316
16 VLEED Capability and Sensitivity
In the practice of verifying the solution uniqueness previously, all the SPB parameters were treated as independent, which has clarified the roles of the structural and
SPB parameters in determining the features in a VLEED spectrum. Here describes
the inspection of the uniqueness of solutions at selected energy values. Except for a
selected pair of SPB parameters, for example, z 0 and λ, the structural and other SPB
parameters were kept unchanged. Optimization parameters for the VLEED spectrum
collected at 70° incidence and 43.5° azimuth were used as next iteration of calculation. The optimal z 0 (E) is used as reference to vary its parameters independently at a
time of calculation. All other parameters were automatically generated with the SPB
functions. The computer was assigned to do loops on the selected pair of variables
with new values.
16.1.1 ReV(z; z 0 , λ) Sensitivity
In fact, the integration of the ReV(z, z 0 , λ) combines the origin of the image plane z 0
and the saturation degree λ of the elastic SPB. The two parameters are correlated to
determine the shape and integration of the ReV(z; z 0 ,λ). VLEED concerns the integration area of the ReV(z) function instead of the exact value of each in determining
the phase shift and the beam reflectivity. Hence, it is essential and realistic to set
the λ(z 0 ) as a function of z 0 to reduce the number of solutions. When the z 0 moves
away of the lattice, the SPB tends to be more saturated because of polarization, as
observed from STM image. If the z 0 moves inward, the SPB will be less saturated
because of ion or atomic vacancy formation.
Figure 16.1 shows the correlation between z 0 and λ at various energies. There
are two groups of correlation curves in each panel corresponding to a 2nπ phase
change of one another. The infinite couples of (z 0i , λ i ) along each curve in a group
provide matching between calculated and experimentally detected intensities I cal /I exp
= 1.00 ± 0.05. In order to establish the correlation between the couple of z 0 and λ
parameters, one can collect data ((z 0i , λ i ), i = 1, 2, …, 30) from each z 0 -λ curve
and then integrate the ReV(z; z 0i , λ i ). The integration ranges from the second atomic
plane (D 12 ) to infinitely far away (in practice the upper limit is chosen as −100 a.u.)
from the surface:
I (Z 0i , λ i ) =
−∞
D12
ReV(z; z 0i , λ i )dz (i = 1, 2, . . . , N = 30)
N = 30 is sufficient for statistical analysis. The average of the integration values
of each curve in Fig. 16.1 is:
I =
1
N
N
i=1
I (Z 0i , λ i ) (N = 30),
16 VLEED Capability and Sensitivity
In the practice of verifying the solution uniqueness previously, all the SPB parameters were treated as independent, which has clarified the roles of the structural and
SPB parameters in determining the features in a VLEED spectrum. Here describes
the inspection of the uniqueness of solutions at selected energy values. Except for a
selected pair of SPB parameters, for example, z 0 and λ, the structural and other SPB
parameters were kept unchanged. Optimization parameters for the VLEED spectrum
collected at 70° incidence and 43.5° azimuth were used as next iteration of calculation. The optimal z 0 (E) is used as reference to vary its parameters independently at a
time of calculation. All other parameters were automatically generated with the SPB
functions. The computer was assigned to do loops on the selected pair of variables
with new values.
16.1.1 ReV(z; z 0 , λ) Sensitivity
In fact, the integration of the ReV(z, z 0 , λ) combines the origin of the image plane z 0
and the saturation degree λ of the elastic SPB. The two parameters are correlated to
determine the shape and integration of the ReV(z; z 0 ,λ). VLEED concerns the integration area of the ReV(z) function instead of the exact value of each in determining
the phase shift and the beam reflectivity. Hence, it is essential and realistic to set
the λ(z 0 ) as a function of z 0 to reduce the number of solutions. When the z 0 moves
away of the lattice, the SPB tends to be more saturated because of polarization, as
observed from STM image. If the z 0 moves inward, the SPB will be less saturated
because of ion or atomic vacancy formation.
Figure 16.1 shows the correlation between z 0 and λ at various energies. There
are two groups of correlation curves in each panel corresponding to a 2nπ phase
change of one another. The infinite couples of (z 0i , λ i ) along each curve in a group
provide matching between calculated and experimentally detected intensities I cal /I exp
= 1.00 ± 0.05. In order to establish the correlation between the couple of z 0 and λ
parameters, one can collect data ((z 0i , λ i ), i = 1, 2, …, 30) from each z 0 -λ curve
and then integrate the ReV(z; z 0i , λ i ). The integration ranges from the second atomic
plane (D 12 ) to infinitely far away (in practice the upper limit is chosen as −100 a.u.)
from the surface:
I (Z 0i , λ i ) =
−∞
D12
ReV(z; z 0i , λ i )dz (i = 1, 2, . . . , N = 30)
N = 30 is sufficient for statistical analysis. The average of the integration values
of each curve in Fig. 16.1 is:
I =
1
N
N
i=1
I (Z 0i , λ i ) (N = 30),