Master’s Thesis
A first principles study of the thermodynamics of phase
separating systems
-The examples RhPd and AlZn-
Jimmy Johansson
LITH-IFM-A-EX--09/2125--SE
Department of Physics, Chemistry and Biology Linköpings universitet
Master’s Thesis
A first principles study of the thermodynamics of phase separating systems
-The examples RhPd and AlZn-
Jimmy Johansson
Supervisors Igor Abrikosov
IFM, Linköpings Universitet Björn Alling
IFM, Linköpings Universitet
Examiner Igor Abriksov
2009-05-25 Theoretical Physics
Department of Physics, Chemistry and Biology Linköping University,SE-581 83 Linköping, Sweden
URL för elektronisk version
http://urn.kb.se/resolve?urn=urn:nbn:s e:liu:diva-18517
ISBN
ISRN: LITH-IFM-A-EX--09/2125--SE
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Serietitel och serienummer ISSN
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Språk Language Svenska/Swedish Engelska/English ________________ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport _____________
Titel A first principles study of the thermodynamics of phase separating systems
-The examples RhPd and
AlZn-Title
Författare Jimmy Johansson
Author
Nyckelord
Keyword Density Functional Theory, Phase Separating Alloy, Coherent Potential Approximation, Exact Muffin Tin Orbital method, Generalized Perturbation Method
Sammanfattning
Abstract
A screened GPM approach in an EMTO-CPA framework was investigated in order to study its ability of describing transition temperatures in phase separating systems, i. e. systems giving either a random or a cluster structure depending on the temperature and the relative concentration of the ingoing atoms of the binary alloy used for the study. A motivation for the study is that the method works well for ordering systems, i. e. systems giving either a random or ordered structure dependent on the temperature and the relative concentration of the components in the binary alloy. Thereby is it of interest to find out the methods capacity in phase separating systems. The so called GPM potentials derived in the approach were applied in statistical Monte Carlo simulations for this purpose. The systems chosen for the investigation were the RhPd and the AlZn binary alloy systems.
For both systems the method showed acceptable accuracy when properties as lattice parameter and mixing enthalpy were calculated. The quality of the derived GPM potentials has also been checked by calculating ordering energy for different ordered structures; directly from first principles calculations and from the GPM approach. The results were in acceptable agreement and thereby indicating that the GPM potentials were reliable. The transition temperatures in the RhPd phase diagram, derived by the statistical Monte Carlo simulations showed anyway deviation from experimental results. The error in the predictions might be due to the existing concentration dependencies in the GPM potentials. The conclusion from this study is that the Monte Carlo scheme might be inconvenient in order to handle the concentration dependencies seen in the GPM potentials.
A screened GPM approach in an EMTO-CPA framework was investigated in order to study its ability of describing transition temperatures in phase separating systems, i. e. systems giving either a random or a cluster structure depending on the temperature and the relative concentration of the ingoing atoms of the binary alloy used for the study. A motivation for the study is that the method works well for ordering systems, i. e. systems giving either a random or ordered structure dependent on the temperature and the relative concentration of the components in the binary alloy. Thereby is it of interest to find out the methods capacity in phase separating systems. The so called GPM potentials derived in the approach were applied in statistical Monte Carlo simulations for this purpose. The systems chosen for the investigation were the RhPd and the AlZn binary alloy systems.
For both systems the method showed acceptable accuracy when properties as lattice parameter and mixing enthalpy were calculated. The quality of the derived GPM potentials has also been checked by calculating ordering energy for different ordered structures; directly from first principles calculations and from the GPM approach. The results were in acceptable agreement and thereby indicating that the GPM potentials were reliable.
The transition temperatures in the RhPd phase diagram, derived by the statistical Monte Carlo simulations showed anyway deviation from experimental results. The error in the predictions might be due to the existing concentration dependencies in the GPM potentials. The conclusion from this study is that the Monte Carlo scheme might be inconvenient in order to handle the concentration dependencies seen in the GPM potentials.
First of all thanks to my supervisor and examiner Professor Igor Abrikosov who let me in to this project and supported me in it. Also thanks to my second supervisor Björn Alling who friendly helped me with anything from the computer program to the conceptual understanding of the physics behind the project. I also found encouraging people on my way. Two guides to mention are Lars Alfred Engström who told us students that it is first when you are unable to touch the seafloor under your feet that you recognize the depths in the ocean of physics. The other was Professor Rolf Riklund who let us students experience the beauty of physics. Also thanks to my mother who gave me patience and to my father who gave me the ability to do mathematics.
Jimmy Johansson Linköping 2009
1 Introduction 1
1.1Background and previous work... 1
1.2Thesis outline... 3
2 Theory 5
2.1Density Functional Theory (DFT)... 5
2.1.1 Introduction... 5
2.1.2 The Hohenberg-Kohn theorems and the Kohn-Sham equations 6
2.1.3 Approximations of the Exchange-Correlation energy... 7
2.1.4 DFT and its limitations... 7
3 Computational methods 9
3.1Muffin thin orbital methods... 9
3.2The Coherent Potential Approximation... 11
3.3Madelung energy... 12
4 Alloy configuration 13
4.1The GPM approach... 13
4.2Monte Carlo simulations... 14
5 Results and calculations 17
5.1Calculation of ground state properties for pure elements... 17
5.2K-point convergence test for pure elements... 17
5.3Equilibrium radius and mixing enthalpy on alloy systems... 18
5.3.1 The RhPd alloy system equilibrium radius and mixing enthalpy 18
5.3.2 The AlZn alloy system equilibrium radius and mixing enthalpy 19
5.4Effective cluster interactions (ECI) for alloy systems... 21
5.4.1 The RhPd alloy system ECI... 21
5.4.2 The AlZn alloy system ECI... 24
5.5Monte Carlo convergence tests for alloy systems... 27
5.5.1 The RhPd alloy system Monte Carlo convergence tests... 27
5.6Monte Carlo phase diagram for alloy systems... 31
5.6.1 The RhPd alloy system phase diagram... 31
6 Discussion and conclusions 33
A Graphs over Effective Cluster Interactions 35
B Graph over Ordering Energy 37
1
Chapter 1
Introduction
1.1
Background and previous work
Alloys are used in many commercial applications and thereby of interest due to describing them in an accurate manner. Binary alloys are generally classified into two generic types of ordering or phase separating character. Powerful tools exist to predict and analyse their behaviour on a theoretical basis using different approaches. In this thesis an EMTO–CPA method is applied in combination with a screened GPM approach together with statistical Monte Carlo simulations in order to calculate phase transition temperatures. One goal with this project is to actually see how well it can describe phase separating systems of binary alloys. In this specific case it is about finding the phase transitions temperature, i.e. the temperature of the transition from random to a cluster structure in the alloy and compare it with values from known studies. A motivation for the study is that the method works well for ordering systems, i. e. systems giving either a random or ordered structure dependent on the temperature and the relative concentration of the components in the binary alloy. Thereby it is of interest to find out the methods ability in phase separating systems. The so called GPM potentials derived in the approach were applied in statistical Monte Carlo simulations for this purpose.
The systems chosen for this purposes is the Rhodium-Palladium system; RhPd, and the Aluminium-Zink system; AlZn. The AlZn alloy is used for example as a protective coating of steel in corrosive environments [1] and also used in studies of spinodal decomposition, a process where components in an alloy separate spontaneously into regions with different concentrations. The RhPd alloy also has industrial applications used for its catalysts abilities [2]. Both systems have been studied in previous works and can be considered as rather well known. See for example ref. [3, 4, 5].
Taking a closer look to the AlZn system gives that it is a phase separating system with positive mixing enthalpy with a wide fcc miscibility gap approximately between 15 and 60 atomic percent Zn seen in the phase diagram below fig. 1.1. In pure form Al has an fcc (face centered cubic) structure and Zn has an hcp (hexagonal close packed) structure. The atomic radii of the two elements are similar with 1.43 Å for Al and 1.37 Å for Zn, and so also the electronegativities of the elements with 1.61 and 1.65 for Al respectively Zn (Pauling scale). Two differences are the equilibrium crystal structure mentioned above (fcc respectively hcp) and the electron per atom ration [3]. As seen in the phase diagram fig. 1.1 the alloy shows a complex behavior. Of particular interest in this study is the so called miscibility gap described above. It gives information about the tendency of phase separation. The interpretation of the diagram in the region between 15 and 60 atomic percent Zn and around 600 K is that if a system is lowered in temperature from a region higher than 624.6 K [3], the top of the monotectoid line, the system will separate in two coexisting fcc solid solutions with different concentration, a_1 and a_2 in the diagram, i.e. one gets a phase separating system. The system thereby goes from a random structure to a cluster structure.
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Fig. 1.1 Experimental AlZn phase diagram. From ref. [3].
The RhPd system is also known as a phase separating system with a simple phase diagram and a wide miscibility gap fig. 1.2 and [4]. Also here will the alloy in a configuration related to temperature and concentration below the curve in fig. 1.2 separate into two fcc solid
components. The components are a_1 and a_2 with different concentration. The RhPd system can be seen as a model system for clustering behaviour. Both elements have an
electronegativity of 2.2 and a lattice constant of 3.80 Å respectively 3.89 Å for Rh and Pd [6].
3
1.2
Thesis outline
This diploma project was carried out within the Theoretical Physics group at the Department of Physics, Chemistry and Biology at Linköping University. This thesis consists of six chapters and a survey of the following chapters is presented below.
Chapter 2: Theory
Here is the Density Functional Theory introduced together with the Hohenberg-Kohn
theorems and the Kohn-Sham equations. Approximations to the Exchange-Correlation energy are also introduced such as the Generalized Gradient Approximation.
Chapter 3: Computational methods
Methods to solve the Kohn-Sham equation are introduced. The Coherent Potential Approximation and Madelung energy are presented.
Chapter 4: Alloy configuration
The GPM approach is discussed and brief overview of the Monte Carlo simulations is taken.
Chapter 5: Results and calculations
The result of the project is presented in the light of results from other studies. Some convergence tests for the calculations are also given.
Chapter 6: Discussion and conclusions
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5
Chapter 2
Theory
2.1 Density Functional Theory (DFT)
2.1.1 Introduction
The problem to deal with in the physics of condensed matter is how to describe the complex many body system of interacting electrons and nuclei. The quantum mechanical way to handle this is by the time independent Schrödinger equation:
Ψ =
Ψ E
Hˆ (2.1)
The Hamiltonian in the equation is given by:
∑
∑
∑
∑
∑
− − − + ∇ − − + ∇ − = ≠ ≠ ik i k i l k k l k k e j i i j j i i i i R r e Z r r e m R R Z Z M H , 2 2 2 2 2 2 2 1 2 ˆ h h (2.2)First and second term describe kinetic energy of nuclei respectively the electrostatic interaction between nuclei. Third and fourth term give kinetic energy of the electron respectively the electrostatic interaction between electrons. The last term deals with the electron–nuclei interaction.
The wave function in the equation above is a function of the positions and spin of all electrons, ri and positions of the nuclei, Rk:
(
r1,r2,r3,...,rN,R1,R2,R3,...,RM)
Ψ =
Ψ (2.3)
In practice the wave function is related to large scale systems dependent on 1023 particles. To simplify the calculations the Born- Oppenheimer approximation is used. The heavy nuclei are considered fixed in space compared to the much faster moving electrons. This gives an approximation where the first term in the Hamiltonian equals zero and the second term equals a constant.
Finally one is left with the main problem to actually solve the Schrödinger equation. The method used in this work is the Density Functional Theory, introduced by Hohenberg and
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2.1.2 The Hohenberg-Kohn theorems and the Kohn-Sham equations
The Hohenberg-Kohn theorems state that the electron density n(r) is fundamental for the system and can be used as a basic variable. Taking the electron density n(r) as a fundamental variable also reduces the problem from dealing with the wave function above depending on all ingoing electron positions in space to dealing with n(r) depending on just one position in space [7,11,12,13].
The first theorem states that the ground state electron density n(r) uniquely gives the external
potential, Vext(r), up to a constant. Hence the electron ground state density uniquely gives the total energy functional E[n(r)] of the system since the potential gives the energy. The second
theorem states that the ground state density n0(r) minimizes the total energy functional E[n(r)]. Thus n(r) and E[n(r)] are of interest to find if ground state properties of the system
are wanted.
Kohn and Sham showed that the total energy functional could be written as:
)] ( [ )] ( [ ) ( ) ( 2 ) ( ) ( )] ( [ 2 r n E r n T r drd r r r n r n e dr r n r V r n E ext ′+ s + xc ′ − ′ + =
∫
∫∫
(2.4)The first term in the functional is related to the Coulomb interaction between electrons and nuclei. The second is the ordinary Coulomb energy of the charge distribution. The third term gives the kinetic energy of a system of none interacting electrons. The last termExc[n(r)] is
the so called exchange-correlation energy of the system. In this term contributions due to exchange energy and correlations between electrons are collected.
Kohn and Sham introduced a set of quasi-particles in the theory. This quasi-electron system depends on non-interaction electrons which gives further advantages in the calculations. The quasi-electron system has the same density n(r) as the real physical system. The independent quasi-electrons are characterized by single-particle wave functions ψi. Kohn and Sham
proposed that the electron density n(r) could be constructed from the wave functions through:
∑
= = N i i r r n 1 2 ) ( ) ( ψ (2.5)The Schrödinger equation for these non-interacting electrons with wave functions ψi , moving
in the so called effective potential Veff(r) is given by:
) ( ) ( ) ( ) ( 2 2 2 r r r V r m∇
ψ
i + effψ
i =ε
iψ
i − h (2.6)The effective potential above is defined as:
) ( )] ( [ ) ( ) ( ) ( 2 r n r n E r d r r r n e r V r Veff ext xc
δ
δ
+ ′ ′ − ′ + =∫
(2.7)These three equations above are known as the Kohn-Sham equations. The system has to be solved iteratively until self-consistency is reached since the effective potential Veff(r) in the
7
Schrödinger equation depends on the density function n(r) but n(r) itself is given by the solutions of Kohn-Sham equation.
The iterative process can be described by the following scheme: First start with an initial qualified guess on n(r). Second step is to construct the Veff (r) from this start function n(r).
Third step is to solve the Kohn- Sham equation for this potential Veff (r) and get the wave
functions ψi. The fourth step is to form a new function n(r) from this wave functions ψi. After
this the procedure starts from step one again with the new density function n(r) and keeps going on until self consistency is reached [14].
2.1.3 Approximations of the Exchange-Correlation energy
In the total energy functional above (2.4) the exchange correlation energy functionalExc[n(r)]
is not exact known. Here one finds the difficulties associated with the original many-electron problem. The contribution to the total energy, E[n(r)], is anyway small [7]. Different approximations are used for the exchange correlation energy. One of them is the Local Density Approximation (LDA).
dr r n r n r n E xc LDA xc [ ( )] [ ( )] ( ) hom
∫
= ε (2.8)Here the n(r) is the electron density in point r and εxchom[n(r)] is the exchange-correlation energy per particle in a homogenous electron gas with density n(r). Within a small volume element n(r) is considered constant and the integral sums up all the energy contributions. However one must remember that electron density in real systems rarely is homogenous [12].
One way to expand the approximation is by the Generalized Gradient Approximation (GGA):
dr r n r n r n r n E xc GGA xc [ ( )]=
∫
ε [ ( ),∇ ( )] ( ) (2.9)Here a gradient correction also is included in the expansion of єxc. There also exist different
types of GGA. Both approximations have their benefits in calculations. Some predictions are better with GGA than LDA and vice versa [15]. In this thesis GGA is used as approximation method for the exchange-correlation energy. Previous studies on the RhPd system have been done in the GGA framework [4, 5].
2.1.4 DFT and its limitations
As mentioned above the LDA and GGA approximation have their shortcomings. For example GGA predicts Fe to be ferromagnetic and have body centered structure, bcc, in its ground state structure as given in experiments. LDA does give not these results [12]. GGA gives for example better structural predictions for 3d-metals, but LDA seems to give better results for 5d-metals. For 4d-metals the two approximations show equal structural results. It should be noted that the advantage of one of the approximations to the other sometimes can be rather accidental [15].
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Chapter 3
Computational methods
3.1 Muffin tin orbital methods
The aim is to solve the one electron Kohn-Sham equation introduced in Chapter 2 [7]. The one-electron wave function is expanded in a basis set:
∑
= j j j i c ϕ ψ (3.1)Using (2.6) one gets:
∑
=∑
j j j j i j j c c H ϕ ε ϕ (3.2)Multiply with ϕk from the left:
∑
∑
= j j k j i j j k j H c c ϕ ϕ ε ϕ ϕ (3.3) Here: kj j k kj j k H ϕ =H resp ϕ ϕ =O ϕ . (3.4)Hkj is the matrix elements of the Hamiltonian and Okj describes the overlap between different
basis functions. Rearranging (3.3):
{
}
∑
− = ∀ j kj i kj j H O k c ε 0 (3.5)Non-trivial solutions to this set of linear algebraic equations exist if and only if:
0 ]
det[Hkj−εiOkj = (3.6)
The advantage of this course of action is that the system of linear equations is easy handled in computations.
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Another problem to deal with is the potential when solving the Kohn- Sham equation. Even after using for example GGA the effective potential Veff is complicated. One way is to use the
so called muffin tin approximation for the potential:
≥ < = mt const mt mt r r if V r r if r V V (r) ( ) (3.7)
Inside a sphere of radii rmt the potential is spherical symmetric and outside the radii rmt the
potential is constant. This pattern runs over all lattice sites. This procedure is applied in the EMTO method (Exact Muffin-Tin Orbital method) used in this thesis where exact orbitals are used instead of linear orbitals as in earlier LMTO methods (Linear Muffin Tin Orbital methods) [12].
Figure 3.1. Schematic picture of the muffin tin spheres and potentials. From ref.[7].
Connected to this there also exists the so called Atomic Sphere Approximation (ASA) related to the electronic structure and used together with some methods. This gives that crystals space is filled by atomic spheres. The Wigner-Seitz cells are replaced by atomic spheres where the volumes equal the muffin tin potential sphere so rmt = rws. The so called ASA+M approach is
a modified technique where multiple moments of the electron charges are included [8]. For example the ASA approach is used together with the LMTO methods [16].
One other approach to solve the Kohn-Sham equations was developed by Korringa and Kohn and Rostocker, called the KKR method. It is a multiple-scattering theory where one regards atoms as scattering centres and where scattered waves at this centres are used to solve the electronic structure problem. One so-called Green function connected to a scattering path operator is introduced in the procedure [15]. As mentioned above the method used in this thesis is the Exact Muffin-Tin Orbital (EMTO) method. It can be considered as an improved screened KKR-method with large overlapping muffin-tin potential spheres and where the method is more accurate with exact and not linearized muffin tin orbitals [17]. The approximation here related to the potential is a Spherical Cell Approximation (SCA) where the rmt ≥rws.
One other method is the Bulk Greens Function Method (BGFM) [12]. This method is a KKR-ASA method similar to EMTO but where KKR-ASA is used for the potential instead. One more
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method is the Local Self-Consistent Greens Function method (LSGF) based on a super cell technique [18]. In ordinary super cell techniques the computational effort increases as N3 where N is the number of atoms in the system. The problem is reduced by considering electronic multiple scattering in a finite spatial region, the local interaction zone (LIZ), placed in a super cell filled with an effective medium. The effective medium can be constructed in the same manner as in the coherent potential approximation (CPA) described below. By placing the (LIZ) in an effective medium the non-local environment effects can be treated. Reducing the (LIZ) to a single site gives an almost equivalence to the conventional CPA but the Madelung energy (also described below) contribution to the total energy is treated properly [18]. The disadvantage of the LSGF to conventional CPA is that it can give time consuming calculations. Advantages are for example that LSGF gives treatment for local environmental effects, such as short range order [7, 8, 18, 19].
3.2 The Coherent Potential Approximation
The theory needs some way to handle the electronic structure in random alloys. The problem one has to face is that alloys form random and disordered structures. The structures investigated in this thesis are random alloys where it exist a fixed underlying lattice but where the atoms in the alloy are randomly distributed on the lattice. One way to do calculations over the alloy is the super cell approach. Here one places the atoms in a random manner and do calculations on the sample. The problem here is fore example that large super cells need a lot of computational time.
Another approach is to place a so-called effective medium on the underlying lattice and by this describe the systems properties by average. One method based on this is the Coherent
Potential Approximation (CPA). The backbone in CPA is scattering of electronic waves in the
system. One constructs the effective medium so the scattering on electrons moving through it equals the concentration averaged scattering from the random alloy components. The systems properties are then given in average by this effective medium.
Fig 3.2. The main idea behind CPA. The random structure is replaced by an ordered effective medium. From ref.[8].
Alloy components are seen as impurities in the effective medium and the scattering to these is determined [12]. The method is arranged so that it becomes a single-site approximation, i.e. only one real atom embedded in the effective medium is studied [8].
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Fig. 3.3. Insertion of real atoms into the effective medium. The inserted atoms are seen as impurities and are colored white and black in the figure. The concentration averaged scattering from components separately equals the scattering from the effective medium. From ref. [12].
3.3 Madelung energy
One of the shortcomings with the CPA is the problem with the charge transfer between different components the in model of the alloy. The CPA single site approach generally neglects local environment effects, and also the charge transfer problem. In the so-called SIM scheme (screened impurity model) one uses the empirical fact that impurity atoms net charges almost completely are screened by the net charges of their nearest neighbors [12, 20]. The approach is to shift the charge distribution inside the atomic sphere and so compensate for what happens outside and postulate that the interaction comes from the boundary between the atomic sphere and the medium [21]. In this way on may use same medium for different alloy component and just modify the alloy component potential. The result is that the SIM scheme gives the Madelung energy contribution part to the total energy. The Madelung energy describes the wanted electrostatic interaction between the components in the crystal and is affected by the charge transfer [7].
The SIM scheme Madelung energy contribution part [15] describes the screened coloumb interaction correction of one alloy component p to the total energy.
S q Escrp p 2 2 αβ − = (3.8)
Here α is of interest, it is the so called one-site screening constant and it deals with the charge transfer problems and is a fitting parameter. The value of α will underestimate the total Madelung energy according to ref. [12]. Another fitting parameter β is introduced to include multipole-multipole interactions. The LSGF method mentioned above is suitable for exploring values on α and β. Other shortcomings with the CPA method are that the single site approximation does not deals with short range order effects in the alloy and local lattice relaxations.
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Chapter 4
Alloy configuration
4.1 The GPM approach
Many alloys show interesting phase behavior as a function of temperature and concentration. The systems studied here, are the RhPd system and the AlZn system (see fig 1.1 and 1.2). For some temperature and concentration the systems have a phase separated appearance where the components of the alloy aggregate in clusters where the local concentration of the composition between the clusters differs; the phases separate.
Taking a deeper look into the theory described so far one needs additional methods to describe the system and also a thermo dynamical approach. It turns out that the Kohn Sham Hamiltonian approach (2.6) is inconvenient with respect to configuration thermodynamics. To reduce the problem one constructs a so called configuration Hamiltonian of Ising type [15] that is more suitable for the purpose. For a homogeneous binary alloy this is defined as:
... 3 1 2 1 , , ) 3 ( , ) 2 ( + + =
∑
∑
∑
∑
∈ ∈ i jk t k j i t t p i j p j i p conf V V H σ σ σ σ σ (4.1)Vp(2) and Vt(3) and so on are so-called effective pair- and three-site interactions and so on for
different clusters. For example V1(3) is the potential for a three site interaction cluster of a
triangle of three nearest neighbors and V2(3) is the potential for a three site interaction cluster
of a triangle of two nearest neighbors and a second nearest neighbor. The task is to determine these effective cluster interactions, the ECI:s. The σi:s above can be seen as a spin like
variable taking values +1 or -1 depending on whether there is an atom A or B at site i in the binary A1-cBc alloy. After the configuration Hamiltonian is found in terms of ECI:s one can
find the configuration energy and by optimizing this by statistical Monte Carlo methods finally find the optimal configuration [12]. The method used in this thesis to derive the ECI:s or the GPM potentials is the Generalized Perturbation Method (GPM).
In the GPM method one considers clusters embedded in an effective CPA medium. The clusters configuration (of totally n atoms in the cluster) is given by the spin like variable above in the following manner: {σ1,σ2,σ3,…σn} there the σi takes value +1 or -1 dependent on
there is an A or an B atom in the specific cluster. It can be shown by taking the product of these spin like variables that the searched effective interaction Vs(n) is given by the total
energy of two different systems. One system is the set of clusters with an even number of B atoms embedded in the effective medium and the other is the same set of clusters but with an odd number of B atoms. The ECI:s are then given by:
odd B even B n s E E V( ) = − − − (4.3)
The so-called called force theorem is then used to find the energy difference between the two sets of cluster. (See for example ref. [22] for a deeper description of the method). Even in the GPM method there exist charge transfer problems and a screening constant is introduced in the description of the potential.
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There exist other approaches for the problem. One of them is the Connolly-Williams or Structure Inversion Method (SIM) [15, 22]. It is based on a technique where one maps energy of predefined ordered structures onto a configuration Hamiltonian (see above). It is a concentration independent method. Short-comings are that it tends to give time consuming calculations and that the number of clusters interactions used for the expansion of the configuration energy is not known a priori.
4.2 Monte Carlo simulations
In the previous section the effective Hamiltonian and the inter-atomic interactions were calculated. The aim of this study is to derive the alloy configuration or state at different non-zero temperatures. The probability of finding a specific system σs = {σ1,σ2,σ3,…σn} in an alloy
configuration s is given by:
∑
− − = s ions configurat all T k E T k E S B s B s e e P (σ )/ (σ )/ (4.4)There the E(σs) is the total energy for the alloy in the configuration s obtained from the
effective Hamiltonian. The denominator in (4.4) is the so-called partition function. From the partition function one can derive the free energy1 of a system by:
∑
− − = s ions configurat all T k E s B e kT F ln (σ )/ (4.5)The free energy is the quantity from which one can obtain different thermodynamic quantities from. The equilibrium configuration is obtained by minimizing the alloy free energy. There exist several analytical techniques for deriving the equilibrium configuration. In this project a numerical Monte Carlo method is used.
The aim is to find a system in equilibrium at each temperature step used in the simulation. The system is built of a box containing in the order of thousands of atoms with periodic conditions at the boundaries. The dynamic under the system is described by altering its states in steps until it reaches the equilibrium state. The approach is to use a so-called Markov Chain. In the Markov process the probability for a state in the chain is independent of all steps except its immediate predecessor.
When the equilibrium state is reached the following condition is fulfilled:
n m m m n n t W P t W P ( ) → = ( ) → (4.6)
The expression above is called the detailed balance for two states n and m (the states also referred to as s and s + 1). The probability P(t) for different configurations or states n and m at a specific time t is given by (5.1). The time variable t can be seen as the (discrete) number of the specific state step of the sequence in the chain. The Wn→m factor gives the transition probability from one state n to another m.
1
15
The probability Ps(t) for a given state s in (5.1) is seldom known exactly because of the
partition function in the denominator of the expression. Using the Markov chain approach reduces this problem by taking the relative probability of one state and its succedor. By dividing Ps+1(t) by Ps(t) in (5.1) one sees that the energy difference between the two states is
the only factor that determines the relative probability:
m
n E
E
E= −
∆ (4.7)
In the Metropolis Monte Carlo sampling scheme the following algorithm is used. To decide whether the atoms are exchanged one calculates the energy Em of a pair of randomly chosen
atoms in the box and thereafter exchange the atoms and recalculates the energy En. If ∆E < 0
the atoms are exchanged or if ∆E > 0 the atoms are exchanged by a probability proportional to exp(-∆E/kbT). In practice the last case gives exchange atoms if a generated random number
chosen uniformly on the interval [0, 1] is less than the quantity exp(-∆E/kbT).
After this step a new test is applied to a new pair of atoms. This procedure gives a Markov chain of atomic configurations. By taking the average over configurations one may calculate equilibrium quantities and find if the system is segregated or random. In practice one calculates the energy average for systems at different temperature. The schematic graph over the average energy < Eav > versus temperature T, when a phase transition occurs is shown in
fig 4.1.
Fig. 4.1. Schematic graph of energy average versus temperature according to phase transition.
Since the heat capacity, c, essential depends on the first derivate of the energy one gets the schematic graph of c versus T as in fig 4.2 where the phase transition occurs at the peak. In this specific case it means that the alloy goes from a random to a cluster structure.
___________________________________________________________________________
16
Fig. 4.1. Schematic graph of heat capacity versus temperature according to phase transition.
In this project the transition temperature was calculated for different concentrations of the alloy in interval of 0.10 from 0 to 1. For a more detailed description of the method in this subchapter see for example ref. [23,24].
17
Chapter 5
Results and calculations
All heavy calculations were carried out at NSC, National Supercomputer Center in Sweden. A program package called EMTO was used for the computations. Briefly it consists of four programs: kstr where structure is defined by vectors, shape related to the structure, kgrn where the main part of the Kohn-sham equations are solved and where concentrations of alloys, Wigner-Seitz radius (see below) etc. is defined and finally kfcd where full charge density calculations are carried out and where energy calculations are obtained.
5.1 Calculation of ground state properties for pure elements
To derive the equilibrium radius for the elements one calculates total energy for different size of the Wigner-Seitz cell (the Wigner-Seitz radius is the radius of a sphere that contains the same volume as the volume per particle in the system) and plots the energy versus equilibrium radius and where the energy minimum correspondents to the equilibrium radius. The calculations were carried out in the GGA framework and the results are seen in table 5.1.
Table 5.1. Wigner-Seitz radius in atomic units. Ref. [25].
This survey Experimental
Rh, fcc, a 2.84 au 2.81 au
Pd, fcc, a 2.92 au 2.87 au
Al, fcc, a 2.99 au 2.99 au
Zn, hcp, c/a 1.73 1.86
Conversion factor from Wigner-Seitz radius in atomic units, au, to lattice parameter in Ångström for fcc structure is 1.3541. The deviation from experimental values is less than 2 % for the fcc structures (table 5.1).
5.2 K-point convergence test for pure elements
One important aspect in electron structure calculations is the sampling of k-space of the Brillouin zone. A large number of k-vectors increase the accuracy in the calculations but also increase the computational time. The number of k-vectors is determined by a parameter in the computer code named NKY.
___________________________________________________________________________ 18 0 10 20 30 40 50 60 70 -10094.14195 -10094.14190 -10094.14185 -10094.14180 -10094.14175 -10094.14170 -10094.14165
k-points convergence test
NKY E n e rg y R y d b e rg
Fig. 5.1 K-points convergence test for Pd in NKY.
As seen in fig 5.1. the energy versus k-vectors converges around a value of 29 of NKY for Pd used in this convergence test. Seen together with the fact that computational time raises fast (an increase in NKY from 30 to 40 gives double computational time) a value of 29 for NKY is used in this project. This NKY value of 29 corresponds to 2304 k-vectors in the Brillouin zone.
5.3 Equilibrium radius and mixing enthalpy on alloy systems
The EMTO program package is also capable to calculate the equilibrium radius i.e. find the lattice parameter for different concentrations like in chapter 5.1 by energy minimization. The mixing enthalpy of a binary AB alloy is also of interest and is defined at component B concentration x by:
Hmix =Halloy(A1−xBx)−xH(B)−(1−x)H(A) (5.1)
Here it also exist experimental values on the mixing enthalpy diagrams for the both systems in this project. In experimental studies, short range order can influence the results and give lower values ref. [12].
5.3.1 The RhPd alloy system equilibrium radius and mixing enthalpy
The derived equilibrium radius versus component concentration was in good agreement with experimental values for pure elements as discussed in chapter 5.1 (table 5.1) and also seen in fig 5.2. Of interest is the mixing enthalpy curve presented in fig 5.3. The calculations were carried out in steps of 0.05 concentration Pd. The calculations gave quite a good agreement with experimental values. A symmetric enthalpy curve was obtained and a maximum is situated around 50% Pd with a value of 75 meV/atom. This value can be compared to experimental results [26, 27]. Ref. [26] gives a experimental value of maximum of 90 meV/atom for the same concentration and ref. [27] gives a value of 105 meV/atom (see fig 5.3). The mixing enthalpy is over all positive and is therefore in agreement with the phase separating tendency of the system.
19 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2.84 2.85 2.86 2.87 2.88 2.89 2.9 2.91 2.92 2.93 RhPd equlibrium radius fraction Pd R W S a .u .
Fig. 5.2. Equilibrium radius versus concentration for RhPd.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Mixing Enthalpy RhPd x (fraction Pd) H (x ) [e V /a to m ]
Fig 5.3. Mixing enthalpy curve for RhPd. Experimental: =from ref. [26]. OOOO=from ref. [27].
5.3.2 The AlZn alloy system equilibrium radius and mixing enthalpy
The AlZn exhibits a equilibrium radius versus concentrations diagram as in fig. 5.4. As discussed in section 5.1 it is in good agreement with experimental results for at least the pure Al fcc state since Zn is hcp in pure form. Of interest is the mixing enthalpy diagram. It exhibits a maximum around 0.40 fraction Zn with a value of 23 meV/atom. It is in good agreement with experimental values in ref. [1] with a value of approximately 22 meV. A positive mixing enthalpy also reflects the phase separating tendency of the system.
___________________________________________________________________________ 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2.9 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99
AlZn equlibrium radius
fraction Zn R W S a .u .
Fig. 5.4. Equilibrium radius versus concentration for AlZn.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.005 0.01 0.015 0.02 0.025
Mixing Enthalpy AlZn
x (fraction Zn) H (x ) [e V /a to m ]
21
5.4 Effective cluster interactions (ECI) for alloy systems
One of the requirements for Monte Carlo simulations is the construction of the Hamiltonian of Ising type in (4.1): ... 3 1 2 1 , , ) 3 ( , ) 2 ( + + =
∑
∑
∑
∑
∈ ∈ ijk t k j i t t p ij p j i p conf V V H σ σ σ σ σTo get the Hamiltonian the ECI2:s are needed (Vp(2)etc..) and were derived by the screened
GPM technique in Sec. 4. Related to equation the (4.1), it is of interest to find out what ECI terms are dominating and thereafter use them in the Monte Carlo simulations. For example is
Vp, the effective pair interaction between a pair of sites p with atoms A and B given such that:
Vp = VpAA + VpBB – 2VpAB (5.1)
If the second order terms of the expansion in (4.1) are dominating and negative, an interpretation of (5.1) indicates a tendency towards phase separation of the system. For the RhPd system has been shown in other studies that the expansion of (4.1) rapidly convergent for terms of order higher than second order (For terms including V(3)t , V(4)f etc.). For the
second order term in (4.1) (the term including V(2)n) gives neighbors with a distance larger
than fourth neighbor a negligible contribution in the RhPd case. ref. [28].
5.4.1 The RhPd alloy system ECI
Fig 5.7. describes how the first two site interactions (V(2)1) are far much stronger than first
three and four site interactions (V(3)1 and V(4)1). Of interest is fig. 5.9 to fig. 5.11. Here the
effective pair interactions (V(2)n) are given as a function of distance related to the coordination
shells. The pair interactions show a rapid convergence according to distance and are mostly negative and are thereby indicating phase separation at T = 0 K. The result of the ECI appearance is in agreement with results from ref. [28]. As referred to above the expansion in (4.1) is rapidly convergent and second order neighbors with a distance larger than fourth neighbor (V(2)4) give a negligible result. Ref. [28]. (For additional graphs over V(3)n and V(4)n
see Appendix A fig. 1 and 2)
2 Effective Cluster Interactions (ECI:s) are also named GPM-potentials here. Two site-interactions are referred to
the notation V(2)n , Three site-interactions are referred to the notation V(3)n etc. The notation ECP is sometimes
___________________________________________________________________________ 22 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 ECI RhPd E C I (m R y ) Pd concentration
Fig. 5.7. ECI for different concentrations Pd. Interactions: =V(2)1 oooo=V
(3) 1
∇∇∇∇=V (4) 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1
ECI- first four ECP- RhPd
E C I (m R y ) Pd concentration
Fig. 5.8. ECI for different concentrations Pd. Interactions: oooo=V(2)1
∇∇∇∇=V
(2) 2 ∆=V (2) 3 =V (2) 4
23 0 5 10 15 20 25 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1
ECP RhPd the first 25, concentration 0.20
E C I (m R y ) ECP number
Fig. 5.9. First 25 two-site interactions for concentration 0.20 Pd. Interactions: =V(2)1 to V(2)25
0 5 10 15 20 25 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5
ECP RhPd the first 25, concentration 0.50
E C I (m R y ) ECP number
___________________________________________________________________________ 24 0 5 10 15 20 25 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5
ECP RhPd the first 25, concentration 0.80
E C I (m R y ) ECP number
Fig. 5.11. First 25 two-site interactions for concentration 0.80 Pd. Interactions: =V(2)1 to V(2)25
5.4.2 The AlZn alloy system ECI
Seen in fig 5.12. are the pair interactions (V(2)1) domination over the three and four site
interactions (V(3)1 and V(4)1). It is unclear to see if the result correlates to similar calculations
in ref. [5]. The magnitude of the interactions are anyway less than the magnitude of the values in the RhPd case. Seen in fig. 5.14 to fig. 5.16 is that the pair interactions (V(2)n) converge
relatively fast according to the nearest neighbor distance of the pair interactions. Also seen in fig. 5.13 and 5.14 to 5.16 is negative value dominance of the pair interactions below concentration 0.50 Zn and the opposite above concentration 0.50 Zn. This should indicate phase separation for at least concentrations under 0.50 Zn and are in some agreement with the miscibility gap seen in fig. 1.1. for concentrations from 0.20 to 0.60 Zn. (For additional graphs over V(3)n and V(4)n see Appendix A fig. 3 and 4)
25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 ECI AlZn E C I (m R y ) Zn concentration
Fig. 5.12. ECI for different concentrations Zn. Interactions: =V(2)1 oooo=V(3)1
∇∇∇∇=V(4)1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
ECI- first four ECP- AlZn
E C I (m R y ) Zn concentration
Fig. 5.13. ECI for different concentrations Zn. Interactions: oooo=V(2)
___________________________________________________________________________ 26 0 5 10 15 20 25 30 35 40 45 50 -0.4 -0.2 0 0.2
ECP AlZn the first 50, concentration 0.20
E C I (m R y ) ECP number
Fig. 5.14. First 50 two-site interactions for concentration 0.20 Zn. Interactions: =V(2)1 to V(2)50
0 5 10 15 20 25 30 35 40 45 50 -0.2 -0.1 0 0.1 0.2 0.3 0.4
ECP AlZn the first 50, concentration 0.50
E C I (m R y ) ECP number
27 0 5 10 15 20 25 30 35 40 45 50 -0.1 0 0.1 0.2 0.3 0.4 0.5
ECP AlZn the first 50, concentration 0.80
E C I (m R y ) ECP number
Fig. 5.16. First 50 two-site interactions for concentration 0.80 Zn. Interactions: =V(2)1 to V
(2) 50
5.5 Monte Carlo convergence tests for alloy systems
The RhPd system was investigated in Monte Carlo simulations and the AlZn system was left for future studies since already the RhPd system showed peculiar results.
Since the screened GPM technique from Sec. 4 was applied to derive the effective interactions in section 5.4 the next step from this was to investigate different aspects of requirements in the Monte Carlo simulations for the RhPd system. Question marks to handle were the size of the box of the sample, the number of pair interactions needed for a convergent result and affection of interactions of higher order.
5.5.1 The RhPd alloy system Monte Carlo convergence tests
According to Sec. 5.4 the 15 first pair interactions should be sufficient to describe the Hamiltonian in (4.1) in an accurate way. The result of the convergence tests was that the final simulations were performed in a box containing 4 times 303 atoms i.e. 108 000 atoms. The calculations were performed in concentration range from 0.00 to 1.00 Pd with steps over 10%. The peek in specific heat (to derive the transition temperature) described in Sec. 4.2 was sought in steps of 10 K in the entire temperature range (below 2500 K). The first 15 pair interaction GPM potentials were used, as well as the two strongest three site interaction and the strongest four site interaction. 6000 Monte Carlo steps were performed per atom at each temperature and the energy was averaged over the last 4000 steps. Below follows a motivation for the values above in the performed convergence tests.
A typical appearance of specific heat versus temperature is show in fig. 5.17. below. The peak is associated with the transition temperature where the alloy goes from random to a cluster structure.
___________________________________________________________________________ 28 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Monte Carlo RhPd 15 ECP
C
Temperature
Fig. 5.17. Monte Carlo simulations for RhPd. Concentration 0.50. Heat capacity c [mRy/K] vs. temperature [K].
First was a test concerning the transition temperature versus box size of the sample performed. Seen in fig. 5.18. is the result for 15 first pair interactions at concentration 0.50 Pd. A box size of side 30 was used according to computational time cost versus accuracy.
15 20 25 30 35 40 45 1400 1450 1500 1550 1600 1650 1700 1750 1800
Monte Carlo RhPd conc 0.50 15 first two site interactions Box size: nxnxn
T
c
K
Box size side n
Fig. 5.18. Monte Carlo simulation for RhPd. Concentration 0.50 Pd. V(2)1 to V(2)15 used. Transition temperature
vs. box size.
Next step was to investigate the number of pair interactions impact on transition temperature given by the Monte Carlo simulation. Fig 5.19. to 5.21. give the transition temperature versus number of pair interactions for three different concentration Pd. 15 pair interactions seemed to give an acceptable accuracy.
29 0 5 10 15 20 25 30 200 400 600 800 1000 1200 1400 1600
Monte Carlo RhPd Tc vs. Number of ECP conc 0.20
T
c
K
Number of ECP
Fig. 5.19. Monte Carlo simulation for RhPd. Concentration 0.20. Transition temperature vs. number of ECP
included (V(2)n). 0 5 10 15 20 25 30 600 800 1000 1200 1400 1600 1800 2000
Monte Carlo RhPd Tc vs. Number of ECP conc 0.50
T
c
K
Number of ECP
Fig. 5.20. Monte Carlo simulation for RhPd. Concentration 0.50. Transition temperature vs. number of ECP
___________________________________________________________________________ 30 0 5 10 15 20 25 800 1000 1200 1400 1600 1800 2000
Monte Carlo RhPd Tc vs. Number of ECP conc 0.80
T
c
K
Number of ECP
Fig. 5.21. Monte Carlo simulation for RhPd. Concentration 0.80. Transition temperature vs. number of ECP
included (V(2)n).
Finally the impact from higher order interactions was examined as seen in fig. 5.22. From this the conclusion was that the transition temperature became slightly altered by use of higher order interaction potentials as seen in fig. 5.22.
1600 1620 1640 1660 1680 1700 1720 1740 1760 1780 1800
Monte Carlo simulations RhPd, conc 0.50 Pd. Different compositions of ECI
T
c
K
Different ECI compositions
A B C
D
Fig. 5.22. Monte Carlo simulation for RhPd. Concentration 0.50. Transition temperature vs. ECI composition.
A=V(2)1 to V(2)15 B= V(2)1 to V(2)15 and strongest V(3)n C=V(2)1 to V(2)15 and two strongest V(3)n
31
5.6 Monte Carlo phase diagram for alloy systems
The aim was to derive phase transition temperatures by the GPM parameters used in the Monte Carlo simulations and from this construct the phase diagram for the alloy.
5.6.1 The RhPd alloy system phase diagram
The phase diagram derived from the GPM-Monte Carlo approach (fig 5.23.) seems to deviate more than acceptable for concentration Pd over 0.20 compared to experimental values in ref. [4]. For concentration 0.80 the transition temperature is twice the experimental value.
Fig. 5.23. RhPd phase diagram. Circles: Monte Carlo simulations from the GPM approach. Curve: Experimental from ref. [4] .
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33
Chapter 6
Discussion and conclusions
The aim of this study was to investigate how well the GPM potentials derived from the EMTO-CPA approach could re-create the temperatures for phase separations in phase separating systems such as the RhPd system and eventually the AlZn system. The method is known to work well for ordering systems and now investigated in phase separating systems.
The first point of view is that lattice parameters are well described by the method. Deviation from experimental values was less than 2% for the fcc structures Rh, Pd and Al. Taking a look at the mixing enthalpy for the RhPd system respectively the AlZn system gives a mixing enthalpy in agreement with experimental values for the AlZn system. The curve is asymmetric with a peak around concentration 0,35 Zn and a maximum of 23 meV/atom. The mixing energy is also positive, indicating phase separating behavior of the system.
The RhPd system mixing energy curve also give acceptable results even if maximum of the curve just reaches around 75 meV/atom while experimental values goes up to at least 90 meV/atom. Due to only short range order effects the experimental curve should give lower values than the theoretical values. Also here the mixing energy is positive, thus indicating phase separating tendencies of the system.
A question is the quality of interaction potentials derived from the GPM approach, the GPM potentials. One way to investigate this is to calculate so called ordering energies for different ordered well known structures in the same manner as in ref. [22]. This has been done in parallel to this diploma work (Appendix B fig.1.). The calculations are arranged so that they are done directly from first principles but also from the GPM potentials. A comparison of the results gave an acceptable agreement and thereby indicating no problems in the quality of the GPM potentials.
Also other conclusions can be drawn from the interaction potentials derived from the GPM approach. Seen in the first pair interactions for RhPd of importance are their negative tendencies thus indicating phase separation of the system. The AlZn system strongest effective interactions give at least for concentrations below 0.50 Zn a negative value and indicating phase separation also here.
The interesting thing is the strange appearance of the phase diagram of the RhPd system in fig. 5.23. Especially since the transition temperatures are almost twice compared to experimental values for concentrations around 0.80 Pd. To construct the phase diagram the GPM potentials are used in a statistical Monte Carlo method. The prediction errors might be due to an inability of the Monte Carlo scheme to handle the existing concentration dependencies seen in the cluster interactions.
A binary alloy in a random state going to an ordered structure by temperature decrease for a particular relative concentration of the ingoing atoms is still described by the same set of GPM potentials since the local concentration of the ingoing atoms is the same before and after the transition. For a phase separating system, in this study RhPd, the transition from a random to a phase separated condition gives in general a different local concentration of the ingoing
___________________________________________________________________________
34
atoms in the clusters formed in the phase separated system. The set of GPM potentials describing the different clusters after transition are not optimal the same as the set of GPM potentials describing the random structure as before the transition.
Suggestion for further work is to see how the concentration independent Connolly-Williams approach handles the systems in this study and compare the results. Other future work would be to investigate alternative approaches in the Monte Carlo scheme.
35
Appendix A
Graphs over Effective Cluster Interactions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 -0.4 -0.3 -0.2 -0.1 0
RhPd Strongest three site interaction vs. concentration
E C I (m R y ) Fraction Pd
Fig.1. Strongest three-site interaction for different concentrations Pd. Interactions: =V(3)8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0
RhPd Strongest four site interaction vs. concentration
E C I (m R y ) Fraction Pd
___________________________________________________________________________ 36 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
AlZn Strongest three site interaction vs. concentration
E C I (m R y ) Fraction Zn
Fig.3. Strongest three-site interaction for different concentrations Zn. Interactions: =V(3)1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
AlZn Strongest four site interaction vs. concentration
E C I (m R y ) Fraction Zn
37
Appendix B
Graph over Ordering Energy
Fig.1. Calculations over ordering energy for some ordered structures. From GPM potentials and direct from first principles.
___________________________________________________________________________
39
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