Radiation Physics for Medical Physiscists - E.B. Podgorsak

1.Heavy charged particles, such as protons, α particles or heavy ions, interact with the target through Coulomb interactions. Typical targets for heavy charged particles are atomic nuclei and atomic orbital electrons.

2.Light charged particles, such as electrons or positrons, interact with the target through Coulomb interactions. Typical targets for light charged particles are either atomic nuclei or atomic orbital electrons.

3.Neutrons interact with the target through direct collisions with the target. Typical targets for neutrons are atomic nuclei.

Targets are either atoms as a whole, atomic nuclei, or atomic orbital electrons. Two-particle collisions are classified into three categories: (1) nuclear reactions, (2) elastic collisions, and (3) inelastic collisions.

1.Nuclear reactions, shown schematically in Fig. 4.1 and discussed in Sect. 4.2, represent the most general case of a two-particle collision of a projectile m1 with a target m2 resulting in two reaction products, m3 and m4, that di er from the initial products m1 and m2.

–In any nuclear reaction a number of physical quantities must be conserved, most notably: charge; linear momentum and mass-energy.

–In addition, the sum of atomic numbers Z and the sum of atomic mass

numbers A for before and after the collision must also be conserved, i.e.,

2.Elastic scattering is a special case of a two-particle collision in which:

–The products after the collision are identical to the products before the collision, i.e., m3 = m1 and m4 = m2.

–The total kinetic energy and momentum before the collision are equal to the total kinetic energy and momentum, respectively, after the collision.

–A small fraction of the initial kinetic energy of the projectile is transferred to the target.

Two-particle elastic scattering is shown schematically in Fig. 4.2 (p. 125). The energy transfer in elastic collisions is discussed in Sect. 4.3; the cross sections for single and multiple elastic scattering of two charged particles are discussed in Sect. 4.4.

3.In inelastic scattering of the projectile m1 on the target m2, similarly to elastic scattering, the reaction products after collision are identical to the initial products, i.e., m3 = m1 and m4 = m2; however, the incident projectile transfers a portion of its kinetic energy to the target in the form

of not only kinetic energy but also in the form of an intrinsic excitation energy E .

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The excitation energy E may represent:

–Nuclear excitation of the target.

–Atomic excitation or ionization of the target.

–Emission of bremsstrahlung by the projectile.

As a result of the various types of projectiles and targets as well as several categories of two-particle collisions, many di erent two-particle interactions are possible. The interactions of interest in medical physics and radiation dosimetry are summarized in Table 4.1.

Table 4.1. Collisions between various projectiles and targets of interest in medical physics and radiation dosimetry

Heavy charged particle interactions with nuclei of the target

(a)Nuclear reaction precipitated by a heavy charged particle projectile m1 striking a nucleus m2 resulting in products m3 and m4.

Example: Deuteron bombarding nitrogen-14

147 N(d, p)157 N

(b)Elastic Coulomb collision of heavy charged particle with atomic nucleus Example: Rutherford scattering of α particle on gold nucleus,

19779 Au(α, α)19779 Au

(c)Inelastic collision of heavy charged particle with nucleus Example: Nuclear excitation,

AZ X(α, α)AZ X AZ X → AZ X + γ

Heavy charged particle interactions with orbital electrons of the target

(d)Inelastic collision of heavy charged particle with atomic orbital electron Example: Excitation or ionization of an atom

Light charged particle interactions with nuclei of the target

(e)Nuclear reaction precipitated by an energetic light charged particle striking the nucleus

Examples: (e,n) and (e,p) nuclear reactions

(f) Elastic collision between a light charged particle and atomic nucleus Example: Non-radiative scattering of electrons on the nuclei of the target

(g)Inelastic collision between a light charged particle and atomic nucleus Example: Bremsstrahlung production by electrons or positrons undergoing

a Coulomb interaction with an atomic nucleus (radiative loss)

Light charged particle interactions with orbital electrons of the target

(h)Elastic collision between a light charged particle and an orbital electron Example: Ramsauer e ect in which an electron of very low kinetic energy

(below 100 eV) undergoes an elastic collision with an atomic orbital electron

(i)Inelastic collision between a light charged particle and atomic orbital electron Example 1: Electron-orbital electron interaction resulting in atomic excita-

tion or ionization (hard and soft collisions)

Example 2: Positron annihilation leaving atom in an ionized state coinciding with emission of two γ annihilation quanta.

Neutron interactions with nuclei of the target

(j)Nuclear reaction caused by neutron colliding with atomic nucleus Example: 1: Neutron capture or neutron activation (see Sect. 8.4)

5927Co(n, γ)6027Co

Example 2: Spallation and nuclear fission for high atomic number targets

(k)Elastic collision between neutron and atomic nucleus (see Sect. 6.2.1)

(l)Inelastic collision between neutron and atomic nucleus (see Sect. 6.2.2) Example: Nuclear excitation

AZ X(n, n)AZ X AZ X → AZ X + γ

4.2 Nuclear Reactions

Two-particle collisions between the projectile m1 and target m2 resulting in products m3 and m4 are referred to as nuclear reactions and are governed by conservation of total energy and momentum laws. As shown in Table 4.1, the projectile can be a heavy charged particle, a light charged particle or a neutron.

The collision leading to a nuclear reaction is shown schematically in Fig. 4.1 with the projectile m1 moving with velocity υ1 and kinetic energy (EK)1 striking a stationary target m2. An intermediate compound entity is produced temporarily that decays into two reaction products, m3 and m4, ejected with velocities υ3 at angle θ and υ4 at angle φ, respectively.

4.2.3 Threshold Energy Ethr for Nuclear Reactions

An exothermic reaction can occur spontaneously; an endothermic reaction cannot take place unless the projectile has a kinetic energy exceeding a threshold energy.

•The threshold is defined as the smallest total energy Ethr or the smallest kinetic energy (EK)thr of the projectile at which an endothermic collision can still occur.

•The threshold energy for an endothermic collision is determined through the use of the so-called invariant

where

E is the total energy before and the total energy after the collision;

pis the total momentum before and the total momentum after the collision;

The invariant is valid for both the laboratory coordinate system and for the center-of-mass coordinate system and, for convenience, the conditions before the collision are written for the laboratory system while the conditions after the collision are written for the center-of-mass system.

The conditions for before and after the collision are written as follows:

– total energy before: Ethr + m2oc2 = m21oc4 + p21c2 + m2oc2 , (4.8) where Ethr is the total threshold energy of the projectile;

–total momentum before: p1;

–total energy after in the center-of-mass system: m30c2 + m40c2;

–total momentum after in the center-of-mass system: 0.

The invariant of (4.7) for before and after the collision then gives

(4.9)

Solving for Ethr = m21oc4 + p21c2 results in the following expression for the total threshold energy

Noting that Ethr = (EK)thr + m1oc2, where (EK)thr is the threshold kinetic energy of the projectile, we get the following expression for (EK)thr

The threshold kinetic energy (EK)thr of the projectile may now be written in terms of the nuclear reaction Q value [(4.5)] as follows:

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– we note that from (4.5) for the Q value we can write

–Inserting the relationship of (4.12) into (4.11) we obtain

where, since Q m2oc2, we can ignore the Q/(2m2oc2) term in (4.13).

In (4.13) the threshold kinetic energy (EK)thr of the projectile exceeds the |Q| value by a relatively small amount to account for conservation of both energy and momentum in the collision.

As a special case the invariant of (4.7) may also be used to calculate the threshold photon energy (Eγ )thr for pair production (see Sect. 7.6.2)

– In the field of nucleus of rest mass mA (pair production) as

– In the field of orbital electron of rest mass me (triplet production) as

4.3 Two-Particle Elastic Scattering: Energy Transfer

Elastic scattering in a two-particle collision is a special case of a nuclear collision between a projectile m1 and target m2:

1.The initial and final products are identical (i.e., m3 = m1 and m4 = m2), however, the projectile changes its direction of motion (i.e., is scattered) and the target recoils.

2.The Q value for the collision, as given in (4.7), equals zero, i.e., Q = 0.

3.A certain amount of kinetic energy (∆EK) is transferred from the projectile m1 to the target m2. The amount of energy transfer is governed by conservation of the kinetic energy and momentum, and depends on the scattering angle θ of the projectile and the recoil angle φ of the target.

Two-particle elastic scattering between projectile m1 moving with velocity υ1 and a stationary target m2 is shown schematically in Fig. 4.2, with θ the scattering angle of the projectile, φ the recoil angle of the target, and b the impact parameter. After the collision particle m1 continues with velocity u1 and the target recoils with velocity u2.

Fig. 4.2. Schematic diagram of an elastic collision between a projectile with mass m1 and velocity υ1 striking a stationary target m2. The projectile is scattered with a scattering angle θ; the target recoils with a recoil angle φ. The impact parameter is b. After the collision the velocity of the projectile m1 is u1; the velocity of the target m2 is u2

4.3.1 General Energy Transfer from Projectile m1 to Target m2 in Elastic Scattering

The kinetic energy transfer ∆EK from projectile m1 to the target m2 is determined classically using the conservation of kinetic energy and momentum laws as follows:

•Conservation of kinetic energy:

where (EK)1 is the initial kinetic energy of the projectile m1.

•Conservation of momentum:

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Example 2: Neutron colliding with lead nucleus (Pb-207):

mneutron mlead nucleus

Since 4mn/mPb ≈ 1/50 = 0.02, we see that in a direct hit only about 2% of the incident neutron kinetic energy is transferred to the lead target. This shows that lead is a very ine cient material for slowing down neutrons; low atomic number materials are much more suitable for this purpose. Of practical importance here is the use of polyethylene as shielding material for doors in high-energy linac bunkers to shield against neutrons produced in the linac.

Example: Interaction between two distinguishable particles such as positron colliding with orbital electron or neutron colliding with hydrogen atom:

In a direct hit between two distinguishable particles of equal mass the whole kinetic energy of the incident particle is transferred to the target in a single hit.

Example: Interaction between two indistinguishable particles such as electron colliding with orbital electron:

After the interaction, the particle with the larger kinetic energy is assumed to be the incident particle; therefore the maximum possible energy transfer is EK/2.

Relativistic Relationships for a Head-On Collision

The relationship for ∆Emax in (4.27) was calculated classically. The relativistic result given below is similar to the classical result, with m1o and m2o standing for the rest masses of the projectile m1 and target m2.