Using p~p~ elastic scattering for
polarization measurements + diagonal
scaling
I tried to evaluate if and how p~p~ elastic scattering in combination with a detector system
consisting of 4 × 2 HERMES modules (1 module = 2 detectors) is usable to measure the
polarization of the hydrogen gas target or the polarized proton beam after spin filtering.
Since the proposed system provides the measurement of all polarization observables it
could pretty much look like the one we need for the AD. For the determination of beam or
target polarization at injection energy (45 MeV) the differential cross sections as well as
the analyzing powers and double-spin asymmetries are known [1].
1 Detector geometry
The assumed detector system with the center of detectors placed at 45%, 135%, 225% and
315% as shown below (Fig: 1) has a φ-acceptance of ≈ 59%. It consists of two layers of
double sided silicon-strip detectors of 97×97 mm2 sensitiv area and a thickness of 300 µm
(HERMES detectors). The distance of the layers (see Fig: 1) to the vertex was chosen to
provide enough space for the BRP and ABS tubes between the detectors.
Fig. 1: left: φ-symmetric arrangement consisting of 4 × 2 HERMES modules: w = 100 mm;
d0 =80 mm; d1 =100 mm (looking in beam direction)
right: detectorsystem + storage cell
1
2 Event generation
”For pp scattering a convenient choice to identify the two protons is to associate the
smaller scattering angle with particle 1. No information is lost, the two protons are not
distinguishable and the angular distributions of all observables extend to 90 degree in the
cm.”
Besides the vertex generation which was done like for the p~d case, Θ and Φ angles of the
first proton were generated according to the complete differential cross-section including
all relevant analyzing powers (Ay ) and spin-correlation coefficients (Cyy , Cxx ).
Fig. 2: left: differential cross section vs. Θlab
right: spin-correlation coefficient Cxx
Fig. 3: Different rates depending on the polarisations of beam and target shown at Θ = 40%
2
3 Event detection
Asking for two protons (one hit in each layer above threshold) in coincidence in opposite
detectors the φ-symmetric detector arrangement placed from z = 0...200 mm (z = 0 at the
center of the storage cell) accepts roughly 16.9% of the generated events. This is factor 2.5
larger than for p~d with LR-detectors.
Fig. 4: left: Φ-distribution of reconstructed events
right: Vertex reconstruction
4 Diagonal scaling
To deduce the polarizations from the measured yields the so called diagonal scaling method,
which is described in [2] will be used.
X = αY β
(1)
There the rows and columns of the 4 × 4 matrix Y which contains the yields for 4 detectors
and 4 polarisation combinations are multiplied in such a way, that the new reduced matrix
X satisfies the following conditions with respect to its row sums ri and column sums ck :
X
ri =
xik
(2)
k
ck =
X
xik .
(3)
i
The task is to simultaneously solve all three equations to find the matrices α, β and
X, where the latter contains the pure polarization informations. By correctly adding the
components together and inserting known analyzing powers one can extract the target
polarization if the beam polarization is known or vice versa. In addition to the mentioned
3
paper there is a mathcad file available, which presents the diagonal scaling analysis of p~p~
scattering using a phi symmetric detector arrangement at the PAXwiki [3].
5 Results
Analyzing the reconstructed p~p~ elastic events with the diagonal scaling method shows
that approximately 7 − 8 · 106 events are needed to reach an absolute accuracy of ∆Q =
0.001 independent of the polarization. To calculate the needed measurement time a target
thickness of dt = 4.8 · 1013 /cm2 and 1 · 1010 injected protons were assumed.
Filtering for x lifetimes
1 2 3 4
Total measurement time, h 30 27 40 75
Table 1: Time needed to measure the polarization with at least 10% relative error after filtering
for different amount of beam lifetimes
Besides the larger geometrical acceptance (factor 2.5) the analyzing power or doublespin asymmetries of p~p~ elastic are larger compared to p~d. Finally the needed measurement
times are factor 3.5 - 4 smaller.
Other positive side effects are, that the detector system could pretty much look like the
one which we need for the AD, there is no need to switch the cyclotron from deuterons
to protons and vice versa and there is no more need to shift the BRP tube and therewith
lower the accuracy of the target polarization measurement. In case we are able to run
3 detectors (2 HERMES + 1PAX TTT or 3 PAX TTT) in one layer the percentage of
accepted events increases from 16.9% to 28%.
For the determination of the optimal detector position
the
figure
P sum of the
P in z-direction
2
2
of merit of all events has been maximized (F OM = i dσdΩ · Cxx + i dσdΩ · Cyy ).
To roughly check the influence of capton, which covers the detector and cell walls I just
implemented a deadlayer of different thickness. Due to the high energetic protons and the
geometrical cuts there is no loss of events until a deadlayer thickness of 250µm because the
low energetic protons are anyway not accepted, since the scattering angle of their partner
to small.
In comparison to the p~d elastic scattering for the beam polarization measurement the p~p~
scattering with the larger detector system is highly preferable.
References
[1] Nn-online: http://nn-online.org/nn/.
[2] H. O. Meyer. Diagonal scaling and the analysis of polarization experiments in nuclear
physics. Phys. Rev. C, 56(4):2074–2079, Oct 1997.
[3] PAX Wiki Page. http://apps.fz-juelich.de/pax/paxwiki/index.php/simulations.
4
A Some details of the simulations
• Detector configuration (Fig: 1)
• Target thickness and luminosity
Fig. 5: Calculation of the event rate including beam current and target density. The example
shows a luminosity of 3.2 ×1028 resulting from a target density of 4.8×1013 atoms/cm2 ,
1×1010 injected protons and 2 lifetimes filtering.
• Definition of the differential cross section function
Fig. 6: Differential cross section including analyzing powers and double-spin asymmetries.
5
• Vertex distribution
Fig. 7: Vertex distribution in z direction
• Event generation
Fig. 8: Generation of events with resulting Θ - and Φ - distribution
6
• Calculation of the scattering angles of the second proton and the straight lines of
both particles
Fig. 9: Calculation of particle parameters
• Calculation of the intersection points with the detetctors
Fig. 10: Intersection of the proton flight path with the detector planes
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• Implementation of Φ - and Z - cut
Fig. 11: Hitdistribution in transvers plane
• Energy loss calculation
Fig. 12: Bethe-Bloch formula for silicon depending on the kinetic energy
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• Energy threshold of the 300µ silicon detectors
Fig. 13: If the deposited energy of the particle in the detector is larger than 0.5 MeV it is above
threshold and will be accepted by the data acquisition
• Distribution of the reconstructed events (1st proton)
Fig. 14: Θ - and Φ - distribution of reconstructed events (1st proton)
• Export of all informations of the reconstructed events to the diagonal scaling analysis
9