

2��K1(��) exp

�i!

c(!)z

�; (8.178)

31

where the following integral Z 1

0

x2J1(bx)

x2 + a2dx = aK1(ab); (8.179)

is noted. (Calculation steps are left for an exercise.) Substituting Ez(r; !) and H�(r; !) into Eq.

(8.164), we �nd

�dEdz= � i�0e

2

(2�)2�

Z 1

�1!

�1� 1

�2

���K0(��)K

�1 (��)d!: (8.180)

In the asymptotic region j�j �� 1; this reduces to

�dEdz=

1

4�"0

e2

c20

Z 1

0!

�1� 1

�2

�d!; (8.181)

in agreement with the earlier result, Eq. (8.146). Eq. (8.180) can be used even when Cherenkov

condition is not satis�ed. In this case, energy loss is through near �eld Coulomb interaction between

a charged particle and ions and electrons in molecules in the dielectric media. We will return to

this problem in Section 8.12.

8.11 Transition Radiation

Transition radiation occurs when a charge crosses a boundary of two dielectric media. No accelera-

tion is required, nor is it necessary for charge to move faster than the speed of light as in Cherenkov

radiation. In this respect, transition radiation is a least demanding radiation mechanism. Radia-

tion emitted from a charge approaching a conductor is an extreme case of transition radiation with

an in�nite permittivity, and may be regarded as the inverse process of radiation accompanying �

decay. Disappearance, rather than creation, of charge is responsible for transition radiation.

We �rst consider a simple case: a charge e approaching normally a conducting plate at a velocity

v (> 0). On impact, the charge is assumed to come to rest. A conducting plate is mathematically

equivalent to an in�nitely permissive dielectric plate. An image charge �e moving in the oppositedirection in the conducting plate can be introduced so that the current density is

Jz(r; t) = �ev [�(z + vt) + �(z � vt)] �(x)�(y); �1 < t < 0; (8.182)

where at t = 0 (or z = 0) the particle is brought to rest. Its Laplace transform is

Jz(r; !) =

Z 0

�1Jz(r;t)e

i!tdt

= �e exp��i!vjzj��(x)�(y): (8.183)

Let the observing point be at P located at (r; �): (The system is symmetric about the axis and thus

32

Figure 8-9: Radiation from a charge impinging on a metal surface. Sudden deceleration at themetal surface is the inverse process of radiation accompanying beta decay.

� is ignorable.) The vector potential at P can be calculated in the usual manner,

Az(r; !) =�04�

ZJz(r

0; !)

jr� r0j eikjr�r0jdV 0 ' �0

4�reikr

ZJz(r

0;!)e�ik�r0dV 0

= ��0e4�r

eikrZ 1

0

hexp

��i!vz0�e�ikz

0 cos � + exp��i!vz0�eikz

0 cos �idz0

= i�0e�

4�reikr

1

k

�1

1 + � cos �+

1

1� � cos �

�; z > 0; (8.184)

and the magnetic �eld from

H�(r; !) ' �ikAz sin �=�0

=e�

4�reikr

�1

1 + � cos �+

1

1� � cos �

�sin �: (8.185)

33

The angular distribution of radiation energy is thus given by

dI(!)

d=

1

2�r2c�0 jH�(r; !)j2

=e2�2c�032�3

�1

1 + � cos �+

1

1� � cos �

�2sin2 �

=1

4�"0

e2v2

2�2c3sin2 �

(1� �2 cos2 �)2; �1 < ! <1: (8.186)

Integration over the solid angle in the region z > 0 (0 < � < �=2) yields

I(!) =1

4�"0

e2

2�c

1

�

��1 + �2

�ln1 + �

1� � � 2��; ! > 0:

Evidently, the radiation energyR10 I(!)d! diverges. This is due to the assumption of a perfect

conductor. In practice, metals cannot be regarded as perfect conductor. The condition that the

surface impedance of metal

Z =

r�i!�0

�i!"0 + �;

be su¢ ciently small compared with the free space impedance Z0 =p�0="0 imposes an upper limit

of the frequency, !"0 � � and a cuto¤ emerges in the integralR !max0 I(!)d!:

Figure 8-10: Transition radiation emitted by a charge passing through a dielectric boundary.

We now analyze the case of a dielectric slab having a relative permittivity "r = "="0: In this

34

case, the particle continues to travel after passing the boundary and the current density is now

Jz(r; t) = �ev�(x)�(y)�(z + vt); �1 < t <1: (8.187)

Its Laplace transform is

Jz(r; !) = �ev�(x)�(y)e�i!v t: (8.188)

The contribution to the radiation �elds from the region z > 0 consists of two parts, one directly

from the charge (as in free space) and the other via re�ection at the dielectric boundary. Denoting

the magnetic re�ection coe¢ cient by � in the Fresnel�s formulae,

� ="r cos � �

p"r � sin 2�

"r cos � +p"r � sin 2�

; (8.189)

and following the same procedure as in the case of conductor plate, we �nd

H�1(r; !) 'ev

4�creikr

�1

1 + � cos �+

�

1� � cos �

�sin �; z > 0: (8.190)

The contribution from the region z < 0 involves refraction at the boundary, and thus additional

retardation because of the longer path length. In Fig.8-10, for z < 0, we observe

l = r + z0 tan �00 sin �; z0 < 0

l0 = �z0= cos �00;sin �0

sin �00=p";

and thus

kl +p�kl0 = kr � kz

p�� sin 2�: (8.191)

Then the contribution from the region z < 0 to the integral becomes

H�2(r; !) = �ev

4�cr(1� �) 1

1 + �p"r � sin 2�

eikr sin � (8.192)

Note that the factor 1� � here indicates the �eld amplitude transmitted into the air region. Thetotal magnetic �eld is H�1 +H�2; and the angular distribution of the radiation energy is given by

dI(!; �)

d=r2c�02�

jH�(r;!)j2 =�0e

2v2

32�3cA2 sin 2�; (8.193)

where

A =1

1 + � cos �+

�

1� � cos � �1� �

1 + �p"r � sin 2�

: (8.194)

In the case of ideal conductor "r !1; � = 1; we recover

dI(!; �)

d=

1

4��0

e2v2

2�2c3

�sin �

1� �2 cos2 �

�2.

35

When "r = 1; � = 0; radiation evidently disappears.

The factor A in Eq. (8.194) does not fully agree with that in the original work by Frank and

Ginzburg,

A0 =1

1 + � cos �+

�

1� � cos � �1 + �

"r(1 + �p"r � sin 2�)

; (8.195)

although this too vanishes when "r ! 1 and reduces to the case of conducting medium when

"r !1:

8.12 Energy Loss of Charged Particles Moving in Dielectrics

The formula derived in Eq. (8.180) yields a physically meaning energy loss rate even when the

Cherenkov condition is not satis�ed, � < 1: A charged particle moving in a dielectric medium

�collides� with atoms and lose its energy through Coulomb interaction with electrons in atoms.

Electrons in an atom are bounded. However, they do respond to electromagnetic disturbance and

absorb energy through the resonance "(!) = 0, where

"(!) = "0

1�

!2p!2 � !20

!: (8.196)

Resonance of the type1

x� x0; (8.197)

can be handled mathematically by introducing an imaginary part,

1

x� x0= P

1

x� x0� i��(x� x0); (8.198)

where P stands for the principal part. This is justi�able because the imaginary part of the function

1

x� x0 + i"=

x� x0(x� x0)2 + "2

� i "

(x� x0)2 + "2; (8.199)

remains �nite even in the limit "! 0;

lim"!0

Z"

(x� x0)2 + "2dx = �: (8.200)

Physically, the resonance leads to absorption of wave energy by charged particles in a material

medium dielectrics, plasmas, etc.

The characteristic scale length of interaction between a charged particle and atoms in a dielec-

tric is evidently of the order of atomic size which indicates that the interaction is of near-�eld,

nonradiating nature dominated by longitudinal (electrostatic) �elds. As we will see, the major

contribution to the energy loss occurs through the pole of the dielectric function, "(!) = 0:

In the near-�eld region ��� 1; the modi�ed Bessel functions K0(��);K1(��) may be approx-

36

imated by

K0(��) ' � ln���2

�� E ; (8.201)

K1(��) '1

��; (8.202)

where E = 0:5772 � �� is the Euler�s constant. The real part of Eq. (8.180) becomes

dEdz' �0e

2

(2�)2Re

Z 1

�1i!

�1� 1

"(!)�0v2

�hln���2

�+ E

id!; (8.203)

with

� =!

c(!)

s1

�2� 1; � < 1: (8.204)

The dielectric function "(!) is in the form

"(!) = "0

1�

!2p!2 � !20

!:

Therefore, the integration can be carried out by evaluating the pole contribution at "(!) = 0; which

occurs at

! = �q!2p + !

20; (8.205)

and exploiting Plemelj�s formula,

lim�!0

1

x� a+ i� = P1

x� a � i��(x� a);

where P indicates the principal part of the singular function 1=(x� a): The result is

�dEdz' 1

4�"0

�e!pv

�2ln

0@ 1:12v

�q!2p + !

20

1A : (8.206)

As the minimum distance �; the intermolecular distance may be substituted because the shell

electrons e¤ectively shield the electric �eld of the charge well inside the atom.

In a plasma, !0 is evidently zero (because electrons in a plasma are free). Then,

dEdz' 1

4�"0

�e!pv

�2ln

�v

�min!p

�; (8.207)

where

�min '�3

4�n

�1=3;

is the average distance between ions. For a Maxwellian electron distribution with a temperature

37

Te; the average energy loss rate may be estimated from

�dEdt' 1

4�"0

(e!p)2

3vTeln

8p2�

3n�3D

!; (8.208)

where

�D =vTe!p; (8.209)

is the Debye shielding length.

8.13 Bremsstrahlung

Whenever a charged particle collides with another charged particle, electromagnetic radiation occurs

due to acceleration by Coulomb force. Collisions between like particles (e.g., electron-electron)

emit quadrupole radiation while collisions between unlike particles (e.g., electron-ion) emit dipole

radiation. Bremsstrahlung is due to collisions between electrons and ions and provides a basic

mechanism for x-ray production.

Let an electron approach an ion having a charge Ze with a velocity v (� c) and impact

parameter b: The acceleration due to Coulomb force is of the order of

a ' � 1

4�"0

Ze2

mb2; (8.210)

and consequent radiation power can be estimated from the Larmor�s formula,

P ' 1

4�"0

2e2a2

3c3: (8.211)

The total energy radiated can in principle be found by integrating the power over time along the

electron trajectory. However, in experiments, one is seldom interested in measuring radiation power

or energy associated with a single electron. What is more relevant is the radiation associated with

a beam of electrons impinging on an ion. In this case, some electrons have impact parameters

vanishingly small. However, the impact parameter has a lower bound imposed by the uncertainty

principle,

bmin '~p; (8.212)

where p is the electron momentum. For an impact parameter b; the time duration in which the

acceleration is signi�cant is

�t ' b

v: (8.213)

38

Therefore, energy radiated by a single electron is

E = P�t

' 1

(4�"0)32

3c3

�Ze3

m

�21

b3v: (8.214)

For an electron beam having a density n; the radiation power can thus be estimated from

P = nev

Z 1

bmin

E2�bdb

=1

(4�"0)34�

3

�Ze3

m

�2nemv

c3~: (8.215)

If the ion density is ni; the quantity Pni de�nes the power density,

1

(4�"0)34�

3

�Ze3

m

�2nenimv

c3~; (W/m3): (8.216)

The frequency spectrum of radiation energy may qualitatively be found as follows. Since the

characteristic time of acceleration

� =b

v; (8.217)

is short, radiation occurs as an impulse and the spectrum is �at in the region 0 < ! < v=b; and

vanishes for ! > v=b: Since the minimum impact parameter is

bmin '~mv

; (8.218)

the upper limit of the frequency spectrum extends to

!max 'mv2

~; or ~! . mv2: (8.219)

This is essentially a statement of energy conservation, that is, the maximum photon energy emit-

ted during bremsstrahlung is limited by the incident electron kinetic energy, which is reasonable.

Therefore, the frequency spectrum of bremsstrahlung is

I(!; b) =

8>>><>>>:1

(4�"0)32

3

(Ze3)2

mc2mv2b2; 0 < ! <

v

b;

0; ! >v

b:

(8.220)

I(!; b) has dimensions of J/frequency. It is convenient to introduce a radiation cross-section �(!)

39

de�ned by�

�(!) =

Z bmax

bmin

I(!; b)2�bdb

=1

(4�"0)34�

3

(Ze3)2

mc2mv2ln

�bmaxbmin

�=

1

(4�"0)34�

3

(Ze3)2

mc2mv2ln

�mv2

~!

�: (8.221)

The integral over the frequency, Z�(!)d!; (8.222)

evidently diverges at the lower end ! ! 0. To remedy this di¢ culty, Bethe and Heitler recognized

that if the velocity v is understood as the mean value of the initial and �nal velocities, i.e., before

and after emission of a photon,

v =1

2(vinitial + v�nal)

=1p2m

�pE +

pE�~!

�; (8.223)

the integral remains �nite,

Z�(!)d! =

1

(4�"0)34�

3

(Ze3)2

mc2mv2

Z !max

0ln

0B@�pE +

pE�~!

�2~!

1CA d!=

1

(4�"0)34�

3

(Ze3)2

mc3~

Z 1

0ln

�1 +

p1� xpx

�dx; x =

~!E ;

=1

(4�"0)34�

3

(Ze3)2

mc3~: (8.224)

The integral in the intermediate step is unity. Multiplying by the ion density ni; we thus obtain

the bremsstrahlung rate per unit length,

dE

dz= ni

Z�(!)d!

=1

(4�"0)34�

3

(Ze3)2nim2c3~

; (8.225)

and radiation power density,

P=V =1

(4�"0)34�

3

(Ze3)2neniv

mc3~; (W m�3): (8.226)

This agrees with the earlier qualitative estimate in Eq. (8.216).

Relativistic correction to the classical bremsstrahlung formulae can be readily found if we move

40

to the electron frame wherein the electron velocity is nonrelativistic. Since the energy and frequency

are Lorentz transformed in the same manner, and the transverse dimensions are Lorentz invariant,

it follows that the radiation cross-section �(!) is Lorentz invariant,

�lab(!lab) = �0(!0);

where the primed quantities are those in the electron frame. The frequencies !lab and !0 are related

through the relativistic Doppler shift,

!0 = !lab(1� � cos �); !lab = !0(1 + � cos �0); (8.227)

where � and �0 are the angles with respect to the electron velocity in each frame. Since in the

electron frame, the radiation is con�ned in a small angle about �0 = �=2; we have

!0 ' !lab : (8.228)

The collision time is shortened by the factor since the transverse �eld is intensi�ed by the same

factor through Lorentz transformation. Therefore, the maximum impact parameter is modi�ed as

bmax = v

!0= 2v

!lab; (8.229)

and the radiation cross-section in the laboratory frame becomes

� (!) = �0(!0) =1

(4�"0)34�

3

e4(Ze)2

m2c3v2ln

� 2mv2

~!

�: (8.230)

It is noted that the minimum impact parameter remains unchanged through the transformation

because it is essentially the Compton length based on the uncertainty principle.

8.14 Radiation due to Electron-Electron Collision

In this case the dipole radiation is absent because in the center of mass frame, two electrons stay

at opposite positions, r1 = �r2; and the dipole moment identically vanishes.. The lowest orderradiation process is that due to electric quadrupole. (The magnetic dipole moment also vanishes.)

The quadrupole moment tensor is

Qij =1

2exixj ; (8.231)

where xi is the i-th component of the relative distance 2r: In Chapter 5, a general formula for the

quadrupole radiation power has been derived. Noting

d3

dt3(xixj) =

:::xixj + 3

::xi:xj + 3

:xi::xj + xi

:::xj ; (8.232)

41

Figure 8-11: Colliding electrons in the center of mass frame. The impact parameter is b:

::xi =

1

4�"0

2e2

mr3xi; (8.233)

:::xi =

1

4�"0

2e2

m

�xir3� 3xivr

r4

�; (8.234)

we �ndd3

dt3(xixj) =

1

4�"0

2e2

mr3

�4(vixj + vjxi)�

6vrxixjr

�; (8.235)

:::Q =

1

4�"0

e3

mr3

�4(vixj + vjxi)�

6vrxixjr

�(8.236)

Substituting this into the quadrupole radiation power,

P =1

4�"0

1

60c5

243Xij

:::Q2

ij � X

i

:::Qii

!235 ; (8.237)

we obtain, after somewhat lengthy calculations,

P =1

4�"0

2

15c5

�e3

m

�2 v2 + 11v2�r4

; (8.238)

where v� is the � component of the velocity related to the initial angular momentum bv0 = r(t)v�(t)

with v0 the velocity at r !1: Energy conservation reads

1

2mv20 =

1

2mv2 +

1

4�"0

e2

2r: (8.239)

42

Then

v2 + 11v2� = v20 �

4e2

4�"0mr+ 11

(bv0)2

r2; (8.240)

and the total radiation energy can be found by integrating the radiation power over time along the

trajectory,

E = 1

4�"0

2

15c5

�e3

m

�2 Z 1

�1

1

r4

�v20 �

4e2

4�"0mr+ 11

(bv0)2

r2

�dt: (8.241)

This can be converted to an integral over the distance r by noting

dt =dr

vr=

drsv20 �

(bv0)2

r2� 4e2

4�"0mr

; (8.242)

E = 1

4�"0

2

15c5

�e3

m

�2� 2

Z 1

rmin

1

r4

v20 �4e2

4�"0mr+ 11

(bv0)2

r2sv20 �

(bv0)2

r2� 4e2

4�"0mr

dr; (J) (8.243)

where rmin is the distance of the closest approach,

rmin =1

2v20

0@4e2m+

s�4e2

m

�2+ 4(bv0)2

1A : (8.244)

However, the radiation energy by a single electron pair is of no practical interest. What is more

relevant is the radiation power emitted by an electron beam impinging on a single electron which

can be evaluated from

P = 2�nv0

Z 1

0E(b)bdb

=1

4�"0

4

15c5

�e3

m

�22�nv0

Z 1

0bdb

Z 1

rmin

1

r4

v20 �4e2

4�"0mr+ 11

(bv0)2

r2sv20 �

(bv0)2

r2� 4e2

4�"0mr

dr; (W). (8.245)

The double integral reduces to

Z 1

0bdb

Z 1

rmin

dr1

r4

v20 �4e2

4�"0mr+ 11

(bv0)2

r2sv20 �

(bv0)2

r2� 4e2

4�"0mr

=25

3v0

Z 1

rmin

1

r5

�r2 � 4re

2

mv20

�3=2dr =

5

6

mv30e2: (8.246)

43

Therefore,

P =1

4�"0

4�

9

ne4v40mc5

; (W). (8.247)

This de�nes a radiation cross-section,

� =P

1

2nmv30

=8�

9

v

cr2e ; (m

2) (8.248)

where 12nmv

30 is the energy �ux density of the beam and

re =1

4�"0

e2

mc2= 2:85� 10�15 m;

is the classical radius of electron.

44

Problems

8.1 A charged particle with charge e and mass m undergoes motion in external electric and

magnetic �elds E and B: Show that the radiation power is given by

P (t0) =1

4�"0

2e2

3c3 6�a2 � (� � a)2

�=

1

4�"0

2e2

3m2c3 2�(E+ v �B)2 � (� �E)2

�;

where a is the acceleration. Since a2 � (� � a)2 is positive de�nite, so is (E+ v �B)2 �(� �E)2.

Note: In an instantaneous rest frame of the particle, v = � = 0 and = 1: The radiation

power in that frame is thus with respect to the proper time,

dE 0d�

= P (�) =1

4�"0

2e2

3m2c3E02;

where E0 is the electric �eld also in that frame. Since the electric �eld is Lorentz transformed

as

E0k = Ek; E0? = (E? + v �B);

we readily see that

P (�) = P (t0);

that is, the proper radiation power is Lorentz invariant. This is not surprising because both

time and energy are Lorentz transformed in a similar manner.

8.2 Carry out the integration in the radiation power due to acceleration perpendicular to the

velocity

P (t0) =1

4�"0

e2��� _�?���24�c

Z1

(1� � cos �)5

�(1� � cos �)2 � 1

2sin2 � cos2 �

�d

to obtain

P (t0) =1

4�"0

2e2��� _����23c

4:

Plot the angular distribution of the radiation intensity

f(�) =1

(1� � cos �)6

�(1� � cos �)2 � 1

2sin2 � cos2 �

�;

for � = 0 and �=2 in a highly relativistic case.

8.3 A nonrelativistic electron with velocity v collides head-on with a heavy negative ion having

a charge �Ze:Find the total energy radiated. What is the dominant frequency component?

45

8.4 A highly relativistic electron passes by a proton at an impact parameter b:Assuming that

the electron trajectory is una¤ected by the proton and radiation, show that the total energy

radiated is

E = 1

4�"0

�e6

12m2ec3b3v

4� �2

1� �2' 1

4�"0

�e6 2

4m2ec4b3:

What is the dominant frequency component contained in the radiation �eld?

8.5 An electron having an initial energy E(t = 0) = 0mc2 undergoes cyclotron motion in a

magnetic �eld B: Show that the rate of energy loss to radiation is given by

�dE(t)dt

=1

4�"0

2e4B2

m4c5�E2 �m2c4

�:

Integrate this and �nd an expression for E(t):

8.6 An electron is placed in a plane wave E0ei(kx�!t): If ~! � mc2; electron recoil may be ignored.

Show that the radiation power is

P = I08�

3r2e ;

where I0 = c"0E20 (W/m2) is the intensity of the incident wave and

re =e2

4�"0mc2= 2:8� 10�15 (m),

is the classical radius of electron.

� =8�

3r2e ;

is known as Thomson scattering cross section.

Note: For ~! � mc2; Klein and Nishina, solving the Dirac equation to take into account

electron recoil, obtained

� ' �r2emc2

~!

�1

2+ ln

2~!mc2

�:

8.7 The light from Crab nebula is believed to be synchrotron radiation emitted by highly energetic

electrons trapped in a weak intergalactic magnetic �eld. If the electron energy is 2� 1011 eVand B = 10�7 T, what is the dominant wavelength component?

8.8 Perform numerically the integration in the radiation power of l-th harmonic of synchrotron

radiation,

Pl =1

4��0

(e!0)2l

�c

Z �

0

sin(2�l sinx)

� sinx(�2 cos 2x� 1) cos(2lx)dx;

for the case � = 0:99 ( ' 7) and plot the result as a function of l:

8.9 Estimate the bremsstrahlung cooling rate (in W/m3) of a hydrogen plasma for an electron

temperature of 10 keV and plasma density of n = 1020 m�3: If the plasma is con�ned by a

magnetic �eld of 5 T, what is the cooling rate due to cyclotron radiation if reabsorption by

46

the plasma is ignored? (Cyclotron radiation can be reabsorbed by a plasma if its density is

high enough and its contribution to energy loss may be ignored.)

8.10 Assuming a permittivity in the form

" (!) =

1�

!2p!2 � !20

!"0;

carry out the integration in the rate of Cherenkov radiation,

�dEdz=

1

4�"0

e2

c20

Z 1

0!

�1� 1

�2

�d!:

8.11 Two identical charges are moving along a circular orbit of radius a at an angular velocity !:

Assume nonrelativistic motion, !a � c: Angular separation of the charges is �0: If �0 = 0;

the problem reduces to a charge 2e rotating at ! which radiates as a dipole. If �0 = �; the

dipole moment vanishes and higher order quadrupole radiation should be considered. At what

angular separation �0 does the dipole radiation power become equal to quadrupole radiation

power? Is it necessary to consider magnetic dipole radiation?

8.12 Four identical charges are placed along a circle of radius a with equal angular spacing �=2:

If the system rotates about the axis at angular velocity !; what is the lowest order radiation

power? What would happen if the number of charges N increases?

8.13 Generalize the quadrupole radiation in electron-electron collision to relativistic velocity.

47