Space charge studies based on beta measurement in J-PARC MR K. Ohmi KEK, Accelerator Lab Dec. 10, 2015, talk at Fermilab K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 1 / 36
Overview Hamiltonian and Resonances 1 Action variable representation Standard model Space Charge force and its Hamiltonian 2 Tune shift and tune slope Resonance terms Lattice nonlinearity 3 Tune shift and tune slppe Resonance terms Superperiodicity 4 Breaking of the superperiodicity Beta function measurement Simulation using the resonance Hamiltonian 5 Without Synchrotron Motion With Synchrotron Motion Summary 6 K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 2 / 36
Hamiltonian and Resonances Introduction Particles move with experience of electro-magnetic field of lattice elements and space charge. Slow emittance growth arising in a high intensity circular proton ring is studied. We assume that the beam distribution is static, and each particle moves in the filed of the static distribution. A halo is formed by the nonlinear force due to the electro-magnetic field of the beam itself. The halo, which consists of small part of whole beam, does not a ff ect the electro-magnetic field. Particle motion is described by a single particle Hamiltonian in the field. This picture is not self-consistent for a distortion of beam distribution due to space charge force. Practical issue in J-PARC MR; the beam loss of 0.1-1% during ⇡ 10 , 000 � 100 , 000 turns. K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 3 / 36
Hamiltonian and Resonances Real issue: J-PARC MR operating point Choice of operating point in J-PARC MR. 1 New operating point (21.3,21.4) is better than present one (22.40,20.75) in simulations and experiments. 2 Qualitative understanding of the reasons is necessary. Figure: Tune scan of beam loss in a space charge simulation (SCTR). K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 4 / 36
Hamiltonian and Resonances Beam loss measureemnt at new operating point (21.23,21.31) Figure: Measured beam loss at ( ν x , ν y ) = (21 . 23 , 21 . 31). K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 5 / 36
Hamiltonian and Resonances How high power is expected at new operating point K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 6 / 36
Hamiltonian and Resonances Action variable representation Hamiltonian for a particle under the space charge force Betatron variables with action variable/angle expression. p x ( s ) = 2 β x ( s ) J x cos( φ x ( s )) q y ( s ) = 2 β y ( s ) J y cos( φ y ( s )) . (1) Hamiltonian, which characterize one turn map, is separated by three parts linear betatron motion ( µ J ) nonlinear component of the lattice magnets ( U nl ) space charge potential ( U ). H = µ J + U nl + U sc . (2) Betatron phase advance per turn, µ x = φ x ( s + L ) � φ x ( s ) = ∂ H = µ x + ∂ ( U nl + U sc ) ˜ (3) ∂ J x ∂ J x K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 7 / 36
Hamiltonian and Resonances Action variable representation Fourier expansion of Hamiltonian X H = µ J + U 00 ( J ) + U m x , m y ( J ) exp( � im x φ x � im y φ y ) (4) m x , m y 6 =0 Tune shift, tune slope First and second terms in RHS characterize shift, spread and slope of tune. µ x = ∂ H = µ x + ∂ U 00 ˜ (5) ∂ J x ∂ J x Resonance Resonance occurs, when m x ˜ µ x + m y ˜ µ y = 2 π n is satisfied at a amplitude ( J x , R , J y , R ); e ff ect of U m is accumulated turn by turn. K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 8 / 36
Hamiltonian and Resonances Standard model Expansion around Resonance J. L. Tennyson, AIP Conference proceedings, 87, 345 (1982). Resonance condition m x ˜ µ x ( J x , J y ) + m y ˜ µ y ( J x , J y ) = 2 π n Above condition gives a fixed point(line) in ( J x , J y ) space for particle motion. Expansion of Hamiltonian around the fixed point ∂ 2 U 00 � ( J � J R ) + ( J � J R ) t 1 � U 00 ( J ) = U 00 ( J R ) + ∂ U 00 � � ( J � J R ) � � 2 ∂ J ∂ J ∂ J � J R � J R (6) Tune slope = ∂ 2 U 00 ∂ν i = ∂ν j (7) ∂ J j ∂ J i ∂ J i ∂ J j K. Ohmi (KEK) Space charge on measured beta Dec. 10, 2015, talk at Fermilab 9 / 36
Hamiltonian and Resonances Standard model Standard Model Resonance term around the fixed point U m ( J ) ⇡ U m ( J R ) m = ( m x , m y ) (8) Standardized Hamiltonian H = Λ 2 P 2 1 + U m ( J R ) cos ψ 1 (9) ∂ 2 U 00 ∂ 2 U 00 ∂ 2 U 00 Λ = m 2 + m 2 + m x m y x y ∂ J 2 ∂ J 2 ∂ J x ∂ J y x y Resonance width (full width) r r U m U m ∆ P 1 = 4 ∆ J x = 4 m x (10) Λ Λ Dec. 10, 2015, talk at Fermilab 10 / K. Ohmi (KEK) Space charge on measured beta 36
Hamiltonian and Resonances Standard model Standard Model . Figure: Relation between ( J x , φ x ) and ( P 1 , ψ 1 ). Dec. 10, 2015, talk at Fermilab 11 / K. Ohmi (KEK) Space charge on measured beta 36
Space Charge force and its Hamiltonian Space Charge Potential Assume Gaussian beam in x,y,z ⇣ � x ( s 0 , s ) 2 x + u � y ( s 0 , s ) 2 ⌘ Z 1 1 � exp U sc ( s 0 , s ) = � λ p r p 2 σ 2 2 σ 2 y + u du (11) β 2 γ 3 q p 2 σ 2 2 σ 2 x ( s 0 ) + u y ( s 0 ) + u 0 x ( s 0 , s ) p 2 β x ( s 0 ) J x cos( ϕ x ( s 0 , s ) + φ x ( s )) + η ( s 0 ) δ ( s ) = q y ( s 0 , s ) 2 β y ( s 0 ) J y cos( ϕ y ( s 0 , s ) + φ y ( s )) . = (12) where ϕ x , y ( s 0 , s ) is the betatron phase di ff erence between s and s 0 and η is the dispersion. δ ( s ) is given function of s , not canonical variable. I ds 0 U sc ( s 0 , s ) U sc ( s ) = (13) Dec. 10, 2015, talk at Fermilab 12 / K. Ohmi (KEK) Space charge on measured beta 36
Space Charge force and its Hamiltonian Tune shift and tune slope U 00 Z 1 U 00 ( J x , J y ) = � λ p r p I dt p 2 + t p 2 r yx + t (14) ds β 2 γ 3 0 1 " # X 1 � e � w x η � w y ( � 1) l I l / 2 ( w x ) I l ( v x ) I 0 ( w y ) . l = �1 where t = u / σ 2 x and r yx = σ 2 y / σ 2 x and w x η = β x J x + η 2 δ 2 β x J x w x = x + u . . (15) 2 σ 2 2 σ 2 x + u v x = 2 p 2 β x J x ηδ β y J y w y = y + u . (16) 2 σ 2 2 σ 2 x + u Dec. 10, 2015, talk at Fermilab 13 / K. Ohmi (KEK) Space charge on measured beta 36
Space Charge force and its Hamiltonian Tune shift and tune slope Tune shift Space Charge Potential 2 π ∆ ν x = ∂ U 00 = � λ p r p I ds β x β 2 γ 3 σ 2 ∂ J x x Z 1 e � w x � w y dt (2 + t ) 3 / 2 (2 r yx + t ) 1 / 2 [( I 0 ( w x ) � I 1 ( w x )) I 0 ( w y )] , (17) 0 ∆ν x ∆ν x ∆ν y ∆ν y 0 0 -0.01 -0.01 -0.02 -0.02 -0.03 -0.03 -0.04 -0.04 -0.05 -0.05 -0.06 -0.06 -0.07 -0.07 -0.08 -0.08 -0.09 -0.09 -0.1 -0.1 100 100 80 80 60 60 J y 40 100 J y 40 100 80 80 20 60 20 60 40 40 20 J x 20 J x 0 0 0 0 Figure: Tune spread ( ∆ ν x , y ( J x , J y )) due to space charge force. Dec. 10, 2015, talk at Fermilab 14 / K. Ohmi (KEK) Space charge on measured beta 36
Space Charge force and its Hamiltonian Tune shift and tune slope Tune footprint 0 (21.39,21.43) (22.40,20.75) -0.05 ∆ν y -0.1 -0.15 -0.2 -0.2 -0.15 -0.1 -0.05 0 ∆ν x Figure: Tune footprint ( ∆ ν x , y ( J x , J y )) due to space charge force. Dec. 10, 2015, talk at Fermilab 15 / K. Ohmi (KEK) Space charge on measured beta 36
Space Charge force and its Hamiltonian Tune shift and tune slope Tune slope Space Charge Potential ∂ 2 U 00 ds β 2 = � λ p r p I x ∂ J 2 β 2 γ 3 σ 4 x x Z 1 e � w x � w y dt ⇢ 3 2 I 0 ( w x ) � 2 I 1 ( w x ) + 1 � � 2 I 2 ( w x ) I 0 ( w y ) , (2 + t ) 5 / 2 (2 r yx + t ) 1 / 2 0 U xx U xx U yy U yy 0 0 -5000 -5000 -10000 -10000 -15000 -15000 -20000 -20000 -25000 -25000 100 100 80 80 60 60 J y 40 100 J y 40 100 80 80 20 60 20 60 40 40 20 J x 20 J x 0 0 0 0 Figure: Tune slope ( U ij = ∂ 2 U 00 / ∂ J i ∂ J j ) due to space charge force. Dec. 10, 2015, talk at Fermilab 16 / K. Ohmi (KEK) Space charge on measured beta 36
Space Charge force and its Hamiltonian Resonance terms Resonance terms Z 1 U m x , m y ( J x , J y ) = � λ p r p I du ds β 2 γ 3 q p 2 σ 2 2 σ 2 x + u y + u 0 1 " X ( � 1) ( m x + l + m y ) / 2 δ m x 0 δ m y 0 � exp( � w x η � w y ) l = �1 I ( m x � l ) / 2 ( w x ) I l ( v x ) I m y / 2 ( w y ) e � im x ϕ x � im y ϕ y ⇤ . (18) |U 30 | |U 30 | |U 40 | |U 40 | 7e-09 2e-07 1.8e-07 6e-09 1.6e-07 5e-09 1.4e-07 1.2e-07 4e-09 1e-07 3e-09 8e-08 6e-08 2e-09 4e-08 1e-09 2e-08 0 0 100 100 90 90 80 80 70 70 60 60 50 50 J y 40 0 10 20 30 40 50 60 70 80 90 100 J y 40 0 10 20 30 40 50 60 70 80 90 100 30 30 20 20 10 10 0 J x 0 J x Figure: U 30 ( δ = σ δ ) and U 40 ( δ = 0) due to space charge force. Dec. 10, 2015, talk at Fermilab 17 / K. Ohmi (KEK) Space charge on measured beta 36
Lattice nonlinearity Tune shift and tune slppe Tune shift due to lattice nonlinearity One turn map is given by Taylar expansion of lattice elements. . Figure: Tune spread ( ∆ ν x , y ( J x , J y )) due to lattice nonlinearity. Dec. 10, 2015, talk at Fermilab 18 / K. Ohmi (KEK) Space charge on measured beta 36
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