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The Trochoidal Monochromater
D B Cassidy
Keck Antimatter Laboratory
Washington State University
The active1 removal of background radiation in magnetically guided positron beam is most effectively
achieved by the use of a Trochoidal monochromater, also known as an ExB filter, a Wien filter (sometimes;
see below) or simply a velocity filter. This device is particularly useful to positron beam users since it
allows the removal of fast, unmoderated positrons from the beamline while simultaneously taking the slow,
moderated beam out of the line of sight of the positron source, thus enabling easy shielding of the target
region. Here we describe the principle of operation of this simple apparatus.
The basic principle of an ExB filter is, as that particular nomenclature implies, that
perpendicular electric and magnetic fields are used to filter out, or indeed, filter in, some
part of a beam. In terms of a typical positron beam system, the magnetic field is axial;
along the direction of motion of the beam (usually defined as the z direction), and the
applied electric field is then perpendicular to this, in the negative x direction (see figure
(1)). This gives rise to a constant drift velocity of the positron gyration centers in the y
direction, thus allowing only those particles with a certain deflection to be transmitted
through an offset pair of apertures. The geometry of a typical ExB filter is shown in
figure (1)
V+
V-
E
B
vz
y
x
z
Figure (1) The geometry of a typical ExB filter. Charged particles (positrons) move along the z direction
with a velocity of vz. when they enter the plates a force is experienced and a constant drift velocity causes,
with the field configurations shown, motion in the positive y direction.
It should be pointed out here that this arrangement, which is sometimes referred to as a
Wien filter, is different from the more common usage of that term, which is more often
used to describe a device in which the perpendicular electric and magnetic fields are both
normal to the particle trajectory. Then, only charged particles with the appropriate
velocity will experience equal and opposite Coulomb and Lorentz forces 2 and be
transmitted through the filter in a straight line. Particles that experience unequal forces
collide with the walls and are not transmitted. This arrangement is used as a mass
analyzer.
In the ExB filter shown in figure (1) the force acting on the particles between the plates
may be written as
F  q E  v  B
(1.1)
Where q refers to the positron charge, E to the electric field, B to the magnetic field and v
is the velocity. Thus, the equation of motion is given by
dv
 dv
dv 
m  x i  y j  z k   qE0i  q  vx i  v y j  vz k   B0k
dt
dt 
 dt
= q  (vy B0  E0 )i  vx B0 j
(1.2)
(1.2) Represents three simultaneous differential equations:
m
dvx
 q(v y B0  E0 )
dt
m
dv y
dt
 qvx B0
dvz
0
dt
(1.3)
(1.4)
(1.5)
Equation (1.5) tells us that the z component of the velocity is unaffected by the ExB
filter, and vz is therefore uniquely determined by the beam energy. That is,
1/ 2
 2T 
vz   B  ,
 m 
where TB is the beam energy.
Differentiating (1.4) with respect to time gives,
(1.6)
2
dvx m d v y

dt qB0 dt 2
and replacing this into (1.3) then gives
2
m d vy q
  v y B0  E0   0 .
qB0 dt 2 m
(1.7)
(1.8)
It is convenient to note here that charged particles in a magnetic field undergo cyclotron
orbits about the field lines. This motion is described simply by
F  ma  mr c2  Bqv  Bqr c
qB
m
where c is the cyclotron frequency. Thus, we write (1.8) as
 c 
d 2vy
dt
2
  c2 v y 
q
 c E0  0
m
(1.9)
(1.10)
To solve this equation we first solve the complementary equation,
d 2vc
(1.11)
  c2vc  0
dt 2
Where vc refers to a complimentary solution. If D is a differential operator this may be
written as
( D 2   c2 )vc  0
 D  i c
(1.12)
The general solution of the complementary equation is then a linear combination of the
exponential roots of the auxiliary equation ((1.12)). That is,
vc  A 'exp(ict )  B 'exp(ict )
 A sin(ct )  B cos(ct )
(1.13)
Here A and B are constants to be determined. In order to obtain a particular solution to
(1.10) we “guess”3 that there is a solution of the form vc = C. Then any time differentials
are necessarily zero and we obtain,
q
 c2v p   c E0
m
(1.14)
qE0 E0
 vp 

m c B0
Here vp refers to a particular solution. Then the general solution to (1.10) is simply the
sum of the complimentary and particular solutions. That is,
v y  A sin( c t )  B cos( ct ) 
E0
B0
(1.15)
Now, for a positron in a magnetic field of 100 Gauss the cyclotron frequency is in the
GHz range. We can see then that the motion in the y direction consists of a fast
oscillatory motion, corresponding to the cyclotron rotations, with a much slower drift
velocity superimposed on top of this. It is only the drift velocity that causes any net
displacement in the y direction and for the purposes of designing an ExB filter we can
say that
E
vy  0
(1.16)
B0
Since from (1.6) we know that the z component of the positron velocity will remain
constant, the time taken to traverse the length of the plates is simply
td 
L
L

vz  2TB 1/ 2


 m 
(1.17)
Also, we know that the drift velocity is constant ((1.16)), so that the deflection d (see
figure (1)) is given by
E
L
(1.18)
d 0
B0  2TB 1/ 2


 m 
Note that the deflection is in the positive y direction (for the field configuration used
here) and that it does not depend on the sign of the charge. Thus, electrons and positrons
would be deflected in exactly the same way in such a system. If the ExB filter and source
end of the beam are floating, constant beam energy may be used for deflection, while
retaining variable beam energy for experiments. This means that the system will not
require frequent tuning. So, if we use some typical values in (1.18) we may determine the
voltage that must be applied to the plates to achieve a required deflection. We do so
because most of the other parameters are constrained, for practical reasons, to a relatively
narrow range. The electric field, E0 is defined as V/s, where s is the distance between the
plates and V is the potential difference between them. Then we have
1/ 2
 2T 
SdB0  B 
 m 
V
L
(1.19)
Table 1 lists some realistic values of the parameters of (1.19). Using these values we find
V = 150 Volts between the plates (i.e. +75 and –75 on each, as shown in figure (1)).
1
Parameter
Typical value
SI value
d (deflection)
B0 (axial field strength)
TB (Beam energy)
S (Plate seperation)
L (Plate length)
1 inch
100 Gauss
100 eV
0.5 inch
15 inch
0.0254 m
0.01 T
1.6 x 10-17 J
0.0127 m
0.381 m
Active as opposed to simply using a long beamline and relying on the inverse square law to reduce
radiation levels in the detection region
2
NB: We use the common convention in which the magnetic contribution to the total force experienced by
a moving charge in magnetic and electric fields is referred to as the Lorentz force.
3
In this case, since we have static fields, such a guess is not unreasonable. However, in no way do we rely
on the assumption that there is a solution in which the velocity in the y direction is constant.