Binary sigma phases

AB INITIO-BASED MEAN FIELD THEORY OF SITE OCCUPATION IN BINARY SIGMA PHASES

E. Kabliman

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1. Introduction: 1. Binary σ-phases

2. Fe-Cr phase diagram

3. Magnetic properties

4. σ-phase structure

5. Aim

2. Ab initio-based mean field theory: 1. Helmholtz‘s free energy

2. Total energy expansion

3. EMTO-CPA

4. Convergence of total energy expansion

5. Entropic contributions

6. Site occupancies

3. Results [FeCr]: 1. Effect of magnetic state

2. Structural variations

3. Final site distribution

4. Results [other σ-phases]: 1. ReW

2. CoCr

3. FeV

5. Summary

3

σ-phase (light plates). Nicrofer 3127. after annealing at 700о С during 6000 hours

σ-phase formation = technological problem

J.L. Garin and R.L. Mannheim. J. Mat. Prog. Tech. 209, 3143 (2009)

J. Vřestal et al. Comp.Mat.Sc. 38, 298 (2006)

σ-phase : • has a lamellar morphology • nucleates near grains boundaries (concentration of elements, e.g. Cr and Mo, is favorable) and on carbide particles

PROBLEM of σ-phase precipitation : • decreases ductility and corrosion resistance • increases brittleness and cracks formation

1. Introduction: 1. Binary σ-phases 2. Fe-Cr phase-diagram 3. Magnetic properties 4. σ-phase structure 5. Aim

2. Ab initio MF theory:

1. Helmholt’z free energy 2. Total energy expansion 3. EMTO-CPA 4. Convergence of expansion 5. Entropic contributions 6. Site occupancies

3. Results [FeCr] : 1. Effect of magnetic state 2. Structural varitations 3. Final site distribution

4. Results [other σ-s] : 1. ReW 2. CoCr 3. FeV

5. Summary

4

σ-phase in binary alloys

• atomic radii of the alloy components must be similar • Fe-Cr system has the smallest VF at room T.

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VF=∆r/rav r – atomic radii

• room T: 8 of 41 systems has VF > 8% • elevated T: 2 of 41 systems has VF > 8%

In binary σ-phases VF ≤ 8 %






5. Summary

5

Fe-Cr phase diagram

HISTORY :

• 1923 Bain: A hard and nonmagnetic phase in Fe-Cr [Chem. Met. Eng. 28 (1923)]

• 1943 Cook, Jones: Crystal structure of σ-phase [J. Iron Steel Inst. London 148

(1943)]

• At present: σ-phase is found in more than 40 systems (Cr-Fe, Cr-Co, V-Fe, etc.)

4

brittle σ-phase

Phase decomposition






5. Summary

6

Magnetic properties

J. Cieslak, M. Reissner et al. Phys. Stat. Sol. (a) 205, No. 8 (2008)

[ TC measured only in Fe-Cr and Fe-V ]

Fe1-xCrx

Fe1-xCrx

Ave

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1. Introduction:

1. Binary σ-phases 2. Fe-Cr phase-diagram 3. Magnetic properties 4. σ-phase structure 5. Aim





5. Summary

7

Site ncoord nmult

(Wyck.pos.) x

at 923 K y

at 923 K z

at 923 K Emprical scheme

A 12 2(a) 0 0 0 (Mn,Fe,Co,Ni)

B 15 4(f) 0,39864(4) 0,39864(4) 0 (Ti,V,Cr,Mn,Mo)

C 14 8(i) 0,46349(4) 0,13122(4) 0 mixed

D 12 8(i) 0,73933(4) 0,06609(4) 0 (Mn,Fe,Co,Ni)

E 14 8(j) 0,18267(3) 0,18267(3) 0,25202(9) mixed

1. H.L. Yakel. Acta Cryst. B 39, 20 (1983) 2. J. S. Kasper, R. M. Waterstrat. Acta Cryst. 9, 289 (1956)

σ-phase structure

Fe1-xCrx for x = 0.495, a = 0.87968 nm, c/a = 0.518. ncoord – site coordination numbers; nmult – site multiplicities






5. Summary

8

Aim

J. Cieslak, M. Reissner et al. Phys. Stat. Sol. (a) 205, No. 8 (2008)

Determine atomic site distribution in disordered σ-phase as a function of T and x

Experiment • lack of data: site occupancies only at few T and x • difficulties:

• long annealing times and purity of a sample • Fe and Cr: similar X-ray scattering factors






5. Summary

Sigma-phase is a substitutionally disordered solid solution in paramagnetic state;

It has a low symmetry structure with undefined „exact“ atom-site correspondence;

How to describe electronic structure and total energy of such a system?

9

SUBSTITUTIONAL DISORDERED SYSTEMS

Methods for total energy calculations

Supercells approach: straightforward method, use of big periodic supercells

Cluster expansion formalism: expansion of total energies in terms of the effective cluster interactions (strucutre inversion method)

Special quasirandom structures: reproduce the most relevant radial correlation functions of an infinite random structure

Methods for electronic structure calculations

Virtual crystal approximation: real disordered system is replaced by an virtual one-component one

Coherent potential approximation: provide the same scattering properties of the one-component effective medium as the average of alloy components, embedded in this effective medium

Locally self-consistent Green‘s function: introduction of local interaction zone around every atom, inside which an atom ‚feels‘ ist real local environment, but outside ‚sees‘ only the CPA effective medium

EMTO-CPA

Some of slides on EMTO-CPA were kindly provided by Prof. A. Ruban (KTH, Sweden)

ATOMIC SPHERES METHODS

• Augmented plane waves (APW)

• Muffin-tin orbitals (MTO) ASA, exact MTO (EMTO)

• Linearized APW and MTO (LAPW and LMTO)

• (Linearized) augmented plane wave method ((L)APW) + local orbitals

• Full-potential LAPW and LMTO methods

• Various Green‘s function method (KKR, KKR-CPA, etc)

Fig. 2.1

13

Partial wave Back-extrapolated partial wave Screened spherical wave

Kinked partial wave or exact muffin-tin

orbital

SSW Partial wave

BPW

They are basis functions of the EMTO method

The solution of the Kohn-Sham equation is then given by

Their linear combination:

A. Ruban

Applications of the EMTO method: advantages

and disadvantages

The EMTO is the method to calculate electronic structure and

total energies of ordered and random alloys as in the bulk as

well as at surfaces and interfaces .

As soon as the total energy is obtained, it can be used for

calculations of different thermodynamic properties, like

enthalpies of formation, solution, segregation, ordering

energies and so on.

It can be also used for calculations of the elastic properties:

in most cases there is very little information about elastic

constants of alloys.

A. Ruban

Metals and alloys in the paramagnetic state

Paramagnetic state in magnets is usually due to randomly

oriented spin magnetic moment on atoms.

Theoretically it can be presented by the so-called disordered

local magnetic moment (DLM) model, in which such a system

considered as equiatomic random alloy of atoms with spin-up

and spin-down orientation.

It is extremely difficult (close to impossible at the present

time) to treat alloys in the DLM state using such methods as

VASP, where a supercell approach is required. However, this

is extremely easy in the case of EMTO, where such a random

alloy is treated using coherent potential approximation (CPA).

A. Ruban

Green’s function

Schrödinger (Kohn-Sham) equation

A. Ruban

Coherent potential approximation (CPA)

(isomorphous single-site approximation)

Since wave function is not self-averaging quantity, the Green’s

technique be used in order to have a consistent theory.

In the CPA random alloy is substituted

by perfectly ordered (CPA) effective

medium, whose scattering properties

satisfy the condition:

These non-linear equations are solved self-consistently A. Ruban

AB INITIO BASED MEAN FIELD THEORY

19

20

Ab initio based mean field theory :

𝐹 = 𝐸𝑡𝑜𝑡 − 𝑇𝑆

𝐻𝑐𝑜𝑛𝑓 = 𝐸0 + 𝑉∝(1)𝛿𝑐𝛼 ,𝑖

𝑖𝛼=1,5

+1

2 𝑉∝𝛽 ,𝑝

(2)𝛿𝑐𝛼 ;𝑖𝛿𝑐𝛽 ;𝑗

𝑖 ,𝑗∈𝑝𝑝𝛼 ,𝛽=1,5

+1

3 𝑉∝𝛽𝛾 ,𝑡

(3)𝛿𝑐𝛼 ;𝑖𝛿𝑐𝛽 ;𝑗𝛿𝑐𝛾 ;𝑘

𝑖 ,𝑗 ,𝑘∈𝑡𝑡𝛼 ,𝛽 ,𝛾=1,5

+ ⋯

𝛿𝑐𝛼 ,𝑖 = 𝑐𝛼 ,𝑖 − 𝑐𝛼 ,𝑤ℎ𝑒𝑟𝑒 𝑐𝛼 ≡ 𝑐𝛼 ,𝑖 𝑎𝑛𝑑 𝑐𝛼 .𝑖 = 1 𝑖𝑓 𝐶𝑟 𝑎𝑡 𝑠𝑖𝑡𝑒 𝑖

𝐸𝑡𝑜𝑡 𝑐𝛼 = 𝐸0 𝑥 + 𝑛𝛼 𝐽𝛼 1 𝑐𝛼

4

𝛼=1

.

𝑛𝛼 − 𝑠𝑖𝑡𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑐𝛼 − 𝐶𝑟 𝑠𝑢𝑏𝑙𝑎𝑡𝑡𝑖𝑐𝑒 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦

𝐽𝛼 1

= 𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝛼

−𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝐸

|𝑐𝛼=𝑥=𝑐𝑜𝑛𝑠𝑡

Helmholtz’s free energy

• Jα(1) are relative effective chemical potentials at given x, which describe

preference of Cr atoms to occupy site α with respect to site E.

• Jα(1) are indeed enough to calculate Cr site occupancies.

Fe1-xCrx

Effective on-site interactions with respect to the E site

IDEA :

site sublattice

pairs






5. Summary

21

Total energy expansion

Problem of CW method: Number of different pair ECI just within a polyhedron of closest sites is 20, while number of non-equivalent multi-site interactions is much larger even within such short distances.

𝐻𝑐𝑜𝑛𝑓 = 𝐸0 + 𝑉∝ 1 𝛿𝑐𝛼,𝑖

𝑖𝛼=1,5

+1

2 𝑉∝𝛽,𝑝

2 𝛿𝑐𝛼;𝑖𝛿𝑐𝛽;𝑗

𝑖,𝑗∈𝑝𝑝𝛼,𝛽=1,5

+1

3 𝑉∝𝛽𝛾,𝑡

3 𝛿𝑐𝛼;𝑖𝛿𝑐𝛽;𝑗𝛿𝑐𝛾;𝑘

𝑖,𝑗,𝑘∈𝑡𝑡𝛼,𝛽,𝛾=1,5

+⋯

Fe1-xCrx

~ 130 atoms

bcc : i = 2-3 fcc : i = 1-2 σ-A : i = 3 σ-B : i = 6 σ-C : i = 8 σ-D : i = 8 σ-E : i = 9






5. Summary

22

Effective pair cluster interactions

Free energy of σ-phase can be quite accurately presented in terms of concentration-independent cluster interactions

Dominating terms in this expansion can be on-site interactions

Single-site mean-field approximation will be sufficient to provide a reasonably accurate description of configurational effects at high temperatures

xCr=0.5 xCr=0.6

SGPM






5. Summary

23

Effective on-site interactions


4

𝛼=1

𝑛𝛼 − 𝑠𝑖𝑡𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑐𝛼 − 𝐶𝑟 𝑠𝑖𝑡𝑒 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦

𝑐5 =30

𝑛5−

1

𝑛5 𝑛𝛼𝑐𝛼

4

𝛼=1

𝐽𝛼 1 =

𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝛼



• Jα(1) are relative effective chemical potentials at given x,

which describe preference of Cr atoms to occupy site α with

respect to site E.

• Etot are found in coherent potential approximation (CPA)

implemented on exact muffin-tin orbitals (EMTO) method

in order to simulate disordered atomic distribution in σ-phase.

Fe1-xCrx






5. Summary

No local environment effects LSGF

No local atomic relaxations Wien2k

24

Check of Coherent Potential Approximation

Locally self-consistent Green‘s function technique

2 supercells (SC1 and SC2) = (4x4x4) of 1920 atoms

Local interaction zone = polyhedron of the nearest neighbors for every site (A,B,C,D,E)

CPA results are in a very good agreement with those of LSGF calculations for supercells






5. Summary

25

Check of EMTO and local atomic relaxations

Etot is calculated by: • full-potential linearized augmented plane wave + local orbitals method (WIEN2k code) • exact muffin-tin orbitals method (EMTO code)

Site Exp. [1] 923 K

Final 0 K

A 0.125 Cr 0.0 Cr

B 0.750 Cr 1.0 Cr

C 0.625 Cr 0.375 Cr

D 0.163 Cr 0.0 Cr

E 0.663 Cr 1.0 Cr

1. H.L. Yakel, Acta Cryst. B 39, 20 (1983) 2. A. Zubkov and B.Mogutnov, Dokl. Ak. Nauk SSSR 311, 388 (1990) 3. D. Read and E. Thomas, J. Phys. Chem. Sol. 29, 1569 (1968)

Calculated (WIEN2k)

Measured

a 0.86433 nm 0.87968 nm [1]

c/a 0.518 0.518 [1]

Ef 5 kJ/mol 6.5 kJ/mol [2]

μ 0.53 μB 0.20 μB [3]

Site relaxations < 0.1 kJ/mol

[Good agreement ]

Fe0.5Cr0.5

limit of CPA

• EMTO and Wien2k obtain the same equilibrium (final) atomic distribution

• Site relaxations in σ-FeCr are negligible






5. Summary

26

Convergence of total energy expansion


4

𝛼=1

+ 𝑛𝛼𝑛𝛽 𝐽𝛼𝛽 2 𝑐𝛼𝑐𝛽

4

𝛼 ,𝛽=1

𝐽𝛼 1



|𝑐𝛼=𝑥=0.5

Mean least-squares fit 𝑱 𝜶 𝟏

: 89 various σ-phase configurations

for x = 0.50 in DLM state

ECI A B C D E

𝐽 𝛼 1

16.0951 0.8670 1.4796 15.4150 0

Jα(1) 16.4062 0.4688 1.7812 14.3438 0

𝐽 𝛼𝛽 2

A B C D E

A -1.1983 -0.3714 -0.6475 -0.4568 0

B -0.4896 -0.3899 -0.4631 0

C -0.5680 -0.4065 0

D -0.6397 0

E 0

• 𝐽 𝛼 1 : root-mean-square error ~ 0.2 mRy/atom

• 𝐽 𝛼 1

+ 𝐽 𝛼𝛽 2 : root-mean-square error ~ 0.2 mRy/atom

• Total energies calculated without/with effective pair interactions 𝐽 𝛼𝛽 2

are similar

• Chemical potentials Jα(1) reproduce well the on-site interactions from

mean least squares fit 𝐽 𝛼 1






5. Summary

𝐹 = 𝐸𝑡𝑜𝑡 − 𝑇(𝑆𝑐𝑜𝑛𝑓 + 𝑆𝑚𝑎𝑔𝑛 + 𝑆𝑣𝑖𝑏𝑟 )

𝑆𝑐𝑜𝑛𝑓 = −𝑘𝐵 𝑛∝ 𝑐∝ ln 𝑐∝ + 1 − 𝑐∝ ln 1 − 𝑐∝

5

𝛼=1

𝑆𝑚𝑎𝑔𝑛 = 𝑘𝐵 𝑛∝[

5

𝛼=1

𝑐∝ ln 𝜇𝛼𝐶𝑟 + 1 + 1 − 𝑐∝ ln 𝜇𝛼

𝐹𝑒 + 1 ]

𝑆𝑣𝑖𝑏𝑟 = 𝑛𝑒𝑔𝑙𝑒𝑐𝑡𝑒𝑑

27

Entropy contributions to free energy

Magnetism:

• In order to simulate paramagnetic state

at finite T disordered local moment

(DLM) state is used.

• Longitudinal spin fluctuations (LSF)

are high temperature magnetic excitations

and induce modified spin magnetic

moments on the alloy components.

{ DLM + LSF }

Site DLM µ(Fe)

DLM µ(Cr)

+LSF µ(Fe)

+LSF µ(Cr)

A 0.0 0 0.89 0.28

B 1.45 0 1.63 0.43

C 1.26 0 1.48 0.34

D 0.04 0 1.06 0.32

E 0.82 0 1.09 0.32






5. Summary

σ 28

P. Korzhavyi et al. Mat. Res. Soc. Symp. Proc. V 842, S4.10.1 (2005)

H at 0 K

H

H – TSconf

H – T(Sconf+Smagn)

at 1000 K

Approach : 32 ordered (end-member) configurations and DLM model

Importance of magnetism 1. Introduction:






5. Summary

𝜕𝐹

𝜕𝑐𝛼=𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝛼

− 𝑇 𝜕𝑆𝑐𝑜𝑛𝑓

𝜕𝑐𝛼+𝜕𝑆𝑚𝑎𝑔𝑛

𝜕𝑐𝛼 = 0

𝑐5 =30

𝑛5−

1

𝑛5 𝑛𝛼𝑐𝛼

4

𝛼=1

𝜕𝐹

𝜕𝑐𝛼= 𝐽𝛼

1 𝑛𝛼 + 𝑘𝐵𝑇𝑛𝛼 𝑙𝑛𝑐𝛼 − 𝑙𝑛 1 − 𝑐𝛼

− 𝑘𝐵𝑇𝑛𝛼 𝑙𝑛𝑐5 − 𝑙𝑛 1 − 𝑐5 +𝑘𝐵𝑇𝑛𝛼 𝑙𝑛 𝜇𝛼𝐹𝑒 + 1

− 𝑘𝐵𝑇𝑛𝛼 𝑙𝑛 𝜇5𝐹𝑒 + 1 − 𝑘𝐵𝑇𝑛𝛼 𝑙𝑛 𝜇𝛼

𝐶𝑟 + 1

+𝑘𝐵𝑇𝑛𝛼 𝑙𝑛 𝜇5𝐶𝑟 + 1

29

Equilibrium site occupancies cα

𝒄∝ =𝝌𝜶

𝟏 + 𝝌∝

where

𝝌∝ = 𝑒 −

1𝑘𝐵

𝐽∝ 1

𝑐5

1 − 𝑐5

(𝜇5𝐹𝑒 + 1)

(𝜇𝛼𝐹𝑒 + 1)

(𝜇𝛼𝐶𝑟 + 1)

(𝜇5𝐶𝑟 + 1)

(by Cr atoms)






5. Summary

30

Effect of magnetic state on site occupations

Fe0.5Cr0.5

B

B (A,D)

Effective on-site interactions Jα






5. Summary

T ≈ 0 K : integer site occupation is defined by Etot and described by the effective on-site interactions Jα. high T : partial site occupation is driven by Sconf, while Smagn corrects cα in (A,D) and B due to proper account of magnetism

31

Structural variations and µα (T)

V expansion : increases available space around (A,D) and occupation by large Cr – not favored – distortion within kagome layers. c/a expansion : decreases available space around E and increases around C.

T dependence of µα(Fe,Cr) Jα

Structural parameters






5. Summary

32

Final site distribution vs T

• Preferences in atomic site distribution according to the empirical scheme of ordering

• Good agreement with experimental data at finite T and results obtained by Wienk at T = 0 K

Fe0.5Cr0.5

Including all effetcs (magnetism, structural changes and T effects)

Site Emprical scheme

A (Mn,Fe,Co,Ni)

B (Ti,V,Cr,Mn,Mo)

C mixed

D (Mn,Fe,Co,Ni)

E mixed

Wie

n2k

1. Introduction:






5. Summary

33

Previous work

32 = 25 end-member compounds (total energy by LMTO-ASA)

24 effective cluster interactions were calculated by Connoly-Williams method

Magnetism was neglected

Results at high T disagree with experiment

Results at T = 0 K disagree with Wien2k

Site occupancy

M.H.F. Sluiter et al. PRL75 , 3142(1995)






5. Summary

34

Final site distribution vs x

• Linear dependence of the site occupancies on composition x • Good agreement with experimental data

EK, P.Blaha, K. Schwarz, A.V. Ruban, B. Johansson, PRB 83, 092201 (2011)

Fe1-xCrx






5. Summary

35

Test : Re0.5W0.5 system ∆(AR) = 2 pm


• Reverse occupation of C and E site in contrast to Fe-Cr • Agreement with the earlier published work






5. Summary


A (Mn,Fe,Co,Ni)

B (Ti,V,Cr,Mn,Mo)

C mixed

D (Mn,Fe,Co,Ni)

E mixed

C. Berne et al. PRB64 ,144103 (2001)

36

Test : Co0.5Cr0.5 system ∆(AR) = 3 pm


• Site occupation similiar to the Fe-Cr system






5. Summary


A (Mn,Fe,Co,Ni)

B (Ti,V,Cr,Mn,Mo)

C mixed

D (Mn,Fe,Co,Ni)

E mixed

37

𝐹 = 𝐸𝑡𝑜𝑡 − 𝑇(𝑆𝑐𝑜𝑛𝑓 + 𝑆𝑚𝑎𝑔𝑛 + 𝑆𝑣𝑖𝑏𝑟 )

𝐸𝑡𝑜𝑡 𝑐𝛼 = 𝐸0 𝑥 + 𝑛𝛼 [ 𝐽𝛼 1

+ 𝐽𝛼 𝑟𝑒𝑙𝑎𝑥

]𝑐𝛼

4

𝛼=1

𝑛𝛼 − 𝑠𝑖𝑡𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑐𝛼 − 𝑉 𝑠𝑖𝑡𝑒 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦

𝐽𝛼 1




𝐽𝛼 𝑟𝑒𝑙𝑎𝑥

= 𝐸𝑟𝑒𝑙𝑎𝑥 /𝑎𝑡𝑜𝑚

Effective on-site interactions Vα(1)

with respect to the E site

Fe0.5V0.5: local atomic relaxations ∆(AR) = 8 pm

ECI A B C D E

Jα(1) 24.0938 -7.7813 0.3750 19.8750 0

Jα(1) +Jα

(relax) 21.9375 -7.4062 0.3750 18.1875 0

EMTO-CPA

• Contributions from local atomic relaxations to effective on-site interactions in Fe-V are small

• Contributions in other systems with ∆(AR) ≤ 8 pm are expected to be also negligible






5. Summary

38

Test : Fe-V system ∆(AR) = 8 pm


• Nearly full occupation of (A,D,) and B in contrast to Fe-Cr system [due to bigger ∆(AR)]


A (Mn,Fe,Co,Ni)

B (Ti,V,Cr,Mn,Mo)

C mixed

D (Mn,Fe,Co,Ni)

E mixed






5. Summary

39

Test : Fe-V system ∆(AR) = 8 pm

• Linear dependence of the site occupancies on composition x • Reasonable agreement with experimental data • With increase of ∆(AR) full occupation of sublattices is more

pronounced






5. Summary

40

Summary :

Ab initio based mean field theory was suggested to calculate the Cr site occupancies in the Fe-Cr σ-phase.

4 effective on-site interaction parameters are enough to simulate the atomic site distribution in σ-phase

The calculated Cr site occupancies are in a good agreement with the results obtained by the FP-LAPW method (WIEN2k) at T = 0K and available experimental data at finite T.

It was shown that the Fe-Cr σ-phase exhibits a non-trivial magnetic behaviour at high T, which affects the site occupation by Fe and Cr atoms. The structural variation (volume and c/a) can lead to an additional atomic redistribution.

The proposed method works also for the other binary σ-phases. In particularly, Re-W, Co-Cr, Fe-V systems have been tested and results agree with empirical scheme of ordering.

The proposed method can be further applied for the ternary σ-phases.






5. Summary

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Prof. A. V. Ruban Applied materials Physics Department of Materials Science and Engineering The Royal Institute of Technology (KTH) Brinellvägen 23, SE-100 44 Stockholm, Sweden

Prof. P. Blaha Prof. K. Schwarz Institute of Materials Chemistry, Vienna University of Technology Getreidemarkt 9/165-TC, A-1060 Vienna, Austria

and WIEN2k group

Binary sigma phases

Technology

Transcript of Binary sigma phases