Side-slipping of a radiating particle
Radiation reaction is revisited, first in a new classical approach, where the physical particle 4-momentum is redefined as the energy-momentum flux across the future light cone and is not parallel to the 4-velocity. Then in a semi-classical approach, it is shown that, when emitting a photon, the par...
Збережено в:
| Дата: | 2001 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
|
| Назва видання: | Вопросы атомной науки и техники |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/79434 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Side-slipping of a radiating particle / X. Artru, G. Bignon, T. Qasmi // Вопросы атомной науки и техники. — 2001. — № 6. — С. 98-102. — Бібліогр.: 7 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-79434 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-794342015-04-02T03:02:01Z Side-slipping of a radiating particle Artru, X. Bignon, G. Qasmi, T. Electrodynamics of high energies in matter and strong fields Radiation reaction is revisited, first in a new classical approach, where the physical particle 4-momentum is redefined as the energy-momentum flux across the future light cone and is not parallel to the 4-velocity. Then in a semi-classical approach, it is shown that, when emitting a photon, the particle "side-slips" transversally to its initial momentum, justifying the non-collinearity between momentum and mean velocity. Side-slipping is finally checked in a pure quantum mechanical treatment of synchrotron radiation. 2001 Article Side-slipping of a radiating particle / X. Artru, G. Bignon, T. Qasmi // Вопросы атомной науки и техники. — 2001. — № 6. — С. 98-102. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 3.50.De 41.60.-m 61.85 http://dspace.nbuv.gov.ua/handle/123456789/79434 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Electrodynamics of high energies in matter and strong fields Electrodynamics of high energies in matter and strong fields |
| spellingShingle |
Electrodynamics of high energies in matter and strong fields Electrodynamics of high energies in matter and strong fields Artru, X. Bignon, G. Qasmi, T. Side-slipping of a radiating particle Вопросы атомной науки и техники |
| description |
Radiation reaction is revisited, first in a new classical approach, where the physical particle 4-momentum is redefined as the energy-momentum flux across the future light cone and is not parallel to the 4-velocity. Then in a semi-classical approach, it is shown that, when emitting a photon, the particle "side-slips" transversally to its initial momentum, justifying the non-collinearity between momentum and mean velocity. Side-slipping is finally checked in a pure quantum mechanical treatment of synchrotron radiation. |
| format |
Article |
| author |
Artru, X. Bignon, G. Qasmi, T. |
| author_facet |
Artru, X. Bignon, G. Qasmi, T. |
| author_sort |
Artru, X. |
| title |
Side-slipping of a radiating particle |
| title_short |
Side-slipping of a radiating particle |
| title_full |
Side-slipping of a radiating particle |
| title_fullStr |
Side-slipping of a radiating particle |
| title_full_unstemmed |
Side-slipping of a radiating particle |
| title_sort |
side-slipping of a radiating particle |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Electrodynamics of high energies in matter and strong fields |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79434 |
| citation_txt |
Side-slipping of a radiating particle / X. Artru, G. Bignon, T. Qasmi // Вопросы атомной науки и техники. — 2001. — № 6. — С. 98-102. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT artrux sideslippingofaradiatingparticle AT bignong sideslippingofaradiatingparticle AT qasmit sideslippingofaradiatingparticle |
| first_indexed |
2025-07-06T03:29:03Z |
| last_indexed |
2025-07-06T03:29:03Z |
| _version_ |
1836866646688399360 |
| fulltext |
E L E C T R O D Y N A M I C S O F H I G H E N E R G I E S I N M A T T E R A N D S T R O N G F I E L D S
SIDE-SLIPPING OF A RADIATING PARTICLE
X. Artru a, G. Bignon b, T. Qasmi
aInstitut de Physique Nucléaire de Lyon, IN2P3-CNRS Université Claude-Bernard, France
e-mail: x.artru@ipnl.in2p3.fr
bÉcole Normale Supérieure de Lyon, France
Radiation reaction is revisited, first in a new classical approach, where the physical particle 4-momentum is
redefined as the energy-momentum flux across the future light cone and is not parallel to the 4-velocity.
Then in a semi-classical approach, it is shown that, when emitting a photon, the particle "side-slips" transversally
to its initial momentum, justifying the non-collinearity between momentum and mean velocity. Side-slipping is
finally checked in a pure quantum mechanical treatment of synchrotron radiation.
PACS: 3.50.De 41.60.-m 61.85
1. RECALL ABOUT RADIATION
An electron submitted to an external field (Ein, Bin)
in vacuum emits radiation with the power:
2
3
2
=
dT
d
dt
dW
cl
rad Xα . (1)
The relativistic 4-vector generalization is
( ) µ
µ
α
τ
XXX
d
dP
cl
rad ⋅=
3
2 . (1')
We take unified dimensions for space and time (c =1)
with the( – + + + ) metric. τdd XX ≡ , where τ is the
proper time. We also use rationalized Maxwell
equations, e.g. ·E=ρ. We keep ħ ≠ 1 and define
αcl ≡ e2/(4π)=ħ/137, where e = –|e| is the charge of the
electron. To account for the loss of the electron energy,
Abraham and Lorentz introduced the dissipative force
Xf clreac α
3
2= . (2)
The non-relativistic equation of motion is then
( ) reacininem fBXEX +×+=
whose relativistic generalization is the Abraham-
Lorentz-Dirac (ALD)
equation:
( ) ( )[ ]XXXXXXFeXm clin
−⋅−⋅= α
3
2 (3)
We use the notation ( ) ν
µ νµ XFXF ≡ . Fin={Ein, Bin}
is the "incoming" (or "external") electromagnetic field,
related to the total, retarded, advanced and outgoing
fields by
outadvretintot FFFFF +=+= , (4)
advretinoutrad FFFFF −=−= , (4')
In the following we shall omit the suffix in. An excellent
review on radiation reaction can be found in Ref. [1].
The "mad electron"
Although mathematically elegant, the ALD equation
is not physically acceptable for the following reasons:
* a third initial condition ( )0X is needed in addition
to ( )0X and ( )0X .
* for almost every ( )0X , the electron eventually
goes into a run-away motion.
* given ( )0X and ( )0X , there may exist one (or a
discrete set of) ( )0X such that the electron avoids run-
away motion, but this value depends on all the fields
Fin(X) that the electron will encounter in the future.
Saying that "nature precisely chooses this ( )0X " con-
stitutes a violation of the causality principle. One may
compare this situation with the following one: In a bus,
a passenger puts a stick vertical on the floor and wants it
to remain standing up in equilibrium during the whole
journey, and also after the bus has stopped. To counter-
act the accelerations of the bus, he or she must give
some initial angular velocity to the stick (Fig. 1). To do
so, the passenger must know exactly in advance the
accelerations of the vehicle during the whole journey.
Fig. 1. Stick standing in equilibrum in a truck
The run-away instability is probably related to the
point-like limit of the classical electron considered by
Lorentz: For a sufficiently small radius, the electrostatic
self-energy is larger than the physical mass. Then the
electron "core" has a negative mass and "likes" to
accelerate, since that lowers its kinetic energy.
It is possible to find approximations of the ALD
equation, valid to first order in αcl, which remove the
arbitrariness of ( )0X and have no run-away
solutions.One of them [2] is obtained by replacing X
and X in the right-hand side of (3) by their values
calculated without radiation reaction, ( ) XFmeX → ,
( ) ( ) XFFmeXFmeX 2+→ ,
with ( )XFXF λ
λ ∂= . One obtains
98 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 98-102.
( ) XFreXXFXFFXFeXm clTh
3
22 +
−+= σ , (5)
where rcl=e2/(4πm) is the classical electron radius and
σTh = (8π/3) rcl
2 – the Thomson cross section. The
second term of (5) can be interpreted as the radiation
pressure of the incoming field.
2. REFORMULATION OF THE ALD
EQUATION AND NEW APPROXIMATIONS
Usually, one identifies Xm with the physical 4-
momentum of the particle. Then Eqs. (1') and (3) do not
seem to conserve the total 4-momentum
instantaneously, because of the Schott term ( ) Xcl
32α .
Redefining the 4-momentum as
µµµ α XXmP cl
3
2−= , (6)
the ALD equation can be replaced by the following
system:
XPXm cl α
3
2+= (7a)
( ) XXXXFeP cl ⋅−= α
3
2 . (7b)
Eqs.(1') and (7b) make the instantaneous conservation
of the total 4-momentum manifest. On the other hand,
the mass is not conserved: P · P ≠ – m2 and P is not
collinear to the electron velocity. These two features are
not physically damning. (6) can be approximated by
( ) ( )32 erXmP −≅ ττ , telling that the electro-magnetic
part of Pμ follows the variations of the core velocity
with some delay.
In what follows, we shall show that (6) is a quite
natural definition of Pμ. As usual, we separate Pμ in core
and electromagnetic contributions:
PXmP c δ+= (8)
with
µ ν
ν
µδ ΘΣ= ∫Σ dP (9)
where µ νΘ is the energy-momentum flux tensor of the
electron field. The latter field is not uniquely defined: it
Fig. 2. World line and future light cone of the particle
can be the retarded one, the advanced one or any linear
combination of the two. Looking at the first
decomposition of Eq. (4), we choose the retarded field.
So we consider that the incoming field does not
contribute to the self 4-momentum and only exerts a
force on the core according to the first term of (3). As
hyper-surface Σ we a priori choose the future light cone
of the electron (Fig. 2). This avoids a contribution from
the radiated field Frad, the 4-momentum of which flows
parallel to the cone and does not cross it.
Let us consider a point Xμ (τ0) of the electron world
line. For a space-time point Yμ of its future light cone,
we define
( ) ( ) ( )rr ˆ,, 10 rrRXY ==≡− τ .
Fig. 3. Truncated lightcone or “flower-pot”
The integrand of (9) can be evaluated most easily in the
electron rest frame, using standard formulas for Eret, Bret,
Θμν and
( ) ( )rr ˆ, −=Σ 13dd ν . (10)
We only give the result:
( )∫ == rr ˆ,1
8 4
3
r
dP cl
π
αδ
( )[ ]∫
−⋅= RRX
r
dcl 4
0
3
8
τ
π
α r . (11)
The second expression is Lorentz invariant and applies
as well in frames where the electron is not at rest. Note
that the acceleration does not enter this formula. It
confirms that there is no contribution of the radiated
field.
The integral diverges at r = 0, recalling that the
classical self-energy of a point-like charge is infinite. In
the following, we will assume that the electron has
somevery small but finite extension rc. To treat the
divergence, we truncate the light cone by a hyperplane
orthogonal to the electron world line at X(τ) where
τ – τ0 = ρ is a small distance, but larger than rc (Fig. 3).
We now take the rest frame of the electron at X(τ)
(not τ0). We close the cone, truncated at r ρ, by the
piece of hyperplane R0 = ρ, |R| ≤ ρ and integrate (9) on
the new hypersurface (in grey on Fig. 3) which we call a
"flower-pot". To first order in ρ,
( ) ( )X ρτ −1 ,=0X ,
( )[ ] [ ]r̂X⋅−=⋅ −−
0
ρτ 41X 44 rr . (12)
The truncated integral of (11) is
99
( ) =
−=> τ
ρ
αδ ρ X,
3
2
2
1
clrP
( ) ( )τατ
ρ
α XX cl
cl
3
2
2
−= , (13)
the second expression being frame-independent. The
hyperplane piece (bottom of the flower-pot) is the
interior the sphere of radius ρ at fixed time. It is
approximately centered at X(τ), the displacement being
of second order in ρ. Its contribution to (9) is
( ) ( )τδδδ ρρρ XmmP rrr
<<< == 0, . (14)
The total 4-momentum at proper time τ (not τ0) is
obtained from (8), (13) and (14). We recover the new
definition
( ) ( ) ( )ταττ XXmP cl
3
2−= (15)
where
ρ
αδ ρ 2
cl
rc mmm ++= <
is the renormalized mass of the electron. The third term
is the Coulomb self-energy at r ≥ ρ whereas the detailed
short range structure of the electron is summarized in
the sum of the first two terms.
Let us make the energy-momentum balance in the
space-time region between two successive "flower-pots"
at proper times τ and τ + dτ:
* P(τ) is coming through the first flower-pot,
* τdXFedP inin
= is brought to the core by Fin,
* ( ) τα dXXXdP clrad
⋅=
3
2 is radiated at infinity
between the two flower-pots,
* ( ) ( ) ττττ dPPdP +=+
is outgoing through the second flower-pot.
Adding the first two quantities and subtracting the last
two ones must give zero. This gives (7b).
The above calculations constitute a new and
relatively simple derivation of the ALD equation,
written in the form (7). From this form on can derive
new types of approximations [3-5], also valid to first
order in αcl. The simplest one to implement in a
computer code is obtained replacing X in the right-
hand sides by (e/m2) F P. One may in addition replace
the last X by P/m. Compared to (5), these
approximations have the advantage of not involving the
field derivatives.
3. SEMI-CLASSICAL APPROACH
Eq. (6) tells that the momentum does not follow the
velocity, but one may see things the other way around
and say that the electron does not follows the direction
of its momentum. We call this phenomenon side-
slipping, by analogy with a skier whose track is not
always tangential to the skis (Fig. 4)
Fig. 4. Side-slipping skier
A discrete side-slipping is naturally obtained in a se-
mi-classical description of the process e
–→e'
– +photon
in an external field. If we consider this process as
instantaneous and local at a definite point X of the
trajectory, it cannot satisfy the conservation of both
momentum,
kPP +′= (16)
and energy
ω ′ (17)
with 22 m+≡ P . However, 4-momentum conser-
vation becomes possible if we assume that the final
electron trajectory starts from a point X' ≠ X aside from
the initial trajectory. In the case of a static electric field,
we replace (17) by
( ) ( ) ω ′′= XX UU (17')
where U(X) is the potential energy. (17') and (17) give
( ) ( )
2
22 θγωδ +
′
≅′=
−
XX UUU (18)
where θ is the angle between P and k. δU is obtained by
a finite displacement δX = X'–X of the electron toward a
lower potential energy. In the ultrarelativistic case, we
take δX perpendicular to the trajectory:
2
22
2
⊥
⊥
− +
′
=
f
fX θγωδ
, (19)
where f is the transverse component of the force. Such
a "side-slipping" was already introduced in channeling
radiation (Eqs. (15)-(17) of Ref. [6]). It contributes to
the decrease of the transverse energy which explains the
very fast energy loss of axially channeled electrons
above hundred GeV.
Let us now consider synchrotron radiation in a
Fig. 5. Photon emission in a synchrotron
uniform magnetic field ẑB B−= derived from the
vector potential
( )00,,A yB= . (20)
The particle hamiltonian is (P2
+ m2)1/2 where
P = p – eA is the mechanical momentum and p the
canonical one. In the gauge (20), the hamiltonian is
invariant under translation in the x and z directions,
therefore px and pz are conserved. We assume that the
photon is emitted when the electron is at x = 0, y = R
(Fig. 5). Then we require the conservation laws (16-17),
100
but with p and p' in place of P and P'. For the x-
component it writes
xxx kByePeyBP +′+′=+ , (21)
where have anticipated a side-slipping y' =y + δy. For |e|
B δy we obtain the same result (18) as for δU and, since |
e|B|f|, δy is given by (19) or
2
22 θγωδ +
′
=
−
Ry
. (22)
The side-slipping has also the virtue of insuring
angular momentum conservation. Let us consider again
the circular trajectory of Fig. 5, but now due to the
spherically symmetric potential U(|X|). Neglecting spin,
the z-component of the angular momenta of the initial
and final electrons are
xzxz PyLPyL ′′−=′−= , . (23)
Here we neglect the quantum recoil effect, i.e. we use
the classical or soft photon approximation (ħω – m).
The source of the radiation – and the radiation itself – is
invariant under a time translation by t times a rotation
by the angle vt/R. For a photon quantum state of
definite angular momentum Jz and frequency ω, this
invariance is expressed as
( )[ ] ( ) 1expexp =∆−×∆− tiJRtvi z ω
therefore
vRJ z ω−= . (24)
The conservation of angular momentum,
zzz JLL +′= . (25)
together with that of linear momentum along x̂ ,
xxx kPP +′= (26)
yield the result (22) again, with '.
Incidentally, identifying (24) with the "classical
photon" result Jz = –ykx implies a "side-slipping" for the
photon also:
RRy z
phot 2
22 θγ +=−
−
, (27)
which could be observed at low-energy synchrotron
machines.
The side-slipping formula (19) can be generalized in
a covariant form, writing the 4-momentum conservation
as
KPQP +′=+ (28)
We assume that Q is provided by the work of the
external field along δX:
XFeQ δ= . (29)
Squaring the two sides of (28), using P2
= P'2 = – m2, K2=0
and neglecting Q2 a priori, we obtain
PKQP ′⋅=⋅
⋅
′
≅ PK
. (30)
Fig. 6. Semi-classical electron trajectory emitting
photons successively
Using PFPPFP ⋅−=⋅ and, to first order in αcl,
XmP = , PXF = , one can verify that
µµδ P
PP
PK
m
X
⋅
′⋅−= (31)
inserted in (29), satisfies (30). The neglect of Q2 has
to be checked a posteriori from (29). We expect it to be
small if the external field varies smooththly, e.g. in
synchrotron or channeling radiations, interpreting Q as
the momentum of the virtual photon(s) taken from the
external field.
In the limit ħ→0, the 4-momenta of the individual
photons goes to zero and their number goes to infinity
so that the total radiated 4-momentum is finite and
given by (1'). Summing all the small side-slippings (31)
during the proper time dτ, approximating K·P' by K·P,
one recovers Eq. (7a), to first order in αcl. This is
illustrated in Fig. 6.
4. FULL QUANTUM DERIVATION
Side-slipping was deduced above from semi-
classical arguments of energy, momentum and angular
momentum conservation. Here we will derive side-
slipping from a full quantum treatment, in the particular
case of synchrotron radiation. Neglecting electron spin,
we start from the Klein-Gordon equation (now ħ =1),
( ) 0][ 222 =Ψ−∂−−∇ mie tA (32)
and consider a wave packet of the form
ψtiipxe −=Ψ (33)
where Pμ = (, p,0,0) is a reference 4-momentum and ψ a
slowly varying function of (t, x, y, z). Using 2 = p2 + m2,
(32) becomes
( )[
( ) ] 02 22
22
=−∂−+
+∂+∂+∂−+∂∂−∂
ψyfipfy
pEii
x
zyx
,
(34)
where ∂± = ∂t ± ∂x and f = |e| B. Assuming p m we
consider ∂+ to be of order
–1, which allows us to neglect
the second and third terms of the square bracket.
Furthermore, we take a wave packet located near
(x,y,z) = 0 at time t = 0 (we change the origin of the
coordinates in Fig. 5). So we neglect the terms in y2 and
y ∂x (but not in yp). We get
( ) ( ) 0]
2
1[ 22 =−∂+∂+∂+∂ ψfyi zyxt
. (35)
Looking for solution of the form
( ) ( ) ( )zyttxzyxt ,,,,, φχψ −= , (36)
we are left with the 2-dimensional Schrödinger equation
for a particle of mass = γm in the linear potential
V(y) = by:
( ) 0]
2
1[ 22 =−∂+∂+∂ φfyi zyt
. (37)
Using the coordinate of the accelerated frame
2
2
tfyya
+= (38)
and setting
101
( ) ( )
−−=
6
exp,,,,
32tbiiftyzytzyt aaφφ (39)
we transform (37) in the free-particle Schrödinger
equation,
( ) ( ) 0,,
2
1 22 =
∂+∂+∂ zyti aazyt a
φ
. (40)
Thus a can be expanded in plane waves:
( )
+−+= ∫ ∫ trqiirziqyq,rdrdq
aaa 2
exp
22
22
φ
ππ
φ
~
(41)
To sum up,
( )
( )
+−++×
×
−−−=Ψ
∫ ∫
−
trqrzqftqyiq,rdrdq
E
tbiiftytxe
a
tiipx
22
exp
22
6
exp
222
32
φ
ππ
χ
~
.
(42)
We consider the transition from the electron state
Ψ=i (in the Schrödinger representation) to the
electron + photon state ak ,f ⊗Ψ ′= where Ψ' is
given by (42) with primed quantities. The wave packets
Ψ and Ψ' are represented by the striated ellipses of
Fig. 5. Taking (without loss of generality) k along the x
axis, the photon vector potential is given by
( ) ( )txiket, −= aXA . (43)
To first order in perturbation, the transition amplitude is
iHfdtiiSf I∫−= (44)
with the interaction hamiltonian HI given by
( ) ( ) ( )∫ Ψ∇⋅Ψ ′= XXAXX ** t,t,t,dieiHf I
3 .
(45)
We now combine Eqs.(42-45). Integrations over y and z
impose q = q' and r = r'. Using the shifted variable x'=x–
t, the integration over the x-dependent factors gives
( ) ( ) ( ) ( )∫ ′′′′ −+′−+′ xxexde tpkpitpkpi *χχ . (46)
We introduce the parameter
( )
′
≅+′′Λ
2
kmpp 2 (47)
and write the remaining 3-fold integral as
( ) ( ) ( )
( )
+−++Λ×
×⋅+⋅′= ∫ ∫ ∫
3
exp
3
22222 tftfqtrqmi
q,rrqq,rdrdqdtI azya φφ
~aa~ ***
, (48)
Shifting the time variable t' = t–q / f, we can decouple the
exponential into
( )
′
+′+Λ
3
exp
3
222 tftrmi (49)
and
( )
++Λ
3
exp
3
22 qqrm
f
i . (50)
The phase factor of (49) is the same as in the semi-
classical radiation formula,
( )[ ]
′⋅−′
′
tti Xkω
exp , (51)
knowing that the transverse components of the velocity
dX /dt' are vy(t') = – ft'/, vz = r/. The factor /' is a
recoil correction.
The factor which interests us is (50). Linearizing the
cubic term about the mean value <q> = vy and
replacing r by <r> = vz, we can rewrite (50) as
( )yiqC δ−⋅ exp (52)
where δy is equal to the right-hand side of (19) or (22).
In (52) we recognize the operator of the y-translation by
δy, written in the momentum space representation. The
maximum transition amplitude is obtained when the
wave packet aφ
~′ is transversaly shifted from aφ
~
by δy.
This confirms the semi-classical derivation of the side-
slipping.
5. CONCLUSION
In this study, we have got new insight in the
radiation mechanism. Using purely classical, semi-
classical and quantum-mechanical approaches, we have
shown that the velocity and the properly defined
momentum of the radiating particle are not parallel, as
illustrated in Fig. 6.
The classical run-away problem still remains
unsolved, but we have obtained a new approximation of
the ALD equation, without run-away and not involving
the field derivatives. It can be easily implemented on a
computer code.
The discrete side-slipping accompanying the
emission of a photon is of the order of the comptom
wavelength, hence hardly detectable. However its
contribution to the decrease of the transverse energy of
a high-energy electron channeled in crystals may be
non-negligible. The "side-slipping of the photon" (27),
much larger than the electron one, may be observed
with precise optics.
The transverse jumping of the particle from the
initial to the final trajectory has no classical
counterpart. It can be viewed as a tunnel effect. A
similar effect should take place in the crossed reaction
γ → e++e – in a strong field (Eq. 2 of Ref. [7]).
ACKNOWLEDGMENTS
Part of this work was supported by the INTAS
contract 97-30392: "Theoretical Investigation of
Propagation of Particles, Ions and X-Rays through
Straight and Bent Nanotubes and Associated
Phenomena". Two of us (G.B. and T.Q.) took part in
this work during training periods at Institut de Physique
Nucléaire de Lyon.
REFERENCES
1. K.T. McDonald. Limits of the applicability of
classical electromagnetic fields as inferred from the
radiation reaction // ArXiv:physics/0003062, 2000.
2. L.D. Landau and E.M. Lifshitz. Course of
Theoretical Physics, v. 2, Classical Theory of Fields,
Pergamon, 1975.
102
3. T. Qasmi. Rayonnement de canalisation dans les
nanotubes (Training work report, unpublished,
1998).
4. G. Bignon. Force de réaction au rayonnement
(Training work report, unpublished, 2001).
5. X. Artru, G. Bignon. A semi-classical approach to
the radiation damping force (NATO Advanced
Research Workshop on Electron-Photon Interaction
in Dense Media, Nor Hamberd, Armenia, 25-29
June 2001) - submitted to NATO Science series 2.
6. X. Artru. Self-amplification of channeling radiation
of ultrarelativistic electrons due to loss of transverse
energy // Phys. Lett. 1988, v. A128, p. 302-306.
7. X.Artru et al. Observation of channeling and
blocking effect in pair creation in a Ge crystal //
Phys. Lett. 1993, v. B313, p. 483-490.
103
SIDE-SLIPPING OF A RADIATING PARTICLE
e-mail: x.artru@ipnl.in2p3.fr
The "mad electron"
|