Investigation of the Frisch-grid inefficiency by means of wave-form digitization

Investigation of the Frisch-grid inefficiency by

means of wave-form digitization

Alf Göök – Fysikprogrammet Institutionen för naturvetenskap

2008-06-23

Examensarbete, 30 högskolepoäng Handledare: Andreas Oberstedt

A

BSTRACT

Frisch grid ionization chambers are commonly used experimental tools for charged particle spectroscopy. In the ideal Frisch grid chamber the anode pulse height is independent of where inside the sensitive volume the charge has been created. This ideal cannot be realized because of imperfect shielding of the anode by the grid. The effect of the imperfect shielding is generally referred to as grid inefficiency. For accurate energy determination the anode pulse height needs to be corrected for this.

At present two opposing explanation for grid inefficiency exist. The first explanation suggests that there is a reduction of the anode pulse height. This is said to arise from positive ions inducing charge on the anode as the electrons are collected. The second explanation suggests that there is a too large anode signal. The addition to the anode signal is said to arise from the drift of electrons.

In this thesis the concept of grid inefficiency is investigated by means of wave form digitization. The use of digital signal processing makes it possible to maintain information on the drift of electrons. This information is lost in charged particle spectroscopy experiments using electronic signal processing networks.

A series of experiments is described in this thesis. The first experiment was performed to find good measuring conditions for the following experiments. For this purpose the drift velocity of electrons was measured in two chamber filling gases, P-10 and CF4. The measured

drift velocities are presented for the two gases. Finally, P-10 was chosen as filling gas for the following experiments.

In the second experiment the grid inefficiency was measured for two different types of shielding electrodes. The method of determining the grid inefficiency is based on the analysis of the shape of digitized charge signals. The measured values are shown to be in good agreement with calculated values.

In the final experiment the effects of grid inefficiency on alpha particle spectroscopy is investigated. It is shown how the correction for grid inefficiency by the two existing models yield equivalent results for energy determination. An attempt to separate the two models is also presented indicating that there is in fact a reduction of the anode pulse height because of grid inefficiency. The thesis is concluded with a theoretical discussion of the anode pulse shape. There grid inefficiency is explained by the drift of electrons. It is shown in this section how explaining grid inefficiency by the drift of electrons should yield the same result as explaining it by the effect of positive ions.

iii

C

ONTENTS

1 Introduction ... 1

1.1 Radiations interaction with matter ... 1

2 Theoretical background... 1

2.1 Ionization by heavy charged particles in gas ... 2

2.2 Transport of ions and electrons... 2

2.3 Recombination and electron attachment ... 3

2.4 Pulse formation principal ... 4

2.4.1 The Ramo-Shockley Theorem for induced charge... 6

2.5 Parallel plate chamber ... 7

2.6 Frisch-grid ionization chamber ... 9

2.6.1 Grid inefficiency... 10

3 Experiments ... 13

3.1 Drift velocity of electrons in chamber filling gas... 13

3.1.1 Setup ... 13

3.1.2 Method ... 14

3.1.3 Results and discussion ... 15

3.2 Determining the value of σ experimentally ... 17

3.2.1 Setup ... 17

3.2.2 Experimental procedure ... 19

3.2.3 Results and discussion ... 20

3.3 Investigation of the grid inefficiency... 23

3.3.1 Setup ... 23

3.3.2 Treatment of data... 24

3.3.2.1 Angular information... 26

3.3.2.2 Pulses for shape analysis... 28

3.3.3 Results and discussion ... 29

3.3.3.1 Explaining the anode pulse shape ... 34

4 Conclusions... 41

5 Acknowledgement... 42

1

1 I

NTRODUCTION

1.1 Radiations interaction with matter

Radiation detectors, such as ionization chambers, Geiger tubes and proportional counters, use interaction of radiation with matter in order to detect radiation. In this thesis the ionization chamber and its properties are investigated. How radiation interacts with matter depends on the type of radiation. Radiation is categorized in three different ones, when speaking about its interaction with matter; heavy charged particles, light charged particles and electromagnetic radiation. I will focus on the interaction of heavy charged particles, which includes alpha particles, protons and fission fragments. The dominant mechanism for energy loss of heavy charged particles is Coulomb scattering by atomic electrons of the matter, which the radiation interacts with. Although collisions with atomic nuclei is of importance to nuclear physics it is of little importance to the energy loss of heavy charged particles, since such a collision is about 1015 _{less probable than a collision with an atomic}

electron [1]. Consider a heavy charged particle, such as an α-particle, of mass M and kinetic energy T, colliding head on with an electron of much smaller mass m (M>>m) initially at rest, the loss of kinetic energy of the heavy particle is given by

      = ∆ M m T T 4 .

An α-particle with a typical energy of 6 MeV would lose about 3 keV per collision. From the foregoing very simplified model one can draw some simple conclusions. It takes several thousand collisions for the heavy particle to lose all its energy. When the collision is not head on, the angle of deflection is, because of its larger mass, negligible for the heavy particle and it will stay on its initial trajectory. As the heavy particle is stopped a number of ion pairs are created in the stopping medium. If the ions created are detected so is the presence of radiation.

2 T

HEORETICAL BACKGROUND

A simple ionization chamber consists of two electrodes at a distance from each other, in fact just a parallel plate capacitor, inside a container filled with a suitable gas. The two electrodes are connected to a voltage supply in order to create an electric field between them. When radiation passes through the chamber and ionizes the gas inside, the created ions and electrons causes a voltage change of the capacitor that can be registered.

2

2.1 Ionization by heavy charged particles in gas

The energy loss of charged particles in a gas is as in any medium mainly due to two different reactions, excitation and ionization. The excitation of an atom is a resonant reaction demanding a specific amount of energy transfer to take place. Ionization of an atom does not require a specific amount of energy but instead has a higher energy threshold. Though the cross section for excitation is about 10 times smaller than for ionization, the high energy threshold makes the ionization probability small, therefore excitation is generally the more dominant reaction in a gas.

The ions created by the incident radiation itself are known as primary ionizations, but there are other ways that ions are created by radiation. An electron freed in ionization can, if it is energetic enough, ionize a second atom. This effect is known as secondary ionization. Another process, known as the Penning effect, occurs when in certain atoms there are meta-stable excited states, which do not deexcite immediately by the emission of a photon. An atom in one of these states may remain excited until it collides with another atom in the gas, resulting in the ionization of the latter. A third important process is the formation of molecular ions. When a molecular ion is formed from the interaction of a positive ion with a neutral atom of the same type, an electron is freed.

Due to the statistical nature of the relative occurrence of the processes that produce ions in a gas two identical incident particles will not in general produce the same amount of ions. It is, however, possible to determine the mean number of ions produced, and from this to calculate the mean energy lost by the incident radiation per ionization. The mean energy loss per ionization in a gas is substantially higher than the mean ionization potential, because of the energy lost to excitations. It is a function of the gas composition, the energy of the incident particle and the type of particle. Empirical results show, however, that the mean energy loss per ionization is not strongly dependent on any of these variables. Typical values of the mean energy lost per ionization are in the range of 25-35 eV/ion-pair.

2.2 Transport of ions and electrons

In an ionized gas electrons and positive ions move randomly with an average energy of 3kT/2, equal to the thermal translation energy of the gas molecules. The velocity of the particles at thermal energy is described by the Maxwell distribution, which gives a mean speed of m kT v

π

8 = ,

where k is the Boltzmann constant, T the absolute temperature and m the mass of the particle. Since the electron mass is much smaller than the positive ion mass they will move

3 much faster. At room temperature the electron speed is in the order of 100 times the positive ion speed [2]. The random motion results in a diffusion of the cloud of electrons and positive ions outward from the point of its creation.

If an electric field is present, the charged particles are accelerated along the field lines. The acceleration is interrupted by collisions, limiting the maximum velocity attained. The average velocity attained by charged particles in the direction of an electric field is known as the drift velocity w. The drift velocityʹs relation to applied field strength and gas pressure can be understood by making the assumption that all particles under consideration have the same random velocity v and that the direction of its motion after a collision is independent of its motion before the collision. The mean free path is the mean distance particles travel between collisions, it is inversely proportional to the pressure of the gas and can be expressed as λ= λ0/p, where λ0 is the mean free path at unit pressure. The average number of collisions per unit time is then v/λ, the average momentum in the direction of the particles drift lost due to a collision is mw. The particle gains an amount Ee of momentum in the direction of its drift per unit time. Once equilibrium has been established the following equations hold mvp eE w Ee mw vp 0 0

λ

λ

= ⇔ = . (1)

Positive ions follow the relation set by Eq. 1 up to relatively strong electric fields. Their drift velocity at a constant pressure is linearly dependent on the electric field strength. The energy distribution of the electrons is strongly affected by electric fields. The average random velocity and the mean free path of electrons, in a given gas, vary with the applied electric field. The drift velocity of electrons is therefore best determined experimentally for each gas. To eliminate the pressure dependence a parameter E/p called the reduced field strength is introduced, which is the ratio of the electric field strength and the gas pressure.

2.3 Recombination and electron attachment

The ionization of a gas is, as described earlier, the removal of one electron from each of a number of gas molecules. While the number of ion pairs created is important for the energy resolution of an ionization chamber, it is equally important that the products of the ionization stay free for a time long enough to be detected. When no electric field is present the electron and the positive ion will generally, after some time, recombine. Under the influence of an electric field this process is, however, only of importance in regions of the gas, where the ionization is very dense [3]. Another process that prevents the created ion pair to be fully detected is called electron attachment. This process involves the capture of a free electron by an electronegative atom forming a negative ion. Electronegative atoms have an almost full outer shell so that the attachment of the electron actually releases energy. To

4 prevent the influence of electron attachment noble gases such as Ar are commonly used as chamber filling gas. Examples of electronegative gases are H2O and O2, this means that if air

is leaking into the chamber, it will severely reduce its efficiency and energy resolution.

2.4 Pulse formation principal

For the sake of describing the physics behind the formation of electric output signals of an ionization chamber an idealized case, where the electrodes are of infinite size, will be considered. One of the electrodes is held at a constant potential V0 and the other at ground potential. When an ionizing particle deposits its energy in the gas it creates a track of electron and positive ion pairs. The electrons and positive ions are acted upon by the electric field between the two electrodes and start to move away from each other. It is the movement of these charged particles that forms the observable output pulses of the chamber.

Fig. 1: Illustration of geometry and motion of a charge carrier in a parallel plate chamber.

To understand the effects of this motion consider first a single charge carrier, of charge q, moving between the two plates of the chamber, as illustrated in Fig. 1. When the charge carrier has moved a distance dx an amount of work dW is performed by the electric field.

x d E q dW = r⋅ r

The drift velocity w in a uniform field is constant due to frictional forces, represented by collision with the molecules of the gas. Keep in mind that this is a vector quantity and has opposite sign for oppositely charged particles in the same electric field. In Fig. 1 w is illustrated as if q is of positive charge, thus

dt w E q dW dt x d wr= r⇒ = r⋅ r . E x D q w V0

5 Due to the conservation of energy the power IV0 supplied by the voltage supply, which is equal to the work performed per unit time by the electric field

w E q dt dW IV0 = = r⋅r.

The strength of the uniform electric field between the plates of the chamber is

D w q I D V E = 0 ⇒ = ,

that is when a charge is moved, it induces a current flow through the circuit. The current flows to the grounded electrode and is positive for both negative and positive charge carriers. The current flow through the positive electrode is always of opposite sign. Integrating the current over time gives a charge; this is considered as the charge induced on the electrode by the motion of q

( )

=

_∫

( )

′ ′= t D wt q t d t I t Q 0

, where tʹ=0 when q is created. (2)

Now consider an ionized atom, i.e. an electron and a positive ion created at a distance d from the negative electrode. Under the influence of the electric field the electron and the positive ion will move away from each other. The electron moves with a velocity w- _{in the direction} opposite to the field vector. The positive ion moves with a velocity w+ _{in the direction of the} field vector. At the point in time when the electron reaches the positive electrode it has travelled a distance w-_{t = D-d and according to Eq. 2 induced a charge Q}e- _{there, which is} given by       − − = − − = − D d e D d D e Qe ₁ .

At the point in time when the positive ion reaches the opposite electrode it has travelled a distance w+_{t = d and according to Eq. 2 induced a charge Q} ion _{on the positive electrode}

D d e Qion ₌₋

.

The sum of the two contributions gives the total charge induced on the positive electrode after both the electron and the positive ion have been collected. At any point in time an equal amount of charge but with opposite sign is induced on the electrode connected to ground. In any real situation a number of electron-ion pairs will form at the same time, the current and therefore the induced charge will be the sum of the contributions from all the charges.

6

2.4.1 The Ramo-Shockley Theorem for induced charge

The above result of the induced charge by the motion of a charge carrier can be achieved using the Ramo-Shockley theorem [4]. The Ramo-Shockley theorem can be used for any detector geometry, and is applicable to gas filled chambers as well as semiconductor radiation detectors [6]. The Ramo-Shockley theorem states that the charge induced on an electrode by the motion of a charge carrier in the chamber is the product of the charge of the carrier and the difference in weighting potential between the points the charge carrier has moved

ϕ

∆

= q

Q

, (3)

where q is the charge of the carrier and ϕ is the weighting potential of the electrode. The weighting potential is not the actual electric potential in the detector, it is the potential created by applying a unit potential to the electrode on which the induced charge is to be calculated and keeping all other electrodes grounded. To calculate the weighting potential of an electrode the Laplace equation

0

2 ₌

∇

ϕ

, (4)

with some special boundary conditions must be solved for the geometry of the chamber. The Laplace operator in orthogonal coordinates is

2 2 2 2 2 2 2 z y x ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ .

The boundary conditions for the Laplace equation are

1. the voltage on the electrode for which to calculate the induced charge on is unity

2. the voltage on any other conductor is zero

3. all charges are removed from the chamber

For the idealized two infinite-electrode chamber described above, Eq. 4 has the simple solution

( )

    _{< ≤} = otherwise , 0 0 , x D D x x

ϕ

7 where x is the distance from the opposite electrode and D is the distance between the two electrodes. Applying this solution to Eq. 3 gives again Eq. 2 for the induced charge, since

x=wt.

2.5 Parallel plate chamber

Consider now a radioactive sample, such as an alpha source, placed on the grounded electrode, which from now on I will refer to as the cathode. The electrodes are connected as in Fig. 2. The voltage developed across the resistor by the current I, caused by moving charge carriers as described above, is the quantity, which is measured.

Fig. 2: Geometry and connection diagram of a parallel plate chamber operated in pulse mode. As long as the voltage V developed across the resistor R is small compared to the voltage V0 applied to the positive electrode (from now on referred to as the anode), application of the energy conservation principle yields

( )

t RI dt dV RC V V V << ₀ ⇒ + = ,

where C is the total capacity of the cathode and the amplifier input. This equation has the general solution

( ) ( )

− = −t RC

_∫

t_e−t′ RC_I

( )

_t′_dt′ C e V t V 0 / / 1 0 . (5)

If RC is chosen to be large, the exponential decay can be neglected as long as a current is present, and the pulse height is given by

D V0

θ

X R

8

( )

( ) ( )

( )

C t Q V t V t P = − 0 = ,

where Q(t) is the integrated current. If the charge is a negative, it is customary to change the sign so that pulse height is an unsigned quantity. If a charge sensitive preamplifier is used, the pulse height dependence on the capacitance of the electrode is eliminated. The voltage pulse is then integrated on a capacitor in the preamplifier and the pulse height will only depend on the properties of the preamplifier. The operation of a chamber under these conditions is usually called pulse mode. As the sample decays it radiates an alpha particle, which is stopped in the gas of the chamber between the two electrodes. When the alpha particle has stopped it has created a track of electron and positive ion pairs in its path. The time for the alpha particle to be stopped is very short in comparison with the motion of electrons and positive ions in the electric field, and the track can be considered as suddenly appearing. After some time the electrons and positive ions are collected by the chamber electrodes. The height of the recorded pulse will, according to Eq. 2, be proportional to the number of electron ion pairs created, which is a reflection of the energy of the incident alpha particle. After the pulse has reached its maximum it decays with the time constant RC, set by the charge sensitive preamplifier.

There are, however, some disadvantages to this setup. Suppose a second decay takes place during the time of collection of electrons and positive ions created by the first decay. There will then be a so called pile up and the pulse height will no longer be proportional to the number of ion pairs created by one decay event. Also, because the time constant is long the setup will be very sensitive to microphonic disturbance. Soundwaves cause the chamber plates to vibrate, which changes the chamber’s capacitance, and hence, since the voltage over the electrodes is constant the charge of the capacitor changes.

To avoid these problems the value of RC is chosen to be large compared to the time for the electrons to be collected, but small compared to the time for positive ions to move. The current caused by moving positive ions can then be neglected in Eq. 5. The time to collect the electrons is ~1000 times shorter than the time to collect the positive ions, because of the electrons higher drift velocity. The probability for two ionizing events to take place during the collection time is thereby reduced by a factor ~1000. The microphonic effects are also limited because of its low frequency. This operation mode is called electron-pulse mode or simply a fast chamber. In the fast chamber the pulse height is a function of the energy and the emission angle of the emitted radiation.

The charge induced on the anode after all electrons, created along the stopping track of an ionizing particle, have been collected is the sum of the charges induced by each electron. Integrating Eq. 2 over the total number of electrons created gives

9       − − = 0 1 cos

θ

D X e n Q ,

where n0 is the number of electrons created and θ is the emission angle measured from the

chamber axis. X is the distance from the origin of the ionizing particle’s stopping track to the centre of gravity of the charge distribution created, given by

( )

∫

∞ = 0 0 1 _{r r dr} n X

ρ

where r is the distance from the origin of the track at the radioactive sample, ρ(r) is the ionization density along the track. The charge induced on the cathode is the same as on the anode but with opposite sign.

2.6 Frisch-grid ionization chamber

It is clear that the fast chamber described above is not a useful tool for particle spectroscopy, since for two identical particles, emitted at different angles, the corresponding pulses will differ in amplitude. Each monochromatic line in a spectrum would show as a square distribution of pulses in the pulse height spectrum. In order to make the pulse proportional to the number of ion pairs created, a third electrode, in form of a grid, is placed in the chamber. The grid is placed close to the anode, far enough from the cathode (where the radioactive sample is) not to be reached by any particles emitted from the sample. The grid is held at an intermediate potential, between the cathode and anode potentials. The grid acts as a shield, ideally creating two separate electric fields, shielding the anode from the effects of charges in the space between cathode and grid. This type of chamber is called a Frisch grid ionization chamber, named after the inventor of the design O. R. Frisch.

In the ideal scenario the region between cathode and grid and the region between grid and anode would work as two independent chambers. In the first chamber, the region between cathode and grid, the ionizations take place and the charge induced on its electrodes (the cathode and the grid) will depend on emission angle as in the fast parallel plate chamber described above. The electrons then drift to the second chamber, the part between grid and anode. In this part of the chamber the electrons always appear at the same distance from the anode and the charge induced on its electrodes will be equal to the charge that has passed the grid. In the ideal case, after the collection of all electrons from an ionizing event the charges induced on the anode, the cathode and the grid, respectively, are

e n Q_A =− ₀ ,

10       − = 0 1 cos

θ

D X e n QC , (6)

(

)

₀ cos

θ

D X e n Q Q Q_G =− _C+ _A = ,

where X is as before the distance from the origin to the centre of gravity of the charge distribution along the ionized track, θ is the emission angle relative to the chamber normal and D is now the distance between cathode and grid.

The ideal scenario cannot be realized, since in reality the cathode-grid field and the grid-anode field are not entirely independent. Because of this the grid-anode pulse heights show a small dependence on where the electrons are created. The anode signals dependence of the ionizing particle’s emission angle is generally referred to as grid inefficiency. Also, for the anode pulse to reflect the number of electrons created none should be caught on the grid. Since the electrons drift along the electric field lines, this obstacle is easily overcome by adjusting the voltages so that no field lines end on the grid. A theoretical approximation of the electric fields in a Frisch grid chamber has been performed by Bunemann [7], where both of these problems are approached. I will return to his calculations regarding the grid inefficiency later. For the grid not to catch any electrons he states a minimum condition on the applied voltages depending on geometric properties of the chamber and the grid.

ρ

ρ

ρ

l D D lp p p V V V V C G G A 2 2 − − + + ≥ − − ;       ₋ =

ρ

ρ

π

4 ln 1 2 2 d l ; d r

π

ρ

=2 , (7)

where VA, VG and VC are the anode, grid and cathode voltages respectively, p is the grid-anode distance, d is the distance between grid wires and r is the grid wire radius. The condition is based on many approximations, it should be taken as a guideline when setting up an experiment and there is no reason not to exceed it.

2.6.1 Grid inefficiency

Different explanations of the grid inefficiency have been used by different authors, leading to different formulas for predicting the charge induced on the anode after an ionizing event. The predicted charges induced on the electrodes are used to make corrections of the experimental pulse height spectra. The correction should be made so that the resulting anode pulse height is proportional to the charge created and the angular dependence eliminated. Here follows a brief description of two widely used explanations of the grid inefficiency, and the corrections of the anode pulse heights that they lead to.

The first explanation, where the concept of the grid inefficiency was introduced, involves the effect of positive ions inducing charge on the electrodes of the chamber. The induction of

11 charge is here not due to the motion of charged particles, but due to the fact that a charge separation is induced on a conductor by charged particles in the vicinity of it. I will, in order to avoid confusion, refer to this as influenced charge. The model is described in detail in Ref. [7] and Ref. [8]. The influenced charge on an electrode in a parallel plate chamber is,

according to Ref. [8], proportional to the difference in potential between the point where the charge is and the potential of the opposite electrode. Because of grid inefficiency, a fraction σ of the charge ideally influenced on the grid, by charges residing in the cathode-grid space, is instead influenced on the anode. During the time it takes electrons from an ionizing event to be collected the positive ions are standing almost still, so the charge influenced on the anode by the positive ions is constant during this time and given by

θ

σ

₀ cos D X e n Q_GI = .

At the time of collection of the last electron from an ionizing event, the charge of the anode is the sum of the total collected charge and the charge influenced by the positive ions. The final charge of the anode is therefore

θ

σ

0 cos 0 D X e n e n QA =− + . (8)

The extent to which the charge influenced on the anode is independent of the field in the cathode grid space has, as mentioned earlier, been calculated by Bunemann [7]. He defines the inefficiency of the grid as

Q A dE dE =

σ

for VA-VG=constant,

where EA is the number of field lines per unit area ending on the anode and EQ is the field from charges in the cathode-grid space. The calculations lead to an expression for calculating the inefficiency of the grid, depending only on geometric properties of the chamber and the grid p l l + =

σ

,

l and p are defined as in Eq. 7. Since the charge detected is less than the charge of the electrons created by the ionizing radiation, the anode pulse height should be corrected by adding an amount of charge equal to the charge influenced by the positive ions. This model and the correction of the anode signal it suggests will be referred to as the adding model. Experimentally the charge influenced by the positive ions can be determined from the

12 difference between measured anode and cathode pulse heights, [9]. In terms of pulse heights this yields

(

A C

)

A C A P P P P = +

σ

−

for the corrected anode pulse height, where PA is the measured anode pulse height and PC is the measured cathode pulse height. Another way of making the same correction would be

σ

σ

− − = 1 C A C A P P P . (9)

The second explanation is described in Ref.[10] and Ref.[11]. The concept of moving charge carriers inducing charge is used in this explanation. Experimental anode signals show a small rise before any electrons have passed the grid. This part of the anode signal is interpreted as an addition to it, arising from charge induced by moving electrons in the cathode-grid space. It is therefore said that the anode signal’s dependence on the ionizing particle’s emission angle is from moving electrons, rather than from positive ions. As electrons drift towards the grid they induce a charge on it, the induced charge is proportional to the distance they have travelled as described in the theory section for a parallel plate chamber. Because the grid is not ideal, a fraction σ of the charge induced on the grid by moving electrons is instead induced on the anode. After electrons have passed the grid they each induce a charge equal to their own on the anode. The total charge induced on the anode is the sum of the charge induced by electrons before they pass the grid and the charge induced by the same electrons after they pass the grid, so that

      − − − = ₀

σ

₀ 1 cos

θ

D X e n e n Q_A . (10)

The value of σ is the one calculated using Bunemann’s formula. Because the charge induced on the anode is more than the charge of the electrons created by the ionizing radiation, the anode pulse height should be corrected by subtracting the charge induced by electrons before they pass the grid. From here on this model and the correction it suggests will be referred to as the subtracting model. In terms of pulse heights this yields

C A C

A P P

P = −

σ

, (11)

for the corrected anode pulse height, where PA is the measured anode pulse height and PC is the measured cathode pulse height.

13

3 E

XPERIMENTS

The experiments described in this thesis were performed to investigate the properties of the Frisch grid ionization chamber and the concept of grid inefficiency in particular. The ultimate goal was to come to a conclusion on which one of the explanations for grid inefficiency is correct. It was also investigated how the use of different corrections affect the final result of particle spectroscopy experiments. In all experiments digitizing techniques were used, making it possible to draw conclusions from the shape of the signals. In principle the same Frisch grid ionization chamber was used in all experiments, but there were small changes in geometry of the chamber and electronics used. The experiments will be described separately.

3.1 Drift velocity of electrons in chamber filling gas

To be able to retrieve as much information as possible from the shape of digitized signals one should, when performing experiments, use conditions that are stable with respect to physical parameters that might influence this shape. Information contained in the time characteristics of signals from an ionization chamber operated as a fast chamber is directly dependent on the drift velocity of electrons in the gas used. The drift velocity must also be high enough to avoid pile up on the rising edge of signals. For the purpose of finding good measuring conditions the drift velocity in two gases was measured as a function of reduced field strength.

3.1.1 Setup

In this experiment the chamber was used as a parallel plate ionization chamber, only applying voltages to create a field between the cathode and the grid. An alpha emitting sample was placed at the centre of the cathode. The distance between the electrodes was 61 mm. The cathode was connected via a charge sensitive preamplifier to a Tektronix digital phosphor oscilloscope (TDS 3034B). A schematic view of the experimental setup is shown in Fig. 3.

14 Fig. 3: Schematic view of the experimental setup. HV=high voltage supply, PA=charge sensitive preamplifier, TFA=timing filter amplifier, OSC=digital phosphor oscilloscope, PrC=pressure controller, FM=flow meter, VP=vacuum pump.

The gases investigated were P-10 (90% Ar + 10% CH4) and CF4. Each measurement series was

taken with constant pressure and gas flow through the chamber controlled by a pressure controller. The drift velocity was measured at p=1,05 bar, varying the voltage to achieve different reduced field strengths.

3.1.2 Method

The charge signals of the electrodes of a parallel plate ionization chamber arise from moving electrons. According to the Shockley-Ramo theorem [4] the current induced on the cathode by moving electrons is described by

D w ne I = ,

where e is the elementary charge, n is the number of drifting electrons and w is the drift velocity of electrons. The charge signal is proportional to the current integrated over time, its time dependence will therefore be governed by two parameters; the electron distribution along the cathode normal and the drift velocity of electrons. For two identical ionization events, where the electron distribution is the same, the only difference in their respective time dependence will be from the drift velocity.

First the relative drift velocity was determined as a function of reduced field strength from the cathode signal rise time. The rise time is defined in terms of constant fraction as the time interval where a signal rises from 10% to 90% of its final amplitude. Before the rise time was

OSC PA PA HV HV Gas PrC FM _Vacuum Pump TFA Ch1 Ext. Trig. D p p θ X

15 measured, for each setting of reduced field strength, signals where averaged over 512 events. This produces signals with improved signal-to-noise ratio, the signals time dependence on the ionizing particles emission angle and energy will always be the same. The rise time of three such averaged signals was measured for each setting of reduced field strength to determine the statistical error in the measurement.

To determine the absolute drift velocity voltage was also applied to the anode and all signals were digitized. This was done at a voltage setting, where the relative drift velocity was stable. The digitized signals were then analyzed to create a two-dimensional pulse height spectrum from which the emission angle for each event could be determined, this procedure is described in detail in section 3.3.2. The digitized signals with emission angles close to perpendicular to the cathode where then averaged. The resulting signal, seen in Fig. 4, is linear in its rise, due to the fact that all electrons arrive at the same point in time at the grid. The linearly rising signal’s rise time is the time for electrons to drift 80% of the distance between the cathode and the grid, from this the drift velocity can easily be deduced. The relative drift velocities already determined was then normalized to this point.

-1,0 -0,5 0,0 0,5 1,0 1,5 2,0 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Pu ls e He igh t [ R el at iv e] Time [µs] Rise Time Averaged Signal at grazing angle

Fig. 4: Cathode charge signal averaged over events at grazing angle with the cathode, taken from the absolute drift velocity determination for the P-10 gas.

3.1.3 Results and discussion

In Fig. 5 and Fig. 6 the measured drift velocities are plotted as functions of reduced field strength for the two gases investigated. The figures can be used to deduce voltage settings, for a given gas pressure and chamber geometry, where the drift velocities are stable.

16 0 100 200 300 400 500 600 700 800 0 1 2 3 4 5 6

7 This Studty _{Kuhmichel, 1995}

Khriachkov, 2003 w

[

cm/ µ s

]

E/p [V/(cm bar)]

P-10

Fig. 5: The measured drift velocity of electrons in P-10 as a function of reduced field strength, comparison with data from Ref. [12] and Ref. [13].

0 100 200 300 400 500 600 700 800 900 0 1 2 3 4 5 6 7 8 9 10 11 12 This study

Schmidt and Polenz, 1988 Christophorou et. al., 1979

w [cm/

µ

s]

E/p [V/(cm bar)]

CF

₄

Fig. 6: The measured drift velocity of electrons in CF4 as a function of reduced field strength,

comparison with data from Ref. [14] and Ref. [15].

The drift velocities measured for P-10 is in good agreement with literature values. This validates that the method used to determine the drift velocity is sufficient. The drift velocities measured for CF4 are not in very good agreement with literature values. This may be

17 or recombination. For these reasons P-10 was chosen as filling gas for the following experiments.

3.2 Determining the value of σ experimentally

Suppose the small rise of the anode signal, before electrons pass the grid, is due to electrons in the cathode-grid space inducing charge there because of grid inefficiency. The value of σ can then be determined experimentally from the shape of the anode signal. Such an investigation has earlier been made by Khriachkov et al. [11], where the experimentally determined values of σ are said to differ very much from the values calculated using Bunemanns formula. This experiment was performed to validate the results in Ref. [11]. In Ref. [16] it was said that the use of a mesh, consisting of crossed wires, instead of the usual parallel wire grid, improves the resolution of the Frisch grid chamber. To deduce values of the grid inefficiency for a mesh, such a shield electrode was also included in this investigation.

3.2.1 Setup

Fig. 7: Schematic view of the experimental setup. D (the cathode-grid distance) was 6.1 cm in all measurements, whereas p (the grid-anode distance) was varied according to Table 2. The abbreviations are HV = High Voltage Supply, PA = Charge Sensitive Preamplifier, PG = Pulse Generator, OSC = Digital Phosphor Oscilloscope, TFA = Timing Filter Amplifier, PrC = Pressure Controller, FM = Flow Meter.

The experimental setup is shown schematically in Fig. 7. A Frisch grid ionization chamber as described in the theory section was used. The main dimensions are also seen in Fig. 7, the

OSC PA PA HV HV Gas PrC FM _Vacuum Pump TFA Ch1 Ch2 PG D p p θ X

18 cathode-grid distance was 6.1 cm in all measurements and the grid-anode distance was varied according to Table 2. Two types of shielding electrodes where used, one grid with parallel wires and one mesh with crossed wires, the dimensions are seen in Table 1.

Table 1: Geometric properties of the shielding electrodes used.

Wire radius r [mm] Wire distance d [mm] Grid 0.05 1 mesh 0.025 1

The chamber was operated with a flow of P-10 gas (90% Ar + 10% CH4) at a constant

pressure of 1,1 bar. The flow and pressure of the gas was controlled with a pressure controller and a vacuum pump.

After charge sensitive preamplifiers the anode and cathode signals were digitized with a Tektronix digital phosphor oscilloscope (TDS 3034B). Before the signals were saved for analysis they were averaged over 512 consecutive events. The cathode charge signal was split and also fed to a timing filter amplifier with high impedance, this signal was used to produce a stable trigger.

The cathode was held at a potential of VC =-2000 V and the grid was grounded in all measurements. In order to keep the ratio of the cathode-grid and grid-anode fields constant, and high enough to keep electrons from being collected by the grid, the voltage VA supplied to the anode was varied with the anode-grid distance. According to Ref [16] the grid-anode field should be three times the cathode-grid field, for best resolution. The voltage supplied to the anode for a given grid-anode distance was the same for the grid and the mesh and can be seen in Table 2.

Table 2: Overview of voltage and electric field strengths applied. E1 = (VG-VC)/D denotes the field

between cathode and grid, E2=(VA-VG)/p the field between grid and anode.

p [mm] VA [V] E2 [V/cm] E2/E1

10 1000 1000 3.05

6 600 1000 3.05

19

3.2.2 Experimental procedure

The signal from the charge sensitive preamplifier is proportional to the charge induced on the electrode it is connected to. Suppose the small rising part of the anode signal is from drifting electrons in the cathode-grid space. The charge induced on the anode by these electrons in the cathode-grid space, as a function of time, is then proportional to the charge induced on the cathode by the same electrons. The relation between the anode and the cathode signals before any electrons have passed the grid is therefore

( )

t P

( )

t , t t1

P_A =

σ

_C < , (12)

assuming equal pulse height for anode and cathode for the same amount of charge induced,

t1 is the time when the first electron passes the grid. By digitizing the pulses they can be analysed as functions of time, and σ can be determined from Eq. 12. In this experiment, to improve the signal-to-noise ratio, the signals were averaged over 512 pulses before saved for analysis. This produces a signal with the shape of an event at the average angle of emission. To determine σ using Eq. 12 it is vital that the relative preamplifier gains for anode and cathode signals are known. This relation was determined by feeding a pulse generator signal to both preamplifiers, and comparing the output pulses.

From the shape of the digitized anode signals t1 is determined as the point in time before the end of the first linear rise. This is done rather arbitrary, but the choice of this time does not strongly affect the final result as long as it is not chosen after any electrons have passed the grid. -4000 -2000 0 2000 4000 6000 0 10 20 30 40 50 60 0 500 1000 -1 0 1 2 3 4 σP_C PC P u lse hei ght [mV ] Time [ns] PA σP_C P_A P_C

Fig. 8: A set of signals recorded with the grid at p =6 mm, the cathode signal scaled with the experimentally determined σ is also shown.

20 Fig. 9 shows a plot of the cathode pulse height vs. the anode pulse height in the time interval from the beginning of the frame to t1. Each point in the plot represents a point in time and its position is determined by the cathode and the anode pulse height at this time. The oscillation around the line is due to the noise of the signals. By the method of least squares a straight line is fitted to the plot, the incline of this line is σ. In Fig. 8 and Fig. 9 the procedure is illustrated for a set of signals taken during measurements, the result is visualized in Fig. 8 by the cathode signal scaled with the obtained σ. This scaled cathode signal represents, as a function of time, the fraction of charge induced on the anode by electrons moving in the cathode-grid space. 0 5 10 15 20 25 -0,2 0,0 0,2 0,4 0,6 0,8 PA (t) [m V] P_C(t) [mV]

Fig. 9: Plot of cathode- vs. anode- pulse height from the signals in Fig. 8. The line is the result of a least square fit of a straight line. The oscillations around the line are due to the noise of the signals.

3.2.3 Results and discussion

In Ref. [16] the small signal on the anode before electrons pass the grid, here presumed to represent the charge induced by electrons before they pass the grid, was explained by a limited reaction time and gain of the grid preamplifier. Before the grid preamplifier reacts, a voltage high enough to drive the current through the circuit is built up on the grid. The change in voltage of the grid charges the capacitor consisting of the grid and the anode, consequently a charge signal is seen on the output of the anode preamplifier. To investigate this conclusion, signals were, in this experiment, recorded using two different ways of grounding the grid; one directly to the ground of the chamber, the other via the bias input of the preamplifier. The latter is the usual way of connecting if one operates the ionization chamber as double and uses the grid signal to extract information on emission angle, [9]. When the grid was connected directly to ground the small rising part of the anode signal was

21 reduced. Further investigation shows a good agreement with the explanation given in Ref. [16]. When the anode-grid distance is small and the capacity between them high, the voltage change of the grid should cause a large change of the charge on the grid-anode capacitor. Fig. 10 shows the anode and cathode charge signals, with the mesh at p=2 mm, for the two ways of grounding. A step-like rise of the anode signal is seen when the mesh is connected via the preamplifier to ground. When the mesh is connected directly to the ground of the chamber this step disappears.

-200 0 200 400 600 800 1000 -1 0 1 2 3 4 5 6 P_C P u ls e he ig ht [ m V ] Time [ns] P_A -200 0 200 400 600 800 1000 -1 0 1 2 3 4 5 6 _P C Pu lse hei ght [mV ] Time [ns] P_A

Fig. 10: Signals recorded with the mesh at p =2 mm, grounded via preamp (top) and grounded directly (bottom).

22 The results of the measured σ can be seen in Table 3, for comparison the value calculated with the Bunemann formula (cf. p. 11) is given for the grid, for the mesh the value of σ calculated for a grid with the same geometric properties (wire radius and spacing) is given. Table 3: Overview of the measured σ, Δσ refers to the corresponding uncertainty from the fitting of the straight line.

p [mm] σ(calculated) σ(measured) direct ground ∆σ(measured) direct ground σ(measured) preamp ground ∆σ(measured) preamp ground 10 0,0185 0,0166 0,0003 0,0256 0,0003 6 0,0304 0,0282 0,0002 0,0454 0,0003 grid 2 0,0860 0,0662 0,0003 0,1137 0,0005 10 0,0287 0,0128 0,0005 0,0207 0,0002 6 0,0470 0,0215 0,0003 0,0311 0,0003 mesh 2 0,129 0,0488 0,0003 0,0825 0,0003

The results from Khriachkovs measurements [11] could not be confirmed in this investigation. As seen in Fig. 11 the values of σ measured at grid-anode distances that are sufficiently large are in good agreement with the calculated values. When the grid-anode distance becomes comparable to the grid wire distance the formula fails. This can be explained by nonuniformity of the field between grid and anode. Uniformity of the field is assumed in the derivation of the formula for calculating σ, the formula will of course fail when the initial assumption is no longer valid. A lot of measured values for σ are given in Ref. [11] at grid-anode distances that are comparable to the grid wire distances, they are not plotted in Fig. 11. The offset in comparison with Khriachkovs measurements in the domain where the assumption of uniformity of the grid-anode field is valid may be explained by the limitations of the grid preamplifier, since it is not clear, if this was taken into account in his measurements.

23 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 -0,02 -0,01 0,00 0,01 0,02 0,03 (1) (2) (3) (4) σ (m eas ur ed )-σ (c alc ulat ed) σ(calculated)

Fig. 11: Comparison of measured and calculated σ. (1) – data from this study with the grid directly grounded, (2) – data from this study with the grid grounded via preamp, (3) and (4) – data from Ref.[11] for two different grids extracted graphically. Where no error bars are presented the uncertainty is smaller than the size of the symbols.

With the assumption that Bunemanns formula holds for grids, a mesh is clearly a better shield than a grid of the same wire radius and wire distance. Exactly to what extent cannot be deduced with confidence from these results, but an estimate of a factor of 2 (by comparing the measured values for the mesh with the calculated for a grid with the same wire distance and radius) can be made.

3.3 Investigation of the grid inefficiency

A series of experiments, using the same experimental setup, was performed to investigate the grid inefficiency. By digitizing signals after charge sensitive preamplifiers the charge induced on the electrodes of the chamber can be determined as functions of time. This can be used to draw conclusions from the shape of the signals. The digitized signals can also be used to create pulse height spectra from which conclusions can be drawn on the effect of different grid inefficiency corrections applied to the raw data.

3.3.1 Setup

The experimental setup is shown schematically in Fig. 12. The grid used was a parallel, stainless steel, wire grid. The chamber was again operated using constant flow of P-10 gas at a pressure of 1,1 bar. The pulses from the chamber were digitized after amplification, preserving information contained in the height and the shape of the pulse. The digitizer was

24 triggered from the timing filter output of the cathode preamplifier. The cathode plate has a hole in the centre, where the radioactive sample is placed. The sample was a calibration source with three alpha lines, consisting of the isotopes 239_Pu, 241_{Am and} 244_Cm.

Fig. 12: Schematic view of the experimental setup. D (the cathode-grid distance) was 6.1 cm, p (the grid-anode distance) was 6 mm. The abbreviations are HV=High Voltage Supply, PA=Charge Sensitive Preamplifier, PG=Pulse Generator, WFD=Wave Form Digitizer, FA=Fast Amplifier, PrC=Pressure Controller, FM=Flow Meter.

3.3.2 Treatment of data

Each digitized pulse consists of 1024 values separated by a time interval Δt = 20 ns. The pulses are analyzed by a computer code that simulates electronics for pulse signal processing.

A common way of displaying pulse amplitudes is the differential pulse height spectra,where the number of pulses dN within an interval of pulse heights [P, P+dP] is displayed as a function of pulse height.

To create a differential pulse height spectrum from the digitized charge signals they are analyzed in order to determine their relative amplitude. The anode pulses are sent through a computer simulated CRRC-cascade, to improve the signal-to-noise ratio. The CRRC-cascade is an electronic circuit that consists of two different components. The first one is the CR-differentiating circuit, schematically illustrated in Fig. 13.

PA HV HV Gas PrC FM Vacuum Pump Ext. trig FA Ch2 Ch1 WFD PA _FA FA D p p θ X PC

25 Fig. 13: CR-circuit.

Working out the voltages from Fig. 13 gives

OUT

IN V

C Q

V = + .

Differentiating both sides of this equation yields

dt dV dt dQ C dt dVIN ₌ 1 ₊ OUT . dQ/dt is the current, so dt dV V RC dt dV R V I dt dQ OUT OUT IN OUT _{⇒ = +} = = 1 .

The basic function of this circuit is that it will differentiate a pulse wider than the time constant RC of the circuit. If the time constant is chosen to be wider than the rise time of the pulse this part will pass without any change, but the decay time of the pulse will be reduced. The purpose of this circuit is to shorten the pulse to prevent pile-ups on the trailing edge. The second component in the CRRC-cascade is the RC-integrating circuit shown schematically in Fig. 14.

Fig. 14: RC-circuit.

Working out the voltages for this circuit yields

OUT IN IR V V = + and C R IN OUT C R IN OUT

26 OUT OUT IN OUT _V dt dV RC V dt dV C dt dQ I = = ⇒ = + .

The RC-integrator performs an integration of signals of smaller width than the time constant, which results in flattening out any high frequency noises of a pulse wider than the time constant, and it also affects the rise time of the signal. In a CRRC-cascade there is first a CR-differentiating circuit followed by a number of RC-integrators. When the input is a step-like function, the output is a smooth signal with approximately Gaussian shape. The height of the Gaussian shaped pulse will be a linear function of the input pulse height. The CRRC-cascade is computer-simulated by solving the equations above numerically for each pulse. Once this is done the height of each resulting pulse is placed into a bin or channel number, each channel number containing a narrow interval of pulse heights. The number of pulses contained in each channel is then plotted against the channel number to produce a differential pulse height spectrum as displayed in Fig. 15. The spectrum shows three distinct peaks that can be identified as the three most intense alpha energies of the sample.

0 1000 2000 3000 4000 0 50 100 150 200 250 300 C oun ts / cha nne l P_A [channels]

Fig. 15: Differential pulse height spectrum of α-particles from the decay of 239_Pu, 241_{Am and} 244_Cm.

3.3.2.1 Angular information

The angular information of emitted ionizing particles is contained in the cathode pulse height PC. To retrieve this information one needs to know the relation between anode pulse height PA and PC for the event in question. The cathode pulses are analyzed with the CRRC-cascade, in the same manner as the anode pulses above. For each event, PA and the corresponding PC are then plotted against each other in a two dimensional differential pulse height spectrum with the number of pulses on the z-axis, as displayed in Fig. 16. The spectrum needs to be corrected for difference in amplification between the anode and

27 cathode pulses. When an ionizing particle is emitted at a grazing angle1 _{with the cathode the}

amount of charge induced by electrons are the same for anode and cathode. A particle emitted at a very grazing angle to the cathode will lose a lot of its initial energy in the sample from which it came so that these events show as a line at low pulse height in the two dimensional spectrum, see the circle marked “1.” in Fig. 16. This line must follow the relation of equal pulse height for anode and cathode. This fact can be used to correct the anode and cathode pulses heights for difference in amplification by fitting a straight line to the above mentioned events and normalizing the pulse heights to this line.

50 100 150 200 250 50 100 150 200 250

P

C

[c

hannels]

2.

P

_A

[channels]

1.

Fig. 16: Two-dimensional pulse height spectrum in scatter-plot representation. The areas marked; “1.” contain events at cos θ ≈ 0, where the alpha-particles have lost most of their energy in the sample, “2.” Corresponds to events at cos θ ≈ 1.

Once the relative pulse height correction is made, the angle of emission for each event can be retrieved from the spectrum. The relation between pulse height and emission angle, neglecting grid inefficiency, is described by

e n PA = 0

28

(

)

_      − =

θ

θ

1 cos cos 0 D X e n PC       − = ⇒ D X e n PC(1) 0 1

( )

1 cos C A C A P P P P − − = ⇒

θ

X is a function of the ionizing particle’s energy, mass and proton number. Since in this

experiment all ionizing particles are alphas, X will only be a function of the energy and therefore constant for one detected anode channel number. A second degree polynomial was fitted to the half height at the minimum cathode pulse height of the three alpha lines, corresponding to the area marked “2.” in Fig. 16. This is done to determine the minimum cathode pulse height PC (1) as a function of PA. The relation is then used to determine the angle of emission for each event with a computer code.

3.3.2.2 Pulses for shape analysis

To improve signal-to-noise ratio, before analyzing the pulses shape, a signal is created by averaging pulses. In this way noise, not time correlated with the signals, is reduced to a large extent. The information on the angle of emission and energy already determined can be used to choose to look at a signal averaged over pulses with a certain energy and emission angle. Before the pulses are averaged the influence of the transient response of the preamplifier on their shape is eliminated. The transient response is defined in terms of the output of the preamplifier due to a sudden change of the input voltage, i.e. the preamplifiers response to a step function. In terms of the ionization current the output signal from the preamplifier is described by

( )

t =

_∫

t I

(

t−t′

) ( )

ht′dt′

P

0

, (13)

where h(t) is the transient response function given by

( )

   ≥ = − otherwise , 0 0 , / _t e t h RC t .

The value of the time constant RC of the preamplifier is determined by fitting an exponentially decaying function to the decay of the pulse by the method of least square. This is done for five random pulses for each preamplifier; RC is then taken as the average of the values obtained.

29 Eq. 13 is solved for I numerically with the computer code, giving the ionization current as a function of time. I(t) is then numerically integrated with respect to time to give the charge induced on the electrode as a function of time.

3.3.3 Results and discussion

The effects of the two different grid inefficiency corrections (cf. Sect. 2.6.1) are investigated by making the two different corrections to the same dataset and using the events at low pulse height (illustrated in Fig. 16 by “1.”) as before to determine the correct relation between anode and cathode pulse height after the correction has been made. The result of this procedure is seen in Fig. 17 and Fig. 18, the resulting channel number for the peak positions differ by less than 0,1% and can only be due to an uncertainty in determining the relation between anode and cathode pulse height and differences arising from binning.

0 1000 2000 3000 4000 0 50 100 150 200 250 300 350 400 Pos FWHM 1. 3393 35 2. 3621 47 3. 3824 37 3. 2. Co un ts / cha n ne l P_A [channels] 1.

30 0 1000 2000 3000 4000 0 50 100 150 200 250 300 350 3. 2. Cou n ts / ch annel P_A [channels] 1. Pos FWHM 1. 3391 35 2. 3618 47 3. 3822 37

Fig. 18: Resulting pulse height spectrum after the correction according to

PA(corrected) = (PA - σPC) / (1- σ).

It has been shown that the two corrections yield the same result after normalizing so that the anode and cathode pulse height ratio for events at grazing angle is unity. The two models can therefore only be separated if the anode and cathode signals are calibrated relatively to each other, meaning that same amount of induced charge leads to the same pulse height. To calibrate the pulse heights, two sets of data were taken with the same experimental setup (as shown in Fig. 12) except for switching the amplification chain for anode and cathode. Anode pulse height spectra were then taken for each data set. The positions of the three alpha lines, displayed in Table 4, were found for each spectrum using the same analysis parameters. The difference between the two amplification chains is then found by the relation

B Ach chi1= i2 + ,

where chij is the position of peak i for amplification chain j, and A and B are constants

depending on the difference in amplification. This relation is then used to correct the anode channel number in the two-dimensional pulse height spectrum, which displayed in Fig. 19.

31 Table 4: Peak position of the three alpha lines from the two different amplification chains.

amp. Chain pos. peak 1 [channels] pos. peak 2 [channels] pos. peak3 [channels] 1 3432,0 3264,3 3864,8 2 3660,6 3481,5 3660,6

Consider now events at grazing angle with the cathode, i. e. cos θ ≈ 0. These events show as a line at low pulse height in the two dimensional spectrum (as described in Sect 3.3.2.1 and illustrated in Fig. 16 by “1.”). This line will in the following be referred to as ʺline 1ʺ.

50 100 150 200 250 50 100 150 200 250 P_A=P_C(1+σ) P C [cha nne ls ] P_A[channels] P_A=P_C σ= 0,0282

Fig. 19: Two-dimensional pulse height spectrum after PA being corrected for the difference in

amplification. To fully illustrate the events at grazing angle each detected event is here plotted as a point. The value of σ is the experimentally determined one, as described in sect 3.2.

The ʺsubtracting modelʺ suggests more charge induced than created on the anode because of grid inefficiency. According to this model, a charge QA(0) was induced on the anode for

events at grazing angle, according to Eq. 10, which leads to

(

)

Q

( ) (

)

n e D X e n e n Q_A cos

θ

₀

σ

₀ 1 cos

θ

_⇒ _A 0 = 1+

σ

₀      − + = .

32 A charge QC(0) is induced on the cathode for the same events; from Eq. 6 follows

(

)

Q

( )

n e D X e n Q_C cos ₀ 1 cos _⇒ _C 0 = ₀      − =

θ

θ

.

According to this ʺline 1ʺ should not follow the diagonal before the correction for grid inefficiency is made. This line should instead follow the relation

( )

( )

=1+

σ

1 0 0 A C Q Q ,

and no events should be detected above the corresponding line in the two-dimensional pulse height spectrum. Fig. 19 clearly shows that this is not the case.

The ʺadding modelʺ suggests less charge induced on the anode than created because of grid inefficiency. The charges QA(0) and QC(0) induced on the anode and the cathode,

respectively, for events at grazing angle are, according to Eq. 8 and Eq. 6 given by

(

)

Q

( )

n e D X e n e n QA cos

θ

= 0 −

σ

0 cos

θ

⇒ A 0 = 0 and

(

)

Q

( )

n e D X e n Q_C cos 0 1 cos ⇒ _C 0 = 0      − =

θ

θ

.

According to this ʺline 1ʺ should follow the diagonal before, as well as after, the correction is made. Fig. 19 shows, however, that ʺline 1ʺ, where the ionizing particles have lost most of its energy in the sample, lies just below the diagonal as predicted by the ʺadding modelʺ.

Fig. 20 and Fig. 21 display the effects of the two different grid inefficiency corrections to the data from Fig. 19. The same value of σ was used in both corrections.

33 50 100 150 200 250 50 100 150 200 250

P

C

[channels]

P

_A

[channels]

P_A = P_C

P

corr_A

=P

_A

-

σ

P

_C

Fig. 20: The effect of applying the correction PA(corrected) = PA-σPC.

50 100 150 200 250 50 100 150 200 250 P_A[channels] PCORR_A =(P_A-σP_C)/(1-σ) P C [c ha nne ls] P_A=P_C

Fig. 21: The effect of applying the correction PA(corrected) = (PA-σPC)/(1-σ).

Fig. 20 displays the effect of applying the correction by subtracting a fraction σ of the cathode pulse height from the anode pulse height, as suggested by the ʺsubtracting modelʺ. The