·
·
Compton Camera operates by
exploiting mainly Compton Effect and it is a combination of at least two detectors;
“scattering” and “absorption” detectors.
·
For a high performance of the
imaging system the scattering detector should have a very high
·
The choice of semiconductor
detectors as scattering detectors comes on the fact that the solid state
materials provide a very high efficiency if compared to gas detectors, in one
hand.
·
In other hand the high density of
those materials and also the available semiconductor technology allows the
manufacturers to increase the performance of such detectors in terms of
efficiency, position and energy resolution.

Figure
I.1: The kinematics of 


Figure I.2:. Illustration of recorded coincident
events by Compton camera 
Figure 1.3Compton camera concept 

Schematic view of a one cell
cylindrical Silicon Drift Detector 


Figure I.4: Anger Camera as
absorption detector 
Figure I.5: 19 Cells SDD as
Scattering Detector 
characteristics

The SDDs
are characterized by ·
good energy resolution values
typically 147 eV for MnKa
at 10°C ·
high count rate capabilities up to
10 ^{6} per second ·
peaktobackground values larger
than 3000 ·
quantum efficiency larger than 80
% for energies between 500 eV and 15 keV 

Figure I.6: 

AutoCad
simulation Figure
1.7: First proposal of the mechanical setup. 

AutoCad
simulation Figure
1.8: Second proposal of the mechanical setup. 
AutoCad simulation Approved on 12.01.2003
Figure 1.9: Third proposal of the mechanical setup.
Experimental
setup. Jun 2003 
Experimental
setup. September 2003 
Experimental
setup. September 2003 

Experimental
setup. September 2003 
Compton
Camera in
In addition to the optical
properties (which are not our main topic), the temperature dependence of the
electrical properties of semiconductors are well known since the discovery of
this material. The temperature effects are much seen on the electrical
conductivity and also on the drift mobility of the charge carriers in
semiconductors.
Electrical
conductivity and Energy band structures.
Figure
I.10: Energy bands of intrinsic (a), ntype (b) and ptype (c) semiconductors.
We have then to underline, that the
Fermi energy level depends strongly on the doping material concentration N_{D}
and N_{A} and also it is closely related to the temperature. The Fermi
energy level is given by:
ntype
semiconductor 
_{} 
I.7 
ptype
semiconductor 
_{} 
I.8 
Intrinsic
semiconductor 
_{} 
I.9 
·
Low Temperature (T < T
s ) or
_{} 
I.15 
_{} 
I.16 
Figure
I.11: Electron Concentration and Temperature n type semiconductor
Temperature Dependence of
the Drift Mobility
At
high temperature regime, temperature effects may be described as the charge
carriers scattering by lattice waves including the absorption or emission of
either acoustical or optical phonons. Since the density of phonons in solid
increases with temperature, the scattering time due to this mechanism will
decrease as well as the mobility of the carriers during the drift.
_{} 
I.17 
At low temperature regime, the drift
mobility is limited by scattering of electrons by ionized impurities and the
mobility varies: _{}

Germanium 
Silicon 
Gallium Arsenide 
Electron mobility 
_{}T^{1.7} 
_{}T^{1.4} 
_{}T^{1.0} 
Hole mobility 
_{}T^{2.3} 
_{}T^{2.2} 
_{}T^{2.1} 


Figure
I.11: Temperature dependences of hole mobility for
different doping levels. 2. High purity Si
(N_{a}~10^{14} cm^{3});[10] 4. N_{a}=2·10^{17}
cm^{3}; N_{d}=4.9·10^{15}
cm^{3};[11] 
Figure I.15: Temperature dependence of the saturation electron
drift velocity [13] 

Figure
1.12: Thermoelectric (TE) cooler /Peltier Module. 
Materials

Thermal
conductivity [W. m1. K1]

Silicon
(Si)

80

Alumina ceramic. (Al2O3).

35

Aluminium
Nitride ceramic

180

Epoxy
glue

0.4

Grease

0.9

Response of the one cell SDD to Fe^{57} source. 



T ~
30 ºC 
T ~
10 ºC 
The energy resolution is
related to the angular resolution of the scattered photon by the following
formula.
_{} 
I.36 
This traduce a certain
uncertainty in the scattered angle which is directly related to the resoltion
of the reconstructed image.
Therefore, a very high energy
resolution in the scarreting detector will give a very small uncertainty on the
scattering angle and then more accurate results.
The main goel of this work is
mainly to minimize the contribution of the different fluctuations (noises)
themaly activated by cooling the silicon drift decetor and realising an
apropriate cooling system for the scattering detector.

·
Implementing
the Geometry ·
Material
properties ·
Loading
sources ·
Initial
and boundary conditions ·
Postprocessing results: 







Figure 1.16: Test setup for cooling the silicon drift detector.

Figure 1.17: Simulation and experimental data of temperature profile on the line L1






Figure 1.17: Configuration 1, Asymmetric.

Figure 1.18: temperature distribution the asymmetric configuration 1 T_{SP}=0°C



Figure
1.19: Configuration symmetricA.

Figure 1.20:
temperature distribution symmetric configuration A T_{SP}=0°C



Figure
1.21: Configuration, symmetricB.

Figure
1.22: temperature distribution the symmetric configuration B T_{SP}=0°C

Single
Figure
1.24: Synoptic scheme of a single loop process for temperature control.
Temperature
control techniques

Figure
1.25: Onoff control in a cooling process. 
Proportional control / PControl
_{} 
2.1 
Where P is known as the proportional gain of the controller, T_{SP} is the setpoint
temperature and T the measured
temperature.
Figure
1.26: Proportional control in a cooling process.
Integral or "Reset Action"/PI control
_{} 
2.2 
When combined with the proportional
function, the power delivered to the cooling device is given by:
_{}

2.3

Where W
is the applied power to the thermocooler, P
the proportioning gain, I the integral gain parameter, T_{SP}, T are the
setpoint temperature and the measured temperature respectively and t is the time.
Figure
1.27: the effect of the Integral parameter I on a PI control of a cooling
process where P=10% and I equal respectively 10,20
and 40 [time unit]. 
_{} 
2.4 
Then, the power W using the proportional function and the derivative function is
given by:
_{} 
2.5 
Where
W is the applied power to the cooler device, P the proportioning factor, D the
derivative Factor, T_{SP}, T are the setpoint temperature and the
measured temperature respectively and t is the time.
_{} 
2.7 
Where
W is the applied power to the cooler component, P the proportional gain, D the
derivative Factor (or damping constant), T_{SP}, T are the setpoint
temperature and the measured temperature respectively, t is the time and I the
integral gain parameter.

Figure
1.28: the effect of P, PI and PID control of a cooling process 
_{} 
2.8 
T_{sp}
is the setpoint temperature.
T(t) is the
measured temperature a time t.
The equation 2.7 becomes then:
_{} 
2.9 


Figure 1.29: Cooling process from
50°C to 10°C 
Figure 1.30: The error function
e(t) 
The integral of the curve e(t) calculated using the trapezium method is then given by:
_{} 
2.11 
However, it is very important to underline, that in a
control process we don’t know the whole error function e(t) at the beginning of
the control, but while processing we can get progressively the value of the
error every Δt (sampling time ) of the
temperature. This means that we calculate a integral
between two point separated by Δt and we have to use the equation 2.10 instead of 2.11.
Then integral is given by:
_{} 
2.12 
Δt=t_{i+1}
– t_{i} is the sampling time
Calculation
of _{}.
The differentiation of the error function e(t) is given by the standard slope formula:
_{} 
2.13 
Where Δt=t_{i+1}
– t_{i}_{ }and _{}a very small sampling time.
Reformulation of
the PID algorithm:
If the sampling time Δt is very small if compared to the whole time needed for the control
process and knowing the differentiation and the integration of the error
function e(t) the equation 2.9 can be written as following:
When Δt→_{}
_{} 
2.14 
Where i=0,1,2,3,….n
P, I and D
are respectively the proportional gain, the automatic reset and the derivative
parameter.
It is obvious from the numerical methods used for the
differentiation and the integration of the error function e(t), that to
calculate the value of W we need a
least two initial errors and we start calculating W from the second error. This means that W_{0}=0 for the first temperature measurement; we can write then:
Index
Value i 
Value of W_{i}_{} 

0 
W_{0}=0
(always) 

1 
_{} 

2 
_{} 

. 
. 

. 
. 

. 
. 

n 
_{} 
2.15 
And this algorithm is applied every _{} sampling time until
the measured temperature and the setpoint temperature coincides. Of course the
P, I and D parameters should be optimized.
The PID Temperature Controller named
TC1700 involves mainly three important modules.
ISP (Insystem Programmable) microcontroller
AT90S8535 from Atmel


Figure
1.31: Synoptic configuration of the TC1700

