Problem Lund/Barnard USPAS 207 Consider a round uniform ion beam with a current of ampere, composed of Hg + ions (atomic mass A =200), a kinetic energy of 2 MeV, a beam radius of 2 cm and normalized emittance of mm-mrad. Calculate for these beam parameters (to or 2 significant figures): a) β = v 0 /c (assume non-relativistic beam) b) n = number density of ions in beam c) kt = transverse temperature (express in ev) d) λ D = transverse Debye length e) Q = generalized perveance f) Λ = plasma parameter g) Δφ = potential difference between center and edge of beam. For reference: e =.6 x 0-9 C [proton charge] k B =.38 x 0-23 J/K [Boltzmann's constant] ε 0 = 8.854 x 0-2 F/m [permittivity of free space] c = 3 x 0 8 m/s [speed of light in free space] m amu =.66 x 0-27 kg [atomic mass unit] m amu c 2 = 93. x 0 6 ev [atomic mass unit in ev]
Lund/Barnard USPAS 207 Problem 2 Show that: where g is any beam quantity that is a function of r only.
Lund/Barnard USPAS 207 Problem 3 Let the equation of motion for a single particle be: Here x is the usual transverse coordinate and s is the longitudinal coordinate. α(s) is a coefficient that depends only on s. Calculate the derivative with respect to s of the square of the emittance:
Problem 4 (TPD ) - Lamor Frame For a uniform solenoidal channel: with no acceleration and an axisymmetric ( / θ = 0) beam with B a z (s) = B 0 = const. γ b β b = const. φ = φ r = φ x x r x r r the particle equations of motion reduce to: x = qb 0 mγ b β b c y y = qb 0 mγ b β b c x r = x 2 + y 2 q φ x mγb 3β2 b c2 r r q φ mγb 3β2 b c2 r y r a) Parralel steps taken in the class notes to transform the quations of motion to a co-rotating frame: x = x cos(k l s) + y sin(k l s) ỹ = x sin(k l s) + y cos(k l s) Find an expression for k l to reduce the equations of motion to the decoupled form: x + ˆk x = ỹ + ˆkỹ = q φ x mγb 3β2 b c2 r r q φ ỹ mγb 3β2 b c2 r r and identify ˆk = const. Hint: The transformation can be carried out directly. But you may find the algreba simpler using complex coordinates as in the class notes: z = x + iy i = e iθ = cos θ + i sin θ z = x + iỹ
b) If the direction of the magnetic field is reversed: how will the dynamics be influenced? c) Neglect space-charge: B 0 B 0 φ = 0 and sketch a typical orbit in the rotating Lamor frame. Will this orbit appear more complicated in the Laboratory frame? Why? Bonus: Sketch the orbit taking advantage of simple choices of inital conditions that can always be made through choice of coordinates. 2
Problem 5 (TPD 4) - Lamor Frame a) From the Lorentz Force equation, show that a static magnetic field B a cannot change the kinetic energy of a particle; E = (γ )mc 2 = const. m d dt ( γβ ) = qβ B a Lorentz Force Equation γ = d x β = β 2 c dt Here γ and β are exact (unexpanded) forms that should not be confusd with γ b and β b ẑ. b) In Class, it was shown for a solenoid magnet with azimuthal symmetry ( θ magnetic field can be expanded in terms of the on-axis field as: = 0), that the Br a ( ) ν (r, z) = ν!(ν )! ν= Bz a (r, z) = B z0 (z) + ν= B z0 (z) B a z (r = 0, z) 2ν B z0 (z) ( r ) 2ν z 2ν 2 ( ) ν 2ν B z0 (z) ( r ) 2ν (ν!) 2 z 2ν 2 on-axis field Take E E b and apply the paraxial equations of motion (see Sec. SG class notes) to show that if nonlinear applied force terms are dropped ( x 2, xy, xy, etc.), the equations of motion are: x + (γ bβ b ) (γ b β b ) x B z0 2[Bρ] y B z0 [Bρ] y = y + (γ bβ b ) (γ b β b ) y + B z0 2[Bρ] x + B z0 [Bρ] x = [Bρ] γ bβ b mc q q φ mγb 3β2 b c2 x q φ mγb 3β2 b c2 y B z0 = B z0(z) z c) Qualitative only: If there are no axial acceleration fields, we take γ b β b = const., [Bρ] = const., are the results of part b) inconsistent with part a)? If so, could they still be OK to use? d) Show if we take B a = A, we can generate the linear field components: from A = ˆθ 2 B z0r A = ˆr Br a = B z0 2 z r Bz a = B z0 ( A z r θ A ) ( θ + z ˆθ Ar z A ) z + ẑ ( (raθ ) A ) r r r r θ
e) Paraxial approximate the canonincal angular momentum [ ( P θ = x p + qa )] z as P θ mγ b β b c(xy yx ) + q B z0 2 Use equation of motion in part b) to show that for a beam with φ = φ(r). d dz P θ = 0 P θ = const. ( x 2 + y 2) 2
Problem 6 (TPD 4) A thin lense changes the angle of a particle trajectory but not the coordinate: This action can be specified by transfer matrices applied at s = s i ( ) x x = M(s s i ) Focusing: Defocusing: M F = ( ) 0 f ( ) 0 M D = f ( xi x i ) f > 0 f > 0 From TPD Problem 2 a), free space drift of length L has a transport matrix: ( ) L M 0 = 0 Consider a lattice of period 2L made up of equally spaced F and D lenses with eqaul values of f. a) Use the transfer matrix analysis developed in class to find the range of f for which the particle orbit is stable. b) Calculate cos σ 0 where σ 0 is the particle phase advance. c) For the case of f chosen to correspond to the stability limit, sketch the motion of a particle initial conditions: lim s s i x(s) = x 0 lim x (s) = x o /L s s i where s = s i is the axial location of a focusing thin lens kick, and s s i is just before the kick. Sketch the particle orbit for focusing strength slightly larger than the stability limit. Superimpose the orbit sketch on a diagram of the lattice (see below):
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