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DLD Anode Position Conversion |
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DLD Anode Position Conversion Kevin Carnes
Derivation of Horizontal Position X This development will be given in terms of the horizontal position signals. Vertical position may be calculated in an analogous manner. The horizontal signals coming from a DLD detector can be written, in units of ns, as
Here,
where
With the LeCroy TDC, we actually measure the position signals in channels, where each channel corresponds to 0.5 ns. Therefore, the full position calculation would be:
where
Published Values of Transverse Velocity The only way to definitively determine the constants
DLD40 0.73 ns/mm DLD80 0.95 ns/mm DLD120 1.25 ns/mm
This number is the inverse of the propagation velocity
DLD40 DLD80 DLD120
Note that, assuming that the conversion from channels to ns has already been made, the position calculation can be made with an explicit factor of 2, as in, for the DLD80,
or the conversion constant can be redefined to incorporate the factor of 2, as in
The second form is equivalent to multiplying the Roentdek single pitch propagation time by 2, giving a value of 1.9 ns/mm for the DLD80, which corresponds to the number listed on page 8 of their manual as “the correspondence between position and time in the 2d image”.
Origin of Transverse Velocity and Consistency Checks The values for the transverse velocities can be understood
by looking at how a delayline is constructed.
We’ll use a DLD80 as an example. The
distance between the outer edges of the ceramic cylinders that hold the
windings are 118 mm and 120 mm. This
works out to straight sections of 110 mm of wire in air on each side and 15 mm curved
sections in contact with each cylinder.
If we consider a single loop of wire, this is 220 mm in air and 30 mm in
contact with the cylinder. The windings
are in pairs, with the two wires of the pair .5 mm apart and a single wire
wrapped 1 mm apart. Therefore, for a
single loop, the pitch is 250 mm to 1 mm.
A signal has to travel 250 mm around the loop to travel 1 mm in the
transverse direction. On the portion of
the loop in air, the signal will travel at the speed of light. However, according to Roentdek, the signal
sees the ceramic on the turns and has an effective relative permittivity there of
This is .978 ns per 1 mm, or 1.02 mm/ns, close to the values of .95 ns and 1.05 mm/ns given in the Roentdek manual. A consistency check on these values from actual data can made using time sum values. We can sum the two position signals to get:
The value we typically plot in our time sum spectra is therefore:
If we take
Conversion examples
Here are a few conversion examples from the sorting code I’m familiar with:
Itzik’s group: Positions are calculated by a Posn function that takes as input a conversion factor Conv and the raw lsig and rsig signals in channels, then calculates
Posn = Conv*(1.0*lsig -1.0*rsig)/2.0
It uses a conversion value of Conv = ch2mmx = .51 mm/ch. This is for a DLD80 detector. So, one of the factors of 2 is included in the conversion constant and the other is explicit in the Posn calculation.
PAW FORTRAN code used
by Lew and Igor’s groups: Their
resort code returns an array, typically called el for electrons and rec for
recoils. It is a two dimensional
array. The second dimension contains the
hit number. The first dimension has
value 1 for the timing signal, 2 for the R – L signal, and 3 for the D – U
signal. In the analyse.f code, these
values are converted to position in mm by multiplying by a conversion factor,
gx or gy, and dividing by 2, as in xr1 = gxr*result(2,1)/2. Note:
in some older code, this explicit factor of 2 was omitted, so that
spectra plotting position in mm were incorrect.
Momenta values were corrected by including another factor of 2 in their
calculation. Instead of using a factor
of
Andre Staudte’s code: He uses a function called makepos, where, for a DLD anode, the position is (tdc(x1) – tdc(x2))/(2.*f), where the tdc values are already in ns, and f is around 1. for a DLD80.
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