Single and Multiple Intrabeam Scattering in Hadron Colliders

Valeri Lebedev

FNAL, P.O. Box 500, Batavia, IL 60510, USA

**Abstract.** Single and multiple intra-beam scattering are usually
considered separately. Such separation works well for electron-positron
colliders but usually yields only coarse description in the case of
hadron colliders. Boltzmann type integro-differential equation is used
to describe evolution of longitudinal distribution due to IBS. The
finite size of the longitudinal potential well, its non-linearity and
*x*-*y* coupling are taken into account. The model predictions for
longitudinal and transverse distributions are compared to the
experimental measurements.

# 

# Introduction {#introduction .INTRODUCTION}

The intrabeam scattering (IBS) is closely related to the particle
scattering in plasma and is well understood conceptually^\[1\]^. In the
most general case IBS is described by Boltzmann type
integro-differential equation but its exact solution is not easy even in
simple cases. In this paper IBS is considered in application to high
energy hadron colliders. Because of large *γ*, the longitudinal velocity
spread is much smaller than the transverse in the beam frame resulting
in a simplified description. First, we consider the emittance growth
rates for Gaussian beams in the case of *x-y* coupled motion. This
approximation works well for transverse degrees of freedom; but due to
longitudinal focusing nonlinearity and finite size of RF bucket it
yields a coarse description of longitudinal distribution. To correct
this problem we build the Fokker-Planck equation^\[2\]^ describing the
longitudinal distribution. Then we generalize equations to include the
single-scattering which dominates particle loss at the beginning of the
process and is slowly diminishing with bunch lengthening. Finally, we
compare the theory predictions with the observed beam parameter
evolution during Tevatron stores.

# IBS Rates for coupled Beam

The evolution of distribution function for particle scattering in the
plasma is determined by Landau collision integral^\[1\]^:

$\frac{\partial f}{\partial t} = - \frac{\partial}{\partial p_{i}}\left( F_{i}f \right) + \frac{1}{2}\frac{\partial}{\partial p_{i}}\left( D_{\text{ij}}\frac{\partial f}{\partial p_{j}} \right)\ \text{.}$

Here the friction and diffusion are

$\begin{array}{r}
F_{i}(p) = - \frac{4\pi\text{ne}^{4}L_{c}}{m}\int_{}^{}{f(p^{'})\frac{u_{i}}{|u|^{3}}}d^{3}p^{'}\ ,\  \\
D_{\text{ij}} = 4\pi\text{ne}^{4}L_{c}\int_{}^{}{f(p^{'})\frac{u^{2}\delta_{\text{ij}} - u_{i}u_{j}}{|u|^{3}}}d^{3}p^{'}\ , \\
u = v - v^{'}\ ,
\end{array}$

*n* is the particle density, *e* and *m* are the particle charge and
mass, and *L~c~* is the Coulomb logarithm. Multiplying Eq. (1) by
*p~x~*^2^, and integrating it with Gaussian distribution,

$f = \frac{1}{(2\pi)^{\frac{3}{2}}\sigma_{\text{px}}\sigma_{\text{py}}\sigma_{\text{pz}}}e^{- \ \frac{p_{x^{2}}}{2\sigma_{\text{px}}^{2}}\  - \ \frac{p_{y^{2}}}{2\sigma_{\text{py}}^{2}}\  - \ \frac{p_{z^{2}}}{2\sigma_{\text{pz}}^{2}}}\ ,$

one obtains

$\frac{d}{\text{dt}}\overline{p_{x^{2}}} = \frac{(2\pi)^{\frac{3}{2}}\text{ne}^{4}\text{mL}\ \Psi\left( \sigma_{\text{px}},\sigma_{\text{py}},\sigma_{\text{pz}} \right)}{\sqrt{\sigma_{\text{px}^{2}} + \sigma_{\text{py}^{2}} + \sigma_{\text{pz}^{2}}}}$,

where

$\begin{array}{r}
\Psi(x,y,z) = \frac{\sqrt{2}}{\pi}\sqrt{x^{2} + y^{2} + z^{2}} \\
\int_{0}^{\infty}{\frac{\ \left( \frac{y^{2} - x^{2}}{y^{2} + t^{2}} + \frac{z^{2} - x^{2}}{z^{2} + t^{2}} \right)t^{3}\text{dt}}{\left( x^{2} + t^{2} \right)^{\frac{3}{2}}\left( y^{2} + t^{2} \right)^{\frac{1}{2}}\left( z^{2} + t^{2} \right)^{\frac{1}{2}}}\ \text{.}}
\end{array}$

The growth rates for two other degrees of freedom are obtained from Eq.
(4) by cyclic substitution. For the case of pancake distribution it can
be approximated as

$\Psi(0,x,y) \approx 1 + \frac{\sqrt{2}}{\pi}\text{ln}\left( \frac{x^{2} + y^{2}}{2\text{xy}} \right) - \text{.}\text{055}\left( \frac{x^{2} - y^{2}}{x^{2} + y^{2}} \right)^{2}$.

This function possesses correct asymptotics and coincides with exact
result within \~1%.

To describe IBS in a ring Eq. (4) needs to be averaged across the bunch.
Below we limit ourselves to the case of pancake distribution,
(∆*p*~\|\|~ /*p*)/*γ* \<\<* θ*~⊥~ and the energy well above the critical
energy, *γ *\>\>*γ~cr~*, which is typical for the high-energy colliders.
Then, there is no significant difference between scattering in plasma
and IBS for longitudinal degree of freedom. Averaging Eq. (4) over the
Gaussian distribution in the 3D space and taking into account that the
energy is equally distributed between potential and kinetic, yields the
momentum spread growth rate^\[3\]^:

$\frac{d}{\text{dt}}\left( \sigma_{p^{2}} \right) = \frac{e^{4}\left\langle \frac{N}{\sigma_{1}\sigma_{2}\sigma_{s}}\frac{\Psi\left( 0,\ \theta_{1},\ \theta_{2} \right)}{\sqrt{\theta_{1^{2}} + \theta_{2^{2}}}}L_{C} \right\rangle_{s}}{4\sqrt{2}m^{2}c^{3}\gamma^{3}\beta^{3}}$.

Here
$\sigma_{p} \equiv \sqrt{\overline{\left( \frac{\Delta p}{p} \right)^{2}}}$is
the rms momentum spread, *σ*~1~, *σ*~2~, *θ*~1~ and *θ*~2~ are the rms
ellipse semi-axis in the *x*-*y* plane and in the plane of local angular
spreads (*θ~x~* --* θ~y~*), *N* is the number of particles in the bunch,
and index *s* denotes averaging over the ring circumference. To find the
ellipse sizes we describe the *x*-*y* coupled betatron motion using an
extension of the Mais-Ripken representation^\[4\]^:

$\hat{x}(s) = \text{Re}\left( {\overset{\sim}{\varepsilon}}_{1}v_{1}(s)e^{- {i\mu}_{1}(s)} + {\overset{\sim}{\varepsilon}}_{2}v_{2}(s)e^{- {i\mu}_{2}(s)} \right)$
,

where **x**(s) = \[*x*, *θ*~x~, *y*, *θ*~y~\]*^T^* is the coordinate
vector, **v**~1~ and **v**~2~ are the eigen-vectors,

$v_{1} = \begin{bmatrix}
\sqrt{\beta_{1x}} \\
 - \frac{i(1 - u) + \alpha_{1x}}{\sqrt{\beta_{1x}}} \\
\sqrt{\beta_{1y}}e^{{i\nu}_{1}} \\
 - \frac{\text{iu} + \alpha_{1y}}{\sqrt{\beta_{1y}}}e^{{i\nu}_{1}}
\end{bmatrix}\ ,\ v_{2} = \begin{bmatrix}
\sqrt{\beta_{2x}}e^{{i\nu}_{2}} \\
 - \frac{\text{iu} + \alpha_{2x}}{\sqrt{\beta_{2x}}}e^{{i\nu}_{2}} \\
\sqrt{\beta_{2y}} \\
 - \frac{i(1 - u) + \alpha_{2y}}{\sqrt{\beta_{2y}}}
\end{bmatrix}$ ,

*β*'s and *α*'s represent the extension of the beta- and alpha-functions
for uncoupled motion, and the parameters *u*, *ν*~1,2~ are determined by
symplecticity condition from *β*'s and *α*'s. The bilinear form
inscribing particle ellipsoid in 4D phase space,

${\hat{x}}^{T}\Xi\hat{x} = 1\ $,

can be expressed through the eigen-vectors and the emittances.
Corresponding formulas can be found in Ref. \[4\]. To obtain the beam
sizes and angular spreads we also need to take into account the
dispersion contribution into the beam sizes and angular spreads. In this
case the distribution function is:

$f(\hat{x},\theta_{||}) \propto \text{exp}\left( - \frac{\left( \hat{x} - {D\theta}_{||} \right)^{T}\Xi\left( \hat{x} - {D\theta}_{||} \right)}{2} - \frac{\theta_{{||}^{2}}}{2\sigma_{p^{2}}} \right)\ $,

where **D** = \[*D~x~*, *D′~x~*, *D~y~*, *D′~y~*\]*^T^* is the vector of
dispersions. Integrating over momentum spread we obtain 4D Gaussian
distribution

$f(\hat{x}) \propto \text{exp}\left( - \frac{{\hat{x}}^{T}\Xi^{'}\hat{x}}{2} \right)\ ,\ \Xi^{'} = \Xi - \frac{\Xi\text{DD}^{T}\Xi}{\sigma_{p^{- 2}} + D^{T}\Xi D}\ $.

That yields the beam sizes and local angular spreads:

$\begin{array}{r}
\sigma_{1,2} = \sqrt{\frac{2\left( 1 - \alpha_{\text{xy}^{2}} \right)}{\sigma_{x^{2}} + \sigma_{y^{2}} \pm \sqrt{\left( \sigma_{x^{2}} - \sigma_{y^{2}} \right)^{2} + 4\alpha_{\text{xy}^{2}}\sigma_{x^{2}}\sigma_{y^{2}}}}}, \\
\theta_{1,2} = \sqrt{\frac{2}{\Xi_{\text{22}}^{'} + \Xi_{\text{44}}^{'} \pm \sqrt{\left( \Xi_{\text{22}}^{'} - \Xi_{\text{44}}^{'} \right)^{2} + 4\Xi_{\text{24}^{2}}^{'}}}}\ ,
\end{array}$

$\sigma_{x} = \sqrt{\varepsilon_{1}\beta_{1x} + \varepsilon_{2}\beta_{2x} + D_{x^{2}}\sigma_{p^{2}}}$
,

$\sigma_{y} = \sqrt{\varepsilon_{1}\beta_{1y} + \varepsilon_{2}\beta_{2y} + D_{y^{2}}\sigma_{p^{2}}}\ $
,

$$\alpha_{\text{xy}} = \frac{\varepsilon_{1}\sqrt{\beta_{1x}\beta_{1y}}\text{cos}\nu_{1} + \varepsilon_{2}\sqrt{\beta_{2x}\beta_{2y}}\text{cos}\nu_{2} + D_{x}D_{y}\sigma_{p^{2}}}{\sigma_{x}\sigma_{y}}$$

For *γ *\>\>*γ~cr~*, the main source of the transverse emittance growth
is related to the excitation of betatron oscillations at regions with
non-zero dispersion. In this case the momentum change and the amplitudes
of particle motion are related as,

$D\frac{\Delta p}{p} = \text{Re}\left( a_{1}v_{1} + a_{2}v_{2} \right) \equiv \text{Va}\ $,

where **V** = \[**v**′~1~, -**v**′′~1~, **v**′~2~, -**v**′′~2~\], **a**
= \[a′~1~, a′′~1~, a′~2~, a′~2~\]. Expressing emittances through
coordinates of vector **a**,

$a = \frac{\Delta p}{p}V^{- 1}D$ ,

and averaging yield the emittance growths rates for two transverse
betatron modes:

$\frac{{d\varepsilon}_{1,2}}{\text{dt}} = \left\langle A_{1,2}\frac{d}{\text{dt}}\overline{\left( \frac{\Delta p}{p} \right)^{2}} \right\rangle\ ,$
.

$\begin{array}{r}
A_{1,2} = D^{T}B_{1,2}D\ ,\ B_{1} = \left( V^{- 1} \right)^{T}\begin{bmatrix}
I & 0 \\
0 & 0
\end{bmatrix}V^{- 1}, \\
B_{2} = \left( V^{- 1} \right)^{T}\begin{bmatrix}
0 & 0 \\
0 & I
\end{bmatrix}V^{- 1}\ ,\ I = \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}\ \text{.}
\end{array}$

For uncoupled case Eqs. (7) and (17) coincide with results presented in
Ref. \[3\].

# IBS in nonlinear well

In the case of non-linear oscillator one needs to make a transformation
to the angle-phase variables and to perform averaging over phase. For
v~\|\|~\<\<v~⊥~ one can neglect friction in Eq. (1) that results in:

$\frac{\partial f}{\partial t} = \frac{1}{2}\frac{\partial}{\partial I}\left( I\frac{D(I)}{\omega(I)}\frac{\partial f}{\partial I} \right)$
.

Here *ω* and *I* are the frequency and action for the dimensionless
Hamiltonian,

$H = \frac{p^{2}}{2} + 2\left( \text{sin}\frac{\phi}{2} \right)^{2}$ .

The diffusion coefficient is determined as:

$\begin{array}{r}
D(I) = 4L_{c}\overset{\sim}{A}\ \left( \oint_{}^{}{n(\phi)\text{pd}\phi}/\oint_{}^{}{\text{pd}\phi} \right), \\
\overset{\sim}{A} = \sqrt{\frac{\pi}{2}}\frac{\pi^{2}\alpha N\ e^{4}q^{2}}{\text{eV}_{0}\text{mc}\text{βγ}^{2}C}\left\langle \frac{1}{\sigma_{1}\sigma_{2}}\frac{\Psi\left( 0,\ \theta_{1},\ \theta_{2} \right)}{\sqrt{\theta_{1^{2}} + \theta_{2^{2}}}} \right\rangle_{s}\ ,
\end{array}$

where *C* is the ring circumference, *α* is the momentum compaction, *q*
is the harmonic number, and *V*~0~ is the RF voltage, *n*(*ϕ*) is the
linear density $\int_{}^{}{n(\phi)d\phi} = 1$.

To take into account single scattering we follow the method developed in
Ref. \[5\]. For high energy collider the maximum longitudinal velocity
determined by the RF bucket size is much smaller than the transverse
ones, *θ*~\|\|*max*~/*γ *\<\< *θ*~⊥~, and one can write:

$\begin{array}{r}
\frac{\partial f}{\partial t} = \left\langle \overset{\sim}{A}\int_{}^{}{n(\phi)\frac{f(p + q) - f(p)}{|q|^{3}}dq} \right\rangle_{\text{period}} = \\
\left\langle \overset{\sim}{A}\int_{}^{}{n(\phi)\frac{f(I^{'}) - f(I)}{\left| p - p^{'} \right|^{3}}\delta\left( \phi - \phi^{'} \right)dI^{'}d\psi d\psi^{'}} \right\rangle_{\text{period}},
\end{array}$

which after averaging yields:

$\frac{\partial f(I,t)}{\partial t} = \overset{\sim}{A}\int_{0}^{\infty}{W(I,I^{'})\left( f(I^{'},t) - f(I,t) \right)dI^{'}}$
.

The kernel is a symmetrical function, *W*(*I*, *I*′) =*W*(*I*′, *I*),
which for *I*′ ≥ *I* can be expressed as following

$\begin{array}{r}
W(I,I^{'}) = \frac{\omega\omega^{'}}{\pi\left( H(I) - H(I^{'}) \right)^{3}} \\
\ \left\lbrack \left( H(I) - H(I^{'}) \right)\int_{0}^{a(I)}{n(\phi)\frac{d\phi}{p}\ } + 2\int_{0}^{a(I)}{n(\phi)p\ d\phi\ } \right\rbrack\ \text{.}
\end{array}$

The kernel divergence at *I* = *I*′ needs to be limited at the minimum
action difference corresponding to the maximum impact parameter usually
determined by the beam size. Numerical solution of Eq. (24) is carried
out similar to Ref. \[5\].

# Discussion

To achieve good agreement between the theory and the experiments it was
critical to include other beam-loss and diffusion mechanisms: the
elastic and non-elastic scattering on the residual gas, particle loss
due luminosity, and the longitudinal diffusion due to RF noise. Figure 1
presents the measured and computed dependencies of proton and antiproton
bunch lengths during one of Tevatron stores. Similar good coincidence
has been obtained for the horizontal and vertical beam emittances.
Measured Tevatron optics with *x-y* coupling taken into account has been
used in the calculations. The saturation of antiproton bunch lengthening
in the second half of the store is related to the beam-beam effects,
which make transverse motion unstable for particles with large
synchrotron amplitudes. At *t* = 28 hour a corrector power supply was
lost. That worsened beam-beam effects with consecutive growth of
particle loss and nullifying bunch lengthening.

![](media/image1.wmf)

FIGURE 1. Measured (dotted line) and computed (solid line) dependence of
rms bunch length on time for Store 3678.

# Acknowledgments

The author thanks A. Burov for fruitful discussions and A. Tollestrup
for help in data analysis.

# References

1\. Landau, L. D. *Phys. Zs. Sowjet,* **10**, p. 154, 1936.

2\. Wei, J. and Ruggiero, A.G., *Proc. 1991 Part. Accelerator
Conference,* San-Francisco, pp. 1871-1873.

3\. Bjorken, J. and Mtingwa, S., *Part. Accel.* **13**, p. 115 (1983).

4\. Lebedev, V. and Bogacz, S., "Betatron motion with coupling of
horizontal and vertical degrees of freedom", JLAB-ACC-99-19,
<http://www.jlab.org/div_dept/admin/>
[publications/papers/99/ACC99-19.pdf]{.underline}, 1999.

5\. Lebedev, V. and Nagaitsev, S., *Proc. 2002 European Part.
Accelerator Conference,* Paris, pp. 1362-1364.