Linear cascade model

Models describing all forms of sputtering other than that described by the linear cascade model are covered in Section 3.2.I.2. 

1 Linear Cascade Model Sputtering resulting from elastic collisions knock-on sputtering) is the most well understood of all the other forms of 

Knock-on sputtering proceeds via a sequence of individual elastic collisions (momentum transfer) occurring between atoms and/or ions as they come into close proximity to each other. How close they approach depends on the energies involved. At and below 100 keV, the distance of closest approach can be defined via the Coulombic potential as  

Similar calculations reveal that 1 -MeV He ions pass right through the electronic cloud of an A1 atom and approach to within 2 X 10 A of the A1 nucleus (the atomic radii of A1 is 1.82 A). The transparency of the solid to He and He ions thus explains the negligible sputter yields resulting on energetic He irradiation. Indeed, these conditions are customarily used in the associated technique called Rutherford Back Scattering (RBS) where interaction of the nuclei is needed and where minimal sputtering is desirable. 

Using this reasoning, the energy picked up by the atom, E2, initially at rest can be expressed as  

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As electronic excitation is not considered within kinetic sputtering, the collisions can be likened to an atomic-scale billiard ball game that is initiated on primary ion impact. The valence electron shells of the atoms/ions involved would thus represent the billiard ball s surfaces. The linear cascade model, which describes the most prevalent form of ion-induced sputtering, at least that from atomic ions and ions comprising small molecules (common examples used in SIMS include 0 , 02 , Cs, Ar" ", Xe" ", and Ga ), assumes a specific form of kinetic sputtering in which a full isotropic collision cascade is produced close to the surface. This is one form of knock-on sputtering. 

When a full isotropic collision cascade is not formed, the sputtering process becomes more anisotropic. This is noted in recoil sputtering, which is another form of knock-on sputtering. As fewer colhsions occur in this form of sputtering, deviations from the trends implied by the linear cascade model are noted. The linear cascade model is covered in Section 3.2.1.1. 

The energy and angular distributions resulting from a full isotropic collision cascade as described by the linear cascade model have been modeled nsing the Sigmund-Thompson relation (Thompson 1968 Sigmund 1969). This predicts the sputtered neutral energy distribution to scale as ... 

In summary, the linear cascade model applies most effectively when using medium to heavy mass primary atomic or small molecular ions (i.e. 0 , O2, Ar, Xe, Ga , Cs, etc.) in the low-to-medium keV energy range (within 0.1-50 keV). These energies are used as sputter yields for such ions peak at between 10 and... 

Although the linear cascade model is extremely effective in describing atomic and certain small molecular emissions, this success does not transfer to ... 

The inability of knock-on mechanisms, inclusive of the linear cascade model, to effectively predict sputter yields in the cases described earlier arises from the fact that such mechanisms describe sputter yields as arising from many individual momentum transfer processes occurring in a linear sequence. However, as outlined in Section 3.2, ejection of atoms/ions or molecules from a solid surface can also occur through ... 

As introduced in Section 3.2.1, potential sputtering, whether kinetically assisted or not, results from inelastic energy transfer processes, with electron-phonon interactions playing a part. Cooperative motion describes a kinetic process in which a single primary ion impact induces the movement of a collective body of atoms within the solid. All of the above result in sputter yields (these are covered in Section 3.2.2) that are greater than that expected based on the linear cascade model (see Section 3.2.1.1). 

Kinetically assisted potential sputtering can take several forms depending on the primary ions, the conditions used, and the matrix examined. For dense atomic and the small molecular ion impact (In ", Bi ", Au , SFj" ", etc.), these generally tend to assume the presence of overlapping collision events within the lattice that occur as a result of the same initial collision event (the linear cascade model assumes individual events). This overlap ensues when momentum transfer is constrained within a more localized volume and/or when multiple atoms from the same impacting ion strike the same region. 

The primary reason why various simulations of sputtering resulting from momentum transfer, as described within the linear cascade model, are successful lies in the fact that the collision events can be treated using classical arguments, that is, Newtonian mechanics as opposed to quantum mechanics. Owing to the insignificant wavelength of ions ( 10" A), quantum mechanics is not needed. In addition, calculations are simplified for isotropic linear cascades as these represent a linear sequence of independent collision events that occur over time scales much shorter than lattice vibrations. 

Linear cascade model A model for describing sputtering... 

Fig. 10.8. Stability diagram established as a function of the reduced Michaelis constants X, of the first cycle of the minimal cascade model of fig. 10.4, versus the reduced Michaelis constants (Kj, K4) of the second cycle. The domain of oscillations corresponds to the domain of instability of the unique steady state admitted by eqns (10.1). The stabiUty properties of the steady state are determined by linear stability analysis. The diagrams are established for (a) equal or (b) unequal values of (X K2) on the one hand, and (X3, 4) on the other. Parameter values are as in fig. 10.6 (Guilmot Goldbeter, 1995).

This linear scheme of signal transduction (Fig. 12) from hypothetical membrane receptors to [Ca " ] and IP3 increases, calcium-calmodulin interaction, kinases activation and gene transcription is clearly an oversimplification of the reality several receptors must exist that are connected to different transduction cascades that activate a series of defense genes. Cross-talking between the pathways further complicates the picture. However, this represents a starting model on which to elaborate more refined hypotheses. 

Fig. 23.7. Dynamics of an enzymatic reaction in lipid nanotube networks with variable topology numeric calculations (bottom)/fluorescence intensity of the reaction product (top) vs. time for three differently chosen network geometries, (a) Reference experiment a static four-vesicle network. The product concentration displays a cascade-like behavior in time and space, (b) Linear-to-circular topology change in the four-vesicle network (c) A model study of the effect of product inhibition as the linear four-vesicle network (top panel) undergoes the same change in structure (bottom panel) as the network in the reference experiment ([28], reprinted with permission)...

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