Study of ultrarelativistic beam emittance
We estimated a minimum radius of a beam under given normalized emittance and a minimum emittance of proton and electron beams. We obtained the simple expression for a minimum radius of a beam xmin = En/1 rad and the value of the minimum for normalized emittance for proton and electron beams En,1 ~ 0...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Schriftenreihe: | Вопросы атомной науки и техники |
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| Zitieren: | Study of ultrarelativistic beam emittance / V.M. Khoruzhiy // Вопросы атомной науки и техники. — 2001. — № 5. — С. 172-173. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-790122015-03-25T03:02:43Z Study of ultrarelativistic beam emittance Khoruzhiy, V.M. We estimated a minimum radius of a beam under given normalized emittance and a minimum emittance of proton and electron beams. We obtained the simple expression for a minimum radius of a beam xmin = En/1 rad and the value of the minimum for normalized emittance for proton and electron beams En,1 ~ 0.01 cm·mrad and En,2 ~ (0.01÷1) cm·mrad respectively. 2001 Article Study of ultrarelativistic beam emittance / V.M. Khoruzhiy // Вопросы атомной науки и техники. — 2001. — № 5. — С. 172-173. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS number: 29.27.Fh http://dspace.nbuv.gov.ua/handle/123456789/79012 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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We estimated a minimum radius of a beam under given normalized emittance and a minimum emittance of proton and electron beams. We obtained the simple expression for a minimum radius of a beam xmin = En/1 rad and the value of the minimum for normalized emittance for proton and electron beams En,1 ~ 0.01 cm·mrad and En,2 ~ (0.01÷1) cm·mrad respectively. |
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Article |
| author |
Khoruzhiy, V.M. |
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Khoruzhiy, V.M. Study of ultrarelativistic beam emittance Вопросы атомной науки и техники |
| author_facet |
Khoruzhiy, V.M. |
| author_sort |
Khoruzhiy, V.M. |
| title |
Study of ultrarelativistic beam emittance |
| title_short |
Study of ultrarelativistic beam emittance |
| title_full |
Study of ultrarelativistic beam emittance |
| title_fullStr |
Study of ultrarelativistic beam emittance |
| title_full_unstemmed |
Study of ultrarelativistic beam emittance |
| title_sort |
study of ultrarelativistic beam emittance |
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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/123456789/79012 |
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Study of ultrarelativistic beam emittance / V.M. Khoruzhiy // Вопросы атомной науки и техники. — 2001. — № 5. — С. 172-173. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT khoruzhiyvm studyofultrarelativisticbeamemittance |
| first_indexed |
2025-07-06T03:08:21Z |
| last_indexed |
2025-07-06T03:08:21Z |
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1836865345152876544 |
| fulltext |
STUDY OF ULTRARELATIVISTIC BEAM EMITTANCE
V.M. Khoruzhiy
NSC KIP, Kharkov, Ukraine
e-mail: khoruzhiy@ kipt.kharkov.ua
We estimated a minimum radius of a beam under given normalized emittance and a minimum emittance of proton
and electron beams. We obtained the simple expression for a minimum radius of a beam min nх Е= /1 rad and the
value of the minimum for normalized emittance for proton and electron beams ,1nЕ : 0.01 cm·mrad and ,2nЕ :
(0.01 ÷ 1) cm·mrad respectively.
PACS number: 29.27.Fh
We considered charge particle beams with accelera-
tion duration τ in the linac which is more than a typical
relaxation time 0τ (the equilibrium state due to close
collision). In this case the beam can be characterized by
a certain value of temperature 0T . Our estimation for
most of proton beams in resonance linacs 0τ = ppτ ~1 s.
Practically this means the existence of a stationary beam
temperature at the next stage of acceleration – circular
accelerators with a minimum output energy ~ 10 GeV.
The stationary electron beam temperature takes place
significantly earlier. We considered properties of such
charge particle beams. Expression for the total particle
velocity β Σ is
2β Σ = 2β + 2
.mβ ⊥ - 2β 2
.mβ ⊥ (1)
where β Σ , β are the normalized total and longitudinal
velocities in a laboratory coordinate system respective-
ly, mβ ⊥ is the transversal normalized velocity in a
moving system of coordinates. From (1) resulting is the
expression for the transversal normalized velocity in the
laboratory system of coordinates
β ⊥ = mβ ⊥ / γ (2)
where γ =1/ 21 β− .
Then the maximal value of the transversal velocity
for given γ is
maxβ ⊥ =1/ γ (3)
For ultrarelativistic beams we have the expression 1- β
= 1, then the maximum value of divergence 'x
=dx/dz= maxβ ⊥ / β practically coincides with the value
maxβ ⊥ (x, z are transversal and longitudinal coordinates
respectively).
Basis for any emittance definition [1, 2] is the inte-
gral invariant xdxdp∫ = const, where х, xp are the
transversal coordinate and momentum of a particle. If
the emittance shape has a canonical form of ellipse, i.e.
its semiaxis coincides with the coordinate axis, then tak-
ing into account (2) it is easy to determine the expres-
sion
( )x m
xp = ( )x l
xp (4)
The indexes l and m are applied to laboratory and mov-
ing systems respectively. From expression (4) it follows
that the integral invariant xdxdp∫ is the relativistic in-
variant too.
If the shape of the normalized emiittance нЕ is as
an ellipse of a canonical form, then
нЕ = 'xx xβ γ β γ⊥= (5)
From expression (5) taking into account (3) it fol-
lows that the minimum radius of ultrarelativistic beams
min нх Е= /1rad. (6)
Thus, the minimum radius of ultrarelativistic beams
(6) takes a place under acceleration of ultrarelativistic
beams with a constant value of emittance нЕ , an effort
to produce the beam with the radius minх x< will
cause losses of charge particles.
Normalized emittances of ultrarelativistic proton ( γ
~80) [3] and electron ( γ ~4000) [4] beams in resonance
accelerators are нЕ ~100 сm·mrad, then from (6) results
that minx : 0.1 cm. The output transversal normalized
velocities of the electron beam for the linac LU-2 [4] are
' 410x β −
⊥ ≤; , the value maxβ ⊥ = 2.5·10-4 for the
output beam energy W = 2 GeV.
Expression (2-6) are valid under low energies too, it
enables to estimate the minimum emiittance from the
gun. From expression (6) results that the estimation of a
minimum emittance as far as possible is connected with
the minimum of beam radius as far as possible. The
minimum of beam radius can be determined using the
quantum mechanics limitation for a transversal dimen-
sion of the wave packet [5]. For a proton beam this min-
imum is minх⊥ : 10-5 сm, for an electron beam it is
minх⊥ : (10-3÷10-5) сm. Then from (6) we can estimate
as far as possible the minimum of a normalized emit-
tance. For a proton beam from the gun it is .minнЕ :
0.01 cm·mrad and for electron beam it is .minнЕ :
(0.01÷1) cm·mrad.
Most of real emittance values for proton and elec-
tron beams from the gun are higher than the minimum
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №5.
Серия: Ядерно-физические исследования (39), с. 172-173.
172
one by an order of magnitude at least. Thus, we estimat-
ed the minimum of beam radius as far as possible (6) for
the given value of a normalized emittance and, next, the
minimum emittance as far as possible for proton and
electron beams for the given minimum of beam radius
as far as possible.
REFERENCES
1.I.M.Kapchinsky. Charge particle dynamics in linear
resonance accelerators. Мoscow: Atomizdat, 1966 (in
Russian).
2.V.M.Khoruzhiy et al. About multi-charge ion beam
emittance // Voprosy Atomnoj Nauki i Tekhniki. Ser.
Tehnika fizicheskogo ehksperimenta (19). 1984, v. 2, p.
58-59 (in Russian).
3.E.G.Komar. Basis of acceleration technique. Moscow:
Atomizdat, 1975, 322 p. (in Russian)
4.V.I.Artemov, G.K.Demyanenko, N.A.Kovalenko,
F.A.Peev. Development of focusing system for LU-2
linac // Voprosy Atomnoj Nauki i Tekhniki, seriya:
Fizika Vysokikh Energij (1). 1972, v. 1, p. 64. (in Rus-
sian)
5.A.M.Shenderovich. About dynamics and formation of
beams in linacs // Zhournal Tekhnicheskoj Fiziki. 1983,
v. 53, # 6, p. 1078-1081 (in Russian).
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №5.
Серия: Ядерно-физические исследования (39), с. 172-173.
173
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