M. LindroosNUFACT06 School Accelerator Physics Transverse motion Mats Lindroos

M. LindroosNUFACT06 School Co-ordinates

M. LindroosNUFACT06 School 2 particles in a dipole field Two particles in a dipole field, with different initial angles

M. LindroosNUFACT06 School The second particle oscillates around the first. This type of oscillation forms the basis of all transverse motion in an accelerator. The horizontal displacement of the second particle with respect to the first

M. LindroosNUFACT06 School Effect of a uniform dipole field of length L and field B The beam is deviated by an angle , B  = magnetic rigidity. Dipole

M. LindroosNUFACT06 School Magnetic Rigidity The force evB on a charged particle moving with velocity v in a dipole field of strength B is equal to it’s mass multiplied by it’s acceleration towards the centre of it’s circular path. where  = radius of curvature of the path remember p = momentum = mv B  is called the magnetic rigidity, and if we put in all the correct units we get :- B  [kG.m] = p (if p is in “GeV/c” ) B  [T.m] = p (if p is in “GeV/c” )

M. LindroosNUFACT06 School Quadrupoles A quadrupole has 4 poles, (2 North and 2 South) arranged symmetrically around the beam. No magnetic field along the central axis Magnetic field BxBx ByBy

M. LindroosNUFACT06 School The “normalised” gradient, k is defined as On the x (horizontal) axis the field is vertical and is given by:- B y  x On the y (vertical) axis the field is horizontal and is given by:- B x  y The field gradient, K is defined as

M. LindroosNUFACT06 School This example is a Focusing Quadrupole (QF) It focuses the beam horizontally and defocuses vertically Rotating the poles by 90 degrees we get a Defocusing Quadrupole (QD) Force on a particle FyFy FxFx

M. LindroosNUFACT06 School FODO Cell We will study the FODO cell in detail during the tutorial!

M. LindroosNUFACT06 School As the particles move around the accelerator or storage ring, whenever their divergence (angle) causes them to stray too far from the central trajectory the quadrupoles focus then back towards the central trajectory. This is rather like a ball rolling around a circular gutter.

M. LindroosNUFACT06 School We characterize the position of the particle in this transverse motion by two things:- Position or displacement from central path, and angle with respect to central path. x = displacement x’ = angle = dx/ds This example is for a constant restoring force

M. LindroosNUFACT06 School These transverse oscillations are called Betatron Oscillations, and they exist in both horizontal and vertical planes. The number of such oscillations/turn is Q x or Q y. (Betatron Tune) (Hill’s Equation) describes this motion If the restoring force (K) is constant in “s” then this is just SHM Remember “s” is just longitudinal displacement around the ring Hill’s equation

M. LindroosNUFACT06 School What happens to the motion of the ball in the circular gutter if we allow the shape of the gutter to vary? The phase advance and the amplitude modulation of the oscillation are determined by the shape of the gutter. The overall oscillation amplitude will depend on the initial conditions, i.e. how the motion of the ball started. K varies strongly with “s”…. Therefore we need to solve Hill’s equation for K varying as a function of “s”

M. LindroosNUFACT06 School To solve it, try:  and  0 are constants, which depend on the initial conditions.  (s) = the amplitude modulation due to the changing focusing strength.  (s) = the phase advance, which also depends on focusing strength. Hill’s equation

M. LindroosNUFACT06 School This will give us: If we plot x.v. x’ as  goes from 0 to 2  we get an ellipse, which is called a phase space ellipse x’ x The area of this ellipse is  NB This area does not depend on  or 

M. LindroosNUFACT06 School x’ x x As we move around the machine the shape of this ellipse will change as  changes under the influence of the quadrupoles However the area of the ellipse (  ) will not change  is called the transverse emittance and is determined by the initial beam conditions

M. LindroosNUFACT06 School x’ x x The projection of this ellipse onto the “x” axis gives the physical transverse beam size Therefore the variation of  (s) around the ring will tell us how the transverse beam size will vary

M. LindroosNUFACT06 School To be rigorous we should define the emittance slightly differently. Observe all the particles at single position on one turn and measure both their position and angle A large number of points on our phase space plot, each corresponding to a pair of x,x’ values for each particle. The emittance is the area of the ellipse which contains all (or certain percentage) of the points or particles beam x’ x emittance

M. LindroosNUFACT06 School This conservation of emittance is an illustration of Liouville’s Theorem. Liouville’s theorem states that if x is the transverse position and p x is the transverse mometum, then for a group of particles:- Transverse velocity NB  =v/c,  =m/m 0 constant emittance   is the normalised emittance and it is conserved even during longitudinal acceleration e is only conserved if there is no longitudinal acceleration Lorenz factors!

M. LindroosNUFACT06 School Matrix formalism Represent the particles transverse position and angle by a column matrix. As we move around the ring will vary under the influence of the dipoles, quadrupoles and empty (drift) spaces. Express this modification in terms of a TRANSPORT MATRIX (M). If we know x 1 and x 1 ’ at some point s 1, After the next element in the accelerator ring at s 2 :-

M. LindroosNUFACT06 School Drift space - No magnetic fields, length = L x 2 = x 1 + L.x 1 ’ L x1’x1’ x1x1

M. LindroosNUFACT06 School Remember B y = K.x and the deflection due to the magnetic field is (Provided L is small) x2’x2’ x1x1 x2x2 x1’x1’ deflection Quadrupole of length, L

M. LindroosNUFACT06 School Define focal length of the quadrupole as f = Define f as positive for a focusing effect then:-

M. LindroosNUFACT06 School Multiply together our drift space and our quadrupole matrices to form Transport Matrices, to describe larger sections of our ring These matrices will move our particles from one point (x(s 1 ),x’(s 1 )) on our phase space plot to another (x(s 2 ),x’(s 2 )) The elements of this matrix are fixed by the elements through which the particles pass from s 1 to s 2. However we can also express (x,x’) as solutions to Hill’s equation:-

M. LindroosNUFACT06 School Substitute for x(s 1 ), x(s 1 )’, x(s 2 ) and x(s 2 )’ into this matrix equation. E.g:- Assume that our transport matrix describes a complete turn around the machine Therefore And  = the change in betatron phase over a complete turn

M. LindroosNUFACT06 School These special functions are called TWISS PARAMETERS Remember  is the total betatron phase advance over one complete turn. Now our transport matrix is:- Number of betatron oscillations/turn

M. LindroosNUFACT06 School It is also interesting to note that  (s) and Q or  are related Where  =  over a complete turn But Over one complete turn Increasing the focusing strength decreases the size of the beam envelope (  ) and increases Q and vice versa.

M. LindroosNUFACT06 School What happens if we change the focusing strength slightly? Tune correction This matrix relates the change in the tune to the change in strength of the quadrupoles. We can invert this matrix to calculate change in quadrupole field needed for a given change in tune