
MM's
G_{en}Picture
Gallery
Version
II 

During the last month, I've set up a simulation
for the electron arm (Target + HMS) of our G_{en}experiment.
The gallery shows some the results for the latest kinematic
settings I have got from Donal. If not stated otherwise
I've assumed a momentum spread of 15% to +15%. The incident electrons
are treated as going straight through a target of a length of 3cm in beam
direction and the beam is rastered across a target surface of 2x2cm^{2
}.
The
Kinematics
The
Acceptance
The
Virtual Image
The
Vertical Beam Offset
The
Reconstruction
Including
the Beam Offset
and
the Target Magnetic Field
The simulations where performed for the following four kinematic
setups (Donal, I hope I got the most recent ones). The simulation
still assumes the first dipol to be near to the target (which would not
be the case in the real experiment, but I've not all the bits of information
ready now to run the simulation with the final arrangement)
Kinematic
Q^{2} 
E_{Beam }[GeV] 
E_{0 }[GeV] 
E_{Beam}E_{0 }[GeV] 
theta_{HMS} [deg] 
theta_{B} [deg] 
0.50 
2.724 
2.474 
0.268 
15.7 
151.60 
1.00 
4.507 
3.971 
0.536 
13.5 
154.26 
1.50 
4.996 
4.193 
0.803 
15.4 
139.44 
1.86 
4.996 
3.926 
1.070 
18.4 
134.29 
The accepted angles (dx/dz and dy/dz)
are determined by the octogonshaped entrancewindow of HMS. The target
field introduces a shift to smaller values of dx/dz and tilts
the octagon of accepted angles slightly. The figure on the right shows
this effect for the Q^{2}=0.5 kinematic and a point target.
Similar plots for all the kinematics studied are shown on this viewgraph
(.ps). 

The effect of the target magnetic field was studied with the so called
"virtual image" methode. The virtual image of a given interaction point
is the point where we have  assuming no field  to start the electron
in order to obtain the same path into HMS (e.g. get the same focal plane
coordinates).
The figure on the right shows the deviation (x_{0}x,
y_{0}y) of the virtual points from their corresponding
interaction points (x_{0},y_{0}) at
Q^{2}=0.5. for a couple of discrete points (x_{0},y_{0})
distributed over the target face covered by the beam raster. As the simulation
includes the whole phase space accepted by HMS we get a cloud of virtual
points for a single interaction point.
Similar plots for all kinematics studied are shown on this viewgraph
(.ps) 

The mean deviation in the x direction increases from 0.75mm at Q^{2}=0.5
to 1mm at Q^{2}=1.86. Also the value of y_{0}
has a strong influence on the shift in the x direction, at Q^{2}=0.5
we the shift increasing from 0mm at y_{0}=1cm to 1.5mm
at y_{0}=1cm. At higer Q^{2}'s the influence
is smaller, but still present.
The influence of a vertical beam offset on the y difference is
rather big, the y shift ranges from 1mm to 1mm for x_{0}=[1cm,1cm]
at Q^{2}=0.5 and about half the value at higher Q^{2}'s.
The effect of the vertiacl beam offset was studied without the target magnetic
field. For HMS moving the electron beam up or down looks like a shift in
delta, the relative momentum displacement.
The figure on the right shows the shift of the the focal plane coordinate
x_{FP} as a function of the vertical beam displacement
x_{0 }for a couple of discrete delta values in the
range 6% to 6% (and with dx_{0}/dz and dy_{0}/dz
= 0). A beam offset of x_{0 }=1cm, if not corrected
for, would introduce an error of about 1% in the delta reconstruction.
As the dependence of x_{FP} from x_{0}
looks pretty linear, it should be possible to apply a correction  based
on a few callibration measurements  to the focal plane x_{FP
}measured to compensate the effect of the vertical beam offset. 


There is a similar effect on the focal plane dx_{FP}/dz
as shown in the figure on the left.
The influence on the focal plane y_{FP} and dy_{FP}/dz
is only minor and can be neglected
The two plots are combined on a single viewgraph
(.ps) 
The methode of correcting the measured focal plane quantities according
to the vertical beam offset has the main advanage that we can use the standard
HMS reconstruction function with full symmetrie and all the knowledge therein
to find the target coordinates.
The figure on the right visualizes the coefficients A_{ijkl }for
the delta reconstruction. The value of delta is calucalated from
the (corrected) focal plane quantities as the
SUM_{ijkl}(A_{ijkl} ^{.}
x^{i} ^{.} (dx/dz)^{j}
^{.} y^{k} ^{.}
(dy/dz)^{l }).
In order to show the most imported coefficients the A_{ijkl}
are weighted with the maximum range of the focal plane quantities to the
power of i, j, k and l, respectively.
To get the ijkl indices you have to combine the numbers in the
axis labels addressing one A_{ijkl} square.


There are also full size viewgraphs for all the quantities being reconstructed,
namely
delta
(.ps), dx/dz
(.ps), y
(.ps)
and dy/dz
(.ps).
The
Reconstruction Including the Vertical Beam Offset
The methode how to treat the vertical beam offset is outlined on the program
page. To test the quality of this methode a simulation was made covering
the whole phase space accepted by HMS and the whole target volume (length
3cm, width and heigth 2cm). Then the resolution was determined by comparing
the target coordinates reconstructed from this sample with the original
ones.
The figure on the right shows that we can obtain a resolution
of about 0.1% (FWHM) in the delta reconstruction. The tail to the
left ist mostly due to events at the edge of the accepted phase space with
a delta below 12% and low values of dx_{0}/dz
(cf. detailed viewgraph
(.ps)). 


The figure on the left shows that we can obtain a resolution of about
0.1^{o} (FWHM) in the dx/dz reconstruction. Again
the tail to the left ist mostly due to events at the edge of the accepted
phase space with a delta below 12% and low values of dx_{0}/dz
(cf. detailed viewgraph
(.ps)). 
The figure on the right shows that we can obtain a resolution
of about 2mm (FWHM) in the y reconstruction. 


The figure on the left shows that we can obtain a resolution of about
0.02^{o} (FWHM) in the dy/dz reconstruction. 
The four plots shown above are combined on a single viewgraph
(.ps).
More details can be found on viewgraphs showing the resolution for delta
(.ps), dx/dz
(.ps), y
(.ps), dy/dz
(.ps)
as a function of the different target coordinates.
The
Reconstruction Including the Vertical Beam Offset and the Target Magnetic
Field
The methode how to treat the target magnetic field is outlined on the program
page. To test the quality of this methode a simulation was made covering
the whole phase space accepted by HMS and the whole target volume (length
3cm, width and heigth 2cm) for the four relevant kinematics as shown in
the table
above. Then the resolution was determined by comparing the target
coordinates reconstructed from this sample with the original ones.
The figure on the right shows that we can obtain a resolution
of about 0.1% (FWHM) in the delta reconstruction (Q^{2}=0.5).
The tail to the left ist mostly due to events at the edge of the accepted
phase space with a delta below 12% and low values of dx_{0}/dz
(cf. detailed viewgraph
(.ps)). 


The figure on the left shows that we can obtain a resolution of about
0.2^{o} (FWHM) in the dx/dz reconstruction (Q^{2}=0.5).
Again the tail to the left ist mostly due to events at the edge of the
accepted phase space with a delta below 12% and low values of dx_{0}/dz
(cf. detailed viewgraph
(.ps)). 
The figure on the right shows that we can obtain a resolution
of about 2mm (FWHM) in the y reconstruction (Q^{2}=0.5). 


The figure on the left shows that we can obtain a resolution of about
0.04^{o} (FWHM) in the dy/dz reconstruction (Q^{2}=0.5). 
The figure on the right shows that we can obtain a resolution
of about 0.1mm (FWHM) in the x reconstruction (Q^{2}=0.5).
This small value does not surprise because x_{0}x_{1}
is the convergence criteria for the iteration process. 


The figure on the left shows that we can obtain a resolution of about
5mm (FWHM) in the z reconstruction (Q^{2}=0.5). 
The six plots shown above and similar ones for the other values of
Q^{2} combined on single viewgraphs are also available and
even more details can be found on viewgraphs showing the resolution for
delta, dx/dz, y , dy/dz, x
and z as a function of the different target coordinates:
Markus Mühlbauer
June 8th 1998