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Projectile's charge Z Projectile's E (MeV)
 


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© The scientific sentence. 2010



Radiation Physics



I. Introduction:

The only way to understand the physical properties of an irradiation of a matter is to treat the related collision problem.

During a collision ion-atom, three processes can occur: capture of target electrons to a shell of the projectile ion or to its continium,
Excitation of target electrons from a shell to a higher one,
Ionization in with electrons leave a shell in the target atom to its continum or outside of the target atom,

In high-energy collisions,in which the ion projectile has a ratio v/c in the range 0.1 - 0.6, the main process is the ionization. Molécules are not formed, but electrons of the target atom undergo a perturbation.



II. High energy collisions



High speed and low speed are relative terms. the ratio Vp/Ve of the projectile speed and the electron target speed is more explicit. If this ratio equals 1, it can be written as:
Vp = Ve → (1/2)meVp2 = (1/2)meVe2 = Bshell; where me is the electrom mass located in a particular shell with Bshell as a binding energy. The kinetic energy of the ion projectile of mass Mp can be written:
Ep = (1/2)mpVp2 = (Mp/me) Bshell = 1836 A Bshell; where A is the atomic mass of the projectile. Per atomic mass unit amu, we have:
Mp = A amu, and E(p,o) = Ep/A = 1836 Bshell.

E(p,o) = Ep/A = 1836 Bshell.

To eject a K shell elctron, from hydrogen target, where the binding energy Bshell is 0.014 Kev. a projectile proton will need 1836 x 0.014(Kev) = 0.026 Mev. From Iron Fe, it will need 7.114(Kev) x 1836 = 13.10 Mev. The more heavy the target, the more is harder to eject electrons from it.
From water, the proton projectile will need 1836 x 70 (ev) ≈ 0.13 Mev to move an electron.
The terms high and low speed involve also the ratio of the charges of the projectile and target atom: Zp/Zt. The criteria for high speed processes is:

(Zp/Vp) . (Ve/Zt) << 1.

Where: Zp, Zc are repectively the charges of the ion projectile and the target atom. Vp and Ve are respectively the speeds of the ion and the target electron.

If We consider 0.3 as the limit value for the inequality, then in water target with Zt = 10, and where the average binding energy is 70 eV, we have for proton:
Ve/Vp << 3 → Ve << 3 Vp → 70 eV x 1836 << 9 Ep → Ep >> 1.43 x 10 -2 Mev.


III. High speed Theories Ionization cross sections:


III.1. PWBA Ionization Cross section:




In The PWBA (Plane Wave Born Approximation) quantum theory, the moving projectile is described by a plane wave, and the target electrons are described by atomic wave functions (hydrogenoid).

Let's consider the diffusion problem where a projectile of mass Mp, speed Vp and charge Zp interacting with a target atom at rest of mass Mt, charge Zt, and initial state i.

The projectile transfers energy to the target atom which becomes in the final state f.
The produced ionization gives rise to the emission of an electron with a kinetic energy toward the target continium.

The hamiltian of the system projectile P - target T, for the collision is written as:
H = Ho(R) + V(R,r)     (1)
Where:
Ho = Hp(R) + Ht(r),     (2)
and V(R,r) is the related perturbation due the projectile.
The two solutions, atomic wave functions, for Ho are:
ψi =(2π)- 3/2 exp[ikiR]φi(r)
ψf =(2π)- 3/2 exp[ikfR]φf(r)
Where ki and kf are respectively the initial and final wave vector related the projectile P.
q = ki - kf is the momentum transfered to the target.

In the Born approximation, the matrix element of the transition of the system from the initial state ψ1 to the final sate ψf due to the projectile perturbation is written as:
T(Born, i, f) = <ψ|V(R,r)|ψ>    (4)
Introducing the Coulombian potential:
V(R,r) = - Z peo2/(R - r), we find:
T(Born, n', n) =
(2Ï€)- 3 &int < n'|- Z peo2/(R - r)|n > exp[ikq.R]d3R
Where |n > and |n'> are respectively the initial and final state of the electron.

The Bethe's integrale &int exp[ikq.R]/(R - r) d3R = (4Ï€/q2) exp[iq.r] leads to:
T(n', n, Born) = (- Z peo2/2Ï€2q2) F(n',n)(q)
Where: F(n',n)(q) = < n'|exp[iq.r]|n > is called the form factor for the inalstic collision.

We can write: q2 = ki2 + kf2 -2kikfcosθ
or qdq = kikf cosθ
The Bethe's expression of the differential cross section associated to the final wave vector of the projectile kf is: dσf = (16 π4/Vp) |Tfi|2 δ(Ef - Ei + Et)d3kf
Where Ef, Ei are respectively the total final and total initial energies of the projectile and Et is the energy transferred to the atom. The delta function that imposes the energy conservation for the collision.

If dΩ is the solide angle related to kf, we have d3kf = k2f(dEf/Vpf),
with dΩ = sin?theta;dθdφ
Integrating over Ef, we find:
dσf/dΩ = (2π)4(kikf/Vp2) |Tfi|2

This expression leads to: dσ(n,n',Born) /dq = 2 (4 Zp2eo4/vp2q2) |F(n',n)(q)|2dφ br> Where φ is the azimutal angle of kf in a perpendicular plan to ki. The first factor 2 is set to take into account of the double occupation of the electron in a shell due to the spin state.
The above expression integrated over the azimutal gives ( Madison and Merzbacher) the formula:
dσ(T,Born) = 16 π Zp2eo4/vp2) |F(T,q)| dTdq/q3
The form factor F(T,q) depends only on q and the kinetic energy T transferred to the electron. It has a developped form and could be computed.

III.2. Excitation Cross section:


In this chapter, we will use the scale law to derive the ratio of the ionization cross section and excitation, since the results for the collision proton-Hydrogen are available:
We have the following formula ( Garcia):
&sigmap (Zp, Zc,E(p,o) = (Zp2/Zt4) σH (1,1,E(p,o)/Zt2),
Where Ep, Zp are the charge of the projectile and the target respectively,
σH the cross section of the proton-Hydrogen, and σp the cross section of any projectile of energy E(p,o).


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