Secreción extracelular de proteasas y fosfolipasas

Kinchin and Pease first presented their model for the calculation of the dam- age function N(E2) in 1954 [78]. The damage function N(E2) gives the average number of displaced atoms due to a recoil at E2 and it also allows an esti- mate of the spatial distribution of the defects produced to be calculated. The collision cascade as described by Kinchin and Pease can be represented dia- grammatically (figures 2.6 and 2.7). To fully understand figures 2.6 and 2.7, the following assumptions must be made [79]:

1. Atoms in collision behave like hard spheres.

2. All collisions are elastic and no energy is dissipated in electron excitation. 3. The cascade proceeds as a series of two-body collisions.

4. These collisions are independent of each other and any spatial correla- tions implied by the periodicity of the crystal structure will be ignored. 5. In a collision such as that illustrated in figure 2.7, when an atom with E

emerges with E’ and generates a new recoil with E” it is assumed that no energy passes to the lattice and

E= E0+E00

6. A stationary atom which receives less than a critical energy Ed is not

displaced, similarly if an incident atom emerges from collision with

Figure 2.6: The Kinchin and Pease model collision cascade.

Figure 2.7: A single branch of the Kinchin and Pease collision cascade model.

The incident particle in figure 2.6 collides with an atom resulting in the for- mation of a cascade. Energy is transferred from the incident particle to the primary recoil atom during the collision. The primary recoil atom is displaced from its original starting position resulting in a vacancy being left in the lattice. As the primary recoil atom continues its journey through the lattice, further collisions occur; the result of these collisions is the formation of a cascade. Once the kinetic energy of the displaced atoms has fallen below the threshold energy, the atoms come to rest either in vacancies or as interstitials.

Figure 2.7 demonstrates the conservation of energy during a collision cascade. A collision cascade is a sequence of collisions between atoms induced by an energetic particle. The energy of the incident particle,E, is equal to the sum of the energy of the two recoil atoms, E’ and E”, equation 2.40 [79].

The above statements along with figures 2.6 and 2.7, form the basis of the Kinchin and Pease definition of the mean displacement energy,Ed. Employing

this theory, the probability of displacing an atom is zero below the energy Ed

and one at E = Ed. It follows that an atom with Ed <E < 2Ed cannot produce

any increase in the number of displaced atoms and N(E) is a step function, defined in equation 2.41 [79].

N(E) =0, if E_≤Ed

N(E) =1, if Ed <E <2Ed

)

(2.41) The probability of finding a scattered atom with energy dE0 at E0 and a recoil atom with energy in dE00 at E00 in a collision with E > Ed is dσ/σ, where σ

anddσare the total and differential cross-sections for the atomic collision [79].

Equation 2.42 gives the probability.

dσ σ = dE00 E or = dE0 E (2.42)

The integral of the product of N(E0) and the probability (equation 2.42) over the energy range, Ed to E, will give the average number of displaced atoms

produced by the scattered atom, statement 2.43.

Z E Ed

N(E0)dE0

E (2.43)

Statement 2.43 is also equivalent to the average number of displaced atoms produced by the recoil atom. This implies the average number of displaced atoms produced by an atom withE: N(E)is equal to equation 2.44.

N(E) = 2

E

Z E Ed

N(x)dx (2.44)

when E>Ed, N(E) =kE,k is a constant whose value may be found by putting

N(2Ed) =1. Hence,k=1/2Ed. Therefore, N(E)can be defined as in equation

2.45.

N(E) = E

2Ed

(2.45) for E> Ed.

Equation 2.45 states an atom would continue to produce cascades until its kinetic energy falls below 2Ed. The number of atoms with 2Ed should be

E2/2Ed [79].

High energy cascades can be employed to calculate the threshold displacement energy using the modified Kinchin and Pease (KP) model and the Norgett- Robinson-Torrens (NRT) model [80]. Both models employ the threshold dis- placement energy, Ed as a single parameter. The number of displacements in

the KP model is computed utilising equation 2.45.

The NRT model is a modified Kinchin and Pease model used to calculate the number of defects generated by a PKA [80] [81]. Equation 2.46 is used to generate the model.

Nd=

kE

2Ed

(2.46) whereNd is the number of defects,Eis the initial PKA energy,Edis the thresh-

old displacement energy and kis the displacement efficiency.

The displacement efficiencykis given the value 0.8 [80]. This value is indepen- dent of the PKA energy, the target material and the temperature. This model was originally fit to MD simulations in copper [11] and iron [82] [83].

Consistent with the experimental and computation literature, table 2.3, a value of Ed = 25 eV was assumed for the two models. Graph 2.8 show plots of the

Figure 2.8: The solid black line shows the number of displaced atoms as a function of PKA energy (Kinchin-Pease model). The solid red line shows the number of stable defects as a function of PKA energy (Norgett-Robinson-Torrens model). Both models assume a thresh- old displacement energy of 25 eV.