d E
R
i( '
)
d d '
E
t
E
dt e
.
ω'
(10.53)
d Ω
Z 0
But the inner integral (over t) is just 2( + ’).16 This delta function kills one of the frequency integrals (say, one over ’), and Eq. (53) gives us a result that may be recast as
2
2
d E
R
I
R
d ,
with I 4
4
*
E E
E E
,
(10.54)
Ω
d
Z
Z
0
0
0
where the evident frequency symmetry of the scalar product EE - has been utilized to fold the integral of I() to positive frequencies only. The first of Eqs. (54) makes the physical sense of the function I() very clear: this is the so-called spectral density of the electromagnetic radiation (per unit solid angle).17
To calculate the spectral density, we can express the function E via E( t) using the Fourier transform reciprocal to Eq. (51):
1
E
E t
( ei t
)
dt
.
(10.55)
2
In the particular case of radiation by a single point charge, we may use here the second (radiative) term of Eq. (19):
1
q
1
n (n β)
β
E
.
(10.56)
3
ei t dt
2
4
cR
1
( β n)
0
ret
Since the vectors n and are more natural functions of the radiation’s emission (retarded) time t ret, let us use Eqs. (5) and (16) to exclude the observation time t from this integral:
q
1 1
n (n β)
β
R ret
E
exp i t
dt .
(10.57)
2
ret
4 2 cR
1
( β
ret
n)
c
0
ret
Assuming that the observer is sufficiently far from the particle,18 we may treat the unit vector n as a constant and also use the approximation (8.19) to reduce Eq. (57) to
15 Note that the expression under this integral differs from d P/ d defined by Eq. (29) by the absence of the term (1 – n) = t ret/ t – see Eq. (16). This is natural because now we are calculating the wave energy arriving at the observation point r during the time interval dt rather than dt ret.
16 See, e.g. MA Eq. (14.4).
17 The notion of spectral density may be readily generalized to random processes – see, e.g., SM Sec. 5.4.
18 According to the estimate (49), for a synchrotron radiation’s pulse, this restriction requires the observer to be much farther than r’ ~ c t ~ R/3 from the particle. With the values R ~ 104 m and ~ 105 mentioned above, r’
~ 10-11 m, so that this requirement is satisfied for any realistic radiation detector.
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1
q
1
i r n (n β)
β
n '
r
E
exp
exp i t
dt .
(10.58)
2
2 4 cR
c
1
(
β n)
c
0
ret
ret
Plugging this expression into Eq. (54), and then using the definitions c 1/(00)1/2 and Z 0 (0/0)1/2, we get19
2
Z q
n (n β) β
n '
r
I
0
2
exp
i t
dt
.
(10.59)
3
2
ret
16
1
( β n)
c
ret
This result may be further simplified by noticing that the fraction before the exponent may be represented as a full derivative over t ret,
n (n β)
β
n (n β) β
d / dt
d n (n β)
,
(10.60)
2
2
1
( β n)
1
( β n)
dt 1 β n
ret
ret
ret
and working out the resulting integral by parts. At this operation, the time differentiation of the parentheses in the exponent gives d[ t ret – nr ’( t ret)/ c]/ dt ret = (1 – nu/ c)ret (1 – n)ret, leading to the cancellation of the remaining factor in the denominator and hence to a very simple general result: 20
2
Relativistic
2
2
Z q
n '
r
radiation:
I
0
n (n β)exp
spectral
i t
dt
.
(10.61)
3
ret
16
c
density
ret
Now returning to the particular case of the synchrotron radiation, it is beneficial to choose the origin of time t ret so that at t ret = 0, the angle between the vectors n and takes its smallest value 0, i.e., in terms of Fig. 5, the vector n is within the [ y, z]-plane. Fixing this direction of the axes so that they do not move, we can redraw that figure as shown in Fig. 8.
y
n
0 βret
x
t
Fig. 10.8. Deriving the synchrotron radiation’s
c ret
spectral density. The vector n is static within
R
z
the [ y, z]-plane, while the vectors r ’( t ret) and
r ' ( t )
ret
ret rotate, within the [ x, z]-plane, with the
0
angular velocity c of the particle.
In this “lab” reference frame, the vector n does not depend on time, while the vectors r ’( t ret) and
ret do depend on it via the angle c t ret:
19 Note that for our current purposes of calculation of the spectral density of radiation by a single particle, the factor exp{ i r/ c} has got canceled. However, as we have seen in Chapter 8, this factor plays a central role in the interference of radiation from several (many) sources. Such interference is important, in particular, in undulators and free-electron lasers – the devices to be (qualitatively) discussed below.
20 Actually, this simplification is not occasional. According to Eq. (10b), the expression under the derivative in the last form of Eq. (60) is just the transverse component of the vector potential A (give or take a constant factor), and from the discussion in Sec. 8.2 we know that this component determines the electric dipole radiation of a system, which dominates the radiation in our current case of a single particle with a non-zero electric charge.
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n ,
0 sin , cos
r ' t
R
R
β
. (10.62)
0
0 ,
ret 1 cos, ,0 sin, ret sin, ,0 cos
Now an easy multiplication yields
n(nβ)
,
(10.63)
ret
sin, sin cos cos, sin2 sin
0
0
0
n '
r
R
exp i t
exp i t cos sin .
(10.64)
ret
0
c
c
ret
As we already know, in the (most interesting) ultra-relativistic limit >> 1, most radiation is confined to short pulses, so that only small angles ~ c t ret ~ – 1 may contribute to the integral in Eq. (61).
Moreover, since most radiation goes to small angles ~ 0 ~ –1, it makes sense to consider only such small angles. Expanding both trigonometric functions of these small angles, participating in parentheses of Eq. (64), into the Taylor series, and keeping only the leading terms, we get
2
3
R
R
R
R
0
c
3
t
cos sin t t
t
t .
(10.65)
ret
0
ret
c ret
c ret
ret
c
c
c 2
c 6
Since ( R/ c)c = u/ c = 1, in the two last terms we may approximate this parameter by 1. However, it is crucial to distinguish the difference between the two first terms, proportional to (1 – ) t ret, from zero; as we have done before, we may approximate it with t ret/22. On the right-hand side of Eq. (63), which does not have such a critical difference, we may be bolder, taking21
sin, sin cos cos, sin2 sin
t
(10.66)
0
0
0
, ,0 0
,
,
c ret
0
0 .
As a result, Eq. (61) is reduced to
2
2
Z q
2
Z q
2
2
I
0
0
a n a n
a
a
,
(10.67)
3
x
x
y
y
3
x
y
16
16
where ax and ay are the following dimensionless factors:
2
i
a t exp
x
( 2
2
)
c
3
t
t
dt ,
c ret 2 0
ret
3 ret
ret
(10.68)
2
i
a exp
y
( 2
2
)
c
3
t
t
dt ,
0 2 0
ret
3 ret
ret
that describe the frequency spectra of two components of the synchrotron radiation, with mutually perpendicular polarization planes. Defining the following dimensionless parameter
2 2
(10.69)
0
3/2,
3c
21 This expression confirms that the in-plane ( x) component of the electric field is an odd function of t ret and hence of t – t 0 (see its sketch in Fig. 7b), while the normal ( y) component is an even function of this difference. Also, note that for an observer exactly in the rotation plane (0 = 0) the latter component equals zero for all times – the fact which could be predicted from the very beginning because of the evident mirror symmetry of the problem with respect to the particle’s rotation plane.
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EM: Classical Electrodynamics
which is proportional to the observation frequency, and changing the integration variable to
2
c t ret/(0 + -2)1/2, the integrals (68) may be reduced to the modified Bessel functions of the second kind, but with fractional indices:
a
x
3
3
2 3 i
2
2
exp
i
d
K
,
0
2
3
c
2 2
0
1/ 2
2 / 3
(10.70)
3
a
3
2 3
exp
i
d
K
y
0 2
2
0
1/2
0
2
2
1/ 3
2
3
c
0
Figure 9a shows the dependence of the Bessel factors defining the amplitudes ax and ay on the normalized observation frequency . It shows that the radiation intensity changes with frequency relatively slowly (note the log-log scale of the plot!) until the normalized frequency defined by Eq. (69) is increased beyond ~1. For the most important observation angles 0 ~ , this means that our estimate (50) is indeed correct, though formally the frequency spectrum extends to infinity.22
(a)
(b)
1
K ( )
2 / 3
0.8
K () d
5 / 3
0.6
0.1
K ( )
1 / 3
0.4
0.2
0.01
0.01
0.1
1
10
0.01
0.1
1
10
Fig. 10.9. The frequency spectra of: (a) two components of the synchrotron radiation, at a fixed angle 0, and (b) its total (polarization- and angle-averaged) intensity.
Naturally, the spectral density integrated over the full solid angle exhibits a similar frequency behavior. Without performing the integration,23 let me just give the result (also valid for >> 1 only) for the reader’s reference:
3
2
I
2
d
q K
d ,
where
.
(10.71)
5 / 3
3
4
3
4
c
Figure 9b shows the dependence of this integral on the normalized frequency . (This plot is sometimes called the “universal flux curve”.) In accordance with the estimate (50), it reaches the maximum at 22 The law of the spectral density decrease at large may be readily obtained from the second of Eqs. (2.158), which is valid even for any (even non-integer) Bessel function index n: ax ay -1/2exp{-}. Here the exponential factor is certainly the most important one.
23 For that, and many other details, the interested reader may be referred, for example, to the fundamental review collection by E. Koch et al. (eds.) Handbook on Synchrotron Radiation (in 5 vols.), North-Holland, 1983-1991, or to a more concise monograph by A. Hofmann, The Physics of Synchrotron Radiation, Cambridge U. Press, 2007.
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Essential Graduate Physics
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c
3
,
3
.
0
i.e.
.
(10.72)
max
max
2
For example, in the National Synchrotron Light Source (NSLS-II) in the Brookhaven National Laboratory near our SBU campus, with its ring’s circumference of 792 m, the electron revolution period T is 2.64μs. With c = 2 π/ T 2.4106 s-1, for the achieved 6103 (E 3 GeV), we get max ~
31017 s-1, i.e. the photon energy max ~ 200 eV corresponding to soft X-rays. In light of this estimate, the reader may be surprised by Fig. 10, which shows the calculated spectra of the radiation that this facility was designed to produce, with the intensity maxima at photon energies up to a few keV.
Fig. 10.10. Design brightness of various synchrotron radiation sources of the NSLS-II facility.
For the bend magnets and wigglers, the “brightness” may be obtained by multiplication of the one-pulse spectral density I() calculated above, by the number of electrons passing the source per second. (Note the non-SI units used by the synchrotron radiation community.) However, for undulators, there is an additional factor due to the partial coherence of radiation – see below.
(Adapted from the document NSLS-II Source Properties and Floor Layout that was available online at https://www.bnl.gov/ps/docs/pdf/SourceProperties.pdf in 2011-2020.)
The reason for this discrepancy is that in the NLLS-II, and in all modern synchrotron light sources, most radiation is produced not by the circular orbit itself (which is, by the way, not exactly Chapter 10
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Essential Graduate Physics
EM: Classical Electrodynamics
circular, but consists of a series of straight and bend-magnet sections), but by such bend sections, and the devices called wigglers and undulators: strings of several strong magnets with alternating field direction (Fig. 11), that induce periodic bending (wiggling”) of the electron’s trajectory, with the synchrotron radiation emitted at each bend.
u
X ray
beam
electrons
Fig. 10.11. The generic structure of the wigglers,
undulators, and free-electron lasers. (Adapted from
http://www.xfel.eu/overview/how_does_it_work/. )
The difference between the wigglers and the undulators is more quantitative than qualitative: the former devices have a larger spatial period u (the distance between the adjacent magnets of the same polarity, see Fig. 11), giving enough space for the electron beam to bend by an angle larger than – 1, i.e.
larger than the radiation cone’s width. As a result, the radiation reaches an in-plane observer as a periodic sequence of individual pulses – see Fig. 12a.
t
2
/ 2 c
u
(a)
(b)
t
t
t
Fig. 10.12. Waveforms of the radiation emitted by
(a) a wiggler and (b) an undulator – schematically.
The shape of each pulse, and hence its frequency spectrum, are essentially similar to those discussed above,24 but with much higher local values of c and hence max – see Fig. 10. Another difference is a much higher frequency of the pulses. Indeed, the fundamental Eq. (16) allows us to calculate the time distance between them, for the observer, as
t
1
t
t
1
,
(10.73)
ret
u
u
u
t
u
2
2
c
c
ret
24 Indeed, the period u is typically a few centimeters (see the numbers in Fig. 10), i.e. is much larger than the interval r’ ~ R/3 estimated above. Hence the synchrotron radiation results may be applied locally, to each electron beam’s bend. (In this context, a simple problem for the reader: use Eqs. (19) and (63) to explain the difference between shapes of the in-plane electric field pulses emitted at opposite magnetic poles of the wiggler, which is schematically shown in Fig. 12a.)
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EM: Classical Electrodynamics
where the first two relations are valid at u << R (the relation typically satisfied very well, see the numbers in Fig. 10), and the last two relations assume the ultra-relativistic limit. As a result, the radiation intensity, which is proportional to the number of the poles, is much higher than that from the bend magnets – see Fig. 10 again.
The situation is different in undulators – similar structures with a smaller spatial period u, in which the electron’s velocity vector oscillates with an angular amplitude smaller than – 1. As a result, the radiation pulses overlap (Fig. 12b), and the radiation waveform is closer to the sinusoidal one. As a result, the radiation spectrum narrows to the central frequency25
2
2
2
c
2
.
(10.74)
0
t
u
For example, for the LSNL-II undulators with u = 2 cm, this formula predicts a radiation peak at phonon energy 0 4 keV, in reasonable agreement with the quantitative calculation results shown in Fig. 10.26 Due to the spectrum narrowing, the undulator’s radiation intensity is higher than that of wigglers using the same electron beam.
This spectrum-narrowing trend is brought to its logical conclusion in the so-called free-electron lasers 27 whose basic structure is the same as that of wigglers and undulators (Fig. 11), but the radiation at each beam bend is so intense and narrow-focused that it affects the electron motion downstream of the radiation cone. As a result, the radiation spectrum narrows around the central frequency (74), and its power grows as a square of the number N of electrons in the structure (rather than proportionately to N
in wigglers and undulators).
Finally, note that wigglers, undulators, and free-electron lasers may be also used at the end of a linear electron accelerator (such as SLAC) which, as was noted above, may provide extremely high values of , and hence radiation frequencies, due to the smallness of radiation energy losses at the electron acceleration stage. Very unfortunately, I do not have time/space to discuss the (very interesting) physics of these devices in more detail.28
10.4. Bremsstrahlung and Coulomb losses
Surprisingly, a very similar mechanism of radiation by charged particles works on a much smaller spatial scale, namely at their scattering by charged particles of the propagation medium. This 25 This important formula may be also derived in the following way. Due to the relativistic length contraction (9.20), the undulator structure period as perceived by beam electrons is ’ = u/, so that the central frequency of the radiation in the reference frame moving with the electrons is 0 ’ = 2 c/ ’ = 2 c/u . For the lab-frame observer, this frequency is Doppler-upshifted in accordance with Eq. (9.44): 0 = 0 ’[(1 + )/(1 – )]1/2 20 ’, giving the same result as Eq. (74).
26 Some of the difference is due to the fact that those plots show the spectral density of the number of photons n =
E/ per second, which peaks at a frequency below that of the density of power, i.e. of the energy E per second.
27 This name is somewhat misleading, because in contrast to the usual (“quantum”) lasers, a free-electron laser is essentially a classical device, and the dynamics of electrons in it is very similar to that in vacuum-tube microwave generators, such as the magnetrons briefly discussed in Sec. 9.6.
28 The interested reader may be referred, for example, to either P. Luchini and H. Motz, Undulators and Free-electron Lasers, Oxford U. Press, 1990; or E. Salin et al., The Physics of Free Electron Lasers, Springer, 2000.
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effect, traditionally called by its German name bremsstrahlung (“brake radiation”), is responsible, in particular, for the continuous part of the frequency spectrum of the radiation produced in standard vacuum X-ray tubes, at the electron collisions with a metallic “anticathode”. 29
The bremsstrahlung in condensed matter is generally a rather complicated phenomenon because of the simultaneous involvement of many particles, and (frequently) some quantum electrodynamic effects. This is why I will give only a very brief glimpse at the theoretical description of this effect, for the simplest case when the scattering of incoming, relatively light charged particles (such as electrons, protons, -particles, etc.) is produced by atomic nuclei, which remain virtually immobile during the scattering event (Fig. 13a). This is a reasonable approximation if the energy of incoming particles is not too low; otherwise, most scattering is produced by atomic electrons whose dynamics is substantially quantum – see below.
y
(a)
(b)
Fig. 10.13. The basic
geometry
of
the
q' , m'
b
βfin
p
bremsstrahlung and the
fin
q
Coulomb loss problems
q, m
βini
'
'
in the (a) direct and (b)
0
x
pini
reciprocal spaces.
To calculate the frequency spectrum of the radiation emitted during a single scattering event, it is convenient to use a byproduct of the last section’s analysis, namely Eq. (59) with the replacement (60):30
2
2
1
q
d n (n β)
n '
r
I
exp
i t
dt
.
(10.75)
2
ret
4 c 4
dt 1 β n
c
0
ret
A typical duration of a single scattering event we are discussing is of the order of a 0/ c ~ (10-10
m)/(3108 m/s) ~ 10-18 s in solids, and only an order of magnitude longer in gases at ambient conditions.
This is why for most frequencies of interest, from zero all the way up to at least soft X-rays,31 we can use the so-called low-frequency approximation, taking the exponent in Eq. (75) for 1 through the whole collision event, i.e. the integration interval. This approximation immediately yields
Brems-
2
strahlung:
1
q
n n β
n n β
single
I
fin
ini 2
.
(10.76)
2
collision
4 c 4
1 β n
1 β n
0
fin
ini
29 Such X-ray radiation had been first observed experimentally (though not correctly interpreted) by N. Tesla in 1887, i.e. before it was rediscovered and studied in detail by W. Röntgen.
30 In publications on this topic (whose development peak was in the 1920s-1930s), the Gaussian units are more common, and the uppercase letter Z is usually reserved for expressing charges as multiples of the fundamental charge e, rather than for the wave impedance. This is why, in order to avoid confusion and facilitate the comparison with other texts, in this section I (while still staying with the SI units used throughout my series) will use the fraction 1/0 c, instead of its equivalent Z 0, for the free-space wave impedance, and write the coefficients in a form that makes the transfer to the Gaussian units elementary: it is sufficient to replace all ( qq’/40)SI with ( qq’)Gaussian. In the (rare) cases when I spell out the charge values, I will use a different font: q Z e, q’ Z ’e.
31 A more careful analysis shows that this approximation is actually quite reasonable up to much higher frequencies, of the order of 2/.
Chapter 10
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EM: Classical Electrodynamics
In the non-relativistic limit (ini, fin << 1), this formula is reduced to the following result: 2
2
1
q
q
I
2
sin
(10.77)
2
4 c
2 2
4
m c
0
(which may be derived from Eq. (8.27) as well), where q is the momentum transferred from the scattering center to the scattered charge (Fig. 13b):32
q p
p Δ
m u
Δ
mc β mc β β
,
(10.78)
fin
ini
fin ini
and (not to be confused with the particle scattering angle ’ shown in Fig. 13!) is the angle between the vector q and the direction n toward the observer – at the collision moment.
The most important feature of the result (77)-(78) is the frequency-independent (“white”) spectrum of the radiation, very typical for any rapid pulses that may be approximated as delta functions of time.33 (Note, however, that Eq. (77) implies a fixed value of q, so that the statistics of this parameter, to be discussed in a minute, may “color” the radiation.)
Note also the “doughnut-shaped” angular distribution of the radiation, typical for non-relativistic systems, with the symmetry axis directed along the momentum transfer vector q. In particular, this means that in typical cases when ’ << 1, i.e. q << p, when the vector q is nearly normal to the vector pini (see, e.g., the example shown in Fig. 13b), the bremsstrahlung produces a significant radiation flow in the direction back to the particle source – the fact significant for the operation of X-ray tubes.
Now integrating Eq. (77) over all wave propagation angles, just as we did for the instant radiation power in Sec. 8.2, we get the following spectral density of the particle energy loss, 2
2
d E
2
q
q
I Ω
d
.
(10.79)
2
2 2
d
3 c
4 m c
4
0
In most applications of the bremsstrahlung theory (as in most scattering problems34), the impact parameter b (Fig. 13a), and hence the scattering angle ’ and the transferred momentum q, have to be 32 Please note the font-marked difference between this variable (q) and the particle’s electric charge ( q).
33 This is the basis, in particular, of the so-called High-Harmonic Generation (HHG) effect, discovered in 1977, which takes place at the irradiation of gases by intensive laser beams. The high electric field of the beam strips valent electrons from initially neutral atoms, and accelerates them away from the remaining ions, just to slam them back into the ions as the field’s polarity changes in time. The electrons change their momentum sharply during their recombination with the ions, resulting in bremsstrahlung-like radiation. The spectrum of radiation from each such event obeys Eq. (77), but since the ionization/acceleration/recombination cycles repeat periodically with the frequency 0 of the laser field, the final spectrum consists of many equidistant lines, with frequencies n0. The classical theory of the bremsstrahlung does not give a cutoff max = n max0 of the spectrum; this limit is imposed by quantum mechanics: max ~ Ep, where the so-called ponderomotive energy Ep =
( eE 0/0)2/4 m e is the average kinetic energy given to a free electron by the periodic electric field of the laser beam, with amplitude E 0. In practice, n max may be as high as ~100, enabling alternative compact sources of X-ray radiation. For a detailed quantitative theory of this effect, see, for example, M. Lewenstein et al., Phys. Rev. A 49, 2117 (1994).
34 See, e.g., CM Sec. 3.5 and QM Sec. 3.3.
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EM: Classical Electrodynamics
considered random. For elastic (ini = fin ) Coulomb collisions we can use the so-called Rutherford formula for the differential cross-section of scattering35
2
2
d
qq' 1
1
.
(10.80)
Ω
d '
4
2 pc sin4 '
0
/ 2
Here d = 2 bdb is the elementary area of the sample cross-section (as visible from the direction of the incident particles) corresponding to their scattering into an elementary body angle36
d '
2 sin ' d ' .
(10.81)
Differentiating the geometric relation, which is evident from Fig. 13b,
'
q 2 p sin
,
(10.82)
2
we may represent Eq. (80) in a more convenient form
2
d
qq'
1
8
(10.83)
2
3
d q
4
u
0
q
Now combining Eqs. (79) and (83), we get
2
d d
16
2
q
qq'
1
E
1
.
(10.84)
d d
q
3 4
4
2
2
mc
c
0
0
q
This product is called the differential radiation cross-section. When integrated over all values of q (which is equivalent to averaging over all values of the impact parameter), it gives a convenient measure of the radiation intensity. Indeed, after the multiplication by the volume density n of independent scattering centers, such integral yields the particle’s energy loss per unit bandwidth of radiation per unit path length, – d 2E/ d dx. A minor problem here is that the integral of 1/q formally diverges at both infinite and zero values of q. However, these divergences are very weak (logarithmic), and the integral converges due to virtually any reason unaccounted for in our simple analysis. The standard (though slightly approximate) way to account for these effects is to write
2
Brems-
2
2
d
strahlung:
E
16
q
qq'
1
q
max
n
ln
intensity
,
(10.85)
2
2
d dx
3
4
4 mc
c
0
0
qmin
and then plug, instead of qmax and qmin, the scales of the most important effects limiting the range of the transferred momentum’s magnitude. In the classical-mechanics analysis, according to Eq. (82), qmax = 2 p
2 mu. To estimate qmin, let us note that the very small momentum transfer takes place when the impact parameter b is very large, and hence the effective scattering time ~ b/ v is very long. Recalling the condition of the low-frequency approximation, we may associate qmin with ~ 1/ and hence with b ~
35 See, e.g., CM Eq. (3.73) with = qq’/40. In the form used in Eq. (80), the Rutherford formula is also valid for the small-angle scattering of relativistic particles, the criterion being << 2/.
36 Again, the angle ’ and the differential d ’, describing the scattered particles (see Fig. 13) should not be confused with the parameters and d describing the radiation emitted at the scattering event.
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EM: Classical Electrodynamics
u ~ v/ . Since for the small scattering angles, q is close to the impulse F ~ ( qq’/4 π0 b 2) of the Coulomb force, we get the estimate qmin ~ ( qq’/4 π0) /u 2, and Eq. (85) should be used with q
Classical
max
3
2 mu
qq'
ln
ln
.
(10.86) brems-
q
4
strahlung
min
0
This is Bohr’s formula for what is called the classical bremsstrahlung. We see that the low momentum cutoff indeed makes the spectrum slightly colored, with more energy going to lower frequencies. There is even a formal divergence at 0; however, this divergence is integrable, so it does not present a problem for finding the total energy radiative losses (- d E/ dx) as an integral of Eq. (86) over all radiated frequencies . A larger problem for this procedure is the upper integration limit,
, at which the integral diverges. This means that our approximate description, which considers the collision as an elastic process, becomes invalid and needs to be amended by taking into account the difference between the initial and final kinetic energies of the particle due to radiation of the energy quantum of the emitted photon, so that
2
2
2
2
p
p
p
p
ini
fin
,
i.e.
ini
fin
E,
E - , .
(10.87)
2 m
2 m
2 m
2 m
As a result, taking into account that the minimum and maximum values of q correspond to, respectively, the parallel and antiparallel alignments of the vectors pini and pfin, we get q
p p
p p
m
E
E
Quantum
max
/ 2
ini
fin
ini
2
fin
1/2
2
1/ 2
ln
ln
ln
ln
,
(10.88) brems-
q
p p
p p
/ 2 m
strahlung
min
ini
fin
2
2
ini
fin
Plugged into Eq. (85), this expression yields the so-called Bethe-Heitler formula for quantum bremsstrahlung.37 Note that at this approach, qmax is close to that of the classical approximation, but qmin is of the order of / u, so that
qmin classical
ZZ '
~
,
(10.89)
q
min quantum
where Z and Z ’ are the particles’ charges in the units of e, and is the dimensionless fine structure (“Sommerfeld”) constant,
2
2
e
e
1
1,
(10.90)
4
SI
Gaussian
c
c
137
0
which is one of the basic notions of quantum mechanics.38 Due to the smallness of the constant, the ratio (89) is below 1 for most cases of practical interest, and since the integral of (84) over q is limited by the largest of all possible cutoffs qmin, it is the Bethe-Heitler formula that should be used.
37 The modifications of this formula necessary for the relativistic description are surprisingly minor – see, e.g., Chapter 15 in J. Jackson, Classical Electrodynamics, 3rd ed., Wiley 1999. For even more detail, the standard reference monograph on bremsstrahlung is W. Heitler, The Quantum Theory of Radiation, 3rd ed., Oxford U. Press 1954 (reprinted in 1984 and 2010 by Dover).
38 See, e.g., QM Secs. 4.4, 6.3, 6.4, 9.3, 9.5, and 9.7.
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Now nothing prevents us from calculating the total radiative losses of energy per unit length:
d E
2
d E
2
16
q
qq'
2
max
1/ 2
1
E
E 1/2
d
n
2
ln
d , (10.91)
2
2
dx
d dz
3
4 c 4 mc
0
0
0
1/ 2
0
where max = E is the maximum energy of the radiation quantum. By introducing the dimensionless integration variable /E = 2/( mu 2/2), this integral is reduced to a table one,39 and we get 2
2
2
2
2
2
d E
16
q
qq'
1 u
16 q'
q
1
n
n
.
(10.92)
2
2
2
dx
3
4 c 4 mc
3
4 c 4
mc
0
0
0
0
Following my usual style, at this point I would give you an estimate of the losses for a typical case; however, let me first discuss a parallel particle energy loss mechanism, the so-called Coulomb losses, due to the transfer of mechanical impulse from the scattered particle to the scattering centers.
(This energy eventually goes into an increase of the thermal energy of the scattering medium, rather than to the electromagnetic radiation.)
Using Eqs. (9.139) for the electric field of a linearly moving charge q, we can readily find the momentum it transfers to the counterpart charge q’:40
qq'
b
qq' 2
p'
p'
.
(10.93)
y
'p dt
q'E dt
y
y
4
0
b 2 2 u 2 2 dt
t 3/ 2
4
bu
0
Hence, the kinetic energy acquired by the scattering particle (and hence to the loss of the energy E of the incident particle) is
Δ
2
2
( p' )
qq'
2
ΔE
.
(10.94)
2 2
2 m'
4
m'u b
0
Such elementary energy losses have to be summed up over all collisions, with random values of the impact parameter b. At the scattering center density n, the number of collisions per small path length dx per small range db is dN = n 2 πbdb dx, so that 2
b
2
max
Coulomb
d E
Δ
qq'
2
db
qq' ln B
b
max
dN
losses
E
n
2
4 n
,
where B
. (10.95)
2
2
dx
4
m'u
b
4
m'u
b
0
b
0
min
min
Here, at the last step, the logarithmic integral over b was treated similarly to that over q in the bremsstrahlung theory. This approximation is adequate because the ratio b max/ b min is much larger than 1.
Indeed, b min may be estimated from ( p’)max ~ p = mu. For this value, Eq. (93) with q’ ~ q gives b min ~
r c (see Eq. (8.41) and its discussion), which, for elementary particles, is of the order of 10-15m. On the other hand, for the most important case when the Coulomb energy absorbers are electrons (which, according to Eq. (94), are the most efficient ones, due to their very low mass m’), b max may be estimated from the condition = b/ u ~ 1/min, where min ~ 1016 s-1 is the characteristic frequency of electron 39 See, e.g., MA Eq. (6.14).
40 According to Eq. (9.139), Ez =0, while the net impulse of the longitudinal force q’Ex is zero, so that Eq. (93) gives the full momentum transfer.
Chapter 10
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EM: Classical Electrodynamics
transitions in atoms. (Quantum mechanics forbids such energy transfer at lower frequencies.) From here, we have the estimate b max ~ u/min, so that
b
u
max
B
~
,
(10.96)
b
r
min
c
min
for ~ 1 and u ~ c 3108 m/s giving b max ~ 310-8 m, so that B ~ 109 (give or take a couple of orders of magnitude – this does not change the estimate ln B 20 too much). 41
Now we can compare the non-radiative Coulomb losses (95) with the radiative losses due to the bremsstrahlung, given by Eq. (92):
d E radiation
m'
1
~
2
'
ZZ
,
(10.97)
d E
m
ln B
Coulomb
Since ~ 10-2 << 1, for non-relativistic particles ( << 1) the bremsstrahlung losses of energy are much lower (that is why I did not want to rush with their estimates), and only for ultra-relativistic particles, the relation may be opposite.
According to Eqs. (95)-(96), for electron-electron scattering ( q = q’ = – e, m = m’ = m e),42 at the value n = 61026 m-3 typical for air at ambient conditions, the characteristic length of energy loss, l
E
,
(10.98)
c
d E / dx
for electrons with kinetic energy E = 6 keV is close to 210-4 m 0.2 mm. (This is why we need high vacuum in electron microscope columns and other vacuum electron devices.) Since l c E2, more energetic particles penetrate to matter deeper, until the bremsstrahlung steps in, and limits this trend at very high energies.
10.5. Density effects and the Cherenkov radiation
For condensed matter, the Coulomb loss estimate made in the last section is not quite suitable, because it is based on the upper cutoff b max ~ u/min. For the example given above, the incoming electron velocity u is close to 5107 m/s, and for the typical value min ~ 1016 s-1 (min ~ 10 eV), this cutoff b max is of the order of ~510-9 m = 5 nm. Even for air at ambient conditions, this is somewhat larger than the average distance (~ 2 nm) between the molecules, so that at the high end of the impact parameter range, at b ~ b max, the Coulomb loss events in adjacent molecules are not quite independent, and the theory needs some corrections. For condensed matter, with much higher particle density n, most collisions satisfy the following condition:
41 A quantum analysis (carried out by Hans Bethe in 1940) replaces, in Eq. (95), ln B with ln(22 mu 2/) – 2, where is the average frequency of the atomic quantum transitions weight by their oscillator strength. This refinement does not change the estimate given below. Note that both the classical and quantum formulas describe a fast increase (as 1/) of the energy loss rate (- d E/ dx) at 1, and its slow increase (as ln) at , so that the losses have a minimum at ( – 1) ~ 1.
42 Actually, the above analysis has neglected the change of momentum of the incident particle. This is legitimate at m’ << m, but for m = m’ the change approximately doubles the energy losses. Still, this does not change the order of magnitude of the estimate.
Chapter 10
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EM: Classical Electrodynamics
3
nb 1 ,
(10.99)
and the treatment of Coulomb collisions as a set of independent events is inadequate. However, this condition enables the opposite approach: treating the medium as a continuum. In the time-domain formulation used in the previous sections of this chapter, this would be a very complex problem, because it would require an explicit description of the medium dynamics. Here the frequency-domain approach, based on the Fourier transform in both time and space, helps a lot, provided that the functions
() and () are considered known – either calculated or taken from experiment. Let us have a good look at this approach because it gives some interesting (and practically important) results.
In Chapter 6, we have used the macroscopic Maxwell equations to derive Eqs. (6.118), which describe the time evolution of electrodynamic potentials in a linear medium with frequency-independent
and . Looking for all functions participating in Eqs. (6.118) in the plane-wave expansion form43
3
i(kr
t)
f (r, t) d k d f e
,
(10.100)
k,
and requiring all coefficients at similar exponents to be balanced, we get their Fourier images: 44
2
k 2
k
,
2
,
k 2 A
j .
(10.101)
k
,
k ,
k
,
As was discussed in Chapter 7, in such a Fourier form, the macroscopic Maxwell theory remains valid even for dispersive (but isotropic and linear!) media, so that Eqs. (101) may be generalized as
2
2
k ()()
k ,
,
k A
j
(10.102)
k ,
2 2 ( ) ( )
( )
,
()
k,
k,
An evident advantage of these equations is that their formal solution is elementary:
Field
() j
potentials
k ,
k ,
A
(10.103)
in a linear
k ,
(
medium
) 2 2
k ()( ,
)
k ,
2 2
k ()( ,
)
so that the “only” remaining things to do is, first, to calculate the Fourier transforms of the functions
(r, t) and j(r, t), describing stand-alone charges and currents, using the transform reciprocal to Eq.
(100), with one factor 1/2 per each scalar dimension,
1
3
i
(kr t
)
f
d r dt f (r, t) e
,
(10.104)
k,
(2 )4
and then to carry out the integration (100) of Eqs. (103).
For our problem of a single charge q uniformly moving through a medium with velocity u ,
(r, t) q (r u t),
(
j r, t) qu (r u t) ,
(10.105)
43 All integrals here and below are in infinite limits unless specified otherwise.
44 As was discussed in Sec. 7.2, the Ohmic conductivity of the medium (generally, also a function of frequency) may be readily incorporated into the dielectric permittivity: () ef() + i()/. In this section, I will assume that such incorporation, which is especially natural for high frequencies, has been performed, so that the current density j(r, t) describes only stand-alone currents – for example, the current (105) of the incident particle.
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the first task is easy:
q
i
t
q
i t
t
q
3
d r dt
q (r u t) (k r )
( k
u
)
e
e
dt
( k u) . (10.106)
k ,
(2 )4
(2 )4
(2 )3
Since the expressions (105) for (r, t) and j(r, t) differ only by a constant factor u, it is clear that the absolutely similar calculation for the current gives
u
q
j
( k u) .
(10.107)
k ,
(2 )3
Let us summarize what we have got by now, by plugging Eqs. (106)-(107) into Eqs. (103):
1
q
( k u)
1
( ) u
q
( k u)
, A
( )
( )u
.(10.108)
k
,
( 3
2 )
( 2
) k 2
( )
( )
k
,
( 3
2 ) 2
k 2
( )
( )
k
,
Now, at the last calculation step, namely the integration (100), we are starting to pay a heavy price for the easiness of the first steps. This is why let us think well about what exactly we need from it.
First of all, for the calculation of power losses, the electric field is more convenient to use than the potentials, so let us calculate the Fourier images of E and B. Plugging the expansion (100) into the basic relations (6.7), and again requiring the balance of exponent’s coefficients, we get
E
i
k
iA
i
( )
( )u k , B
ik A
i
( )
( )k u
, (10.109)
k
,
k
,
k
,
k ,
k
,
k
,
k
,
so that Eqs. (100) and (108) yield
kr
iq
u k k u
3
i(
t)
3
( ) ( )
(
)
E(r, t)
i k
t
d k d E e
d k d
e
r
. (10.110)
k ,
(2 )3
( ) 2
2
k ()()
(
)
This formula may be rewritten as the temporal Fourier integral (51), with the following r-dependent complex amplitude:
ikr
u k k u
3
iq
( ) ( )
(
)
E r
ik r 3
E e
d k
e
d k
(10.111)
k,
(2 )3
( ) 2
2
k ()(
.
)
Let us calculate the Cartesian components of this partial Fourier image E, at a point separated by distance b from the particle’s trajectory. Selecting the coordinates and time origin as shown in Fig. 3, we have r = {0, b, 0} and u = { u, 0, 0}, so that only Ex and Ey are different from zero. In particular, according to Eq. (111),
iq
()( u
) k
( E )
dk dk
dk
x ( k u) exp
. (10.112)
x
3
x
y
z
2
2
x
iky b
(2 ) ()
k ()()
The delta function kills one integral (over kx) of the three, and we get
iq
dk
( E )
z
u
ik b dk
. (10.113)
x
3
( ) ( )
exp y y
(2 ) () u
u
2
/ 2
u 2
k 2
k 2
()()
y
z
Chapter 10
Page 28 of 40
EM: Classical Electrodynamics
The internal integral (over kz) may be readily reduced to the table integral d/(1 + 2) in infinite limits, equal to ,45 and result represented as
2
i q
exp ik b
y
( E )
dk
(10.114)
x
(2 )3 () 2
2
k
,
1/ 2
y
y
where the parameter (generally, a complex function of frequency) is defined as46
2
2 1
Function ()
()() .
(10.115)
2
u
The last integral may be expressed via the modified Bessel function of the second kind:47
2
iqu
( E )
K ( b
) .
(10.116)
x
(2 )2 () 0
A very similar calculation yields
q
( E )
K ( b
) .
(10.117)
y
(2 )2 () 1
Now, instead of rushing to make the final integration (51) over to calculate E( t), let us realize that what we need most is the total energy loss through the whole time of the particle’s passage over an elementary distance dx. According to Eq. (4.38), the energy loss per unit volume is d
E j dt
E ,
(10.118)
dV
where j is the current of the bound charges in the medium, and should not be confused with the stand-alone incident-particle current (105). This integral may be readily expressed via the partial Fourier image E and the similarly defined image j., just as it was done at the derivation of Eq. (54): d
E dt
d e i t
d 'e i 't j E
2
d
d ' j E
( ')
2
j E
. (10.119)
'
'
d
dV
Let us incorporate the effective Ohmic conductivity ef() into the complex permittivity () just as this was discussed in Sec. 7.2, using Eq. (7.46) to write
j E i
.
(10.120)
ef
( )E
As a result, Eq. (119) yields
d
E 2 i E E
d
2
E
4 Im
d .
(10.121)
dV
0
(The last step was possible due to the property (–) = *(), which was discussed in Sec. 7.2.) 45 See, e.g., MA Eq. (6.5a).
46 The frequency-dependent parameter () should not be confused with the dc low-frequency dielectric constant
(0)/0 that was discussed in Chapter 3.
47 As a reminder, the main properties of these functions are listed in Sec. 2.7 – see, in particular, Fig. 2.22 and Eqs. (2.157)-(2.158).
Chapter 10
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Finally, just as in the last section, we have to average the energy loss rate over random values of the impact parameter b:
d E
d E 2
d E
2
2
d b 2
bdb
2
8 bdb E E
d . (10.122)
x
y
Im
dx
dV
b
dV
b
0
min
min
Due to the (weak) divergence of the functions K 0() and K 1() at 0, we have to cut the resulting integral over b at some b min where our theory loses legitimacy. (On that limit, we are not doing much better than in the past section). Plugging in the calculated expressions (116) and (117) for the field components, swapping the integrals over and b, and using the recurrence relations (2.142), which are valid for all Bessel functions, we finally get:
d E
2
d
2
q Im ( *
b ) K ( *
b ) K ( *
b )
.
(10.123) Radiation
min
1
min
0
min
dx
()
intensity
0
This general result is valid for a linear medium with arbitrary dispersion relations () and ().
(The last function participates in Eq. (123) only via Eq. (115) that defines the parameter .) To get more concrete results, some particular model of the medium should be used. Let us explore the Lorentz-oscillator model that was discussed in Sec. 7.2, in its form (7.33) suitable for the transition to the quantum-mechanical description of atoms:
2
nq'
f
()
j
, with f ;1
. (10.124)
0
j
2
2
0
m
j (
) 2 i
j
j
j
If the damping of the effective atomic oscillators is low, j << j, as it typically is, and the particle’s speed u is much lower than the typical wave’s phase velocity v (and hence than c!), then for most frequencies Eq. (115) gives
2
1
1
2
2
,
(10.125)
2
2
2
u
v () u
i.e. * / u is virtually real. In this case, Eq. (123) may be reduced to Eq. (95) with 1.
u
123
b
.
(10.126)
max
The good news here is that both approaches (the microscopic analysis of Sec. 4 and the macroscopic analysis of this section) give essentially the same result. The same fact may be also perceived as bad news: the treatment of the medium as a continuum does not give any new results here.
The situation somewhat changes at relativistic velocities, at which such treatment provides noticeable corrections (called density effects), in particular reducing the energy loss estimates.
Let me, however, leave these details for special topic courses and focus on a much more important effect described by our formulas. Consider the dependence of the electric field components on the impact parameter b, i.e. on the closest distance between the particle’s trajectory and the field observation point. At b , we can use, in Eqs. (116)-(117), the asymptotic formula (2.158),
1/2
K
( )
e
,
at
,
(10.127)
n
2
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EM: Classical Electrodynamics
to conclude that if 2 > 0, i.e. if is real, the complex amplitudes E of both components Ex and Ey of the electric field decrease with b exponentially . However, let us consider what happens at frequencies where 2() < 0, 48 i.e.
1
1
1
()()
.
(10.128)
2
2
2
0
0
v ()
u
c
(This condition means that the particle’s velocity is larger than the phase velocity of the waves at this particular frequency.) In this case, the parameter () is purely imaginary, so that the functions exp{ b}
in the asymptotes (127) of Eqs. (116)-(117) become just phase factors, and the field component amplitudes fall very slowly:
1
E () E ()
.
(10.129)
x
y
1/ 2
b
This means that the Poynting vector drops as 1/ b, so that its flux through a surface of a round cylinder of radius b, with its axis on the particle trajectory (i.e. the power flow from the particle), does not depend on b at all. This is an electromagnetic wave emission – the famous Cherenkov radiation.49
The direction n of its propagation may be readily found taking into account that at large distances from the particle’s trajectory, the emitted wave has to be locally planar and transverse (nE), so that the so-called Cherenkov angle between the vector n and the particle’s velocity u may be simply found from the ratio of the electric field components – see Fig. 14a:
E
x
tan
.
(10.130)
Ey
Ey
E
(a)
(b)
y
n
v t
0
ut
Ex
u
0
x
Fig. 10.14. (a) The Cherenkov radiation’s propagation angle , and (b) its interpretation.
The ratio on the right-hand side of this relation may be calculated by plugging the asymptotic formula (127) into Eqs. (116) and (117) and calculating their ratio:
48 Strictly speaking, the inequality 2() < 0 does not make sense for a medium with a complex product ()(), and hence complex 2(). However, in a typical medium where particles can propagate over substantial distances, the imaginary part of the product ()() does not vanish only in very limited frequency intervals, much more narrow than the intervals that we are discussing now – please have one more look at Fig. 7.5.
49 This radiation was observed experimentally by Pavel Alekseevich Cherenkov (in older Western texts,
“Čerenkov”) in 1934, with the observations explained by Ilya Mikhailovich Frank and Igor Yevgenyevich Tamm in 1937. Note, however, that the effect had been predicted theoretically as early as 1889 by the same Oliver Heaviside whose name was mentioned in this course so many times – and whose genius I believe is still underappreciated.
Chapter 10
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Essential Graduate Physics
