Leroy C., Rancoita P.-G. Principles Of Radiation Interaction In Matter And Detection

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580 Principles of Radiation Interaction in Matter and Detection

a distribution of upset rates corresp onding to the variation of cross section as a

function of LET. The IRPP approximation uses the diﬀerential upset cross section,

d σ

ion

/d LET, the ratio

= 2

1 +

of the total surface area of the sensitive volume to the surface area, the ratio

d/l

max

of the total height (d) of the sensitive volume to the maximum length (l

max

)

that can be traveled in the sensitive volume and the diﬀerential LET spectrum,

d φ/d LET. The SEU rate is then:

SEU

LET

i,max

LET

i,min

d σ

ion

d LET

(7.64)

with

LET

max

(d/l

max

)LET

d Φ(LET)

d LET

[> l(LET)] d LET. (7.65)

LET

i,min

and LET

i,max

are the lower bin limit and the upper bin limit in

d σ

ion

/d LET, resp ectively.

The path-length distribution used in the RPP approximation can be replaced

by an eﬀective LET spectrum, d φ/d LET

eﬀective

calculated by a Monte-Carlo

method. This eﬀective LET spectrum is combined with the ion upset cross sec-

tion, σ

SEU

(LET ) to produce the upset rate. The eﬀective LET of the incident ion

is deﬁned by

LET

eﬀective

dep

ρ d

, (7.66)

where d is the eﬀective thickness of the device component. The upset rate is given

by [Inguimbert and Duzelier (2004)]

SEU

4σ

sat

LET

max

LET

min

d Φ(LET)

d LET

eﬀective

SEU

(LET) d LET, (7.67)

where σ

sat

is the ion upset cross section at saturation (Fig. 7.12). N

SEU

[Eq. (7.67)]

was calculated in [Inguimbert and Duzelier (2004)] using GEANT4. The re-

sults of [Inguimbert and Duzelier (2004)] are in agreement with CREME96 re-

sults [Nymmik, Panasyuk, Pervaja and Suslov (1992)]. However, the use of GEANT4

opens the possibility of accurate modeling of device geometry, structure, material

composition and functions of the sensitive volume while including the various me-

chanisms of energy deposition, toward an accurate estimate of the deposited energy

and then upset rate. For accelerator data, where one has the choice to select speciﬁc

ion beams of well deﬁned energy, it is possible to measure the cross section as a

function of LET. Then, the measured cross section is reported in a plot as a function

of LET (an example is shown in Fig. 7.12).

This plot is characterized by two parameters: i) the threshold LET (LET

)

which is the minimum LET required to cause the speciﬁc SEE; ii) the saturation

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Displacement Damage and Particle Interactions in Silicon Devices 581

cross section, σ

sat

, which is approached at very high LET value and corresponds

to a limit, at which all the sensitive nodes of the device have b een upset. σ

sat

associated to the sensitive volume size and can be interpreted as the projection of

the sensitive volume in the direction normal to the semiconductor chip. σ

sat

(cm

)

is given by the product of x and y. Conversely, x and y are determined by the

measurement of σ

sat

[see Eq. (7.60)], taking into account the number of information

bits in the chip. In practice, the cross section is measured at several LET values

and the ﬁt to the data is done with a Weibull curve, after correction for geometric

eﬀects. This function has the form:

F (LET) = 1 − exp

−

LET − LET

, (7.68)

where LET

is the LET threshold for the device, such that F (x) = 0 for LET <

LET

; W is the width parameter and s is a dimensionless shape parameter. LET

applies to the overall chip response reﬂecting the composition of sensitive volumes

and their respective thresholds. It diﬀers from LET

, deﬁned before, which was

related to an individual sensitive volume. Then, the direct ionization (heavy ion)

SEE cross section is expressed as

σ = σ

sat

× F (LET). (7.69)

7.2.8.2 Calculation of SEU Rate for Protons and Neutrons

Protons and neutrons may generate SEEs via indirect ionization. As observed be-

fore, the complexity of the nuclear reactions breaks down the application of the

concept of LET and the SEE rate is determined from the proton or neutron ﬂuxes

and SEE nucleon cross sections:

SEU

max

min

d Φ(E)

d E

nucleon

(E) d E, (7.70)

where d Φ(E)/d E is the diﬀerential nucleon (proton or neutron) ﬂux spectrum as a

function of the proton or neutron energy, E; E

min

and E

max

are the minimum and

maximum energy of the diﬀerential nucleon energy spectrum, respectively; σ

nucleon

is the nucleon SEE cross section as a function of the nucleon energy. The proton (in-

direct ionization) SEE cross section is expressed with a 1-parameter or 2-parameter

Bendel function. The 1-parameter Bendel cross section as a function of the proton

energy is practically expressed in units of proton

−1

−2

bit

−1

and given by [Bendel

and Petersen (1983)]:

σ(A, E) = 1 × 10

−12

1 − exp

−0.18

Y (E)

´i

(7.71)

with

Y (E) = (E − A)

1/2

if E > A, (7.72)

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582 Principles of Radiation Interaction in Matter and Detection

Y (E) = 0 otherwise. (7.73)

σ is the upset cross section depending on E and A which are the proton energy

and apparent threshold in MeV, respectively. Cross section data are often better

described with the 2-parameter Bendel cross section which is given by [Stapor,

Meyers, Langworthy and Petersen (1990)] in units of proton

−1

−2

bit

−1

σ(A, B, E) =

× 10

−12

1 − exp

−0.18

Y (E)

(7.74)

with Y (E) deﬁned as in Eq. (7.72). E is the proton energy, the parameters A and

B are in MeV. The ratio B/A is the limiting cross section or the ﬁtted cross section

at inﬁnite proton energy, see for instance [Oberg et al. (1994)] where A = 1.5 MeV,

B/A = 0.953 have been used to ﬁt 148, 200 and 500 MeV proton SEU data for

87C51FC microcontroller at bias voltages of 4.5 and 5.5 V for 148 MeV beam, 4.0

and 5.0 V for 500 MeV beam, and only 4.0 V for 200 MeV beam. There is another

two parameter ﬁt parametrization [Shimano et al. (1989)] which has the advantage

to separate the energy dependence from σ

sat

as in Eq. (7.69):

σ(A, σ

sat

, E) = σ

sat

1 − exp

−0.18

Y (E)

(7.75)

with Y (E) given by Eq. (7.72), σ

sat

corresponds to (24/A)

[Eq. (7.71)] or (B/A)

[Eq. (7.74)] but is determined independently of A. Using one of these parametriza-

tions, Eqs. (7.71–7.75), it is possible to ﬁnd the SEE rate in any proton environment

from the measurement of the SEE cross section at one proton energy.

If the parameter A and σ

sat

are given, the error ∆σ(E

) for the proton energy

is expressed as [Bendel and Petersen (1983)]

∆ σ(E

) = σ(E

) −

∞

σ(A, σ

sat

, E) f

(E) dE, (7.76)

where σ(E

) is a measured cross section, f

(E) is a energy distribution function for

proton of energy

. If a Gaussian distribution is assumed for

(

), with

∞

f(E

) dE = 1,

A and σ

sat

can be determined by minimizing the sum of the square of relative errors,

i.e., by applying the formula

∆σ(E

)

σ(E

)

. (7.77)

However, one can use heavy-ion data, when available, to predict proton induced

upsets. There exist empirical models for proton induced SEU based on heavy-ion

data in terms of device cross section as a function of ion LET in order to deter-

mine an expression of the proton device cross section as a function of the proton

energy. An example of such approach is given with the PROFIT model applied to

calculate proton induced SEU rates in orbit (due to trapped protons) for various

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Displacement Damage and Particle Interactions in Silicon Devices 583

digital circuits parts (silicon-based) [Calvel et al. (1996)]. This model is based on

heavy-ion data. Establishing a relationship between upset rates due to heavy ions

and upset rates due to nucleons requires several strong assumptions because of the

diﬀerent mechanisms of energy deposition, direct and indirect ionizations. To rely

on heavy-ion data, one has to assume that all sensitive cells have the same surface

size and depth for both types of data. As observed above they have diﬀerent thresh-

old LETs (or critical charge). This distribution of threshold LETs can be ﬁtted by a

Weibull function. Another strong assumption is to consider proton interacting with

the silicon crystal through elastic interaction and ionization produced by the silicon

recoil atoms in the silicon crystal, i.e., inelastic interactions are considered as elastic

collisions, for the purpose. With this assumption, all proton induced events are due

to recoiling silicon atoms, having their LET calculated with TRIM [Ziegler, Biersack

and Littmark (1985b); Ziegler (2001, 2006)]. These recoil Si-atoms in Si have a maxi-

mum LET value be 15 MeV cm

−1

, as calculated from TRIM. Therefore, only

cells with a threshold LET ≤ 15 MeV cm

−1

will be sensitive to SEU induced

by protons. The ions experimental data can be ﬁtted using a Weibull function. The

ion device cross section is:

ion

= Σ

1 − exp

LET − L

, (7.78)

where Σ

is the heavy-ions saturated device cross section in cm

, L

is the LET

threshold in MeV cm

−1

, W and S are the Weibull parameters obtained from

the ﬁt of the curve Σ

ion

as a function of LET in MeV cm

−1

. Then in the

framework of the PROFIT model, the proton device cross section, Σ

, as a function

of the proton energy E

can be expressed as a function of the ion device cross

section:

= Σ

(

1 − exp

LET[E

)] − L

)

c N

nuclear

), (7.79)

with c = 2 × 10

−4

cm (typical cell depth), N

= 5 × 10

Si atoms per cm

and

) =

2 M

+ M

)

(1 − cos θ) E

, (7.80)

with M

and M

are the masses of proton and silicon atom, respectively. The

angle θ is the average scattering angle of protons (θ = 52

, as used in [Calvel et al.

(1996)]), LET[E

)] (calculated with TRIM) is the LET for recoil Si atoms in

silicon for E

proton energy. In Eq. (7.79), σ

nuclear

) is the proton-silicon nuclear

cross section as a function of the proton energy.

At saturation, the maximum LET of recoil Si-atoms is 15 MeV cm

−1

and

Eq. (7.79) becomes

sat

) = Σ

ion

(LET = 15 MeV cm

−1

) c N

hσ

nuclear

i, (7.81)

where Σ

sat

is the saturation proton device cross section and hσ

nuclear

i is the average

proton–silicon nuclear cross section in cm

(hσ

nuclear

i = 6 × 10

−26

for pro-

ton energies between 100 and 200 MeV). Previous works [Rollins (1990); Petersen

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584 Principles of Radiation Interaction in Matter and Detection

(1992)] on SEU rates in orbit induced by trapped protons used the two-parameter

Bendel cross section [Eq. (7.74) instead of Eq. (7.81)] with a diﬃculty to ﬁt the

lower energy experimental data. The PROFIT model is used to estimate the pro-

ton sensitivity of devices based on heavy-ion measurement but conversely it can

also be used to estimate heavy-ion sensitivity based on proton data. Examples of

application of the PROFIT model to estimate device sensitivity for heavy ions can

be found in [McDonald, Stapor and Henson (1999); Henson, McDonald and Stapor

(2006)].

7.2.9 SEE Mitigation

SEE mitigation or methods of reducing SEE impacts on data responses of a device

and on control of a device or system are developed by device designers. Examples

can be found in [LaBel (1996)]. For instance, for mitigation of memories and data-

related devices, parity checks may be used [Carlson (1975)]. Parity is a bit added

to the end of a data structure, which can state whether an odd or even numb ers

of ”ones” were in that structure. This method allows the detection of an error

if an odd numb er of bits are in error. If an even numb er of errors occurs, the

parity remains correct. This method detects, but does not correct errors. Other

methods such as cyclic redundancy check (CRC) coding [Short (1987)] or hamming

code [Carlson (1975)] are used for error detection. These mitigation technique may

be also applied to some types of control devices, such as microprocessor program

memory. In practice one has to add additional hardware and software to the system

design in order to perform an eﬀective mitigation. This is an iterative process, since

these additions have to be tested by the study of the device response to radioactive

sources, particle (pion, proton, neutron) and ion beams.

In the case of the radiation environment created by the operation of a particle

accelerator, SEE mitigation of electronic devices may be partially done by suppres-

sing or, at least, by reducing components of the harmful radiation ﬁeld. Installation

of dedicated shieldings may achieve that goal. Tuning of accelerator beam con-

ditions may also help with the attenuation or the removal of the radiation ﬁeld

locally. Components failure rates may be decreased by modifying their operation

conditions. For instance, the reduction of the operating voltage of a failing device

may restore its normal function without compromising its performances.

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Chapter 8

Ionization Chambers

Ionization chambers are detectors ﬁlled with a gas or a noble liquid in which the ioni-

zing particle creates ion–electron pairs. The ions and electrons then drift under the

applied electric ﬁeld to the electrodes where they are collected. If the electric ﬁeld

is increased, the electrons have enough energy to ionize the gas themselves. The de-

tectors using this mode of operation are called proportional counters. If the electric

ﬁeld is further increased, for example in a Geiger–Mueller counter, the electrons are

so energetic that UV photons are emitted, when they reach the anode. The principle

of operation of the ionizing chamber will be explained in detail in this chapter. To

demonstrate the usefulness of this type of detector, the study of pollution in liquid

argon using an α-cell will be discussed extensively. The chapter will conclude with

the principles behind proportional and Geiger–Mueller counter.

8.1 Basic Principle of Operation

This typ e of detector has been used for many years and its principle of operation is

relatively simple. An ionization chamber, in its simplest form, consists of two parallel

metallic electrodes (copper for instance) separated by a distance D. High voltages

(V ), up to several thousand kV, are applied to the anode. This voltage is maintained

by an external circuit, characterized by a resistance R and a capacitor C. The gap

D between the two plates is ﬁlled with a gas or a noble liquid and deﬁnes the

sensitive volume of the chamber. Ionizing particles traversing the sensitive volume

will ionize the gas or noble liquid and produce ion–electron pairs. Positive ions

and electrons are the charge carriers. The electric ﬁeld, E = V/D, created by the

potential diﬀerence across the gap, will cause electrons and positive ions to drift in

opposite directions toward the anode and cathode, respectively, where the charge

produced by ionizing particles is collected. The number of electrons produced in the

gap by n minimum ionizing particles (mip) is:

−

Dρ

, (8.1)

585

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586 Principles of Radiation Interaction in Matter and Detection

where W is the average energy-loss necessary to create an ion-electron pair in the

gas or noble liquid. For example, in argon gas, W = 26 eV, ρ ≈ 1.8 ×10

−3

g/cm

STP is the argon gas density, D is the gap thickness, and dE/dx ≈ 1.52 MeV is the

energy loss per g/cm

. Using these numbers, one ﬁnds for argon gas in the gap:

−

∼ 105 cm

−1

× nD, (8.2)

and, consequently, an electric charge

Q ∼ (105 cm

−1

× nD) × 1.6 × 10

−19

C. (8.3)

If one considers the case of an isolated ion-electron pair created at some distance

from the anode, the electron drift velocity is

(e) =

= µ

E = µ

, (8.4)

where µ

is the electron mobility. In the case of a gas-ﬁlled parallel plate chamber,

the electron drift velocity also depends on the gas pressure, p, i.e.,

(e) =

E =

V µ

, (8.5)

in which case the mobility is expressed in bar cm

−1

The collection of these charges produces a drop, ∆V, in voltage across the ca-

pacitor of the external circuit. Then, a pulse of height given by

∆V =

∆Q

is produced across the resistor and recorded by the external electronic readout

circuit.

In the case of liquid argon, only electrons contribute in practice to the charge col-

lection since the positive ion drift velocity [v

(ion)] is found to be three to ﬁve orders

of magnitude smaller than the electron drift velocity [v

(e)]. The electron mobility

in liquid argon (T = 87 K) is µ

≈ 500 cm

−1

compared to the positive ion

mobility µ

ion

≈ 6 ×10

−4

−1

[Gruhn and Edmiston (1978)]. The number of

electrons reaching the anode dep ends on the details of the chamber design, the na-

ture of the sensitive volume and its purity, and on the applied voltage. The ionization

electrons are aﬀected by recombination with parent ions (germinate recombination)

or with positive ions on their way to the anode (columnar recombination) and by

capture by electronegative impurities present in the active medium. The electrons

escaping recombination or capture eventually make their way to the anode, where

they are collected. The probability for an electron to escape recombination and cap-

ture increases with voltage and therefore, the number of electrons collected at the

anode increases with voltage up to a saturation value of voltage, V

sat

, for which the

charge created in the sensitive volume by the incoming radiation is all collected. No

further increase in the collected charge takes place, when the voltage is increased

beyond V

sat

. V

sat

is called the saturation voltage. The region, where the charge col-

lected remains roughly constant with voltage, is called the ionization region, i.e.,

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Ionization Chambers 587

the region in which the ionization chambers are operated. We will see in another

section that increasing the voltage beyond this saturation voltage leads to the mul-

tiplication region, i.e., a region where the electrons are accelerated enough to ionize

molecules of the gas and pro duce secondary electrons.

Taking into account recombination or capture eﬀects, one can write the time

decay of the concentration of electrons n

−

as:

−

= −k

− k

−

− k

−

, (8.6)

where, in standard notations, k

is the neutralization factor weighting the neutral-

ization of electrons at the anode, n

is the concentration of positive ions, k

is the

recombination rate constant, n

is the concentration of electronegative impurities

and k

is the capture rate constant (or attachment rate factor). Recombination and

attachment phenomena will be discussed in the next section, keeping in mind spe-

ciﬁc applications using ionization chambers with liquid argon as sensitive volume.

8.2 Recombination Eﬀects

This section treats the various recombination eﬀects, including diﬀusion, which

aﬀect the electron charge collection at the electrodes of the chamber. In absence

of impurities in the liquid ﬁlling the gap, the charge collection is aﬀected by two

processes called germinate and columnar recombinations.

8.2.1 Germinate or Initial Recombination

Germinate or initial recombination takes place when an electron produced by ioniza-

tion recombines with its parent ion. This recombination happens unless the electron

becomes subjected to an electric ﬁeld larger than the Coulomb ﬁeld which is main-

taining it in the vicinity of the ion. This initial recombination mainly occurs in the

case of heavily ionizing particles, such as low energy α particles, and is minimal

for mip’s. Obviously, the application of large electric ﬁelds minimizes the initial

recombination.

The ratio of the amount of charge collected after initial recombination, Q

rec

, to

the amount of initial charge, Q

, has been calculated by Onsager (1938):

rec

= exp

−

¶·

1 + E

2²k

¶¸

, (8.7)

where r

= e

/²kT is the Onsager length or radius, r

is the thermalization length,

k is the Boltzmann constant, T is the temperature, E is the applied electric ﬁeld,

and ² is the electric p ermittivity of the medium.

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588 Principles of Radiation Interaction in Matter and Detection

8.2.2 Columnar Recombination

The ionization electrons which escape germinate recombination are free to drift un-

der the inﬂuence of the applied electric ﬁeld. During ionization, specially by heavy

particles, ions are created close to the ionizing particles trajectory, within a column

along the trajectory. Drifting electrons and positive ions can then recombine. This

columnar recombination depends on the particle emission angle with the applied

electric ﬁeld. A strong recombination takes place for large positive ion density along

the electrons path as it is the case in liquid. However, this recombination remains

small for large values of the applied electric ﬁeld. The theory of columnar recom-

bination has been formulated by Jaﬀe (1913) and Kramers (1952), based on the

assumptions that the ionization is uniformly distributed along the line of motion of

the ionizing particle (z -direction), that the ion density is Gaussian distributed in a

column of radius b, and that all ionization is in the form of positive and negative

ions. The initial carrier concentrations, n

(t = 0), are:

(t = 0) =

πb

exp

−

+ y

(8.8)

with N

the initial density of ions along the column axis (z-direction). Then, the

evolution of the ion, n

, and electron, n

−

, concentrations is described in this model

by the equations:

= ∓µ

E sin φ

∂n

∂x

+ D

∆n

− k

−

(8.9)

and

, (8.10)

where µ

−

= µ

(µ

= µ

ion

) is the electron (ion) mobility, φ is the emission angle

of ionizing particles with respect to the electric ﬁeld, D

−

) is the diﬀusion

coeﬃcient for electrons (ions) and k

is the columnar recombination factor. The

ratio of the amount of charge collected after columnar recombination, Q

rec,c

, to the

amount of initial charge Q

has been calculated by Kramers (1952), assuming that

the diﬀusion term is negligible compared to displacement and recombination terms,

in an external ﬁeld (ζ = y

rec,c

√

+∞

√

+ 1

dζ (8.11)

with

f =

Eb sin φ

√

πQ

. (8.12)

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Ionization Chambers 589

8.2.3 The Box Model

For electric ﬁelds up to a few hundred V/cm in liquid argon, the electron yield, as

calculated using Onsager’s model, is a linear function of the ﬁeld and the slope is

determined by the temperature T and the electric permittivity ², as used in Eq. (8.7)

which can be rewritten as [Buckley et al. (1989)]:

Q(E) = Q(E = 0) (1 + βE), (8.13)

where Q(E = 0) is the number of electrons that escape recombination with posi-

tive ions in the absence of any external electric ﬁeld. The slope β is given by (see

Eq. (8.7)):

β =

2kT

. (8.14)

The comparison of Eq. (8.14) with data shows large discrepancies and indicates

that Onsager’s model is inappropriate in the case of liquid argon: the measured

slope/intercept ratio is about a factor 4 larger than the predicted one [Buckley et

al. (1989)]. The explanation for the discrepancy is that even if an electron escape

recombination with its parent ion, it may still recombine with another close ion, a

possibility not accounted for by Onsager’s model which assumes isolated electron–

ion pairs [Buckley et al. (1989)].

The defects in the Onsager model has led Thomas and Imel (1987) to propose

the box model, based on the earlier works of Jaﬀe (1913) and Kramers (1952). In

the box model, the diﬀusion factor of electrons is negligible since the measured

diﬀusion factor is very small, D

−

= (18 ± 2) µm/

√

mm [Deiters et al. (1981)] for a

drift ﬁeld of 10 kV/cm and D

−

= (28 ± 2) µm/

√

mm [Derenzo et al. (1974)] for a

lower drift ﬁeld of 2.7 kV/cm. In addition, the p ositive ion mobility is set to zero

in the box model as a result of the ion drift velocity being three to ﬁve orders of

magnitude smaller than the electron drift velocity (see above). As a result of these

approximations, the evolution of the ion, n

, and the electron, n

−

, concentrations,

is described by the equations:

= k

−

(8.15)

and

−

= −µ

E sin φ

−

− k

−

. (8.16)

These equations are solved imposing the boundary condition that each electron–ion

pair is isolated (box model) and using the initial condition n

= n

−

. Then, the

fraction of charge collected, Q/Q

, is given by:

ln(1 + ξ) (8.17)

with

ξ =

. (8.18)