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\begin{document}
\title{METHODS FOR FINDING STATIONARY LONGITUDINAL \\
PHASE SPACE DISTRIBUTIONS}
\author{M.~D'YACHKOV\\
\it Physics Department, UBC, 6224 Agricultural Road, \\
\it Vancouver, B.C. V6T 1Z1, Canada\\
\\[0mm]
and \\
\\[0mm]
R.~BAARTMAN \\
\it TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C.
V6T 2A3, Canada\\
}
\maketitle
\begin{abstract}
Longitudinal instability thresholds for broadband and space charge
impedance depend on the fact that the stationary distribution changes
with intensity. Non-iterative methods are described and employed to find
stationary phase space distributions and line densities under the
influence of a constant imaginary $Z/n$. These methods have some
advantages over iterative ones; they take less computational time and no
relaxation technique is required. In the `negative mass', or
bunch-shortening regime (eg. space charge impedance above transition),
no stationary distributions can be found beyond a certain threshold
intensity. It was found that for certain types of hollow distributions
a threshold should also exist for the `positive mass' regime.
\end{abstract}
\section{Introduction}
The stability of a bunched beam should be defined with respect to a
stationary distribution. In the case of broadband or space charge
impedance the stationary distribution changes significantly with
intensity \cite{bart,oide}. As has been shown recently \cite{oide} the
thresholds calculated ignoring the potential well distortion differ from
those obtained in self-consistent calculations so the problem of finding
stationary distributions is very important for stability analysis. The
first self-consistent theory developed for electron beams led to a
nonlinear integral equation which is known as the ``Ha\"{\i}ssinski
equation"\cite{hass}, but this equation is valid for the
Maxwell-Boltzmann distribution only. A review of self-consistent
integral equations used to find stationary distributions in electron
bunches can be found in ref.\,\cite{zott}. It has been shown that these
formulations are equivalent to the Ha\"{\i}ssinski equation. Different
distribution functions were studied in ref.\,\cite{bart} for the case of
space charge impedance, for which the self-force is proportional to the
derivative of the beam's line density. However, the iterative technique
employed in that paper diverges in some cases.
The method described in this paper can be used to find the
self-consistent distribution and line density in the case of space
charge impedance for any distribution function. (This feature of this
method is important for proton beams.) It can also be used to determine
threshold intensities beyond which no stationary distribution can exist.
In the iterative method \cite{bart}, thresholds are difficult to find
because the numerical technique becomes unstable near threshold. The
method can also be inverted to recover the distribution function from a
given stationary line density.
\section{Space Charge Impedance}
In the case of space charge impedance the self-force is proportional to
the derivative of the line density and one can choose the normalization
in such a way that the Hamiltonian for the particle in the bunch can be
written as
\begin{eqnarray}
H(p,q) = {p^2 \over 2} + V(q),
\label{1}
\end{eqnarray}
where $p$ is a longitudinal momentum and $V(q)$ is a potential given by
\begin{eqnarray}
V = {q^2 \over 2} + I (\lambda(q) - \lambda_0)\,.
\label{2}
\end{eqnarray}
where $q$ is a longitudinal coordinate and $\lambda(q)$ is a line
density. The constant $\lambda_0 = \lambda(0)$ is added in \refeq{2} to
keep the minimum value of the Hamiltonian \refeq{1} constant
(min$[H(p,q)] = 0$). If this is not done then the minimum value of the
Hamiltonian \refeq{1} may change with intensity and some important
features of the distribution may be lost (e.g. one can lose the `hole'
in a hollow distribution as the intensity is raised).
If the line density in \refeq{2} has been normalized as $\int\lambda
dq=1$, the intensity parameter $I$ in terms of machine parameters is
\begin{eqnarray}
I={2\pi hI_0\over V\cos\phi_s}{\rm Im}\left({Z \over n}\right) \,,
\label{3}
\end{eqnarray}
where $h$, $V$ and $I_0$ are resp. harmonic number, rf voltage and
average current. The impedance $Z/n$ is imaginary in the case of
space charge. We allow $I$ to have either sign: for space charge it is
positive below transition and negative above.
The problem we study is how the phase space trajectories ($H = {\rm
const}$) and line density depend upon the beam intensity and on the
distribution of particles in the beam. A distribution is stationary if
it can be written as a function of the Hamiltonian \refeq{1}. For an
arbitrary distribution function $f(H(p,q))$ (well-behaved in the sense
that it is positive, smooth, non-singular, and $f(\infty)=f'(\infty)=
0$), the line density $\lambda(q)\propto\int f(H(p,q))dp$ cannot be
found because by \refeq{2} $H$ itself depends upon $\lambda$.
Nevertheless, we can find $\tilde\lambda(V)\propto\lambda(q(V))$ by
\begin{eqnarray}
\tilde{\lambda}(V)=2\int_0^\infty f(V+p^2/2)dp=
2\int_0^\infty{f(V+\xi)\over\sqrt{2\xi}}
d\xi\,.
\label{4}
\end{eqnarray}
Proper normalization is retained if we define a parameter $\tilde I$ so
that
\begin{eqnarray}
I=\tilde{I}\int_0^\infty\tilde{\lambda}(V)\,{dq\over dV}\,dV\,.
\label{5}
\end{eqnarray}
We can see that $I\lambda(q)=\tilde{I}\tilde{\lambda}(V)$ and from
\refeq{2}
\begin{eqnarray}
q=\pm\sqrt{2\left[V+\tilde{I}\left(\tilde{\lambda}_0-
\tilde{\lambda}(V)\right)\right]}.
\label{6}
\end{eqnarray}
Since $V(q)$ is a symmetric function, we need only consider the
interval $q\geq 0$. A significant feature of self-consistent stationary
distributions with space charge is that they have only one fixed point:
one can see from \refeq{6} that $V(q_1)=V(q_2)$ only if $q_1=\pm q_2$,
so the only fixed point is $q=0$.
Integrating \refeq{5} by parts and taking into account \refeq{6} we
find
\begin{eqnarray}
I=-2\tilde{I}\int_0^\infty\tilde{\lambda}'(V)
\sqrt{2\left[V+\tilde{I}\left(\tilde{\lambda}_0-
\tilde{\lambda}(V)\right)\right]}\,dV\,.
\label{13}
\end{eqnarray}
Therefore, we can find $I$, the line density and the potential well
as a function of the parameter $\tilde{I}$. The integral \refeq{4}, and
therefore also $q(V)$ \refeq{6}, can
be calculated analytically for a number of different distribution
functions.
Unfortunately, it is not always possible to find an analytical
expression for $V(q)$, but for many applications this is not very
important. The integral \refeq{13} can be easily evaluated numerically.
Of especial importance is the fact that the described method allows us
to determine threshold conditions, beyond which no stationary
distribution exists.
\section{Threshold}
We can easily find the threshold condition by differentiating \refeq{6}.
No stationary distribution can be found if
$\tilde{I}\tilde{\lambda}'(V)=1$, because in this case
$dV/dq \rightarrow \infty$. Therefore,
to define thresholds for an arbitrary distribution
function we need to find extremes of $\tilde{\lambda}'(V)$. Results obtained
for some distribution functions are summarized in Table\,1.
\begin{table}
\begin{center}
\begin{tabular}{|c|l|c|c|}
\hline
$f(H)$&Type&$I_{th}^+$ &$I_{th}^-$\\
\hline\hline
$\sqrt{1-H}$&local ell.
&$\infty$ &0 \\
$(1-{H \over 2})^{3 \over 2}$& binomial
&$\infty$
&$-{2 \sqrt{2} \over 3} = -0.94$ \\
$(1-{H \over 3})^{5 \over 2}$ & binomial
&$\infty$
&${64-27\sqrt{6}\over 15/8} = -1.14$ \\
$e^{-H}$ &gaussian
&$\infty$ &$-1.55$ \\
$H \sqrt{1-H}$ &hollow
&$\sqrt{3}$
&$\sqrt{3 \over 2} \left({1-{\pi \over 2}}\right) = -0.70$ \\
$H e^{-H}$ &hollow
&$5.47$ &$-8.82$ \\
$H^2 e^{-H}$ &hollow
&$16.72$
&$-14.58$ \\
\hline
\end{tabular}
\caption{Threshold parameters for different distribution functions}
\end{center}
\end{table}
Some qualitative conclusions can be drawn from this procedure (here and
later we assume that $\tilde{\lambda}'(V)$ is a smooth function,
otherwise the self force in \refeq{2} cannot exist).
\begin{itemize}
\item There is always a `negative mass' threshold ($I_{\rm th}<0$)
because for any realistic distribution function there is always a region
at the outside edge where $f'(H)<0$.
\item If $f(H)$ is monotonic there is no `positive mass' threshold
($I_{\rm th}>0$).
\item Hollow distributions with $f(0)=0$ always have a `positive mass'
threshold.
\end{itemize}
The last item is not obvious, but one can show that
$\tilde{\lambda}'(0)>0$ for such a distribution.
We illustrate it for the case of hollow distributions which have
only one maximum. We also assume that $f(H)$ and $f'(H)$ have no breaks
and singularities at $H > 0$, and $f(0) = f(\infty) = 0$.
Since the distribution has only one maximum there is a point
$H_m > 0$ for which $f'(H) \geq 0$ for $0 < H < H_m$ and $f'(H) \leq 0$
for $H_m < H$.
Therefore, we can see that
\begin{eqnarray}
{f'(\xi)\over\sqrt{2\xi}}>{f'(\xi)\over\sqrt{2H_m}}
\label{a1}
\end{eqnarray}
for any $\xi > 0$ where $f'(\xi) \not = 0$. Differentiating \refeq{4}
using \refeq{a1} we get
\begin{equation}
\tilde\lambda'(0)=2\int_0^\infty{f'(\xi)\over\sqrt{2\xi}}d\xi>
{2\over\sqrt{2H_m}}\int_0^\infty f'(\xi)d\xi=0
\label{a2}
\end{equation}
The threshold found for the gaussian distribution $e^{-H}$ by the method
described here ($I_{\rm th}$ = 1.551) is in good agreement with the
result obtained in ref.\,\cite{bart} by solving the Ha\"{\i}ssinski integral
equation iteratively ($I_{\rm th}$ = 1.552). Line densities at the
thresholds for some distributions from Table\,1 are shown in
Fig.\,1 and Fig.\,2 (dashed lines correspond to the points where
$d\lambda/dV \rightarrow \infty$)
\begin{figure}\centering
\insertplot{HEACC92B.PL1}{1.0in}
\caption{Line densities at the `negative mass' threshold.}
\end{figure}
\begin{figure}\centering
\insertplot{HEACC92B.PL2}{1.0in}
\caption{Line densities at the `positive mass' threshold.}
\end{figure}
The dependence of $I_{\rm th}$
on the parameter $\mu$ for the binomial distributions
is shown in Fig.\,3. The binomial distribution has been choosen in such a
form that it converges to $e^{-H}$ if $\mu \rightarrow \infty$.
\begin{figure}\centering
\insertplot{HEACC92B.PL3}{2.0in}
%\fbox{\begin{picture}(162,108)\end{picture}}
\caption{The dependence of $I_{\rm th}$ on the parameter $\mu$ for the
binomial distributions $f(H)=\left(1-{H\over\mu+1/2}\right)^{\mu}$}
\end{figure}
\section{Recovering the Distribution Function}
In this section we describe an algorithm which allows us to recover a
stationary distribution from the line density in the case of space
charge. This algorithm is useful, for example, for finding an initial
distribution of particles to use in a multi-particle tracking code,
given the line density. It can be derived from \refeq{2} and \refeq{4}.
Let us choose two points $V_1$ and $V_2$ so that $V_1 < V_2$. We can
rewrite \refeq{4} in the form
\begin{equation}
\lambda(V_1) =
2\int_{V_1}^{V_2} {f(\xi) \over \sqrt{2(\xi-V_1)}}d\xi+
2\int_{V_2}^\infty{f(\xi) \over \sqrt{2(\xi-V_1)}}d\xi\,.
\label{61}
\end{equation}
If we choose a linear approximation for $f(H)$ in the interval
$V_1q_n$, then one can define
$\lambda_i = \lambda(q_i)$ and
$V_i = q_i^2/2 + I \lambda_i$,
%\begin{eqnarray}
%\lambda_i &=& \lambda(q_i),
%\nonumber \\
%V_i &=& {q_i^2 \over 2} + I \lambda_i \,,
%\end{eqnarray}
%were $i=0 ,\ldots n$
so the recursion formula for $f_{i-1}$ can be
obtained by substituting \refeq{7} into \refeq{61}
\begin{eqnarray}
f_{i-1} &=&- {f_i\over 2}+{3 \lambda_{i-1} \over 4 \sqrt{2 (V_i-V_{i-1})} }
\times
\nonumber \\
& & \left(\lambda_i-2\int_{V_i}^{V_n}
{f(\xi)\over \sqrt{2(\xi-V_{i-1})}}d\xi \right)\,.
\label{8}
\end{eqnarray}
With $f_n = 0$ and using the recursion formula \refeq{8} we can find
$f_{i-1}$ for $i = n,n-1,...1$. The integral in \refeq{8} can be
evaluated by any standard numerical method.
\begin{thebibliography}{99}
\bibitem{bart}R.\,Baartman {\it Stationary Longitudinal Phase Space
Distributions with Space Charge} Proc. 1991 IEEE Part. Acc. Conf.
p.1731.
\bibitem{oide}K.\,Oide and K.\,Yokoya {\it Longitudinal Single-Bunch
Instability in Electron Storage Rings} KEK Preprint 90-10 (1990).
\bibitem{hass}J.\,Ha\"{\i}ssinski {\it Exact Longitudinal Equilibrium
Distribution of Stored Electrons in the Presence of Self-Fields}
Nuovo Cimento {\bf 18B}, p.72 (1973).
\bibitem{zott}B.\,Zotter {\it A Review of Self-Consistent Integral
Equations for the Stationary Distribution in Electron Bunches} Proc.
4$^{\rm th}$ Advanced ICFA Beam Dynamics Workshop, KEK Report 90-21
(1990).
\end{thebibliography}
\end{document}