Betatron Oscillations

Physics 598ACC Accelerators: Theory and Applications
Instructors: Fred Mills, Deborah Errede
Lecture 4: Betatron Oscillations
19 Jun 2007 Accelerators: Theory and
Applications
1

Summary
A. Mathieu-Hill equation
B. Transfer matrix – properties
C. Floquet theory solutions
D. CSL invariant and emittance
19 Jun 2007 Accelerators: Theory and
Applications
2

Betatron Oscillations
(see “Theory of the Alternating-Gradient Synchrotron”, E.D. Courant and H.S. Snyder)
The equations of motion for betatron oscillations were found to be of the type
4.1
y ′ + Ks ( )y = 0
where K(s) is a periodic function with period C = 2πR, or higher periodicity if the
accelerator "lattice" has higher periodicity, as will usually be the case. In terms of the
magnetic field, its gradient and the "magnetic rigidity” Bρ = P 0c
, K = B 2
⎛
⎜ 0
⎞
⎟
B ′
+
e ⎝ Βρ⎠
Bρ
for the horizontal plane, and K =− B ′ for the vertical plane.
Bρ
Then 4.1 is a Mathieu-Hill equation for y, and Floquet's theorem applies to the
solutions, that is that the two linearly independent solutions can be written in the form
exp[±iψ(s)]w(s), where w(s) is a periodic function of s. We will use this fact later, but
first let us look at some general properties of the solutions.
19 Jun 2007 Accelerators: Theory and
Applications
note
B
'
∂B
= ∂ x
z
3

Let N be the number of periods per revolution, and
be the length of one period.
Since the y equation is linear, the solutions at two points s and s 0 are linearly related. If
Y is the column vector we can write
4.2
⎡ y ⎤
⎢
⎣ y ′ ⎥
⎦
Y()= s M s| s 0
( )Ys 0
( )
L= C N
( ) M( s 2
s 1 )
where M s 2
s 1 is a 2x2 matrix . The determinant of is unity, since the
*
Wronskian determinant is constant because the coefficient of y' is zero in 4.1. We note
that the transformations M form a group, since the identity Μ( s|s)= I exists, the
inverse M −1 ( s | s 0 )= M( s 0
| s) exists because det Μ ≠ 0 , and M( s| s 0 )= M( s| s 1 )M ( s 1
| s 0 ).
Some useful examples of M are when K is constant. For K=0 (drift),
4.3
M( ⎡
s| s 0 )= 1 s− s ⎤
0
⎢
⎣ 0 1 ⎥
⎦
19 Jun 2007 Accelerators: Theory and
Applications
*see notes
4

Wronskians: (see, for example, Morse and Feshbach p. 524)
for a differential equation of the form
" '
y + f(x)y + g(x)y=
0
y y
W(y ,y ) = = y y −y y
1 2 ' '
1 2 ' ' 1 2 2 1
y1 y2
(y 1 ,y 2 solutions to the eqn above)
for f(x), g(x) continuous on an open interval I, two solutions y 1 , y 2 are linearly
independent if their Wronskian W is nonzero for any range of x in I.
Also Abel’s Theorem states that for the above differential eqn the Wronskian of the two
solutions is
−∫f (z)dz
W(y ,y )(x) = W(y ,y )(x )e = ce
1 2 1 2 0
note that if f(x)=0, the Wronskian is constant. (canonical Poisson brackets = 1, and components of the
brackets almost correspond to the elements of the matrix M. The Poisson bracket is the determinant of M)
x
−
0 0
x
∫
f (z)dz
19 Jun 2007 Accelerators: Theory and
Applications
5

When K is positive (F or focusing),
4.4
M( s| s 0 )=
⎡
cosφ
⎢
⎢
⎣ − Ksinφ
sinφ ⎤
K
⎥
⎥
cosφ⎦
while if K is negative (D or defocusing),
4.5
M( s| s 0 )=
⎡
⎢
⎢
⎣
coshψ
-Ksinhψ
sinhψ ⎤
⎥
-K
⎥
coshψ⎦
( ) ψ= -K( s − s ) 0
Here φ= K s − s 0 , and .
19 Jun 2007 Accelerators: Theory and
Applications
6

Now let us define the matrix for one period
4.6
⎡a
b⎤
M(s + L | s ) = M(
s)
= ⎢
c d ⎥
⎣ ⎦
The matrix for one revolution is Μ N ( s) , and for k revolutions is Μ kN ( s)
. The motion
will remain bounded if the matrix elements remain bounded as kN-> ∞. Consider the
eigenvalues λ of M
4.7 Μ Y =λY
Solutions exist if
4.8
The equation for λ becomes
4.9
det Μ − λI = 0
λ 2 −λ( a + d)+ 1 = 0
19 Jun 2007 Accelerators: Theory and
Applications
7

Let
4.10 cosμ = 1 2 Tr M = a + d
2
The two solutions of 4.8 become
µ is real if
4.12
λ=cosμ ± isinμ
4.11 *see homework
a + d ≤ 2
Now define α, β, and γ in the following way;
a − d = 2αsinμ
4.13 b =βsinμ
c =−γsinμ
det Μ =1
implies that
4.14
γ = 1+α 2
β
19 Jun 2007 Accelerators: Theory and
Applications
8

for transfer matrix on p.10
cosμ=
αsin
μ=
a+
d
2
a−
d
2
b and c are given in 4.13.
a+ d a−d
a = + = cosμ+αsinμ
2 2
d= cosμ−αsinμ
19 Jun 2007 Accelerators: Theory and
Applications
9

The transfer matrix is now
* see notes p.9
4.15
⎡
Μ()= s
a b ⎤
⎣
⎢ c d⎦
⎥
⎡ cosμ + αsinμ
= βsinμ
⎣ ⎢ −γsinμ
⎤
cosμ - αsinμ⎦
⎥ = I cosμ+Jsinμ
⎡
4.16 J = α β ⎤
,
⎣ ⎢ −γ −α⎦
⎥ det J =1
4.17
M Nk = ( Icosμ +Jsinμ) Nk = IcosNkμ +JsinNkμ
Then if µ is real, the matrix elements of M Nk remain bounded, and the motion is stable.
We note that M()= 0 I,
4.19
M −1 ()= μ M( -μ),
M( μ 1
+μ 2 )= M( μ 1 )M( μ 2 )
19 Jun 2007 Accelerators: Theory and
Applications
10

The definition of µ does not depend on the point s where the matrix is defined, for, if
we calculate the matrix between s 1 and s 2 +L, first by going from s 1 to s 2 , then by going
from s 2 to s 2 +L, if M s 2
s 1 is the matrix connecting s 1 and s 2 ,
while if we first go from s 1 to s 1 +L, and then from s 1 +L to s 2 +L,
Then
4.20
4.21
( )
M( s 2
+ Ls 1 )= M( s 2
)M( s 2
s 1 )
( ) ( )
(see definition 4.6)
M(s + L | s) = M s + L| s + L M s = M(s | s) M(s)
2 1 2 1 1 2 1 1
M( s 2
)= M( s 2
s 1 )M( s 1
)M −1 s 2
s 1
( )
Thus the two matrices are related by a similarity transformation. Thus if M(s )Y = λY
,
M
' '
(s
2)Y =λY
( ) ′
also, where Y = M −1 s 2
s 1
Y . If det M( s 2
)−λI = 0 then det M( s 1
)− λI = 0
also, and the two matrices have the same eigenvalues, hence the same value of µ.
1
19 Jun 2007 Accelerators: Theory and
Applications
11

Modern accelerators have lattices which are composed of successive regions of
constant values of K (which might include curvature). The matrix for each region is
one of the forms 4.3 to 4.5. Then we can find the one period matrix M(s) by matrix
multiplication, and find µ from the trace of that matrix, and α, β, γ by using 4.13. On
the other hand, to find α, β, γ at every point in the lattice by this method is tedious and
time consuming. We will develop better means to calculate these functions, but first we
need to learn more about them.
Let us attempt a solution to 4.1 in the phase-amplitude form
4.22
where w and ψ are real, and w is periodic. Then
4.23 y ′ + Ky = [ w ′ ± i2 ( w ′ ψ ′ + w ψ ′ )− w ( ψ ′) 2 + Kw]exp( iψ)= 0
The exponent is not zero, in general, so both the real and imaginary parts of the bracket
must vanish.
y = ws ()exp[±iψ( s)]
19 Jun 2007 Accelerators: Theory and
Applications
12

For the imaginary part,
( ) ′ = 0 ′
4.24 w 2 ψ ′ , or
ψ = 1 w 2
where we have taken the arbitrary constant of integration to be 1 by absorbing it into the
definition of w. For the real part,
w ′′
+ Kw − 1 = 0
w 3
4.25 (using 4.24)
We are now in a position to express M in terms of w and ψ. Any solution can be
written as a linear combination of the two linearly independent solutions
4.26
y = Awcosψ + Bwsinψ
⎛
y ′ = A w ′ cosψ− sinψ ⎞
⎝ w ⎠ + B ⎛
w ′ sinψ+cosψ ⎞
⎝ w ⎠
19 Jun 2007 Accelerators: Theory and
Applications
13

We can evaluate A and B at s 1 , let ψ=0, y=y 1 , and y ' =y 1‘ . Then A = y 1
,
and B = w 1
y ′ 1
− w ′ 1
y 1
. Introduce the values of A and B into 4.26 and
collect coefficients of y 1 and y 1
'
to find
4.27
M( s 2
s 1 )=
⎡
⎢
⎢ ⎛
cosψ⎜
⎣
⎢ ⎝
w ′ 2
−
w 1
cosψ w 2
− w 2
w 1
′ sinψ
w 1
w 1
′
w 2
⎞
⎟ − sinψ
⎠
w 1
⎤
w 1
w 2
sinψ
⎥
⎛
⎜
1 ⎞
+ w 1
′ w ′ 2
⎟ cosψ w 1
⎥
+ sinψ w 1
′ w
2′
⎝ w 1
w 2
⎠ w 2
⎥
⎦
Now we evaluate M in the case s 2 = s 1 +L, and require w 1 = w 2 = w
4.28
M( s 2
)=
⎡ cosψ − w w ′ sinψ w 2 sinψ ⎤
⎢
− ⎛ 1 ⎝ w + ( w ′ ⎞
2 )2
⎠
sinψ cosψ+w w ′
⎣
sinψ ⎥
⎦
19 Jun 2007 Accelerators: Theory and
Applications
14

Now we compare 4.28 with 4.15, and we can find the following relations
4.29
w 2 = β
w w ′ =−α
α=− β ′
2
μ=ψ( s + Ls)−ψs
()=
α ′ = Kβ−γ
1
γ= +
2
w
s + L
∫
s
ds
β
'
( w ) 2
s+
L
= ∫
s
dψ
ds
ds
(see 4.24)
(use 4.25)
The last equation follows from 4.25 and the relations between w, α, and β. Another
useful differential relationship exists for γ, although α, β, and γ are related by 4.14,
4.30
γ ′ =2Kα
19 Jun 2007 Accelerators: Theory and
Applications
15

We can now use 4.29 and 4.30 in 4.27 to find a general transfer matrix between any
two points 1 and 2 in the lattice,
4.31
⎡
⎢
M( s 2
s 1 )= ⎢
⎢
⎣
β 2
β 1
( cosψ +α 1
sinψ ) β 1
β 2
sinψ
⎛
cosψ α 1
−α 2
⎞ ⎛
⎜ ⎟ − sinψ 1+α 1α 2
⎞
⎜ ⎟
⎝ β 1
β 2 ⎠ ⎝ β 1
β 2 ⎠
β 1
β 2
⎤
⎥
⎥
( cosψ−α 2
sinψ)
⎥
⎦
We will use these differential relationships together with matrix properties to solve for
lattice parameters. First we wish to define an important quantity, ν, which is related to
the phase advance µ, but is a property of the whole accelerator.
4.32
ν= Nμ
2π =
19 Jun 2007 Accelerators: Theory and
Applications
∫
s
s+ C
ds
2πβ
ν is the total number of
betatron oscillations in one
revolution (in the y coordinate).
see 4.29
16

We can also find a constant of the linear motion, W,
called the CSL (Courant, Livingston, and Snyder) invariant.
Any solution can be written, by virtue of 4.29,
4.33
y= Wβcos( ψ+δ)
W
β
[ ]
'
y = sin( ψ+δ) −αcos( ψ+δ)
where W and δ are constants. Solving for cos and sin, squaring and adding,
4.34 W = y2 +β′ ( y +αy) 2
β
=γy 2 + 2αy y ′ +β y ′
2
19 Jun 2007 Accelerators: Theory and
Applications
17

This is the equation of an ellipse of area πW in y,y ' space. The ellipse is upright when
α = 0 and tilted otherwise. The maximum value of y at a given point in the lattice is
βW
because of 4.33. If βW is the maximum amplitude of betatron oscillation
of particles in a beam, then the beam "emittance", which is the area enclosing all the
beam particles, is πW. Note that πWP is the projected area of the orbit in phase space,
and will tend to remain constant as the particles are accelerated (adiabatic invariant).
19 Jun 2007 Accelerators: Theory and
Applications
18

End of Lecture
19 Jun 2007 Accelerators: Theory and
Applications
19

19 Jun 2007 Accelerators: Theory and
Applications
20