Radiation Interactions with Matter: Energy Deposition

Biological effects are the end product of a long series of phenomena, set in motion by the passage of radiation through the medium.

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Interactions of Heavy Charged Particles

Energy-Loss Mech anisms

 The basic mechanism for the slowing down of a moving charged particle is Coulombic interactions between the part icle and electrons in the m e diu m . This is common to all charged particles

 A heavy charged particle traversing m a tter loses energy prim arily through the ioniz a tion and excitation of ato m s.

 The m oving charged particle exerts electromagnetic forces on a t om i c electrons and im parts energy to them . The energy transferred may be sufficient t o knock an electron out of an atom and thus ioniz e i t , or it may leave th e atom in an excited, nonioniz ed state .

 A heavy charged particle can transfer onl y a small fraction of its energy in a single electronic collision. Its deflection in the collision is negligible .

 All heavy charged partic les travel essentially straight paths in matter.

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[Tubiana, 1990]

Maximum Energy Transfer in a Single Collision

The maxim u m energy tran sfer occurs if the collision is head-on.

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Assum p tions:

 The particl e moves r a pidly compared with the el ectr on.

 For maxim u m energy transfer, th e collision is head-on.

 The energy transferred is large co mpared with th e binding energy of the electron in the ato m .

 Under these conditions the electron is co nsidered to be initially free and at rest, and the collision is elastic.

Conservation of ki netic energy:

1 MV 2 = 1 MV 2 + 1 mv 2

2 2 1 2 1

Conservation of m o mentum :

M V = M V 1 + mv 1 .

Q ma x =

1 MV 2 -

1 MV 2 =

4 mME ,

2 2 1 ( M  m ) 2

Where E = MV 2 /2 is the initial kinetic energy of the incident particle.

Q ma x values for a range of prot on energies.

Except at extrem e relativistic energies, the maxi mu m fractional energy l o ss for a heavy charged partic le is small.

Maxi mu m Possible Energy Transfer, Q max , in Proton Collision with Electron

Proton Kinetic		Maxim u m Percentage
Energy E	Q ma x	Energy Transfer
(MeV)	(MeV)	100Q ma x /E
0.1	0.00022	0.22
1	0.0022	0.22
10	0.0219	0.22
100	0.229	0.23
10 3	3.33	0.33
10 4	136	1.4
10 5	1.06 x 10 4	10.6
10 6	5.38 x 10 5	53.8
10 7	9.21 x 10 6	92.1

Q max

 4 mME ( M  m ) 2

Single Collision Energy Loss Spectra

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 The y axis represents the calcul ated proba bility that a given co llision will result in an energy loss Q.

 N.B. , the maxim u m ener gy loss calculated above for the 1 MeV proton, of

21.8 keV is off the scale.

 The m o st probable energy loss is on the order of 20 eV.

 N.B. , ener gy loss spectra for fast charged particles are very sim i lar in the range of 10 – 70 eV.

 Energy loss spectra f o r slow charged particles differ, the m o st probable energy loss is closer to the Q ma x .

Stopping Power

 The average linear rate of en ergy loss of a heavy charged p a rticle in a medium (MeV cm -1 ) is of fundam ental im portance in radiation physics, dosim etry and radiation biol ogy.

 This quantity, designated –d E /d x , is call e d the stopping pow e r of the mediu m for the particle.

 It is also referred to as the linear energy transfer (LET) of the particle, usually expressed as keV  m -1 i n water.

 Stopping pow e r and LET are closely associated wi th the dose and with t h e

biological effectiven ess of different kinds of radiation.

Calculations of Stopping Power

In 1913, Niels Bohr derived an explicit form ula for the stopping power of heavy charged particles.

Bohr calcu lated the energy loss of a heavy charged p a rticle in a collision with an electron, then averag ed over all possible distances an d energies.

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The B e t h e Form ula for S t opping Power.

Using relativistic quantum mechanics, Be the derived the following expression for the stoppi ng power of a uniform medi um for a heavy charged particle:

- dE dx

= 4  k 2 z 2 e 4 n

mc 2  2

 ln



2 mc 2  2

I ( 1   2 )

  2  .



k o = 8.99 x 10 9 N m 2 C -2 , (the Boltzman constant)

z = atom ic num ber of the heavy particle,

e = magnitude of the electron charge,

n = num b er of electrons per unit volum e in the medium,

m = el ectr on rest mass,

c = speed of light in vacuu m ,

 = V / c = speed of the particle relative to c,

I = mean excitation energy of t h e medium.

 Only the charge ze and velocity V of the heavy char ge d particle enter the expression for stoppi ng power.

 For the medium , only the electron density n is im portant.

Tables for Computation of Stopping Powers

If the constants in the Bethe equation fo r stopping power, dE/dX, are com b ined, the equation reduces to the following form :

 dE

dx

 5 . 08 x 10  31 z 2 n

 2

[ F (  )  ln I ev ]

MeV cm -1

where,

F (  )  ln

1 . 02 x 10 6  2

1   2

  2

[Turner ]

Conveniently,…..

For a given value of β , the kinetic energy of a partic le is proportional to the rest mass,

Table 5.2 can also be used for other heavy particles.

Example:

The ratio of kinetic energies of a deuteron and a proton trave ling at the same speed is

1 2

2 M d V

1 M V 2

 M d  2

M p

2 p

Therefore the value of F( β ) of 9.972 for a 10 MeV proton, also applies to a 20 MeV deuteron.

Mean Excitation Energies

Mean exci tation energies, I , have been calculated us ing the quantum mechanical approach or measured in experiments. The following approxi m ate em pirical form ulas can be used to estim a te the I v a lue in eV for an elem ent with at omi c num ber Z:

I ≈ 19.0 eV; Z = 1 (hydrogen)

I ≈ 11.2 eV + (11.7)(Z) eV; 2 ≤ Z ≤ 13 I ≈ 52.8 eV + (8.71)(Z) eV; Z > 13

For com pounds or m i xtures, the contributi ons from the individual com ponents m u st be added.

In this way a co m p o s ite lnI value can be obtained that is weighted by the electron densities of the various elements.

The following exam ple is for water (and is proba bly sufficient for tissue ).

n ln I = 

i

N i Z i lnI i,

Where n is the total num ber of el ectrons in the materi al (n = Σ i N i Z i )

When the material is a pure com pound, the electron densities can be replaced by the electron num bers in a single m o lecule.

Example:

Cal culate t h e mean excitation energy of H 2 O

Solut ion:

I values are obtained from the em pirical relations above. For H, I H = 19.0 eV, for O, I O = 11.2 + 11.7 x 8 = 105 eV.

Only the ratios, N i Z i / n are need ed to cal cu la te the co m posite I.

Since H 2 O has 10 electrons, 2 from H and 8 from O, the equation becom e s

lnI =

2 x 1 ln 19 . 0  1 x 8 ln 105  4 . 312

giving I = 74.6 eV

10 10

Stopping power versus di stance: the Bragg Peak

 dE

dx

 5 . 08 x 10  31 z 2 n

 2

[ F (  )  ln I ev ]

MeV cm -1

 At low energies, the f actor in front of the bracket increas es as   0, causing a peak (called the Br ag g peak) to occur.

 The linear rate of energy loss is a maxi mu m as the p a rticle energy approaches 0.

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Rat e of energ y loss along an alpha particle track.

 The peak in energy loss at low energies is exem plified in the Figure, above, which plots -dE/dx of an alpha partic le as a function of distance in a materi al.

 For m o st of the alpha particle track, the charge on the alpha is two electron charges, and the rate of energy loss in creases roughl y as 1/E as predicted by the equation for st opping power.

 Near the end of the track, the charge is reduced through electron pickup and the curve f a lls off.

Stopping Pow e r of Water for Protons

 dE

dx

 5 . 08 x 10  31 z 2 n

 2

[ F (  )  ln I ev ]

MeV cm -1

What is needed to cal c ulate stopping pow e r , - dE/ d X? n the electron density

z the atom ic num ber

lnI the m e an excitation energy For protons, z = 1,

The gram m o lecular weight of water is 18 .0 g/m o le and the num b er of electrons per m o lecule is 10.

One m 3 of water has a mass of 10 6 g.

The density of electrons, n, is:

n = 6.02 x 10 23 m o lecules/m o le x

3.34 x 10 29 electrons/m 3

10 6 g m  3

18 . 0 g / mole

x 10 e - /m olecule =

As found above, for water, ln I ev = 4.312. Therefore, eq (1) gives

 dE  0 . 170 [ F (  )  4 . 31 ]

MeV cm -1

dx  2

At 1 MeV, from Table 5.2, β 2 = 0.00213 and F( β ) = 7.69, therefore,

 dE 

dx

0 . 170

0 . 00213

[ 7 . 69  4 . 31 ]

= 270 MeV cm -1

The stoppi ng power of water for a 1 MeV proton is 270 MeV cm -1

Mass Stopping Pow e r

 The mass stopping pow e r of a material is obtaine d by dividing the stopping power by the density ρ .

 Comm on units for mass stopping power, -dE/ ρ dx, are M e V cm 2 g -1 .

 The mass stopping power is a useful quan tity because it expres ses the rat e of energy loss of the charged particle per g cm -2 of the mediu m tr aversed.

 In a gas, f o r example, -d E /dx depends on pressure, but –d E / ρ dx does not, because dividing by the density exactly comp ensates for the pressure.

 Mass stopping power does not differ greatly for m a terials with sim ilar atom ic com position.

 Mass stopping powers for water can be sc aled by density and used for ti ssue, plastics, hydrocarbons, and other materi als that consist prim arily of light elements.

For Pb (Z =82), on the other hand, -d E / ρ dx = 17.5 MeV cm 2 g -1 for 10-MeV protons. (water ~ 47 MeV cm 2 g -1 for 10 MeV protons)

**General ly, heavy atoms are l e ss efficient on a g cm -2 basis for slowing down heavy charged particles, be cause many of their electrons are too tightl y bound i n the inner shells to participate eff ectively in the absorption of energy.

Range

The range of a ch arged particle is the distance it travels before co ming t o rest. The range is NOT equal to the energy divided by the stopping power.

Table 5.3 [Turner] gives th e mass stopping power and range of protons i n water. The range is expressed in g cm -2 ; that is, the range in cm multiplied by the density of water ( ρ = 1 g cm -3 ).

Like mass stopping powe r, the range in g cm -2 applies to all materials of sim i lar atom ic com position.

A useful relationship…..

For two heavy charged particles at the same initial speed  , the ratio of their ranges is sim p ly

R 1 (  )

z 2 M

= 2 1 ,

R 2 (  )

z 2 M

where:

R 1 and R 2 are the ranges

M 1 and M 2 are th e res t masses and z 1 and z 2 are the charges

If particle num ber 2 is a proton ( M 2 = 1 and z 2 = 1), then the range R of the other particle is given by:

R(  ) =

M R (  ) ,

z 2 p

where R p (  ) is the proton range .

Figure 5.7 shows the ranges in g cm -2 of protons, alpha particles, and electrons i n water or muscle (virt u ally the same), bone, and lead.

For a given proton energy, the range in g cm -2 is greater in Pb than in H 2 O, consistent with the smalle r mass stopping power of Pb.

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References:

This m a terial was taken largely from J.E. Turner, Atom s , Radiation , an d Radiatio n P r otection , W iley, New York, 1995, Chapter 5 “Interaction of heavy charged particles with m a tter”.

Additional reading:

E.L Alpen, Radiatio n Biophysics , Prentice Hall, Englewood Cliffs, New Jersey, 1990.

M. Tubiana, J. Dutrice, A. W a mbersie, Introductio n t o Radiobiology , Taylor and Francis, New York, 1990, Chapter 1.