Estilos de aprendizaje El concepto de estilo en el lenguaje pedagógico suele utilizarse para señalar una serie de distintos

Signals arise in HPGe detectors due to the motion of charge carriers after they have been formed by incident radiation, see Section 2.1.3. The output signal forms immediately after the incident particle deposits its energy and liberates electrons in the active region of the

Figure 2.4: This energy spectrum from Ref. [73] illustrates the advantages using a PPC over a coaxial configuration in terms of energy threshold and resolution at low-energy [76].

detector. Once the last of the charge carriers arrives at the electrode the process of charge induction ends and the pulse is fully developed.

Electric Field and Potential

The geometry of a HPGe detector determines the electric field and capacitance. The electric field directly determines the drift velocity of charge carriers, therefore the configuration is very important in considerations of pulse shape, timing behavior and the overall completeness of charge collection. In each configuration, to determine the electric potential within the detector, we need to solve Poisson’s equation,

∇2ϕ=−ρ

, (2.5)

whereϕis the electric potential in the presence of the charge density ρin a dielectric with a dielectric constant. For p-type germanium,ρ=−eNA, whereeis the electronic charge and

NA is the density of acceptor atoms. Once the electric potential is calculated, the electric field is simply,

E=−∇ϕ. (2.6)

If we neglect diffusion, the charge carriers generated within the detector will follow the electric field lines (or direction of maximal gradient in the potential) from their point of origin to the electrode. Holes get collected at the p+contact while electrons get collected at the n+contact.

Induced Charge

The Shockley-Ramo Theorem [92, 93] provides a method for calculating the induced charge on the collecting electrodes due to the motion of charge carriers. The concept of a weighting field and weighting potential are crucial to the theorem. The instantaneous current induced on a given electrode is given by,

i=qv·E0, (2.7)

whereq is the charge of the carrier,vis the velocity andE0 is the weighting field. Similarly, the induced charge on the electrode is the product of the charge on the carrier multiplied by the difference in the weighting potential (unit-less),ϕ0 from the start to the end of the charge

carrier path length [87]:

Q=q∆ϕ0. (2.8)

The weighting potential is calculated by solving the Laplace equation (∇2_ϕ

0 = 0) for the

geometry of the detector with artificial boundary conditions [87]:

• The voltage on the electrode for which the induced charge is to be calculated is set to unity.

• The voltages on all other electrodes are set to zero.

• Ignore trapped charges (i.e.use Laplace not Poisson equation).

The solution under these conditions yields the weighting potential and the gradient of which is the weighting field. The weighting potential serves as a convenience that allows simple

determination of the induced charge and is not the actual potential within the detector. The induced charge is calculated by taking the differences in the weighting potential at the start and end of the carrier motion. The path of the carrier is determined by the electric field lines. After mapping out the position of a carrier as a function of time, the induced charge as a function of time can be traced out to determine the shape of the output pulse. Figure 2.5 shows the weighting potential for a modified BEGe detector and charge carrier (holes) drift paths.

Figure 2.5: The weighting potential and charge carrier (holes) drift paths of a p-type modified BEGe detector is shown here (the one used in this dissertation). Notice that the weighting potential is maximum near the point contact and essentially zero everywhere else. Induced charge will not contribute to the output signal until the carriers are near the point contact. Figure generated with them3dcr [94] and siggen[95] software packages.

In order to solve for the potential in complex geometries numerical techniques must be used. The Poisson equation is solved by applying a relaxation algorithm. Once the potential is known throughout the bulk, the electric field can be calculated. For example, them3dcr

software package [94] was developed to calculate weighting potential, capacitance and electric fields for complex geometries. Thesiggen software package [95] was developed to calculate

the induced charge and signals from complex geometries based on them3dcr output.

Capacitance

In general, to minimize noise, the capacitance must be as small as possible. In the case of PPC or BEGe detectors, this translates into minimizing the size of the p+contact. According to Ref. [88], if the p+ contact on a PPC is taken to be hemispherical in shape with radiusr, the capacitance is given by,

C ≈2π 0r (2.9)

where0 = 8.85×10−12farad/m is the free space permittivity andis the dielectric constant

of the material. For reference, the capacitance of planar and coaxial detectors is > 10 pF, whereas PPC and BEGe detectors have capacitances<2 pF.

2.2.3 ‘Dead’ Layers

The n+and p+contacts do have an appreciable thickness and cannot be neglected, especially the n+ contact. The most common technique used to create the n+ contact is to diffuse lithium onto the crystal. In the case of both PPC and BEGe detectors, this is on the outside of the detector and will have an effect on the detector response to low-energy gamma-rays (<200 keV). Chapter 6 will discuss the effects of the n+ layer in detail.