Binary sigma phases

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Transcript of Binary sigma phases
AB INITIOBASED MEAN FIELD THEORY OF SITE OCCUPATION IN BINARY SIGMA PHASES
E. Kabliman
OU
TL
INE
2
1. Introduction: 1. Binary σphases
2. FeCr phase diagram
3. Magnetic properties
4. σphase structure
5. Aim
2. Ab initiobased mean field theory: 1. Helmholtz‘s free energy
2. Total energy expansion
3. EMTOCPA
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 σphases]: 1. ReW
2. CoCr
3. FeV
5. Summary
3
σphase (light plates). Nicrofer 3127. after annealing at 700о С during 6000 hours
σphase 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)
σphase : • has a lamellar morphology • nucleates near grains boundaries (concentration of elements, e.g. Cr and Mo, is favorable) and on carbide particles
PROBLEM of σphase precipitation : • decreases ductility and corrosion resistance • increases brittleness and cracks formation
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
4
σphase in binary alloys
• atomic radii of the alloy components must be similar • FeCr 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 σphases VF ≤ 8 %
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
5
FeCr phase diagram
HISTORY :
• 1923 Bain: A hard and nonmagnetic phase in FeCr [Chem. Met. Eng. 28 (1923)]
• 1943 Cook, Jones: Crystal structure of σphase [J. Iron Steel Inst. London 148
(1943)]
• At present: σphase is found in more than 40 systems (CrFe, CrCo, VFe, etc.)
4
brittle σphase
Phase decomposition
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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 FeCr and FeV ]
Fe1xCrx
Fe1xCrx
Ave
rage
µ p
er F
e at
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1. Introduction:
1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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)
σphase structure
Fe1xCrx for x = 0.495, a = 0.87968 nm, c/a = 0.518. ncoord – site coordination numbers; nmult – site multiplicities
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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 σphase 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 Xray scattering factors
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
Sigmaphase is a substitutionally disordered solid solution in paramagnetic state;
It has a low symmetry structure with undefined „exact“ atomsite 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 onecomponent one
Coherent potential approximation: provide the same scattering properties of the onecomponent effective medium as the average of alloy components, embedded in this effective medium
Locally selfconsistent 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
EMTOCPA
Some of slides on EMTOCPA were kindly provided by Prof. A. Ruban (KTH, Sweden)
ATOMIC SPHERES METHODS
• Augmented plane waves (APW)
• Muffintin orbitals (MTO) ASA, exact MTO (EMTO)
• Linearized APW and MTO (LAPW and LMTO)
• (Linearized) augmented plane wave method ((L)APW) + local orbitals
• Fullpotential LAPW and LMTO methods
• Various Green‘s function method (KKR, KKRCPA, etc)
Fig. 2.1
13
Partial wave Backextrapolated partial wave Screened spherical wave
Kinked partial wave or exact muffintin
orbital
SSW Partial wave
BPW
They are basis functions of the EMTO method
The solution of the KohnSham 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 socalled disordered
local magnetic moment (DLM) model, in which such a system
considered as equiatomic random alloy of atoms with spinup
and spindown 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 (KohnSham) equation
A. Ruban
Coherent potential approximation (CPA)
(isomorphous singlesite approximation)
Since wave function is not selfaveraging 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 nonlinear equations are solved selfconsistently 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.
Fe1xCrx
Effective onsite interactions with respect to the E site
IDEA :
site sublattice
pairs
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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 nonequivalent multisite 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
+⋯
Fe1xCrx
~ 130 atoms
bcc : i = 23 fcc : i = 12 σA : i = 3 σB : i = 6 σC : i = 8 σD : i = 8 σE : i = 9
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
22
Effective pair cluster interactions
Free energy of σphase can be quite accurately presented in terms of concentrationindependent cluster interactions
Dominating terms in this expansion can be onsite interactions
Singlesite meanfield 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 σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
23
Effective onsite interactions
𝐸𝑡𝑜𝑡 𝑐𝛼 = 𝐸0 𝑥 + 𝑛𝛼 𝐽𝛼 1 𝑐𝛼
4
𝛼=1
𝑛𝛼 − 𝑠𝑖𝑡𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑐𝛼 − 𝐶𝑟 𝑠𝑖𝑡𝑒 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦
𝑐5 =30
𝑛5−
1
𝑛5 𝑛𝛼𝑐𝛼
4
𝛼=1
𝐽𝛼 1 =
𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝛼
−𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝐸
𝑐𝛼=𝑥=𝑐𝑜𝑛𝑠𝑡
• 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 muffintin orbitals (EMTO) method
in order to simulate disordered atomic distribution in σphase.
Fe1xCrx
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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 selfconsistent 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 σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
25
Check of EMTO and local atomic relaxations
Etot is calculated by: • fullpotential linearized augmented plane wave + local orbitals method (WIEN2k code) • exact muffintin 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 σFeCr are negligible
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
26
Convergence of total energy expansion
𝐸𝑡𝑜𝑡 𝑐𝛼 = 𝐸0 𝑥 + 𝑛𝛼 𝐽𝛼 1 𝑐𝛼
4
𝛼=1
+ 𝑛𝛼𝑛𝛽 𝐽𝛼𝛽 2 𝑐𝛼𝑐𝛽
4
𝛼 ,𝛽=1
𝐽𝛼 1
= 𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝛼
−𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝐸
𝑐𝛼=𝑥=0.5
Mean leastsquares fit 𝑱 𝜶 𝟏
: 89 various σphase 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 : rootmeansquare error ~ 0.2 mRy/atom
• 𝐽 𝛼 1
+ 𝐽 𝛼𝛽 2 : rootmeansquare 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 onsite interactions from
mean least squares fit 𝐽 𝛼 1
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
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
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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 (endmember) configurations and DLM model
Importance of magnetism 1. Introduction:
1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
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)
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
30
Effect of magnetic state on site occupations
Fe0.5Cr0.5
B
B (A,D)
Effective onsite interactions Jα
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
T ≈ 0 K : integer site occupation is defined by Etot and described by the effective onsite 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 σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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
Wie
n2k
1. Introduction:
1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
33
Previous work
32 = 25 endmember compounds (total energy by LMTOASA)
24 effective cluster interactions were calculated by ConnolyWilliams 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 σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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)
Fe1xCrx
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
35
Test : Re0.5W0.5 system ∆(AR) = 2 pm
• Preferences in atomic site distribution according to the empirical scheme of ordering
• Reverse occupation of C and E site in contrast to FeCr • Agreement with the earlier published work
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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
• Preferences in atomic site distribution according to the empirical scheme of ordering
• Site occupation similiar to the FeCr system
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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 + 𝑆𝑚𝑎𝑔𝑛 + 𝑆𝑣𝑖𝑏𝑟 )
𝐸𝑡𝑜𝑡 𝑐𝛼 = 𝐸0 𝑥 + 𝑛𝛼 [ 𝐽𝛼 1
+ 𝐽𝛼 𝑟𝑒𝑙𝑎𝑥
]𝑐𝛼
4
𝛼=1
𝑛𝛼 − 𝑠𝑖𝑡𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑐𝛼 − 𝑉 𝑠𝑖𝑡𝑒 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦
𝐽𝛼 1
= 𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝛼
−𝜕𝐸𝑡𝑜𝑡𝜕𝑐𝐸
𝑐𝛼=𝑥=𝑐𝑜𝑛𝑠𝑡
𝐽𝛼 𝑟𝑒𝑙𝑎𝑥
= 𝐸𝑟𝑒𝑙𝑎𝑥 /𝑎𝑡𝑜𝑚
Effective onsite 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
EMTOCPA
• Contributions from local atomic relaxations to effective onsite interactions in FeV are small
• Contributions in other systems with ∆(AR) ≤ 8 pm are expected to be also negligible
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
38
Test : FeV 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 FeCr 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 σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
39
Test : FeV 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 σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 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 FeCr σphase.
4 effective onsite interaction parameters are enough to simulate the atomic site distribution in σphase
The calculated Cr site occupancies are in a good agreement with the results obtained by the FPLAPW method (WIEN2k) at T = 0K and available experimental data at finite T.
It was shown that the FeCr σphase exhibits a nontrivial 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 σphases. In particularly, ReW, CoCr, FeV systems have been tested and results agree with empirical scheme of ordering.
The proposed method can be further applied for the ternary σphases.
1. Introduction: 1. Binary σphases 2. FeCr phasediagram 3. Magnetic properties 4. σphase structure 5. Aim
2. Ab initio MF theory:
1. Helmholt’z free energy 2. Total energy expansion 3. EMTOCPA 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 σs] : 1. ReW 2. CoCr 3. FeV
5. Summary
AC
KN
OW
LE
DG
EM
EN
TS
41
Prof. A. V. Ruban Applied materials Physics Department of Materials Science and Engineering The Royal Institute of Technology (KTH) Brinellvägen 23, SE100 44 Stockholm, Sweden
Prof. P. Blaha Prof. K. Schwarz Institute of Materials Chemistry, Vienna University of Technology Getreidemarkt 9/165TC, A1060 Vienna, Austria
and WIEN2k group