Physics Letters B 765 (2017) 276–279
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Volume reflection efficiency for negative particles in bent crystals
V.M. Biryukov
Institute for High Energy Physics in National Research Center “Kurchatov Institute”, 142281 Protvino, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
11 September 2016
Received
in revised form 23 November 2016
Accepted
14 December 2016
Available
online 15 December 2016
Editor: M.
Doser
Keywords:
Bent
crystals
Channelling
Volume
reflection
We suggest a formula for the efficiency of a single volume reflection of negatively charged particles
in bent crystal planes and compare it to recent experiments at SLAC, MAMI and CERN with electrons
and negative pions in the energy range from 0.855 to 150 GeV in Si crystals. We show that Lindhard
reversibility rule provides sufficient basis for quantitative understanding of these experiments.
© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The increased interest to interaction of electrons with bent
crystals has lead over the recent years to several important ex-
periments
on the so-called volume reflection [1–6]. While the full
understanding of the process may be gained by the use of com-
puter
simulations taking into account both the electron motion in
the field of bent crystal atomic planes and the scattering on the
crystal constituents, it would be quite useful to obtain simple ap-
proximate
formulas describing basic features of electron volume
reflection in bent crystals.
The
volume reflection is caused by interaction of an incident
particle with the potential of the bent crystal atomic planes, which
give the particle an angular kick of the order of a critical channel-
ing
angle θ
C
in the direction opposite to the crystal bending. Such
a reflection is not 100%-efficient as some particles “stick” to the
atomic planes (so called volume capture caused by scattering) in-
stead
of bouncing back. The “stuck” particles are trapped with the
bent atomic planes in the channeled states and thus are steered
away.
The
purpose of this paper is to understand the efficiency of
volume reflection of electrons. We will show that the Lindhard re-
versibility
rule [7] provides sufficient basis for its understanding.
First, we follow the ideas of refs. [8,9] and repeat the derivation of
the efficiency of particle reflection off the bent crystal plane.
According
to the Lindhard’ reversibility rule, the probability for
a channeled particle to be scattered from a certain channeled state
to certain unchanneled state equals the probability for the opposite
E-mail address: valery.biryukov@ihep.ru.
process. This rule was proven in the experiments with GeV-beams
[10–12], where the state of each particle was detected by means
of solid state detectors placed along the crystal length. In 1982
a PNPI experiment observed the effect of 1 GeV protons volume
capture into channeling mode in the depth of the bent crystal [12],
in the region where the particles’ trajectories are tangent to the
crystallographic planes.
2. Formula for the reflection efficiency
The reversibility rule allows to consider the feed-in (volume
capture or rechanneling) and feed-out (dechanneling) processes
from the unified point of view. The rate of particle transitions from
the channeled to random states is set by the dechanneling length
L
D
. Over the distance dz, the transition probability equals dz/L
D
.
According to the reversibility rule, the rate of the opposite tran-
sitions
from random (over-barrier) to channeled (under-barrier)
states is set by the same quantity L
D
.
In
a crystal bent with radius R, the non-channeled particle stays
near the channel (on the phase plane) along the length of the or-
der
of Rθ
C
. Therefore, the probability of transition to channel over
the all interaction time amounts to about Rθ
C
/L
D
. Respectively, the
efficiency of volume reflection f
VR
is reduced by this value:
1 − f
VR
≈
Rθ
C
L
D
, (1)
This formula, which is in fact the result of the reversibility rule, has
agreed with the experiments where the volume-capture probabil-
ities
were measured at 1 GeV and 70 GeV [13]. Also, the 70-GeV
experiment has confirmed the linear dependence of the probability
on the crystal bending radius.
http://dx.doi.org/10.1016/j.physletb.2016.12.032
0370-2693/
© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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