Download Gain calculations

University of Leicester
PLUME Ref: PLM-PAY-GainCalcs-013-3
Date: 28/10/2009
MCP detector gain calculations
P. Peterson
Date
Updated Reference Number
Change
02/10/2008
28/10/2009
PLM-PAY-GainCalcs-013-2
PLM-PAY-GainCalcs-013-3
second version issued
Calculation clarifications,
converted format. This is a
newer version of this
document written nine
months ago but only
uploaded now.
MCP detectors have been used for all manner of remote sensing applications for years. The
use of MCPs as electron detectors is very widely known, but our final detector design will be
hunting for micrometeoroids that will seed the channel with many more electrons. This
document contains derivations for both x-ray gain for our particular setup and expected
signal sizes and micrometeorite gain.
X-ray Gain
For unsaturated operation, we can derive a decent approximation for the gain of our plate
when it is detecting x-rays. Provided the secondary emission co-efficient remains constant,
the total gain of the plate G is given by the secondary emission co-efficient e raised to the
power of the number of times an electron starting at the top of the tube will strike the walls of
the channel on the way down N (1)

N
G  e

(1)

The secondary electron emission co-efficient for the kind of glass our MCP is made out of is
a function of the kinetic energy of an incoming electron, and the shape of this function is
givenin figure 1. [1]
e


e  1
VC
V1
Figure 1: Secondary electron yield versus collision potential

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PLUME Ref: PLM-PAY-GainCalcs-013-3
University of Leicester
Date: 28/10/2009
This co-efficient falls to zero for low collision energies for obvious reasons, but it also drops
off to zero when the collision energy gets very high – the electron penetrates so far into the
channel wall that the secondaries cannot escape. There are thus two energies for which the
emission coefficient is equal to one, and they’re called the crossover potentials. When the
collision energy is less than about twice the lower crossover potential, we can approximate
e as being linearly proportional to the collision energy e Vc (2). For lead glass the first
crossover potential V1 averaged over various collision conditions is about 30V [8].
e 


Vc
V1
(2)

If we assume that the electron is emitted normal to the channel wall with energy e V0 , we
can use a ballistic model of the electron’s flightpath down the channel to derive Vc as a
 of channel diameter D , length L and accelerating plate voltage V plate(3). For lead
function
glass, the average emission energy is 1eV.
2
plate
2
V
Vc 
4V0 L
D


(3)


2
Similar ballistic model calculations give us the number of collisions N (4).

2
N
4V0 L
D2 (4)
V plate
Substituting equations 3 and 4 into equation 1 gives (5). Since we assumed that the
secondary electrons will be launched perpendicular from the channel walls and that each
collision
 produces an entirely predictable number of secondaries, the gain we calculate will
be the ‘peak gain’, an ideal case. The actual gain of the plate will vary from event to event.
4V0 L


 V plate2 
G peak  

L2 2 
4V
V

0
1

D 
2
D2
V p la te
(5)
The output gain of a channel scales with increased numbers of starting electrons ne to the
power of 0.4 [9]. This empirical law allows us to modify the gain equation to include

potentially
large numbers of starting electrons. (6)
4V0 L


2

V plate
0.4 
G peak  n e 

2
L 2 
4V
V

0
1

D 

2
D2
V p la te
(6)
Lab setup x-ray gain
Our lab MCP is a 12.5µm plate 0.5mm thick. Saturation voltage for this plate is

calculated
at just over 800V, and an x-ray event in the front of the tube will produce about
16,000 electrons at this voltage with a collective charge of 2.6x10-15 C. Through the 0.5pF
capacitor in the preamp, this corresponds to a preamp voltage input of 5.17mV. If the pulse
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University of Leicester
PLUME Ref: PLM-PAY-GainCalcs-013-3
Date: 28/10/2009
generator is set to give pulses with this voltage level, then if they are not visible in the
oscilloscope on the other side of the electronics chain then the signals from the plate will not
get through either.
Prof. Fraser estimates that we would get about 50,000 electrons out of the MCP at
this voltage, however. This would give us a signal voltage of 16.15mV, a little more
optimistic.
Saturation
Saturation has many guises. The primary saturation effect is called wall-charge
saturation. Each collision excites electrons from the glass walls of the channel, and the
emitted electrons fly away down the tube, accelerated by the plate voltage. However,
removing those negative electrons gives the wall a positive charge, and this positive charge
produces an electric field counteracts the applied field. This reduces the collision energy of
the electrons, and with it the electron emission co-efficient (2).
A theoretical model of the progress of an electron cloud down a channel undergoing
this saturation effect shows that this collision energy reduction occurs quite sharply at a
certain point down the channel [2]. This means that there will be a fairly well defined cutoff
point beyond which the calculations made in the previous section will no longer be valid.
40:1 L/D channels are not long enough to produce saturation from x-ray events.
A good empirical for the plate voltage at which saturation effects become significant is given
in equation (7) [2]. This saturation voltage Vsat is independent of channel diameter, and
differing bias angles will not affect it too much. This equation is given in the context of x-ray
detection producing a single electron.
Vsat  8.94 L D  450
 (7)
By using the gain calculation equation (5) and setting the voltage term to that given in
equation (7) we can get an estimate for the maximum unsaturated gain for a particular plate
geometry
 (8). This is important, because equation 8 is independent of the initial electron
number.
4V0 L
2
D2

8.94 L D 450
(8.94 L D  450) 2 
Gsat  

2
L
4V
V


2
0 1

D 
(8)
To get reasonable meteoroid size resolution from our detector, we should set the plate
voltage so that the largest meteoroid size we can detect gives us a peak gain of Gsat .

[1] A. Authinarayanan and R W Dudding (1976)
[2] GW Fraser, JP Pearson, GC Smith et al (1983)
[3] P Moleneux SRD_v2

[4] JD Carpenter, TS Stevenson, GW Fraser, JC Bridges, AT Kearsley, RJ Chater and S. V.
Hainsworth: Nanometer Hypervelocity Dust Impacts In Low Earth Orbit
[5] D Brandt, JD Carpenter (2005)
[6] GW Fraser: The ion detection efficiency of microchannel plates (2002)
[7] Gemological Institute of America: GIA Gem reference guide (1995)
[8] GW Fraser, GP Price: Calculating the output charge cloud of a microchannel plate (2001)
[9] GW Fraser’s empirical equation
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