Spohn H. Dynamics of Charged Particles and their Radiation Field
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8.5 Larmor’s formula 103
R B(q
ε
(t) + R
ω, t + R)
∼
=
√
ε
d
3
y
e
4π
ϕ
ε
(y − q
ε
)
− (1 −
ω · v
ε
)
−1
(
ω ×˙v
ε
)
−(1 −
ω · v
ε
)
−2
(
ω ·˙v
ε
)(
ω × v
ε
)
t+
ω·(y−q
ε
(t))
=
ω × RE(q
ε
(t) + R
ω, t + R), (8.52)
where we used the property that t + R −|q
ε
(t) + R
ω − y|=t +
ω ·
(y − q
ε
(t)) + O(1/R) for large R. Inserting in (8.49) yields
lim
R→∞
I
R,ε
(t) = I
ε
(t)
=− lim
R→∞
d
2
ω(1 −
ω · v
ε
(t))
R E(q
ε
(t) + R
ω, t + R)
2
(8.53)
=−ε
d
2
ω(1 −
ω · v
ε
(t))
×
e
4π
d
3
yϕ
ε
(y − q
ε
)(1 −
ω · v
ε
)
−2
(
ω ·˙v
ε
)
2
−
e
4π
d
3
yϕ
ε
(y − q
ε
)(1 −
ω · v
ε
)
−1
˙v
ε
+ (1 −
ω · v
ε
)
−2
(
ω ·˙v
ε
)v
ε
2
t+
ω·(y−q
ε
(t))
. (8.54)
I
ε
(t) is the energy radiated per unit time at ε fixed. As argued before, it is indeed
of order ε. From the expression (8.53) it can be seen that I
ε
(t) ≤ 0.
Equation (8.54) is not yet Larmor’s formula. To obtain it we have to go to
the adiabatic limit ε → 0. Then q
ε
(t) → r(t).Since ϕ
ε
(x) → δ(x),wehave
y
∼
=
q
ε
(t)
∼
=
r(t) in (8.54). From the d
3
y volume element we get an additional
factor of (1 −
ω · v
ε
)
−1
. Thus
lim
ε→0
ε
−1
I
ε
(t) = I (t) =−e
2
d
2
ω(1 −
ω · u(t))
4π(1 −
ω · u(t))
−3
2
×
(
ω ·˙u(t))
2
− [(1 −
ω · u(t)) ˙u(t) + (
ω ·˙u(t))u(t)]
2
=−(e
2
/6π)
γ
4
˙u(t)
2
+ γ
6
(u(t) ·˙u(t))
2
=−(e
2
/6π)γ
6
˙u(t)
2
− (u(t) ×˙u(t))
2
, (8.55)
which is the standard textbook formula of Larmor. Note that the same energy loss
per unit time was obtained already in (8.6) using only the energy balance for the
comparison dynamics.
104 Comparison dynamics
Starting from (8.49) one could alternatively first take the limit ε
−1
I
R,ε
(t) →
I
R,0
(t),which is the change of energy in a ball of radius R centered at the particle’s
position r(t) in the adiabatic limit. As before the irreversible energy loss is isolated
through
lim
R→∞
I
R,0
(t) = I (t). (8.56)
The energy loss does not depend on the order of limits, as it should be.
We recall that in Larmor’s treatment the trajectory of the charge, taken as a
point charge, is prescribed. In our case the charged particle is guided by external
fields and interacts with its own Maxwell field, which is physically somewhat more
realistic. Since the charge distribution is extended, by necessity, Larmor’s formula
holds only in the adiabatic approximation.
Notes and references
Section 8
The radiation damped harmonic oscillator is discussed in Jackson (1999) with a
variety of physical applications. The asymptotic condition was first stated in Dirac
(1938). It has been reemphasized by Haag (1955) in analogy to a similar condition
in quantum field theory.
Section 8.1
Singular, or geometric, perturbation theory is a standard tool in the theory of dy-
namical systems. Sakamoto (1990) presents the theory at the level of generality
needed here. We refer to Jones (1995) for a review with many applications. In the
context of synergetics (Haken 1983) one talks of slow and fast variables and the
slaving principle, which means that fast variables are enslaved by the slow ones.
Within our context this would correspond to an attractive critical manifold. The
renormalization group flows in critical phenomena have a structure similar to that
discovered here: the critical surface corresponds to critical couplings which then
flow to some fixed point governing the universal critical behavior. The critical sur-
face is repelling, and slightly away from that surface the trajectory moves towards
either the high- or low-temperature fixed points.
Section 8.2
Particular cases have been studied before, most extensively the one-dimensional
potential step of finite width and with linear interpolation (Haag 1955; Baylis and
Notes and references 105
Huschilt 1976; Carati and Galgani 1993; Carati et al. 1995; Blanco 1995; Ruf and
Srikanth 2000), head-on collision in the two-body problem (Huschilt and Baylis
1976), the motion in a uniform magnetic field (Endres 1993), and motion in an
attractive Coulomb potential (Marino 2003). These authors emphasize that there
can be several solutions to the asymptotic condition. From the point of view of
singular perturbation theory such behavior is generic. If ε is increased, then the
critical manifold is strongly deformed and is no longer given as a graph of a func-
tion. For specified q(0), v(0) there are then several ˙v(0) on
C
ε
, which means that
the solution to the asymptotic condition is not unique. However, these authors fail
to emphasize that the nonuniqueness in the examples occurs only at such high
field strengths that a classical theory has long lost its empirical validity. At mod-
erate field strengths the worked-out examples confirm our findings. The general
applicability of singular perturbation theory was first recognized in Spohn (1998).
Sections 8.3, 8.4, and 8.5
The discussion is adapted from Kunze and Spohn (2000a, 2000b).
9
The Lorentz–Dirac equation
We return to the Lorentz model and add slowly varying external potentials. On a
formal level one can carry out the expansion in ε just as for the Abraham model.
The net result is that the rotational degrees of freedom decouple from the transla-
tional degrees of freedom, and the latter are governed by
m
0
˙
u = (e/c)F · u + (e
2
/6πc
3
)(
¨
u − c
−2
(
˙
u ·
˙
u)u), (9.1)
which includes radiation reaction. Equation (9.1) is the Lorentz–Dirac equation,
written in microscopic units. m
0
is the experimental rest mass of the particle. We
reintroduced the speed of light, c. F is the electromagnetic field tensor of the ex-
ternal fields, where for better readability we omit the subscript “ex” in this sec-
tion. The scaling parameter ε has been reabsorbed into the definition of F, which
amounts to setting ε = 1. It should be kept in mind that the radiation reaction is a
small correction to the Hamiltonian part.
If one fixes an inertial frame of reference and goes over to three-vectors, then
the time component of the Lorentz–Dirac equation reads
d
dt
m
0
c
2
γ(v) + eφ(q) − (e
2
/6πc
3
)γ
4
(v ·˙v)
=−(e
2
/6πc
3
)γ
4
(˙v · κ(v)˙v),
(9.2)
and the space part becomes
m
0
γκ(v)˙v = e(E(q) + c
−1
v × B(q))
+(e
2
/6πc
3
)γ
2
κ(v) [¨v + 3γ
2
c
−2
(v ·˙v) ˙v], (9.3)
where as before κ(v) = 1l + c
−2
γ
2
v ⊗ v. Equation (9.3) differs from its semirel-
ativistic sister (8.1) only through the proper relativistic kinetic energy. Equation
(9.2) is identical to the energy balance (8.6), again with proper adjustment of the
kinetic energy. Thus we can follow the blueprint of section 8.2 to establish the
106
9.1 Critical manifold, the Landau–Lifshitz equation 107
existence of the critical manifold and to derive an effective second-order equation
for the motion on the critical manifold.
The Lorentz–Dirac equation makes definite predictions about the orbit of a
charged particle, including the effects of radiation losses, and one would expect
that these predictions can be verified experimentally. Of course, if radiation damp-
ing is neglected, there is a multitude of laboratory set-ups. The real challenge is to
observe quantitatively the minute changes in the Hamiltonian orbit due to radiation
losses. We will discuss two proposals in section 9.3. The first one is the motion of
an electron in a Penning trap. In the quadratic approximation for the quadrupole
field, this problem can still be handled analytically, which is done in section 9.2
along with a few other examples of independent interest. The second proposal is
the motion of an electron when hit by an ultrastrong laser pulse. In this case the ex-
ternal potentials are time dependent and one has to rely on a numerical integration
of the effective second-order equation.
9.1 Critical manifold, the Landau–Lifshitz equation
We write (9.3) in the standard form of singular perturbation theory; compare with
section 8.2. Then
˙x = f (x, y), ε ˙y = g(x, y,ε) (9.4)
with
f (x, y) = (x
2
, y), (9.5)
g(x, y,ε) = (6π c
3
/e
2
)
m
0
γ
−1
y − e γ
−2
κ(x
2
)
−1
(E(x
1
) + c
−1
x
2
× B(x
1
))
−3εγ
2
c
−2
(x
2
· y)y. (9.6)
To conform with (8.1) we reintroduced the small parameter ε.Atze-
roth order the critical manifold is {(x, h(x))|x ∈
R
3
× V} with h(q, v) =
(e/m
0
)γ
−1
κ(v)
−1
E(q) + c
−1
v × B(q)
. Linearizing (9.5), (9.6) at y = h(x)
the repelling eigenvalue is (6π c
3
/e
2
) m
0
γ
−1
+ O(ε), which vanishes as |v|/c →
1. Thus we have to rely on the construction of section 8.2, which ensures that
for given maximal velocity ¯v one can choose ε small enough such that the orbit
remains on the critical manifold for all times.
To order ε the effective second-order equation is given by (8.31), except that
now m(v) = m
0
γκ(v).Wework out the various terms and switch back to mi-
croscopic units. Then the motion on the critical manifold of the Lorentz–Dirac
108 The Lorentz–Dirac equation
equation is governed by
˙q = v,
m
0
γκ(v)˙v = e(E + c
−1
v × B) +
e
2
6πc
3
e
m
0
γ(v ·∇)(E + c
−1
v × B)
+
e
m
0
2
c
−1
(E × B) + c
−1
(v · E)E + c
−1
(v · B)B
+
− E
2
− B
2
+ c
−2
(v · E)
2
+ c
−2
(v · B)
2
+2c
−1
v · (E × B)
γ
2
c
−1
v
. (9.7)
While singular perturbation theory provides a systematic method, Eq. (9.7)
can also be derived formally. In (9.3) we regard m
0
γκ(v)˙v = e (E + c
−1
v × B)
as the unperturbed equation, differentiate it once, and substitute ¨v inside the
square brackets of (9.3). Resubstituting ˙v from the unperturbed equation results
in Eq. (9.7). This argument is carried out more easily and in greater general-
ity, because it allows for time-dependent potentials, in the covariant form of the
Lorentz–Dirac equation. The unperturbed equation is
m
0
˙
u = (e/c)F(q) · u. (9.8)
One differentiates with respect to the eigentime,
(m
0
c/e) ¨u = (u ·∇
g
)F(q) · u + F(q) ·
˙
u. (9.9)
Substituting (9.9) in (9.1) and resubstituting (9.8) yields
m
0
˙
u = (e/c)F ·u +
e
2
6πc
3
(e/m
0
c)(u ·∇
g
)F · u + (e/m
0
c)
2
F · F · u
−(e/m
0
c
2
)
2
(F · u) · (F · u)u
. (9.10)
In three-vector notation the space part of Eq. (9.10) coincides with (9.7), except
for the additional term (e/m
0
)γ (∂
t
E + c
−1
v × ∂
t
B) because of the time depen-
dence of the fields. As usual, the time component of (9.10) provides the energy
balance.
Equation (9.10) and its formal derivation appeared for the first time in the sec-
ond volume of the Course in Theoretical Physics by Landau and Lifshitz. Hence
it seems to be appropriate to call Eq. (9.10) the Landau–Lifshitz equation. The er-
ror in going from (9.1) to (9.10) is of the same order as that in the derivation of
the Lorentz–Dirac equation itself. Thus we regard the Landau–Lifshitz equation as
9.2 Some applications 109
the effective equation governing the motion of a charged particle in the adiabatic
limit.
9.2 Some applications
(i) Zero magnetic field, one-dimensional motion.Weassume B = 0 and φ
ex
to vary only along the 1-axis. Setting v = (v, 0, 0), q = (x, 0, 0), and E =
(−φ
, 0, 0), the Landau–Lifshitz equation becomes
m
0
γ
3
˙v =−eφ
(x) −
e
2
6πc
3
e
m
0
γφ
(x)v. (9.11)
The radiation reaction is proportional to −φ
(x)v, which can be regarded as
a spatially varying friction coefficient. For a convex potential, φ
> 0, such as
an oscillator potential, this friction coefficient is strictly positive and the result-
ing motion is damped until the minimum of φ is reached. In general, how-
ever, φ
will not have a definite sign, like in the case of the double well
potential φ(x) (x
2
− 1)
2
or the washboard potential φ(x) −cos x.Atlo-
cations where φ
(x)<0 one has antifriction and the mechanical energy in-
creases. This gain is always dominated by losses as can be seen from the energy
balance
d
dt
m
0
γ + eφ +
e
2
6πc
3
e
m
0
γφ
v
=−
e
2
6πc
3
e
m
0
2
φ
2
−
1
m
0
e
2
6πc
3
e
m
0
2
γφ
φ
v. (9.12)
The last term in (9.12) does not have a definite sign. But its prefactor is down by
the factor e
2
/m
0
c
3
and therefore it is outweighed by −φ
2
.
Equation (9.11) has one peculiar feature. If φ(x) =−Ex, E > 0, over some
interval [a
−
, a
+
], then φ
= 0over that interval and the friction term vanishes.
The particle entering at a
−
is uniformly accelerated to the right until it reaches
a
+
. From Larmor’s formula we know that the energy radiated per unit time equals
(e
2
/6πc
3
)(e/m
0
)
2
E
2
. Since the mechanical energy is conserved, the radiated en-
ergy must come entirely from the Schott energy stored in the near field. The
same behavior is found for the Lorentz–Dirac equation. If, locally, E = const. and
B = 0, then the Hamiltonian part is solved by hyperbolic motion, i.e. a constantly
accelerated relativistic particle. For this solution the radiation reaction vanishes
which means that locally the critical manifold happens to be independent of ε. The
radiated energy originates exclusively from the Schott energy.
110 The Lorentz–Dirac equation
(ii) Zero magnetic field, central potential. Forzero magnetic field the Landau–
Lifshitz equation simplifies to
m
0
γκ(v)˙v = e E +
e
2
6πc
3
e
m
0
γ(v ·∇)E
+
e
m
0
c
2
(v · E)E − γ
2
E
2
v + γ
2
c
−2
(v · E)
2
v
. (9.13)
We take E =−∇φ
ex
and assume that φ
ex
is central. Let us set q = r, |r|=
r, ˆr = r /|r|, φ
ex
(q) = φ(r) which implies E =−φ
ˆr. Then (9.13) becomes
m
0
γκ(v)˙v =−e φ
ˆr +
e
2
6πc
3
e
m
0
γ
− (v · ˆr)φ
ˆr −
1
r
(v − (v · ˆr )ˆr )φ
+
e
m
0
c
2
φ
2
(v · ˆr )ˆr − γ
2
v + γ
2
c
−2
(v · ˆr )
2
v
. (9.14)
The angular momentum L = r × m
0
γ v satisfies
˙
L =
e
2
6πc
3
−
e
m
0
1
r
φ
−
e
m
0
c
2
γ
2
1 − c
−2
(v · ˆr )
2
φ
2
L. (9.15)
Thus the orientation of L is conserved and the motion lies in the plane perpendic-
ular to L.Nofurther reduction seems to be possible and one would have to rely
on numerical integration. Only for the harmonic oscillator, φ(r ) =
1
2
m
0
ω
2
0
r
2
, can
a closed form solution be achieved.
(iii) Zero electrostatic field and constant magnetic field. We set B = (0, 0, B)
with constant B. Then (9.7) simplifies to
m
0
γκ(v)˙v =
e
c
(v × B) +
e
2
6πc
3
e
m
0
c
2
(v · B)B −γ
2
B
2
v +γ
2
c
−2
(v · B)
2
v
.
(9.16)
We multiply by κ(v)
−1
to obtain
m
0
γ ˙v =
e
c
(v × B) +
e
2
6πc
3
e
m
0
c
2
(v · B)B − B
2
v
. (9.17)
The motion parallel to B decouples with ˙v
3
= 0. We set v
3
= 0 and v =
(u, 0), u
⊥
= (−u
2
, u
1
). Then the motion in the plane orthogonal to B is governed
by
γ ˙u = ω
c
(u
⊥
− βω
c
u), (9.18)
with the cyclotron frequency ω
c
= eB/m
0
c and β = e
2
/6πc
3
m
0
. Equation (9.18)
holds over the entire velocity range. For an electron βω
c
= 8.8 × 10
−18
B [gauss].
Thus even for very strong fields the friction is small compared to the inertial terms.
9.2 Some applications 111
Equation (9.18) can be integrated as
d
dt
γ =−βω
2
c
(γ
2
− 1) (9.19)
with solution
γ(t) =
γ
0
+ 1 + (γ
0
− 1)e
−2βω
2
c
t
γ
0
+ 1 − (γ
0
− 1)e
−2βω
2
c
t
−1
, (9.20)
which tells us how u(t)
2
shrinks to zero. To determine the angular dependence we
introduce polar coordinates as u = u(cos ϕ, sin ϕ). Then
du
dϕ
=−βω
c
u,
dϕ
dt
= γ
−1
ω. (9.21)
Thus u(ϕ) shrinks exponentially,
u(ϕ) = u(0) e
−βω
c
ϕ
. (9.22)
Since βω
c
= 8.8 × 10
−18
B [gauss] for an electron, even for strong fields the
change of u over one revolution is tiny.
To obtain the evolution of the position q = (r, 0), |r|=r,weuse the fact that
for zero radiation reaction, β = 0,
r =
u
ω
c
γ. (9.23)
By (9.22) this relation remains approximately valid for non-zero β. Inserting u(t)
from (9.20) one obtains
r(t) = r
0
e
−βω
2
c
t
1 + ((γ
0
− 1)/2)(1 − e
−2βω
2
c
t
)
−1
(9.24)
with r
0
the initial radius and u(0)/c = (γ
0
− 1)
1/2
/γ
0
the initial speed which are
related through (9.23). In the ultrarelativistic regime, γ
0
1, and for times such
that βω
2
c
t 1, (9.24) simplifies to
r(t) = r
0
1
1 + γ
0
βω
2
c
t
(9.25)
and the initial decay is according to the power law t
−1
rather than exponential.
Foranelectron βω
2
c
= 1.6 × 10
−6
(B [gauss])
2
s
−1
. Therefore if one chooses
a field strength B = 10
3
gauss and an initial radius of r
0
= 10 cm, which corre-
sponds to the ultrarelativistic case of γ = 6 × 10
4
, then the radius shrinks within
0.9stor(t) = 1
µmbywhich time the electron has made 2 × 10
14
revolutions.
(iv) The Penning trap.Anelectron can be trapped for a very long time in the
combination of a homogeneous magnetic field and an electrostatic quadrupole po-
tential, which has come to be known as a Penning trap. Its design has been opti-
mized towards high-precision measurements of the gyromagnetic g-factor of the
112 The Lorentz–Dirac equation
electron. Our interest here is that the motion in the plane orthogonal to the mag-
netic field consists of two coupled modes, which means that the damping cannot be
guessed by pure energy considerations using Larmor’s formula. One really needs
the full power of the Landau–Lifshitz equation.
An ideal Penning trap has the electrostatic quadrupole potential
eφ(x) =
1
2
mω
2
z
−
1
2
x
2
1
−
1
2
x
2
2
+ x
2
3
, (9.26)
which satisfies φ = 0, superimposed with the uniform magnetic field
B = (0, 0, B). (9.27)
The quadrupole field provides an axial restoring force whereas the magnetic field
is responsible for the radial restoring force, which however could be outweighed
by the inverted part of the harmonic electrostatic potential.
We insert E =−∇φ and B in the Landau–Lifshitz equation. The terms propor-
tional to (v ·∇)E, E × B,(v · B)B, and B
2
v are linear in v, respectively q. The
remaining terms are either cubic or quintic and will be neglected. This is justified
provided
|v|
c
1 (9.28)
and
(m
0
ω
2
z
/e) r
max
B, i.e. r
max
c(ω
c
/ω
2
z
), (9.29)
if r
max
denotes the maximal distance from the trap center. With these assumptions
the Landau–Lifshitz equation decouples into an in-plane motion and an axial mo-
tion governed by
˙u =
1
2
ω
2
z
r + ω
c
u
⊥
− β
ω
2
c
−
1
2
ω
2
z
u +
1
2
ω
c
ω
2
z
r
⊥
, (9.30)
¨z =−ω
2
z
z − βω
2
z
˙z. (9.31)
Here q = (r, z), v = (u, ˙z), (x
1
, x
2
)
⊥
= (−x
2
, x
1
).
The axial motion is just a damped harmonic oscillation with frequency ω
z
and
friction coefficient
γ
z
= βω
2
z
. (9.32)
The in-plane motion can be written in matrix form as
d
dt
ψ = ( A + βV )ψ (9.33)