Laves A-B AB 2 MgCu 2 (C14) MgZn 2 (C15) MgNi 2 (C36) Laves VASP ZrCr 2 Laves VASP(Vienna Ab-initio Simulation Package) Laves Energy-Volume Quasi-Harm
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- まいえ いんそん
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1 ZrCr 2 Laves
2 Laves A-B AB 2 MgCu 2 (C14) MgZn 2 (C15) MgNi 2 (C36) Laves VASP ZrCr 2 Laves VASP(Vienna Ab-initio Simulation Package) Laves Energy-Volume Quasi-Harmonic Energy-Volume Phonon-DOS Phonon-DOS Energy-Volume Phonon-DOS 2 Phonon-DOS
3 MedeA Phonon-DOS Morse B(bulk modulus) D(θ D ) F Energy-Volume Phonon-DOS F
4 1 Laves A-B AB 2 Laves 200 Laves MgCu 2 (C14) MgZn 2 (C15) MgNi 2 (C36) FrankKasper [2] 1.1 C14 AB C36 ABAC A C15 ABC C14 12 C15 C36 24 [3] C15 C14 [4] Laves 1.1: C14 C15 C36 prototype MgCu 2 MgZn 2 MgNi AB ABC ABAC 2
5 1.1: ZrCr 2 Laves 3
6 2 2.1 VASP(Vienna Ab-initio Simulation Package) PAW VASP 2.2 MedeA VASP MedeA MedeA 1 Windows Phonon-DOS Phonon MedeA Phonon 4
7 Morse Moruzzi[5] quasi-harmonic a b c λ E(r) = a + be λr + ce λr (2.1) Morse E(r) = A 2De λ(r r 0) + De 2λ(r r 0) (2.2) a = A (2.3) b c = 2e λr 0 (2.4) D = b2 4c (2.5) B(bulk modulus) 2.5 B(bulk modulus) x = e λr (2.6) 2.1 E(x) = a + bx + cx 2 (2.7) 5
8 P P = de (2.8) dv V P = de/dx dv/dx (2.9) V = 4π 3 r3 (2.10) 2.6 r V r = ln x λ (2.11) V = 4π 3 ( ln x λ )3 (2.12) x 2.9 P P = xλ3 (b + 2cx) (2.13) 4π(ln x) 2 B P B = V dp dv B = dp/dx dv/dx (2.14) (2.15) B = xλ3 2 [(b + 4cx) (b + 2cx)] (2.16) 12π ln x ln x B θ D 6
9 2.6 v s g(ω) ω = v s k (2.17) g(ω) = V k2 2π 2 dk dω = V ω2 2π 2 v 3 s (2.18) v s ω k 1/v 3 s g(ω) = V ω2 2π 2 ( 1 v 3 l + 2 ) (2.19) vt 3 longitudinaltransverse v t v l 1 v 3 s = 1 v 3 l + 2 v 3 t (2.20) v s ω D ωd g(ω)dω = 3 (2.21) 0 ( V 1 6π 2 vl v 3 l 3 v 3 = 2 v 3 t ) + 1 v 3 l ω 3 D = 3 (2.22) (2.23) 7
10 2.22 V 6π 2 v 3 ωd 3 = 1 (2.24) v ρ B ρ v = B ρ ρ = M V (2.25) (2.26) V V = 4 3 πr3 (2.27) v = πr3 B (2.28) ω 3 D = 6π 2 3 4πr 3 v3 = 6π 2 ( 4 3 πr3 ) 1 ( 4πr3 3 ) 3 2 ( B M ) 3 2 (2.29) ω D = (6π 2 ) ( 3 πr3 ) 1 B 6 M (2.30) k B θ D = h 2π ω D (2.31) k B ω D k B Boltzmann s constants h Plank s cpnstants ω D Debye frequency 2.30 θ D = h (6π 2 ) ( 2πk B 3 πr3 ) 1 B 6 M rb θ D = M v t v l ρ v t = S ρ (2.32) (2.33) (2.34) 8
11 v l = L ρ (2.35) Molzzi S = 0.30B L = 1.42B B v = (2.36) ρ v 2.30 θ D (θ D ) 0 = r0 B M (2.37) r 0 [a.u.] M B r 0 [kbar] 2.7 D(θ D ) D(y) = 3 y 3 y 0 e x x 4 dx (2.38) (e x 1) 2 D(y) y 0 D 1 y = θ D 2.8 F E(r, T ) = E(r) + E D (r, T ) T S D (r, T ) (2.39) T E D S D E D (r, T ) = E 0 + 3k B T D ( ) θd [ ( ) ] 4 S D (r, T ) = 3k B 3 D θd ln(1 e θ DT ) T T (2.40) (2.41) E 0 = 9 8 k Bθ D (2.42) 9
12 E(r, T ) = E(r) k B T [ D ( θd T ) 3 ln(1 e θ DT ) ] k Bθ D (2.43) 2.9 1K β = α 3 (2.44) 1/3 r 0 [a.u.] T[K] α(t ) = 1 r 0 dr 0 dt (2.45) 10
13 3 ZrCr 2 C14 C15 C36 3 Laves VASP C14 C15 C36 Energy-Volume Phonon-DOS Energy-Volume Moruzzi 3.1 Energy-Volume ZrCr 2 Energy-Volume E-V x y y () x y Zr1 Cr // 2 5 //C14 infile1:=[[0.925, ], [0.95, ], [0.975, ], [1.0, ], [1.025, ]] //C15 11
14 infile2:=[[0.95, ], [0.975, ], [1.0, ], [1.025, ], [1.05, ]] //C36 infile3:=[[1.0, ], [1.025, ], [1.05, ], [1.075, ], [1.1, ]] //MedeA C14 V1:=Vector([ , , ]): V2:=Vector([ , , ]): V3:=Vector([ , , ]): //C15 V4:=Vector([ , , ]): V5:=Vector([ , , ]): V6:=Vector([ , , ]): //C36 V7:=Vector([ , , ]): V8:=Vector([ , , ]): V9:=Vector([ , , ]): // ; V1.(CrossProduct(V2,V3)); V4.CrossProduct(V5,V6); V7.CrossProduct(V8,V9); //3 ^3* 0 *3 for i from 1 to nops(infile1) do infile1[i,1]:=infile1[i,1]^3* /12*3; end do: for i from 1 to nops(infile2) do infile2[i,1]:=infile2[i,1]^3* /6*3; 12
15 end do: for i from 1 to nops(infile3) do infile3[i,1]:=infile3[i,1]^3* /24*3; end do: for j from 1 to nops(infile1) do infile1[j,2]:=infile1[j,2]/12*3; end do: for j from 1 to nops(infile2) do infile2[j,2]:=infile2[j,2]/6*3; end do: for j from 1 to nops(infile3) do infile3[j,2]:=infile3[j,2]/24*3; end do: //4 fitting data1:=convert(transpose(convert(infile1,array)),listlist): fit1:=fit[leastsquare[[x,y], y=c0+c4*x+c1*x^2+c2*x^3+c3*x^4, {c0,c1,c2,c3,c4,c5}]](data1): fit_c14:=unapply(rhs(fit1),x); data2:=convert(transpose(convert(infile2,array)),listlist): fit2:=fit[leastsquare[[x,y], y=c0+c4*x+c1*x^2+c2*x^3+c3*x^4, {c0,c1,c2,c3,c4,c5}]](data2): fit_c15:=unapply(rhs(fit2),x); data3:=convert(transpose(convert(infile3,array)),listlist): fit3:=fit[leastsquare[[x,y], y=c0+c4*x+c1*x^2+c2*x^3+c3*x^4, {c0,c1,c2,c3,c4,c5}]](data3): fit_c36:=unapply(rhs(fit3),x); // d1:=display(p1,plot(fit_c14(x),x=30..60,color=red)); d2:=display(p3,plot(fit_c15(x),x=30..60,color=blue)); d3:=display(p5,plot(fit_c36(x),x=30..60,color=green)); display(d1,d2,d3,view=[40..52, ]); 13
16 3.1: E-V fitting 3.1 C15 C36 C14 Pavlu [4] 3.2 Phonon-DOS Phonon-DOS Phonon-DOS VASP A(T) Phonon-DOS C14 Laves
17 Cv : vibrational heat capacity at constant volume. E(T)-E(0) : the change in vibrational internal energy from 0 K. E(0) is the zero point energy (ZPE). S(T) : the vibrational entropy at temperature T. -(A(T)-E(0)) : the change in the vibrational Helmholtz free energy from 0 K. E(T) : the electronic plus vibrational energy of formation. A(T) : the electronic plus vibrational Helmholtz free energy. 3.1: C14 Phonon-DOS T Cv E(T)-E(0) S(T) -(A(T)-E(0)) E(T) A(T) K J/K/mol kj/mol J/K/mol kj/mol kj/mol kj/mol
18 // pho_c14:=[[0., ], [1., ], [10., ], [100., ], [200., ], [300., ], [400., ], [500., ], [600., ], [700., ], [800., ], [900., ], [1000., ], [1100., ], [1200., ], [1300., ], [1400., ], [1500., ], [1600., ], [1700., ], [1800., ], [1900., ], [2000., ]]; pho_c15:=[[0., ], [1., ], [10., ], [100., ], [200., ], [300., ], [400., ], [500., ], [600., ], [700., ], [800., ], [900., ], [1000., ], [1100., ], [1200., ], [1300., ], [1400., ], [1500., ], [1600., ], [1700., ], [1800., ], [1900., ], [2000., ]]; pho_c36:=[[0., ], [1., ], [10., ], [100., ], [200., ], [300., ], [400., ], [500., ], [600., ], [700., ], [800., ], [900., ], [1000., ], [1100., ], [1200., ], [1300., ], [1400., ], [1500., ], [1600., ], [1700., ], [1800., ], [1900., ], [2000., ]]; //0 Zr2 Cr1 //[Ry] [ev] for i from 1 to nops(pho_c14) do pho_c14[i,2]:=(pho_c14[i,2] )*1000/4.1855/23060; pho_c15[i,2]:=(pho_c15[i,2] )*1000/4.1855/23060; pho_c36[i,2]:=(pho_c36[i,2] )*1000/4.1855/23060; end do: pho_c14; pho_c15; pho_c36; // poi1:=pointplot(pho_c14,color=red,legend="c14_phonon",symbol=box): 16
19 p1:=pointplot(pho_c14,color=red,connect=true): poi2:=pointplot(pho_c15,color=blue,legend="c15_phonon",symbol=circle): p2:=pointplot(pho_c15,color=blue,connect=true): poi3:=pointplot(pho_c36,color=green,legend="c36_phonon",symbol=cross): p3:=pointplot(pho_c36,color=green,connect=true): display(p1,p2,p3,poi1,poi2,poi3,view= , labels=["temperature[k]","free Energy[eV/ZrCr2 atoms]"], labeldirections=[horizontal,vertical]); 3.2: Phonon-DOS Phonon-DOS 0K 2000K 100K 3.2 C15 C K Pavlu 17
20 3.3 Energy-Volume ZrCr 2 C14 C15 C36 3 Laves 0K 2000K 100K r C14 C15 C36 // // 5 //C14 infile1:=[[0.925, ], [0.95, ], [0.975, ], [1.0, ], [1.025, ]] //C15 infile2:=[[0.95, ], [0.975, ], [1.0, ], [1.025, ], [1.05, ]] //C36 infile3:=[[1.0, ], [1.025, ], [1.05, ], [1.075, ], [1.1, ]] //C14 ; V1.(CrossProduct(V2,V3)); //C15 ; V4.CrossProduct(V5,V6); //C36 ; V7.CrossProduct(V8,V9); //MedeA 18
21 r_c14:=3.031; r_c15:=2.910; r_c36:=2.817; // C14 r [ ] [a.u.] for i from 1 to nops(infile1) do infile1[i,1]:=infile1[i,1]*r_c14*1.89: end do: //C14 E (3 )[ev] [Ry] for j from 1 to nops(infile1) do infile1[j,2]:=infile1[j,2]/12*3/ ; end do: //4 fitting data1:=convert(transpose(convert(infile1,array)),listlist): fit1:=fit[leastsquare[[x,y], y=c0+c4*x+c1*x^2+c2*x^3+c3*x^4, {c0,c1,c2,c3,c4}]](data1): fit_c14:=unapply(rhs(fit1),x): fit_c14(x); // p1:=pointplot(infile1,color=red,legend="c14",symbol=box): p2:=pointplot(infile1,color=red,connect=true): display(p1,p2); d1:=display(p1,plot(fit_c14(x),x= ),color=black): display(d1,view= ,labels=["[a.u.]"," [Ry]"], labeldirections=[horizontal,vertical]); C15 C36 Fitting 19
22 3.3: C14 Laves 3.4: C15 Laves 20
23 3.5: C36 Laves [KBar] [Ry/(a.u.) 3 ] [ev/ 3 ] [GPa] [kbar] [1] //r0( ) r0:=fsolve(diff(fit_c14(x),x)=0,x= ); //r1(medea 0 r[ ] r[a.u.] ) r1:=3.031*1.89; //MedeA 3 V1:= /12*3*1.89^3; //V0 V //r 1/3 //dv V0:=(r0/r1)^3*V1: 21
24 V:=r->r^3*V0/(r0)^3: r:=v->(v/ )^(1/3): dv:=unapply(1/(diff(v(r),r)),r); // B(C14):=unapply(V(r)*dV(r)*diff(dV(r)*diff(fit_C14(r),r),r) * /( )^3* ,r); // B1:=display(plot(B(C14)(x),x=1..5),color=black): display(b1); C15 C // B(C14)(r0); B(C15)(r0); B(C36)(r0);
25 3.6: //ZrCr2 M Zr1 Cr2 M= *2; // thetad_c14:=unapply(41.63*(r2*b(c14)(r2)/m)^(1/2),r2): thetad_c15:=unapply(41.63*(r2*b(c15)(r2)/m)^(1/2),r2): thetad_c36:=unapply(41.63*(r2*b(c36)(r2)/m)^(1/2),r2): // B10:=display(plot(thetaD_C14(r2),r2= ,color=red,legend="C14")): B11:=display(plot(thetaD_C15(r2),r2= ,color=blue,legend="C15")): B12:=display(plot(thetaD_C36(r2),r2= ,color=green,legend="C36")): display(b10,b11,b12,labels=[" [a.u.]"," [K]"], 23
26 labeldirections=[horizontal,vertical]); 3.7: 24
27 // f1:=unapply(exp(x)*x^4/(exp(x)-1)^2,x): Debye:=unapply(3/y^3*Int(f1(x),x=0..y),y): Df_C14:=unapply(Re(evalf(Debye(thetaD_C14(r)/T))),r,T): Df_C15:=unapply(Re(evalf(Debye(thetaD_C15(r)/T))),r,T): Df_C36:=unapply(Re(evalf(Debye(thetaD_C36(r)/T))),r,T): // B100:=display(plot(Df_C14(r,300),r= ,color=red,legend="C14")): B110:=display(plot(Df_C15(r,300),r= ,color=blue,legend="C15")): B120:=display(plot(Df_C36(r,300),r= ,color=green,legend="C36")): display(b100,b110,b120,view= ,labels=[" [a.u.]"," D( D)"], labeldirections=[horizontal,vertical]); 25
28 3.8: F E-V 2.43 //(kb/ry) Ry // Ry ev >FreeEnergy_C14:=unapply((fit_C14(r)-(8.617*10^(-5)/ ) *T*(Df(r,T)-3*ln(1-exp(-thetaD(r)/T))) +(9/8)*(8.617*10^(-5)/ )*thetaD(r))* ,r,T): // [K] p11:=plot(freeenergy(r5,1),r5= ): p12:=plot(freeenergy(r5,100),r5= ,color=blue): p13:=plot(freeenergy(r5,200),r5= ,color=green): p14:=plot(freeenergy(r5,400),r5= ,color=black): p15:=plot(freeenergy(r5,800),r5= ,color=yellow): 26
29 // p1:=evalf(freeenergy(r0,1)); p2:=evalf(freeenergy(r0,100)); p3:=evalf(freeenergy(r0,200)); p4:=evalf(freeenergy(r0,400)); p5:=evalf(freeenergy(r0,800)); p30:=pointplot([evalf(fsolve(diff(freeenergy(x,1),x)=0,x= )),p1],symbolsize=20); p31:=pointplot([evalf(fsolve(diff(freeenergy(x,100),x)=0,x= )),p2],symbolsize=20); p32:=pointplot([evalf(fsolve(diff(freeenergy(x,200),x)=0,x= )),p3],symbolsize=20); p33:=pointplot([evalf(fsolve(diff(freeenergy(x,400),x)=0,x= )),p4],symbolsize=20); p34:=pointplot([evalf(fsolve(diff(freeenergy(x,800),x)=0,x= )),p5],symbolsize=20); display(p11,p12,p13,p14,p15,p30,p31,p32,p33,p34,labels=[" [a.u.]"," [ev/zrcr2 atoms]"],labeldirections=[horizontal,vertical]); 3.9 T = [K] C
30 3.9: T = [K] C14 [ev/zrcr 2 atoms] 3.2: C14 T [K] r 0 [a.u.] F [ev/zrcr 2 atoms]
31 C15 C36 T = [K] 3.10: T = [K] C15 [ev/zrcr 2 atoms] 3.3: C15 T [K] r 0 [a.u.] F [ev/zrcr 2 atoms]
32 3.11: T = [K] C36 [ev/zrcr 2 atoms] 3.4: C36 T [K] r 0 [a.u.] F [ev/zrcr 2 atoms]
33 //0K 2000K 100K tmp:=[]: ttmp:=[]: tttmp:=[]: for i from 0 to 2000 by 100 do tmp:=[op(tmp),[i,0]]: ttmp:=[op(ttmp),[i,0]]: tttmp:=[op(tttmp),[i,0]]: end do: //0K tmp[1,2]:= ; ttmp[1,2]:= ; tttmp[1,2]:= ; for j from 2 to nops(tmp) do tmp[j,2]:= ((tmp_c14[1,2]-tmp_c14[j,2])); ttmp[j,2]:= ((tmp_c15[1,2]-tmp_c15[j,2])); tttmp[j,2]:= ((tmp_c36[1,2]-tmp_c36[j,2])); end do: // ppp1:=listplot(tmp,color=red,legend="c14"); ppp2:=listplot(ttmp,color=blue,legend="c15"); ppp3:=listplot(tttmp,color=green,legend="c36"); display(ppp1,ppp2,ppp3,labels=[" [K]"," [ev/zrcr2 atoms]"], labeldirections=[horizontal,vertical]); Energy-Volume 3.12 Phonon-DOS K 31
34 3.12: 3.4 0K 1500K 100K ZrCr 2 Laves //0[K] 1500[K] 100[K] tmp2:=[[0,evalf(fsolve(diff(freeenergy(x,1),x)=0,x= ))]]: for i from 100 to 1500 by 100 do tmp2:=[op(tmp2),[i,evalf(fsolve(diff(freeenergy(x,i),x)=0,x= ))]]; end do: //0[K] 1500[K] 100[K] tmp:=[]: for i from 0 to 1500 by 100 do tmp:=[op(tmp),[i,0]]: end do: 32
35 ///100[K] for j from 2 to nops(tmp) do tmp[j,2]:=(((tmp2[j,2])/tmp2[j-1,2])^3-1)/100; end do: //5 fitting data11:=convert(transpose(convert(tmp,array)),listlist): fit11:=fit[leastsquare[[x,y], y=c0+c1*x+c2*x^2+c3*x^3+c4*x^4+c5*x^5]](data11): fit_b1:=unapply(rhs(fit11),x): // pp:=plot(fit_b1(x),x= ,labels=[" [K]"," [10^-5/K]"], labeldirections=[horizontal,vertical]); 3.13: 33
36 3.14: Kellow C C36 C15 C14 Kellow C [6] 34
37 4 ZrCr 2 Laves Phonon-DOS Quasi- Harmonic Phonon-DOS Energy-Volume Phonon-DOS 35
38 [1] 2006 [2] 1975 [3] F. Stein, M. Palm, G. Sauthoff, Structure and stability of Laves phases, Part I. Critical assessment of factors controlling Laves phase stability, Intermetallics 12, (2004), [4] J. Pavlu, J. Vrest al, M.Sob, Stability of Laves Phases in the CrZr System, CALPHAD-COMPUTER COUPLING OF PHASE DIAGRAMS AND THERMOCHEMISTRY, [5] V. L. Moruzzi, J. F. Janak, Calculated thermal properties of metals, The American Phisical Society, [6] A. Kellow, T. Grosdidier, C. Coddet, H. Aourag, Theoretical study of structural, electronic, and thermal properties of Cr2(Zr,Nb) Laves alloys, Acta Meterialia Inc,
39 37
phonopy phonon-dispersion phonon-dos MedeA phonopy phonopy, phonopy MedeA phonopy MedeA Moruzzi Quasi-harmonic MedeA,Quasi-harmonic phonopy phonopy.
phonopy phonon 1513 2015 3 phonopy phonon-dispersion phonon-dos MedeA phonopy phonopy, phonopy MedeA phonopy MedeA Moruzzi Quasi-harmonic MedeA,Quasi-harmonic phonopy phonopy. 1 3 1.1 phonopy.....................................
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers