The invention relates to a method of determining parameters
of a sample by X-ray scattering comprising the steps of
- a) exposing the sample to X-rays and measuring scattered X-ray intensity;
- b) generating a parameterized model of the sample which is used for numerical
simulation of scattered X-ray intensity on the basis of a physical scattering theory;
- c) comparing the experimental and simulated X-ray scattering data to generate
an error value;
- d) modifying the parameters of the model by means of a genetic algorithm involving
an amount of individuals each with an equal number N of encoded parameters forming
a generation and applying the genetic operators of "selection", "crossover" and
"mutation" used for composing successive generations of evolving individuals.
Such a method has been disclosed in A. Ulyanenkov, K. Omote,
J. Harada; Physica B 283 (2000), 237-241.
X-ray analysis is applied in numerous ways in biology,
chemistry, physics and materials science. The interaction of x-rays with a sample
can reveal information about the sample that would be difficult to obtain by other
means, if at all. Moreover, x-ray analysis is a non-destructive method, making it
One way to investigate unknown parameters t1...
tN of complex samples such as multilayer structures by X-ray analysis
are the so called trial and error methods. These methods start with the measurement
of an experimental scattering spectrum of the sample. Then, numerous test structures
(each with a different set of parameters defining a respective test structure) are
assumed and simulated scattering spectra for these test structures are calculated
on the basis of a physical model (e.g. kinematic scattering theory). The parameters
defining the test structure whose theoretical spectrum has the best match with the
experimental scattering spectrum are considered to be approximately the true parameters
of the sample. The best match is typically identified as the global minimum of an
error value, in particular a &khgr;2 difference function of the experimental
spectrum and the simulated spectra. Typical parameters thus approximated include
film or layer thickness, lattice strain, contamination or doping level and the like.
In general, it is impossible to calculate simulated spectra
for all possible test structures due to the finite capacity of computers. To the
contrary, it is necessary to provide a decision algorithm determining the test structures
to be calculated. There is an arbitrarily chosen test structure (or a set of test
structures) to begin with, and the next test structures to be calculated are chosen
by the decision algorithm on the basis of the quality of match of the previous test
The most common decision algorithms are gradiental methods.
If for a certain parameter ti the difference function &khgr;2
decreases when changing the parameter ti slightly into one direction,
then the next set of parameters is chosen such that ti is shifted slightly
in this direction. If the difference function &khgr;2 increases again,
however, the parameter ti is no more changed into said direction. Typical
gradiental methods include the Simplex algorithm, the Levenberg-Marquardt algorithm
and the Newton algorithm. However, gradient methods run the risk of getting trapped
in local minima not identical with the global minimum of the difference function,
and no useful approximation for the true parameters of the sample can be found in
such a case.
One type of decision algorithm overcoming this problem
is the so called genetic algorithm. This decision algorithm works as follows:
In the beginning, there is an arbitrarily chosen starting
set of so called individuals (or chromosomes). Each individual represents a test
structure. These individuals all carry an equal number of so-called encoded parameters
(or genes). The encoded parameters are each correlated to a physical parameter of
the test structure, e.g. an impurity level or a layer thickness. Thus the number
of encoded parameters corresponds to the number of physical parameters of the modeled
sample to be investigated. All individuals as a whole represent a population. A
typical population comprises 20 to 50 individuals.
During the genetic algorithm, the population is altered.
The starting population is transformed into a new population, and this new one is
altered into another one and so on. The populations at different steps of alteration
are called generations. A generation to be altered is called a parent generation,
and the generation newly created out of the parent generation is called an offspring
generation. Thus, the starting population is the first parent generation in the
algorithm. Subsequent generations of individuals are intended to contain an individual(s)
approaching the global minimum of the error value.
For the creation of an offspring generation, at first the
error value of each individual in the parent generation is calculated. The elite,
i.e. a number of typically one or two individuals with the smallest error value
in the parent generation, are copied without any alteration into the offspring generation.
Then the individuals of the parent generation are subjected to genetic operators.
These operators include a selection operator which determines
individuals of the parent generations which are used to produce the offspring generation.
Another operator is the mutation operator. It decides about
changes in single genes of an offspring individual, in particular which genes and
to what degree these genes are altered. In the state of the art, these mutations
are random changes in the genes.
A third operator determines the crossover of the genes
of two mating parent individuals (crossover operator). It decides what gene is taken
from which parent. In the state of the art, it is known to cut the parent chromosomes
(=individuals) in half, and the upper part of the offspring individual is chosen
as the upper part of the first parent individual, and the lower part of the offspring
individual is chosen as the lower part of the second parent individual.
Through application of these genetic operators, an unlimited
number of subsequent generations can be created. Typically, about 10 to 100 generations
are calculated. In the final (youngest) offspring generation, the individual with
the lowest error value corresponds to the test structure with the best match with
the true sample structure. The genetic algorithm is very reliable and gives a good
approximation to the true sample structure, with only a low risk of running into
a local minimum of the error value function.
In order to increase the final accuracy of the determined
sample parameters, a gradiental method may subsequently be applied, with the set
of parameter determined in the genetic algorithm as a starting set of parameters
in the gradiental method. This starting set of parameters should be close enough
to the global minimum to avoid trapping in a non-global minimum.
However, the genetic algorithm requires a lot of calculations,
determined by the amount of individuals per generation, the number of generations
and the complexity of the genetic operators.
It is the object of the present invention improve the genetic
algorithm such that it can approximate the true sample parameters faster and with
a higher accuracy.
This object is achieved by a method as described in the
beginning, characterized in that after a given number k of successive generations
the genetic operator of "mutation" is no longer applied in evolution of further
In other words, in the last few generations calculated,
the mutation genetic operator is deactivated. By this means, random changes in the
genes are avoided when the encoded parameters and the respective test structure
are already close to the corresponding desired true structure. During creation of
the last generations, random changes most often worsen the quality of match of the
corresponding test structure and lead away from the wanted global minimum of the
error value function.
When dispensing with the random changes during the last
generations in accordance with the invention, the exact global minimum can be approximated
by the genetic algorithm with a higher accuracy, since only genes already close
to this minimum are available for new combinations in the offspring generations.
The inventive genetic algorithm during the last generations can replace a subsequent
application of a gradiental method and thus save calculation power and time.
When a total number of G generations is created by the
inventive genetic algorithm, and after k generations the mutation operator is no
more applied, then k may be chosen such that a minimum number of generations G-k
is performed without the mutation operator, for example with G-k ≥ 5, or
such that a minimum fraction of generations (G-k)/G is performed without the mutation
operator, for example (G-k)/G ≥ 20%. Alternatively or in addition, it is
also possible that the transition generation (i.e. the determination of k) can be
coupled to a minimum quality of match of the best fitting individual of the transition
generation: as soon as the quality of match of the best fitting individual of a
generation reaches a limiting value, subsequent generations are generated without
the mutation operator.
In a particularly preferred variant of the inventive method,
from one generation to the next a "movement" genetic operator is applied which moves
at least some of the encoded parameters of at least some of the individuals towards
the respective encoded parameters of the individual with the smallest error value.
The proposed movement genetic operator is somewhat similar
to the mutation operator, but the change in the gene (=encoded parameter) or genes
is not random, but intentional and directed. The alteration of an encoded parameter
affected by the inventive movement operator is such that the difference between
the encoded parameter of the parent individual and the encoded parameter of the
best matching individual of the parent generation is larger than the encoded parameter
of the offspring individual and the encoded parameter of said best matching individual
of the parent generation. In most cases, the inventive movement will lead to an
offspring individual with a lower error value than a randomly mutated offspring
The inventive movement of an encoded parameter may be set
to the digital minimum, i.e. the encoded parameter is moved only one digit towards
the value of the best matching individual, or it may comprise a certain percentage
of the encoded parameter difference, such as 20%. Preferably, the percentage is
less or equal to 50%.
It is possible to affect all or just a fraction of the
individuals of a generation by the movement operator. Moreover, it is possible to
alter all or only a part of the genes of an affected individual. Preferably, each
affected individual is subjected to the movement operator only with respect to a
few, in particular only one encoded parameter.
Note that the inventive movement operator can be applied
to a parent individual immediately before creating an offspring individual, or directly
at the creation of the offspring individual.
In a preferred variant, at least some of the offspring
individuals of every successive generation evolve from more than two parent individuals
of the respective previous generation. In particular, genes from three parent individuals
can be combined to generate a new offspring individual. By this means, the genes
of selected individuals can be mixed in less generations, and a wider parameter
space can be explored.
Further preferred is a variant of the inventive method,
wherein at least some of the offspring individuals of every successive generation
evolve from only one parent individual of the respective previous generation (="cloning").
Genetic operators such as mutation and movement may still apply, however. Thus,
successful gene combinations can be kept even when they do not belong to a kept
elite, and can be slightly changed with a high chance of being an individual of
low error value in the offspring generation again.
In an alternative and advantageous variant of the inventive
method, the "crossover" genetic operator is modified such that the encoded parameters
of each offspring individual is selected randomly from the respective encoded parameters
of the parent individuals of the previous generation. The vector describing the
random selection of the parent individual for each gene is called the random chromosome.
The random choice of the genes accelerates the mixing of selected successful genes
and promotes the exploration of a wide parameter space.
It is particularly advantageous for a variant if from one
generation to the next at least the individual with the smallest error value is
transferred unchanged (="elite"). By this means, the individual with the lowest
error value in the offspring generation always has a lower or equally low error
value as the individual with the lowest error value in the parent generation. Therefore
the matching of the best matching individuals can only improve in the course of
A preferred variant of the inventive method provides that
the error value is generated by means of a weighting function, in particular based
on logarithmic values of the scattering data. By means of the logarithmic scale,
an overestimation of regions of high absolute values in the error value function
is avoided. The weighting function is preferably a logarithmically scaled instead
of absolute x-ray intensity within the &khgr;2 difference function.
Further advantages can be extracted from the description
and the enclosed drawings. The features mentioned above and below can be used in
accordance with the invention either individually or collectively in any combination.
The embodiments mentioned are not to be understood as exhaustive enumeration but
rather have exemplary character for the description of the invention.
The invention is shown in the drawings:
- Fig. 1
- shows a flowchart of a genetic algorithm (=CGA, Classical GA) according to the
state of the art;
- Fig. 2
- shows a flowchart of an inventive extended genetic algorithm (=XGA);
- Fig. 3
- shows a graphic representation of an analytical function used to test the inventive
- Fig. 4
- shows a convergence chart for the CGA, CGA + movement operator, and XGA for
the analytical function of Fig. 3;
- Fig. 5
- shows an X-ray reflectivity diagram of a Au/Fe3O4/MgO
sample, with experimental results dotted and XGA simulation results as a continuous
- Fig. 6
- shows the convergence of the &khgr;2 function for CGA, XGA, Simulated
Annealing and Simplex Method when fitting the experimental x-ray reflectivity data
of Fig. 5;
- Fig. 7
- shows an X-ray diffraction diagram of a Si1-xGex/Si(004)
Bragg reflection diagram, with experimental results dotted and XGA simulation results
as a continuous line;
- Fig. 8
- shows the convergence of the &khgr;2 function for XGA and Simulated
Annealing when fitting the experimental x-ray diffraction data of Fig. 7.
In the following, the inventive extended genetic algorithm
and its application to x-ray analysis are explained in detail.
The classic scheme of genetic algorithm is extended to
improve the robustness and efficiency of the method. New genetic operators implemented
in this work are shown to increase the convergence speed and reliability of the
optimization process. A complex model function with multiple local extrema and real
x-ray reflectivity and diffraction data have been used to testify the modified algorithm.
The effectiveness of the new technique is compared to other optimization methods.
X-ray metrological methods in science and industry have
been proved to be efficient techniques for sample characterization and growth process
development and control. A large variety of sample structures can be probed by x-rays
to examine film thickness, interface roughness, crystallographic lattice strain
and distortion, material contamination, etc.. Despite the measured x-ray data can
be used directly for evaluation of some sample parameters, the detailed knowledge
about the sample structure can be obtained using special data treatment procedures
only. These procedures usually utilize a trial-and-error technique, which uses a
parameterized sample model for the simulation of the x-ray scattering process, and
then the difference &khgr;2 between calculated and measured intensities
is minimized in view of the sample parameters. Thus, an effective and robust optimization
algorithm is required for accurate data interpretation. Moreover, in most of experimental
setups, x-ray measurements provide the magnitude of scattered x-ray intensity, whereas
the amplitude and the phase of x-rays are lost, and that is why no procedures are
available for reconstruction of sample physical parameters directly from measured
intensities. The loss of x-ray phase information during the measurements causes
the ambiguity of data interpretation results, e.g. when several sample models result
in a comparably small difference &khgr;2 between measurements and simulations.
This situation requires from the optimization technique to be able to find not only
a single global minimum, but also a set of deepest minima on the &khgr;2
hypersurface, which may contain real physical solutions. Genetic algorithms (=GA)
[Ref. 1], widely used nowadays, seem to be most successful in solving all mentioned
problems in x-ray data analysis. They combine a stochastic search and strategy aimed
features that help to find the global minimum along with other local minima of the
cost function &khgr;2 comparable on magnitude.
In the present work, we propose specific modifications
of the classic GA scheme, which improve the efficiency of the algorithm. These new
implementations into standard GA have been testified and the results are compared
with results obtained by classic GA (CGA). As the test objects, a complex model
function possessing multiple minima, and real x-ray diffraction and reflection measurements
have been used. In section II, we describe and qualitatively ground the implemented
modifications of the genetic algorithm. Section III presents the test results of
the proposed method, called further extended GA (XGA) for model multi-minima function
and the comparison with CGA. The real x-ray diffraction and reflection measurements
are fitted by using extended GA and various methods, and the performance of XGA
is compared with other optimization techniques [Ref. 4].
II. Modifications of classic GA
An explanation of the details of genetic algorithms can
be found in numerous publications (see, e.g. [Ref. 1] and citations therein), and
in this section only the principle construction of a classic GA is described. Genetic
algorithms exploit the Darwin's evolutionary idea and the principle of survival
of most adapted individuals in a population. The first step in any GA application
is the formalization of the parameters to be investigated in order to unify all
the operations with them. Usually, the binary representation of the whole parameter
space is used, that allows to easily code the problem. Every point in the parameter
space then represents a unique physical state of the investigated system, and the
goal is to explore this space to find the point which gives the best fit according
to the pre-defined fitness criteria. Since the fitted parameters are formalized,
any set of them describing comprehensively the system is considered as an
individual; the limited number of individuals compose a population,
which is evolving on the basis of some genetic rules. The procedure of parameter
formalization is called encoding, whereas reverse operation of obtaining
the physical parameters is a decoding.
The typical GA search procedure consists of the following
steps, depicted in Figure 1. Firstly, the random population of individuals,
which are the points in the parameter space or, the same, the single set of the
system fitted parameters, is created (the size of the population is an internal
GA parameter). Then the population begins to evolve by the production of new generations,
i.e. the creation of the successive populations on the basis of primary one. The
most principle rules (called also GA operators) used for the creation of
a new generation of individuals are selection of parents, crossover
and mutation operators. The first operator regulates the parents selection
procedure for the production of offsprings for the new generation. The second rule
describes how the parents hand down their features (single parameters, genes) to
children. The mutation operator provides irregular changes of offsprings to strengthen
a statistical nature of GA. Applying these basic operators along with other optional
rules (for example, elitism, i.e. the transfer of the fittest individuals
from the current generation to the successive generation without any changes), after
a certain number of generations GA delivers the fittest individual, i.e. the set
of parameters at which the system satisfies the fitness criteria in the best way.
The probability to find the best existing set of parameters by GA increases with
the number of generations evolved, i.e. with the length of evolution.
All of the operators described above have various implementations
and internal parameters, which are used to optimize the GA efficiency for different
applications. The genetic algorithm, being correctly tuned for the concrete problem,
demonstrates excellent results as an optimization and search procedure [Ref. 2].
In this work, we propose further modifications of basic GA operators and general
rules to increase the efficiency and reliability of the algorithm. These modifications
- [1.] One more basic GA operator is implemented, the movement operator.
Before the selection of parent individuals for mating, a limited number of parents
are moved towards the best individual in the population, i.e. the parameters of
the moved parents are changed closer to the parameters of the fittest individual.
This operator improves the convergence of GA, by increasing the amount of individuals
in the vicinity of the fittest one.
- [2.] Each offspring can have not only two parents but arbitrary number of parents.
In our case, we particularly propose a number of parents n = 1,2,3. This multiple
choice combines the classic two-parental approach with single "cloning" (n = 1)
of parents, which improves the convergence of GA by increasing an amount of individuals
with better fitness, and multi-parental options (n = 3), which increase the variety
of a population by wider exploration of the parameter space.
- [3.] A new crossover principle, providing a better mixture of parents' features
in children, is implemented by using a random sequence of bits, the so called
random chromosome. When parents mate, the offspring inherits the parameter
from that parent, to which the corresponding bit in the random chromosome points
to. This modification results in a more efficient parameter mixture in the crossover
procedure, optimizes the source code and accelerates the algorithm because only
simple boolean operations are involved (in particular in the case of two parents).
- [4.] A more effective principle for the formation of new generations is developed.
The sequence of formation is the following: firstly, the limited number of elite
individuals is moved to the successive population; then the population is filled
by the individuals from the previous generation and by new offsprings randomly,
with a probability depending on the individual's fitness and the number of children
left. This improvement prevents individuals with good fitness from being replaced
by new children and replaces the individuals with rather bad fitness, i.e. increasing
the convergence and decreasing the probability for the best solutions to be skipped.
- [5.] The mutations are forbidden in a given number of last generations. This
modification permits to use solely GA for a fine parameter fitting cycle. Because
of their stochastic nature, conventional GA schemes rarely deliver exactly the point
with the best fitness in the parameter space. Usually, the gradiental optimization
methods have to be further applied in the vicinity of the solution found by GA to
refine the parameters [Ref. 2] (see flowchart in Figure 1). For example, if the
fitness criterion is based on some cost function (typically, the difference between
experimental and simulated data), and the aim of GA is to find the global minimum
of this cost function, the final solution is usually found by GA near this minimum
point. A gradiental method like Levenberg-Marquardt has to be used to "slide down"
the solution onto the bottom of the function &khgr;2 .Our fifth modification
allows to precisely localize the final solution with an accuracy suitable for data
interpretation, what reduces the whole optimization time in comparison with the
cascade use of GA and gradiental techniques.
Thus, our modified XGA version of genetic algorithm has
the following principle scheme (flowchart in Figure 2):
- (i) generation of a random population consisting of S individuals (population
- (ii) evaluation of fitness;
- (iii) movement of M individuals towards the fittest one;
- (iv) creation of C children by using one parent (p1, cloning), two
parents (p2, crossover through random gene), and three parents (p3),
C = p1 + p2 + p3;
- (v) if mutations are permitted in current generation, m mutations of parameters
are carried out, each being an inversion of a random bit in an corresponding children's
- (vi) a new population is created.
To optimize a source code, the new generation is constructed
on the basis of the previous one, firstly by selection of unchangeable elite E individuals,
then by substitution of old individuals by C children in dependence on the fitness
of replaced individuals and the number of remained children. The evolution proceeds
by repeating the cycle in Figure 2 either G times (number of generations) or until
a requested fitness tolerance is reached. In the last F generations, the mutations
are forbidden to refine the fit results.
III. XGA vs. other optimization techniques
To study an effectiveness of above-mentioned XGA implementations,
we used firstly a smooth two-dimensional analytical function
possessing single global maximum at (x;y) = (0.75; 0.625) and multiple local maxima
(Figure 3); the function is defined within the interval
Figure 4 shows the convergence diagram for CGA, CGA with only movements,
and full-featured XGA. Evidently, XGA finds global maxima faster than other modifications
at equal conditions (the population consists of 200 individuals). We also carried
out the tests with a more complicated model function, viz. linear combination
of several functions similar to the depicted one in Fig. 3, but with different values
of the constants. Then the statistical error of trapping in the wrong maxima has
been evaluated for CGA and XGA by performing multiple runs of both algorithms. The
error for CGA is found to be 28% against 4% for XGA, and thus XGA demonstrates higher
reliability in comparison with a classic GA scheme.
XGA has been also applied for the fitting of real experimental
data from x-ray reflectivity (XRR) and high-resolution x-ray diffraction (HRXRD),
the two most commonly used x-ray techniques. The convergence and speed of XGA are
compared with other known methods, simulated annealing (SA) and the simplex method
(SM). Figure 5 shows measured (open dots) and simulated (solid lines) x-ray
reflectivity from gold and magnetite thin solid films on MgO substrate at a wavelength
&lgr; = 0.154056 nm. The sample model consisting of the sequence Au/Fe3O4/MgO
with the nominal thicknesses 55nm/120nm/substrate and some roughness at the interfaces
between the layers has been used for fitting of reflectivity data simulated by Parratt's
formalism [Ref. 3] to the measured curves. The fitted parameters were the thicknesses
of the layers tAu and tFe3O4 and the roughnesses of the sample
surface and the interfaces. All the methods (CGA, SA, SM, XGA) resulted in an acceptable
fitness of curves with slight differences in the refined parameters which, however,
are within the precision of experimental data. The values of the thicknesses and
roughnesses are found to be tAu ≅ 53.8 nm, tFe3O4≅
146.3 nm, &sgr;surf ≅ 0.78 nm, &sgr;Au/Fe3O4 ≅
0.1 nm, &sgr;Fe3O4/MgO ≅ 0.3 nm. However, the effectiveness
of the methods is evidently different, as follows from the Figure 6 showing
the &khgr;2 convergence diagrams for each used technique. To adjust
the time scales of algorithms, the iteration scales of SA and SM are brought into
correspondence to GA's generations scale by multiplying them by the ratio of computation
times tGA/SA and TGA/SM. Diagrams show that XGA finds the
best available solution faster than the other methods. Figure 6 demonstrates also
the advantage of XGA due to the fifth implementation from the Section II, i.e. the
prohibition of mutation in the final stage of evolution. After 100 generations,
CGA is still localized in the vicinity the global minimum of &khgr;2,
whereas XGA has already found the point, recognized by simplex method as an exact
global minimum (with certain tolerance), due to ban of mutations after the 60th
generation. Thus, not necessarily gradiental methods have to be used to refine a
X-ray diffraction measurements (open dots) from a Si1-xGex/Si
(004) Bragg reflection at wavelength &lgr; = 0.154056 nm are shown in
Figure 7. Two parameters, the thickness of the Si1-xGex/Si
layer and the concentration of germanium x, have been fitted (solid line in Fig.
7). The discrepancy between the curve simulated by dynamical diffraction theory
and the experimental data on the right wing of the substrate peak are caused by
a diffuse x-ray scattering, which is not taken into account in the present calculations.
Both SA and XGA methods used for minimization of the &khgr;2 function
lead to close values of thickness tSiGe≅ 65.7 nm and concentration
x ≅ 9.2%. However, the &khgr;2 convergence charts in the
Figure 8 clearly demonstrate the advantage of XGA in speed (SA iterations
are rescaled by time ratio factor).
The inventive extended genetic algorithm is shown to be
the most effective and robust optimization technique in comparison with the classic
genetic algorithm, simulated annealing and the simplex method. Phenomenological
tests with complex model functions possessing multiple extrema, as well as with
real experimental x-ray data, both reflectivity and diffraction, have shown the
advantages of XGA. In view of large computer resources required for fitting of x-ray
data, the robustness and quickness of XGA plays an essential role for precise data
analysis in x-ray analytical software.
- [Ref. 1] D.E.Goldberg, Genetic Algorithms in Search, Optimization and Machine
Learning, Addison-Wesley, Reading, MA, 1989.
- [Ref. 2] A.Ulyanenkov, K.Omote, J.Harada, Physica B 283 (2000) 237.
- [Ref. 3] L. G. Parratt, Phys. Rev. 95, 359 (1954)
- [Ref. 4] F. H. Walters, L.R. Parker, S. L. Morgan and S. N. Deming, Sequential
Simplex Optimization, CRC Press, Boca Raton, FI, 1991; H. L. Aarts and J. Kost,
Simulated Annealing and Bolzman Machines, John Wiley, New York, 1989
- Fig. 1: Flowchart of the classic GA scheme;
- Fig. 2: Flowchart of the modified XGA scheme;
- Fig. 3: The model function has numerous local maxima along with a hardly recognizable
- Fig. 4: Convergence charts of classic GA, partly modified classic GA and full-featured
XGA recorded for the model function from Figure 3;
- Fig. 5: Measured (open dots) and fitted (solid line) x-ray reflectivity for
- Fig. 6: Convergence of the &khgr;2 function for various optimization
techniques recorded during the fitting of the x-ray reflectivity data on Figure
5. The iteration scale of Simulated Annealing and Simplex Method techniques is reduced
to the GA generations scale on the principle of equal computation time;
- Fig. 7: Measured (open dots) and fitted (solid line) (004) Bragg x-ray diffraction
for Si1-xGex/Si sample;
- Fig. 8: Convergence of &khgr;2 function for XGA and simulated annealing
recorded during the fitting of x-ray diffraction data on Figure 7. The iteration
scale of the Simulated Annealing method is reduced to the GA generations scale on
the principle of equal computation time.