· Compton camera as a medical imaging system.

· 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 Compton scattering cross section where the best candidates for this purpose are mainly silicon and germanium.

· 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 Compton scattering


*Figure I.2:. Illustration of recorded coincident events by Compton camera*	*Figure 1.3*Compton 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 Sensitive Material NaI(Tl) Diameter 20 inch Thickness ¾ inch Energy resolution 10% at 662keV Position resolution 3-4 mm (With collimator)	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 · peak-to-background values larger than 3000 · quantum efficiency larger than 80 % for energies between 500 eV and 15 keV

Figure I.6: Compton camera schematic in Siegen.

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.

Compton Camera in Siegen.

Experimental setup.

Jun 2003

Compton Camera in Siegen:

Experimental setup.

September 2003

Compton Camera in Siegen:

Experimental setup.

September 2003

Compton Camera in Siegen:

Experimental setup.

September 2003

Compton Camera in Siegen: Experimental setup. September 2003

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), n-type (b) and p-type (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:

n-type semiconductor N_D >> N_A, N_D >> n_i		I.7
p-type semiconductor N_A >> N_D, N_A >> n_i		I.8
Intrinsic semiconductor n_o = p_o		I.9

· Low Temperature (T < T s ) or IONIZATION RANGE: the increase in temperature ionizes donors. Donor ionization continues until a temperature T_s (saturation temperature) where all the donors are ionized. For n-type semiconductors at very low temperature the electron concentration in the conduction band is given by:

I.15

Medium Temperature (T s < T < T i ) or EXTRINSIC RANGE: nearly all donors have been ionized, n= N_D . This remains until a temperature T_i when n_i becomes equal to N_D. Common semiconductors utilize the n- doping properties in this temperature range.

I.16

High Temperature (T > T i ) or INTRINSIC RANGE: In this temperature range the concentration of electrons generated by thermal excitation across the band gap, n_i , is much larger than N_D; the electron concentration is n= n_i (T). Excitation occurs from the VB to the CB and the situation is similar to the intrinsic semiconductor with p= n.

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.
1. High purity Si (N_a = 10¹² cm^-3); [9]

2. High purity Si (N_a~10¹⁴ cm^-3);[10]
3. N_a=2.4·10¹⁶ cm^-3; N_d=2.3·10¹⁵ cm^-3;[11]

4. N_a=2·10¹⁷ cm^-3; N_d=4.9·10¹⁵ cm^-3;[11]

Figure I.15: Temperature dependence of the saturation electron drift velocity [13]

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

Figure 1.13: Synoptic schem of the silicon drift detector mounted on an Aluminuim Nitride ceramic support

Figure 1.14: Housing of the silicon drift detector.

Figure 1.15: Cooling setup of the silicon drift detector.

Materials	Thermal conductivity [W. m-1. K-1]
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 Fe⁵⁷ 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.

For the thermal analysis of our setup (Peltier elements + Ceramic support + silicon drift detector) we have used QuickField software. QuickField can perform linear and nonlinear thermal analysis for 2D, 3D and axisymmetric models. Using finite element calculation techniques and the program is based on heat conduction equations with convection and radiation boundary conditions.

	· 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 symmetric-A.	Figure 1.20: temperature distribution symmetric configuration A T_SP=0°C

Figure 1.21: Configuration, symmetric-B.	Figure 1.22: temperature distribution the symmetric configuration B T_SP=0°C

Figure 1.23: temperature profile in the three configurations (Asymmetric, Symmetric-A and Symmetric-B) on the Line L2 at T_Peltier=0°

Single Loop Temperature Controllers

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

Temperature control techniques

Good thermal transfer between the cooler device, the system to be cooled and the temperature sensor. This can be achieved by using some thermal couplers.
Adequate “thermal mass” of the system to minimize its sensitivity to external perturbations loaded from ambient atmosphere or from other external disturbance source.
The temperature sensor is located within reasonable “thermal distance” from the cooler device such that it will respond to any changes in temperature and it has to be representative of the system temperature (the “Object” to be cooled).
Adequate power for the cooler device should be available.

On-Off Control

Figure 1.25: On-off control in a cooling process.

Proportional control / P-Control

2.1

Where P is known as the proportional gain of the controller, T_SP is the set-point 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 thermo-cooler, P the proportioning gain, I the integral gain parameter, T_SP, T are the set-point 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].

Derivative (Automatic Rate)

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 set-point temperature and the measured temperature respectively and t is the time.

Proportional + Integral + Derivative (PID control)

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 set-point 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 set-point 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_iand 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 W₀=0 for the first temperature measurement; we can write then:

*Index Value i*	*Value of W_i*
0	W₀=0 (always)
1
2
.	.
.	.
.	.
n		2.15

And this algorithm is applied every sampling time until the measured temperature and the set-point temperature coincides. Of course the P, I and D parameters should be optimized.