Physics Letters A 314 (2003) 272–277
www.elsevier.com/locate/pla
The radiation reaction force on an electron reexamined
José A. Heras
Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Apartado Postal 21-081,
04021 México Distrito Federal, Mexico
Received 16 December 2002; accepted 21 May 2003
Communicated by V.M. Agranovich
Abstract
According to classical electrodynamics an accelerated electron emits radiation not only from its electric charge but also from
its magnetic moment. Therefore, the radiation reaction theory of the electron should also consider the radiation emitted by its
magnetic moment. In this Letter a radiation reaction force acting on a suddenly accelerated, non-relativistic point electron is
derived from energy conservation considering both the radiation from its electric charge and the radiation from its magnetic
moment. The associated equation of motion is a third-order differential equation for the acceleration which predicts nonrunaway
accelerations provided suitable initial conditions are imposed. It is pointed out that the self-accelerations of the electron are
predominantly caused by the radiation from its magnetic moment.
2003 Elsevier B.V. All rights reserved.
PACS: 03.20.De; 03.50.De; 41.20.Bt; 41.60.Ap
The classical radiation reaction theory of accel-
erated point charges continues being discussed [1],
mainly because it predicts unphysical runaway solu-
tions. However, it has been recently pointed out that
the theory predicts nonrunaway solutions when it is
applied to accelerated point dipoles [2–4]. Taking into
account the fact that an electron possesses both an
electric charge and a magnetic moment, the question
naturally arises: can the radiation from the magnetic
moment of the electron eliminate the runaway solu-
tions?
At first sight, the electron relation
e/µ
≈
5
×
10
10
cm seems to imply that the radiation from the
magnetic moment of the electron
µ
would be so small
E-mail address: heras@toluca.podernet.com.mx (J.A. Heras).
that its reactive effects would be unimportant com-
pared to those caused by its electric charge
e
. How-
ever, the radiation fields of the magnetic moment de-
pend also on the time derivative of the acceleration
˙
a.
For sudden processes
|
a
|
/
|˙
a
| ∼
T
1 s, where
T
is
a time scale over which the acceleration appreciably
changes. This result can compensate the significant
difference between the values of
e
and
µ
. In this Letter
a radiation reaction force acting on a suddenly accel-
erated, non-relativistic electron considering the radia-
tions from its magnetic moment and its electric charge
is derived from energy conservation, emphasizing that
the equation of motion predicts nonrunaway acceler-
ations which are essentially caused by the radiation
from its magnetic moment.
The radiation reaction force on a suddenly acceler-
ated, non-relativistic particle with a magnetic moment
0375-9601/$ – see front matter
2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0375-9601(03)00907-1
J.A. Heras / Physics Letters A 314 (2003) 272–277
273
µ
will be first derived. The radiation electric field is
[4]:
(1)
E
rad
=
n
×
µ
(
n
· ˙
a
)
Rc
3
ret
.
Notation of Jackson’s book [5] is used. The associated
Poynting vector S
µ
=
(c/
4
π )
|
E
rad
|
2
n is specifically
given by
(2)
S
µ
=
n
(
n
×
µ
)
2
(
n
· ˙
a
)
2
4
π R
2
c
5
,
which is used into the instantaneous radiated power
per unit solid angle:
dP
µ
/dΩ
=
(
S
µ
·
n
)R
2
to obtain,
after an integration over all solid angle, the total
instantaneous radiated power
(3)
P
µ
=
4
µ
2
15
c
5
˙
a
2
−
2
15
c
5
(
µ
· ˙
a
)
2
.
Via the usual argument based on energy conservation
(see Ref. [5]), Eq. (3) implies the radiation reaction
force
(4)
F
µ
= −
4
µ
2
15
c
5
...
a
+
2
15
c
5
(
µ
·
...
a
)
µ
.
In the case in which
µ
and
...
a are perpendicular, Eq.
(4) reduces to
(5)
F
µ
= −
4
µ
2
15
c
5
...
a
,
which originates the following equation of motion:
(6)
a
+
τ
3
µ
...
a
=
0
,
where
τ
µ
is the characteristic time defined by
(7)
τ
µ
=
4
µ
2
15
mc
5
1
/
3
.
The general solution of Eq. (6) is a linear combination
of runaway and nonrunaway solutions. However, by
assuming the initial conditions: a
(
0
)
=
a
0
,
˙
a
(
0
)
=
−
a
0
/τ
µ
and
¨
a
(
0
)
=
a
0
/τ
2
µ
the solution of Eq. (6) is
nonrunaway
(8)
a
(t)
=
a
0
e
−
t /τ
µ
.
The existence of runaway solutions in classical elec-
trodynamics is usually connected with the point par-
ticle approximation [6]. For extended charged parti-
cles the theory is free of runaway solutions whenever
the size of the particles exceeds its classical radius [7].
However, when obtaining the nonrunaway solution in
Eq. (8) the point particle approximation was used. This
means that the existence of runaway solutions lies on
the structure of the equation of motion rather than on
the point particle approximation. For linear equations
involving derivatives of acceleration of order greater
than two, the runaway solutions can be eliminated by
an appropriate choice of initial conditions.
For an electron (
µ
=
e
¯
h/(
2
m
e
c)
=
9
.
27
×
10
−
21
stc cm) the characteristic time
τ
µ
in Eq. (7) is given
by
(9)
τ
µ
=
1
.
01
×
10
−
22
s
,
which can be approximated by
τ
µ
≈
0
.
08
¯
λ
e
/c
, where
¯
λ
e
is the reduced Compton wavelength of the electron.
This result seems to indicate that Eq. (8) is not
suitable for electrons. The argument is that quantum
effects enter at scale times
¯
λ
e
/c
and thus classical
electrodynamics seems to appropriate only for scale
times
>
¯
λ
e
/c
. However, two subtle points should be
stressed:
(i) According to Eq. (8) the acceleration is not en-
tirely completed at the time
τ
µ
but at
t
= ∞
, that
is, lim
t
→∞
a
(t)
=
0. The time
τ
µ
is defined as the
characteristic time during which the acceleration
diminishes to 37% of its initial value. Although
with less and less intensity, the acceleration con-
tinues acting for times
> τ
µ
. For effective times in
the classical domain (
τ
eff
>
¯
λ
e
/c
) the value of the
acceleration is
<
0
.
0003% of its initial value.
(ii) The derivation of Eq. (8) requires the specification
of a
(
0
)
,
˙
a
(
0
)
and
¨
a
(
0
)
in terms of a
0
and
τ
µ
. But
how to obtain a
0
? The initial data are not part
of the theory; they should be provided externally.
In other words: the time
τ
µ
is meaningful only
when the initial data are specified. Otherwise,
τ
µ
is meaningless. In practice the initial data may
be provided by an external force acting on the
electron at least during a finite time.
Having in mind the above considerations, consider
now that an electron is urged by a constant external
force F during
t
0. In this case the equation of
motion is
(10)
a
+
τ
3
µ
...
a
=
F
m
e
.
274
J.A. Heras / Physics Letters A 314 (2003) 272–277
Under the initial conditions: a
(
0
)
=
0,
˙
a
(
0
)
=
F
/(m
e
τ
µ
)
and
¨
a
(
0
)
= −
F
/(m
e
τ
2
µ
)
, the solution of
Eq. (10) is
(11)
a
(t)
=
F
m
e
1
−
e
−
t /τ
µ
.
This is integrated with v
(
0
)
=
0 to yield the velocity
(12)
v
(t)
=
F
m
e
t
−
τ
µ
+
τ
µ
e
−
t /τ
µ
.
For the effective time
τ
eff
=
kτ
µ
the velocity takes
the value v
(τ
eff
)
=
F
τ
µ
(k
−
1
+
e
−
k
)/m
e
. If, for
instance,
k
=
100 then v
≈
99F
τ
µ
/m
e
.
In the absence
of the radiation reaction force, the velocity is v
=
100F
τ
µ
/m
e
. Therefore, for this specific
k
the effect of
the radiation reaction force results in the reduction of
1% of the velocity caused by the external force (for
k
=
13 the reduction is of 7
.
7%). The point is that
the effective time
τ
eff
=
100
τ
µ
≈
10
−
20
s
≈
7
.
8
¯
λ
e
/c
is
in the classical domain. In other words: for electrons
the time
τ
µ
should not necessarily be in the classical
domain since the associated acceleration is not entirely
completed in such a time. The relevant time is
τ
eff
=
kτ
µ
which should be in the classical domain:
τ
eff
>
¯
λ
e
/c
. This implies that
k >
13. Such a restriction
imposes limits on the applicability of Eq. (12), namely,
this equation is “classically” plausible only for times
τ
eff
>
13
τ
µ
. This means that the electron velocity
caused by the external force reduces less than 7.7% on
account of the action of the radiation reaction force.
Actually, Eq. (11) can be written as:
m
e
a
=
F
+
F
res
, where F
res
= −
Fe
−
t /τ
µ
is an equivalent radiation-
resistance force, which is derived from the radiation
reaction force F
µ
= −
m
e
τ
3
...
a and Eq. (11). For times
τ
eff
>
13
τ
µ
it follows that F
res
<
Fe
−
13
and so for
an electron the radiation-resistance force F
res
is small
con respect to the external force F. A similar conclu-
sion was given by McDonald [8] in the context of the
Abraham–Lorentz equation. Classical electrodynam-
ics is then appropriate for describing the reactive ef-
fects of the magnetic moment of the electron whenever
a constant external force acting on the electron during
times
τ
eff
>
¯
λ
e
/c
is introduced. If an external force
could be applied on the electron during times
¯
λ
e
/c
then the classical theory given here is no longer ap-
plicable.
Similar results can be given for the case of a time-
dependent external force. For instance, an electron
inside the electric field E
(t)
=
E
0
e
−
t /T
,
where
T
is
the associated characteristic time satisfying
T >
¯
λ
e
/c
,
is urged by the force F
(t)
=
e
E
0
e
−
t /T
and so the
equation of motion is
(13)
a
+
τ
3
µ
...
a
=
e
E
0
m
e
e
−
t /T
.
Under the initial conditions: a
(
0
)
=
0,
˙
a
(
0
)
=
e
E
0
×
T
2
/σ
,
¨
a
(
0
)
= −
e
E
0
(T
2
+
T τ
µ
)/(σ τ
µ
)
, where
σ
=
m
e
τ
µ
(T
2
+
T τ
µ
+
τ
2
µ
)
, the solution of Eq. (13) is given
by
(14)
a
(t)
=
e
E
0
T
3
m
e
(T
3
−
τ
3
µ
)
e
−
t /T
−
e
−
t /τ
µ
.
This is integrated with v
(
0
)
=
0 to give the velocity
(15)
v
(t)
=
e
E
0
T
3
[
T (
1
−
e
−
t /T
)
−
τ
µ
(
1
−
e
−
t /τ
µ
)
]
m
e
(T
3
−
τ
3
µ
)
,
which reduces to
(16)
v
≈
e
E
0
m
e
(T
−
τ
µ
),
for
t
T
,
t
τ
µ
and when
T
is significantly
greater than
τ
µ
so that
T
3
τ
3
µ
.
In general, for
sudden external forces the difference
T
−
τ
µ
is a
relative measure of the effect of the radiation reaction
force on the velocity. The key to test this effect
deals with the possibility of applying an external
force with a characteristic time
T
sufficiently small
(whenever
T >
¯
λ
e
/c
) to achieve that the difference
T
−
τ
µ
is appreciable. Such external forces could
be produced by ultrashort electromagnetic pulses.
If an electromagnetic pulse lasting on the scale of
attoseconds
(
10
−
18
s) could be applied to the electron
then a small reduction of 0
.
01% in the velocity may
be detected. For electromagnetic pulses on the scale of
zeptoseconds
(
10
−
21
s) the reduction of the velocity
will reach the “classical” limit of 7.7%. Pulses at
the scale of attoseconds are now possible [9] via
ultra-intense lasers and optical scientists are now
striving to produce, via the lasetron, pulses at the scale
of zeptoseconds [10]. The damping of the electron
motion in a ultra-intense laser field has theoretically
been investigated by Keitel et al. [11] in the context
of the Lorentz–Dirac equation. On the other hand,
it is pertinent to note that Smirnov [12] has made a
quantum mechanical derivation of a force similar to
Eq. (5) emphasizing that such a force can act on the
J.A. Heras / Physics Letters A 314 (2003) 272–277
275
motion of a neutron in an atomic nucleus resulting in
the radiation broadening of excited energy levels.
Consider now a suddenly accelerated, non-relativ-
istic particle having both an electric charge and a
magnetic moment. The electric radiation field is given
by
(17)
E
rad
=
n
×
e(
n
×
a
)
Rc
2
+
µ
(
n
· ˙
a
)
Rc
3
ret
,
from which the associated Poynting vector is obtained
S
eµ
=
n
e
2
(
n
×
a
)
2
4
π R
2
c
3
+
n
(
n
×
µ
)
2
(
n
· ˙
a
)
2
4
π R
2
c
5
(18)
+
n
e
{
n
·
(
a
×
µ
)
}
(
n
· ˙
a
)
2
π R
2
c
4
.
This yields the total instantaneous radiated power
P
eµ
=
2
e
2
3
c
3
a
2
−
2
e
3
c
4
µ
·
(
a
× ˙
a
)
(19)
+
4
µ
2
15
c
5
˙
a
2
−
2
15
c
5
(
µ
· ˙
a
)
2
.
If a and
˙
a are collinear and
µ
and
˙
a are perpendicular
then Eq. (19) reduces to
(20)
P
eµ
=
2
e
2
3
c
3
a
2
+
4
µ
2
15
c
5
˙
a
2
.
Via the usual argument based on energy conservation,
this power implies the radiation reaction force acting
on a particle with an electric charge and a magnetic
moment
(21)
F
eµ
=
2
e
2
3
c
3
˙
a
−
4
µ
2
15
c
5
...
a
.
The associated equation of motion is given by
(22)
a
−
τ
e
˙
a
+
τ
3
µ
...
a
=
0
.
The general solution of Eq. (22) is a linear combi-
nation of three complex solutions exhibiting runaway
and nonrunaway behaviors. However, under the ini-
tial conditions a
(
0
)
=
a
0
,
˙
a
(
0
)
= −
a
0
/τ
and
¨
a
(
0
)
=
a
0
/τ
2
, where the time
τ
is given by
(23)
τ
=
ξ
2
/
3
+
4
τ
2
e
−
2
τ
e
ξ
1
/
3
6
ξ
1
/
3
,
with
ξ
=
108
τ
3
µ
−
8
τ
2
e
+
12
3
τ
3
µ
(
27
τ
3
µ
−
4
τ
3
e
),
the solution of Eq. (22) is nonrunaway:
(24)
a
(t)
=
a
0
e
−
t /τ
.
Notice that if
τ
e
=
0 then Eq. (22) reduces to Eq. (6)
and
τ
in Eq. (23) reduces to
τ
µ
. Nevertheless, if
τ
µ
=
0 then Eq. (22) reduces to the Abraham–Lorentz
equation but
τ
in Eq. (23) does not reduces to
τ
e
but to
a complex time. For arbitrary values of
τ
µ
and
τ
e
, the
time
τ
in Eq. (23) is complex and consequently Eq.
(24) is a complex solution. However, if either of the
equivalent conditions
(25)
τ
µ
τ
e
(
4
/
27
)
1
/
3
or
µ
e
(
40
/
243
)
1
/
2
r,
where
r
=
e
2
/(mc
2
)
is the classical radius of the
particle, is satisfied then the time
τ
in Eq. (23) is real
and so Eq. (24) becomes real. In particular, an electron
satisfies Eq. (25):
τ
µ
/τ
e
≈
16 or
µ/e
≈
68
.
5
r
e
, where
r
e
is the classical radius of the electron.
The existence of unphysical runaway solutions in
the Abraham–Lorentz equation has recurrently led
to proposals of alternative equations of motion of
a point charge. For example, Ford and O’Connell
[1], Jackson [5], Rohrlich [13] and Spohn [14] have
recently claimed that the equation
m
e
a
=
F
ext
+
τ
e
˙
F
ext
,
which has been attributed to Landau and Lifshitz [15],
is the appropriate equation of motion without runaway
solutions. Nevertheless, Baylis and Huschilt [16] have
pointed out that solutions of this equation are not
generally consistent with Maxwell’s equations. It is
surprising to find that in all of previous proposals
for the “correct” equation of motion, a fundamental
property of the electron producing electromagnetic
radiation, namely, its magnetic moment has been
overlooked. By introducing the term connected with
the magnetic moment (for example, in the form given
in Eq. (22)) and assuming suitable initial conditions,
the runaway accelerations of the point electron are
naturally eliminated.
For an electron the time
τ
in Eq. (23) is given by
(26)
τ
=
0
.
99
×
10
−
22
s
,
which differs about 2% from the time
τ
µ
given in
Eq. (9), that is,
τ
=
0
.
98
τ
µ
. This result indicates
that in sudden processes the self-accelerations of the
electron are predominantly caused by the radiation
from its magnetic moment. Whenever Eq. (25) is
satisfied, Eq. (24) becomes Eq. (8) when
τ
µ
is replaced
276
J.A. Heras / Physics Letters A 314 (2003) 272–277
by
τ
. This remarkable result means that in the absence
of external forces the reactive effects on a suddenly
accelerated particle caused by its electric charge and
its magnetic moment can equally well described by
considering only its magnetic moment and making
τ
µ
→
τ
. A similar conclusion follows for a constant
external force F. The equation of motion is
(27)
a
−
τ
e
˙
a
+
τ
3
µ
...
a
=
F
m
e
.
Under the initial conditions: a
(
0
)
=
0
,
˙
a
(
0
)
=
F
/(m
e
τ )
and
¨
a
(
0
)
= −
F
/(m
e
τ
2
)
, the solution of Eq. (27) is
(28)
a
(t)
=
F
m
e
1
−
e
−
t /τ
.
This is integrated with v
(
0
)
=
0 to give the velocity
(29)
v
(t)
=
F
m
e
t
−
τ
+
τ
e
−
t /τ
.
By making
τ
µ
→
τ
the magnetic equations (11) and
(12) become the electric–magnetic equations (28) and
(29). If for example, F is applied during
t
=
100
τ
µ
then the magnetic equation (11) gives v
≈
99F
τ
µ
/m
e
and the electric–magnetic equation (29) gives v
≈
99
.
02F
τ
µ
/m
e
. These results confirm that the radiation
reaction effects on a suddenly accelerated electron
are essentially due to the magnetic moment. The
theory may be extended to the case of time-dependent
external forces. If for example, the electric field E
(t)
=
E
0
e
−
t /T
is applied to the electron then the equation of
motion is
(30)
a
−
τ
e
˙
a
+
τ
3
µ
...
a
=
e
E
0
m
e
e
−
t /T
.
Under suitable initial conditions the solution is
(31)
a
(t)
=
e
E
0
T
3
m
e
(T
3
+
τ
e
T
2
−
τ
3
µ
)
e
−
t /T
−
e
−
t /τ
.
This is integrated with v
(
0
)
=
0 to give the velocity
(32)
v
(t)
=
e
E
0
T
3
[
T (
1
−
e
−
t /T
)
−
τ (
1
−
e
−
t /τ
)
]
m
e
(T
3
+
τ
e
T
2
−
τ
3
µ
)
.
This becomes v
≈
(e
E
0
/m
e
)(T
−
τ )
for
t
T
,
t
τ
,
T
τ
e
and when
T
is significantly greater than
τ
µ
so that
T
3
τ
3
µ
.
Moreover, if
τ
µ
is significantly
greater than
τ
e
then
τ
≈
τ
µ
and so the discussion
given for Eq. (16), including the limit of applicability
and the possible testing of the equation, is essentially
translated to the case of a point electron with an
electric charge and a magnetic moment.
In conclusion, a non-relativistic equation of motion
for a suddenly-accelerated point electron including the
radiation reaction forces caused by its magnetic mo-
ment and its electric charge was derived and compared
with a similar equation of motion including only the
radiation reaction force caused by its magnetic mo-
ment. A first conclusion was that both equations have
nonrunaway solutions provided suitable initial condi-
tions are imposed. This means that the expected con-
sistency of the classical radiation reaction theory of a
point electron is naturally provided by the inclusion
of the radiation from its magnetic moment. A second
conclusion was certainly surprising: the characteristic
time of the acceleration predicted by the first equa-
tion differs only in 2% of the characteristic time of the
acceleration predicted by the second equation, which
means that in sudden processes the self-accelerations
of the electron are essentially caused by the radiation
from its magnetic moment.
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