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15 Methodology: Parameterization
The following summarizes the numerical processing of calculation:
• The SPB parameters were fixed first by using the known identities of the missingrow structure model [4] and the conventional SPB.
• To set the constants in the single-variable SPB functions, we calculated the VLEED
data at a few energies by adjusting the most sensitive parameters.
• Uncertainty of the solutions was overcome by introducing the correlation among
the SPB parameters as well as the local work function. The infinite number of
solutions due to independently varying λ and z 0 were reduced to finite ones by the
function of λ(z 0 ). Introducing the local work function then ascertain the unique
solution.
15.1.3 Calculation Methods
Conventionally, one can only change one parameter at a time in calculations.
Although the computer can automatically do the numerical processing, this method
is tedious and fruitless. The orthogonal optimizing method is widely used in optimization for experimental design, not only for reducing the time span of experiments
but also for predicting the trends of results and correlation among the variables.
Complementing the orthogonal optimization, z 0 -scanning and z 0 -optimizing was
carried out. After the SPB and geometrical parameters have been fixed, calculations were performed based on the single-variable parameterization by varying z 0
over −2.5 [refer to pure Cu(001)] ± 1.25 with 0.25 (a.u.) steps. Contour plot of z 0
versus E is then drawn with results satisfying I c (z 0i , E)/I e (E) = 1.0 ± 0.05. The z 0 (E)
contour plot is the unique yield of the z 0 -scanning method, which gives the best fit of
the measurement and shows all the possible solutions as well. This z 0 -scanning and
z 0 -optimizing method is also convenient to compare different models and to refine
the SPB shapes and the bond geometries.
Once the refinement of the z 0 (E) plot is completed, the program automatically fits
the value of the z 0 (E i ) to give a desired level of agreement (normally I c (E i )/I e (E i )
= 1.00 ± 0.01 ~ 0.03). In contrast to the z 0 scanning method, the step of z 0
automatically varies from 0.25 to 0.0005 depending on the result of κ = I c (E i )/I e (E i ).
If κ reaches the required precision, calculation will automatically turn to the next
step E i+1 . This method yields simply the geometry-dependent z 0 (E) profile and a
replication of the measured VLEED spectrum. Quantities such as the bond geometry,
work function, barrier shapes and energy band structure are automatically given as
the product of data processing. Therefore, careful calibration of the experimental
data is essential.
15 Methodology: Parameterization
The following summarizes the numerical processing of calculation:
• The SPB parameters were fixed first by using the known identities of the missingrow structure model [4] and the conventional SPB.
• To set the constants in the single-variable SPB functions, we calculated the VLEED
data at a few energies by adjusting the most sensitive parameters.
• Uncertainty of the solutions was overcome by introducing the correlation among
the SPB parameters as well as the local work function. The infinite number of
solutions due to independently varying λ and z 0 were reduced to finite ones by the
function of λ(z 0 ). Introducing the local work function then ascertain the unique
solution.
15.1.3 Calculation Methods
Conventionally, one can only change one parameter at a time in calculations.
Although the computer can automatically do the numerical processing, this method
is tedious and fruitless. The orthogonal optimizing method is widely used in optimization for experimental design, not only for reducing the time span of experiments
but also for predicting the trends of results and correlation among the variables.
Complementing the orthogonal optimization, z 0 -scanning and z 0 -optimizing was
carried out. After the SPB and geometrical parameters have been fixed, calculations were performed based on the single-variable parameterization by varying z 0
over −2.5 [refer to pure Cu(001)] ± 1.25 with 0.25 (a.u.) steps. Contour plot of z 0
versus E is then drawn with results satisfying I c (z 0i , E)/I e (E) = 1.0 ± 0.05. The z 0 (E)
contour plot is the unique yield of the z 0 -scanning method, which gives the best fit of
the measurement and shows all the possible solutions as well. This z 0 -scanning and
z 0 -optimizing method is also convenient to compare different models and to refine
the SPB shapes and the bond geometries.
Once the refinement of the z 0 (E) plot is completed, the program automatically fits
the value of the z 0 (E i ) to give a desired level of agreement (normally I c (E i )/I e (E i )
= 1.00 ± 0.01 ~ 0.03). In contrast to the z 0 scanning method, the step of z 0
automatically varies from 0.25 to 0.0005 depending on the result of κ = I c (E i )/I e (E i ).
If κ reaches the required precision, calculation will automatically turn to the next
step E i+1 . This method yields simply the geometry-dependent z 0 (E) profile and a
replication of the measured VLEED spectrum. Quantities such as the bond geometry,
work function, barrier shapes and energy band structure are automatically given as
the product of data processing. Therefore, careful calibration of the experimental
data is essential.