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

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68 Nuclear Medicine Physics

59. R. Wideröe. Über ein neues prinzip zur herstellung hoher spannungen. Arch. f.

Elektotech. 21, 387, 1929.

60. E.O. Lawrence, N.E. Edlefsen. On the production of high speed protons. Science

72, 376, 1930.

61. M.S. Livingston. The production of high-velocity hydrogen ions without the use

of high voltages. Ph.D. thesis, University of California, 1931.

62. E.O. Lawrence, M.S. Livingston. The production of high speed protons without

the use of high voltages. Phys. Rev. 38, 834, 1931.

63. E.O. Lawrence, M.S. Livingston. The production of high speed light ions without

the use of high voltages. Phys. Rev. 40, 19, 1932.

64. R.R. Wilson. Magnetic and electrostatic focusing in the cyclotron. Phys. Rev.

53,

408, 1938.

65. D.W. Kerst, R. Serber. Electronic orbits in the induction accelerator. Phys. Rev. 60,

53, 1941.

66. M.S. Livingston. The cyclotron. J. Appl. Phys. 15, 2, 1944.

67. E.O. Lawrence, D. Cooksey. On the apparatus for the multiple acceleration of

light ions to high speeds. Phys. Rev. 50, 1131, 1936.

68. P. Debye. Nährungsformeln für die zylinderfunktionen für große werte des argu-

ments und unbeschränkt veränderliche werte des index. Math. Ann. 67, 535,

1909.

69. M.E. Rose. Focusing and maximum energy of ions in the cyclotron. Phys. Rev. 53,

392, 1938.

70. B.L. Cohen. The theory of the ﬁxed frequency cyclotron. Rev. Sci. Instr. 24, 589,

1953.

71. J.J. Livingood. Principles of Cyclic Particle Accelerators. D. Van Nostrand Company,

New York, 1961.

72. E.M. McMillan. The synchrotron—A proposed high energy particle accelerator.

Phys. Rev. 68, 143, 1945.

73. J.R. Richardson, K.R. MacKenzie, E.J. Lofgren, B.T. Wright. Frequency modulated

cyclotron. Phys. Rev. 69, 669, 1946.

74. D. Bohm, L. Foldy. Theory of the synchro-cyclotron. Phys. Rev. 72, 649, 1947.

75. L.H. Thomas. The paths of ions in the cyclotron. Phys. Rev. 54, 580, 1938.

76. E.D. Courant, M.S. Livingston, H.S. Snyder. The strong focusing synchrotron—a

new high energy accelerator. Phys. Rev. 88, 1190, 1952.

77. L.J. Laslett. Fixed-ﬁeld alternating-gradient accelerators. Science 124, 781, 1956.

78. M.S. Livingston, J.P. Blewett. Particle Accelerators. McGraw-Hill, New York, 1962.

Positron Physics

Adriano Pedroso de Lima and Paulo M. Gordo

CONTENTS

3.1 Positron and Positronium: General Remarks ................. 69

3.1.1 History and General Concepts ....................... 69

3.1.2 Properties of Symmetry ............................ 71

3.1.3 Positronium .................................... 72

3.2 Interaction of Positrons with Matter ........................ 74

3.2.1 Energy Loss and Thermalization ..................... 74

3.2.2 Range and Implantation Proﬁle ...................... 76

3.2.3 Diffusion and Mobility ............................ 78

3.2.4 Annihilation .................................... 79

3.3 Fundamentals of the Experimental Techniques ................ 83

3.3.1 Positron Lifetime ................................ 84

3.3.2 Momentum Distribution ........................... 87

References .............................................. 93

3.1 Positron and Positronium: General Remarks

In this chapter,the different processes involving the interaction of the positron

with matter and the way that information about the annihilation site can be

transported by the emitted radiation are brieﬂy described. The experimental

setups used to extract this information will also be discussed.

At present, the medical applications using positrons are mainly concen-

trated in positron emission tomography (PET), where the annihilation and

subsequent detection of emitted radiation are used to deﬁne tissue or organ

images, as described in detail in Chapter 6. However, we will show that the

possibility of combining information on the chemical environment at the

position where the annihilation takes place may offer new and important

opportunities for applications in clinical diagnostics.

3.1.1 History and General Concepts

The history of the positron starts in 1928 with Dirac’s prediction of the exis-

tence of the antiparticle to the electron [1]. The ﬁrst experimental evidence

70 Nuclear Medicine Physics

of the existence of the positron was reported by Anderson in 1932 through

cosmic ray pictures taken in a cloud chamber [2]. Studies based on the annihi-

lation of the positrons with the electrons in matter were initiated in 1949 after

the discovery by DeBenedetti et al. [3] that, when the positron annihilates in

solid matter, the two annihilation gamma rays are not exactly collinear. This

was interpreted as resulting from the effect of the momentum of the electron

taking part in the annihilation. It was then suggested that, for the annihilation

process, both the momentum and energy conservation laws of the electron–

positron pair could be used to provide information on the properties of the

matter.

The ﬁrst experimental studies using positrons on the electronic structure

of matter were devoted to the identiﬁcation of Fermi surfaces in metals. The

dramatic developments in nuclear spectroscopy equipment that took place

in the two decades starting 1945 allowed the unequivocal establishment, by

the end of the 1960s, that

•

Positron annihilation parameters were sensitive to crystal lattice

imperfections.

• Positrons could annihilate after being captured in a defect, which

means that their wave functions were conﬁned in the defect location

(with the positron described by a localized state).

This positron behavior was clearly demonstrated in various papers, for

example, thermal holes studies in metals [4] and ionic crystals [5], and elastic

deformations in semiconductors [6].

Until the mid-1980s, positron studies of defects in solids were performed,

preferentially, in metals and metal alloys. The experience obtained in this

ﬁeld was then applied to semiconductor studies, and soon the majority of

published papers using positrons were directed toward studies on simple

and compound semiconductors.

In parallel with the progress on the understanding of the physics involved

in the positron annihilation in matter, important developments were also

madewith the experimentaltechniques.Around1980, theﬁrst variable energy

positron beams specially developed for material studies were constructed.

These systems have opened new ﬁelds on depth proﬁle defect studies (from

the surface to the bulk material), interfaces, and multilayer systems, which are

of crucial importance on metallic and semiconductor ﬁlms used in modern

devices. More recently, a new generation of pulsed positron beams, being

able to focus the beam in a micrometer scale spot on the target, have become

the most advanced positron beams. These systems can also scan the material

surface, allowing studies of well-deﬁned regions of the material.

Very recently, already in the new millennium, research groups have started

using positrons in matter studies in biological systems. The ﬁrst results are

very promising, as variations in the annihilation radiation characteristics in

biological systems have been conﬁrmed.

Positron Physics 71

3.1.2 Properties of Symmetry

The annihilation of the electron–positron pair is the process through which a

particle–antiparticle pair is transformed into energy (photons). Since the total

energy of the system involved in this annihilation (the pair formed by the free

positron and one electron of the solid) has a value 2 ×m

−E

∼

1022 keV,

where m

represents the rest mass of the electron and the positron and E

represents the binding energy of the bulk electron, the energy of the emit-

ted photons are in the same energy range as the γ photons emitted by

atomic nuclei. Since this process is invariant under charge conjugation, a gen-

eral selection rule can be applied to the annihilation process. In particular,

depending on the relative orientations of the electron and positron spins, the

annihilation is only allowed with formation of an even number of photons

when spins have antiparallel orientations (S = 0) or with an odd number of

photons when spins are parallel (S = 1). Following this selection rule, the

number of photons emitted during the annihilation of a randomly oriented

electron–positron pair is not completely deﬁned. However, in practice, only

the annihilations involving the emission of 2γ or 3γ photons have an impor-

tant role, as the cross sections for emission of higher number of photons are

several orders of magnitude smaller than the annihilations with emission of

2γ or 3γ. The single photon annihilation of the electron–positron pair is forbid-

den by the momentum conservation law and can only occur in the presence

of a third body that absorbs the recoil momentum, as was experimentally

observed in 1991 by Palathingal et al. [7]. The probability for occurrence of

this annihilation process is, in general, ﬁve orders of magnitude lower than

the 2γ emission probability. The ratio between the probabilities for 3γ and

2γ annihilation for collisions between electrons and positrons with random

orientations is given by the cross-section ratio σ

3γ

/σ

2γ

= 1/372, showing that

the 2γ annihilation is, indeed, the dominant process.

The probability rate, per unit time, for annihilation of the positron with

an electron from the medium Γ, is given by the product of the collision cross

section σ, with the electron speed v, and the probability of ﬁnding one electron

at the positron position, which is expressed by the overlap of the electron

and positron wave functions. For 2γ annihilations in the nonrelativistic limit

(since thermalized positrons have very low kinetic energy), the collision cross

section, as calculated by Dirac [1], is given by

2γ

πr

, (3.1)

and the probability rate is given by

Γ ≡ Γ

2γ

= σ

2γ

νn

= πr

(ns

−1

), (3.2)

72 Nuclear Medicine Physics

where r



= e

= 2.8 ×10

−15



is the classical radius of the electron

or positron, c is the speed of light in vacuum, n

is the electronic density at

the annihilation site, m

is the rest mass of the electron or positron, and r

the electronic density parameter given by



4πn

. (3.3)

The expected values for the average positron lifetime are naturally depen-

dent on the electronic density of the medium. Lifetimes of the order of 100 ps

areexpected for positronsin metals, but the average lifetime will increase with

the decrease of the r

parameter (semiconductors, polymers, and liquids).

3.1.3 Positronium

Positronium (Ps) is a quasistable bound system of a positron and an electron,

similar to the hydrogen-like atom (and can be described in the same way),

with a reduced mass m

/2 (where m

is the electron mass). The Schrödinger

equation for Ps is then identical to that for the hydrogen atom, when we

replace the reduced mass of the hydrogen by one half of the electron mass.

The Ps binding energy in vacuum is 6.8 eV and its diameter is approximately

1.6 Å.

The angular momentum sum for the electron and the positron spins can

have two possible results leading to two fundamental Ps states: the singlet

state,

, named as para-positronium, p-Ps (conﬁguration with antiparallel

spins); or the triplet state,

, referred to as ortho-positronium, o-Ps (conﬁg-

uration with parallel spins). In the absence of external magnetic ﬁelds, the

formation probabilities for these two states are 25% and 75%, respectively. The

difference in energy between the different spin states, the so-called hyperﬁne

structure, is only 8.4 ×10

−4

eV, which leads to all the possible states having

approximately the same formation probability. As a consequence, there will

be, on average, three o-Ps for each p-Ps formed.

Following the parity conservation rules, the Ps annihilation in vacuum from

the p-Ps state can only be through the emission of an even number of photons,

whereastheo-Ps state annihilation occurswith the emission of an odd number

of photons.

As was previously referred to, if we consider the collisions between

positrons and electrons with randomly oriented spins, the number of annihi-

lations with emission of three photons is considerably less than the number of

annihilations with emission of two photons. However, the electron–positron

bound states, p-Ps and o-Ps, where the spins have deﬁnite relative orienta-

tions before annihilation, present quite different annihilation rates. With no

external perturbation, the p-Ps and the o-Ps cannot be modiﬁed from their

formation states before their annihilations and, consequently, the ratio of 2γ

Positron Physics 73

to 3γ Ps annihilation should be 1/3. The lower annihilation probability, per

unit time, for the 3γ emission is also indicated by the longer lifetime of the

o-Ps state. The experimental values of p-Ps and o-Ps lifetimes, 125 ps and

142 ns [8], respectively, are in excellent agreement with the values previously

calculated by West [9].

Nevertheless, within a material where Ps formation is possible, the fre-

quent collision of o-Ps with atoms and molecules of the medium results

in a fast exchange of its spin state. This spin exchange can occur either

through the change of the electron in the Ps by an electron from a neigh-

boring molecule, followed by annihilation (pick-off process); or, in special

cases such as in the presence of paramagnetic molecules, through the change

of spin of the Ps electron (spin conversion). In this way, the initial 1/3 ratio

of the formation of the two possible positron states is not preserved in time;

and the ratio of the 2γ to 3γ emission is, in this case, larger. As a conse-

quence of this, a substantial reduction in the o-Ps lifetime in materials is

observed (typically, the reduction is two orders of magnitude, for exam-

ple, reaching values of few nanoseconds). The o-Ps lifetime then reﬂects

the ortho- to para-transition probability and is one of the basic properties

used to study the Ps interactions with neighboring atoms or molecules inside

materials.

Four models are presently used to describe Ps formation in matter:

• The Ore model [10, 11]: The Ps may be formed from the ionization

produced by the energetic positron with the subsequent binding to

the free electron produced. This process can occur if the positron

kinetic energy satisﬁes the condition: (E

–6.8 eV) ≤ E ≤ E*, where E

represents the ionization energy of the molecule and E* represents

the energy needed for electronic excitation of the molecule.

• Resonant mechanism [12]: The Ps formation process involving the

last inelastic collision of the positron with bound electrons result-

ing, through energy resonant absorption, in the formation of an

intermediary excited state of the positron–molecule system.

• The Spur mechanism [13]: through the capture of one of the many

electrons, created by ionization, during the last steps of the positron

energy loss process.

• Blob model [14] is an extension of the Spur mechanism, taking into

account the distribution of freeelectrons formed by ionization during

all the paths of the positron inside the medium, in order to explain the

change of Ps formation probability under the inﬂuence of an applied

external electric ﬁeld [15].

The ﬁrst two models are applied to the gaseous state whereas the last two

are valid for condensed matter (liquids and solids).

74 Nuclear Medicine Physics

Radioactive decay

Free positron

annihilation (70%)

99.73%

0.27%

Slow down

o-Ps

p-Ps

τ = 0.125 ns

τ = 142 ns

Positronium

formation (30%)

75%

25%

2γ

3γ

511 keV

Conversion

Pick-oﬀ

Chemical

reactions

98.7%

1.3%

FIGURE 3.1

History of a positron emitted by a radioactive source since its creation until annihilation in

water.

Positronium formation frequently occurs in gases, liquids, solid polymers,

and living tissues, that is, in materials with low electronic density and with

enoughfreespace to contain the Ps. In ioniccrystals, its formation is associated

with the existence of defects. Positronium formation in metals and semicon-

ductors is inhibited by their compact structure, by the high electronic density

in the interstitial space between the atoms, and by the Ps atom dimensions. In

metals and semiconductors, the Ps formation can only occur on the surface,

either the external surface of the material or the surface of large dimension

cavities inside the material.

In Figure 3.1, a schematic representation is shown of the physical processes

as seen by the positron implanted in water, from which it can be observed

that annihilation occurs preferentially through the emission of two gamma

rays; nevertheless, the 3γ annihilation is also present, although with a small

percentage (0.5%) and is mainly due to the formation of Ps.

3.2 Interaction of Positrons with Matter

3.2.1 Energy Loss and Thermalization

The different interaction processesthat occur when energetic positronsenter a

solid areschematically represented in Figure 3.2. Some positrons are backscat-

tered in elastic collisions at the surface, and those entering into the solid

rapidly lose their kinetic energy in a set of ionizations and excitations with

Positron Physics 75

incident

Scattering

diﬀraction

Fast

Slow

Ps decay

Surface states

–

ermal diﬀusion

Nonthermal e

Annihilation

Energy loss :

– Core excitations

– Plasmons

– Electron–hole pairs

– Phonons

Channelling

FIGURE 3.2

Positron interactions with a solid and a surface.

the electrons of the solid. In the case where the radioactive source is placed

inside the material, the backscattering process cannot take place.

The energy loss rates are typically 1 MeV/ps when the positron energy is

in the range from 100 keV to 1 MeV and 100 keV/ps in the energy range

from 100 eV to 100 keV, independent of the material where the positron is

moving [16]. For kinetic energies less than 100 eV, the positron energy loss

rate strongly depends on the environmental characteristics:

•

In metals, the excitation of free electrons, with the creation of

plasmons and electron–hole pairs, is the dominant mechanism for

positron energy loss. In the energy range below 1 eV, the domi-

nant process is phonon dispersion. The total positron thermalization

time, as calculated at room temperature, is of the order 1–3 ps, much

smaller than the typical lifetime of a positron in a metal, 100–200 ps

[16,17].

• In semiconductors, when the positron energy is lower than the semi-

conductor energy gap, electron–hole excitations are not possible,

and the positron energy loss is due to phonon dispersion. This

76 Nuclear Medicine Physics

dispersion is already effective for energies around 1 eV. The calcu-

lated thermalization times are comparable with those obtained in

metals [18].

• In liquids, the ionizations and electronic excitations represent the

main role for energies greater than the ionization limit. For lower

energies,the positron reaches thermal equilibrium primarily through

intra- and intermolecular excitations and through rotational excita-

tions of the molecules. The total duration of the process is less than

1 ps [19,20].

• In gaseous media, particularly in noble gases, the time scale needed

for the positron to reach thermal equilibrium is much longer, from

1 to 100 ns. In the low energy range (<100 eV), through chemi-

cal processes, various interactions of the positron with atoms and

molecules of the medium may be considered, including Ps formation

[21,22].

We can then conclude that, in condensed matter, the thermalization time of

the positron is relatively small (a few picoseconds) when compared with the

average time the particle lives in the medium and so can be neglected.

After thermalization is achieved, the positron attains thermal equilibrium

with the medium, and its path can be described through a diffusion process

until annihilation with an electron of the medium.

Depending on the distance to the solid surface when it gets thermalized,

the positron can reach the surface during the diffusion and escape as a free

particle or as Ps, or be captured as a positron or Ps surface state.

3.2.2 Range and Implantation Proﬁle

The implantation proﬁle (or stopping proﬁle) of the positron in matter

depends on various aspects, in particular the material in which it is implanted

and, especially, on its energy. However, two distinct conditions can be

considered:

• Positrons emitted by radioactive sources having a continuous energy

spectrum, characteristic of the β

decay (maximal energy of the order

of the MeV for most radioactive isotopes commonly used).

•

Positrons with well-deﬁned energy produced by monoenergetic

beam systems.

The stopping proﬁle of energetic positrons emitted by a radioactive source

can be described by an exponential function:

P(z) = αe

−αz

, (3.4)

Positron Physics 77

where P(z) represents the probability of positron penetration to the depth

z (measured from the surface in the case where the positron is implanted in

matter from outside or the average distance from the radioactive source when

the source is located inside matter), with the absorption coefﬁcient α being

given by

α = 16

1,4

max

−1

, (3.5)

where ρ (g/cm

) is the material density and E

max

(MeV) is the maximal

energy of the emitted positrons. For positrons emitted by a

Na radioactive

source (which is the isotope most used on material studies), the characteristic

penetration length, 1/α, is 0.11 mm in silicon, 0.050 mm in germanium, and

0.014 mm in tungsten. In the case of positrons emitted by

O, and

F nuclei (with nuclear medicine applications), the characteristic penetration

length in water varies between 0.5 and 1.4 mm.

For monoenergetic positrons obtained from variable energy positron sys-

tems with energies up to some tens of keV, the positron implantation proﬁle

of a positron with energy E, as simulated by Valkealahti and Nieminen [23],

to the form originally suggested for electrons by Makhov [24], obeys the

relationship:

I(z, E) =−

[

P(z, E)

]

=−



−(z/z

)



m−1

−(z/z

)

, (3.6)

wherem is a dimensionless parameter and z

is related to the average stopping

depth,

z,by

(

1 +(1/m)

)

and

z =

, (3.7)

where n and A are empirical parameters, ρ is the material density, Γ is

the gamma function, and P(z, E) represents the probability for the incident

positron with energy E to stop (thermalize) inside the region between z and

z +dz (Makhov distribution). The more commonly used values for these

parameters, which are considered to give a better description of the material

behavior, are [25]

A = 4.0 μgcm

−2

keV

−n

, m = 2 and n = 1.6.

The average positron stopping depth varies from the nanometer scale up

to a few micrometers when its energy is up to

∼

30 keV.

This position selectivity along the average stopping depth allows the

monoenergetic positrons the possibility of depth investigation from the sur-

face until the bulk material. Using a variable energy positron system, it is then

possible to obtain the depth proﬁle of defects close to the surface region and

to perform the analysis of interfaces of thin ﬁlm and multilayer systems.