Quantum field methods in the electron cooling
Quantum field methods are proposed to describe the electron cooling process. The influence of the magnetic field and the anisotropy of the temperature distribution of the electron gas are considered in the framework of quantum field theory. It is shown the longitudinal component of the thermal motio...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1118652017-01-16T03:02:52Z Quantum field methods in the electron cooling Khelemelya, O.V. Kholodov, R.I. Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Quantum field methods are proposed to describe the electron cooling process. The influence of the magnetic field and the anisotropy of the temperature distribution of the electron gas are considered in the framework of quantum field theory. It is shown the longitudinal component of the thermal motion of electrons acts the main role in the electron cooling of heavy ions in the presence of a sufficiently strong magnetic field. Quantum effects in the electron cooling are estimated. Запропоновано використовувати квантово-польовi методи для опису процесу електронного охолодження. Розглянуто в рамках квантової теорiї поля вплив магнiтного поля з урахуванням анiзотропiї температурного розподiлу електронного газу. Показано, що в присутностi сильного магнiтного поля головну роль в електронному охолодженнi важких iонiв вiдiграє повздовжня компонента теплового руху електронiв. Проведено оцiнку квантових ефектiв в електронному охолодженнi. Предложено использовать квантово-полевые методы для описания процесса электронного охлаждения. Рассмотрено в рамках квантовой теории поля влияние магнитного поля при учете анизотропии температурного распределения электронного газа. Показано,что в присутствии достаточно сильного магнитного поля главную роль в электронном охлаждении тяжелых ионов отыгрывает продольная составляющая теплового движения электронов. Проведена оценка квантовых эффектов в электронном охлаждении. 2013 Article Quantum field methods in the electron cooling / O.V. Khelemelya, R.I. Kholodov// Вопросы атомной науки и техники. — 2013. — № 3. — С. 53-57. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 52.30.-q, 52.25.Xz http://dspace.nbuv.gov.ua/handle/123456789/111865 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина |
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Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Khelemelya, O.V. Kholodov, R.I. Quantum field methods in the electron cooling Вопросы атомной науки и техники |
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Quantum field methods are proposed to describe the electron cooling process. The influence of the magnetic field and the anisotropy of the temperature distribution of the electron gas are considered in the framework of quantum field theory. It is shown the longitudinal component of the thermal motion of electrons acts the main role in the electron cooling of heavy ions in the presence of a sufficiently strong magnetic field. Quantum effects in the electron cooling are estimated. |
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Khelemelya, O.V. Kholodov, R.I. |
| author_facet |
Khelemelya, O.V. Kholodov, R.I. |
| author_sort |
Khelemelya, O.V. |
| title |
Quantum field methods in the electron cooling |
| title_short |
Quantum field methods in the electron cooling |
| title_full |
Quantum field methods in the electron cooling |
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Quantum field methods in the electron cooling |
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Quantum field methods in the electron cooling |
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quantum field methods in the electron cooling |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина |
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| citation_txt |
Quantum field methods in the electron cooling / O.V. Khelemelya, R.I. Kholodov// Вопросы атомной науки и техники. — 2013. — № 3. — С. 53-57. — Бібліогр.: 10 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T02:49:18Z |
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2025-07-08T02:49:18Z |
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| fulltext |
QUANTUM FIELD METHODS IN THE ELECTRON
COOLING
O.V.Khelemelya∗, R.I.Kholodov,
Institute of Applied Physics NAS of Ukraine, 40000, Sumy, Ukraine
(Received Marth 22, 2013)
Quantum field methods are proposed to describe the electron cooling process. The influence of the magnetic field and
the anisotropy of the temperature distribution of the electron gas are considered in the framework of quantum field
theory. It is shown the longitudinal component of the thermal motion of electrons acts the main role in the electron
cooling of heavy ions in the presence of a sufficiently strong magnetic field. Quantum effects in the electron cooling
are estimated.
PACS: 52.30.-q, 52.25.Xz
1. INTRODUCTION
In 1967 G.I. Budker [1] proposed the electron cool-
ing method of charge massive particle beam (damp
volume space).
One combines of moving “hot” ion and “cold”
electron beams at a some location of the storage ring.
Due to the Coulomb interaction beams temperatures
equalized then ”warmed up” electrons are removed
from the system drive. The cooled ion beam contin-
ues to move in a storage ring.
In modern facilities, such HESR (High Energy
Storage Ring, Germany, FAIR Collaboration), elec-
tron cooling method is a necessary part of the process
of accumulation of antiprotons. Charmonium spec-
troscopy, which is one of the main items in the exper-
imental program of HESR, requires antiproton mo-
mentum up to 8.9 GeV/c with a resolution ∆p/p ∼
10−5. This can be achieved only with electron cool-
ing.
Originally Coulomb theory of binary collisions is
used to describe the process of electron cooling. The
advantage of this method is evident in the construc-
tion of the theory, but the total result can be obtained
only by means of numerical simulations [2, 3, 4].
Also, the friction force can be obtained by the
methods of plasma physics (dielectric model). Here
the properties of the medium are given by the di-
electric constant of the plasma, which determines the
form of the expression for the energy loss of ions mov-
ing in the electron gas [5].
The theory of binary collisions and the dielectric
model have a significant disadvantage associated with
the introduction of empirical values such as the cutoff
parameters in the Coulomb logarithm.
The quantum field theory method is devoid of this
disadvantage, the Coulomb logarithm contains values
determined from the first principles.
The question of energy loss of the interaction
charged particles with a plasma without a magnetic
field was investigated in the framework of quantum
field theory by Larkin [6].
The interaction of a nonrelativistic charged par-
ticle with a plasma in a magnetic field was described
within quantum field theory in Akhiezer‘s work [7].
The energy losses were obtained with approximation
of zero temperature.
In this paper the effect of the anisotropic electron-
gas temperature on the energy losses are taken into
account in the linear approximation.
2. ENERGY LOSSES OF PARTICLE IN A
MAGNETIZED ELECTRON GAS
Let consider the basic steps of quantum field theory
to the description of electron cooling.
The Hamiltonian of the system of charged parti-
cles interacting by Coulomb’s law with the passing
through of the particles, can be written as
H = H0 + H ′(t), (1)
where H0 is the main Hamiltonian of nonperturbative
system of plasma particles, the second term H ′(t) ,
the Hamiltonian of interaction, describes the pertur-
bation introduced by the projectile particle:
H ′(t) =
∫
d~rJ0(~r, t)a0(~r, t), (2)
a0(~r, t) = (4π)−1
∫
d~r′ |~r − ~r′|−1
j0(~r′, t).
a0 is operator of a scalar potential, j0 and J0 are
operators of the charge density of the system and of
the projectile particle, respectively, the operators are
taken in the interaction representation. The elements
∗Corresponding author E-mail address: xvdm@mail.ru
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2013, N3(85).
Series: Nuclear Physics Investigations (60), p.53-57.
53
of the scattering matrix
S = T
−i
∞∫
−∞
H ′ (t) dt
(3)
connects the various states of the original system and
external particle. These states are characterized by
the quantum numbers α, n, where α ≡ (ν, pz, q) are
quantum numbers of the projectile particle in a mag-
netic field ~H and n is set of quantum numbers de-
scribing the state of the environment with a certain
energy En and a certain number of particles Nn.
We assume the speed of the moving particle V
is large enough that
(
e2V −1h̄−1 ¿ 1
)
its interaction
with the particles of the medium can be considered by
perturbation theory. In the linear approximation in
H ′ (t) probability of transition from the initial state
α, n to the final state α′, n′ has the form
Wif = 2πδ (Ei − Ef ) |H ′|2 (4)
or
Wif = 2πδ (Ei − Ef )
∫
d~rd~r′〈α′
∣∣∣Ĵ0 (~r)
∣∣∣ α〉, (5)
〈α
∣∣∣Ĵ0 (~r′)
∣∣∣ α′〉〈n′ |ϕ̂ (~r)|n〉〈n |ϕ̂ (~r′)|n′〉 ,
where En − En′ + εα − εα′ εα ≡ εν,pz =
η (ν + 1) m/M + pz/2M is the energy of projectile
particle, M is its mass, η = eH/mc is the Larmor
frequency of electron in magnetic field ~H (α̂0, Ĵ0 are
operators in the Schrodinger representation).
Let sum expression for the probability of the final
states environment and average over the initial states
of the density matrix ρ0 = exp {β (Ω + µN − En)} to
get the full transition probability of a particle from a
state with energy εν,pz in a state of the energy εν′,p′z
Wα,α′ =
∑
n
exp {β (Ω + µN − En)}
∑
n′
Wif . (6)
Passing to Fourier components
Wα,α′ = 2π
∫
d3kΦ
(
~k, εα − εα′
)
U
(
~k
)
, (7)
where Φ
(
~k, εα − εα′
)
and U
(
~k
)
are components of
Fourier function,
Φ (~r1 − ~r2, ω) =
∑
n,n′
exp {β (Ω + µN − En)}× (8)
×〈n′ |α̂0(~r1)|n〉〈n |α̂0(~r2)|n′〉δ (En − En′+ω) ,
U (~r1 − ~r2) =
∑
q,q′
〈α′
∣∣∣Ĵ0(~r1)
∣∣∣ α〉〈α
∣∣∣Ĵ0(~r2)
∣∣∣ α′〉.
The energy losses per unit of time of the particle is
connected with the probability
−dE
dt
=
∑
α′
(εα − εαprime) Wα,α′ . (9)
The final form of this equation can be written as
−dEν,p
dt
=
2e2mωB
(2π)2
∑
ν
∞∫
−∞
ωdω
(1− e−βω)
× (10)
×
∫
d3k
k
Λν,ν′
(
kt√
2mωB
)
×
×Im
κ
(
~k, ω
)
1 + κ
(
~k, ω
)δ (εν,p − εν′,p−kz − ω) ,
where the dielectric susceptibility κ
(
~k, ω
)
can be de-
termine through polarized operator:
κ
(
~k, iω
)
= −k−2P
(
~k, iω
)
. (11)
The graphic technique is applied to calculate the
polarization operator. In the one-loop approxima-
tion (first Born approximation) the polarization op-
erator are described by the Feynman diagram shown
in Fig. 1.
P (~r − ~r′, iω) =
2e2
β
∑
p4
G (~r, ~r′, p4) (~r′, ~r, p4 − iω) ,
(12)
where G (~r, ~r′, p4) is Green’s function of an electron
in a magnetic field that has the form
G (~r, ~r′, p4) =
∑
α
Ψα(r1)
1
εα − µ + ip4
Ψ∗α(r2), (13)
where Ψα(r) is the wave function of a par-
ticle in a magnetic field. The factor of in-
verse temperature β = 1/T are contained in
the resulting formula for the energy loss (10).
Fig.1. The Feynman diagram of the polarization op-
erator in the one-loop approximation
Special function Λν,ν′ (a) is
Λν,ν′ (a) =
∞∫
0
dsJ0
(
2
√
as
)
Lν(s)Lν′(s) exp(−s),
(14)
where a = (h̄kt)
2
2mh̄ωB
is the ratio of the transverse en-
ergy (h̄kt)
2
2m and the distance between adjacent Lan-
dau levels h̄ωB , Lν(x) = ex
n!
dn
dxn (e−xxn) is Laguerre
polynomial, J0(x) is Bessel function.
In the case of strong magnetic fields (a ¿ 1)
54
Λ′ν,ν′ ≈ δν,ν′ +
a [(ν + 1)δν+1,ν′ − (2ν + 1)δν,ν′ + νδν−1,ν′ ] +
a4
4
[
(ν + 1)(ν + 2)δν+2,ν′ − 4(ν + 1)2δν+1,ν′+
2(3ν2 + 3ν + 1)δν,ν′ − 4ν2δν−1,ν′ +
ν(ν − 1)δν−2,ν′ ] , (15)
where δ is the delta function [10].
In the case of weak magnetic fields (a À 1):
Λ′′ν,ν′ ≈
mh̄ωB
π∆
, (16)
where ∆ is the area of the triangle which
was built on the segments h̄k⊥, p⊥ =
2mh̄ωB(ν + 1/2), p′⊥ = 2mh̄ωB(ν + 1/2).
Fig.2. The function Λνν′(a) (solid curve), its as-
ymptotic for a ¿ 1 (dash–dotted curve) and for
a À 1 (dotted curve) with numbers of Landau lev-
els ν = 3 and ν′ = 2
In Fig. 2. the explicit form of special functions
Λν,ν′(a) (14) and its approximations in the case of
strong Λ′ν,ν′(a) (15) and weak Λ′′ν,ν′(a) (16) magnetic
fields for the numbers of Landau levels ν = 3 and
ν′ = 2 are presented [10].
3. THE APPROACHING OF ZERO
TEMPERATURE OF ELECTRON GAS
If the electron temperature is T = 0 in [7], a sim-
ple expression for an arbitrary angle of the projectile
particle relative to the external magnetic field of ar-
bitrary strength was obtained.
−dEp
dt
=
e2ω2
P
4πV
[
LC − f
(
α,
ωP
ωB
)]
, (17)
where LC = ln
(
2mMV 2
(M+m)ωB
)
is Coulomb logarithm,
the addition term to the friction force relating to the
influence of the magnetic field is written as
f
(
α,
ωP
ωB
)
=
1
π
(
ωP
ωB
)2
z1∫
0
g(z)dz −
z3∫
z2
g(z)dz
,
g(z) =
z(1− z)√
z(z − z1)(z − z2)(z3 − z)
,
z1,2 =
1
2
(
z3 + sqrtz2
3 − 4
(
ωP
ωB
)2
sin2 α
)
,
z3 = 1 +
(
ωP
ωB
)2
.
The resulting analytical expression corresponds ex-
pression derived in the framework of the clas-
sical theory of binary collisions and the di-
electric model (Such result is shown in Fig. 3.)
Fig.3. Dependence f(α, u), on the angle
0 ≤ α ≤ π/2 of a projectile particle with respect to
the external magnetic field 0 ≤ u = ωP /ωB
When the magnetic field is turned off the addition
part f (α, ωP /ωB) and we get a supplement known
expression for the polarization losses without a mag-
netic field when the temperature is neglected, that is,
the initial velocity of the incoming particle is much
greater than the speed of random motion of particles
in the plasma.
4. THE ACCOUNTING OF THE
ANISOTROPIC TEMPERATURE OF THE
ELECTRON GAS IN THE LINEAR
APPROXIMATION
Taking into account of the temperature of the
electron gas is essential problem of electron cooling.
The temperature is met in the expression for the
energy loss of the projectile particle in the form βεα
βεα = βεα⊥ + βεα‖ =
ωB
T
(
ν +
1
2
)
+
p2
2mT
. (18)
An important point in accelerator technology is
the effect of ”plated” of the electrostatically accel-
erated beam of charged particles. This effect is a
consequence of Liouville’s theorem. After acceler-
ating the electron velocity distribution is essentially
anisotropic. The transverse temperature is a thou-
sand times greater than the longitudinal component
55
(for the characteristic values T⊥e ≈ Tcath ∼ 1000K,
U ∼ 10kV , T‖ = T 2
cath/E ∼ 1K.
Let use replacement to account for the tempera-
ture anisotropy
βεα = β⊥εα⊥ + β‖εα‖ =
ωB
T⊥
(
ν +
1
2
)
+
p2
2mT‖
.
(19)
In the linear approximation of the temperature
the dielectric susceptibility one can write as [10]
κ (ω, k, T ) = κ (ω, k, 0) + AT‖ + BT⊥ + C, (20)
where
κ (ω, k, 0) = −ω2
P
k2
(
k2
z
ω2
+
k2
⊥
ω2 − ω2
B
)
. (21)
The first term in (20) is the well-known expression
in the plasma physics obtained in hydrodynamics ap-
proximation [5]
A = −ω2
P k2
z
mk2
(
3k2
z
ω4
+ k2
⊥
3ω2 + ω2
B
(ω2 − ω2
B)3
)
, (22)
B = −ω2
P k2
⊥
mk2
× (23)
(
k2
z
ω2
3ω2 − ω2
B
(ω2 − ω2)2
+
3k2
⊥
(ω2 − 4ω2
B) (ω2 − ω2
B)
)
.
Next two terms in (20) give a correction for tem-
perature in the linear approximation [10].
C = −ω2
P
k2
(
h̄2k2
z
4mω2
+
h̄2k4
⊥k2
z
8m2ω2ω2
B
(
1 +
3T⊥
h̄ωB
))
−
(24)
−ω2
P
k2
(
h̄2k4
zk2
⊥
4m2
(
3ω2 + ω2
B
ω2 − ω2
B
))
.
Expression (24) takes into account the quantum
corrections [10]. Let consider the case of longitudi-
nal motion of the particles in a magnetic field. In
the weak and strong magnetic fields one can obtain
simple analytical expressions for the energy loss. En-
ergy losses are written in approximation of a weak
magnetic field (ωB/ωP ¿ 1) as
−dE
dt
=
q2ω2
P
Vi
[
LC − τ
2
− 1
2
(
ωB
ωP
)]
. (25)
where τ = 3v2
eT
V 2
i
, LC is Coulomb logarithm (17)
If the value of the magnetic field one comes to
the result obtained in the case without the magnetic
field [6] ~H = 0 that means the correspondence prin-
ciple is performed.
In approximation of a weak magnetic field they
are (ωB/ωP À 1)
−dE
dt
=
q2ω2
P
Vi
[(
1− 4τ‖
3
(
ωB
ωP
)4
− 2τ⊥
3
(
ωB
ωP
)2
)
LC−
ln
√
1 +
(
ωB
ωP
)2
+
2τ‖
3
(
ωB
ωP
)4
− τ⊥
3
(
ωB
ωP
)2
.
(26)
The term with the Coulomb logarithm yields main
contribution to the expression (26). It is proportional
(
1− 4τ‖
3
(
ωB
ωP
)4
− 2τ⊥
3
(
ωB
ωP
)2
)
. (27)
The value equal to the ratio of the second and the
third term of (27) is
ξ = 2
T‖
T⊥
(
ωB
ωP
)2
. (28)
It is an experimentally proved fact that with in-
creasing values of the external longitudinal magnetic
field the effect of the ”fast” electron cooling are ob-
served [3, 4]. The effect is observed in strong external
magnetic fields which locks the transverse motion of
the electrons. So the energy exchange is possible only
through the longitudinal component of the tempera-
ture which is several orders of magnitude low than
transverse one. The cooling of the beam of heavy
charged particles is several times better as result.
In early experiments term was only a fraction of a
unit [9]. In modern installations, for example, project
HESR (High Energy Storage Ring), ξ reaches values
(ξ ≈ 10). The contribution of the longitudinal com-
ponent of the thermal motion of electrons is much
higher than the perpendicular one.
5. QUANTUM EFFECTS IN THE
ELECTRON COOLING
In the electron gas in a magnetic field two types
of quantum effects are possible: 1) if the temperature
of the electron gas below the degeneracy temperature
then all of the electron gas behaves as a quantum ob-
ject; and 2) the energy of particles is characterized
by the Landau levels due to the motion of particles
in a magnetic field. If the transverse temperature
of the electron gas is less than the distance between
adjacent Landau levels then there will be quantum
effects.
There are characteristic density of the electron gas
N ≈ 3 · 107cm−3, plasma ωP ≈ 2.9 · 108c−1 and cy-
clotron ωB ≈ 3.5 · 108c−1 frequencies in the electron
cooling. For such parameters the degeneracy temper-
ature of the electron gas is
T0 =
h̄2
2m
(
3π2N
)2/3 ∼ 10−10eV. (29)
56
The temperature below the temperature of the
electrons is assumed while technically impossible.
In the second case with the Landau levels the tem-
perature of the electron T0 gas are imposed less strin-
gent conditions
h̄ωB
T⊥
=
eBh̄
2mcT⊥
∼ 10−5. (30)
So on devices such HESR quantum effects are not
important. However, it should be noted that quan-
tum electron cooling can be observed in the labora-
tory if we increase the magnetic field and decrease the
transverse temperature by two orders of magnitude,
that is technically possible.
In conclusion, the main advantage of the quan-
tum approach to the description of electron cooling
is avoiding of the disadvantages associated with the
empirical determination of the Coulomb logarithm
which is typical for the theory of binary collisions
and for the plasma model. In quantum field theory
the Coulomb logarithm is defined from the first prin-
ciples.
The authors are grateful to Miroshnichenko V.I.
for useful discussions
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КВАНТОВО-ПОЛЕВЫЕ МЕТОДЫ В ЗАДАЧЕ ЭЛЕКТРОННОГО ОХЛАЖДЕНИЯ
А.В.Хелемеля, Р.И.Холодов
Предложено использовать квантово-полевые методы для описания процесса электронного охлажде-
ния. Рассмотрено в рамках квантовой теории поля влияние магнитного поля при учете анизотропии
температурного распределения электронного газа. Показано, что в присутствии достаточно сильного
магнитного поля главную роль в электронном охлаждении тяжелых ионов отыгрывает продольная со-
ставляющая теплового движения электронов. Проведена оценка квантовых эффектов в электронном
охлаждении.
КВАНТОВО-ПОЛЬОВI МЕТОДИ В ЗАДАЧI ЕЛЕКТРОННОГО ОХОЛОДЖЕННЯ
О.В.Хелемеля, Р.I.Холодов
Запропоновано використовувати квантово-польовi методи для опису процесу електронного охолоджен-
ня. Розглянуто в рамках квантової теорiї поля вплив магнiтного поля з урахуванням анiзотропiї темпе-
ратурного розподiлу електронного газу. Показано, що в присутностi сильного магнiтного поля головну
роль в електронному охолодженнi важких iонiв вiдiграє повздовжня компонента теплового руху елек-
тронiв. Проведено оцiнку квантових ефектiв в електронному охолодженнi.
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