SiC SiC QMAS(Quantum MAterials Simulator) VASP(Vienna Ab-initio Simulation Package) SiC 3C, 4H, 6H-SiC EV VASP VASP 3C, 4H, 6H-SiC (0001) (11 20) (1 1
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- しょうこ ふじがわ
- 5 years ago
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1 QMAS SiC
2 SiC SiC QMAS(Quantum MAterials Simulator) VASP(Vienna Ab-initio Simulation Package) SiC 3C, 4H, 6H-SiC EV VASP VASP 3C, 4H, 6H-SiC (0001) (11 20) (1 100) MedeA SiC QMAS - C Si (0001) VASP (0001) QMAS VASP
3 SiC (Lely-method) (MSE:Metastable Solvent Epitaxy) (first principles calculations) Schrödinger QMAS(Quantum MAterials Simulator) qmas-atom qmas-band QMAS SiC
4 1 SiC VASP(Vienna Ab-initio Simulation Package) SiC 3C, 4H, 6H SiC (0001) (11 20) (1 100) VASP SiC (0001) SiC VASP QMAS(Quantum MAterials Simulator) QMAS SiC VASP 2 SiC SiC 3 QMAS 4 5 2
5 2 2.1 (SiC) Si C 19 SiC 1955 (Lely ) SiC Lely 10mm 1981 Tairov Lely SiC Lely SiC (2000 ) 1980 SiC SiC (Metastable Solvent Epitaxy) SiC MSE [1] VASP 2 SiC 2.2 SiC (Lely-method) SiC Lely Lely 3
![Si C 2.1 Lely (2200 2400) 2500 [2] 2.1: Lely SiC [2] 2.2.2 (MSE:Metastable Solvent Epitaxy) MSE SiC 2.](/docs-images/92/110562382/images/6-0.jpg "2 MSE MSE TaC 2 SiC Si MSE (feed) 3C-SiC 4H-SiC 4")
6 Si C 2.1 Lely ( ) 2500 [2] 2.1: Lely SiC [2] (MSE:Metastable Solvent Epitaxy) MSE SiC 2.2 MSE MSE TaC 2 SiC Si MSE (feed) 3C-SiC 4H-SiC 4
7 (seed) SiC 1800 SiC Si C µm Lely MSE 2.3 MSE Si C C S-SiC 2.3(a) 4H 3C 3C 4H 2.3(b) 3C-SiC 4H-SiC SiC µ 2.3(b) 3C µ c 4H µ c 4H 4H-SiC 4H-SiC C C C 2.3(c) Si C 4H-SiC 3C-SiC MSE [2] 2.2: MSE SiC [2] 5
8 2.3: MSE C C (a) 4H 3C (b) Si-C (c) [2] 6
9 SiC SiC C-SiC 4H-SiC 6H-SiC (111) (0001) (1 10) (11 20) (11 2) (1 100) 4H 6H-SiC (0001) 3C-SiC (111) C-SiC (1 10) (11 2) 4H 6H-SiC (11 20) (1 100) Si C 3C-SiC (111) 4H 6H-SiC (0001) Si (111) (0001) Si C Si C Si C [2] 7
10 図 2.4: 3C-SiC の表面 青色で示したのが Si 原子 黒色で示したのが C 原子 図 2.5: 4H-SiC の表面 青色で示したのが Si 原子 黒色で示したのが C 原子 8
11 図 2.6: 6H-SiC の表面 青色で示したのが Si 原子 黒色で示したのが C 原子 この構造では (0001) 面と (0001 ) 面の終端面の構成元素の組み合わせにより 表 面エネルギーが変わってくる この様子を模式的に表すと図 2.7 のようになり 表 面を覆う Si 原子の割合 (被覆率 coverage ratio)θsi によって 全く違った構造を 取れる事が分かる 本研究では 図 2.7 のようなスラブモデルを作成し それぞれ の表面エネルギーを計算した θsi = 0.5 でないモデルでは モデル中の Si 原子数 と C 原子数の比が 1:1 でなくなる この状態での表面エネルギー計算のためには 化学ポテンシャルに依存した計算法を利用する必要がある そこで 西谷研究室では VASP を用いて化学ポテンシャルに依存した計算手法 を用いて表面エネルギー計算してきた しかし この計算手法だけでは電場勾配 による影響を考慮するという面で不安を残している そこで新たな計算手法での 再計算が必要である 9
12 2.7: Si θ Si Si-rich C-rich θ Si =0.5 a-type b-type [2] SiC 2.8 SiC C (a) 3C-SiC (b) 6H-SiC 2.8 (a) (b) 3C 6H SiC 3C 6H Si 1414 (a)
13 2.8: SiC [3] (a) Olenski Abbaschian 1996 (b) Fromm Gebhardt 1976 SiC VASP SiC 6H > 4H > 3C VASP 11
14 3 3.1 (first principles calculations) Schrödinger (first principles calculations) Schrö dinger Schrödinger Hψ = ɛψ (3.1) ( d2 + V )ψ = ɛψ (3.2) dx2 (Hamiltonian:H) (wave function:ψ) (energy Eigen value:ɛ) (Kinetic Energy) (d 2 ψ/dx 2 ) (Vψ) (potential:v ) (nuclearpotential) ( exchange-correlation interaction) ψ self consistent loop [4] Full Potential pseudo potential 2 PseudoP 12
15 ultra soft norm [4] (Atomic Orbital) (Plane Wave) 2 AO (Linear Combination) LCAO PW (augument) APW (Linearlized) FP All Electron AE Projector Augmented Wave Pseudo P [4] (Local Density Approximation:LDA) (Generalized Gradient Approximation:GGA) version Perdew-Wang 91 [4] PW (K-space) (Fast Fourier Transformation) (real space) PW (periodic boundary condition) [4] 3.2 QMAS(Quantum MAterials Simulator) QMAS(Quantum MAterials Simulator) PAW(Project Augmented-Wave) 13
16 QMAS PAW FLAPW(Full-potential Linear Augmented Plane Wave) FLAPW PAW [5] 3.1: FLAPW PAW FLAPW PAW PAW 3.1 QMAS 3 (qmas-atom qmas-band qmas-gamma) qmas-atom qmas-band qmas-atom qmas-atom V qmas-band (total number of electrons) (principal quantum number) (azimuthal quantum number) LDA GGA GGA 3.1 qmas-atom 14
17 3.1: qmas-atom cdir z 3.1 C 6.0 nc C 1 K 2 L nv 3.1 C 2 1 2p 2 15
18 iexc LDA GGA 1 LDA 2 GGA nn 1,2,3.. K L M.. ll 0,1,2.. s p d f C 3 5 occ C icv C qmas-band qmas-band Schrödinger ɛ qmas-atom self consistent loop QMAS (rydberg:ry) 16
19 (hartree:e h ) qmas-atom 4H-SiC : qmas-band title 8 unit 1 2 Bohr number atom Si 4 C
20 number element 4H-SiC Si C 2 lattice v angle v (Primitive Vector) a,b,c 4H-SiC a : b : c = : : [ ] α : β : γ = 90:90:120[ ] element list qmas-atom qmas-band 3.3: qmas-band number space H-SiC P6 3 mc
21 number wycoff pos number atom number space 4H-SiC 4 wycoff position 2 (1/3,2/3,1/4) a 1/3 b 2/3 c 1/4 (1/3,2/3,1/4) (0,0,0) (1.309,0.755,2.531) 3 Primitive Vector P 0,P 1,P 2 (x, y, z) (X, Y, Z) P 0 = p 0,0 p 0,1 p 0,2,P 1 = p 1,0 p 1,1 p 1,2,P 2 = p 2,0 p 2,1 p 2,2 (3.3) x B = y (3.4) z X Y = P 0 x + P 1 y + P 2 z (3.5) Z
22 3.4: qmas-band 20
23 cutoff wf QMAS Ry VASP number max iteration number max relax 1 2 th charge 1 cal stress
24 3.5 (force) cell [4] 3.5: [4] 3.3 QMAS VASP QMAS 2 Si C E SiC(surface) E H SiC(slab) 22
25 E SiC(bulk) E H(bulk) E SiC(surface) = E H SiC(slab) E SiC(bulk) E H(bulk) (3.6) QMAS QMAS SiC 2.8 QMAS SiC VASP 23
26 4 4.1 SiC QMAS 3C 4H 6H-SiC 2.8 QMAS E-V E-V x y y 2 E-V x y [4] Qmas SiC 24
27 4.1: SiC lattice constant polytype length[ ] angle[ ] symmetry c/a a b c 3C F-43m 4H P6 3 mc 6H P6 3 mc polytype 4.2: atomic coordinate Si C x y z x y z 3C H H (a) 3C 4H 6H-SIC E-V (B) 4.3 3C > 6H > 4H 3C E-V 0K
28 4.1: 3C,4H,6H-SiC (a) (E-V) (b) E-V 4.3: lattice constant polytype length[ ] V[ ] E[eV/SiC] c/a a b c 3C H H VASP SiC 3C 4H SiC 4.2 3C 4H Qmas SiC Qmas 26
29 4.2: SiC 4.2 SiC QMAS QMAS SiC VASP VASP 2 SiC Lely C C-rich SiC (0001) 4.3(a) MSE TaC 2 3C-SiC Si H-SiC 4H-SiC Si Si-rich 4.3(b) 2 SiC Si-rich (0001) 4.5 C-rich 27
30 (0001) Si-rich C-rich Si-rich C-rich Si-rich MSE SiC C-rich Lely SiC 4.3: SiC [2] 28
![SiC [2] 4.](/docs-images/92/110562382/images/31-1.jpg "5: C-rich")
31 4.4: Si-rich SiC [2] 4.5: C-rich SiC [2] 29
32 MedeA SiC 2. QMAS 1 3. QMAS Maple Maple VESTA(Visualization for Electronic and STructural Analysis) 4H-SiC (11 20) Si C 4.6 (1 100) 4.7 (0001) 4.8 (11 20) - - Lely - MSE - 30
33 4.6: VESTA 4H-SiC (11 20) (1 100) Si C 4.7: VESTA 4H-SiC (11 20) (0001) Si C 31
34 4.8: VESTA 4H-SiC (11 20) (11 20) Si C Si C SiC 3C, 4H, 6H-SiC 4H, 6H-SiC (0001) (11 20) (1 100) 3C-SiC (111) (1 10) (11 2) QMAS E Si(bulk) E C(bulk) E SiC(bulk) E SiC(slab) E Si(bulk) E C(bulk) 4.1 QMAS E SiC(bulk) E SiC(slab) E = E SiC(slab) E SiC(bulk) (4.1) 4.2 ( E) (S [m 2 ]) (E SiC(surface) ) E SiC(surface) = E S (4.2) [2] 32
35 4.9: (0001) (11 20) (1 100) (111) (1 10) (11 2) [2] 4.10: [2] 33
36 4.4: SiC lattice constant polytype length[ ] angle[ ] symmetry c/a a b c 3C F-43m 4H P6 3 mc 6H P6 3 mc SiC (0001) (000-1) θ Si =0.5 Si C 1:1 4.2 µ E i(bulk) n i 4.3 µ i = E i(bulk) n i (4.3) SiC µ SiC(bulk) Si µ Si C µ C A B AB AB=A+B+ H f SiC 4.4 H f µ Si + µ C = µ SiC(bulk) = µ Si(bulk) + µ C(bulk) + H f (4.4) 4.4 SiC (µ Si µ C ) H f 0 (4.5) µ Si(bulk) + H f µ Si µ Si(bulk) (4.6) µ C(bulk) + H f µ C µ C(bulk) (4.7) 34
37 µ Si µ C 4.4 µ Si = µ Si(bulk) µ C = µ C(bulk) + H f C-rich Si-rich E SiC(suface) (S [m 2 ]) E SiC(suface) = E SiC(slab) n Si (µ Si(bulk) + H f ) n C µ C(bulk) S (C rich) (4.8) E SiC(suface) = E SiC(slab) n Si µ Si(bulk) n C (µ C(bulk) + H f ) (Si rich) (4.9) S Si-C Si100 % C100 % SiC 1:1 Si C SiC SiC Si SiC Si µ Si(bulk) SiC µ Si SiC µ C SiC Si SiC µ Si = µ Si(bulk) µ C = µ C(bulk) + H f C-rich µ C = µ C(bulk) µ Si = µ Si(bulk) + H f [2] 4.11: Si-C Si100 % C100 % SiC 1:1 [2] 35
38 4.2.5 QMAS Maple E-V C-SiC(1 10) Maple C-SiC(111) θ Si = H, 6H-SiC H-SiC error Maple 36
39 4.12: 3C-SiC(1 10) Maple 4.13: 3H-SiC(111) Si-rich Maple 37
40 4.2.6 SiC : SiC [n i ] Cutoff-E k-mesh polytype Energy [ev] n i Cutoff-E [ev] k-mesh 3C-SiC E+03 n Si :n C =1: H-SiC E+04 n Si :n C =2: H-SiC E+04 n Si :n C =3: C-Si E+03 n Si :n C =1: C-C E+03 n Si :n C =0: (0001) (11 20) (1 100) : (0001) (11 20) (1 100) [n i ] Cutoff-E k-mesh polytype Surface Energy [ev] n i Cutoff-E [ev] k-mesh 3C-SiC (1 10) E+05 n Si :n C =16: (11 2) E+05 n Si :n C =24: H-SiC (11 20) E+05 n Si :n C =32: (1 100) E+05 n Si :n C =32: H-SiC (11 20) E+05 n Si :n C =48: (1 100) E+05 n Si :n C =48: (0001) 4.7 polytype a-type b-type 2.7 θ Si = 1/2 38
41 4.7: (0001) (θ Si ) [n i ] Cutoff-E k-mesh a-type b-type 2.7 θ Si = 1/2 polytype coverage (θ Si ) Energy [ev] n i Cutoff-E [ev] k-mesh 3C-SiC θ Si = E+05 n Si :n C =48: a-type θ Si = 1/ E+05 n Si :n C =48: b-type θ Si = 1/ E+05 n Si :n C =48: θ Si = E+05 n Si :n C =52: H-SiC θ Si =0 error n Si :n C =64: a-type θ Si = 1/2 error n Si :n C =64: b-type θ Si = 1/2 error n Si :n C =64: θ Si =1 error n Si :n C =68: H-SiC θ Si =0 error n Si :n C =48: a-type θ Si = 1/ E+05 n Si :n C =48: b-type θ Si = 1/ E+05 n Si :n C =48: θ Si = E+05 n Si :n C =52: : [J/m 2 ]. 1[eV]= [J] Surface 3C-SiC 4H-SiC 6H-SiC (11 20) (1 100)
42 4.9: [J/m 2 ]. 1[eV]= [J] environment coverage (θ Si ) 3C-SiC 4H-SiC 6H-SiC Si-rich θ Si = error error a-type θ Si = 1/ error b-type θ Si = 1/ error θ Si = error 9.30 C-rich θ Si = error error a-type θ Si = 1/ error b-type θ Si = 1/ error θ Si = error
43 5 4 3C-SiC (1 10), (11 2) VASP QMAS 5.1 3C-SiC (111) Si-rich C-rich 5.2 QMAS VASP (1 10) VASP (11 2) 5.1: 3C-SiC (1 10), (11 2) VASP QMAS 41
44 5.2: 3C-SiC (111) Si-rich C-rich VASP QMAS QMAS VASP (111) VASP Maple QMAS SiC QMAS QMAS 42
45 [1] SiC (2009) [2] (2011) [3] R.W.Olensinski, G.J.Abbaschian The C-Si System (1984) [4] VASP (2011) [5] ProjectorAugmented Wabe 43
46 44
Si SiO 2. Si 1 VASP Si 1,, Si-Si 0.28Å Si Si-Si 0.19Å Si 166 Si Å Si
8615 2012 3 Si SiO 2. Si 1 VASP Si 1,, Si-Si 0.28Å Si 160 1 2 Si-Si 0.19Å Si 166 Si 4 0.85Å Si 118 2 4 1 3 1.1 SiO 2... 3 1.2 Si... 3 1.3 Cz... 4 1.4 SiO 2... 4 1.5... 5 2 6 2.1... 6 2.2 VASP(Vienna Ab-initio
SiC Si Si Si 3 SiC Si SiC Metastable Solvent Epitaxy MSE MSE SiC MSE SiC SiC 3C,4H,6H-SiC 4H-SiC(11-20) 4H-SiC (0001) Si Si C 3 C 1
SiC 4718 20 2 21 SiC Si Si Si 3 SiC Si SiC Metastable Solvent Epitaxy MSE MSE SiC MSE SiC SiC 3C,4H,6H-SiC 4H-SiC(11-20) 4H-SiC (0001) Si Si C 3 C 1 1 4 1.1.................... 4 1.2 SiC........................
4/15 No.
4/15 No. 1 4/15 No. 4/15 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = m ψ(r,t)+v(r)ψ(r,t) ψ(r,t) = ϕ(r)e iωt ψ(r,t) Wave function steady state m ϕ(r)+v(r)ϕ(r) = εϕ(r) Eigenvalue problem
02-量子力学の復習
4/17 No. 1 4/17 No. 2 4/17 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = 2 2m 2 ψ(r,t)+v(r)ψ(r,t) ψ(r,t) Wave function ψ(r,t) = ϕ(r)e iωt steady state 2 2m 2 ϕ(r)+v(r)ϕ(r) = εϕ(r)
1 1.1,,,.. (, ),..,. (Fig. 1.1). Macro theory (e.g. Continuum mechanics) Consideration under the simple concept (e.g. ionic radius, bond valence) Stru
1. 1-1. 1-. 1-3.. MD -1. -. -3. MD 1 1 1.1,,,.. (, ),..,. (Fig. 1.1). Macro theory (e.g. Continuum mechanics) Consideration under the simple concept (e.g. ionic radius, bond valence) Structural relaxation
講 座 熱電研究のための第一原理計算入門 第1回 密度汎関数法による第一原理バンド計算 桂 1 はじめに ゆかり 東京大学 2 密度汎関数理論 第一原理 first-principles バンド計算とは 結晶構造 Schrödinger 方程式は 量子力学を司る基本方程式で 以外の経験的パラメータや
講 座 熱電研究のための第一原理計算入門 第1回 密度汎関数法による第一原理バンド計算 桂 1 はじめに ゆかり 東京大学 2 密度汎関数理論 第一原理 first-principles バンド計算とは 結晶構造 Schrödinger 方程式は 量子力学を司る基本方程式で 以外の経験的パラメータや任意パラメータを使わず 基 ある 定常状態において電子 i の状態を定義する波動 本的な物理方程式のみを用いて行う電子状態計算であ
positron 1930 Dirac 1933 Anderson m 22Na(hl=2.6years), 58Co(hl=71days), 64Cu(hl=12hour) 68Ge(hl=288days) MeV : thermalization m psec 100
positron 1930 Dirac 1933 Anderson m 22Na(hl=2.6years), 58Co(hl=71days), 64Cu(hl=12hour) 68Ge(hl=288days) 0.5 1.5MeV : thermalization 10 100 m psec 100psec nsec E total = 2mc 2 + E e + + E e Ee+ Ee-c mc
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13 12 30 2 1 3 1.1... 3 1.2... 4 1.3 Bravais... 4 1.4 Miller... 4 2 X 5 2.1 Bragg... 5 2.2... 5 2.3... 7 3 Brillouin 13 3.1... 13 3.2 Brillouin... 13 3.3 Brillouin... 14 3.4 Bloch... 16 3.5 Bloch... 17
PowerPoint プレゼンテーション
実空間差分法に基づく 第一原理電子構造 量子輸送特性計算 09/05/15 大阪大学大学院工学研究科小野倫也 Real-space first-principles calculation code The grand-state electronic structure is obtained by solving the Schrödinger (Kohn-Sham) equation 1 2
27011559 2018 3 1 3 2 4 2.1........................ 4 2.2.................................. 5 2.3 pseudovasp................................... 7 3 8 3.1 EAM potential.................................
Chebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
QMI_10.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx
QMI_09.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 3.1.2 σ τ 2 2 ux, t) = ux, t) 3.1) 2 x2 ux, t) σ τ 2 u/ 2 m p E E = p2 3.2) E ν ω E = hν = hω. 3.3) k p k = p h. 3.4) 26 3 hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.5) h
研修コーナー
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200708_LesHouches_02.ppt
Numerical Methods for Geodynamo Simulation Akira Kageyama Earth Simulator Center, JAMSTEC, Japan Part 2 Geodynamo Simulations in a Sphere or a Spherical Shell Outline 1. Various numerical methods used
* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *
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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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EPSON LP-8900ユーザーズガイド
3 4 5 6 7 8 abc ade w p s 9 10 s s 11 p 12 p 13 14 p s 15 p s A B 16 w 17 C p 18 D E F 19 p w G H 20 A B 21 C s p D 22 E s p w 23 w w s 24 p w s 25 w 26 p p 27 w p s 28 w p 29 w p s 30 p s 31 A s B 32
原子・分子架橋系の量子輸送の理論 現状と展望
1cm 10 21 1 m 1nm 10 9 1 10-4 N Maxwell DFT E.Tsuchida and M.Tsukada Phys.Rev.B52(1995)5573-5578 Phys.Rev.B54(1996)7602-7605 J.Phys.Soc.Jpn.67(1998)3844-3858 Chem.Phys.Lett.311(1999)236-240 N.Watanabe
19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional
19 σ = P/A o σ B Maximum tensile strength σ 0. 0.% 0.% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional limit ε p = 0.% ε e = σ 0. /E plastic strain ε = ε e
[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)
![[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)](/thumbs/47/23763429.jpg "[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)")
. D............................................... : E = κ ............................................ 3.................................................
卒業研究報告 題 目 Hamiltonian 指導教員 山本哲也教授 報告者 汐月康則 平成 14 年 2 月 5 日 1
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講 座 熱電研究のための第一原理計算入門 第2回 バンド計算から得られる情報 桂 1 はじめに ゆかり 東京大学 が独立にふるまうようになる 結晶構造を定義する際に 前回は 第一原理バンド計算の計算原理に続いて 波 アップスピンの原子 ダウンスピンの原子をそれぞれ指 のように自由な電子が 元素の個性のない一様な周期的 定することで 強磁性体や反強磁性体など さまざまな ポテンシャルに置かれたときに
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2003/12/04 TASK PAF A. Fukuyama et al., Comp. Phys. Rep. 4(1986) 137 A. Fukuyama et al., Nucl. Fusion 26(1986) 151 TASK/WM MHD ψ θ ϕ ψ θ e 1 = ψ, e 2 = θ, e 3 = ϕ ϕ E = E 1 e 1 + E 2 e 2 + E 3 e 3 J :
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Structural Design for Curved Panels by Laminated Composite Materials (Identification of Lamination Parameters Using Modal Testing Method ) Tetsuya NARISAWA, Shohei IWATA Abstract - Using a modal testing
δf = δn I [ ( FI (N I ) N I ) T,V δn I [ ( FI N I ( ) F N T,V ( ) FII (N N I ) + N I ) ( ) FII T,V N II T,V T,V ] ] = 0 = 0 (8.2) = µ (8.3) G
![δf = δn I [ ( FI (N I ) N I ) T,V δn I [ ( FI N I ( ) F N T,V ( ) FII (N N I ) + N I ) ( ) FII T,V N II T,V T,V ] ] = 0 = 0 (8.2) = µ (8.3) G](/thumbs/91/106215968.jpg "δf = δn I [ ( FI (N I ) N I ) T,V δn I [ ( FI N I ( ) F N T,V ( ) FII (N N I ) + N I ) ( ) FII T,V N II T,V T,V ] ] = 0 = 0 (8.2) = µ (8.3) G")
8 ( ) 8. 1 ( ) F F = F I (N I, T, V I ) + F II (N II, T, V II ) (8.1) F δf = δn I [ ( FI (N I ) N I 8. 1 111 ) T,V δn I [ ( FI N I ( ) F N T,V ( ) FII (N N I ) + N I ) ( ) FII T,V N II T,V T,V ] ] = 0