Spin crossovers in the Earth mantle. Spin crossovers in the Earth mantle

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1 Spin crossovers in the Earth mantle Spin crossovers in the Earth mantle Renata M. Wentzcovitch Dept. of Chemical Engineering and Materials Science Minnesota Supercomputing Institute

2 Collaborators Han Hsu University of Minnesota Koichiro Umemoto University of Minnesota Matteo Cococcioni University of Minnesota Peter Blaha Vienna University of Technology Stefano de Gironcoli SISSA (Trieste)

3 Earth s lower mantle Lower Mantle: Ferrosilicate perovskite + ferropericlase Low iron concentration (~ 0.10) High-temperatures and high pressures (Mg 1-y Fe y )(Si 1-x Fe x )O 3 perovskite (Mg 1-x Fe x )O ferropericlase +

4 Outline Spin crossovers Thermodynamics model of a spin crossover: (Mg,Fe)O (Mg,Fe)SiO 3 (It is not what it seems ) Spin crossover is in (Mg,Fe)(Si,Fe)O 3 Summary

5 What is a spin crossover? d-electrons in crystal field M m+ [core] 3d n

6 What is a spin crossover? d-electrons in crystal field M m+ [core] 3d n E C

7 What is a spin crossover? d-electrons in crystal field M m+ [core] 3d n E C E X 3d 6 S=2 High Spin (HS) compression

8 What is a spin crossover? d-electrons in crystal field M m+ [core] 3d n E C E X 3d 6 S=1 S=2 Intermediate High Spin Spin (HS) (IS) compression

9 What is a spin crossover? d-electrons in crystal field M m+ [core] 3d n E C E X 3d 6 S=2 High Spin (HS) S=1 Intermediate Spin (IS) compression S=0 Low Spin (LS)

10 Spin crossover observed in (Mg,Fe)O and (Mg,Fe)SiO 3 by XES (Mg,Fe)SiO 3 (100 GPa = 1 Mbar) Badro et al., Science (2003) Badro et al., Science (2004)

11 First Principles Calculations Density Functional Theory (DFT and DFT+U) (Cococcioni and de Gironcoli, PRB, 2005) Plane waves + Pseudopotential (Troullier-Martins, PRB, 1991, Vanderbilt, PRB, 1990) Structural relaxation in all configurations (Variable cell shape MD, Wentzcovitch, PRB, 1991) Density Functional Perturbation Theory (Baroni et al., RMP, 2001) Quantum ESPRESSO distribution

12 Ferropericlase

13 Structural Models of Magnesiowüstite X Fe =3.125% X Fe =12.5% 2a 1 x2a 2 x2a 3 supercell of MgO (a: conventional lattice vector) Fe Mg O X Fe =18.75% Exp: X Fe =17% (Badro et al., 2003)

14 ρ el around Fe 2+ (Isosurface:ρ el =0.3e/Å 3 ) O A. HS Maj C. LS Maj O O O O O Fe Fe O O O O O O B. HS Min O V oct ~-8% O Fe O +1% -1% -2% O O +2% -1% -3% O

15 H=H LS -H HS (kj/mol) Static HS-to-LS transition 3.125% 12.5% 18.75% P (GPa) P calc = GPa

16 H=H LS -H HS (kj/mol) Static HS-to-LS transition 3.125% 12.5% 18.75% P (GPa) P exp = GPa

17 X Fe =18.75% (6 irons in supercell) Static H (kj/mol) Fe HS 0Fe LS 4Fe HS 2Fe LS 2Fe HS 4Fe LS 0Fe HS 6Fe LS P (GPa) P T does not depend on HS/LS ratio

18 V (cm 3 /mol) B 0 50 n=n LS /(n LS +n 100 HS ) P (GPa) n=0 X Fe =17% n=1/3 n=2/3 n=1 Exp. (Lin et al., 2005) V ~-4.2% Static equation of state 8 C X Fe =18.75% P (GPa) V HS-LS = nx Fe cm 3 /mol

19 Thermodynamics of ferropericlase

20 Ideal solid solution of HS and LS ferropericlase n = n LS /(n HS +n LS ) G = (1-n)G HS ng LS + G mix

21 Ideal solid solution of HS and LS ferropericlase n = n LS /(n HS +n LS ) G = G HS n(g HS - G LS ) + G mix G HS/LS = F HS/LS + PV HS/LS G HS/LS = F HS/LS (stat+vib) + F HS/LS mag + PV HS/LS F HS/LS mag = - TS HS/LS mag G mix = - TS ideal

22 Ideal solid solution of HS and LS ferropericlase n = n LS /(n HS +n LS ) G = G HS n(g HS - G LS ) + G mix G HS/LS = F HS/LS + PV HS/LS G HS/LS = F HS/LS (stat+vib) + F HS/LS mag + PV HS/LS F HS/LS mag = - TS HS/LS mag G mix = - TS ideal

23 Thermodynamics (0 th order) 1) Magnetic entropy S HS/LS mag = RX Fe n HS/LS ln[m(2s+1)] = X Fe (1-n)Rln15 (S=2, m=3, HS) = 0 (S=0, m=1, LS)

24 Thermodynamics (0 th order) 1) Magnetic entropy S HS/LS mag = RX Fe n HS/LS ln[m(2s+1)] = X Fe (1-n)Rln15 (S=2, m=3, HS) = 0 (S=0, m=1, LS) 2) G mix = -TS ideal n = n LS /(n HS +n LS ) S ideal = -X Fe R[nln(n)+(1-n)ln(1-n)]

25 Thermodynamics (0 th order) 1) Magnetic entropy S HS/LS mag = RX Fe n HS/LS ln[m(2s+1)] = X Fe (1-n)Rln15 (S=2, m=3, HS) = 0 (S=0, m=1, LS) 2) G mix = -TS ideal n = n LS /(n HS +n LS ) S ideal = -X Fe R[nln(n)+(1-n)ln(1-n)] 3) Vibrational energy and entropy is insensitive to iron spin state F HS/LS (stat+vib) = F HS/LS QHA

26 Thermodynamics (0 th order) 1) Magnetic entropy S HS/LS mag = RX Fe n HS/LS ln[m(2s+1)] = X Fe (1-n)Rln15 (S=2, m=3, HS) = 0 (S=0, m=1, LS) 2) G mix = -TS ideal n = n LS /(n HS +n LS ) S ideal = -X Fe R[nln(n)+(1-n)ln(1-n)] 3) Vibrational energy and entropy is insensitive to iron spin state F HS/LS (stat+vib) = F HS/LS QHA 4) Fe/Mg configuration entropy is insensitive to iron spin state

27 Free energy: G(P,T,n)= G HS (stat+vib) (P,T) nδg HS-LS (stat+vib) ( (P,T) Tk B X Fe (1-n)ln[m(2S+1)] Tk B X Fe [nlnn + (1-n)ln(1-n)] G(P,T,n) is minimized when: npt (, ) = 1 H HS LS 1 + m(2s+ 1) exp XFekT B

28 LS fraction n(p,t) (Tsuchiya, Wentzcovitch, de Gironcoli, PRL, 2006) X Fe =18.75% Exp Geotherm (Boehler, RG, 2000)

29 LS fraction n(p,t) X Fe =18.75% Exp + Lin et al., Science (2007)

30 Thermodynamics (1 st order) 1) Magnetic entropy S HS/LS mag = RX Fe n HS/LS ln[m(2s+1)] = X Fe (1-n)Rln15 (S=2, m=3, HS) = 0 (S=0, m=1, LS) 2) G mix = -TS ideal n = n LS /(n HS +n LS ) S ideal = -X Fe R[nln(n)+(1-n)ln(1-n)] 3) Vibrational energy and entropy is sensitive to iron spin state F HS/LS (stat+vib) = F HS/LS QHA 4) Fe/Mg configuration entropy is insensitive to iron spin state

31 Free energy minimization npt (, ) = 1 st+ vib G HS LS 1 + m(2s+ 1) exp XFekT B

32 + + = qj B qj B qj qj T k V T k V V U T V F ) ( exp 1 ln 2 ) ( ) ( ), ( ω ω PV TS F G + = T V F P = V T F S = Static +vibrational free energy VDoS and F(T,V) within the quasiharmonic approximation

33 Vibrational Virtual Crystal Model Replace Mg mass by the average cation mass of the alloy Replace some inter-atomic force constants of MgO to reproduce the that the static elastic constants of the alloy

34 LS fraction n(p,t) (Wentzcovitch et al., PNAS, 2009) X Fe =18.75% Exp + Lin et al., Science (2007)

35 Free energy: Ideal solid solution GnPT (,, ) = ng ( PT, ) + (1 ng ) ( PT, ) + G LS HS mix Volume: V( PT,, n) = nv ( PT, ) + (1 nv ) ( PT, ) Compressibility: LS HS V ( n) K( n) = n V K LS LS + (1 n) V K HS HS ( V LS V HS n ) P T

36 Volume V(P,T,n(P,T)) for x Fe = 18.75% x Fe = 18.75% + 300K (exp.) + 300K (exp.) + Experiments (Lin et al., Nature, 2005) (x Fe =17%, RT) o and Δ (Fei et al., GRL, 2007) (X Fe =20%, RT)

37 Crowhurst et al. Science, 2006 x Fe exp = 0.06 Wentzcovitch, Justo, Wu, da Silva Yuen, Kohlstedt, PNAS 2009 x Fe calc =

38 Thermodynamics properties x Fe = 18.75% + 300K (exp.) Experiments (o x Fe =0 and += 40%) Wu et al, PRB 2009

39 Future investigations Elasticity and mantle velocities Real solid solution Possible effect on mantle rheology (diffusion creep behavior sub-solidus viscosity) Mantle convection

40 Spin Crossovers in Perovskite (Fe +2 ) (Fe +3 )

41 Displacement of A-site Fe+2 atom by spin state change HS Bicapped trigonal prism (8-coordinated Fe) LS Distorted octahedron (6-coordinated Fe) : Fe : O

42 New species of Fe 2+ : IS? detector 57 Fe +2 γ ray E=14.4 kev 57 Co 57 Fe* 57 Fe source V (cm/sec) McCammon et al. Nature Geosciences (2008)

43 New species of Fe 2+ : IS? detector 57 Fe +2 γ ray E=14.4 kev 57 Co 57 Fe* 57 Fe source V (cm/sec) I = 3/ kev I = 1/2 QS m = ± 3/2 m = ± 1/2 m = ± 1/2 McCammon et al. Nature Geosciences (2008)

44 New species of Fe 2+ : IS? detector 57 Fe +2 γ ray E=14.4 kev 57 Co 57 Fe* 57 Fe source V (cm/sec) I = 3/ kev QS m = ± 3/2 m = ± 1/2 E ~ Eo + eqv zz / 4 E ~ Eo - eqv zz / 4 I = 1/2 QS ~ eqv zz /2 m = ± 1/2 V zz 2 V = z z. Spin state charge distribution V zz QS. Spin state cannot be derived using just QS McCammon et al., Nature Geosciences (2008)

45 New species of Fe 2+ : IS? At 0 GPa: HS state with QS = 2.4 mm/sec New Fe 2+ (QS = 3.5 mm/s) for P > 30 GPa Fe 2+ QS = 3.5 mm/s increases at the expense of the HS Fe 2+ (QS = 2.4 mm/s) The two sets of peaks merge at P ~ 60 GPa McCammon et al. Nature Geosc. (2008)

46 HS and LS configurations at 0 GPa x Fe = 0.25 and Hsu, Umemoto, Blaha, and Wentzcovitch, EPSL 2009

47 Effect of E XC and Hubbard U 24 GPa 15 GPa (Mg Fe )SiO 3 The low U HS further stabilize HS Fe. QS improved by U No HS-to-IS QS = 2.4 mm/s QS = 3.5 mm/s 7 GPa 4 GPa The two competing HS indistinguishable in GPa in LDA(+U). No HS-to-IS Hsu et al., EPSL 2009

48 HS and LS configurations at 0 GPa Main difference: Fe position Different ligand field different orbital occupancy different QS Hsu, Umemoto, Blaha, and Wentzcovitch, EPSL 2009

49 Spin Crossover in Perovskite (Fe +3 )

50 Previous Calculations Experiments Calculations XES Mössbauer (QS: ~0.5 ~3.0 mm/s) HS LS (P T ~ GPa) 50% HS LS 50% remains HS (P T ~ 150 GPa) GGA Ground state: (A-HS, B-LS) (A-HS, B-LS) (A-LS, B-LS) P T > 75 GPa Zhang & Oganov, EPSL (2006) Stackhouse et al., EPSL (2007) Inconsistent with Exp P T too high Fraction of HS Fe 3+ too low Catalli et al., EPSL (2010)

51 (Mg 1-x Fe x )(Si 1-x Fe x )O 3 40-atom supercell (x = 0.125) Parallel and anti-parallel spin DFT+U (U = 4 ev and U SC ) Self-consistent U (U SC ) View along [001] Hsu et al., PRL (2011).

52 Ground state: (A-HS, B-HS) GGA+U (U = 4 ev) The ground state has HS iron at both A and B sites. HS-LS crossover occurs in the B-site iron. The A-site iron remains HS. Predicted P T = 29 GPa P T insensitive to spin alignment. Hsu et al., PRL (2011).

53 Relative Enthalpies (U SC ) LDA+U SC P T = 41 GPa GGA+U SC P T = 70 GPa P T observed in Mössbauer spectra : GPa Hsu et al., PRL (2011).

54 Quadrupole Splitting (QS) (Mg,Fe)SiO 3 (Mg,Fe)(Si,Fe)O 3 Hsu et al., EPSL (2010). Exp McCammon et al., Nature Geosci. (2008); Jackson et al., Am. Mineral. (2005); Catalli et al., EPSL (2010). Hsu et al., PRL (2011).

55 Equation of state Experimental data: (A-HS;B-HS) (A-HS; B-LS). If there were spin-state crossover in the A-site iron, the associated volume reduction would be barely noticeable. Hsu et al., PRL (2011).

56 Anomaly in bulk modulus At 300 K: significant softening in bulk modulus. At 2000 K: anomaly is still prominent. Fe 3+ in Pv has greater effect on the lower mantle than Fe 2+ in Pv. Hsu et al., PRL (2011).

57

58 Summary HS-LS crossover in (Mg 1-x Fe x )O is well reproduced computationally The bulk modulus softens throughout the mixed spin state. This effect broadens and decreases with temperature The crossover in Fe +2 in perovskite is actually a displacement transition The spin state change happens in Fe +3 in the perovskite B site only

59 Acknowledgements NSF/EAR NSF/ATM (VLab) Computations performed at the MSI-UMN