Lima J.J.Pedroso, de (ed.). Nuclear Medicine Physics

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78 Nuclear Medicine Physics

3.2.3 Diffusion and Mobility

After reaching thermal equilibrium inside the medium, the positron behaves

as a quantum-free particle and its movement can be described through the

diffusion equation Boltzmann equation. For thermal energies, the phonon

dispersion is the dominant process in solids and the diffusion coefﬁcient, as

calculated using the relaxation time approximation; and the Nernst–Einstein

relation is given by

T =

∗

rel

, (3.8)

where m* is the effective positron mass, k

the Boltzmann constant, T the

temperature, e the positron charge, and τ

rel

the relaxation time, which is the

inverse of the dispersion rate.

The mobility and the average free path of the positron are, respectively,

given by

eτ

rel

∗

, (3.9)

and,



∗



1/2

rel

. (3.10)

It has been experimentally conﬁrmed through the observation of a T

−1/2

dependence on the diffusion coefﬁcient [26] that, in metals and semicon-

ductors, the dominant dispersion process is due to acoustical phonons,

leading to

= D



300 K



1/2

. (3.11)

The values of the positron diffusion coefﬁcient in metals and semicon-

ductors at 300 K is within the interval D

= 1–3 cm

−1

. Combining these

values of the diffusion coefﬁcient with the typical positron lifetime values

(τ

≈ 100 −200 ps), the calculated average diffusion length of the positron is



≈ 100 −200 nm. (3.12)

These values of the positron diffusion length are, usually, much larger

than the average free path (Equation 3.10), l

≈ 5 −10 nm, and the Broglie

wavelength, λ

term

= 6.2(300 K/T)

1/2

nm, of the thermal positrons, which are

the main reasons that justify the application of the diffusion theory to

describe the motion, after thermalization, of the positron inside metals and

semiconductors.

Positron Physics 79

In liquids, the positron diffusion length is also of the same order of mag-

nitude and is still much greater than interatomic distances, so that diffusion

theory also remains valid for these materials.

In summary, the thermalized positrons in condensed matter can be consid-

ered as waves, and the typical values of the characteristic lengths are within

the following scale:

1/α ≥ 50 μm > L

≈ 100 nm > λ

term

≈ 10 nm > l

≈ 5 nm. (3.13)

Fromthe analysis of these values, the followingconsiderations canbe made:

• In the so-called conventional positron systems, when positrons emit-

ted from radioactive sources are used, the implications of positron

diffusion are not so important, as the implanted positrons are

distributed through a large region (L

• In the variable energy positron systems, and particularly in the low

energy region (≤10 keV), where the average implantation depth is

small and the distribution is sharp, the motion of the positron after

thermalization has to be considered and, in this case, is well described

by the diffusion theory.

3.2.4 Annihilation

The positron annihilates preferentially after thermal equilibrium with the

medium has been reached. Under this condition, the positron wave function

in a solid can be calculated through the Schrödinger equation:

−



∇

(



r) + V(



r)Ψ

(



r) = EΨ

(



r). (3.14)

In a perfect periodic lattice, the wave functions of thermalized positrons are

described by delocalized states, that is, Bloch wave functions. The potential

has two terms:



r) = V

Coul

(



r) + V

corr

(



r), (3.15)

where the ﬁrst term represents the Coulomb electrostatic potential and the

second term takes into consideration the electron–positron correlation effects

within the local density approximation.

Due to the strong Coulomb repulsion of positive ions inside the solid, the

positronwave function is concentrated in the interstitial space between atoms.

The positron wave function changes drastically in the neighborhood of a

lattice site where the crystal periodicity has a gap, for example, due to the

presence of one vacancy in the lattice (one atom is missing): this can be easily

generalized to more complex defects. If an atom is removed from the periodic

80 Nuclear Medicine Physics

crystal lattice, the local electrostatic potential changes and the local electronic

density decreases, as the core electrons are missing. The delocalized electrons

of the solid that are present in the lattice vacancy create a negative charge in

thisregion, which means an attractive potential for the positron.This potential

can be strong enough to create a positron bound state (localized state) in the

lattice vacancy site. The positron transition to localized states is referred to as

trapping, and the defects are called trapping centers. In principle, positrons are

trapped by any type of lattice defect that has an attractive electronic potential.

In metals and semiconductors, the more general cases are open-volume-type

defects: vacancies, vacancy aggregates, impurity–vacancy complexes, dislo-

cations, grain boundaries, cavities, interfaces, surfaces, etc. Impurities and

precipitates can also give rise to attractive potentials, in this way forming

trapping centers to the positron. The trapping rate and the degree of local-

ization of the positron are strongly dependent on the defect type and on its

physical properties.

The high values of trapping rates and the dependence of the annihilation

gamma ray characteristics on the defect type where the positron is trapped

are the main reasons for the high sensitivity and selectivity of positron anni-

hilation techniques in the characterization of defects in condensed matter. As

examples, single vacancy concentrations in the range 0.1–200 at. ppm can be

detected by this technique, whereas cavities with radius of some micrometers

can also be observed at concentrations as low as 0.02 at. ppm. The dependence

of both the positron trapping and the annihilation characteristics on the size of

the defects, the charge states, and electronic conﬁgurations of defects makes

possible, in most cases, the identiﬁcation of the type and concentration of the

defects present in the matter. These sensitivity and selectivity properties have

already awarded positron annihilation techniques an important role in defect

detectionin condensed matter studies. The technique is nowadaysestablished

as positron annihilation spectroscopy.

In the majority of cases, as was previously indicated, the electron–positron

pair annihilates mostly through the emission of 2γ photons. The study of the

positron annihilation characteristics starts with the calculation of the proba-

bility per unit time, Γ(



p), that an electron–positron pair annihilates through

the emission of a photon pair with total angular momentum



p = 



Equation 3.2 provides a ﬁrst step to calculate this probability. However, in

thisequation, neither the interaction ofthe positronwiththe electronsfromthe

medium nor the momentum of the pair involved in the annihilation is taken

into consideration. One frequently used model is the independent particle

model where an n particle system is described by n systems of individual

particles, with each of them seeing the average ﬁeld created by all the other

particles. The Coulomb attraction between the electron and positron (as well

as the correlations between all the electrons of the system), which implies

an increase of the annihilation rate due to the higher electronic density in the

neighborhoodof the positron,is taken into account separately by the inclusion

of an additional term.After these approximations, the probability density per

Positron Physics 81

unit time for the annihilation of the electron–positron pair by two photons of

momentum



p = 



k, usually referred to as momentum density, is given by

2γ

(



p) =

πr

(2π)



i,j

−







r ·e

−i



p·



(



r)Ψ

−

(





[



]



, (3.16)

where r

is the Bohr electron radius, c is the speed of light in vacuum, Ψ

(



and Ψ

−

(



r) are the wave functions of the positron and the electron, respec-

tively; n

and n

−

are the distribution functions describing the occupation

of the positron and electron states, respectively; and γ[n(



r)] represents the

electronic density ampliﬁcation factor at the annihilation site. The summa-

tion includes all the states occupied by the positron and the electron. For

valence electrons in a crystalline solid, n

−

is the Fermi–Dirac distribution

function. In the positron system generally used, n

is, to a good approxi-

mation, described by a delta function, as only one positron with negligible

kinetic energy (thermalized positron) is present in the sample at a particular

time. The ampliﬁcation factor, γ[n(



r)], describes the effect of the positron–

electron attractive force and of the resulting electron cloud accumulated

around the positron. This has the effect of changing the potential seen by

the positron and, consequently, the wave functions of both positron and elec-

tron. The effect of the higher electron density around the positron, due to

the electron cloud, is to produce an increase of the annihilation probabil-

ity and, obviously, a decrease in the positron lifetime. Different expressions

for γ[n(



r)] based on many body calculations can be found in literature

[27,28]. Naturally, electron–positron correlations are different for metals and

semiconductors [29].

Equation 3.16 shows that positron annihilation characteristics in matter are

stronglyrelated to the electronic state at the site where the annihilation occurs.

Since, in the majority of cases, the positron annihilates from thermalized

states, the total momentum of the process contains information of the momen-

tum of the electron at the annihilation site. The experimental measurement of

the electronic density and of the electronic momentum distribution obtained

from the information transported by the annihilation gamma photons is then

possible.

Using the expression (Equation 3.16) for the momentum density, one

can calculate the measurable physical quantities of positron annihilation

experiments:

•

The integration of ρ

2γ

(



p) over a single momentum component gives

the two-dimensional angular distribution of the annihilation gamma

rays emitted, the so-called 2D-ACAR (two-dimensional angular

correlation of the annihilation radiation):

N(θ, ϕ) ∝ N(p

) ∝



2γ

(



p) dp

; (3.17)

82 Nuclear Medicine Physics

• The integration of ρ(



p) over two momentum components gives the

one-dimensional angular distribution, the so-called 1D-ACAR, and

the broadening of the annihilation radiation energy distribution due

to the Doppler effect:

N(θ) ∝ N(E) ∝



2γ

(



p) dp

; (3.18)

• The integration of ρ(



p) over all three momentum components gives

the annihilation probability, per unit time, of a photon pair of arbi-

trary total momentum. This value corresponds to the inverse of the

positron lifetime, τ:

−1

= λ ≡ Γ =



2γ

(



p) dp

. (3.19)

The positron lifetime is, therefore, inversely proportional to the electronic

density sampled by the positron at the annihilation site. For a delocalized

positron, its lifetime reﬂects the average electron density in the interstitial

space between atoms. For a localized positron, the annihilation can occur

from different localized states, for each state i, the positron experiences a

particular value of the electronic density corresponding to a particular value

of the lifetime, τ

= 1

≡ 1

These three measurable observables constitute the basic quantities for the

positron spectroscopy in material studies, using the techniques

• 2D-ACAR

• 1D-ACAR and Doppler broadening measurements

• Lifetime measurements

In the media where Ps formation can take place (e.g., in liquids, polymers,

porous materials, living tissues, and organs), it is mainly the information

related to this particle (lifetime and intensity) that can lead to useful infor-

mation concerning the chemical surrounding at the Ps trapping sites. In

particular, in polymers and porous materials, the Ps behavior constitutes

a powerful way of obtaining information, for example, on porosimetry

(measurements of dimensions and size distribution of porous and testing

connectivity between porous) and on surface properties of porous materials

including thin ﬁlms. The Ps formed inside the bulk of polymeric or porous

materials may reach, by diffusion, the surface of an open volume or porous

and get trapped there. Inside the porous material, the Ps loses its initial kinetic

energy, of some eV, through collisions with the cavity wall. Since it has a nega-

tive work function, the Ps, after losing its energy excess, cannot go back to the

bulk material and stays localized, that is, trapped inside the porous material.

In this state, the Ps can have different interactions:

Positron Physics 83

1. In addition to self annihilation, it can annihilate with one porous

surface electron by a pick-off process. The pick-off annihilation rate

is proportional to the probability of Ps surface overlap. In this way,

the Ps lifetime decreases when the porous dimension is reduced. Due

to this lifetime dependence on the dimensions of the porous, the Ps

lifetimemeasurementsare,indeed,a veryimportant technique forthe

measurements of average porous dimensions and size distributions

of porous materials with dimensions from some tenths until a few

tens of nm (mainly in the range 0.2–50 nm).

2. If the porous material is partially or completely occupied with liquid

or gas molecules, the Ps annihilation rate by the pick-off process with

electrons of the molecules ﬁlling the porous, a signiﬁcant reduction of

the Ps lifetime is then observed, and the value obtained may be used

to identify chemical adsorption or molecular occupation of the pores.

3. If there are negative ions or free electrons inside the porous mate-

rial, Ps may be involved in chemical reactions or spin conversion

processes, respectively.

These situations can quite often be differentiated, which brings to Ps spec-

troscopy unique characteristics in surface chemistry, especially in catalysis

investigations.

3.3 Fundamentals of the Experimental Techniques

The different positron techniques mentioned above are based on the anal-

ysis of the annihilation radiation due to the mass energy transformation

of the electron–positron pair, which, in general, produces 2γ photons, each

with energy approximately 511 keV. In special cases, a bound state electron–

positron is formed, the so-called Ps atom and, in this case, the transformation

can occur with the emission of either two or three photons, as previously

discussed in Section 3.1.3.

The basic principles of each technique are shown, schematically, in

Figure 3.3. These can be classiﬁed in two main groups that can be distin-

guished either by the positron sensitivity to the electronic density (positron

lifetime measurements) or to the distribution of the electron momentum

(Doppler broadening and ACAR measurements) in the sample to be studied.

Positrons from a radioactive source (e.g.,

Na) enter the sample, ther-

malize in a few picoseconds, and diffuse through the medium before ﬁnal

annihilation. The maximum implantation depth is of the order 100 μm, which

is much larger than the typical value of the positron diffusion length (of the

order 100 nm). The positron lifetime can be measured as the temporal differ-

ence, Δt, between the emission of the 1.27 MeV photon, which corresponds

84 Nuclear Medicine Physics

ermalisation

(10

–12

~ 100 μm

Δt

Birth γ-ray

1.27 MeV

Annihilation γ-rays

0.511 MeV

Diﬀusion

≈ 100 nm

x, y

= 

x, y

Positron lifetime

Angular correlation

Sample

Doppler broadening

0.511 MeV ± ΔE, ΔE = p

c/2

FIGURE 3.3

Schematic representation of the different experimental techniques in positron studies.

to the “birth” of the positron and the 511 keV annihilation photon, which

corresponds to its “death.” The Doppler broadening technique records the

broadening ΔE of the 511 keV line. ΔE can be calculated through the electron

momentum component along the propagation direction, p

, ΔE = p

c/2. The

angular deviation Θ is measured by the angular correlation of the annihila-

tion radiation method. The angular deviation in the x–y plane, Θ

x,y

is related

to the components p

x,y

of the electron momentum, with Θ

x,y

= p

x,y

/(m

c),

where m

is the electron rest mass and c is the light velocity in vacuum.

3.3.1 Positron Lifetime

The probability that a positron annihilates in the time interval between t and

t +dt is given by

p(t) dt =

[1 −e

−t

]dt =

−t

dt, (3.20)

wheree

−t

represents the probability that in the instant t, after its production,

the positron exists in a state with average lifetime τ.

In some cases, the positron can exist in different possible states, each with a

characteristic lifetime, τ

, and a probability, I

, to be in that state. The lifetime

spectrum is then a weighted average (with the respective probabilities) of the

Positron Physics 85

different exponential terms:

P(t) =



−t

, (3.21)

and, this way, is equivalent to the absolute value of the time derivative of the

probability that the positron exists at the instant t, D(t):

D(t) =



−t

. (3.22)

In an experimental lifetime system, the measured spectrum, N(t),isa

convolution of P(t) with the time resolution of the system, R(t):

N(t) =

+∞



−∞

P(t −t



) R(t



) dt



. (3.23)

From this expression, and knowing the resolution function of the spec-

trometer, R(t), it is possible to obtain, experimentally, the τ

and I

values

by performing a numerical ﬁt of Equation 3.23 to the measured time spec-

trum. However, it should be noted that, due to the correlations between the

parameters to be adjusted, decomposition up to two components (i = 2) is

only possible when the lifetimes are relatively short (100–200 ps), as in the

case of metals and semiconductors; and up to four components when the

lifetimes present in the material being studied are longer (>1 ns), as in the

case of materials where Ps formation can occur (as in polymers). Even so, in

metals and semiconductors, the unambiguous identiﬁcation of two lifetimes

in experimental time spectra is only possible if τ

/τ

> 1.5.

Since the average lifetime is calculated through the weighted average of the

observed lifetimes,

¯τ =



i=1

, (3.24)

is then a parameter that can be obtained with high statistical precision (vari-

ations of 1–2 ps may be observed) and related to the time spectrum centrum;

variations in this parameter of the order of a few picoseconds can be consid-

ered as a reasonably strong indication that some change has occurred in the

region seen by the positron (e.g., a new type of defect has appeared in the

material under study).

The experimental measurement of the average positron lifetime can be

made if we obtain good precision the instants the positron enters the material

and it annihilates and if we measure the time gap between these two events.

The signal corresponding to the annihilation can easily be obtained through

86 Nuclear Medicine Physics

2.602 years

3.7 ps

(0.1 %)

(90.4 %)

EC (9.5 %)

(1.274 MeV)

FIGURE 3.4

Schematic diagram of the

Na radioactive decay.

the detection of one of the two 511 keV emitted in the positron annihilation.

The signal corresponding to the instant the positron enters the material can

be obtained in different ways depending on the type of experimental system

in use.

In the so-called “conventional systems,” that is, those using the positrons

directly from a radioactive source, such as

Na, the signal corresponding to

the positron emission is obtained by the detection of the 1.27 MeV gamma

photon due to the decay of the excited state in

Ne, which was populated by

the

Na positron emission (see Figure 3.4 for the schematic diagram of the

Na radioactive decay).

In the pulsed beam positron systems, the signal indicating the positron

arrival to the sample can be electronically generated by the beam control

system.

In Figure 3.5a, a system for positron lifetime measurements is schemat-

ically represented; and in Figure 3.5b, typical positron lifetime spectra are

compared.

100

1000

Counts

10000

Detect.1

(a) (b)

Detect.2

Clock

Memory

Δt

–1

Time (ns)

Silicon

Silicon with vacancies

345

FIGURE 3.5

(a) Block diagram of a system for positron lifetime measurements. (b) Simulated lifetime spectra

of positrons in crystalline Si and in Si with vacancy-type defects.

Positron Physics 87

3.3.2 Momentum Distribution

In the annihilation process of the positron–electron pair, the energy and

momentum conservation laws imply that, in the mass center reference frame,

the 2γ photons are emitted in exactly opposite directions with energy 511 keV

per gamma. However, in the laboratory frame, this pair has a nonvanishing

momentum that has to be transported by the annihilation photons. As a con-

sequence, a deviation of colinearity is observed, with a value θ, and also a

Doppler shift of the 2γ photons in relation to the energy m

A geometric interpretation of the momentum conservation law in the

positron–electron annihilation as viewed in the laboratory frame is repre-

sented in Figure 3.6, where



p represents the momentum of the positron–electron pair at the instant

of annihilation.



and



represent the momentum of the annihilation photons.



and



represent the transversal and longitudinal components of



p in

relation to the direction taken as reference.

The energy and linear momentum conservation imply that

c +p

c = 2m

(3.25)

and



p, (3.26)

where m

is the rest mass of the electron and of the positron.

The angular shift from the colinearity θ depends on the magnitude of



, whereas the magnitude of



determines the energy shift (Doppler

broadening) of the photons.

Since the kinetic energy of the annihilating pair is a few eV, a simple

calculation shows that, in the laboratory frame, the colinearity shift is

∼

(3.27)

•

−

FIGURE 3.6

Vector diagram of the momentum conservation law in the annihilation of the positron–electron

pair as viewed in the laboratory frame.