Binary sigma phases

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

OUTLINE
2
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
5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy
3. EMTO-CPA
4. Convergence of expansion
6. Site occupancies
3. Results [FeCr] :
2. Structural varitations
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.
V.N.BykovandV.A.Solov’ev:Zhurn.Strukt.KhimiiVol.5No.2,288(1964)
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 %
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
AverageµperFeatomCurietemperature
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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)
𝛿𝑐 𝛼,𝑖
𝑖𝛼=1,5
+
1
2
𝑉∝𝛽,𝑝
(2)
𝛿𝑐 𝛼;𝑖 𝛿𝑐𝛽;𝑗
𝑖,𝑗∈𝑝𝑝𝛼,𝛽=1,5
+
1
3
𝑉∝𝛽𝛾 ,𝑡
(3)
𝛿𝑐 𝛼;𝑖 𝛿𝑐𝛽;𝑗 𝛿𝑐 𝛾;𝑘
𝑖,𝑗,𝑘∈𝑡𝑡𝛼,𝛽,𝛾=1,5
+ ⋯
𝛿𝑐 𝛼,𝑖 = 𝑐 𝛼,𝑖 − 𝑐 𝛼 , 𝑤ℎ𝑒𝑟𝑒 𝑐 𝛼 ≡ 𝑐 𝛼,𝑖 𝑎𝑛𝑑 𝑐 𝛼.𝑖 = 1 𝑖𝑓 𝐶𝑟 𝑎𝑡 𝑠𝑖𝑡𝑒 𝑖
𝐸𝑡𝑜𝑡 𝑐 𝛼 = 𝐸0 𝑥 + 𝑛 𝛼 𝐽 𝛼
1
𝑐 𝛼
4
𝛼=1
.
𝑛 𝛼 − 𝑠𝑖𝑡𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑐 𝛼 − 𝐶𝑟 𝑠𝑢𝑏𝑙𝑎𝑡𝑡𝑖𝑐𝑒 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦
𝐽 𝛼
1
=
𝜕𝐸𝑡𝑜𝑡
𝜕𝑐 𝛼
−
𝜕𝑐 𝐸
| 𝑐 𝛼 =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 :
sitesublattice
pairs
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
𝛿𝑐 𝛼,𝑖
𝑖𝛼=1,5
+
1
2
𝑉∝𝛽,𝑝
2
𝛿𝑐 𝛼;𝑖 𝛿𝑐 𝛽;𝑗
𝑖,𝑗∈𝑝𝑝𝛼,𝛽=1,5
+
1
3
𝑉∝𝛽𝛾,𝑡
3
𝛿𝑐 𝛼;𝑖 𝛿𝑐 𝛽;𝑗 𝛿𝑐 𝛾;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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary

23
Effective on-site interactions
1
𝑐 𝛼
4
𝛼=1
𝑛 𝛼 − 𝑠𝑖𝑡𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑐 𝛼 − 𝐶𝑟 𝑠𝑖𝑡𝑒 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦
𝑐5 =
30
𝑛5
−
1
𝑛5
𝑛 𝛼 𝑐 𝛼
4
𝛼=1
𝐽 𝛼
1
=
𝜕𝑐 𝛼
−
𝜕𝑐 𝐸
• 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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary

26
Convergence of total energy expansion
1
𝑐 𝛼
4
𝛼=1
+ 𝑛 𝛼 𝑛 𝛽 𝐽 𝛼𝛽
2
𝑐 𝛼 𝑐𝛽
4
𝛼,𝛽=1
𝐽 𝛼
1
=
𝜕𝑐 𝛼
−
𝜕𝑐 𝐸
| 𝑐 𝛼 =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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary

𝐹 = 𝐸𝑡𝑜𝑡 − 𝑇(𝑆𝑐𝑜𝑛𝑓 + 𝑆 𝑚𝑎𝑔𝑛 + 𝑆 𝑣𝑖𝑏𝑟 )
𝑆𝑐𝑜𝑛𝑓 = −𝑘 𝐵 𝑛∝ 𝑐∝ ln 𝑐∝ + 1 − 𝑐∝ ln 1 − 𝑐∝
5
𝛼=1
𝑆 𝑚𝑎𝑔𝑛 = 𝑘 𝐵 𝑛∝[
5
𝛼=1
𝑐∝ ln 𝜇 𝛼
𝐶𝑟
+ 1 + 1 − 𝑐∝ ln 𝜇 𝛼
𝐹𝑒
+ 1 ]
𝑆 𝑣𝑖𝑏𝑟 = 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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary

𝜕𝐹
𝜕𝑐 𝛼
=
𝜕𝑐 𝛼
− 𝑇
𝜕𝑆𝑐𝑜𝑛𝑓
𝜕𝑐 𝛼
+
𝜕𝑆 𝑚𝑎𝑔𝑛
𝜕𝑐 𝛼
= 0
𝑐5 =
30
𝑛5
−
1
𝑛5
𝑛 𝛼 𝑐 𝛼
4
𝛼=1
𝜕𝐹
𝜕𝑐 𝛼
= 𝐽 𝛼
1
𝑛 𝛼 + 𝑘 𝐵 𝑇𝑛 𝛼 𝑙𝑛𝑐 𝛼 − 𝑙𝑛 1 − 𝑐 𝛼
− 𝑘 𝐵 𝑇𝑛 𝛼 𝑙𝑛𝑐5 − 𝑙𝑛 1 − 𝑐5 + 𝑘 𝐵 𝑇𝑛 𝛼 𝑙𝑛 𝜇 𝛼
𝐹𝑒
+ 1
− 𝑘 𝐵 𝑇𝑛 𝛼 𝑙𝑛 𝜇5
𝐹𝑒
+ 1 − 𝑘 𝐵 𝑇𝑛 𝛼 𝑙𝑛 𝜇 𝛼
𝐶𝑟
+ 1
+𝑘 𝐵 𝑇𝑛 𝛼 𝑙𝑛 𝜇5
𝐶𝑟
+ 1
29
Equilibrium site occupancies cα
𝒄∝ =
𝝌 𝜶
𝟏 + 𝝌∝
where
𝝌∝ = 𝑒
−
1
𝑘 𝐵
𝐽∝
1 𝑐5
1 − 𝑐5
(𝜇5
𝐹𝑒
+ 1)
(𝜇 𝛼
𝐹𝑒
+ 1)
(𝜇 𝛼
𝐶𝑟
+ 1)
(𝜇5
𝐶𝑟
+ 1)
(by Cr atoms)
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary

30
Effect of magnetic state on site occupations
Fe0.5Cr0.5
B
B
(A,D)
Effective on-site interactions Jα
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
Wien2k 1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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)
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary
Site Emprical
scheme
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary
Site Emprical
scheme
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 + 𝑆 𝑚𝑎𝑔𝑛 + 𝑆𝑣𝑖𝑏𝑟 )
𝐸𝑡𝑜𝑡 𝑐 𝛼 = 𝐸0 𝑥 + 𝑛 𝛼 [ 𝐽 𝛼
1
+ 𝐽 𝛼
𝑟𝑒𝑙𝑎𝑥
]𝑐 𝛼
4
𝛼=1
𝑛 𝛼 − 𝑠𝑖𝑡𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑐 𝛼 − 𝑉 𝑠𝑖𝑡𝑒 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦
𝐽 𝛼
1
=
𝜕𝑐 𝛼
−
𝜕𝑐 𝐸
𝐽 𝛼
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary

38
Test : Fe-V system ∆(AR) = 8 pm
• Preferences in atomic site distribution according to the empirical
scheme of ordering
• Nearly full occupation of (A,D,) and B in contrast to Fe-Cr system
[due to bigger ∆(AR)]
Site Emprical
scheme
A (Mn,Fe,Co,Ni)
B (Ti,V,Cr,Mn,Mo)
C mixed
D (Mn,Fe,Co,Ni)
E mixed
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
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.
1. Introduction:
1. Binary σ-phases
5. Aim
3. EMTO-CPA
6. Site occupancies
3. Results [FeCr] :
1. ReW
2. CoCr
3. FeV
5. Summary

ACKNOWLEDGEMENTS
41
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

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