14.4 Energy-Dependent 3D-SPB
287
monotonically:
ImV(E) = γ
1 + E
φ
δ , (γ = −0.26, δ = 1.7)
the φ is the work function of the crystal of concern.
The spatial-decay of the inelastic damping, ImV(z), was often expressed typically
with a step and a Gaussian-type function:
ImV(z) = β × exp[−α|z − z 1 |], (z < z 1 )
= η
(z < z 1 < z SL )
(14.6)
where β, α, η and z 1 are adjustable parameters to be fixed in calculations. β and η
were employed to modify the intensity of damping at various regions and z SL is the
atomic positions of the second atomic layer.
The ReV(z), ImV(z), and ImV(E) are traditionally independent from one another.
The independent treatment is indeed convenient in examining contributions made
by the individual component but renders the correlation among the parameters. In
the one-dimensional approach, there are all together seven independent parameters
to be optimized in VLEED calculations with z being the variable of integration:
ReV(z; λ, z 0 , V 0 )and,
ImV(z, E; z 1 , α, β, η, γ, δ)
The independent treatment of such a huge number of correlated parameters causes
troublesome in the uniqueness of solutions. Figure 14.9 shows the schematic diagrams of the (a) ReV(z), (b) the pseudo-Fermi z function, (c) the ImV(z) in a
step-Gaussian wise and, (d) the Fermi-z decay ρ(z) defined in [42].
14.4.4 Energy Dependent 3D-SPB
VLEED integrates over large areas of a surface, which effects z 0 (x, y) to be z 0 (E). As
it is known that at z = z 0 , the Poisson equation approaches zero. Therefore, z 0 is the
boundary of the region occupied by electrons. If we permit z 0 to vary as a function
of the surface coordinates, then z 0 (x, y) provides a contour describing the spatial
distribution of electrons. The surface coordinate (x, y) relates to the k // in reciprocal
lattice (k x ∝ 1/x). In reality, the SPB parameters are functions of energy E, lateral
wave vector k // and surface coordinates (x, y) but the resultant z 0 (E, k // (x, y)) is so
complicated that it is impractical to attempt to generalize the functional dependence
of a mixture of the real and reciprocal spaces in the theory models.
On the other hand, the term k // (x, y) is the contribution of multiple beams in
the VLEED calculation integrated over large areas as each beam has a specific k // .
VLEED calculation can only provide the average effect of the z 0 (E), which is the
287
monotonically:
ImV(E) = γ
1 + E
φ
δ , (γ = −0.26, δ = 1.7)
the φ is the work function of the crystal of concern.
The spatial-decay of the inelastic damping, ImV(z), was often expressed typically
with a step and a Gaussian-type function:
ImV(z) = β × exp[−α|z − z 1 |], (z < z 1 )
= η
(z < z 1 < z SL )
(14.6)
where β, α, η and z 1 are adjustable parameters to be fixed in calculations. β and η
were employed to modify the intensity of damping at various regions and z SL is the
atomic positions of the second atomic layer.
The ReV(z), ImV(z), and ImV(E) are traditionally independent from one another.
The independent treatment is indeed convenient in examining contributions made
by the individual component but renders the correlation among the parameters. In
the one-dimensional approach, there are all together seven independent parameters
to be optimized in VLEED calculations with z being the variable of integration:
ReV(z; λ, z 0 , V 0 )and,
ImV(z, E; z 1 , α, β, η, γ, δ)
The independent treatment of such a huge number of correlated parameters causes
troublesome in the uniqueness of solutions. Figure 14.9 shows the schematic diagrams of the (a) ReV(z), (b) the pseudo-Fermi z function, (c) the ImV(z) in a
step-Gaussian wise and, (d) the Fermi-z decay ρ(z) defined in [42].
14.4.4 Energy Dependent 3D-SPB
VLEED integrates over large areas of a surface, which effects z 0 (x, y) to be z 0 (E). As
it is known that at z = z 0 , the Poisson equation approaches zero. Therefore, z 0 is the
boundary of the region occupied by electrons. If we permit z 0 to vary as a function
of the surface coordinates, then z 0 (x, y) provides a contour describing the spatial
distribution of electrons. The surface coordinate (x, y) relates to the k // in reciprocal
lattice (k x ∝ 1/x). In reality, the SPB parameters are functions of energy E, lateral
wave vector k // and surface coordinates (x, y) but the resultant z 0 (E, k // (x, y)) is so
complicated that it is impractical to attempt to generalize the functional dependence
of a mixture of the real and reciprocal spaces in the theory models.
On the other hand, the term k // (x, y) is the contribution of multiple beams in
the VLEED calculation integrated over large areas as each beam has a specific k // .
VLEED calculation can only provide the average effect of the z 0 (E), which is the