transverse beam transfer functio ns via the vlasov equation

NAPAC13 1

Transverse Beam Transfer Functions via the Vlasov Equation

M. Blaskiewicz, V.H. Ranjbar C-AD BNL • Introduction • Calculating BTFs using particle simulations • Calculations with the Vlasov equation • Comparison

Introduction • Stability and Response properties are encapsulated in Beam

Transfer Functions (BTFs). • Measurements are generally straightforward but can be

challenging. • The data can be quite rich. • Reliable predictions and improvements to the model can

result from an adequate theory. • The problem is important enough to warrant more than one

approach.

NAPAC13 2

Calculating BTFs using simulations Starter problem: Consider a kicker giving a kick f(τ) as

the bunch passes on turn 0. Let Df(k, τ) be the beam response (dipole moment) to this kick on turns k=0,1,…

Now take the forcing function Assuming linear response Hence, only one simulation is needed for all Q. NAPAC13 3

gnefnQnF )()2cos(),( ��� �

���

�

�

��

��

�

�

�

�

���

�

�

� �

0

22

)(

0

),(Re

)](2cos[),(

)2cos(),(),(

m

gmjQmf

gnjQn

mng

mf

k

gkf

emDe

emnQmD

ekQknDnD

�� �

��

���

Calculating BTFs using simulations In a real BTF with |ω1|< ω0/2=π/Trev Hence we need the impulse response to a sine kick and a cosine

kick to get the response for all frequencies. Also, BTF is one complex number and not a function of τ. Take the Fourier component of the beam response at the drive

frequency.

NAPAC13 4

)sin()sin()cos()cos(]cos[

2/||],cos[)])(cos[(),(

0101

10

1110

10

���������

������������

nkTAnkTAkTnA

hkTnAkTnAkF

revrev

rev

rev

rev

����

�������

� � �����

�

���� dmjDmDeeBTFb

b

rev jn

m

mTj ),(),()( sincos0

)(1

01 �� ���

�

�

��

Calculations with the Vlasov equation Single particle equations

Vlasov

NAPAC13 5

� ���

�����

�

�

b

dDWPqF

xxQQxQddp

pddx

e

sc

�

��

�������

��

�����

�

2

0111

00

2

),()(2

),(

)],()[(2)(

0),,,,(�

��

���

���

��

�����

�� FQpF

ddp

xF

ddxapxF

s

���� cossin aa ��

Take moments

NAPAC13 6

),,,,(1)(

),,(),,(

������

apxFpx

dxdpaaPaX

�����

�

�

���

!�

���

�

�

���

!

"

� �

� �

� �

� �

��

�"

�"�

"�"���

���

���

�

�

)sin(),,(

),()(2

),(

2)(

2

0111

00

2

2

��#����

��������

��

��$��

���

�

��

aaXadadXd

dXdWPqF

XddXQXQddP

PXQXddX

b

e

s

Head-tail phase First order terms Extend head-tail phase. (new?) Equation for X2 is a linear ODE with “constant” coefficients.

NAPAC13 7

QQQiiQXX ������ 001 )/exp( %�&��

])[,(~)(),()sin( 11 XFaaXaQiQQ scs ����

"����

�

����

��

��'�

�(�

���),()()sin( aaaQsc )/exp(21 sQiXX (�

])[,(~)()/exp(),()( 12 XFaQiaXaiQQ ss ��'�

"(�����

�

���

��

Integrating factor Integrate from ψ to ψ+2π employing periodicity of X2. Back substitute X1 and integrate over ε.

NAPAC13 8

]/)(exp[ sQQi '� ���

dttFtGD )(),(ˆ)(1 �� ��

),(

0

2

0

2),( )sin()()sin(),(ˆ )

� ��

�

� )#)���#� aiGaiG eatdaKeadadatG � �

��� � � �

* +ss

QQaiQaiaK

/))((2exp1/)()(

����"

�'�

����

�� ,,�0

)sin(),(s

sc

QaQQdaG

%�&�%�&� �����$� /11

/1

01

010 )()()(ˆ)()( ie

i eFDeWDF ���� �

�

BTF with f(τ) for n=2. Space charge and a Q=5 resonator wake with frτb=6. Solid lines are tracking results, crosses from Vlasov.

Zero chrom, sc and hi freq imped, comp3.plt

NAPAC13 9

Low frequency resolution BTF taken at frequency equal to inverse bunch length (n=2).

NAPAC13 10

low frequency BTF (n=0). Has a step function wake for various chromaticities.

NAPAC13 11

BTF with same wake and chrom but with f=1/4τb.(n=1) upper and lower sidebands differ.

NAPAC13 12

Like previous but now including space charge.

NAPAC13 13

Conclusions • Transverse beam transfer functions can be calculated reliably

using either simulations or the Vlasov equation. • Only 2 tracking runs are needed to get the full TBTF for

bunches short compared to the machine circumference. • The Vlasov approach requires more CPU power than tracking

for comparable results. This is especially true with large space charge.

NAPAC13 14

Discretize with linear interpolation The matrix element is left as a 2D integral. The driving terms for upper and lower sidebands are

NAPAC13 15

���

���N

Nnnm nFMmD )()( ,1

))� ��

�

)� danTeeaduKMm

p

aiGaiGnm

b p

p

p )sin()(222

0 2,1

2),(),(

, ��� � � �

��

�

��

amxxxTmua p /sin,1||for ||1)(,222 ��-����� �

)exp(),( 00 ����� iniQF ���

16

BTF with f(τ) for n=1. Solid lines are tracking results. Crosses are from Vlasov. Space charge is the collective effect.