Download Superconductive Stabilization System for Charging Particle

SUPERCONDUCTIVE STABILIZATION SYSTEMS FOR CHARGED PARTICLE ACCELERATORS
O.M.Pignasty, D.V.Popovich,V.M.Rashkovan, N.S.Repalov, Yu.D.Tur
NSC-KPTI,Kharkov, Ukraine
(1)
Abstract
For the development of accelerators the key problem is the
maintenance of radial and phase stability of the flow. Wide opportunities
in this direction are opened by the application of superconductive
systems with using the effects of magnetic potential well.Without
restricting the generality of the problem, as a particular case, we have
considered the stability and stationary motion conditions for charged
particles in the electromagnetic fields of ideally conductive rings.
Results of theoretical analysis, based on Lyapunov’s stability theory, of
the radial and longitudinal particles motion in the superconductive
magnetic system are presented. The estimation of criterion and
conditions of beam stability is given.
where m i , v i , qi - mass, velocity and charge of the
i-particle,Ψ qi - flow, created by i-charge through contour,
I - current in SC ring,R ij - distance between two charges,ε 0 -
qi and
physical constant,nij - normal vector in the direction between
qj
charges.
The stationary movement on circular orbit imposes the necessary
conditions of stability on the system:
 ∂L

 ∂ρ j
Introduction
It is generally recognized that simultanous attainment of the intense
beam stability along all the coordinates is not possible without involving
into interaction the external stabilizing magnetic fields. On our opinion
the new principal method of approach to decision of the question of
stabilization and concentration of the charged particle beams cosists in
the development of superconducting (SC) control system based on the
use of the effect of the magnetic potential well (MPW) [1].
The
(2)
0
given conditions should be executed for any orbit. Assuming
v

<< 1 we receive the Lagrange
c

function in the form:
S
mi ⋅ v i2
+
2
∑
i=1
As a particular case, let us study the possibility to use this effect for
formation of the electron ring in the collective ion accelerator. Consider
the dynamics and stability conditions for particles of the electron ring in
the field of the SC magnetic contour (see Fig.1).
0
= 0.
that the particles move slowly 
L=
Model, Formulas and Results
∂L
∂z j
= 0;
S
∑
I ⋅ Ψqi + L11
i =1
I2
1
−
⋅
2 4π ⋅ ε ⋅ ε 0
S
∑
i =1
qi ⋅ qj
.
R ij
(3)
For analysis of the stationary movement we enter the dimensionless
parameters of the system:
x 1j =
ρj
r1 - the dimensionless radius of the j-particle orbit;
x 2 j = ϕ j - the dimensionless angular coordinate of the
j-particle;
x 3j =
zj
r1
- the dimensionless deviation from the plane of
orbit at stationary movement;
∂x
x&1j = 1j ;
∂τ
r1 - radius is SC contour.
Expressions:
1 

χ = ⋅ϕ − ϕ 
j S  1
j
Fig.1.
For the case of the movement of S-charge particles in the
electromagnetic field of the ideally electroconductive ring the Lagrange
v
2
function to members of the  order has the form[2]:
c
 
s
L=
∑
i =1
m i ⋅ v 2i
+
2
s
∑
i =1
m i ⋅ v 4i
2⋅c
2
s
+
∑
i =1
[
)]
ql ⋅ q j 
l
 1 − v i ⋅ v j + v i ⋅ n ij ⋅ v i ⋅ n ij  ,
−
⋅

4 ⋅ π ⋅ ε ⋅ ε 0 L =1 R ij 
s
∑
(
)(
1 S
⋅∑ ϕ j
S j= 1
S
ϕ1 = ∑ χj
;
ϕ j = ϕ 1 − S⋅ χ j ;
j= 1
(4)
represent the speeds of the appropriate dimensionless
coordinates. τ = ω ⋅ t , where ω has the dimension 1/c, and t
represents the time coordinate.
We enter new parameters:
γ0 =
l2
l ⋅ Ψ qi + L 11 ⋅ −
2
;
χ1 =
γ1 =
L1 1
β
⋅
⋅
Ψ
m r1 ,
Dj =
x 1j
kj
(
)
⋅  2 − k 2j ⋅ K j − 2 ⋅ E j 


S ⋅ q 2 L11
1
S ⋅ q⋅ µ 0
⋅
⋅ 2 ,γ =
,
2
4π ⋅ ε ⋅ ε 0
r1
Ψ
2π ⋅ L 11 ⋅ m
where Ψ - is the magnetic flux frosted in the SC ring;
(5)
K, E - are the elliptic integrals of the first and second kinds of the
modulus k j .
The new constant of the particle interaction (Coulomb) γ 1 and the
dimensionless constants γ 0 and γ 2 as well as the dimensionless
current and speed of the particles determine the final version of
equations.
These cyclic coordinates are connected with two first integrals of the
motion which are the law of conservation of momentum and the law of
conservation of magnetic flow:
S
S
∂L
 S


= m1 ⋅ ρ 12 ⋅ ∑ χ&j + ∑ m j ⋅ ρ 2j ⋅  ∑ χ i − S ⋅ χ&j  +
 i =1

 ∂χ&1
j= 1
j= 1

S

+ I⋅ ∑ f j + L S ⋅ χ&1 = β;
(6)

j= 1


S
S
S
 ∂ L = f ⋅ χ& + f ⋅  χ& − S ⋅ χ&  + L ⋅ I = Ψ ,
∑
∑ i
1 ∑ j
j
11
j
 ∂ I
 i =1

j= 1
j =1
Fig.2.
Condition for the stationary orbit
γ 0 = γ 0 (x1 ) be existing.
or after correspoding transformations:
N


2
χ&1 ⋅ a11S + I⋅ a12 = β − ∑ χ i ⋅ a11 − N⋅ m j ⋅ ρ j
i= 2



N
χ& ⋅ a + I⋅ a = Ψ − χ& ⋅ a − N⋅ f .
∑
i ( 12
1
12
22
j )


i=2



(
β
);
xk = x k 0 + yk
and consider the
i
i
i
criteria of electron ring stability in the plane of the ideal conductive
contour. Then it is possible to expance the W function in the Taylor
series in the vicinity of origin of coordinates:
Let perturb the system
(7)
W−W
- is the moment of momentum of system;
r r
N
N q ⋅q
n j ⋅ ni ⋅ ρ j ⋅ ρ i
1
j
i
.
LS =
⋅∑∑
⋅
2πεε 0 j =1 i =1 R ij
c2
Ψ
f j = qj ,
ϕ&j
We transit from the Lagrange function to the Hamiltonian. The
division is proceeded accurate to the second member expansion in
theTaylor series concerning the position of the balance:
2
2
2
S m i ⋅ ρ&i
S m i ⋅ z&i
S a11⋅ χ&i
+
+ ∑
H= W+ ∑
+ ∑
2
i =1 2
i=1 2
i=2
(8)
2
a
S
S


+ 11 ⋅ ∑ ∑  χ − χ  ,
i
4 j = 2i = 2 j
where W =
β 2 ⋅ a 22 − 2 ⋅ β ⋅ Ψ ⋅ a 12 + Ψ 2 ⋅ a 11 S
.
2⋅ ∆
0
=
1
⋅
2
S
∑∑
j=1 i =1
S
+
S
∑
j=1
∂W 2
∂ x 1i ∂x 1j
∂W 2
∂x 23j
s
S
S
∑∑
j=1 i =1
j= 1
a 22 = L 11 ,
S
S
j= 1
j= 1
∆ = a 11 ⋅ a 22 − a
2
22
= L 11 .
According with Lyapynov’s stability theory, if we want to stabilize
the system, we need to satisfy conditions:
∂W
∂x1j
0
=0,
∂W
∂x 3 j
0
=0.
(9)
For considering case conditions (9) for the stationary orbit be existing
are submitted on Fig.2.
( )
(10)
+ 0 y3 .
S
∂W 2
∂x 1i ∂x 1j
0 ⋅y 1i ⋅ y 1j +
∂W 2
∑ ∂x
j =1
2
3j
2
0 ⋅y 3 j
.
(11)
According with Silvestr’s criterion, the real square-law form ( 11 )
will be definitely positive, if all main minor of its determinant are
positive, than it is provided by the completion of the inequalities:
∂ 2W
f = ∑ f j = a 12 , a 11 = ∑ m j ⋅ ρ 2j ,
⋅ y 1j +
Last expression represents the Lyapunov function for the system
of charges and the ideal conductive ring. The linear members of
the expansion vanish proceeding from the necessary conditions of
the stability ( 9 ).
Here we have the Lapunov's function in the square-law form:
∂2W
> 0;  2
 ∂x 1j
  ∂2W
0 ⋅
  ∂x 12i
0
It is clear, that the members not included in the W function are
∂x 12j
definitely positive accurate to the second in finitesimal order at
expansion in the Taylor series in the vicinity of the equilibrium orbit.
concerning to the radial stability;
We enter the designations:
∑ m j ⋅ ρ 2j + L s = a 11 ,
2
0 ⋅y 3j
0 ⋅y 1i
∂2w
∂x 23 j
0
>0
  ∂2W
0−
  ∂x 1i ∂x 1j
2

>0
0

( 12 )
(13)
concerning to the longitudinal stability
.
The numerical experiments show, that the inequalities(12,13) are
fulfilled in a wide range of magnetic and geometrical parameters of the
system, that is seen from the results of numerical analysis submitted on
Fig. 3,4,5.
∂2W
= f (x 1 , γ 2 ) .
∂x 12
Fig. 5. Space of stability as a function
Fig.3. Sufficiently stability condition by
x
1 when
∂2W
= f (x 1 ) .
γ =const,
2
∂x 12
Conclusion
So, the above analysis shows that the system for formation and
accompaniment of electron bunches of toroidal geometry with included
SC elements guarantees the stable process run in the parameter range
sufficient for the practical achievement.
References
[1] V.S.Mihalevich, V.V.Kozorez, V.M. Rashkovan
and etc.
"Magnetic potential well - effect of stabilization superconductive
dynamic systems", Kiev, Naukova Dumka, 336 p (1991)
[2] A.M.Korsunsky,N.S.Repalov, Yu.D.Tur. "Control of the Parameters
of Electromagnetic Field in the Linear Accelerators”. Particle
Accelerators ,vol. 11, p11(1980)
Fig.4. Sufficiently stability condition by
x
1 when
x 1 =const,
∂2W
= f (γ ) .
2
∂x 12