Towards a methodology for large-sample prompt-gamma neutron-activation analysis

(1)

(2)

(3)

prompt-gamma neutron-activation analysis

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 23 november 2004 te 10.30 uur

door

Ingeborg Heleen DEGENAAR doctorandus in de natuurkunde

(4)

Prof. dr. ir. T.H.J.J. van der Hagen Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. ir. J.J.M. de Goeij, Technische Universiteit Delft, promotor Prof. dr. ir. T.H.J.J. van der Hagen, Technische Universiteit Delft, promotor Prof. dr. ir. C.W.E. van Eijk, Technische Universiteit Delft

Prof. dr. M.J.A. de Voigt, Technische Universiteit Eindhoven

Prof. dr. W. Görner, German Academy of Sciences, Berlin, Germany

Prof. dr. N.M. Spyrou, University of Surrey, United Kingdom

Prof. dr. R.F. Fleming, University of Michigan, United States

Dr. M. Blaauw heeft als begeleider in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.

Published and distributed by: DUP Science

DUP science is an imprint of Delft University Press P.O. Box 98 2600 MG Delft The Netherlands Telephone: +31 15 27 85 678 Telefax: +31 15 27 85 706 Email: info@library.tudelft.nl ISBN: 90-407-2509-8

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the publisher: Delft University Press

(5)

(6)

List of symbols and abbreviations

Greek symbols

*

α specific angle (0, 45, 135, 180 degrees)

∆d m average neutron distance to CM in d-direction

∆r m average neutron distance to CM in r-direction

∆x m translation along X-axis

∆y m translation along Y-axis

ε full-energy peak detection efficiency

εref efficiency of a reference geometry

εx efficiency of counting geometry x

ϑ ° rotation around X-axis

θ isotopic abundance

λ s-1 decay constant

µ _m-1 _{gamma-attenuation coefficient} t

µ _m-1 _total_{gamma-attenuation}_coefficient

ρ _{kg m}-3 _density

ρ _m-3 _{atom density}

σ standard deviation

σa m2 microscopic absorption cross section

σa,0 m2 microscopic absorption cross section at v0

σ’a (m2) m2 effective microscopic absorption cross section

σs m2 microscopic scattering cross section

a th

σ _m2 _{microscopic flux-averaged absorption cross section}

Σa m-1 macroscopic absorption cross section

Σa,0 m-1 macroscopic absorption cross section at v0

Σs m-1 macroscopic scattering cross section

Σt m-1 macroscopic total cross section

τ s dead time

τM s MCA dead time

τS s system dead time

*_{In this list the symbols in the polynomials in Chapter 5 are not given. The symbols used only in the theory} section of Chapter 6 or the Appendix are also not given.

(10)

ϕ ° rotation around Y-axis MB

ϕ _m-2_s-1 _{total Maxwell Boltzmann flux} th

ϕ _m-2_s-1_{total thermal flux}

ϕ _m-2_s-1 _{neutron flux} Φ m-2 _{neutron fluency} 2 χ _chi-square_value 2 r

χ _reduced_chi-square_value

Ω sterad solid angle

Ωb sterad illumination brightness

Ω sterad effective solid angle

Latin symbols

*

A net area of the full-energy peak, corrected for dead time

b number of gammas emitted per absorbed neutron

C s-1 _{counting rate}

d conversion parameter

d m depth with respect to rin

dCM m d-coordinate of CM

Df number of degrees of freedom

Eγ eV gamma energy En eV neutron energy f s-1 frequency f self-shielding factor fP pulser factor F coupling factor

F equivalent irradiated mass fraction

i specific element

i specific gamma energy

i specific isotope

Iγ (relative) gamma intensity

k J K-1 _{constant of Boltzmann}

m kg mass

M kg mol-1 _{molar mass}

M atom mass

Me effective atom mass

M(v) s m-1 Maxwell-Boltzmann velocity distribution

n m-3 neutron density

na total number of absorbed neutrons

N m-3 atom density

N m-3 isotope density

NAV mol-1 Avogadro number

N0 total number of atoms

Np number of pulses seen in PGNAA spectrum

Nγ number of gammas

*_{In this list the symbols in the polynomials in Chapter 5 are not given. The symbols used only in the theory} section of Chapter 6 or the Appendix are also not given.

(11)

P probability of absorbing a neutron in a copper foil

P pile-up constant

Q quality of a fit

r m radius

r m distance to the central axis of the beam

r m position of the neutron absorption

r m position in the sample

rc m radius of collimator hole

rin m position where the neutron beam enters the sample

R m-3 s-1 capture-rate density R m-3_s-1 _{interaction-rate density}

S m2 cross section of object

S m2 _{cross section of neutron beam}

tFM4 m-2 s-1 value of MCNP’s FM4 tally ti s irradiation time tw s waiting time tc s counting time tL s live time tr s clock time

T (probability of) transmission

T K temperature

T0 K reference temperature, conventionally taken as 293.6 K

x m sample thickness

x m depth in material

xd m sample-detector distance

v specific voxel

v m s-1 neutron velocity

v0 m s-1 reference neutron velocity conventionally taken as 2200 m s-1

V m3 sample volume

w element mass fraction

w m-2 s-1 normalised density distribution

Z atomic number

Abbreviations

CM centre of mass

DE double-escape peak

DGNAA Delayed-gamma neutron-activation analysis

FE full-energy peak

INAA Instrumental neutron-activation analysis

LS-INAA Large-sample instrumental neutron-activation analysis LS-PGNAA Large-sample prompt-gamma neutron-activation analysis

NAA Neutron-activation analysis

PGNAA Prompt-gamma neutron-activation analysis

(12)

(13)

Chapter 1 Introduction

Abstract

The aim of the study described in this thesis is the development of a methodology so that element analysis in large samples can be performed practically using prompt-gamma neutron-activation analysis. In this chapter first an introduction is given into the physical principles of neutron-activation analysis. Secondly, the two subjects are described that are coming together in large-sample prompt-gamma neutron-activation analysis, viz. small-sample prompt-gamma neutron-activation analysis and large-sample instrumental neutron-activation analysis. This is done by giving an overview of important aspects in both types of analysis. Finally, the aims and criteria are given for the research described in this thesis, together with an outline.

(14)

1.1 Introduction

The aim of the study described in this thesis is the development of a methodology for performing element analysis in large samples using prompt-gamma neutron-activation analysis. This method will be complementary to the already existing analysis methods: small-sample prompt-gamma neutron-activation analysis (PGNAA) and large-sample delayed-gamma neutron-activation analysis (LS-DGNAA).

The methodology for large-sample prompt-gamma neutron-activation analysis (LS-PGNAA) was developed to determine mass fractions of elements in large samples that can be analysed with PGNAA, like hydrogen, boron and cadmium. Potential large samples are drill cores, (polluted) ditch soil and (polluted) garbage, as in LS-DGNAA.

In this chapter attention is given to the physical principles of activation analysis. Secondly, some aspects of the research done with small-sample PGNAA and LS-DGNAA are given. Finally, the aims and criteria of the study are given together with an outline. A further introduction to LS-PGNAA is given in Chapter 2.

1.2 Physical principles

The principle of neutron-activation analysis can be explained by considering Figure 1.1. Sample material to be analysed consists of a number of nuclei. When a neutron is impinging on sample material a reaction may take place between the neutron and a target nucleus, leading to a compound nucleus. After emitting one or more prompt gammas, directly after neutron capture, the compound nucleus can either become stable or turn into a radioactive nucleus. The energy of the prompt gammas is associated with the binding energy of the neutron in the newly formed nucleus. The radioactive product nucleus will generally decay by emission from the nucleus of (negatively charged) electrons, typically accompanied by photons. To a lesser extent emission of (positively charged) positrons or capture of an orbital electron takes place, again typically accompanied by photons. Electrons and positrons emitted in nuclear decay are known as β-_{and β}+_{radiation and photons as γ-radiation. Due to}

secondary processes, electrons and photons may be emitted from the electron shells as well. In this case the photons are called X-rays.

(15)

Figure 1.1 Reaction scheme of an incident neutron impinging on a target nucleus, yielding a radioactive nucleus.

Element analysis based on the observation of prompt-gamma radiation is called “prompt-gamma neutron-activation analysis” (PGNAA). When based on the observation of gamma radiation from decay of the radioactive product formed it is called “delayed-gamma neutron-activation analysis” (DGNAA). Small-sample DGNAA can be applied with and without chemical treatment of the irradiated sample material. When the sample material is chemically treated between irradiation and measurement, the method is known as radiochemical neutron-activation analysis (RCNAA or RNAA). Otherwise DGNAA is referred to as instrumental neutron-activation analysis (INAA).

1.2.1 Interaction of neutrons with matter

Sample material consists of a number of atoms per unit of volume (N in m-3_{) in which each}

atom is an isotope of an element. The number of stable isotopes per element ranges from 1 to 10. The number of atoms of each isotope i per unit of volume is given by Ni (in m-3). An atom

of an isotope has its own scattering and absorption characteristics for neutrons, given by the microscopic scattering cross section (σs in m2) and the microscopic absorption cross section

(σa in m2). Microscopic cross sections represent the probabilities that a neutron has a specific

interaction with a target nucleus and are dependent on the energy of the incoming neutron. Different types of reactions can take place, like scattering, absorption and fission.

For the development of a methodology for LS-PGNAA the characteristics of the sample material are important. Therefore the macroscopic scattering cross section (Σs in m-1)

(16)

are commonly employed. In these equations the summation is carried out over all isotopes i. The sum of both cross sections is the total macroscopic cross section (Σt in m-1) (1.3).

(

)

s s,i i i

N

Σ

=

∑

σ

(1.1)

(

)

a a,i i i

N

Σ

=

∑

σ

(1.2) t s a

Σ

=

Σ

+

Σ

(1.3)

Σs = macroscopic scattering cross section (m-1) σs = microscopic scattering cross section (m2)

N = atom density (m-3₎

Σa = macroscopic absorption cross section (m-1) σa = microscopic absorption cross section (m2) Σt = macroscopic total cross section (m-1) index i for specific isotope

Neutron absorption and scattering are neglected in small-sample activation analysis, but do play an important role in the analysis of large samples, where they may become substantial and thus cannot be neglected anymore.

1.2.2 Interaction of gammas with matter

When gammas (and also X-rays) interact with matter, the interactions are different from those of neutrons with matter. Gammas noticeably interact with electrons. The relevant interactions between gammas and matter are the photo-electric effect, the Compton scattering and the pair production.

The photo-electric effect is the transfer of all energy of a gamma to an electron in one of the inner electron shells. After absorption of the energy the electron is ejected from the atom with an energy equal to the incident gamma minus the binding energy of the electron ejected. The vacancy in the inner shell is filled with an electron from an outer shell and X-rays and Auger electrons are emitted as the result. Due to their low energies they are readily absorbed in matter, and thus almost do not escape.

Compton scattering is the partial transfer of the energy of a gamma (Eγ ) to an electron

in a collision. After the collision the direction of the gamma is changed. The Compton gamma may subsequently escape from matter or may have more interactions. The energy transfer is dependent on the angle of incidence and may range from 0 keV to (Eγ -255) keV for

(17)

Figure 1.2 Sketch of a gamma spectrum for a radionuclide, emitting only one specific gamma energy, larger than 1022 keV. (SE = single-escape peak, DE = double-escape peak, FE = full-energy peak).

Pair production is the conversion of 1022 keV from the energy of a gamma into two particles, occurring when the gamma is close to the nucleus of an atom. These two particles are a positron and an electron. Each particle needs 511 keV to be created and the remainder of the transferred energy is divided as kinetic energy between the two particles. After thermalisation, the positron will annihilate again with an electron within matter, yielding two 511 keV gammas that may escape from matter.

The types of interactions mentioned above occur also in detector material and are registered in a gamma spectrum as given in Figure 1.2 for one radionuclide with a single gamma transition. The energy of the gammas absorbed via the photo-electric effect with the detector material is registered in the full-energy peak (FE), since the entire energy from the gamma is absorbed.

Compton scattering will only transfer a part of the energy to the detector. When no energy is absorbed further after the collision in the detector material, the partially transferred energy to the electron from the incident gamma is registered in the Compton continuum, i.e. the region between zero and almost the full-energy peak. When after the collision the remainder of the energy, viz. the Compton gamma, is fully absorbed in the detector material, the energy of the Compton electron and subsequent absorption(s) of energy are registered in the full-energy peak.

Compton continuum FE SE DE Co unt s Energy

(18)

When gammas have a pair interaction with the detector material a positron and an electron are created leading to two 511 keV gammas. These can escape from the detector material resulting in the so-called single-escape peak (SE) or the double-escape peak (DE) in the spectrum. When both 511 keV gammas transfer their energy fully to the detector material all energy is registered in the full-energy peak (FE).

Normally, only gammas that have deposited all their energy in the detector are useful. The gamma will then show up in the full-energy peak. A partial and variable energy deposit due to Compton scattering contributes to the continuum in the spectrum that yields no useful information in neutron-activation analysis and hampers detection of other gamma peaks in the same energy region. Single-escape peaks and double-escape peaks are also useful in activation analysis due to their discrete amounts of energy.

For each radionuclide emitting a delayed gamma the energy of the gamma emitted is rather specific for that radionuclide. This also applies to the prompt gamma, being specific for the compound nucleus. The radionuclide or compound nucleus can be identified through the energies observed in the gamma spectrum, and quantified through the observed intensity of the radiation.

In INAA the gamma energies range mostly from 100 keV through 3 MeV, in PGNAA the gamma energies range mostly from 1 up to 10 MeV. Gamma attenuation is usually neglected in small-sample activation analysis. However, in large-sample activation analysis gamma attenuation cannot be neglected anymore.

1.2.3 Transport of particles

Different entities are used to express the transport of neutral particles, i.e. neutrons or gammas, through materials. In this thesis the entities used are fluency, flux and density. The symbols given in this paragraph are used to describe the transport of neutrons in this thesis and will not be used to describe the transport of gammas. All neutron entities have a distribution related to the energy of the neutron (En in MeV). Since En is related to the velocity

of the neutron (v in m s-1_{) with} 1 2 2 n

E = mv , a distribution of a neutron entity can be described

(19)

The neutron density _{n , ,t dr}

(

_{r v}

)

3_{(in m}-3_{) is defined to be the expected number of}

neutrons with velocity v in dr3 about r at a time t. The neutron flux ϕ r v( , ,t ) (in m-2s-1) is defined to be the product of neutron density and neutron velocity (v in m s-1):

(

)

( , ,t )

n , ,t

ϕ

r v

≡

v

r v

(1.4)

The neutron fluency ( , )dΦ r v v (in m-2) is defined to be the time-integrated neutron flux:

( )

(

)

0

,

,t dt

Φ

r v

≡

∞

∫

ϕ

r,v

(1.5)

Of more importance in experiments and simulations is the rate at which reactions are occurring at any point in material. The interaction-rate density R ,t

( )

r (in m-3_s-1_{) is defined to}

be the expected rate at which interaction are occurring in dr3about r at time t.

(

)

3

( ) (

)

3

R , ,t dr

r v

≡

Σ

v

ϕ

r v

, ,t dr

(1.6)

R = interaction-rate density (m-3_s-1₎ Σ = macroscopic cross section (m-1₎ ϕ = neutron flux (m-2_s-1₎

A phenomenon like neutron self-shielding and/or gamma attenuation occurs, because of the dimensions of the sample material. Nuclei deeper in the material would tend to be shielded from the incident neutrons by the nuclei that are nearer to the surface of the material, since interactions remove neutrons from the incident flux. To account for such effects the flux of particles that did not encounter an interaction at depth x, ϕ

( )

x , in the sample material is introduced as a differential equation:

( )

t

x

ϕ

_{Σ ϕ}

∂

= −

∂

for neutrons (1.7)

( )

t

x

ϕ

_{µ ϕ}

∂

_{= −}

∂

for gammas (1.8) x = depth in material (m)

Σt = macroscopic total cross section (m-1) µt = total gamma-attenuation coefficient (m-1)

When Equations 1.7 and 1.8 are integrated over the layer thickness they result in respectively Equations 1.9 and 1.10, in which ( )ϕ x is the flux at layer thickness x.

( )

_x

( )

0 _e

Σtx

ϕ

₌

ϕ

− _for_neutrons_(1.9)

( )

_x

( )

0 _e

µtx

(20)

1.3 Absolute and relative standardisation

The amount of element (m in kg) present in the sample in INAA is related to the net area of the full-energy peak (A) observed in the delayed gamma spectrum (Equation 1.11).

(

) (

)

(

1 2

)

AV

N

_a

₁

ti tw

₁

tc

AM

m

f C ,C

I

_γ

_e

λ

_e

λ

_e

λ

ϕ ε

σ θ

− − −

=

−

(1.11)

m = amount of an element in the sample (kg) A = net full-energy peak area

M = molar mass (kg mol-1₎ ϕ = neutron flux (m-2_s-1₎

Iγ = absolute gamma intensity

ε = full-energy peak detection efficiency

NAV = Avogadro number (mol-1)

σa = microscopic absorption cross section (m2) θ = isotopic abundance

λ = decay constant (s-1₎

ti = irradiation time (s)

tw = waiting time (s)

tc = counting time (s)

f (C1,C2) = correction for neutron self-shielding and gamma attenuation

The process of correlating the readings of any instrument with a standard is usually called “calibration”, whereas “standardisation” is the process of making conform to a standard. However, in NAA, where many calibration processes are involved, the determination of the relation between m and A is called standardisation. A distinction is made between absolute and relative standardisation.

In the absolute standardisation method, the nuclear and other constants, i.e. M, Iγ, NAV,

σ, θ and λ, are taken from literature. Other variables have to be measured or calculated, i.e. A, ϕ,ε, ti, tw, tc and f(C1, C2).

In the relative standardisation method, the unknown sample is compared to a calibration sample containing a known amount of the element of interest. The calibration sample is irradiated and measured under the same conditions as the sample, so that M, ϕ, Iγ,ε,

Nav,σ andθ cancel out. The ratio of A’s corresponding to the element of interest in the two

measured spectra is used to calculated the mass fractions (Equation 1.12).

(

)

(

)

(

)

(

)

(

)

(

)

1 2 1 2

1

y ,i y ,w y ,c x,i x,w x ,c t t t x, x, x x t t t y y _y, _y,

e

f C ,C

m

A

m

A

_e

_{f C ,C}

λ λ λ λ λ λ − − − − − −

−

=

−

(1.12)

(21)

In the case of NAA with small samples, the correction for neutron self-shielding and gamma attenuation can be neglected in Equation 1.11 and 1.12.

In the case of PGNAA the gamma spectrum is measured during irradiation. Therefore all terms with tλi disappear and the equation to use is 1.13.

(

)

(

11 22

)

x, x, y,i x x y y x,i y, y,

f C ,C

t

m

A

m

=

A t

f C ,C

(1.13)

1.4 Small-sample prompt-gamma neutron-activation analysis

PGNAA for small samples with a mass of a few hundred milligrams has been introduced in the sixties of the previous century (Isenhour and Morrison, 1966). With respect to the already existing INAA some additional elements could be determined in the samples, e.g. H, B and N. In PGNAA the neutrons can be delivered to the sample material by more or less isotropically emitting neutron sources or by (guided) neutron beams (Alfassi and Chung, 1995). Radio-isotopic neutron sources (252_Cf,241_{Am-Be or}238_{Pu-Be) and neutron generators}

have been applied to in situ analysis of bulk coal (e.g. Wormald and Clayton, 1983) and to in vivo analysis of humans (e.g. Matthews and Spyrou, 1982). Neutron beams are mainly used for the elemental analysis of a variety of objects that can be easily transported to the irradiation facilities (e.g. Mackey et al., 1996).

In small-sample PGNAA neutron self-shielding can be neglected. When neutron beams are used with small-sample PGNAA the most suitable temperature of the neutrons is colder than thermal. That allows for more efficient neutrons, since the capture cross section is typically inversely proportional to the velocity of the neutron, so the “colder” the neutrons impinging on the sample material at constant flux, the more effective the detection and the better the sensitivity of the element analysis. Also, the further away from the neutron source the colder the neutron temperature and the lower the gamma background at the sample position.

In INAA the delayed gamma spectrum is recorded at another position than at the position at which the sample is irradiated. Therefore two optimum positions can be chosen: one position at which the neutron flux is as high as possible and another position, where the gamma background is as low as possible and where the detector efficiency is optimised, i.e. the count rate is low enough not to result in spectrum distortions and high enough to ensure a low detection limit and/or a reasonable counting time.

(22)

In the case of PGNAA the position of irradiation and that of the measurement are the same. To optimise the neutron flux, the sample can be positioned nearby the reactor core, but the gamma background is rather high at that position. Therefore the sample is situated at the end of a neutron guidance system, where the neutron flux is much lower than the neutron flux at the position of a sample in INAA. The detector is situated such that the count rate is optimised versus dead time.

Since after almost every neutron capture one or more prompt gammas are emitted, theoretically one could count (almost) all neutron capture reactions, also those leading to stable products. This is not the case in INAA, where only decays are recorded, that take place in the measurement period.

In practice, due to the energies of prompt gammas of several MeV’s, most of the gammas will end-up in the Compton continuum and not in the full-energy peak in PGNAA. The probability of absorbing all energy of the gamma in the detector material is smaller when the energy of the gamma is higher. Therefore the detection in the full-energy peak is much more likely in INAA, where the energies of the gammas are much lower.

1.5 Large-sample instrumental neutron-activation analysis

An advantage of using large samples in INAA is that inhomogeneous samples can be analysed and no representative sub-samples need to be taken. On the other hand, problems arise like neutron self-shielding and gamma attenuation. In small sample INAA the neutron self-shielding and gamma attenuation can be commonly neglected. But when larger samples are analysed both have to be taken into account.

Overwater (1994) developed a methodology to perform INAA of unknown samples of up to 1 m long and 0.15 m diameter (35 kg at a ρ = 2000 kg m-3_{). Included were}

computational methods to calculate neutron self-shielding (Overwater and Hoogenboom, 1994) and the gamma attenuation at different positions in the sample (Overwater et al., 1993). Bode et al. (1998) have tested the methodology and others have applied it, particularly for materials with an inhomogeneity that causes difficulties in sub-sampling (Blaauw et al., 1997; Lakmaker and Blaauw, 1997).

The methodology developed in this thesis will be used for a facility complementary to the facility developed by Overwater. Both facilities accept samples of the same size, but use different gammas, i.e. delayed versus prompt, in their analysis. Other differences are that in LS-PGNAA the gamma energy will be higher, that the neutrons are not isotropically

(23)

approaching the sample, but from one side, and that the spectrum will be recorded during the irradiation.

Also other institutes have plans for new facilities for large sample neutron-activation analysis that are based on the work done by Overwater (e.g. Lin and Henkelmann, 2002; Stamatelatos, 2002).

1.6 Aims and criteria

In the development of the methodology for LS-PGNAA some aims and criteria were set to guide the design process. Firstly, the large-sample analysis method should be non-destructive, including the determination of correction factors for neutron self-shielding and gamma attenuation, since non-destructivity is one of the advantages of small-sample PGNAA with respect to other analysis methods. Secondly no a priori knowledge about the composition of the sample material was to be used for the correction methods, just like in the earlier correction method for LS-INAA (Overwater and Hoogenboom, 1994).

Until now, no information from literature is available of a non-destructive method to correct for neutron self-shielding and gamma attenuation in large or bulk samples using PGNAA. In other research a suited calibration material was used, chosen based on a priori knowledge about the matrix composition, dimensions and density of the sample material. In the correction method, described in this thesis, neutron fluxes measured at different positions outside the sample are used to derive the neutron-density distribution inside the sample material. The gamma attenuation was determined by Moens’s method (Moens et al., 1981) and the calculation of effective solid angles, using the dominant Compton effect in PGNAA.

Thirdly, the inaccuracy of the analysis with the method should be lower than 10 %. Therefore, the inaccuracy of all correction methods together should be lower than 10 %. The value of 10 % gives an optimum between a small inaccuracy and a large variety of sample materials. It should be noted that the percentage of 10 % does not include errors due to imprecision, e.g. due to counting statistics.

It was aimed that the LS-PGNAA methodology should complement the LS-INAA methodology. Therefore a criterion was set about the geometry of the sample container. The maximum container size was to be the same as in LS-INAA, i.e. cylinders of up to 1 m long and up to 15 cm diameter.

A criterion was set about the composition matrix of the sample material. This matrix has to be homogeneous in its density, and more specifically in its neutron absorption, neutron scattering and gamma attenuation. This criterion was set to simplify the development of a

(24)

methodology for LS-PGNAA. Notice was taken during the development that in future the methodology can be extended for the use of inhomogeneous samples.

Another important point was the choice of neutron source. It was chosen to use a neutron beam with thermal neutrons with a Maxwell-Boltzmann energy spectrum. An argument for the choice of a neutron beam over an isotropic source was to find an optimum between a low gamma background and a high neutron flux. A neutron beam has a better optimum in a lower gamma background and a higher neutron flux than an isotropic neutron source, since the sample can be positioned further away from the source without losing too much neutron flux.

An argument for the thermal energy of the beam was found in the relation between the macroscopic absorption cross section (Σa) of the sample material and the energy of the

neutrons. The lower the energy of the neutrons, the higher Σa and the larger the amount of

neutron self-shielding in the large sample. In small-sample PGNAA it is aimed to absorb as many neutrons as possible. But in LS-PGNAA a quantitative analysis of the sample is possible, only when the neutron-density profile inside the sample can be determined non-destructively. When the neutrons would all be absorbed inside the sample material, the quantitative analysis would only be possible if the internal matrix of the sample material was known, which would violate the “no a priori knowledge” boundary condition. Therefore the energy of the neutrons has to be higher than the energies of the neutrons in small-sample PGNAA, so that the neutron density behind the sample material will not be zero.

An argument for the choice of the Maxwell-Boltzmann spectrum of the neutron beam is that by using neutrons with a Maxwell-Boltzmann spectrum moderation effects in the sample material are minimal and simpler models for neutron transport are likely to be applicable.

1.7 Outline of this thesis

As stated in Paragraph 1.5, some problems must be solved that arise when the sample size is enlarged from a few hundreds of milligrams to a few kilograms. Neutron self-shielding and gamma attenuation play a substantial role in the analysis of large samples. In Chapter 2 some examples are given about how in other research these problems are dealt with and why the research described in this thesis is necessary, and also different.

(25)

Note that in this thesis the term sample is used for the sample container and the sample material together. When sample material is used, only the sample material is meant, i.e. without the sample container.

To correct for neutron self-shielding and gamma attenuation, the neutron density inside the sample material must be determined together with the gamma attenuation of the sample material. For each volume element of the sample material, both corrections must be known. After integrating the correction factors over the volume elements, the correction factor of the entire sample material is known. In order to calculate the correction factors for the volume elements, the (local) sample parameters with respect to gamma absorption and neutron self-shielding must be determined.

When a small sample is irradiated isotropically, mainly Σa affects the neutron-density

profile, as in small sample INAA. But when a sample is irradiated unilaterally with a neutron beam, both Σs and Σa must be taken into account. LS-INAA is performed in an almost

isotropic neutron flux irradiating the entire sample, so the neutron behaviour can be approximated by diffusion theory. Since the use of a neutron beam is intended, the neutron transport phenomena could not be approximated with diffusion theory, and a new correction method was required.

Simulation methods, i.e. Monte Carlo codes, were used in the determination of the influence of the sample parameters on the neutron density. An introduction to Monte Carlo codes is given in Chapter 3. To determine whether the Monte Carlo codes could be used as research tools, a comparison is carried out between the irradiation of a large sample in an experimental setting and with a Monte Carlo simulation. This is described in Chapter 4.

For the neutron self-shielding Σs, Σa (Chapter 5) and the influence of the effective

mass of the atoms in the sample material (Me) on the neutron density must be known

(Chapter 6). A methodology to determine these sample parameters is developed and its boundary conditions are determined in these chapters.

For gamma attenuation, the total attenuation coefficient (µt) as a function of gamma

energy and sample composition must be known (Overwater et al., 1993). In LS-PGNAA higher gamma energies are dealt with than in INAA, so the methodology described by Overwater (1993) could not be used, since Overwater uses a transmission experiment with gamma energies up to 1408 keV from a radionuclide source. In Chapter 7 the problem of gamma attenuation is overcome using the approach of Moens (Moens et al., 1981), the interaction characteristics of high energy gammas and the XCOM database (Berger et al.,

(26)

1999). Boundary conditions are determined within which the method meets the criterion of accuracy.

In Chapter 8 all comes together. Both correction methods for neutron self-shielding and gamma attenuation were tested in a LS-PGNAA experiment carried out at JAERI, Japan. A Monte Carlo code is used to integrate the correction factors over the volume elements, and to calculate the correction factor of the entire sample material for selected prompt gamma energies.

In Chapter 9 the methodology is discussed. The aims and criteria from this chapter are used as guiding principles.

1.8 References

Alfassi, Z.B., Chung, C., Prompt gamma neutron activation analysis, CRC Press, Boca Raton, Florida, USA (1995).

Berger, M. J., Hubbel, J. H., Seltzer, S. M., Coursey, J. S., and Zucker, D. S., XCOM: Photon cross-section database, NIST Standard Reference Database 8, National Institute of Standards and Technology, Gaithersburg, USA (1999).

Blaauw, M., Lakmaker, O., van Aller, P., The accuracy of instrumental neutron activation analysis of kilogram-size inhomogeneous samples, Anal. Chem. 69, 2247-2250 (1997).

Bode, P., Lakmaker, O., van Aller, P., Feasibility studies of neutron activation analysis with kilogram-size samples, Fres. J. Anal. Chem. 360, 10-17 (1998).

Isenhour, T.L., Morrison, G.H., Modulation technique for neutron capture gamma ray measurements in activation analysis, Anal. Chem. 38, 162-167 (1966).

Lakmaker, O., Blaauw, M., The use of big sample INAA for inhomogeneous materials: Homogenizations no longer needed?, J. Radioanal. Nucl. Chem. 216, 69-74 (1997).

Lin, X., Henkelmann, R., Instrumental neutron activation analysis of large samples: A pilot experiment, J. Radioanal. Nucl. Chem. 251, 197-204 (2002).

Mackey, E.A., Anderson, D.L., Chen-Mayer, H., Downing, R.G., Greenberg, R.R., Lamaze, G.P., Lindstrom, R.M., Mildner, D.F.R., Paul, R.L., Use of neutron beams for chemical analysis at NIST, J. Radioanal. Nucl. Chem. 203, 413-427 (1996).

Matthews, I.P., Spyrou, N.M., Multielemental analysis of bulk matrices by measurement of prompt and delay gamma-rays as well as cyclic activation using an isotropic neutron source, Int. J. Appl. Radiat. Isot. 33, 61-68 (1982).

Moens, L., De Donder, J., Lin, X., De Corte, F., De Wispelaere, A., Simonits, A., Hoste, J., Calculation of the absolute peak efficiency of gamma-ray detectors for different counting geometries, Nucl. Instr. and Meth. 187, 451-472 (1981).

Overwater, R. M. W., The physics of big sample instrumental neutron activation analysis, PhD thesis, Technological University Delft, Delft, The Netherlands (1994).

Overwater, R.M.W., Bode, P., De Goeij, J.J.M., Gamma-ray spectroscopy of voluminous sources - corrections for source geometry and self-attenuation, Nucl. Instr. and Meth. A 324, 209-218 (1993).

Overwater, R.M.W., Hoogenboom, J.E., Accounting for the thermal flux depression in voluminous samples for the instrumental neutron activation analysis, Nucl. Sci. Eng. 117, 141-157 (1994).

Stamatelatos, I., Personal Communication, (2002).

Wormald, M.R., Clayton, C.G., In-situ analysis of coal by measurement of neutron-induced prompt gamma-rays, Int. J. Appl. Radiat. Isot. 34, 71-82 (1983).

(27)

Chapter 2 Prompt-gamma neutron-activation analysis

with voluminous samples

Abstract

An overview is given of aspects of research towards PGNAA with voluminous samples. First some aspects in bulk-sample PGNAA are discussed, where irradiations by isotropic neutron sources or neutron generators are used mostly for in-situ or on-line analysis. Since virtually all of this research is carried out in a comparative and/or qualitative way or by using a priori knowledge about the sample material, this research does not meet the criteria set in Chapter 1 for large-sample PGNAA. In the remainder of the chapter attention is paid to research with neutron beams for irradiation of large objects in a laboratory setting. Shortcomings of these methods are the need of a priori knowledge of the sample material or the assumption that the mass fractions of all determined elements add up to 1.

(28)

2.1 Bulk-sample PGNAA with isotropic sources or generators

In the field of PGNAA with voluminous samples isotropic neutron sources, e.g. 252_Cf, 239_{Pu/Am and}241_{Am/Be, as well as neutron generators, are used for bulk-sample analysis.}

These sources can be taken to the object to be analysed in situ. However, most of these sources yield a lower neutron flux than neutron beams. To obtain useful information from a PGNAA spectrum with a lower flux of neutrons the sample to analyse should be large and/or the concentrations of the elements to analyse not too low.

The average neutron energy of sources for bulk-sample analysis is usually higher than the (thermal) neutron energy of a neutron beam. To have a significant activation in (bulk) samples irradiated with isotropic neutron sources or neutron generators the neutrons have to be thermalised. Neutrons can be thermalised using for instance graphite or water between the neutron source and the bulk sample. This will of course reduce the neutron flux at the spot of the elements to analyse. Another solution is to have the neutrons thermalised in the bulk-sample material itself.

When a bulk sample is irradiated with a neutron source, analysis can take place based on the measurement of scattered neutrons, delayed gammas and/or prompt gammas. Since this thesis is about large-sample PGNAA, below the focus is on prompt gammas only.

In situ bulk-sample PGNAA can be used for a variety of purposes. Therefore a lot of research is done towards bulk-sample PGNAA and its applications. Because of the large extent of research it was not possible to give a short but complete view of its research and only some examples are indicated with some references for further reading. Examples of bulk-sample PGNAA are the determination of the quality of oil in an oil field (e.g. Yonezawa et al., 2001; Jonah et al., 1997; Tittle, 1989), the determination of the quality of coal in mining (Borsaru et al., 2001; Goodarzi and Swaine, 1994; Salgado and Oliveira, 1992), the detection of nitrogen-containing explosives for airport safety and demining (Farsoni and Mireshgi, 2001; Pesente et al., 2001) and the measurement of nitrogen in body (Chichester and Empey, 2004).

In situ bulk-sample PGNAA has some advantages over other analysis methods. Firstly, the object to analyse does not need to be taken to a laboratory, but information is available in situ, and often also on-line. Secondly, the object to analyse does not need to be homogenised, but can be analysed as a whole. Thirdly, minor element mass fractions can be measured with in situ bulk-sample PGNAA, due to the large volume of the object to analyse. And finally, when the method is used on-line, the signal ratios of different elements can be

(29)

determined easily and the variation in element mass fraction ratios can be determined in time, which is often the main goal of the analysis.

Although the neutron source and the object of study are not the same in large-sample analysis as in bulk-sample analysis, there are some similarities in the methodology. For example in both types of analysis the problems of neutron self-shielding and gamma attenuation have to be solved. However, there is a less pressing need to develop corrections for neutron self-shielding and gamma attenuation, when bulk-sample PGNAA is carried out in a comparative or semi-quantitative way.

When a quantitative determination is carried out in bulk-sample PGNAA the results are compared with a calibration curve. For this calibration curve, prompt-gamma spectra are recorded of calibration samples with known amounts of specific elements that will be analysed in sample materials. The shape of the sample to be analysed, and implicitly the matrix of the sample material to be analysed, has to be the same as the shape and the matrix of the sample material used to determine the calibration curve to be able to circumvent the problems of neutron self-shielding and gamma attenuation. Information about the sample material to be analysed has to be known in this method.

Because one of the aims in the development of a methodology for LS-PGNAA is that material is to be analysed quantitatively and without a priori information on the composition, the methods for bulk-sample PGNAA cannot be used for LS-PGNAA, as a whole or partially. Therefore in the remainder of the chapter the focus is on LS-PGNAA with neutron beams in a laboratory setting.

2.2 Large sample PGNAA with a neutron beam using practical

calibration methods

Sueki et al. (1996) performed LS-PGNAA on an earthen vessel about 15 cm in diameter, 10 cm in width and 0.5 cm in thickness. The dimensions of the neutron beam are 2x2 cm2, i.e. smaller than those of the vessel. The problems of neutron self-shielding and gamma attenuation were overcome using the internal mono-standard method. With this method, the ratio of photo peak areas for the detection of the gamma from sample element x and comparator element y can be written as:

(30)

( )

(

)

( )

(

)

( )

(

)

( )

(

)

x x γ,x n ,x n γ,x n x y y y γ,y n ,y n γ,y n

,

a a

N b E

E

dE d

A

N b E

E

dE d

ϕ

σ

ε

ϕ

σ

ε

=

∫∫

r

(2.1)

A = net area of the full-energy peak, corrected for dead time N = density of isotope of interest (m-3₎

σa = microscopic absorption cross section (m2)

b = number of gammas of interest emitted per absorbed neutron

ϕ = neutron flux (m-2_s-1₎

ε = full-energy peak detection efficiency

E = energy (MeV)

r = position of the neutron absorption (m)

Subscript x for sample element Subscript y for comparator element Subscript γ for gamma

Subscript n for neutron

On the first approximation the equation can be simplified by defining an effective microscopic absorption cross section σ’a, whose ratio σ’a,x/σ’a,y is expected to become

approximately constant at each irradiation facility. The product ϕσ can than be represented by w_x

( )

r σ'

( )

E_γ,x . The position vector of this product is w r_x

( )

, which is equivalent to the normalised density distribution. When the product ϕσ is integrated solely over the energy spectrum, Equation 2.2 results.

( )

(

)

( )

(

)

x a,x x γ,x x γ,x

x

y y a,y y γ,y y γ,y

'

,

'

,

N

b E

w

E

d

A

N

b E

w

E

d

σ

ε

σ

ε

=

∫

r

(2.2)

σ’a = effective microscopic absorption cross section (m2) w(r) = normalised density distribution (m-2_s-1₎

The relative full-energy peak efficiency

∫

w_x

( )

r ε

(

E_γ,x,r

)

dr has to be determined for each irradiation position of the sample against the incident neutron beams. The ratio

σ’a,xbx/σ’a,yby has to be determined for all elements of interest. Then the relative element mass

fractions (Nx/Ny) can be determined using Equation 2.2.

To determine absolute element mass fractions in earthen ware Sueki et al. assumed that all elements are oxides and that the oxides together constitute 100 % of the sample material. This may be generally true in the case of the earthen ware, if all elements as oxides can be determined. The internal mono-standard method cannot be used in a general methodology for LS-PGNAA, in which it cannot be assumed that the constituents determined of the sample material add up to 100 %, since not all elements can be detected, even by PGNAA and INAA combined and not all constituents will be oxidised elements.

(31)

The applicability of the internal mono-standard method in a combination of PGNAA and INAA is described by Nakahara (2000) and Oura (1999a). To determine absolute element mass fractions they used a comparator element inside the sample material of which the mass fraction is determined by another method than PGNAA. Archaeological earthen wares (Sueki et al., 1998) and bronze mirrors (Oura et al., 1999b) were analysed, as well as stony and iron meteorite samples (Latif et al., 1999; Oura et al., 2002) and organs of rats (Oura et al., 2000).

2.3 Large sample PGNAA with a neutron beam using simulations for

calibration

2.3.1 Monte Carlo library linear least-squares approach

Shyu et al. (1993) developed a computational method with which they could determine quantitatively element mass fractions from a LS-PGNAA spectrum. The method consists of five steps.

Generate the complete spectral response for a sample of an assumed composition by Monte Carlo simulation irradiated by a neutron beam. This implies that a Monte Carlo code exists that describes the physics and the experimental set-up accurately.

Within this Monte Carlo code, arrange to keep track of the individual spectral response for each isotope within the sample to provide the necessary library gamma spectra of each isotope. With this step the problem of neutron self-shielding and gamma attenuation is taken into account, which is non-linear in the specific isotope fractions.

Repeat step 1 and step 2 for other compositions with a composition like the one in step 1, but each time having a small change in the relative amount of a isotope.

Use the linear least-squares (LLS) method to obtain the elemental amounts in any unknown sample or samples for which the complete spectral response has been measured experimentally. The different library spectra from step 2 and step 3 are fitted to the experimental measured PGNAA spectrum.

If the calculated elemental amounts for the sample are not close enough to those originally assumed for the Monte Carlo calculation so that a linear relation exists, then another iteration of steps 1 through 4 is necessary by starting with an assumed sample composition closer to that of the unknown sample.

The LLS-method applied is a linear method solving the problem due to neutron self-shielding and gamma attenuation in the sample, which is non-linear. The problem is circumvented in step 2, but only when the assumed values are close to the ‘real’ values,

(32)

because then the problem can be approached as a linear problem, since the difference in neutron self-shielding and gamma attenuation is small between the ‘assumed’ sample and the ‘real’ sample.

When the ‘assumed’ values are not close enough to the ‘real’ values a ‘local’ optimum is reached with the LLS-method in step 5. To determine whether a ‘local’ or a ‘real’ optimum is reached a large change in the relative amounts of a few elements has to be given. The “local” optimum found is worse than the “real” optimum.

The sensitivity of the library spectra is studied by Shyu et al. (1993). They found that a small change in the amount of hydrogen is largely influencing the detector response for other elements. Hydrogen has a large microscopic scattering cross section and will therefore change the neutron-density distribution inside the sample material substantially, resulting in other sensitivities for all elements. To overcome this problem a guideline is given for the use of the LLS-method: the amount of hydrogen in the sample material to analyse has to be known prior to the PGNAA analysis or has to be below a certain value.

Some other elements cannot be determined sensitively, since a small change in the amount of that element hardly changes the detector response, e.g. magnesium and oxygen. These elements have isotopes with small microscopic cross sections and will hardly change the neutron density in the sample material. Therefore the precise mass fractions of these elements cannot be determined with the LLS-method, so they have to be determined otherwise.

Shortcomings of the LLS-method are the prior determination of isotopes that largely influence the library spectra of isotopes and the problems with the isotopes that hardly influence the library spectra of other isotopes. Since one of the aims in the methodology for LS-PGNAA is the ability to analyse sample materials with an unknown matrix, the LLS-method cannot be applied in the methodology for LS-PGNAA.

2.3.2 Fixed-point iteration method

A general formulation of the non-linear problem of determining the composition of a large homogeneous sample from a prompt-gamma measurement is given by Akkurt et al. (2001). From this formulation an associated iterative method is developed that solves the problem using a minimal number of gamma measurements. With a neutron-gamma transport code it is tried to find the composition of a sample material from the measured photo-peak areas. Required data are the measured ratios of full-energy peak areas, sample density, the geometry

(33)

of the experimental set-up, the neutron-source spectrum, the efficiency of the detector and the cross-section data. The two formulas solved with a forward iteration are:

( )

1 m c N m c 1 j i j i i j i j

A

w

A A

− =





= 









∑



w

(2.3) N

1

i i

w

=

∑

(2.4) w = mass fraction

A = net area of the full-energy peak, corrected for dead time N = total number of isotopes in the sample material

superscript m = measured in an experiment superscript c = calculated with transport model index i,j = specific isotope

The fixed point iteration uses the solution found as a starting point for the next iteration, until the imprecision is smaller than wanted, between the nth_{solution and the (}_n+1)th

solution. For the first iteration an assumption has to be made about the constituents in the sample material.

The difference between the method described in Paragraph 2.3.1 and this method is that in Paragraph 2.3.1 the contribution of single elements to the spectral response is taken into account in the LLS-method, while the method described in this paragraph deals with all elements together, most of them with two or more isotopes, using the fixed point iteration method. The method described in Paragraph 2.3.1 only gives a good solution if the ‘assumed’ values are close to the ‘real’ values, because it is a linear method; the fixed point iteration method gives also good solutions if the ‘assumed’ values are not close to the ‘real’ values, because it can deal better with neutron self-shielding and gamma attenuation.

The fixed-point iteration method uses a combination of both measurement and simulation (Holloway and Akkurt, 2000), just like the LLS-method described in Paragraph 2.3.1. The idea of combining measurements and simulation will be used in the development of a methodology for LS-PGNAA. However, since Akkurt et al. (2001) used a boundary condition that the mass fractions of all elements determined in the sample material add up to 1, their complete methodology was not used.

(34)

2.4 Neutron self-shielding and gamma attenuation

Some problems must be solved that arise when the sample size is considerably larger than a few hundreds of milligrams, up to a few kilograms. Neutron self-shielding and gamma attenuation play a substantial role in the analysis of such voluminous samples. Above some examples are given about how others tried to argument or solve the problem of neutron self-shielding and gamma attenuation. Most of them used either prior knowledge, a fixed composition matrix or defined the constituents determined to be 100 % of the mass of the sample material.

In the development of a methodology for LS-PGNAA one of the aims is not to use a priori information. The sample materials for future analysis will have different matrix compositions. Finally, it cannot be assumed in the methodology for LS-PGNAA, that the constituents of the sample material add up to 100 %, since not all elements can be detected, even by PGNAA and INAA combined.

The main task in the methodology to be developed for LS-PGNAA is that the problem of neutron self-shielding and gamma attenuation has to be solved. The neutron-density distribution outside the sample is studied as a function of the macroscopic scattering cross section (Σs), the macroscopic absorption cross section (Σa) and the effective atomic mass (Me).

The relations found in this study were inverted to derive the characteristics of the sample material as function of the neutron density outside the sample. Using these characteristics the neutron-density distribution inside the sample is calculated, solving the problem of neutron self-shielding (Chapter 5 and 6).

The problem of gamma attenuation is solved by assuming that the gamma-attenuation coefficient as a function of energy is roughly the same for a large range of sample materials when the average atomic number and the density of the sample material are within certain boundaries. The detector efficiency is calculated according to the assumptions of Moens et al. (1981). These state that for any counting geometry the detection efficiency can be determined from a measured detection efficiency in another geometry (the “reference” geometry) and the calculated effective solid angles for the two geometries (Chapter 7).

(35)

2.5 References

Akkurt, H., Holloway, J.P., Smith, L.E., A fixed point iteration for large sample prompt gamma activation analysis, Transactions of the American Nuclear Society 85, 438-440 (2001).

Borsaru, M., Biggs, M., Nichols, W., Bos, F., The application of prompt-gamma neutron activation analysis to borehole logging for coal, Appl. Radiat. Isot. 54, 335-343 (2001).

Chichester, D.L., Empey, E., Measurement of nitrogen in the body using a commercial PGNAA system-phantom experiments, Appl. Radiat. Isot. 60, 55-61 (2004).

Farsoni, A.T., Mireshgi, S.A., Design and evaluation of a TNA explosive-detection system to screen carry-on luggage, J. Radioanal. Nucl. Chem. 248, 695-697 (2001).

Goodarzi, F., Swaine, D.J., The influence of geological factors on the concentration of boron in Australian and Canadian coals, Chem. Geol. 118, 301-318 (1994).

Holloway, J.P., Akkurt, H., Some aspects of the mathematical modeling of prompt gamma neutron activation analysis, Proceedings of the PHYSOR 2000, (2000).

Jonah, S.A., El-Megrab, A.M., Varadi, M., Csikai, J., An improved neutron reflection setup for the determination of H and (O+C)/H in oil samples, J. Radioanal. Nucl. Chem. 218, 193-195 (1997).

Latif, S., Oura, Y., Ebihara, M., Kallemeyn, G.W., Nakahara, H., Yonezawa, C., Matsue, T., Sawahata, H., Prompt gamma-ray analysis (PGA) of meteorite samples, with emphasis on the determination of Si, J. Radioanal. Nucl. Chem. 239, 577-580 (1999).

Moens, L., De Donder, J., Lin, X., De Corte, F., De Wispelaere, A., Simonits, A., Hoste, J., Calculation of the absolute peak efficiency of gamma-ray detectors for different counting geometries, Nucl. Instr. and Meth. 187, 451-472 (1981).

Nakahara, H., Oura, Y., Sueki, K., Ebihara, M., Sato, W., Latif, S.A., Tomizawa, T., Enomoto, S., Yonezawa, C., Ito, Y., Some basic studies on non-destructive elemental analysis of bulky samples by PGAA, J. Radioanal. Nucl. Chem. 244, 405-411 (2000).

Oura, Y., Ebihara, M., Yoneda, S., Nakamura, N., Chemical composition of the Kobe meteorite; Neutron-induced prompt gamma ray analysis study, Geochem. J. 36, 295-307 (2002).

Oura, Y., Enomoto, S., Nakahara, H., Matsue, H., Yonezawa, C., Prompt gamma-ray analysis of rats, J. Radioanal. Nucl. Chem. 244, 311-315 (2000).

Oura, Y., Nakahara, H., Sueki, K., Sato, W., Saito, A., Tomizawa, T., Nishikawa, T., Completely non-destructive elemental analysis of bulky samples by PGAA, Czech. J. Phys. 49, 311-321 (1999a).

Oura, Y., Saito, A., Sueki, K., Nakahara, H., Tomizawa, T., Nishikawa, T., Yonezawa, C., Matsue, H., Sawahata, H., Prompt gamma-ray analysis of archaeological bronze, J. Radioanal. Nucl. Chem. 239, 581-585 (1999b).

Pesente, S., Cinausero, M., Fabris, D., Fioretto, E., Lunardon, M., Nebbia, G., Prete, G., Viesti, G., Effects of soil moisture on the detection of buried explosives by radiative neutron capture, Nucl. Instr. and Meth. A 459, 577-580 (2001).

Salgado, J., Oliveira, C., Corrections for volume hydrogen content in coal analysis by prompt gamma neutron activation analysis, Nucl. Instr. and Meth. B 66, 465-469 (1992).

Shyu, C.M., Gardner, R.P., Verghese, K., Development of the Monte Carlo-library least-square method of analysis for neutron capture prompt gamma-ray analyzers, Nucl. Geophys. 7, 241-267 (1993).

Sueki, K., Kobayashi, K., Sato, W., Nakahara, H., Tomizawa, T., Nondestructive determination of major elements in a large sample by prompt gamma-ray neutron activation analysis, Anal. Chem. 68, 2203-2209 (1996).

Sueki, K., Oura, Y., Sato, W., Nakahara, H., Tomizawa, T., Analysis of archaelogical samples by the internal monostandard method of PGAA, J. Radioanal. Nucl. Chem. 234, 27-31 (1998).

Tittle, C.W., A history of nuclear well logging in the oil industry, Nucl. Geophys. 3, 75-85 (1989).

Yonezawa, C., Matsue, H., McKay, K., Povinec, P., Analysis of marine samples by neutron-induced prompt gamma-ray technique and ICP-MS, J. Radioanal. Nucl. Chem. 248, 719-725 (2001).

(36)

(37)

Chapter 3 Monte Carlo simulation methods in neutron

and gamma transport calculations

*

Abstract

A brief introduction is given about what Monte Carlo methods in general do in the field of neutron and gamma transport and which specific methods are used in the research described in this thesis. The methods used for neutron-transport calculations are described more in detail. It is made clear why a certain code is used in which chapter. A comparison experiment was described between the neutron-transport codes used, with emphasis on the description of the Maxwell-Boltzmann velocity distribution.

*_{Part of this chapter was published as: Degenaar, I.H., Blaauw, M., The neutron energy distribution to use in} Monte Carlo modeling of neutron capture in thermal neutron beams, Nucl. Instr. and Meth. B 207, 131-135 (2003).

Towards a methodology for large-sample prompt-gamma neutron-activation analysis

prompt-gamma neutron-activation analysis

Table of contents

List of symbols and abbreviations

Greek symbols

Latin symbols

Abbreviations

Chapter 1

Introduction

Abstract

1.1 Introduction

1.2 Physical principles

(

)

N

Σ

=

∑

σ

(

)

N

Σ

=

∑

σ

Σ

=

Σ

+

Σ

(

)

(

)

( , ,t )

n , ,t

ϕ

r v

≡

v

r v

( )

(

)

,

,t dt

Φ

r v

≡

∫

ϕ

r,v

( )

(

)

( ) (

)

R , ,t dr

r v

≡

Σ

v

ϕ

r v

, ,t dr

( )

( )

x

x

ϕ

Σ ϕ

∂

= −

∂

( )

x

x

ϕ

µ ϕ

_{Σ ϕ}

_{µ ϕ}

_{= −}

_x

_e

₌

_x

_e

₁

₁

_e

_e

_e

_e

_e

_e

_{f C ,C}