Accelerator School Transverse Beam Dynamics-2. V. S. Pandit

Transcription

1 Accelerator School 8 Transverse Beam Dnamics- V. S. Pandit

2 Equation of Motion Reference orbit is a single laner curve. Diole is used for bending and quadruole for focusing We use coordinates (r, θ, ) Diole field along direction and no field comonent along θ. B B g B B g Reference orbit has curvature s as indeendent variable along the curve (svt) unit vectors r ˆ, ˆ, θ ˆ move with the article What we want? To know the motion of articles awa from Ref. orbit.

3 A beam consists of man charged articles with different divergence and location from the defined trajector. The also reel each other. In transort beam centroid is set to move on defined ref trajector. All other articles are forced eternall to move about the centroid and remain confined. Oscillations of a articles around the centroid in: Transverse lanes - Betatron oscillations Longitudinal lane - Snchrotron oscillations.

4 Equation of Motion Position of a article can be eressed as R rˆ + ˆ r + For small deviations dθ we have ˆ θ. ˆ θ ˆ θ θ. ˆ Interest is the behavior of the and from the reference orbit. R ( r r θ ) ˆ + (r θ + r θ ) ˆ θ + ˆ The equation of motion of an ion is mr F ˆ ˆ θ ˆ q( v B) q r r θ B r B We take B B r B B g g

5 Equation of Motion Equations of motion are m( r θ ) qr θ ( B m. q. r θ. g. qb r mv qg mv + ( + k k) qg mv g) s vt v v θ << v v v r θ r ( ) << Let the designed momentum be o, so k qg / qb mv + ( + ) ( + ) ( ) r + v

6 Equation of Motion + + ( k) k k qg / In a QM central orbit is straight line so k is the momentum deendent quadruole strength is the weak focusing of a BM Generall / and k are functions of length s. ( s) + K( s) ( s) ( s) when, K( s) k( s) ( s) when, ( s) k( s) K

7 Weak Focusing Let consider the case when. + ( k) + k Both the equations reresent simle harmonic oscillation rovided < k < < k < s.sin( n ) s.sin( n ) qg B k r B r n is the field inde. s qb vt mv eb m These are betatron oscillations. ω t t. n

8 Weak Focusing. sin( nω t).sin( nω t Betatron frequencies. Betatron tunes: Q ω ω n ω ω / ) ω ω n r B r Particle eecute simle harmonic motion in horizontal and vertical lanes - focusing in both lanes. Quadruole strength is ositive but ver small. Weak gradient is roduced b decreasing the magnetic field slowl as radius increases. It has been used successfull in betatron, cclotrons, snchrotrons etc. Betatron tunes (Q,Q z ) <. Betatron oscillation wavelength is larger than the circumference of the machine large deviations from the designed orbit big magnet aerture. n Q ω / ω n B n

9 Strong Focusing B n r B r In weak focusing magnets n.5 In strong focusing case n 5 < n < n >> n << - A magnet with n>> focuses in the vertical lane and defocuses in the horizontal lane A magnet with n<<- defocuses in the vertical lane and focuses in the horizontal lane A series of focusing and defocusing magnets in each lane leads to an overall focusing due to alternating gradient focusing.

10 Alternating Gradient Focusing To kee beam size with a certain limit i.e within the inner diameter of vacuum chamber, we need high horizontal and vertical tunes (Q, Q > ) : Concets of Strong Focusing. QM gives the focusing onl in one lane. But continuous doublets can give horizontal and vertical focusing

11 . Piecewise method of solution Equations of motion with are ( s) + K( s) ( s) K( s) k s ( s) when, ( ) when, K ( s) k( s) Assumtion: Strength arameters are constant within individual magnet. Field dros abrutl to zero at the end. + K This is called hard edge model and here K(s)K. The solutions for equation of motion in terms of initial values For K> (focusing): For K< (defocusing): cos K sin sin K cos s K cosh sinh K cosh K sinh

12 Piecewise method of solution. Qudruole magnet K k d sd be the length of the quadruole For k >, quadruole will roduce focusing in -lane and defocusing in -lane. The corresonding transfer matrices are M F M cos k sin sin k cos M D Under thin lens aroimations (f/(kd)) M F f M D f M cosh k k sinh sinh k cosh For k < the above matrices will be interchanged

13 Piecewise method of solution. Drift Sace: For a drift sace of length sd, k and M M d 3. Diole magnet : For a ure diole magnet k. For horizontal motion / K. For arc length equal to l in the magnet we have l M l cos sin sin cos M

14 Piecewise method of solution 4. Sector magnet : Focusing at the edge: α> : horizontal defocusing, vertical focusing α< : horizontal focusing, vertical defocusing α : no focusing Rectangular diole Different dioles

15 Edge Focusing Wedge is a kind of focusing or defocusing quadruole magnet α α α tan tan f f l M tan α M tan α w s w M M M M M

16 Transfer Matri with disersion, ) ( + sin cos sin ) cos ( sin cos l l

17 Transfer Matri with Field inde, ) ( + + n n sin cos sin ) cos ( sin cos cos sin sin cos For the diole with field inde n d db B d db B n l n n

18 Eamle: Mass sectrometer: articles are searated according to their energ and focused due to the / effect of the diole n M cos sin sin cos ( cos) sin

19 n π Eamle: ± For + Beam broadening 4

20 Solenoid focusing The field of an ideal magnetic solenoid is invariant under transverse rotations about it's ais of smmetr (z). Field for linear dnamics is B( z) ( B ( z) / )( iˆ + ˆ) j + B ( z) kˆ z Derivative is with resect to z. + H + k ( z) + qbz( z) k( z) P P is mechanical momentum Canonical momenta and are k + k z k ( z) Couling term + k( z)( )

21 Solenoid focusing Equations of motion of a article k( z) k ( z) + k( z) + k ( z) k( z) qb z P ( z) Couling between and motions can be removed b transforming coordinates to the rotating Larmor frame of reference ~ + k ( z) ~ ~ ~ + k ( z) ~ ~ Equations of motion ehibit focusing in both the lanes. Solutions will be like a focusing quadruole QP. cosθ sinθ θ ( z) z z sinθ cosθ k( z) d( z)

22 Solenoid focusing The transfer matri needs three stes. (a) Transformation of laborator coordinates to Larmor frame at entr of the solenoid ( R(z ) ). (b) Solution of equations of motion in the Larmor frame (G(z,z)) (c) Transformation of Larmor coordinates to laborator coordinates at the eit of the solenoid (R - (z)) R )G ~ M( z, z) R ( z ( z, z) R( z ) θ k z z ) cosθ k( z)sinθ sinθ k( z)cosθ ( cosθ sinθ kl sinθ k( z)cosθ cosθ k( z)sinθ G sinθ cosθ cosθ k sinθ sinθ k cosθ cosθ k sinθ sinθ k cosθ

23 Solenoid focusing The transfer matri of a hard edge solenoid of length L is C M( L) KSC SC KS SC K C S K SC C cos(kl) S sin(kl) SC KS C KSC S K SC SC K C k z) K qb / P ( θkl is the rotation angle about z ais.

24 Stabilit Criterion The transformation of osition and divergence from one oint to another oint is done through the matri M Y MY Y Y For series of elements such as DP, QP, drift sace etc. having matrices M, M, M 3,..M n Y Mn...M 3M MY f MY o What is the condition for stable motion? Eigen values are the measure for the magnitude of matri elements. Finite eigen values indicate that the transformation matri remains finite as well

25 Stabilit Criterion The eigenvalue equation is MY λ Y Magnitude of MY is changed but direction is same as Y Eigenvalues are the solution of ( M λi) m m λ m m λ λ λ( m + m ) + ( mm mm) Since M is unimodular i.e. det(m), we have λ. λ.cos µ + We get two eigenvalues λ cos µ ( m + m ) i µ e, λ Tr e iµ Both the eigenvalues remain finite for real value of µ (M)

26 Stabilit Criterion MY λ Y λ e i µ, λ e iµ Thus condition for stabilit real µ cos µ TrM What we have learn?

27 Summar What we have learn?. Calculate the transfer Matri M of each element.. Multil all to get final transfer matri of a eriod. 3. Find the eigen values of the final transfer matri. 4. Calculate Phase shift er eriod. 5. A hase shift less than π stabilit of the beam