4. Statistical description of particle beams

Transcription

1 4. Statistical description of particle beams 4.1. Beam moments

2 4. Statistical description of particle beams 4.1. Beam moments In charged particle beam dynamics, we are commonly not particularly interested in the phase-space location of individual particles statistical mechanics approach is useful. We already studied the ellipsoid transformation approach that maps sets of particles. The phase-space volume associated with this set was described in terms of the beam matri S. In D this matri is εβ() s εα() s Ss () =, dets= ε A= π dets εα() s εγ() s with α(s), β(s), and γ(s) the Courant-Snyder beam functions that characterize the beam ellipse. We can give these functions a statistical interpretation.

3 4. Statistical description of particle beams 4.1. Beam moments For a given distribution of beam particles with coordinates i, we define the moments of this distribution as 1 s () = i() s : first moment in i= 1 n 1 n ( s) = ( i( s) ( s) ) : nth central moment in i= 1 Of course, the same definition applies for all other coordinates. Of particular importance are the second beam moments: 1 1 s () = ( i ) = i ( ) i= 1 i= () s = = ( )( ) i i i i i= 1 i= 1 n ( i ) i ( ) 1 1 () s = = i= 1 i= 1

4 4. Statistical description of particle beams 4.1. Beam moments We now ask: are the moments related to the Courant-Snyder functions? To answer this question, we consider a uniform density of points in the (, ) plane and calculate the second central moments. we make the transition to a continuous description! Assuming a centered distribution (i.e. zero first moments), the second moment in is given by: dd, : populated area of the (, ) ellipse A A = A = dd ρ(, ) dd ρ(, ) dd For a uniform density ρ within an elliptical boundary, this means

5 4. Statistical description of particle beams 4.1. Beam moments To ease the calculation, we can transform the ellipse into a circle by means of the Floquet transformation. the uniform density is maintained. ρ(, ) = 1, = r cos ϕ, = r sin ϕ, dd = rdrdϕ X π X 3 r cos ϕrdrdϕ π rdr X = = = rdrdϕ π r dr 4 r= 0 ϕ= 0 r= 0 X π X r= 0 ϕ= 0 r= 0 On the other hand, we know from Sect. 3.3 that the beam width X is correlated with the Courant-Snyder function β and the emittance ε by Xs () = εβ() s = 1 εβ 4 The factor ¼ is a characteristic of the uniform density!

6 4. Statistical description of particle beams 4.1. Beam moments The same result is obtained for the other second moments. We can write the beam matri S equivalently as S 6 S = c with c = 4 for a uniform density of points in the, plane. The general 6 6 (symmetric) beam matri is thus given by y y λ δ y y λ δ y y y yy yλ yδ = c 6 y y yy y y λ y δ λ λ yλ y λ λ λδ δ δ yδ y δ λδ δ

7 4. Statistical description of particle beams 4.. Moment equations

8 4. Statistical description of particle beams 4.. Moment equations If the motion of the particle set is determined by Hill s equation + K( s) = 0, i = 1,, - number of particles i i then we can set up the equations of motion for the second moments: d ds d ds i =, i i i i = Ks () + i i i i i i i Inserting the second equation into the first yields the second order equation for the second central moment in : d K s + ( ) = 0 ds This form is not useful as it contains two kinds of second moments.

9 4. Statistical description of particle beams 4.. Moment equations Yet, we can epress the unwanted moment in terms of the invariant of Hill s equation D 1 (see Sect. 3.4): D d = = D ds We thus obtain an equation that only contains constants and functions of the second central moment in : d ds 1 d + K( s) D 1 + = 4 d s This equation can be simplified making use of the identity d ds 1 d d = ds ds 0

10 4. Statistical description of particle beams 4.. Moment equations Defining the abbreviation we obtain the final form for the equation of motion of the variance of the set of beam particles in : D + () = 0 d 1 K s 3 ds def = This rms envelope equation is more general than the envelope equation as it applies arbitrary phase-space distributions of the beam particles. In this contet, the invariant D 1 of Hill s equation is commonly referred This equation has the same form as the envelope equation of Sect. 3.3 to as the square of the rms emittance ε rms D ε = 1 rms The rms emittance is thus given by the determinant of the beam matri S.

11 *$ -3 *$ y' 1.5 ' 1.5 y' ' *$ -3 *$ Statistical description of particle beams 4.. Moment equations y *$ *$ y y' y y' 0.5 y ' *$ ' Uniform (left) and non-uniform densities in the D projections of 4D phase-space distributions 1.5 *$-3

12 4. Statistical description of particle beams 4.3. Analysis of emittance measurements

13 4. Statistical description of particle beams 4.3. Analysis of emittance measurements Schlitz Gitter d h Strahlachse Ionenstrahl g a Principle and setup of a slit and collector emittance measurement device

14 4. Statistical description of particle beams 4.3. Analysis of emittance measurements Simple graphical 3D representation of the raw data of a slit-andcollector emittance measurement

15 4. Statistical description of particle beams 4.3. Analysis of emittance measurements Calculation of beam moments from the raw data, given by the current matri i (, ) i, = n, = m nm n = 1,, n, m = 1,, m ma ma with i nm the collector current as a function of n and m, and : step size of the slit position, where > slit width. This defines the spatial resolution of the device. : angle between neighboring collector stripes, which defines the angular resolution.

16 4. Statistical description of particle beams 4.3. Analysis of emittance measurements The first beam moments are now 1 1 = ni n = ni nm inm nm, inm nm, nm, nm, 1 1 = mi m = mi nm inm nm, inm nm, nm, nm, The second moment follow as ( ) ( ) = inm ( n n), = inm ( m m) i i = inm ( n n)( m m) i nm nm, nm nm, nm, nm, nm, nm nm, nm nm

17 4. Statistical description of particle beams 4.3. Analysis of emittance measurements The higher order moments can be calculated correspondingly. The rms emittance is then directly given by ε = rms Properties of this emittance calculation: no assumption is made with respect to the phase-space distribution of the beam particles there is usually no need to define a cut-off current i min, so that i nm =0 for i nm,raw <i min the calculated moments (and hence the rms emittance) can directly be compared to the moments of multi-particle simulations and to the moments continuous (model) distributions.

18 6. Beam Optics with space charge 6.1. Single particle equations of motion

19 6. Beam optics with space charge 6.1. Single particle equations of motion So far, we have only treated single particle dynamics, i.e., the motion of each particle is not influenced by other beam particles. This is an approimation as charged particles always produce a self field! What changes if we take into account the self fields? The description of the system s dynamics is now more complicated: Problem of self-consistency arises: the (changes of the) particle positions depend on the total force, but at the same time, the total force depends on the particle positions. The self-field forces induce additional resonance effects. We get additional sources of emittance growth. We must distinguish a fine-grained description where all individual particles are taken into account from a smoothed description where the self fields are continuous functions of the spatial coordinates.

20 6. Beam optics with space charge 6.1. Single particle equations of motion Simple model of an unbunched beam: homogeneously filled moving cylinder of radius a. We have two components of electromagnetic forces: I Repulsive electric force F B E e φ r dφ r φ Attractive magnetic force F m da a Smooth description: div EdV = EdS, curl BdA = Bds to the beam model of constant density both longitudinally and transversely: ρ I div E =, curl B = µ 0j, ρ =, j = ρβce z ε πa βc 0 y

21 6. Beam optics with space charge 6.1. Single particle equations of motion For the simplified electric and magnetic fields E = Er er, B = Bϕeϕ we find in cylindrical coordinates the forms dv = rdrdϕdz, EdS = Er rdϕdz, E S Bds = B rdϕ, jda = ρcβ rdrdϕ, s B, j A ϕ and hence the field components for a beam of radius a µ = ε c ρ I r I r Er = r = B = cr = ε πε β πε 1, ϕ µρβ c a 0c a ( 1/ ) 0 0

22 6. Beam optics with space charge 6.1. Single particle equations of motion The Lorentz force equation F = e E + v B takes on the particular form ( ) ei Fr() r = e E cb = 1 πε0βc ei r = πε βcγ a ( r β ϕ ) ( β ) 0 Generalization to uniform elliptic beams of half aes X,Y: r a The magnetic field yields the factor 1/γ F = ei, F = ei y y πε βγ + c X X Y πε cβγ Y X + Y ( ) ( ) 0 0

23 6. Beam optics with space charge 6.1. Single particle equations of motion Drift transformation with space charge without space charge beam width X(s) / cm s / cm

24 6. Beam optics with space charge 6.1. Single particle equations of motion Single particle equations of motion that include space-charge forces: 1 + k,et () s F = (, y, s) 0 mγc β 1 y + ky,et () s y F (,, ) = 0 y y s γ β m c We insert the linear space-charge forces of the simplified model: K ei + k () s = = ( + ) 0, K X X Y 4πε mc β γ,et K y + ky =,et ( s) ( + ) y 0. K : " generalized perveance" Y X Y K is a dimensionless measure for the strength of the space charge. K can be regarded as the scaled current.

25 6. Beam optics with space charge 6.1. Single particle equations of motion Linear model: oscillation frequency is the same for all particles,et y,et k = k = k = const. σ = k S Def K k = k σ = ks : depressed tune 0 : zero current phase advance a The reduction of the phase advance σ 0 σ due to space charge forces is often referred to as the Laslett tune shift. The ratio σ/σ 0 is a good measure for the relative strength of the space charge compared to the eternal focusing strength. For synchrotrons, we generally use the terms tune Q 0,,y = σ 0,,y /π i.e., the number of oscillations per turn and the corresponding incoherent tune shift Q,y. We will now derive a formula to estimate Q,y in a synchrotron in terms of the beam parameters.

26 6. Beam optics with space charge 6.1. Single particle equations of motion Continuous solenoid channel, sigma-0=60 deg., zero current kv Continuous solenoid channel, sigma-0=60 deg., sigma=15 deg. kv (cm) 1.0 (cm) y (cm) 1.0 y (cm) Cells Cells Particle oscillation frequency for zero current of σ 0 = 60º (left), depressed to σ = 15º due to space charge (right).

27 6. Beam optics with space charge 6.1. Single particle equations of motion K k = k 0 = ( k 0 k) a k0 k0 k, k k0, Qy, Q0, y, k Qy, = 1 K : relative tune shift k Q ka 0 0, y, 0 For a continuous focusing system, we know from Sec. 3 a S = εβ, β = = σ With R the radius of the synchrotron, this yields 1 R Qy, K, I = ε y, R Q ecβ πr or, in terms of the number of particles along the ring circumference: r0 e y,, r 3 0 πβ γ εy, 4πε0mc Q =

28 6. Beam optics with space charge 6.1. Single particle equations of motion General, nonlinear case (simulation codes): e + k,et() s E = 3 (, y, s) 0 mc βγ e y + ky,et() s y E (,, ) = 0 3 y y s mc βγ with the space-charge fields as the solution of Poisson s equation E E + y 1 = ρ (,,) ys y ε0 ρ(,y,s) may be either the finegrained field of all particles or the smooth field due to a phase space probability density General problem for analytic treatment: the functional form of the charge density ρ is usually not conserved and hence unknown! problem of stationary phase space distributions (treated later).

29 6. Beam Optics with space charge 6.. Moment equations

30 6. Beam optics with space charge 6.. Moment equations Usually, we not particularly interested in single particles. moment equations Discrete description: if the complete information on particles is given, the first moment and the second central moment in are calculated via: 1 1 () s = (), s () s = i() s () s i i= 1 i= 1 ( ) If we succeed in deriving closed equations of motion for the second moments, the system s description is considerably simplified. The equations of motion for the moments are obtained by calculating their time derivatives and inserting the single particle equations of motion.

31 6. Beam optics with space charge 6.. Moment equations In the particle description, the other second moments are defined as 1 1 s () = ( i ) = i ( ) i= 1 i= () s = = ( )( ) i i i i i= 1 i= 1 n ( i ) i ( ) 1 1 () s = = i= 1 i= 1 The equation of motion of the first moment is simply the single particle equation for the center-of-mass motion. We are interested in the equations of motion of the second moments: d s s 1 ( )( ) i i () = = () (M1) ds = i

32 d ds d ds 6. Beam optics with space charge 6.. Moment equations Same procedure for the derivatives of the other second moments: e () s = () s k,et() s () s + E () 3 s mc βγ (M) e () s = k,et() s () s + E () 3 s mc βγ (M3) Here, we have replaced '' according to the single particle equation of motion. ote that the moments that are related to the electric field E involve higher than second moments in general. In that case, the moment equations lead to an infinite hierarchy. For the linear model, only second moments appear closed system of equations.

33 6. Beam optics with space charge 6.. Moment equations Smooth description: based on a phase space probability density f(,,y,y,s), i.e., f dτ is the probability for finding a particle inside a small volume dτ =dd dydy around (,,y,y ) at s. This is the smooth version of the Liouville theorem df = 0 f f f f f + + y + + y = ds s y y 0 Inserting the smooth single particle equation of motion + e smooth k βγ =,et() s E (,, ) 0 3 y s mc we obtain the Vlasov equation, i.e. the closed equation of motion for f f f f e smooth f + + y k,et() s E (,, ) 3 y s s y mc βγ e f 1 k s y E y s = E = fd dy smooth smooth y,et() (,, ) 0, div 3 y mc βγ y ε 0

34 6. Beam optics with space charge 6.. Moment equations Usually, we not particularly interested in the detailed information contained in f (,,y,y,s) moment equations. The first moment and the second central moment in are given by: s () f (,, yy,, sd ) τ, () s ( ) f (,, yy,, sd ) = = To derive the equations of motion of the moments, we evaluate the s-derivatives similar to: d = () s ( ) f (,, y, y, s ) d τ ds s The moments are then calculated inserting the Vlasov equation. The resulting set of equations agrees with single particle approach. Second moment analysis of the Vlasov equation. τ

35 Def 6. Beam optics with space charge 6.. Moment equations Applications of the moment equations We insert equation (M) into equation (M1): 1 ( ) ( ) Def ε i j j i ij, = 1 s () = (), s () s = d e E() s ε () s s () + k,et() s = ds mc βγ In this form, the last equation does not seem to be of much help since it contains two unknown functions of s. Let us study them now. For a beam with elliptical symmetry in real space, we can show that I = ρ = ρ y E + () s (, y) πε β + 4 c y X Y 0 RMS emittance

36 6. Beam optics with space charge 6.. Moment equations The equation for is now obtained as () s 1 d K ε () s s () + k,et() = 0 s 3 ds + y As epected, the space charge couples the - and the y-planes. RMS K-V equation: basis of concept of equivalent beams The only remaining unknown function is the RMS emittance. In order to study its time behavior, we calculate its derivative: d e ( ε () ) s = E 3 E ds mc βγ using equations (M1), (M), and (M3). We immediately conclude: E = 0 " no" space charge field ε () s = const. E linear space charge field

37 For beams with 6. Beam optics with space charge 6.. Moment equations negligible space charge forces, or uniform density and elliptic symmetry in transverse real space, the equation for () s is closed, i.e. does not contain an unknown function. We can then integrate the coupled system 1 d K ε + = s () k,et() s 0 3 ds + y 1 d K ε y () +,et() = 0 ys k y s y 3 ds + y y We often want to know the beam edge. What is the ratio Xs ()/ s ()?

38 π 6. Beam optics with space charge 6.. Moment equations For a -dimensional uniform density ρ(, y), we have ρ(, y) ddy =, ρ(, y) = 1, = rcos ϕ, ddy = rdrdϕ ρ(, y) ddy R 3 r cos ϕrdrdϕ π rdr r= 0 ϕ= 0 r= 0 R = R π = R = rdrdϕ π r dr 4 r= 0 ϕ= 0 r= 0 and hence for an elliptic beam of uniform density transversely: 1 = X R We will later treat second and higher order moments for other phase space density functions. Problem: does an elliptic beam of uniform density stay elliptic and uniform? We will answer this question later.

39 6. Beam optics with space charge 6.. Moment equations In terms of X(s), the RMS K-V envelope equations are simply referred to as the K-V envelope equations d K ε + Xs () k,et() s X 3 ds X + Y X = 0 d K εy () +,et() Ys k y sy 3 ds X + Y Y = 0 with the total emittances defined as ε = 4 ε, ε = 4ε y y For uniform density, the total emittance is the area of the -dim., - and y,y - phase-space projections. The K-V equations are an important tool for the first-order design of high-current lattices. Let us study some eamples.

40 6. Beam optics with space charge 6.. Moment equations After having solved the K-V envelope equations, the X(s) and Y(s) are known, and hence the electric fields for a homogeneous density I I y E( ys,, ) =, E( ys,, ) = πε cβ X( X + Y) πε cβ Y( X + Y) y 0 0 This enables us now to solve the single particle equations of motion K + k () s = 0 Xs ()[ Xs () + Ys ()] K y + ky () s y = 0 Ys ()[ Xs () + Ys ()] The collective character of space-charge dynamics is reflected by the fact that the single particle equations of motion cannot be solved without knowing the global beam transformation.

41 6. Beam optics with space charge 6.. Moment equations Mismatched beam in a periodic quadrupole channel

42 6. Beam optics with space charge 6.. Moment equations Matched beam in a periodic quadrupole channel

43 6. Beam optics with space charge 6.. Moment equations Unstable (chaotic) beam in a periodic quadrupole channel

44 6. Beam optics with space charge 6.. Moment equations The following questions arise: Under which conditions is the beam transformation stable? How do we obtain the matched solution? (remember: the Courant-Synder theory of alternating-gradient synchrotrons of Sec. 3.1 applies for zero current only!) We will see that both questions are closely related. Thus, the net topic will be the stability analysis of the K-V envelope equations.