c2 4 PDF
c2 4 PDF
c2 4 PDF
In case of charged particle motion in an external field, one of the most fruitful
approaches allowing to calculate the characteristics of radiation, generated by a
particle with charge e, is an approach, where a trajectory r(t) of the particle in the
given field has been found at first, and then the electric and magnetic components
of the electromagnetic field are defined according to the rules of classical elec-
trodynamics [1]:
h h : ii
e 1 b2 ð n b Þ e n ð n b Þ b
EðtÞ ¼ þ ; ð2:1:1aÞ
R2 ð 1 n b Þ 3 cRð1 n bÞ3
dI cR2
¼ jEðxÞj2 : ð2:1:6Þ
dx dX 4p
As a rule, the radiation is formed by a source with a finite area S, moreover, this
source can emit the electromagnetic waves (the photons) anisotropically. In this
case the radiation is characterized by brightness
dP W
L¼ ð2:1:7Þ
dX dS sr m2
For the radiation with frequencies from optical and above ones the spectral
brightness is often assigned through the number of photons. Using the Planck’s
2.1 Radiation Characteristics in the Classical and Quantum Electrodynamics 7
law e ¼
hx in semi-classical approach, the energy characteristics are expressed
through the number of photons N:
dN
dP ¼ e : ð2:1:9Þ
dt
Then instead the spectral brightness one may use the brilliance
dL dN photon
¼B¼ : ð2:1:10Þ
de dt dX dS de=e s sr m2 de=e
The spectral flux (spectral density) is calculated after the integration over a
solid angle
Z
photon
US ðeÞ ¼ B dx dy dX : ð2:1:12Þ
s De=e
And finally, the radiation flux is received via the integration over a spectrum:
Z
photon
U¼ US ðeÞ de=e : ð2:1:13Þ
s
The field strength of the monochromatic electromagnetic wave (for example, the
laser radiation) is characterized by the dimensionless parameter:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2e2 hA2 i e E0
a0 ¼ ¼ : ð2:1:14Þ
ðmc2 Þ2 mc x
Hereinafter, the usage of the term ‘‘the photon beam’’ supposes that it concerns the
electromagnetic radiation propagating along the fixed direction with a negligibly
small angular divergence, the characteristics of which (intensity, polarization,
position of maximum in spectrum, temporal modulation, etc.) are possible to
adjust in a rather large range.
2.2 Polarization Characteristics of Radiation 9
is a degree of polarization;
u0 ¼ ð1=2Þ arctg ðn1 =n3 Þ ð2:2:3Þ
present a partly polarized photon beam (for which 0\n21 þ n22 þ n23 \1) as
superposition of completely polarized and non-polarized beams with various
intensities.
In the classical electrodynamics, the Stokes parameters are calculated as
follows:
The components of the field are calculated in a system, where the third axis
coincides with the direction of a wave vector. If the task has any chosen plane, the
coordinate system is assigned via basis vectors. If in problem there is a chosen
plane, then the coordinate system is
Ei Ek ¼ dX Ei Ek ; i; k ¼ 1; 2: ð2:2:8Þ
DX
Generally speaking, the averaging similar to (2.2.8) can be carried out not only by the
angular variables but also by any other non-observable kinematic ones. Thus, during
the calculation of polarization characteristics of coherent bremsstrahlung, the
averaging similar to (2.2.8) is carried out by the momentum of a final electron [3].
Ter-Mikaelyan in his monograph [4] considering the spatial region, in which the
bremsstrahlung is generated by ultrarelativistic electron moving in a medium, has
shown that the longitudinal size of this region (along the direction of the initial
electron) sharply increases with the growth of the electron Lorentz-factor and with
decrease of the photon energy. This spatial scale, which was named ‘‘formation
length’’ ‘f, can have macroscopic sizes greatly exceeding the wavelength of the
2.3 The Formation Length of Radiation by a Charged Particle 11
bremsstrahlung photon. After the passage of the length ‘f, the electron and emitted
photon can be considered as independent particles.
The estimation of this spatial scale can be found from classical electrodynamics
(see, for example, [5]). In this approach, the charge, which passes through a rather
small area and where external fields are concentrated, is emitted an electromag-
netic wave with the length k without appreciable distortion of a charge trajectory
and the change of its energy (see Fig. 2.1).
The determination of the formation length follows from the phase relationships:
on the length ‘f, which a charge passes after the area of a field at velocity b, the
front of a wave, emitted in angle h, should ‘‘lag behind’’a charge for a wave length:
‘f
‘f cos h ¼ k; ð2:3:1Þ
b
and (2.3.1) directly results in the formula for the formation length:
k
‘f ¼ : ð2:3:2Þ
1=b cos h
‘f ¼ 2c2 k: ð2:3:3Þ
If the following area of a field concentration is located along a trajectory on the
distance L\‘f (see Fig. 2.1), then in this case the electromagnetic waves, emitted
by a charge in two areas of an external field, will interfere in a destructive manner,
i.e. the intensity of resulting radiation will be less than the sum of intensities from
two independent sources.
Let carry out the quantum consideration of the formation length problem on an
example of bremsstrahlung, following to Ter-Mikaelyan [4].
We shall estimate the minimal value of a longitudinal recoil momentum ql,
which is transferred to a nucleus, during the process of bremsstrahlung of the
ultrarelativistic electron with energy e1. Such situation is realized for collinear
geometry, when the final electron with energy e2 and a photon with energy hx
move along the direction of the initial electron:
ql min ¼ p1 p2 k: ð2:3:4Þ
Here p1, p2, k are momenta of initial and final electrons and photons, accordingly.
Neglecting the energy transferred to a nucleus (i.e. in case of fulfillment of a
condition e1 ¼ e2 þ hx), momentum pi in the ultrarelativistic approach becomes
e1 1 e1 hx 1
p1 ¼ 1 2 ; p2 ¼ 1 2 ;
c 2c1 c 2c2
h e2
‘¼ ¼ 2c1ke ; ð2:3:6Þ
ql min hx
where ke is the Compton wavelength of an electron. It is clear that for the case
x e1 ; e2 (i.e. e2 e1 ) from the formula (2.3.6) follows the expression (2.3.3):
h
‘ ¼ 2c2 k ¼ ‘f
that illustrates the generality of the concept of the formation length both for
quantum consideration, where recoil effects are important, and for classical one.
The concept of the formation length plays an important role in considering of
various physical effects (see in detail the review [6]). With regard to the radiation
in periodic structures, where a constructive interference is the reason of mono-
chromaticity of the radiation spectrum (for the fixed radiation angle h), the
wavelength corresponding to the spectral line with minimal frequency (so-called
‘‘fundamental’’ harmonic), is defined from the relationship
‘f ¼ d; ð2:3:7Þ
where d is a period of the structure.
Expression (2.3.7) does not depend on the radiation mechanism and is appli-
cable both in classical electrodynamics (for instance, for undulator radiation or
Smith–Purcell radiation), and in quantum one (the typical example is the coherent
bremsstrahlung). The mentioned mechanisms, as well as some others, are con-
sidered in the following chapters of this book.
passing with velocity bk c through the first period; DtK ¼ d cos h=c is time of the
wave front passing from the first period till identical position on the second period.
The phase difference of two wave packages generated by electron on the first
and second periods are the follows:
!
2p d d cos h d
U ¼ xðDte DtK Þ ¼ bjj c ¼ 2p 1 bk cos h : ð2:4:1Þ
k bjj c c k
Thus, the field of radiation on the second period is defined by the expression
E2 ðkÞ ¼ E1 ðkÞ expðiUÞ: ð2:4:2Þ
Reasoning by analogy, it is possible to express the radiation field for the nth
period as:
En ðkÞ ¼ E1 ðkÞ expðiðn 1ÞUÞ: ð2:4:3Þ
Then the total field from the periodic structure containing N elements is repre-
sented as the sum
Having designated (expðiUÞ ¼ q), we shall receive an expression for the total
intensity of the field:
ER ðkÞ ¼ E1 ðkÞ 1 þ q þ q2 þ þ qN1
1 qN 1 expðiNUÞ
¼ E1 ðkÞ ¼ E1 ðkÞ ; ð2:4:5Þ
1q 1 expðiUÞ
d 2 WR
¼ const jER ðkÞj2
dx dX
j1 exp ðiNUÞj2 d2 W
¼ const jE1 ðkÞ j2 ¼ FN : ð2:4:6Þ
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} j1 exp ðiUÞj2 dx dX
d2 W
dx dX
2 2
d W
Here dx dX ¼ const jE1 ðkÞj describes the radiation ‘‘collected’’ from one period of
a trajectory, and a multiplier
1 expðiNUÞ2
FN ¼ ð2:4:7Þ
1 expðiUÞ
sin2 ðNU=2Þ
FN ¼ : ð2:4:8Þ
sin2 ðU=2Þ
The function FN has a set of sharp maxima for the values of an argument, which
makes a denominator zeroth:
U d
¼p 1 bk cos h ¼ m p; m is an integer:
2 km
The last formula is reduced to the following expression for the case b bk
d
km ¼ ð1 b cos hÞ; ð2:4:9Þ
m
which was received regardless to any fixed radiation mechanism and can be
applied to any type of radiation, which is characterized by the periodic disturbance
of a trajectory. The received relationship is generalization of the resonance con-
dition (2.3.7) for m 6¼ 1.
Frequently, the index m ¼ 1; 2; 3; . . . refers to harmonic number. The harmonic
m = 1 for ultrarelativistic particles with frequency
4pc2 c 2c2 x0
x1 ¼ ¼ ð2:4:10Þ
d 1 þ c 2 h2 1 þ c 2 h2
2.4 Interference Factor and the Resonance Condition 15
The diagram of the function FN is presented in Fig. 2.3 for h = 0 at N ¼ 5 and 10.
As expected, the function FN differs from zero in a small range of frequencies
close by xm, and the width of this range is defined by a number of the periods:
Dxm 1
: ð2:4:12Þ
xm N
As it follows from the picture, the maximal value of the function is
References
3. Diambrini Palazzi, G.: High-energy Bremsstrahlung and electron pair production in thin
crystals. Rev. Mod. Phys. 40, 611 (1968)
4. Ter-Mikaelyan, M.L.: High-Energy Electromagnetic Processes in Condensed Media. Wiley-
Interscience, New York (1972)
5. Akhiezer, A.I., Shulga, N.F.: High Energy Electrodynamics in Matter. Gordon and Breach,
Amsterdam (1996)
6. Baier, V.N., Katkov, V.M.: Concept of formation length in radiation theory. Phys. Rep. 409,
261 (2005)
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