Physics 598ACC Accelerators: Theory and Applications

Transcription

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

2 Summary A. Mathieu-Hill equation B. Transfer matrix properties C. Floquet theory solutions D. CSL invariant and emittance 2

3 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. note B ' B = x z 3

4 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 *see notes 4

5 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 ' ' 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 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 5

6 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. 6

7 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 7

8 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 8

9 for transfer matrix on p.10 cosμ= αsin μ= a+ d 2 a d 2 b and c are given in a+ d a d a = + = cosμ+αsinμ 2 2 d= cosμ αsinμ 9

10 The transfer matrix is now * see notes p Μ()= s a b c d cosμ + αsinμ = βsinμ γsinμ cosμ - αsinμ = I cosμ+jsinμ 4.16 J = α β, γ α det J = 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 ) 10

11 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 ( ) 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) 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 11

12 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 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)] 12

13 For the imaginary part, ( ) = 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 (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 13

14 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 14

15 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 15

16 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 ν= Nμ 2π = s s+ C ds 2πβ ν is the total number of betatron oscillations in one revolution (in the y coordinate). see

17 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 17

18 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 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). 18

19 End of Lecture 19

20 20