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.