Application of Nuclear Physics to Medicine

E E

µ µ E µ

µ µ

γ γ

e →

= <

=

ρ ρ E ρ E ρ ρ

γ γ

en,Comp Comp Comp Comp en,Comp

ˆ Pair:

−

E 2m

µ µ E µ µ

γ e

e ≃

= =

ρ ρ E ρ E ρ

γ γ

en,pp pp pp pp

4.2 Photoelectric effect

The photon interacts with an e- of the atomic shell, it’s an interaction in

the first orbital of the atom, near to the nucleus. When the photon arrives

Figure 16. Photoelectric effect

it’s completely absorbed by the electron, which escapes; then an electron

of the outer shell can fill the empty place. In generally, when you have

photoelectric effect you have also other radiation emitted by the atom, due

17

Figure 17. Spectrum of Photoelectric effect

to the fact we have electron that move between orbital. We could have also

another effect: this photon can escape the atom or can be taken from en

electron that escape the atom, this is the Auger effect. The spectrum is

the sum of:

ˆ ⇒

e- of the outer (M, L, K) shell fills the free shell Emission of X ray;

ˆ Sometimes energy released not emitted as X ray but absorbed by e-

⇒ Emission of Auger e-;

This is an inelastic interaction: γ kinetic energy used also to free the e- so

we have an Energy threshold. −

E = hν E

e b

Where hν is the energy of the photon, E is the kinetic energy of the electron

e

and E is the e-binding energy.

b

4.3 Compton scattering

Photon interacts with an electron of the atomic shell, it’s an interaction

which belongs to the outer shells of the atom, because e- is considered “free”

so this is an Elastic interaction and we don’t have no Energy threshold.

Photon survives after the interaction and Almost all energy is transferred

to e-. The energy of the emitted electron has a particular distribution: for

an incident photon of about 0.5 MeV you produce electron with a peak of

a bit less than 0.5 MeV; tipically the e- produced are closed to the nergy of

the photon. 18

Figure 18. Compton scattering

Figure 19. Spectrum of Compton scattering

4.4 Pair production

The photon arrives close the the nucleus, feels a strong field and disappears

produce a couple e+e-. To have this process you need the nucleus: it’s

impossibile having a gamma that disappers in vacuum alone and producing

a pair, due to energy momentum conservation. In this case we have an

Figure 20. Pair production

· ·

energy threshold equal to 2 m = 2 511 KeV = 1.022 M eV . Of course, as

e

we said there must be Square Quadri-momentum conservation: photon=0,

19

e + e− > 0 so we need something else in initial state (nucleus recoil).

4.5 Cross section Figure 21. Cross section

The cross section is the probability of interaction between the beam and

the target. The dimension of the cross section is an area, so we can imagine

that it’s the area superimposed between the beam and the target. The unit

−24 2

is the barn = 10 cm .

The cross section strongly depends on the material (on the Z) and it deacrease

with the energy. Let’s see the cross section for these 3 effects:

ˆ 4 3.5

∝ →

Photoelectric: σ Z /E Important in heavy metal and in low

5

∝

energy range; (σ Z /A)

ˆ ∝ → ∝

Compton: σ Z/E linear attenuation wrt energy;(σ Z/A)

ˆ 2 2

∝ → ∝

pair: σ Z /lnE almost constant at high energy (σ Z /A)

In the energy range of 1 MeV, for example, we have most only photoelectric

effect, and at 10 MeV we have 50% of pair and 50% of Compton. If we

have Compton effect, the photon survives so after this maybe we can have

photoeletric effect.

The human body is similar to the water, so it’s useful to remember some

number: for 1 MeV the cross section is near 1 barn; tipically a good value

of σ for a photon in water for energy conventional for hadrotherapy isn’f far

from 1 barn. 20

Figure 22. Cross section vs energy

5 Charged Particle: interaction with matter

The interaction can happen with the atomic eletrons, with the nucleus,

passing near the nucleus or interaction with eletrons and nucleus. All these

Figure 23. Charged particle type of interaction

processes are EM process so we know all the characthistcs of them. These

are the main interactions in the hadrontherapy. We can have also nuclear in-

teraction and this is a problema because at that energy not all the properties

of nuclear interaction are known.

5.1 EM interaction on atomic electrons

This is Inelastic Collision: the proton arrives, it gives some energy to the

e, one part of the energy is used to break the bond (ionization of atom)

and the other one is given to the eletron in kinetic energy. Infact the is no

21

conservation of kinetic energy (part of the energy loss by proton is to free the

6

≃

electron). This is a frequent process because σ 10 barn: we have a big

particle (p) that gives its energy to a small particle (e), it gives only a small

part of its energy, energy loss but negligible deviation (mass projectile >>

electron mass) (γ give all its energy in just one shot). We have Hundreds of

collisions, it makes no sense to speak of cross section, but of energy loss of

the all processes. We can have 2 types of collisions:

ˆ ⇒

Soft collision: Atom excitation disexcitation emitting photons;

ˆ ⇒

Hard collision: Atom ionization electron emission (δ ray)

As we can see in the figure we have an ion that ionizes a lot of time the

material and each time produces some δ rays.

Let’s see why the cross section is so big. We said that Cross section (inter-

action probability) is almost equal to geometrical surface, so we can write:

2

∼

σ π(R + R )

1 2

−24 −2

Remember that 1barn = 10 cm . What is the ratio of probability

between this process and the other one?

σ S

atom atom 8−10

= = 10

σ S

nucleus nucleus

The interaction with the electron (atom) is much more probable than with

the nucleus, because of course the dimension of the atom is much bigger.

−8 −16

2 2

σ = π(10 ) = 10 cm = 100 M barn

atom −12 −24

2 2

σ = π(10 ) = 10 cm = 1 barn

nucelus −13 −26

2 2

· −

σ = π(2 10 ) = 10 cm = 10 100 mbarn

pp 22

Bethe-Bloch formula Due to the fact that this process has a probability

in practical equal to 1, it’s Make no sense to speak of cross section, instead

we speak about energy deposited dE/dx. It’s important to evaluate the

energy loss in each dx, because it’s important to deposit a lot of energy in a

very little region in order to create a not reparable damage in the dna. The

Bethe-Block Formula or stopping power describes the energy deposited by

the passege of a charged particle in a medium and can be written as:

2

2 2 2 2

2

dE 4πN m c e 2m c β δ

ρZ z C

e e

A 2

− − −

ln β

= .

2 2 2

−

dx A M 4πϵ m c β I(1 β ) 2 Z

0 e

U

When δ is density correction, important at high energy and C is shell cor-

rection, important for low energy. We can divide this formula into 3 parts:

some medium properties (red), some general constant (green) and the beam

characteristics (z, β) (blue). This formula depends on the density, because

of course the beam encouters more material if the density is high,; it depends

on the ratio of number of charged proton over A, this is not very important

because this ratio is more or less 0.5. So the importat thing of the medium is

the density. This formula depends also on the beam properties: it depends

on the z of the beam, it’s not true that is the z is larger it’ better because

larger z of the beam implies also side effects from nuclear interaction. This

Figure 24. Behaviour of Bethe-Bloch

formula has a behaviour like this in this plot: at very low energy dE/dx is

very high, because the kinetic energy is low so also β is low; then we have

the minimum and then a relativistic rise but and a Fermi plateau. Notice

that if the energy of the particle is zero i expect that dE/dx is zero, because

i can’t create something from zero, but de/dx is infinite. This description

can’t be good for low energy and we need Cell coorection in which at very

low energy the hypothesis that the atomic e- is at rest falls (in bethe-bloch

formula we have the hypothesis that the e are at rest, because their velocity

23

is very low with respect to the velocity of the beam). The most part of the

energy of the proton is released when it has low energy, at the end of its

range. We can have also some other correction like Barkas effect: if the

Figure 25. Low energy range

particle that enters in the matter is a positive particle it’s attracted by the

electron of the atom and proton can takes one of these and in this case its

effective charge decreases; on the other hand, if a negative particle enters in

the material, it rejects all the e of the atom clouds, so the interaction can

2

be different. There are also Minor correction which depends on the Z , that

are important at low energy. 2 2

Let’s see how it’s important the ratio z /β : let’s take 3 different particles,

all with z=1 at E = 30 MeV, but different masses.

kin

ˆ dE 1

→ ⇒ − ∼ −

proton (m=939 MeV) β = 0.25 [14.4 0.06] =

dx 0.06

240 a.u

ˆ dE 1

→ ⇒ − ∼ −

pion (m=140 MeV)= β = 0.57 [14.4 0.32] = 44 a.u

dx 0.32

ˆ dE 1

→ ⇒ − ∼ −

muon (m=105 MeV) β = 0.63 [14.4 0.39] = 36 a.u

dx 0.39

All the part in the bracket is very similar. Having the same β doens’t mean

having the same energy, but the same enrgy per nucleon (proton with 10

24

MeV, the same beta is in Helium with 40 MeV).

Bethe-bloch depends n the crossed matter in these parameters: ρ (density,

can vary orders of magnitude), Z/A: range from 0.5 to 0.42 and I, that is

mean excitation energy (ionization potential): range from 19 eV for H, to 820

eV for Pb. In a low density material we have Few collisions, some with high

→

energy transfer so the Energy loss showa large fluctuation Landau tail;

in a high density material we have many collisions so the Energy loss show

→

low fluctuation Gaussian shape (e-producedby ionizationare called δ

(ray)).

We need to unterstand how much interaction has the proton crossing the

matter: if we take a proton of 200 MeV, the e