- Email: [email protected]
Initial studies in the modelling of position resolving cryogenic detectors
Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
Initial studies in the modelling of position resolving cryogenic detectors J.V. Ashby*, R.F. Fowler, C. Greenough Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, UK
Abstract In this paper, we describe some results in the modelling of a Cryogenic Detector. These detectors use the heat generated from an X-ray event to determine the event’s time and position. The model makes the basic assumption that the heat transport can be represented through by linear diffusion process and that the times at which the temperature changes reach the edge sensors can be used to determine the position of the event. The paper develops a finite element model of the device and performs a series of numerical experiments. The results of these experiments are compared with a simple analytic model. Two methods of determining the event position are presented: one based on an analytic solution and a second using neural network. r 2002 Elsevier Science B.V. All rights reserved. PACS: 95.55.Aq; 95.75.Pq; 84.35.+i Keywords: Cryogenic detectors; Numerical modelling; Finite elements; Neural networks; Position resolving
1. Introduction There are many applications of detectors in science and engineering research. Two large areas are in detectors for high energy physics experiments and in space telescopes of different types. To date considerable use has been made of CCD devices in these areas. However, there is a need to improve these types of imaging system and to design new generations of devices. For future applications highly advanced cryogenic detectors, capable of providing accurate position and timing information on the incoming X-rays, in addition to (more importantly) high resolution spectroscopy of a B2 eV FWHM at *Corresponding author. E-mail address: [email protected] (J.V. Ashby).
2 keV will be necessary. An achievable technology of this type is the multi-node readout of an extended single absorber (perhaps up to 10 10 mm2 ). Here, the design and engineering of the X-ray absorber, plus the coupled solution of the temperature sensor (TES) operated under negative feedback, is seen as a substantial task with many challenges. Computational modelling and prototype simulation is viewed as a key aid to reducing the development time of these devices.
2. The physical model We assume that the heat transfer process is purely one of diffusion. This may not be true: for example, if the dominant transport mechanism is
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 8 7 4 - 5
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
ballistic. Clearly this assumption may well be inadequate but it serves as a starting point. The transport process will be governed by a timedependent heat transport equation [1] of the form @T H 1 ¼ þ r ðkrTÞ @t Cp rCp
ð1Þ
where H is the heat production per unit mass of any source, Cp is the specific heat, r the density and k the thermal conductivity. In general, the physical properties of conductors such as gold and silver will be nonlinear over temperature ranges being considered 1–5 K: In this model, the three important properties are: r; the density; k; the thermal conductivity and Cp ; the specific heat capacity. For a sufficiently small temperature change (B0:1 K) these can be assumed constant. Given this and assuming the initial temperature distribution is a delta function, a solution to (1) on a infinite plane will be a Tð~ r; tÞ ¼ expðr2 =2btÞ t
using (2) we have 2 @u a r 2 ¼ 3 ð2bt r Þexp ¼0 ð3Þ @t 2bt 2bt pffiffiffiffiffiffiffi thus 0 ¼ 2bt r2 ) r ¼ 2bt where r is the distance to the event and t time taken for the temperature peak generated by the event to reach the sensor. This expression will be used later in deriving an analytic expression for the position of an event.
4. The computational model The diffusion equation will be solved computationally using the finite element method. The detector plate is subdivided into a number of finite elements with associated nodes. The temperature over each element is approximated by Te ¼
n X
ð2Þ
where b ¼ 2k=Cp r and a is a constant dependent on the initial conditions [1]. We will use this in the numerical experiments in later sections.
3. Temperature spike transmission time An X-ray event will generate a temperature pulse that will spread out through the device. The sensors at the detector edges will measure a rise in temperature as the pulse reaches them: the pulse will diffuse from the impact point with circular symmetry. The differences in arrival time of the pulse at the sensors will provide sufficient information to determine the event position (~ r0 ) and time (t0 ). In fact, we take the arrival (ti ) to be the arrival time of the temperature maximum. Two approaches to determine ~ r0 and t0 will be discussed in later sections. We can derive a simple relationship between the peak arrival time and the distance to the event using (2). At the temperate peak @u=@t ¼ 0: So
233
ð4Þ
Ni Ti
i¼1
where Ni are the interpolation functions (shape functions) and Ti are the nodal value of the temperature. Eq. (1) is solved by using a standard Galerkin approach. So (1) becomes for each element: Z 1 H @Te Nj r krTe þ dO ¼ 0: ð5Þ rCp Cp @t Oe Assuming that k is independent of position and that the heat sources, H; can be represented in the same manner as T; we can construct a system of ordinary differential equations for each element Ke Te þ Me
@Te ¼ Se He @t
ð6Þ
where Te ¼ ½Ti ; Tj ; Tk ; Tl and He ¼ ½Hi ; Hj ; Hk ; Hl are element vectors and Z k Kije ¼ rNi rNj dO ð7Þ rCp Oe Mije
¼
Z
Ni Nj dO; Oe
Sije
1 ¼ Cp
Z Oe
Ni Nj dO ð8Þ
234
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
are element matrices. These elemental equations are now assembled into a system of equations for the whole problem domain. The solution of the time-dependent system (6) is performed using an implicit y method in which two time levels, n and n þ 1; are linked through a weight y: The time derivatives are approximated by finite differences: a forward difference at n and a backward difference at n þ 1: This leads to 1 1 nþ1 Ke y þ Me Te þ ð1 yÞKe Me Tne Dt Dt ¼ Se ½yHnþ1 þ ð1 yÞHne e
ð9Þ
where n þ 1 and n indicate that the expression is evaluated at times tnþ1 ¼ ðn þ 1ÞDt and tn ¼ nDt; Dt being the step size in time. The value of y can be varied to provide different schemes. Two common values are y ¼ 12; the Crank–Nicholson scheme, and y ¼ 1; the fully implicit scheme.
5. The computational experiments In these experiments, the values of r; k and Cp were taken to be: 19500 kg=m3 ; 2:07 kW=m K and 2:9 104 kJ=kg K: The initial background temperature of the device was 1 K and the event strength set so that the initial temperature rise was around 0:1 K: The boundary conditions used were @T=@n ¼ 0 over the majority of the boundary and near the sensors T was fixed to 1 K: The mesh density and time step size were chosen to produce mesh and time step independent results. In all 121 nodes and 100 elements were used with a time step of 1:0 107 s ð10 msÞ: The results of the simulation are best displayed through temperature response graphs at the sensors. These graphs show the temperature rise at the four sensors and in the case where the event is equi-distant from the sensors these are identical. Fig. 1 shows the sensor response graphs for an event at an arbitrary point on the detector surface. The response graphs are quite different and their peaks reach the sensors at differing times making it
possible to characterise an event. These computational results were compared against the analytic solution (2) which represents the diffusion of a temperature pulse in a infinite plate. The agreement is very good as one might expect from such a simple device geometry. The second set of computational results produced were a complete time and temperature response field for the device. Figs. 2 and 3 show the temperature and time response surfaces at Sensor 1 for any event occurring in the device. It has been generated through a sequence of 361 events spaced evenly on a regular 19 19 grid. The castellation in the time response surface reflects the 1:0 107 time step being used in the computations. These results are the basic data in determining an events position.
6. Analytic position detection The outputs of the model detector are the four arrival times t1 ; t2 ; t3 and t4 ; at which the temperature peaks arrive at a particular sensor. By using analytic solution (2), the event position can be found. Using three of the arrival times the position of the event ~ rðt0 Þ satisfies the following relations: r0 j2 ¼ bðt1 t0 Þ; j~ r1 ~
j~ r2 ~ r0 j2 ¼ bðt2 t0 Þ;
j~ r3 ~ r0 j2 ¼ bðt3 t0 Þ
ð10Þ
t0 is eliminated between the three and this leads to r0 j2 j~ r1 ~ r0 j2 ¼ bðt2 t1 Þ; j~ r2 ~ j~ r2 ~ r0 j2 j~ r3 ~ r0 j2 ¼ bðt2 t3 Þ
ð11Þ
two equations, in the two unknowns x0 and y0 (using ~ rk ¼ fxk ; yk g). Expansion of these and expressing them in a matrix form lead to ! 1 bðt2 t1 Þ x22 y22 þ x21 þ y21 2 bðt2 t3 Þ x22 y22 þ x23 þ y23 ! ! x 1 x2 y 1 y 2 x0 ¼ : ð12Þ x 3 x2 y 3 y 2 y0 The positions of the sensors are given by the vectors ~ r1 ; ~ r2 and ~ r3 : For the simple configuration being considered the vectors are:
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
235
Fig. 1. Sensor response curve for event at an arbitrary point (0.2,0.4).
~ r2 ¼ l½1; 1 and ~ r3 ¼ l½1; 1 where r1 ¼ l½1; 1 ; ~ 2l is the edge distance between sensors. Clearly, " # " #" # 0 2 x0 1 bðt2 t1 Þ ð13Þ ¼l 2 bðt2 t3 Þ 2 0 y0 giving x0 ¼ ð1=4lÞbðt1 t2 Þ and y0 ¼ ð1=4lÞbðt2 t3 Þ: The event time can easily be recovered given x0 and y0 :
7. General position detection For a general device its shape and nonuniformity in material properties will affect the transmission time of the temperature pulse.
A more general approach of determining the event time and position from the four outputs, t1 ; t2 ; t3 and t4 is by using a neural network trained to simulate the detector’s response. We have used a multilayer perceptron (or MLP) as it is able to model both simple and very complex functional relationships [2]. Fig. 4 shows the general form of an MLP network. The basic ideas and software used are from the Neural Network Based System Identification TOOLBOX of Norgaard [1]. The network used was a two-layer model in which there was one hidden layer with four units. The activation functions ðf ; FÞ were chosen to be the hyperbolic tangent and the linear activation. The two-layer MLP can be characterised by the
236
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
Fig. 2. Temperature response surface.
Fig. 3. Time response surface.
following [3,4]: y#i ðw; WÞ ¼ Fi
q X j¼0
Wij fi
m X l¼1
! wjl zl þ wj0
! þ Wi0 : ð14Þ
The weights w and W are the parameters of the network and they are determined from the set of sample data through the training process. The sample data are sets of inputs, uðtÞ and corresponding desired outputs, yðtÞ: The objective of training is to determine the weights that will
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
# produce predictions yðtÞ which are close to the true output yðtÞ: A large number of training algorithms exist, each of which is characterised by the way in which the search direction and step size are selected. For these computations, the method due to Levenberg and Marquardt is used the details of which are given in Ref. [5]. The computational model was used to generate a grid of 361 data points consisting of the event position, ðxi ; yi Þ: The response times and temperature peaks for each sensor, ðtji ; Tij Þ; i ¼ 1; 4; j ¼ 1; N were generated.
Fig. 4. A fully connected two-layer MLP-network.
237
The data was separated into the training (181 points) and test (180 points) sets. The data was suitably scaled and 500 iterations were used to train the MLP network. The final error was around 1 103 : Fig. 5 shows an example of the final fit of the network to the test data. These weights can then be used the calculate the event positions from (14) and a set of timing values.
8. Conclusion In this paper, we have described some initial attempts at modelling a position-resolving cryogenic detector. Although this work is based on the assumption that the heat transfer process can be adequately represented by a linear diffusion, comparison will be needed against experimental data to validate this approach. We have shown that the computational results compare well with the simple analytic solution on an infinite plate and that the position of an event can be recovered via either an analytical expression (using the analytic solution) or an appropriately trained neural network.
Fig. 5. Fitting of output 1 (x coordinate) to computational data.
238
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
Future work will include a more detailed physical model of the detector and a better representation of the heat flow into the temperature bath on which the detector sits. Also some representation of the heat losses from the device surfaces will required to develop a more realistic computational model.
References [1] F. Mandl, Statistical Physics, Wiley, New York, 1978.
[2] H. Demuth, M. Beale, Neural Network Toolbox, The MathWorks Inc., 1987. [3] M. Norgaard, Neural network based system identification toolbox, Department of Automation, Technical University of Denmark, Technical Report 97-E-851, 1997. [4] M.T. Hagan, H.B. Demuth, M. Beale, Neural Network Design, PWS Publishing Company, Boston, MA, 1995. [5] R. Fletcher, Practical Methods of Optimisation, Wiley, New York, 1987.
Initial studies in the modelling of position resolving cryogenic detectors J.V. Ashby*, R.F. Fowler, C. Greenough Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, UK
Abstract In this paper, we describe some results in the modelling of a Cryogenic Detector. These detectors use the heat generated from an X-ray event to determine the event’s time and position. The model makes the basic assumption that the heat transport can be represented through by linear diffusion process and that the times at which the temperature changes reach the edge sensors can be used to determine the position of the event. The paper develops a finite element model of the device and performs a series of numerical experiments. The results of these experiments are compared with a simple analytic model. Two methods of determining the event position are presented: one based on an analytic solution and a second using neural network. r 2002 Elsevier Science B.V. All rights reserved. PACS: 95.55.Aq; 95.75.Pq; 84.35.+i Keywords: Cryogenic detectors; Numerical modelling; Finite elements; Neural networks; Position resolving
1. Introduction There are many applications of detectors in science and engineering research. Two large areas are in detectors for high energy physics experiments and in space telescopes of different types. To date considerable use has been made of CCD devices in these areas. However, there is a need to improve these types of imaging system and to design new generations of devices. For future applications highly advanced cryogenic detectors, capable of providing accurate position and timing information on the incoming X-rays, in addition to (more importantly) high resolution spectroscopy of a B2 eV FWHM at *Corresponding author. E-mail address: [email protected] (J.V. Ashby).
2 keV will be necessary. An achievable technology of this type is the multi-node readout of an extended single absorber (perhaps up to 10 10 mm2 ). Here, the design and engineering of the X-ray absorber, plus the coupled solution of the temperature sensor (TES) operated under negative feedback, is seen as a substantial task with many challenges. Computational modelling and prototype simulation is viewed as a key aid to reducing the development time of these devices.
2. The physical model We assume that the heat transfer process is purely one of diffusion. This may not be true: for example, if the dominant transport mechanism is
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 8 7 4 - 5
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
ballistic. Clearly this assumption may well be inadequate but it serves as a starting point. The transport process will be governed by a timedependent heat transport equation [1] of the form @T H 1 ¼ þ r ðkrTÞ @t Cp rCp
ð1Þ
where H is the heat production per unit mass of any source, Cp is the specific heat, r the density and k the thermal conductivity. In general, the physical properties of conductors such as gold and silver will be nonlinear over temperature ranges being considered 1–5 K: In this model, the three important properties are: r; the density; k; the thermal conductivity and Cp ; the specific heat capacity. For a sufficiently small temperature change (B0:1 K) these can be assumed constant. Given this and assuming the initial temperature distribution is a delta function, a solution to (1) on a infinite plane will be a Tð~ r; tÞ ¼ expðr2 =2btÞ t
using (2) we have 2 @u a r 2 ¼ 3 ð2bt r Þexp ¼0 ð3Þ @t 2bt 2bt pffiffiffiffiffiffiffi thus 0 ¼ 2bt r2 ) r ¼ 2bt where r is the distance to the event and t time taken for the temperature peak generated by the event to reach the sensor. This expression will be used later in deriving an analytic expression for the position of an event.
4. The computational model The diffusion equation will be solved computationally using the finite element method. The detector plate is subdivided into a number of finite elements with associated nodes. The temperature over each element is approximated by Te ¼
n X
ð2Þ
where b ¼ 2k=Cp r and a is a constant dependent on the initial conditions [1]. We will use this in the numerical experiments in later sections.
3. Temperature spike transmission time An X-ray event will generate a temperature pulse that will spread out through the device. The sensors at the detector edges will measure a rise in temperature as the pulse reaches them: the pulse will diffuse from the impact point with circular symmetry. The differences in arrival time of the pulse at the sensors will provide sufficient information to determine the event position (~ r0 ) and time (t0 ). In fact, we take the arrival (ti ) to be the arrival time of the temperature maximum. Two approaches to determine ~ r0 and t0 will be discussed in later sections. We can derive a simple relationship between the peak arrival time and the distance to the event using (2). At the temperate peak @u=@t ¼ 0: So
233
ð4Þ
Ni Ti
i¼1
where Ni are the interpolation functions (shape functions) and Ti are the nodal value of the temperature. Eq. (1) is solved by using a standard Galerkin approach. So (1) becomes for each element: Z 1 H @Te Nj r krTe þ dO ¼ 0: ð5Þ rCp Cp @t Oe Assuming that k is independent of position and that the heat sources, H; can be represented in the same manner as T; we can construct a system of ordinary differential equations for each element Ke Te þ Me
@Te ¼ Se He @t
ð6Þ
where Te ¼ ½Ti ; Tj ; Tk ; Tl and He ¼ ½Hi ; Hj ; Hk ; Hl are element vectors and Z k Kije ¼ rNi rNj dO ð7Þ rCp Oe Mije
¼
Z
Ni Nj dO; Oe
Sije
1 ¼ Cp
Z Oe
Ni Nj dO ð8Þ
234
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
are element matrices. These elemental equations are now assembled into a system of equations for the whole problem domain. The solution of the time-dependent system (6) is performed using an implicit y method in which two time levels, n and n þ 1; are linked through a weight y: The time derivatives are approximated by finite differences: a forward difference at n and a backward difference at n þ 1: This leads to 1 1 nþ1 Ke y þ Me Te þ ð1 yÞKe Me Tne Dt Dt ¼ Se ½yHnþ1 þ ð1 yÞHne e
ð9Þ
where n þ 1 and n indicate that the expression is evaluated at times tnþ1 ¼ ðn þ 1ÞDt and tn ¼ nDt; Dt being the step size in time. The value of y can be varied to provide different schemes. Two common values are y ¼ 12; the Crank–Nicholson scheme, and y ¼ 1; the fully implicit scheme.
5. The computational experiments In these experiments, the values of r; k and Cp were taken to be: 19500 kg=m3 ; 2:07 kW=m K and 2:9 104 kJ=kg K: The initial background temperature of the device was 1 K and the event strength set so that the initial temperature rise was around 0:1 K: The boundary conditions used were @T=@n ¼ 0 over the majority of the boundary and near the sensors T was fixed to 1 K: The mesh density and time step size were chosen to produce mesh and time step independent results. In all 121 nodes and 100 elements were used with a time step of 1:0 107 s ð10 msÞ: The results of the simulation are best displayed through temperature response graphs at the sensors. These graphs show the temperature rise at the four sensors and in the case where the event is equi-distant from the sensors these are identical. Fig. 1 shows the sensor response graphs for an event at an arbitrary point on the detector surface. The response graphs are quite different and their peaks reach the sensors at differing times making it
possible to characterise an event. These computational results were compared against the analytic solution (2) which represents the diffusion of a temperature pulse in a infinite plate. The agreement is very good as one might expect from such a simple device geometry. The second set of computational results produced were a complete time and temperature response field for the device. Figs. 2 and 3 show the temperature and time response surfaces at Sensor 1 for any event occurring in the device. It has been generated through a sequence of 361 events spaced evenly on a regular 19 19 grid. The castellation in the time response surface reflects the 1:0 107 time step being used in the computations. These results are the basic data in determining an events position.
6. Analytic position detection The outputs of the model detector are the four arrival times t1 ; t2 ; t3 and t4 ; at which the temperature peaks arrive at a particular sensor. By using analytic solution (2), the event position can be found. Using three of the arrival times the position of the event ~ rðt0 Þ satisfies the following relations: r0 j2 ¼ bðt1 t0 Þ; j~ r1 ~
j~ r2 ~ r0 j2 ¼ bðt2 t0 Þ;
j~ r3 ~ r0 j2 ¼ bðt3 t0 Þ
ð10Þ
t0 is eliminated between the three and this leads to r0 j2 j~ r1 ~ r0 j2 ¼ bðt2 t1 Þ; j~ r2 ~ j~ r2 ~ r0 j2 j~ r3 ~ r0 j2 ¼ bðt2 t3 Þ
ð11Þ
two equations, in the two unknowns x0 and y0 (using ~ rk ¼ fxk ; yk g). Expansion of these and expressing them in a matrix form lead to ! 1 bðt2 t1 Þ x22 y22 þ x21 þ y21 2 bðt2 t3 Þ x22 y22 þ x23 þ y23 ! ! x 1 x2 y 1 y 2 x0 ¼ : ð12Þ x 3 x2 y 3 y 2 y0 The positions of the sensors are given by the vectors ~ r1 ; ~ r2 and ~ r3 : For the simple configuration being considered the vectors are:
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
235
Fig. 1. Sensor response curve for event at an arbitrary point (0.2,0.4).
~ r2 ¼ l½1; 1 and ~ r3 ¼ l½1; 1 where r1 ¼ l½1; 1 ; ~ 2l is the edge distance between sensors. Clearly, " # " #" # 0 2 x0 1 bðt2 t1 Þ ð13Þ ¼l 2 bðt2 t3 Þ 2 0 y0 giving x0 ¼ ð1=4lÞbðt1 t2 Þ and y0 ¼ ð1=4lÞbðt2 t3 Þ: The event time can easily be recovered given x0 and y0 :
7. General position detection For a general device its shape and nonuniformity in material properties will affect the transmission time of the temperature pulse.
A more general approach of determining the event time and position from the four outputs, t1 ; t2 ; t3 and t4 is by using a neural network trained to simulate the detector’s response. We have used a multilayer perceptron (or MLP) as it is able to model both simple and very complex functional relationships [2]. Fig. 4 shows the general form of an MLP network. The basic ideas and software used are from the Neural Network Based System Identification TOOLBOX of Norgaard [1]. The network used was a two-layer model in which there was one hidden layer with four units. The activation functions ðf ; FÞ were chosen to be the hyperbolic tangent and the linear activation. The two-layer MLP can be characterised by the
236
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
Fig. 2. Temperature response surface.
Fig. 3. Time response surface.
following [3,4]: y#i ðw; WÞ ¼ Fi
q X j¼0
Wij fi
m X l¼1
! wjl zl þ wj0
! þ Wi0 : ð14Þ
The weights w and W are the parameters of the network and they are determined from the set of sample data through the training process. The sample data are sets of inputs, uðtÞ and corresponding desired outputs, yðtÞ: The objective of training is to determine the weights that will
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
# produce predictions yðtÞ which are close to the true output yðtÞ: A large number of training algorithms exist, each of which is characterised by the way in which the search direction and step size are selected. For these computations, the method due to Levenberg and Marquardt is used the details of which are given in Ref. [5]. The computational model was used to generate a grid of 361 data points consisting of the event position, ðxi ; yi Þ: The response times and temperature peaks for each sensor, ðtji ; Tij Þ; i ¼ 1; 4; j ¼ 1; N were generated.
Fig. 4. A fully connected two-layer MLP-network.
237
The data was separated into the training (181 points) and test (180 points) sets. The data was suitably scaled and 500 iterations were used to train the MLP network. The final error was around 1 103 : Fig. 5 shows an example of the final fit of the network to the test data. These weights can then be used the calculate the event positions from (14) and a set of timing values.
8. Conclusion In this paper, we have described some initial attempts at modelling a position-resolving cryogenic detector. Although this work is based on the assumption that the heat transfer process can be adequately represented by a linear diffusion, comparison will be needed against experimental data to validate this approach. We have shown that the computational results compare well with the simple analytic solution on an infinite plate and that the position of an event can be recovered via either an analytical expression (using the analytic solution) or an appropriately trained neural network.
Fig. 5. Fitting of output 1 (x coordinate) to computational data.
238
J.V. Ashby et al. / Nuclear Instruments and Methods in Physics Research A 477 (2002) 232–238
Future work will include a more detailed physical model of the detector and a better representation of the heat flow into the temperature bath on which the detector sits. Also some representation of the heat losses from the device surfaces will required to develop a more realistic computational model.
References [1] F. Mandl, Statistical Physics, Wiley, New York, 1978.
[2] H. Demuth, M. Beale, Neural Network Toolbox, The MathWorks Inc., 1987. [3] M. Norgaard, Neural network based system identification toolbox, Department of Automation, Technical University of Denmark, Technical Report 97-E-851, 1997. [4] M.T. Hagan, H.B. Demuth, M. Beale, Neural Network Design, PWS Publishing Company, Boston, MA, 1995. [5] R. Fletcher, Practical Methods of Optimisation, Wiley, New York, 1987.