11a Lesson Interactions of Charged Particles with Matter MP200 Radiation Physics - 2010 Duke Medical Physics Graduate Program
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Introduction
Charged particles lose their energy in a manner that is distinctly different from that of uncharged radiations ( photons or neutrons). A charged particle is surrounded by its Coulomb electric force eld that interacts with orbital electrons (collision loss) and the nucleus (radiative loss) of all atoms it encounters as it penetrates into the medium. The energy transfer from the charged particle to matter in each individual interaction is very small. Therefore, a charged particle undergoes a large number of interactions before its kinetic energy is spent.
Types of Charged-particle Coulomb force Interactions Charged particle Coulomb force interactions can be divided into three categories depending on the size of the classical impact parameter b compared to the classical atomic radius a. 1. Soft Collisions ( b >>> a ). Coulomb force interaction of the charged particle with orbital electron for b >>> a. The Coulomb force eld aects the atom as a whole. It can excite the atom to a higher energy level, and sometimes ionize the atom by ejecting a valence electron. The net eect is the transfer of a small amount of energy to an atom of the absorbing medium. Because large values of b are the most probable than near hits on individual atoms, soft collisions are most numerous type of
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charged particle interactions. Half of the energy transferred to the medium is due to soft collisions.
2. Hard Collisions ( b a ) Coulomb force interaction of the charged particle with orbital electrons, when b is of the order of a atomic radius. Charged particle interacts with atomic electron which is then ejected from the atom with considerable kinetic energy. This electron is called -ray. If this -ray is energetic enough, it can undergo additional Coulomb interactions, in a separate track (called spur). Hard collisions are very few compared to the soft collisions. The probability for hard collisions depends upon quantum-mechanical spin and exchange eects. Because of that, the eect of hard collisions depend on particle type. ( electrons vs. heavy particles ) It should be noted that whenever an inner shell electron is ejected from an atom by a hard collision, characteristic X -rays/Auger electrons will be emitted just as it had been done by photons. 3. Coulomb force interaction of the charged particle with the external nuclear eld ( bremsstrahlung production)
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b << a . When b << a, Coulomb force interaction of charged particle takes place with nucleus. Only light charged particles ( electrons and positrons) experience large amount of energy losses through these interactions. In all but 2 3% of such encounters, the electron is scattered elastically, and it is not a mechanism for the transfer of energy to the medium.But it deects the electrons. For high Z media, electron backscatter increases with Z . The other 2 3% of the cases, the electron passes near the nucleus and inelastic radiative interaction occurs in which the X -rays are emitted. Electrons are deected in this process and give a signicant fraction of energy to the photon. These X -rays are referred to as bremsstrahlung radiation. The interaction has a dierential cross section which is proportional to Z 2 and inversely proportional to A. It is insignicant for heavy charged particles. 4. Nuclear Interactions by Heavy Charged Particles A heavy charged particle having a high kinetic energy (100 M eV ) and an impact parameter b, less than nuclear radius may interact inelastically with the nucleus.
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When one or more nucleons ( p or n) are struck, they may be driven out of the nucleus in an internuclear cascade process, collimated in the forward direction. The highly excited nucleus decays from its excited state by the emission of evaporation particles and rays. Thus, the spatial distribution of dose is changed when nuclear interactions are present, since some of its kinetic energy that would otherwise be deposited as local, is carried away by neutrons and rays.
Energy transfer in a single collision
Consider a heavy particle of mass M , velocity V , collides with an electron ( mass me ). Assumptions : heavy particle moves rapidly compared to electrons orbiting speed ( electron is considered free). Energy transfer is larger than the binding energy of the electron ( collision is elastic) The conservation of linear momentum M V = M V1 + me ve Elastic Collision, conservation of kinetic energy 1 1 1 2 M V 2 = M V12 + me ve 2 2 2
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(1)
(2)
Solving (1) and (2),
V1 =
(M me ) V (M + me )
The max energy transfer to the electron, 1 1 4me M E Qmax = M V 2 M V12 = 2 2 (M + me )2
1 where, E = 2 M V 2 is the kinetic energy of the heavy particle
Qmax
4me M 1 2me M 2 V 2 2 . MV = = (M + me )2 2 (M + me )2 2m3 V 2 1 e = = me V 2 = E 2 4me 2 1 E 500
If the incident particle, electron M = me , Qmax
If the incident particle is proton, M = 1836me Qmax =
Relativistic case
Qmax =
2me 2 V 2 2me m2 M2 + M + 1
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If me <<<< M
Qmax = 2me 2 V 2 = 2 2 me c2 2 2me c2 2 Qmax = (1 2 )
Stopping Power
The charged particle experiences a large number of interactions before its kinetic energy is expended. In each interaction, the particle kinetic energy is transferred to the medium ( collision loss ) or to photons ( radiative loss). In addition, the particles path may be altered due to collisions ( elastic or inelastic scattering).
Denition: Linear Stopping Power: ( dT ) dx The rate of energy loss per unit path length by a charged particle in a medium is called the linear stopping power. S= Units: M eV.cm1 Denition: Mass Stopping Power: (
dT dx )
dT dx
Dividing the linear stopping power by the density of the medium, is called the mass stopping power.
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Mass stopping power = Units: M eV.cm2 .g 1 There are two types of stopping powers. Radiative stopping power Collision ( ionization) stopping power
dT dx
dT Radiative stopping power. ( Srad = ( dx )r ) This results from charged particle Coulomb interactions with the nuclei of the absorbing medium. Only light charged particles ( e or e+ )experience energy losses through these interactions and are referred to as bremsstrahlung interactions. dT Collision stopping power. ( Scol = ( dx )c ) This results from charged particle Coulomb interactions with orbital electrons of the medium. Both heavy and light charged particle experience these interactions that result in energy transfer from charged particles to orbital electrons. ( excitation and ionization of atoms).
Total Stopping Power
The sum of radiative and collision stopping powers. Stot = Scol + Srad Radiative Stopping Power For light charged particles, Mass radiative stopping power,
dT dx r
is given by,
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Srad = Units: M eV.cm2 .g 1 Here,
dT dx
= Na rad Ei
r
NA M rad = cross section for bremsstrahlung production Ei = initial total energy of the charged particle, Ei = Ek + mc2 Na = Number of atoms per units mass Na = Values for rad for bremsstrahlung productions are given in the table. Energy Range Non-relativistic Relativistic (Ei me c2 ) High-relativistic(me c2 << Ei << extreme relativistic ( Ei >>
me c2 1) Z 3
(cm2 .nucleon1 ) 16 22 3 re Z complicated power series 2 me c E 2 Z 8re 2 ln( meic2 ) 1 1) 6
Z 3 2 4re Z 2 ln 183 + 1 Z3 1 18
Collision Stopping power for heavy charged particles
The energy transfer from a heavy charged particles to a medium occurs through Coulomb interactions of the charged particle with orbital electrons of the absorber atoms. These collision interactions fall into two categories depending on the size of impact parameter b and atomic radius a. soft collisions ( b >> a) hard collisions ( b a)
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The derivation of mass collision stopping power, Stot for a heavy charged particle is based on the calculation of momentum change p of the heavy particle colliding with orbital electrons.
Momentum Transfer from Heavy Charged Particle to Orbital Electrons
The momentum transfer p along the line than bisects the angle ( ). p = Coulomb force, ze2 Fc = 4 0 r2
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Fp dt =
Fc . cos dt
ze2 p = 4 0
( ) 2 ( ) 2
cos dt d r2 d
where is the angle between r and the bisector. The angular momentum, L = M v b = M r2 M = Mass of heavy charged particle b = impact parameter v = initial velocity of particle ( before interaction) = d angular velocity dt Substituting,
( ) ze2 1 2 p = ( ) cos d 4 0 bv 2 2 ( ) ze 1 = [sin ] (2 ) 4 0 bv 2 2 ze 1 cos p = 2 4 0 bv 2
For heavy charged particles 0 ze2 1 p = 4 0 bv The energy transfer to the orbital electron from the heavy charged particle for a single interaction with impact parameter b, (p)2 E (b) = 2me 2 z2 e2 E (b) = 2 2 4 0 me v b2
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Linear Collision stopping power,
dT dx
The total energy loss by a charged particle in the medium per unit length. It is calculated by integrating E (b) over all b ranging from bmin to bmax , and accounting for all electrons available for interactions. dT = dx
bmax bmin
E (b)
n x
n = number of electrons per unit path length in a cylinder x with inner radius b and outer radius b + db
There are two physical limitations aecting the energy transfer from heavy charged particle to electrons; The minimum possible energy transfer is governed by the ionization and excitation potentials of orbital electrons resulting in bmax , beyond which energy transfer become impossible.
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The maximum energy transfer in a head-on collision between heavy charged particle and orbital electrons and result in a minimum impact parameter, bmin The number of electrons between b and b + db, n = Ne dm ZNA )dm =( A where Ne = number of electrons per unit mass( and dm = mass between b, b + db ZNA ) A
dm = dV = (b + db)2 b2 x = 2bdbx Substituting above, n ZNA = 2( )bdb x A Mass collision stopping power, Scol = dT dx
bmax bmin
dT e2 z 2 = 4Ne 2 dx 4 0 me v
2
2
db b
e2 z 2 bmax = 4Ne ln 2 4 0 me v bmin ZNA e2 z 2 bmax = 4 ln 2 A 4 0 me v bmin
2
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Mass collision stopping power z 2 and Mass collision stopping power
1 2 v
Since ( Z ) varies from substance to substance within a narrow range ( A 0.5 for low Z elements and 0.4 for high Z elements, except hydrogen )
dT ( dx ) varies slightly from substance to substance.
i.e. energy losses of a given charged particle passing through layers of equal thickness in g.cm2 are about the same for all substances.
Minimum energy transfer and Mean Ionization Potential For large b, the energy transfer E (b) is smaller than the binding energy of the orbital electrons or smaller than the minimum excitation potential for orbital electrons. There is no energy transfer for b > bmax where bmax corresponds to a minimum energy transfer Emin . Emin = mean ionization excitation potential I of the medium I depends on the stopping medium, not on the type of the charged particle. Using equation for E ,
Emin i.e. bmax
1 I
e2 z2 =I =2 2 4 0 me v b2 max
2
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Maximum energy transfer For small b, the energy transfer is governed by Emax that is in a headon collision. For a head-on collision between M and me , Emax = me 4M me Ek 4 Ek (M + me )2 M 2 m e M v 2 = 2me v =4 M2
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Therefore, Emax i.e. bmin
1 Emax 2 1 2
z2 e2 2 = 2me v =2 2 b2 4 0 me v min
Relativistically, Emax = 2m0 c2
Classical Mass Collision stopping power
The energy transfer E (b) from a heavy charged particle to an electron ranges from Emin (bmax ) to Emax (bmin ).
2 i.e. I E (b) 2me v
The ratio
bmax bmin
=
Emax Emin
=
2 2me v I
dT ZNA e2 z 2 bmax = 4 ln 2 dx A 4 0 me v bmin
1 Assume, k = 4 0 , C= dT Substitute in, dx ,
2 NA Zr0 , A
2
r0 =
k e2 m0 c2 ,
v = c
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dT ZNA k e2 z 2 bmax =4 me c2 ln dx A me c2 2 bmin z2 bmax = 4C 2 m0 c2 ln bmin
2
If, k =
2Cm0 c2 z 2 2
dT bmax = 2k ln dx bmin
2 2me v = 2k ln I
Bethe relativistic quantum mechanical mass collision stopping power
The mass collision stopping power ( contains both soft and hard collision terms), dT dx
Soft collision term
=
c
dTs dx
+
c
dTh dx
c
Using Born approximation (which assume heavy particle velocity much greater than maximum Bohr-orbit velocity of the atomic electron)
dTs dx In this equation,
c
2Cm0 c2 z 2 2m0 c2 2 H ln 2 2 = 2 2) I (1
H = arbitrary energy boundary between soft and hard collision C= NA Z 2 Z r0 = 0.150 cm2 .g 1 A A
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NA Z = number of electrons in one gram of medium A k e2 = 2.818 1013 cm = classical electron radius r0 = 2 m0 c and, 2Cm0 c2 z 2 Zz 2 k= = 0.1535 2 M eV.g 1 .cm2 2 A Then, dTs dx
Hard Collision Term
c
2m0 c2 2 H = k ln 2 2 2) I (1
Assuming that the incident particle is a heavy particle ( mass greater than electron) ) and H << Emax , dTh dx = k ln
c
Emax 2 H
Total Mass Collision Stopping Power Combining both soft and hard collision terms for a heavy particle, dT dx 2m0 c2 2 Emax ln =k 2 2 2 (1 2 ) I
c
Emax Therefore, dT dx
2 2 = 2m0 c 1 2
c
2m0 c2 2 ln = 2k 2 I (1 2 )
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Since, Zz 2 k = 0.1535 2 M eV.cm2 .g 1 A dT dx 2 Zz 2 2 ln I M eV.cm2 .g 1 = 0.3071 2 13.8373 + ln 2 A 1
c
Note: The value of I should be in eV . z = atomic number of the particle Z = atomic number of the medium A = mass number of the medium
Important features of this formula
The mean ionization potential I (eV ) (excitation energy) is dened as the geometric mean value of all ionization and excitation potential of an atom in the absorbing medium. I is calculated using the equation for if all other quantities in
dT dx c dT dx c .
Also it can be measured
are known.
The logarithm of I is entered in the above formula. Therefore following approximate empirical values can be used to estimate I in eV .
I = 19.0 (eV ) Z = 1 = 11.20 + 11.7Z (eV ) = 52.8 + 8.71Z (eV ) I for a compound or mixture
2 Z 13 Z > 13
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Consider the individual contribution from with Zi and each element then, n ln(I ) = Zi Ni ln(I ) where, Ni = atoms.cm3 for ith element with Zi and Ii n= Ni Zi = total number of electrons in the mixture
For a pure compound n, ni zi are replaced by electrons numbers. Example: Calculate I for H2 O. IH = 19 eV IO = 11.2 + 11.7 8 = 105 eV n = 10, NH = 2 NO = 1 n ln I = Ni Zi ln I
10 ln I = 2 1(ln 19.0) + 8 ln 105 I = 74.6 eV Example : Calculate the mass collision stopping power of water for 1 M eV protons. Solution: dT dx Zz 2 2 = 0.3071 2 13.8373 + ln 2 ln I M eV.cm2 .g 1 2 A 1
c
For proton, z = 1,
for water Z = 10 ,
A = 18 , ln(I ) = 4.312 eV
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dT dx
c
10 = 0.3071 18 2
2 13.8373 + ln 2 4.312 2 1
at 1 M eV , 2 = 0.00213
dT dx
= 270.93 M eV.cm2 .g 1
c
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