Interazioni delle particelle cariche G. Battistoni INFN Milano
Collisional de/dx of heavy charged particles A particle is heavy if m part >>m electron. The energy loss is due to collision with atomic electrons. During the interaction particle-electron the b of the projectile can be assumed constant T max = 4 e β<<1 m e M (m e + M ) T i 4 m e M T i M, T i Soft (distant) collision: the electron is promoted to an excited level in the atom Hard (close) collision : the atom is ionized. The knocked-out electron can be quite energetic (δ electron) m e θ Te T max m e = e c β γ 1+ γ m e M +! m $ e # & " M % T f Maximum energy transfer is small if M>>m
Main references for de/dx: Ø E.A. Uehling, Annual Review of Nuclear Science, 4, 315 (1954) Ø Nuclear Science Series, Report n. 39, Studies if penetration of charged particles in matter, National Academy of Sciences National Research Council, (1964) Ø H.A. Bethe, J. Ashkin, Passage of Radiations Through Matter, Experimental Nuclear Physics, Vol. I, Ed. E. Segre` (1960) Ø R.D. Birchoff, The passage of fast electrons through matter, Encyclopedia of Physics, Vol. XXXIV, Springer-Verlag (1958) Ø ICRU Report n. 37, Stopping Powers for Electrons and Positrons, (1984) Ø ICRU Report n. 49, Stopping Powers and Ranges for Protons and Alpha particles, (1993) Ø http://www.nist.gov/physlab/data/index.cfm Ø http://www.nist.gov/physlab/data/star/index.cfm 3
Collisional de/dx of heavy particles (IV) The precise QM calculation ( Bethe-Bloch) is given by (for spin 0 and spin ½ particle The formula breaks down when β οf projectile is comparable to β e of electron (β e ~ α em ~ 1/137). The de/dx has a minimum at βγ~4, i.e. β~0.96 (de/dx ~ MeV/cm) + + = δ β β β β β β π Z C L z zl I T m c z m c r n dx de e e e e ) ( ) ( ) (1 ln 1 max 0 ( ) + + + + = δ β β β β β β π Z C L z zl Mc T T I T m c z m c r n dx de e e e e ) ( ) ( 4 1 ) (1 ln 1 0 max max 1 Hans Bethe 1906-005
The mean excitation energy I For an actual atom there will several ionization/excitation levels,i i, to be considered, therefore the formula must be extended in order to account for all of them. The mean excitation energy, I, can be defined as a suitable (logarithmic) average over all possible atomic levels: A useful approximation is given by: I Z =1 + 7 Z ev I Z = 9.76 + 58.8Z 1.19 ev ln I = i f i =1 i I <163eV I 163eV f i ln( I i ) 5
Average Ionisation Energy <I> The I parameter is crucial. For the water (water!!!) up to 011 the choice was oscillating between 75 (standard ICRU), and 81 ev. Now almost fixed to 78.5.
Il problema H O 7
Low β behaviour : z eff At β ~ 10 - the electrons have the same velocity of the projectile: energy transfer mechanism is no more efficient, reducing LET of stopping ions! zè z eff This effect cause a sudden decrease of the de/dx below the Bragg Peak Z=9 Z=36 Z=18 Z=10 Z=6 Z eff = Z(1 exp 15βZ /3 ) Z=4
Corrections to de/dx: High energy: δ is the so called density correction, extensively discussed in the literature and connected with medium polarization Low energies: C is the shell correction, which takes into account the effect of atomic bounds when the projectile velocity is no longer much larger than that of atomic electrons and hence the approximations under which the Bethe-Bloch formula has been derived break down. This correction becomes important at low energies Higher order: L 1 is the Barkas (z 3 ) correction responsible for the difference stopping power for particles-antiparticles, L is the Bloch (z 4 ) correction 9
Density effect: The density effect δ(η) is the reduction of the energy losses due to the polarization of the medium. It becomes important when the particle is relativistic, the sooner in more condensed media. It can be approximated by ln βγ X = ln10 η = βγ ( ) = 0 X < X 0 δ η δ ( η) = lnη + C + a(x 1 X) m X 0 > X > X 1 δ ( η) = lnη + C X > X 1 The parameters C, X 0, X 1, a and m depend on the material and on its physical state (density etc). It is important to stress that for large energies δ ( η) ln( η ) therefore partially suppressing the relativistic rise of de/dx.
Shell corrections: The quantity C is the sum of the corrections for each electron shell of the atom to the Bethe-Bloch expression: The variation of C i s with velocity and atomic number can be computed. Each term is large and negative at very low velocities, but, as the velocity increases, the sign changes. Each C i passes through a maximum and then goes down. Furthermore, each C i should approach zero as η for large η=βγ. C i s also vary rapidly with the mean ionization energy I. A practical fit for C, valid for η > 0.13 can be expressed by: where I is given in ev. 11
Actual de/dx of heavy particles. Bragg Peak 1/β
And if the material is a compound?
Range of charged particles The energy loss of charge particles can be translated in a maximum range R (different from photons: attenuation) E 0 = R = 0 R R de dx dx The range R can be written as dx = 0 E 0 mc de dx z β ln γ β m e c +... I = z f β (β) = z f E (E ) 0 de = mc z f E (E ) z 0 β 0 0 βdβ ( ) 3 f β 1 β de = mc βdβ ( 1 β ) 3 Some useful scaling laws and behaviour can be obtained: R a (β) R b (β) = m z a b m b z a de 1 dx nonrel v R = 1 E 0 de E 0 de 0 dx
Range e + /e - in Water RadioTherapy electrons R E kin β decay electrons 15
Range and dose release: different projectiles the range & energy release by charge particles has very attractive features Kinetic energy (AMeV)
de/dx and range examples: 17
Range fluctuations The de/dx is a stochastic process: fluctuation of de/dx and range is observed experimentally. The larger the de/dx, the smaller the fluctuation Mean range or CSDA range: Distance where I = I 0 / Extrapolated Range: position where the tangent to the trasmission curve cross the x axis
Energy loss fluctuations energy ΔE deposited in a layer of finite thickness dx. For thin layers or low density materials: Few collisions, some with high energy transfer. e - δ electron ΔE most probable <ΔE> Energy loss distributions show large fluctuations towards high losses: Landau tails For thick layers and high density materials: Many collisions. Central Limit Theorem Gaussian shaped distributions. e - ΔE m.p. <ΔE> DE DE
Energy loss: examples 1
Fluctuation in Energy Loss (straggling) Energy losses of massive charged particles are a statistical phenomenon and in each interaction different amounts of kinetic energy can be transferred to atomic electrons. The energy loss then has a distribution function. A possible parameterization is given by the Landau function: f 1 1 λ ( λ) = exp ( λ+ e ) π with λ = C ΔE ΔE mc e Zz β A ρδx
δ-rays and energy straggling Energy loss distribution is not Gaussian around mean (Landau dist.), because in some cases a lot of energy is transferred to a single electron: δ-rays. If the particle cross thick material than the energy distribution function gets Gaussian. If we assume that the particle looses ΔE in the Δx step in the material, the ΔE pdf is given by: F(ΔE) = 1 ΔE ΔE exp πσ σ ΔE = de dx Δx The width of the gaussian depends both on material and projectile σ = 4π z eff Ze 4 Nγ (1 β )Δx
δ ray production and de/dx fluctuations Let us assume for the δ-ray production cross section the simplified expression: m e c dσ dt π z r e β T The average number of δ-rays with energy between T min and T max produced in a pathlength t such that the resulting energy loss is negligible compared with the initial particle energy, is given by: < n δ >= Σ δ t = n e t T max T min dσ dt dt = πn ez r e m e c β $ 1 t& 1 % T min T max ' ) ( and the corresponding energy loss: T max < ΔE δ >= n e t T dσ dt dt T min = πn e z r e m e c β # t ln T & max % ( $ ' T min 4
δ ray production and de/dx fluctuations The straggling due to the δ-ray energy loss distribution can be evaluated making use of a very general property of Poisson distributed variables. Given a Poisson distributed number of events n, each one described by a distribution x, with given <n>, <x>, and <x >, the following relations hold for the statistical variable y: y = n x i < y >=< n >< x > i=0 σ y =< y > < y > =< n >< x > Therefore for the energy straggling iy can be obtained: T max σ ΔEδ =< T >< n δ >= n e t T dσ dt dt T min = πn ez r e m e c β t ( T max T min ) 5
δ ray production and de/dx fluctuations It is useful to introduce the energy ξ such that the probability of emitting one δ-ray with energy ξ along t is one (it is customary to assume T max >> ξ in the definition). A parameter which is indicative of the skewness of the ΔE distribution can be built as: ξ = πn e z r e m e c β T max m e = e c β γ 1+ γ m e M +! m e # " M t $ & % κ = ξ = π z r e n e T max β 4 γ ' t 1+ γ m e M +! m e ) # () " M $ & % *, +, Ø Ø Ø For ξ << 1 the distribution approaches a Landau one with very long tails For ξ with finite value the distribution is given by a Vavilov function (very complex..) For ξ >> 1 the distribution is roughly gaussian 6
Energy loss fluctuations: examples Straggling functions in silicon for 500 MeV pions, normalized to unity at the most probable value δ p /x. The width w is the full width at half maximum (from PDG) 7
Linear Energy Transfer (LET) Linear energy transfer (LET) is a measure of the energy transferred to material as an ionizing particle travels through it. Typically, this measure is used to quantify the effects of ionizing radiation on biological specimens or electronic devices. Linear energy transfer is closely related to stopping power. Whereas stopping power, the energy loss per unit distance, focusses upon the energy loss of the particle, linear energy transfer focuses upon the energy transferred to the material surrounding the particle track, by means of secondary electrons Since one is usually interested in energy transferred to the material in the vicinity of the particle track, one excludes secondary electrons with energies larger than a certain value Δ Hence, linear energy transfer (also called "restricted linear electronic stopping power") is defined by where refers to the energy loss due to electronic collisions minus the kinetic energies of all secondary electrons with energy larger than Δ. When Δ approaches infinity, there can be no electrons with higher energy, and linear energy transfer becomes identical to the linear electronic stopping power
Linear Energy Transfer The LET is the rate at which energy is transferred to the medium and therefore the density of ionisation along the track of the radiation. LET is expressed in terms of kev/mm or MeV/cm de = energy lost by radiation dx = length of track LET = de dx Radiation that is easily stopped has a high LET and vice versa Radiation 1 MeV γ-rays 100 kv p X-rays 0 kev β-particles 5 MeV neutrons 5 MeV α-particles 1 GeV muon LET kev/mm 0.5 6 10 0 50 0. Table from: P. Dendy & B. Heaton, Physics for Diagnostic Radiology, nd Edition
Linear Energy Transfer (II) The linear energy transfer or restricted linear electronic stopping power, L Δ for charged particles, is defined as: L Δ = de Δ dx de Δ is the energy lost by a charged particle due to electronic collisions in a step dx, minus the sum of the kinetic energies of all the electrons released in the step with kinetic energy in excess of Δ. Generally is intended LET = L, i.e. the ratio between the total energy released to the medium electrons and the step dx
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LET of ions nearby the BP in water. Related to dose imparted in hadrontherapy
Li Be B N RANGE VS. LET mm kev/mm NSRL BEAMS Brookhaven National Laboratory Adam Rusek 015
Fluence (particles/cm ) Fluence characterizes the number of particles in a beam. In beam line design (eg protons) particles always travel in the same direction ± a few degrees so we don t need the general definition of fluence in terms of a sphere. Instead, we can just use a plane element of area and define fluence by where da is an infinitesimal element of area perpendicular to the beam and dn is the number of protons passing through it. The fluence rate is sometimes denoted by lowercase φ and earlier called flux. Fluence and fluence rate are scalar fields: they are directionless quantities that may, and usually do, vary with x, y, z and t.
The Fundamental Formula N protons area A The equation relating dose to fluence and stopping power is the starting point of most beam line design problems. From the figure : Δx dose = fluence mass stopping power
D = Φ S/ρ in Practical Units J/Kg = Gy is a perfectly good unit of dose but (protons/m ) for fluence and J/ (Kg/m ) for mass stopping power are not convenient. A better form is where Φ is in Gp/cm and S/ρ is in MeV/(g/cm ). The gigaproton = 1 Gp = 10 9 protons is a handy unit for proton therapy. If we use charge areal density instead of fluence that gets rid of the constant and we find where q/a is the total proton charge divided by the area through which it passes (nc/cm ) (nc = nanocoulomb). Finally, taking the time derivative where i P /A is the proton current density (na/cm ). Therapy doses are of the order of 1 Gy, target areas are of the order of cm, and S 5 MeV/(g/ cm ) (160 MeV protons in water)