Using of undulators for low energy ion linac
The possibility of using undulators for focusing and acceleration of charged particles in RF field is discussed. There are suggested that the RF field does not have harmonics in synchronism with the beam. The accelerating force is produced by a combination of RF and undulator fields. Examples illust...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Using of undulators for low energy ion linac / E.S. Masunov // Вопросы атомной науки и техники. — 2001. — № 3. — С. 62-64. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-792372015-03-31T03:02:44Z Using of undulators for low energy ion linac Masunov, E.S. The possibility of using undulators for focusing and acceleration of charged particles in RF field is discussed. There are suggested that the RF field does not have harmonics in synchronism with the beam. The accelerating force is produced by a combination of RF and undulator fields. Examples illustrating the efficiency of the proposed method acceleration are given for low energy ion beams. In the undulator accelerator (UNDULAC) an electrostatic, a magnetic and radio frequency undulators can be used. The focusing conditions of the beam are studied. Methods for increasing of the ion beam intensity are discussed. 2001 Article Using of undulators for low energy ion linac / E.S. Masunov // Вопросы атомной науки и техники. — 2001. — № 3. — С. 62-64. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS numbers: 41.75.L, 41.85.E, 29.27.F http://dspace.nbuv.gov.ua/handle/123456789/79237 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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The possibility of using undulators for focusing and acceleration of charged particles in RF field is discussed. There are suggested that the RF field does not have harmonics in synchronism with the beam. The accelerating force is produced by a combination of RF and undulator fields. Examples illustrating the efficiency of the proposed method acceleration are given for low energy ion beams. In the undulator accelerator (UNDULAC) an electrostatic, a magnetic and radio frequency undulators can be used. The focusing conditions of the beam are studied. Methods for increasing of the ion beam intensity are discussed. |
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Article |
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Masunov, E.S. |
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Masunov, E.S. Using of undulators for low energy ion linac Вопросы атомной науки и техники |
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Masunov, E.S. |
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Masunov, E.S. |
| title |
Using of undulators for low energy ion linac |
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Using of undulators for low energy ion linac |
| title_full |
Using of undulators for low energy ion linac |
| title_fullStr |
Using of undulators for low energy ion linac |
| title_full_unstemmed |
Using of undulators for low energy ion linac |
| title_sort |
using of undulators for low energy ion linac |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
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http://dspace.nbuv.gov.ua/handle/123456789/79237 |
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Using of undulators for low energy ion linac / E.S. Masunov // Вопросы атомной науки и техники. — 2001. — № 3. — С. 62-64. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT masunoves usingofundulatorsforlowenergyionlinac |
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2025-07-06T03:17:10Z |
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2025-07-06T03:17:10Z |
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1836865899182686208 |
| fulltext |
USING OF UNDULATORS FOR LOW ENERGY ION LINAC
E.S. Masunov
Moscow Engineering Physics Institute,
115409, Kashirskoe shosse, 31, Moscow, Russia, masunov@dinus.mephi.ru
The possibility of using undulators for focusing and acceleration of charged particles in RF field is discussed. There
are suggested that the RF field does not have harmonics in synchronism with the beam. The accelerating force is
produced by a combination of RF and undulator fields. Examples illustrating the efficiency of the proposed method
acceleration are given for low energy ion beams. In the undulator accelerator (UNDULAC) an electrostatic, a mag-
netic and radio frequency undulators can be used. The focusing conditions of the beam are studied. Methods for in-
creasing of the ion beam intensity are discussed.
PACS numbers: 41.75.L, 41.85.E, 29.27.F
1 INTRODUCTION
In a conventional RF linac the beam is accelerated
by a synchronous wave. Another method to accelerate
ions – in the fields without a synchronous wave – was
suggested in Ref. [1] in which case the accelerating
force is to be driven by a combination of two non-syn-
chronous waves (two undulators). In undulator linac in
question, one of the undulators must be of the RF type
(it drives non-synchronous RF wave field), the second
one being, optionally, of magnetic, electrostatic, or ra-
dio frequency types. The 3D dynamics of the ion beam
in undulator linear accelerator (UNDULAC) will be de-
termined by the type of undulator and transverse struc-
ture of its field. In this paper a theoretical aspects of the
method of ion acceleration and focusing in the fields
without synchronous wave and possible realization un-
dulator accelerator for ion are investigated. The results
are compared with a conventional ion RF linac.
2 PARTICLE MOTION EQUATION
The motion equation of a particle in the field of two
waves can be written using Lagrangian function, as
)-Av(P
ΣΣ Φ⋅∇= e
dt
d
, (1)
where ApP e+= is the generalized momentum,
)(rΣΦ is the electrostatic potential,
])r(A)r(A[A lnRe)( ϕ⊥ϕ⊥
Σ += i
l
i
n eer is the total
vector potential of the two fields with amplitudes l,nA ,
phases ln,ln,ln,ln, tzk αωd +−=ϕ ∫ and wave numbers
l,nk . When the phase velocities ln,ln,ln,ph, k/v ω= dif-
fer significantly from the average velocity of the parti-
cles bv , the trajectories of the individual particles of the
beam in general have a complicated shape but can al-
ways be represented as the sum of a slow variation r
and rapid oscillation r~ . Accordingly, the beam momen-
tum p can be represented as the sum of a slowly vary-
ing and a rapidly oscillating component, ppp ~+= . Af-
ter an average is taken over the rapid oscillations from
(1) we can obtain the time-averaged equation of nonrel-
ativistic ion motion if the beam velocity
))/(( lnlncb vv kkωω ±±≡≈ :
effU
d
d
td
d
r
r −=
2
2
, (2)
where 3D effective potential
)r()r( ψ,UUU 21eff
⊥⊥ += , (3)
( )22
2
2
4
ln1
m
eU AA += , (4)
( )ψ⋅= i*
ln2 e
m
eU AA
2
2
Re
2
, (5)
⊥r and )α(α)ω(ω)(ψ lnlnln tzkk ±+±−±= are
slowly varying transverse coordinate and phase.
We see from (3) that the longitudinal bunching and
accelerations of the beam are provided by a combined
wave with phase velocity cv which is close to the aver-
age particle velocity. The choice of the functions
)( ⊥rnA and )( ⊥rlA is not arbitrary because simulta-
neously to acceleration it is necessary to keep up the
transverse focusing of the beam. Equilibrium trajecto-
ries can exist for all particle phases if the following con-
ditions hold in the injection plane:
01 =
∂
∂
⊥
U
r
, .U 2 0=
∂
∂
⊥r
(6)
Conditions (6) are necessary but not sufficient con-
ditions for focusing of the beam. In the absence of reso-
nance there can be transverse stability only if the «effec-
tive potential» Ueff has a minimum in the transverse
plane (X,Y) This imposes a limitation on the amplitudes
)( ⊥rnA , )( ⊥rlA and confine configuration of the
fields. The necessary condition of simultaneous trans-
verse and longitudinal focusing is the existence of an
absolute minimum for effU . For low energy ion accel-
erator it is difficult to create resonator system with two
generators, when ln ωω ≠ . It is interesting to consider
some versions of the linear undulator accelerator:
1. UNDULAC-E(M) that employs a combination of
RF field ( /λ2πω cn = , λβ= nph,n ck /2π ) and
static periodical electric (or magnetic) field of an
undulator ( 0ω =l , /λ2π/λ lk 00l +µ= ), where
l=0,1,2...; 0λ is slowly varying period of structure,
0μ is a phase advance of field per period;
2. UNDULAC-RF that employs a combination of two
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 62-64.
62
space nonsynchronous harmonics of RF field in the
periodical resonator structure ( /λ2πωω cln == ,
ln kk ≠ ), where 00vn n2k /λπ/λμ += , vμ is
phase advance per period of RF structure, l,
n=0,1,2,... and n≠l.
3 ION BEAM FOCUSING AND ACCELERA-
TION IN UNDULAC
In UNDULAC-E (M) the rate of acceleration is pro-
portional to the amplitudes of the RF field Ev and undu-
lator fields Eo (Bo). Still, increase of the beam energy
occurs due to RF field only. As is shown in [2], the en-
ergy gain is found by cvme,c ψcosEeT/dzWd = where
Te,m are the acceleration efficiency factors for UNDU-
LAC-E and UNDULAC-M. The choice of RF field am-
plitude Ev and undulator field amplitude Eo (Bo) is not
independent, since it is necessary to realize beam focus-
ing and to provide a large transmission coefficient K.
For example, if the RF field and the undulator electric
field have only single space harmonics ( 00 == l,n );
phase advance per period of structure 0=µ v for RF
field and π=µ 0 for undulator, focusing and accelera-
tion occur for all paraxial particles, when λλ=β /c 02
and 2/3/χ ≥= v0 EE . In this case the acceleration effi-
ciency factors
s
v
e mc
eET
βπ
λ
χ= 24
. For a plane magnetic
undulator the function effU has minimum, and the fo-
cusing takes place for all phase ψ if 0cv BB β= . This
condition connects values vB , 0B and cβ and permits to
express the factor mT through the amplitude of RF field,
c
2
v
m mc
eET
βπ4
λ
= . This value is close by eT .
The combined acceleration field can be driven with-
out use of a magnetic or an electrostatic undulator. In-
deed, consider ion beam dynamics in the periodical RF
structure without synchronous space harmonics. Interac-
tion of the beam with each harmonic can be treated as
ion interaction with a radio frequency undulator (UN-
DULAC-RF). The combined field of two harmonics
would accelerate the beam if ccb ck/ωββ =≈ where kc
is a wave number of a combined wave field,
0,1,2,... ,( 2)/( ,nkkkkkk pncpnc =≠≠±=
)0,1,2,... =p . The rate of energy gain
2ψsin d ⋅= v,1rfc ΕeΤ/dzW . The acceleration and phase
stability of the beam are possible when the phase of a
synchronous particles ψc in the combined field is in an
intervals [ ]π/2π/4, and [ ]/23 /4,5 ππ . In this case the
frequency of ion beam bunching is double ( 2ωω =b ).
For undulator, where RF field has a phase advance
µ=0 per a period, and only two harmonics n=0, p=1 are
taken into acount, the transverse focusing of the beam is
possible if the amplitude of the first harmonic is larger
than zero one (E1 > E0) and the beam velocity
/λ2λββ 0cb =≈ . For this case the acceleration effi-
ciency factor is
s
2
v,0
rf βmc4
λeE
T
π
= . The rate of energy
gain is the same as in UNDULAC-E(M).
For the undulator, where the RF field has a phase ad-
vance per period πµ =v and beam velocity
/λλββ 0cb =≈ , value of the acceleration efficiency
factor
s
2
v,0
rf βmc2
λeE
T
π
= . Besides, the choice of harmonic
amplitudes 0E and 1E are independent because the con-
dition of focusing in UNDULAC-RF can be fulfilled for
any value of E0/ 1E . The maximum values of 0E and
1E can be found from RF characteristic of the res-
onator, the transverse acceptance and beam current.
4 UNDULAC AND RF FOCUSING IN LINAC
It is interesting to compare the methods of accelera-
tion in UNDULAC and in linac under consideration
with axisymmetric RF focusing (ARF). The main prin-
ciples of APF can be described by means of two wave
approximation method: in a periodical RF structure the
beam is accelerated by a synchronous space harmonic of
the wave while another, nonsynchronous, harmonic is
only responsible for focusing the particles. By means of
Hamiltonian analysis in the 4-dimensional phase space
it is possible to find the relationship between the defined
longitudinal acceptance and the limit value of transverse
beam emittance, which provides the maximal transmis-
sion coefficient K≈1 [3]. For two- wave approach the
rate of energy gain cnv,sc ψcosEeT/dzWd = . The beam
focusing takes place and K≈1 if the amplitude of non-
synchronous harmonic nE >> sE . For two fundamental
harmonics 10 == n,s and a phase advance per period
πµ =v , the efficiency factor for axisymmetric RF fo-
cusing
s
2
n
s βmc4
λeET
π
≈ . This magnitude is similar to.
me,T , but it is twice as low than rfT .
The using of undulators for acceleration and focus-
ing of ion beams is extremely promising. First, the prob-
lem of design of UNDULAC RF system is simplified
considerably, since focusing and acceleration of parti-
cles is possible for both transverse (TE or TEM) and
longitudinal (TM) RF fields without any external focus-
ing elements and dedicated slow-wave systems. No drift
tubes are required for TEM wave. Second, an efficient
bunching and a large transmission coefficient of parti-
cles can be achieved solely by changing the amplitude
and period of the static undulator field. This eliminates
the serious problems involving adjustment and matching
of the RF system since the latter can be made uniform.
Third, an UNDULAC can be used for acceleration high
intensity ion beams [4]. Indeed, the main factor limiting
beam intensity in ion accelerator is space charge forces.
There exist, at least, three way to increase ion beam in-
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (34), с. 63-64.
63
tensity in a linear undulator accelerator: (i) to enlarge
beam cross-section; (ii) to accelerate several beams in a
channel of RF structure; (iii) to compensate for the
space charge by accelerating ions with opposite charge
signs within the same bunch.
(i) In a UNDULAC where there are no drift tubes, a
ribbon or a hollow beams having large cross-sections
can be accelerated. Acceleration of a ribbon ion beam
with the current J>1A in a plane electrostatic undulator
was studied in [5]. It was shown that a large cross-sec-
tion and electrostatic shielding of the space charge field
decreases Coulomb defocusing of the particles in the
narrow accelerating channel.
(ii) In the new accelerator one can accelerate the
several beams in a single channel of RF structure since
there are no drift tubes involved. The problem is to
choose a dedicated symmetry of the transverse radio-
frequency and periodic magnetic field. The RF system
must be a small transverse size. It is preferable to use a
shielded multielectrode line where transverse electro-
magnetic waves (TEM) can propagate. Configuration of
the RF field and magnetic undulator field must be such
as to maintain several equilibrium trajectories simulta-
neously [4].
(iii) Study of feasibility of simultaneous acceleration
of both positive and negative ions with identical charge-
to-mass ratio within the same bunch is of great interest.
The current limit of the ion beam can be increased sig-
nificantly by using the space charge compensation of
positively H+ (D+) and negatively H- (D-) charged ions
accelerated in the same bunch. This conclusion can be
drawn from Eq.(1). Indeed, the effective potential effU
depends on the particle charge squared, i.e. averaged
motions of positive and negative charged ions are iden-
tical. It allows to increase the beam current limit.
All possible methods of focusing and acceleration in
undulator linear accelerator would be effective for low
energy ion, when 2
n
b mc4
λeEβ
π
≅ . The acceleration effi-
ciency factor rf,e,mT decreases with growth of the beam
velocity like in the RFQ accelerator. Therefore UNDU-
LAC can be used as an initial part of the high intensity
linear accelerator (buncher) or as aninjector for the neu-
tron generators and nuclear fusion reactor.
5 CONCLUSION
Theoretical studies of using undulators in the accel-
eration system showed a possibility to create a new type
of ion linear accelerator (UNDULAC). Three types of
undulator for ion acceleration linac are suggested. The
requirement of particle focusing imposes a limitation on
the field amplitudes in undulators of all types besides
UNDULAC-RF with π=µ v , where the focusing con-
dition holds for any relation between harmonic ampli-
tudes. The rate of energy gain in UNDULAC is compa-
rable with analogous value for RFQ and conventional
linac, where RF focusing is realized. But the new accel-
erator has a number of advantages. In this accelerators it
is possible to use not only a longitudinal (TM) but also a
transverse (TE or TEM) radio frequency field. For the
TEM wave the drift tubes are absent. The beam intensi-
ty can be increased in these RF structures by means of
construction of a multibeam channel or enlarging the
beam cross-section. The other important way to increase
the beam intensity is using the space charge compensa-
tion. In UNDULAC positive and negative ions would be
within the same bunch and the current limit of the ion
beam can be substantially increased.
REFERENCES
1. E.S.Masunov // Sov. Phys.-Tech. Phys. 1990, v. 35,
No. 8, p. 962-965.
2. E.S.Masunov // Proc. of the XVIII LINAC Confer-
ence, Geneva, Switzerland, CERN 96-07, 1996.
V. 2, pp. 487 – 489
3. E.S.Masunov, N.E.Vinogradov // Phys. Rev. ST Ac-
cel. Beams. 2001, No 7, 070101.
4. E.S.Masunov // Proc of the 1995 Particle Accelera-
tor Conference and International Conference on
High-Energy Accelerator, Dallas, 1995. v. 2, p
. 1149.
5. E.S.Masunov, A.S.Roshal // Proc of the 1997 Par-
ticle Accelerator Conference, Vancouver, 1997. v.
4, p. 2835.
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