Title\thanksref{label1}
Title\thanksref{label1}
We discuss the radiation generated by positrons channeling in a crystalline undulator. The undulator is produced by periodically bending a single crystal with an amplitude much larger than the interplanar spacing. Different approaches for bending the crystal are described and the restrictions on the parameters of the bending are discussed. We also present numeric calculations of the spontaneous emitted radiation and estimate the conditions for stimulated emission. Our investigations show that the proposed mechanism could be an interesting source for high energy photons and is worth to be studied experimentally.
fel2k-we-4-07.tex
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\begin{frontmatter}

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\title{Spontaneous and stimulated undulator radiation by an
ultra-relativistic positron channeling in a periodically bent
crystal\thanksref{contrib}\thanksref{ack}
}

\thanks[contrib]{long version of our contribution to the 22nd International Free
Electron Laser Conference, Durham, NC, USA, 13-18 August 2000
}
\thanks[ack]{Peprinted from Nuclear Instruments and Methods A, Volume
474, 1--3, in press, Copyright 2001, with permission from Elsevier
Science. http://www.elsevier.com/locate/nima.}

% use optional labels to link authors explicitly to addresses:
% \author[label1,label2]{}
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\author{W. Krause\thanksref{email}}
\author{A. V. Korol}
\author{A. V. Solov'yov}
\author{W. Greiner}

\address{Institut f\"ur Theoretische Physik der Johann Wolfgang
Goethe-Universit\"at, Postfach 11 19 32, 60054 Frankfurt am Main, Germany}

\thanks[email]{E-mail: krause@th.physik.uni-frankfurt.de}

\begin{abstract}
We discuss the radiation generated by positrons channeling in a
crystalline undulator. The undulator is produced by periodically
bending a single crystal with an amplitude much larger than the
interplanar spacing. Different approaches for bending the crystal are
described and the restrictions on the parameters of the bending are
discussed. We also present numeric calculations of the spontaneous
emitted radiation and estimate the conditions for stimulated
emission. Our investigations show that the proposed mechanism could be
an interesting source for high energy photons and is worth to be
studied experimentally.
\end{abstract}

\begin{keyword}
% keywords here, in the form: keyword \sep keyword
undulator radiation \sep channeling positrons \sep
periodically bent crystal
% PACS codes here, in the form: \PACS code \sep code
\PACS 41.60
\end{keyword}
\end{frontmatter}

% main text
\section{Introduction}
\label{sec:intro}

We discuss a new mechanism, initially proposed in
\cite{Korol98,Korol99}, for the generation of high energy photons
by means of the planar channeling of ultra-relativistic positrons
through a periodically bent crystal. In this system there appears, in
addition to the well-known channeling radiation, an undulator type
radiation due to the periodic motion of the channeling positrons which
follow the bending of the crystallographic planes. The intensity and
the characteristic frequencies of this undulator radiation can be
easily varied by changing the positrons energy and the parameters of
the crystal bending.

\begin{figure}[ht]
\includegraphics*[width=10cm]{fel2k-we-4-07-fig1_pbm.eps}
\caption{Schematic representation of spontaneous and stimulated
radiation due to positrons channeling in a periodically bent
crystal. The $y$- and $z$-scales are incompatible!}
\label{fig:bent_crystal}
\end{figure}

The mechanism of the photon emission by means of the crystalline
undulator is illustrated in Figure \ref{fig:bent_crystal}.  It is
important to stress that we consider the case when the the amplitude
$a$ of the bending is much larger than the interplanar spacing $d$
($\sim 10^{-8}\;\mathrm{cm}$) of the crystal ($a \sim 10\ d$), and,
simultaneously, is much less than the period $\lambda$ of the
bending ($a/\lambda \sim 10^{-5} \dots 10^{-4}$).

In addition to the spontaneous photon emission by the crystalline
undulator, the scheme we propose leads to a possibility to generate
stimulated emission. This is due to the fact, that photons emitted at
the points of the maximum curvature of the trajectory travel almost
parallel to the beam and thus, stimulate the photon generation in the
vicinity of all successive maxima and minima of the trajectory.



% If certain conditions are provided (see section
% \ref{sec:bent_crystal}), the positrons will still channel within the
% bent crystal. Thus we have created a crystal undulator that
% parameters differ much from those in the case of the straight channel.
% Two types of radiation are produced: the channeling radiation due to
% the motion of the interplanar oscillations and the undulator radiation
% due to the motion of the positron, following the bending of the crystal.



\section{The bent crystal}
\label{sec:bent_crystal}

The bending of the crystal can be achieved either dynamically or
statically. In \cite{Korol98,Korol99} it was proposed to use a
transverse acoustic wave to dynamically bend the crystal. The
important feature of this scheme is that the time period of the
acoustic wave is much larger than the time of flight $\tau$ of a bunch
of positrons through the crystal. Then, on the time scale of $\tau$,
the shape of the crystal bending doesn't change, so that all particles
of the bunch channel inside the same undulator. One possibility to
couple the acoustic waves to the crystal is to place a piezo sample
atop the crystal and to use radio frequency to excite oscillations.

The usage of a statically and periodically bent crystal was discussed
in \cite{Uggerhoj00}. The idea is to construct a crystalline undulator
based on graded composition strained layers.
 
We now consider the conditions for stable channeling.  The channeling
process in a periodically bent crystal takes place if the maximum
centrifugal force in the channel, $F_{\mathrm{cf}}\approx m \gamma
c^2/R_{\mathrm{min}}$ ($R_{\mathrm{min}}$ being the minimum curvature
radius of the bent channel), is less than the maximal force due to the
interplanar field, $F_{\mathrm{int}}$ which is equal to the maximum
gradient of the interplanar field (see \cite{Korol99}). More
specifically, the ratio $C=F_{\mathrm{cf}}/F_{\mathrm{int}}$ has to be
smaller than 0.1, otherwise the phase volume of channeling
trajectories is too small (see also \cite{Korol00}). Thus, the
inequality $C<0.1$ connects the energy of the particle, $\varepsilon=m
\gamma c^2$, the parameters of the bending (these enter through the
quantity $R_{\mathrm{min}}$), and the characteristics of the
crystallographic plane.

A particle channeling in a crystal (straight or bent) undergoes
scattering by electrons and nuclei of the crystal. These random
collisions lead to a gradual increase of the particle energy
associated with the transverse oscillations in the channel. As a
result, the transverse energy at some distance $L_d$ from the
entrance point exceeds the depth of the interplanar potential well,
and the particle leaves the channel. The quantity $L_d$ is called the
dechanneling length \cite{Gemmel74}. To estimate $L_d$ one may follow
the method described in \cite{Biryukov96,Krause00a}. Thus, to
consider the undulator radiation formed in a crystalline undulator, it
is meaningful to assume that the crystal length does not exceed $L_d$.

Let us demonstrate how one can estimate, for given crystal and
energy $\varepsilon$, the ranges of the parameters $a$ and $\lambda$
which are subject to the conditions formulated above.  For doing this
we assume that the shape of the centerline of the periodically bent
crystal is $a\,\sin(2\pi\,z/\lambda)$.  Figure \ref{fig:parameters}
illustrates the above mentioned restrictions in the case of
$\varepsilon=500$ MeV positrons channeling in Si between the
(110) crystallographic planes.  The diagonal straight lines correspond
to various values (as indicated) of the parameter $C$.  The curved
lines correspond to various values (as indicated) of the number of
undulator periods $N$ related to the dechanneling length $L_d$
through $N=L_d/\lambda$.  The horizontal lines mark the values of the
amplitude equal to $d$ (with $d=1.92\cdot 10^{-8}\;\mathrm{cm}$ being
the (110) interplanar distance in Si) and to $10\,d$.  The vertical
line marks the value $\lambda = 2.335 \cdot 10^{-3}$ cm, for which the
spectra (see section \ref{sec:spectra}) were calculated.


\begin{figure}[ht]
\includegraphics*[width=8cm]{fel2k-we-4-07-fig2.eps}
\caption{The range of parameters $a$ and $\lambda$ for a bent
Si(110) crystal at $\varepsilon=500$ MeV.}
\label{fig:parameters}
\end{figure}


\section{Spectra of the spontaneous emitted radiation}
\label{sec:spectra}

We calculated the spectra of the spontaneous emitted radiation using
the quasiclassical method \cite{Baier98}.  The trajectories of the
particles were calculated numerically and then the spectra were
evaluated \cite{Krause00a}.  The latter include both radiation
mechanisms, the undulator and the channeling radiation.

The spectral distributions of the total radiation emitted in forward
direction for $\varepsilon=500$ MeV positrons channeling in Si between
the (110) crystallographic planes are plotted in figure
\ref{fig:spectra}. The wavelength is fixed at
$\lambda=2.335\cdot10^{-3}$ cm, while the ratio $a/d$ is changed from
0 to 10. The length of the crystal is $L_d=3.5\cdot10^{-2}$ cm and
corresponds to $N=15$ undulator periods.

\begin{figure}[ht]
\includegraphics*[width=10cm]{fel2k-we-4-07-fig3_pbm.eps}
\caption{Spectral distributions of the total radiation emitted in
forward direction for $\varepsilon=500$ MeV positrons channeling in Si
between the (110) crystallographic planes for different $a/d$
ratios.}
\label{fig:spectra}
\end{figure}


The first graph in figure \ref{fig:spectra} corresponds to the case of
the straight channel ($a/d=0$) and, hence, presents the spectral
dependence of the ordinary channeling radiation only.  The spectrum
starts at $\hbar\omega\approx 960$ keV, reaches its maximum value at
$1190$ keV, and steeply cuts off at $1200$ keV.  This peak corresponds
to the radiation into the first harmonic of the ordinary channeling
radiation, and there is almost no radiation into higher harmonics.
The latter fact is consistent with general theory of dipole radiation
by ultra-relativistic particles undergoing quasiperiodic motion.  The
dipole approximation is valid provided the corresponding undulator
parameter $p_c = 2\pi \gamma(a_c/\lambda_c)$ is much less than 1.  In
this relation $a_c$ and $\lambda_c$ stand for the characteristic
scales of, correspondingly, the amplitude and the wavelength of the
quasiperiodic trajectory.  In the case of 500 MeV positrons channeling
between the (110) planes in Si one has $p_c\approx 0.2 \ll 1$ and all
the channeling radiation is concentrated within some interval in the
vicinity of the energy of the first harmonic.

Increasing the $a/d$ ratio leads to modifications in the spectrum of
radiation.  The changes which occur manifest themselves via three main
features, (i) the lowering of the ordinary channeling radiation peak,
(ii) the gradual increase of the intensity of undulator radiation due
to the crystal bending, (iii) the appearing of additional structure
(the sub-peaks) in the vicinity of the first harmonic of the ordinary
channeling radiation.  A more detailed analysis of these spectra can
be found in \cite{Krause00a}.


\section{Discussion of stimulated photon emission}
\label{sec:stimulated}

The scheme illustrated by figure \ref{fig:bent_crystal} allows to
discuss the possibility to generate stimulated emission of high energy
photons by means of a bunch of ultra-relativistic positrons moving in
a periodically bent channel. Indeed the photons emitted in the nearly
forward direction at some maximum or minimum point of the trajectory
by a group of particles of the bunch stimulate the emission of photons
with the same energy by another (succeeding) group of particles of the
same bunch when it reaches the next maximum/minimum.

In \cite{Korol99} estimates for the gain factor for the spontaneous
emission in crystalline undulators were obtained. It was demonstrated
that to achieve a total gain equal to 1 on the scale of the crystal
length (equal to the dechanneling length), one has to consider volume
densities $n$ of the channeling positrons on the level of
$10^{20}\dots 10^{21}$ cm$^{-3}$ for positron energies within the
range $0.5\dots 5$ GeV.  These magnitudes are high enough to be
questioned whether they can be really reached.

Let us estimate the volume density $n$ of a positron bunch which can
be achieved in modern colliders.  To do this we use the data presented
in \cite[see p. 142]{PDG98}  for a beam of 50 GeV positrons available
at SLC (SLAC, 1989).  The bunch length is 0.1 cm and the beam radius
is 1.5 $\mathrm{\mu m}$ (H) and 0.5 $\mathrm{\mu m}$ (V), resulting in
the volume of one bunch $V=2.4\cdot 10^{-9}\;\mathrm{cm^3}$.  The
number of particles per bunch is given as $4.0\cdot 10^{10}$.
Therefore, one obtains $n=1.7\cdot 10^{19}\;\mathrm{cm^{-3}}$.

This value, although being lower by an order of magnitude than the
estimates obtained in \cite{Korol99}, shows that the necessary
densities should be reachable in the future with accelerators
optimized for high particle densities.

Finally let us discuss the required transverse emittance of the
beam. For doing this we need to consider the angle between the
particle's trajectory and the crystal plane. If this angle is larger
than the Lindhard angle $\Psi_{\mathrm{P}}$, the particle will not be
captured into the channeling mode and leaves the channel immediately
\cite{Lindhard}. For 5 GeV positrons channeling between the (110) planes
of silicon, we have $\Psi_{\mathrm{P}}=72\;\mathrm{\mu rad}$ and for
50 GeV positrons it equals to $\Psi_{\mathrm{P}}=23\;\mathrm{\mu
rad}$.

One can compare these values with the divergence of the SLC beam,
which transverse emittance in vertical direction is given as $0.05\,
\pi \;\mathrm{rad\ nm}$ \cite{PDG98}. With a vertical beam radius of
0.5 $\mathrm{\mu m}$ we get for the vertical beam divergence
$\Psi=100\;\mathrm{\mu rad}$. Thus the divergence of the beam is about
four times higher than the acceptance of the channel and so only a
quarter of all particles will participate in the channeling
process. Evidently it is necessary to reduce the divergence of the
beam, for example by increasing the beam radius. But then it is also
necessary to reduce the bunch length, to keep the particle density
high enough. Fortunately, like for the particle density, the values
achievable today differ only about one order of magnitude from the
values estimated above for the stimulated emission.


\section{Conclusion}
\label{sec:conclusion}

To conclude we point out that the use of crystalline undulators is a
new interesting way to produce high energy radiation. As we have shown
above, the present accelerator technology is nearly sufficient to
achieve the necessary conditions. Also, it looks like that by means of
modern technology it is possible to construct the required crystalline
undulator, which, as compared to the undulators based on magnetic
fields, will allow to generate photon emission of much higher
energy.

In our opinion the effect described above is worth experimental study
which, as a first step, may be concentrated on measurements of the
spontaneous undulator radiation spectra.

The related problems not discussed in this paper, but which are under
consideration, include the investigation of the crystal damage due to
the acoustic wave, photon flux and beam propagations.

\ack
The research was supported by DFG, GSI, and BMBF.
AVK and AVS acknowledge the support from the 
Alexander von Humboldt Foundation.


% END of main text

%%% for bibtex \bibliography{main}

\begin{thebibliography}{10}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi

\bibitem{Korol98}
A.~V. Korol, A.~V. Solov'yov, W.~Greiner, J.\ Phys.\ G. 24 (1998) L45.

\bibitem{Korol99}
A.~V. Korol, A.~V. Solov'yov, W.~Greiner, Int.\
  J.\ Mod.\ Phys. E 8~(1) (1999) 49.

\bibitem{Uggerhoj00}
U.~Mikkelsen, E.~Uggerh{\o}j, Nucl. Inst. and Meth. B 160
  (2000) 435.

\bibitem{Korol00}
A.~V. Korol, A.~V. Solov'yov, W.~Greiner, Int.\ J.\ Mod.\ Phys. E 9~(1) (2000) 77.

\bibitem{Gemmel74}
D.~S. Gemmel, Rev.\ Mod.\ Phys. 46~(1) (1974) 129.

\bibitem{Biryukov96}
V.~M. Biruykov, Y.~A. Chesnokov, V.~I. Kotov, Crystal Channeling and its
  Application at High-Energy Accelerators, Springer, Berlin, 1996.

\bibitem{Krause00a}
W.~Krause, A.~V. Korol, A.~V. Solov'yov, W.~Greiner, J. Phys.~G:
Nucl. and Part. Phys. 26 (2000) L87.

\bibitem{Baier98}
V.~N. Baier, V.~M. Katkov, V.~M. Strakhovenko, High Energy Electromagnetic
  Processes in Oriented Single Crystals, World Scientific, Singapore, 1998.

\bibitem{PDG98}
C.~Caso et~al, Review of Particle Physics, Eur.~Phys.~J. C 3
  (1998).

\bibitem{Lindhard}
J.~Lindhard, K.\ Dan.\ Viddensk.\ Selsk.\ Mat.\ Phys.\ Medd. 34~(1) (1965) 14.

\end{thebibliography}


\end{document}


physics0109041.extracted_bib
\begin{thebibliography}{10}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi

\bibitem{Korol98}
A.~V. Korol, A.~V. Solov'yov, W.~Greiner, J.\ Phys.\ G. 24 (1998) L45.

\bibitem{Korol99}
A.~V. Korol, A.~V. Solov'yov, W.~Greiner, Int.\
  J.\ Mod.\ Phys. E 8~(1) (1999) 49.

\bibitem{Uggerhoj00}
U.~Mikkelsen, E.~Uggerh{\o}j, Nucl. Inst. and Meth. B 160
  (2000) 435.

\bibitem{Korol00}
A.~V. Korol, A.~V. Solov'yov, W.~Greiner, Int.\ J.\ Mod.\ Phys. E 9~(1) (2000) 77.

\bibitem{Gemmel74}
D.~S. Gemmel, Rev.\ Mod.\ Phys. 46~(1) (1974) 129.

\bibitem{Biryukov96}
V.~M. Biruykov, Y.~A. Chesnokov, V.~I. Kotov, Crystal Channeling and its
  Application at High-Energy Accelerators, Springer, Berlin, 1996.

\bibitem{Krause00a}
W.~Krause, A.~V. Korol, A.~V. Solov'yov, W.~Greiner, J. Phys.~G:
Nucl. and Part. Phys. 26 (2000) L87.

\bibitem{Baier98}
V.~N. Baier, V.~M. Katkov, V.~M. Strakhovenko, High Energy Electromagnetic
  Processes in Oriented Single Crystals, World Scientific, Singapore, 1998.

\bibitem{PDG98}
C.~Caso et~al, Review of Particle Physics, Eur.~Phys.~J. C 3
  (1998).

\bibitem{Lindhard}
J.~Lindhard, K.\ Dan.\ Viddensk.\ Selsk.\ Mat.\ Phys.\ Medd. 34~(1) (1965) 14.

\end{thebibliography}