Coordinate Calculation

The signals from the detectors can be combined to give three coordinates, U, V and W by calculating the relative time difference between the arrival time of the MCP signal, tM CP, and the delay line signals tU1,U2...W2:

U = (tM CP −tU1)−(tM CP −tU2)

V = (tM CP −tV1)−(tM CP −tV2)

where the arrival times have been corrected for additional delays as discussed in section 5.4.4.

Each of the delay lines have a slightly different length dependent upon the detector wrapping, this is reflected by the differences in the peak of the time sums as shown earlier in figure 5.4.3b. In order to account for this the coordinates U, V, W are multiplied by scale factors SU, SV and SW,

SU =T SU/T SU

SV =T SU/T SV

SW =T SU/T SW (5.4.3)

where T S refers to the peak time sum for each layer. The coordinates U, V and W are thus

U0 =SU ·U

V0 =SV ·V

W0 =SW ·W (5.4.4)

Positron/Ion Detector

The conversion from U’, V’, W’ to cartesian coordinates (x, y) for the positron/ion detector is relatively simple compared to the electron detector as there are no gaps in the delay line windings. The (x, y) coordinates from each pair of layers, UV, UW and VW are xU V =U0 yU V = 1 √ 3(U 0₋ 2V0) xU W =U0 yU W = 1 √ 3(2W 0₋ U0) xV W =V0 +W0 yV W = 1 √ 3(W 0 ₋ V0) (5.4.5)

5.4. Reaction Microscope Data Analysis 107

The three layers can be combined for this detector by averaging the contribut- ing x and y coordinates for each pair of layers.

Electron Detector

The gap in the electron detector windings means that there are significant detector dead areas as when the electron cloud from the MCP hits the gap, there is no pulse generated. Unlike the positron/ion detector there is now no linear relationship between the difference in signal propagation times and the position on the delay line. To correct for this, the coordinates U’, V’, W’ calculated from equation 5.4.4 are shifted depending upon which side of the gap the cloud hit. Figure 5.4.4) defines the dimensions of the electron detector gap where d is the total side length of the delay line,h is the size of the gap andl1,2 are the lengths either side

of the gap.

The position of the gap itself, is found by converting the bins from time into millimetres and using the geometric measurements for each layer of the detector which were given in section 3.3.3 in figure 3.3.5. Given this information the coordinates are shifted, for example, for the U layer where the gap is located at

Ugap, U_{shif ted}0 =      U0 −h/2 if U0 < Ugap U0 +h/2 if U0 > Ugap (5.4.6)

likewise for the V and W layers.

If the centroid of the MCP electron cloud hits the gap, then the pulse which is detected no longer corresponds closely to the centroid’s location, therefore hits on the electron detector which are within 3 mm of the edge of the gap are removed (where the electron cloud distribution is estimated using Lapington and Edgar (1989)). The shifted coordinates are then converted into cartesian coordinates using 5.4.5 where U0 is replaced byUshif ted and so on. The impact of these shifts

is shown in figures 5.4.5a and 5.4.5b.

Finally, for both positron/ion and electron data the coordinate matching is tested. Good coordinate matching means that thexandycoordinates calculated from each pair of layers is similar, in this case the coordinates are considered to be well matched if the difference is ≤2 mm, the full width half maximum of the

h w l1 l2 h l1 l2 Top View Side View d

5.4. Reaction Microscope Data Analysis 109

l1 l2

(a) Example data for the U layer before correction for the delay line gap

h

l1 l2

d

(b)The example data from a) with shifts accounting for the delay line gap

Figure 5.4.5: Example of the correction applied to data to account for the delay line gap in the electron detector.

y_uv− y_vw(mm) yuv −30 −20 −10 0 10 20 10 20 30 40 50 60 70 80

Figure 5.4.6: Comparison of the electron detector y coordinate calculated from the UV layers with that from the VW layers. The black box indicates events which would be considered well matched under the criterion −2 ≤ yuv −yvw ≤ 2 mm

i.e. the difference in the y coordinate calculated from the two different pairs of delay lines is within 2 mm. The high intensity point in the centre of the image is simply indicative of a positron beam focused onto the detector, producing a lot of events in one location.

spatial spread of the positron beam. Figure 5.4.6 compares the matching of the

y coordinates calculated from the UV and VW layers. 76% of coordinates are within acceptable limits in this example, therefore 24% of events are removed from the data set and a final check for triple coincidences is performed. Events where there is poor coordinate matching are indicative of a ‘bad’ U, V or W coordinate which can be due to noise, for example on the detector and associated electronics.

Following calculation of the (x, y) coordinates for each event, detector images can be constructed (see figure 5.4.7). The positron/ion detector has a character- istic hole in the centre due to the hole in the MCP and the electron detector has a star shaped hole in the centre due to the combination of the gaps in each layer. Momentum reconstruction requires information about the deviation of the detected particle’s position from the ionisation event position. When the positron

5.4. Reaction Microscope Data Analysis 111

a) b)

c)

Figure 5.4.7: Examples of typical detector images a) Positrons on the positron/ion detector b) Ion candidates on the positron/ion detector c) Electrons on the electron detector

beam is directed through the centre of the positron/ion detector, the ionisation event location is assumed to be the centre of the detector image. However, when the unscattered positron beam is not central on the positron/ion detector, as in figure 5.4.7a), the centre is found by taking a histogram of thexandycoordinates. This gives an approximately Gaussian distribution and fitting to this allows the peak position to be determined. The scattered particle’s detected position is then related to the peak position of the unscattered beam in order to find the difference. This method is also applied to the electron detector data and the final output coordinates for both detectors, (x0, y0) are the differences between (x, y) and the central position. Due to the cyclotron motion of charged particles in a magnetic field, the positions (x0, y0) describe a circle with a cyclotron radius

rc=

√

x02 _+y02_.

The output from this analysis is a list of potential events with the positron- electron time, t_e+_e−, (x, y) coordinates and cyclotron radii for the positron,r_c,e+,

and electron, rc,e−, where there was a triple coincidence with an ion candidate in

an appropriate time window where all particles have sufficient coordinate infor- mation.