# particle diffusion due to coulomb scattering

V. Lebedev and S. Nagaitsev, FNAL, Batavia, IL 60510, USA

Abstract

Conventionally, the multiple and single particle scattering in a storage
ring are considered to be independent. Such an approach is simple and
often yields sufficiently accurate results. Nevertheless, there is a
class of problems where such an approach is not adequate and the single
and multiple scattering need to be considered together. This can be
achieved by solving an integro-differential equation for the particle
distribution function, which correctly treats particle Coulomb
scattering in the presence of betatron motion. A derivation of the
equation is presented in the article. A numerical solution for one
practical case is also considered.

## 1 diffusion equation

Let *f(x,θ, t)* be the one-dimensional beam transverse phase-space
distribution function at time *t*. In the presence of damping and
diffusion the evolution of the function *f* in a ring can be described
by a Fokker-Planck equation:

$\frac{\partial f}{\partial t} + v_{0}\theta\frac{\partial f}{\partial x} + \kappa(t)\frac{\partial f}{\partial\theta} = \lambda(t)\frac{\partial}{\partial\theta}\theta f + \frac{D_{\theta}(t)}{2}\frac{\partial^{2}f}{\partial\theta^{2}}$
. (1)

Here the periodic functions *κ*(*t*), *λ*(*t*) and *D~θ~*(*t*) describe
the focusing, damping and diffusion in the ring, and v~0~=*β*~0~*c* is
the average beam velocity. Making a transition to the action-phase
variables (*I, φ*) and performing averaging over the ring circumference
one obtains,

$\frac{\partial f}{\partial t} - \lambda\frac{\partial}{\partial I}\left( \text{If} \right) = \frac{D}{2}\frac{\partial}{\partial I}\left\lbrack I\frac{\partial f}{\partial I} \right\rbrack$
. (2)

We define the action to be equal to the single particle emittance,

$I = \frac{1}{2}\left( \frac{1 + \alpha(s)^{2}}{\beta(s)}x^{2} + 2\alpha(s)x\theta + \beta(s)\theta^{2} \right)\ ,$
(3)

where *β*(*s* ) and *α*(*s* ) are the beta- and alpha functions of a
ring, and *s* is the longitudinal beam coordinate.

For the residual gas scattering the diffusion coefficient is well known
\[1\]. In the case of scattering in a ring it can be expressed in the
following form,

$D = 4{\pi r}_{e}^{2}v_{0}\left( \frac{m_{e}c^{2}}{{pv}_{0}} \right)^{2}\sum_{i}^{}{Z_{i}(Z_{i} + 1)L_{c}^{i}\oint_{}^{}{n_{i}\beta(s)\frac{\text{ds}}{L}}}\ ,$
(4)

where the summing is performed over partial densities of residual gas
atoms, and the integration averages the gas density weighted by the
*β*-function. For the high energy scattering (*β*~0~\>α*Z~i~*,
*α* ≈1/137) the Coulomb logarithm is

$L_{c}^{i} = \text{ln}\left( \frac{\theta_{\text{max}}^{i}}{\theta_{\text{min}}^{i}} \right)\ ,\ \theta_{\text{min}}^{i} \approx {\alpha Z}_{i^{\frac{1}{3}}}\frac{m_{e}c}{p}\ ,\ \theta_{\text{max}}^{i} \approx \frac{\text{274}}{A_{i^{\frac{1}{3}}}}\frac{m_{e}c}{p}\ ,$
(5)

where the minimum and maximum angles are determined by the field
screening due to atomic electrons and by the diffraction on nuclei.
Usually the ring acceptance, *ε~m~*, is small and does not fit particles
with *θ*~max~ of Eq. (5). In this case *θ*~max~ is determined by
acceptance $\theta_{\text{max}} = \sqrt{\frac{\varepsilon_{m}}{\beta}}$.

The solutions of Eq. (2) are commonly used to describe an emittance
growth in particle accelerators due to various random diffusion
processes. This equation describes well the core of the beam
distribution, but completely fails to describe its tail \[2\]. Far away
tails can be sufficiently well estimated using a single scattering
approximation, but in many applications a prediction of tails behavior
in vicinity of the core is required. It is possible to computer-model
the distribution function by Monte-Carlo methods. However, we found it
beneficial to advance the analytical treatment of the Coulomb scattering
process to a point, where, for a given residual gas pressure, the
corrected distribution function can be obtained with the help of a
simple computer code. Similar approach was used in Ref. \[3\] to analyze
the longitudinal diffusion due to intrabeam scattering (IBS) in a ring
with laser cooling. In what follows we consider the Coulomb scattering
on the residual gas, but the developed theory can be easily adapted to
other Coulomb scattering phenomena.

## 2 collision integral

To simplify formulas we omit below the summation over different gas
species and the contribution of atomic electrons. Then one can write the
differential small angle cross-section in the following form:

$\frac{d\sigma}{d\Omega} \approx 4Z^{2}r_{e}^{2}\left( \frac{m_{e}c^{2}}{v_{0}p} \right)^{2}\frac{1}{\left( \theta_{}^{2} + \theta_{\text{min}}^{2} \right)^{2}}$,
(6)

where *θ*~⊥~^2^=*θ*~x~^2^+*θ*~y~^2^. and
\<$\theta_{} < \theta_{\text{max}}^{l}$ After integrating this over
*θ*~y~ one obtains the one-dimensional cross-section,

$\frac{d\sigma}{{d\theta}_{x}} \approx 2{\pi Z}^{2}r_{e}^{2}\left( \frac{m_{e}c^{2}}{v_{0}p} \right)^{2}\frac{1}{\left( \theta_{x}^{2} + \theta_{\text{min}}^{2} \right)^{\frac{3}{2}}}$
, (7)

and the total cross-section

$\sigma_{\text{tot}} \approx \frac{4{\pi Z}^{2}r_{e}^{2}}{\theta_{\text{min}}^{2}}\left( \frac{m_{e}c^{2}}{v_{0}p} \right)^{2}$
. (8)

For a combined treatment of both the small- and large-angle scattering
one has to write the right-hand side of Eq. (2) in a general form of the
collision integral \[4\]:

$\frac{\partial f}{\partial t} - \lambda\frac{\partial\left( \text{If} \right)}{\partial I} = \left\langle \int_{- \infty}^{\infty}{\left( \frac{d\sigma}{{d\theta}_{x}}|_{\left( \theta - \theta^{'} \right)} - \sigma_{\text{tot}}\delta(\theta - \theta^{'}) \right){nv}_{o}fd\theta^{'}} \right\rangle_{\varphi,s}$(9)

where *δ*(...) is the Dirac's delta-function. Expressing particle angles
and coordinates through the action-phase variables, neglecting *θ*~min~
in the cross-section, and denoting

$B = 4{\pi r}_{e}^{2}v_{0}\left( \frac{m_{e}c^{2}}{{pv}_{0}} \right)^{2}Z^{2}\oint_{}^{}{n\beta\frac{\text{ds}}{L}}$
. (10)

one obtains for the collision integral of Eq (9),

$\begin{array}{r}
\left( \frac{\partial f}{\partial t} \right)_{c} \equiv \left\langle {nv}_{0}\int_{- \infty}^{\infty}{\frac{d\sigma}{{d\theta}_{x}}|_{\left( \theta - \theta^{'} \right)}fd\theta^{'}} \right\rangle|_{\varphi s} = \\
\ \frac{B}{4\pi}\int_{0}^{2\pi}{d\varphi\int_{0}^{2\pi}{d\varphi^{'}}}\int_{0}^{\infty}{dI^{'}}f\left( I^{'} \right)\frac{\delta\left( x(I,\varphi) - x(I^{'},\varphi^{'}) \right)}{\left| \theta(I,\varphi) - \theta(I^{'},\varphi^{'}) \right|^{3}\beta}\ \text{.}
\end{array}$ (11)

Integration over $\varphi$ and $\varphi^{'}$ yields,

$\left( \frac{\partial f}{\partial t} \right)_{c} = \frac{B}{4}\int_{0}^{I_{b}}\frac{I + I^{'}}{\left| I - I^{'} \right|^{3}}f\left( I^{'} \right)dI^{'}\ \text{.}$
(12)

Neglecting *θ*~min~ in transition from Eq.(9) to Eq.(11) causes
divergence of the integral at $I \approx I^{'}$. Instead of performing
an exact integration one can fix this divergence by modifying the kernel
of Eq. (12) similar to the method used to limit divergence in Eq.(6).
Combining Eqs. (9) and (12) one obtains the equation describing the
particle scattering:

$\frac{\partial f}{\partial t} - \lambda\frac{\partial}{\partial I}\left( \text{If} \right) = \int_{0}^{I_{b}}{W(I,I^{'})f(I^{'},t)dI^{'}}$,
(13)

where the kernel *W* is:

$\begin{array}{r}
W(I,I^{'}) = \frac{B}{4}\frac{I + I^{'} + \frac{I_{\text{min}}}{2}}{\left( (I - I^{'})^{2} + (I + I^{'})I_{\text{min}} + \frac{I_{\text{min}}^{2}}{4} \right)^{\frac{3}{2}}} - \\
 \\
 - \sigma_{\text{tot}}\overline{n}v_{o}\delta(I - I^{'})\ ,
\end{array}$ (14)

and $\overline{n}$ is the average gas density in the ring. The particle
conservation requires that

$\int_{0}^{\infty}{W(I,I^{'})dI^{'} = 0}$ . (15)

That yields a relationship between the minimum action and the total
interaction cross-section

$I_{\text{min}} = \frac{B}{2\sigma_{\text{tot}}\overline{n}v_{o}} = \frac{\overline{n\beta}}{\overline{n}}\frac{\theta_{\text{min}^{2}}}{2}$.
(16)

Similarly to Eq. (2), Eq. (13) has a logarithmic accuracy, but unlike
Eq. (2) it correctly describes the large angle scattering. Note that the
form of the kernel, $W(I,I^{'})$, presumes that the angles in the beam
are smaller than the maximum scattering angle,
$\theta_{\text{max}}^{i}$, which is well justified in most practical
cases. Otherwise θ~max~ has to be explicitly taken into account in Eq.
(6).

The accelerator aperture is always finite. Therefore the upper limit in
the integral of Eq. (13) is set by the boundary action value, *I*~b~.
That also yields that the distribution function at the boundary is zero,
*f(I*~b~,*t)*=0.

It is now quite trivial to obtain a Fokker-Planck equation from Eq. (13)
by expanding the function *f* in series at *I′=I*, *f(I′,t)≈
f(I,t)+f′(I,t)(I′-I)+1/2 f′′(I,t)(I′-I)*^2^, and integrating to
$I_{\text{max}} = \overline{\beta}\frac{\theta_{\text{max}^{2}}}{2}$,
where $\overline{\beta}$ is the average ring β-function. The integration
yields:

$\frac{\partial f}{\partial t} - \lambda\frac{\partial}{\partial I}\left( \text{If} \right) \approx \frac{B}{4}\text{ln}\left( \frac{I_{\text{max}}}{I_{\text{min}}} \right)\frac{\partial}{\partial I}\left\lbrack I\frac{\partial f}{\partial I} \right\rbrack$
. (17)

Recalling that ln(*I*~max~/*I*~min~)=2*L~c~* we arrive at Eq. (2).

## 3 Numerical Method

For a numerical solution we use a finite difference algorithm. We split
the total range of the action variable, \[0,*I*~b~\], into *N* equal
size cells, *δI* =*I*~b~/*N*. Then, Eq. (13) can be rewritten as

${\delta f}_{n} = \frac{\delta t}{\delta I}\left( \lambda\frac{f_{n + 1}I_{n + 1} - f_{n - 1}I_{n - 1}}{2} + \frac{B}{4}\sum_{m = 0}^{N - 1}{\overset{\sim}{W}(n,m)}f_{m} \right)$
(18)

Taking into account that the cell size is much larger than the minimum
action, *I*~min~, we can write the probability of a particle exchange
for two distant cells

$\overset{\sim}{W}(n,m) = \frac{n + m}{|n - m|^{3}}\ ,\ n \neq m,\ m \pm 1\ \text{.}$
(19)

Finding the probability of the particle exchange for nearby cells is
complicated by the fact that the kernel of Eq. (13) experiences strong
variations on one cell size. In this case the probability of the
particle exchange between two neighboring cells is equal to,

$\begin{array}{r}
\overline{\delta I\frac{\text{df}}{\text{dt}}}|_{n + 1 \rightarrow n} \approx \int_{I_{n} - \delta I}^{I_{n}}{\text{dI}\int_{I_{n}}^{I_{n} + \delta I}{dI^{'}\left\{ W(I,I^{'})\left( f_{n} + \left( I^{'} - I_{n} \right)f^{'} \right) - \right.\ }} \\
\left. \  - W(I^{'},I)\left( f_{n} + \left( I - I_{n} \right)f^{'} \right) \right\} = f^{'}\int_{I_{n} - \delta I}^{I_{n}}{\text{dI}\int_{I_{n}}^{I_{n} + \delta I}{W(I,I^{'})\left( I^{'} - I \right)dI^{'}}}\text{.}
\end{array}$ (20)

where we presumed that the particle density changes linearly across the
cells, and *I~n~*=*nδI*. Integrating and taking into account that
$\delta I\text{>>}I_{\text{min}}$we obtain

$\begin{array}{r}
\overline{\delta I\frac{\text{df}}{\text{dt}}}|_{n + 1 \rightarrow n} = 2f^{'}I_{n}L_{n}\ ,\  \\
L_{n} = \text{ln}\left( \frac{\left( \delta I + \sqrt{{\delta I}^{2} + 2I_{n}I_{\text{min}}} \right)^{2}}{\left( 2\delta I + \sqrt{4{\delta I}^{2} + 2I_{n}I_{\text{min}}} \right)\sqrt{2I_{n}I_{\text{min}}}} \right)\ \text{.}
\end{array}$ (21)

That yields the probability of particle exchange for nearby cells:

$\overset{\sim}{W}(n,n \pm 1) = L_{I}(2n \pm 1)\ $ . (22)

The probability $\overset{\sim}{W}(n,n)\ $is determined by the particle
conservation so that,

> $\sum_{m = 0}^{\infty}{\overset{\sim}{W}(n,m)} = 0\ $ . (23)

The index *m* in this sum is running to infinity. It takes into account
that a particle can be scattered outside of the accelerator aperture.
Consequently, the particle number is not conserved in a finite aperture
of a ring. Note that the probability of Eq.(20) expresses the difference
analog of Eq. (2) with diffusion coefficient equal to
$\frac{\text{DL}_{n}}{L_{c}}$.

## 4 Experimental results

The above method has been applied to the particle scattering in
Tevatron. First, using the beam scraping it was verified that the
particle distribution of a 150 GeV proton beam, injected into Tevatron,
is very close to a Gaussian one. Second, a new beam was injected. It was
unbunched to exclude the beam heating by the RF noise and the intrabeam
scattering. The beam intensity was sufficiently small to make sure that
the coherent effects did not affect the beam dynamics. Third, the beam
was scraped horizontally and vertically. The scraping time of a few
minutes is much shorter than a characteristic time of the beam
evolution. That and knowing the number of particles removed by the
vertical scraping (\~25%), allowed us to know the initial vertical
particle distribution function. Fourth, the beam scrapers were removed
and the beam was left untouched for one hour. Then, we moved the
vertical scraper in, while recording the beam intensity as a function of
the scraper position. The vertical scraper was chosen so that the beam
momentum spread would not affect the measurements.

![](media/image1.wmf)

Figure 1: Dependence of the beam current on the vertical scraper
position for the beam core (top) and beam tails (bottom); solid line --
measurements, dashed line -- com­puter simulations for *L~I\ ~*= 8.6,
dotted line -- the dependence which would be measured with the initial
distribution; x0 -- the final scraper position at the initial scraping.

The results of the measurements and the comparison with numeric
simulations are shown in Figure 1. Taking into account that only one
fitting parameter, the unknown average Tevatron vacuum, is used there is
a good agreement between the theory and the measurements. Note that
although the Coulomb logarithm is not a well-determined value and
depends on *Z* its uncertainty does not exceed 10-20%. The
experimentally determined value, *L~i~*, coincides with the theoretical
one within 5% for *Z*=7. The measured 5 mm×mrad/hour emittance growth
rate (95%, normalized) corresponds to an average Tevatron vacuum of
3⋅10^-9^ Torr (N~2~ equivalent). Note that high accuracy of the beam
current measurements allowed us to measure tiny tails of the
distribution function, which could not be seen by regular beam profile
monitors. If the large angle scattering is switched off in the
simulations, so that particle scattering is described by diffusion only,
there is large difference between calculations and measurements as
presented in Figure 2.

A good agreement between the observed and the predicted distribution
function tails yields an important practical conclusion that, presently,
the gas scattering is the major beam heating mechanism in the Tevatron.
If other than the Coulomb scattering heating mechanisms were present,
the tails population would be smaller than predicted. Therefore the
planned improvement of Tevatron vacuum should significantly improve the
beam emittance lifetime (currently about 30 hours).

![](media/image2.wmf)

Figure 2: The same as Figure 1 but the large angle scattering is
switched off in the model

## 5 acknowledgments

We are thankful to the entire Tevatron department for helping us perform
the experiment. We are thankful to A. Burov for useful discussions.

## 6 References

\[1\] J.D. Jackson, Classical Electrodynamics, Ch. 13, John Wiley &
Sons, Inc., 1967.

\[2\] A.W. Chao and M. Tigner (editors), "Handbook of Accelerator
Physics and Engineering", p.214, World Scientific, 1999.

\[3\] V.A.Lebedev et al., NIM, A 391 (1997) 176-187.

\[4\] E.M. Lifshits and L.P. Pitaevskii, Physical Kinetics, Pergamon
Press.