水素原子 - 組織・特性計算グループ(Microstructure y z r r y yz ry yz rr z r r z r r zr z rz z rrrr 233 233 22 2 3 2 3 112 r r 1 (4.2) また、 1 tan cos cos sin sin sin cos cos sin sin cos cos cos sin cos cos sin cos φ φ φφ φ θφ θφ φ φ θ φ φφ φ φ θφ φ θ φ φφ = ∂ ∂ =− ∴ ∂ ∂ =− =− =− ∂

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  • by T.Koyama

  • +LNM

    OQP =

    22

    2mV r E( ) ( ) ( ) r r (1)

    x ry rz r

    ===

    sin cossin sincos

    (2)

    r x y z zr

    yx

    2 2 2 2= + + = =, cos , tan (3)

    r x y z

    r rx

    x rx

    xr

    rr

    r ry

    y ry

    yr

    rr

    r rz

    z rz

    zr

    rr

    2 2 2 2

    2 2

    2 2

    2 2

    = + +

    =

    = = =

    =

    = = =

    =

    = = =

    sin cos sin cos

    sin sin sin sin

    cos cos

    (4.1)

    cos

    sinsin

    sin cos cossin

    cos cos

    sinsin

    sin sin cossin

    cos sin

    sinsin sin

    cossin

    sin

    =

    =

    =

    = = =

    =

    =

    = = =

    =

    =

    = =

    =

    zr

    xzr

    rx

    xzr x

    xzr r

    yzr

    ry

    yzr y

    yzr r

    zrr

    zr

    rz r

    zr z

    zr r r r

    2 3 3

    2 3 3

    2 2

    2

    3

    2

    3

    21 1

    r

    r1

    (4.2)

    1

  • tan

    coscos sin sin

    sin coscos sin

    sin

    coscos cos

    sin coscossin

    cos

    =

    =

    = = =

    =

    = = =

    =

    =

    yx

    xyx x

    yx

    rr r

    yxx x x r r

    z z

    1

    1 1

    1 0 0

    2 2 22

    2 2 22

    2 22

    2

    2

    (4.3)

    x, y, z

    =

    +

    +

    =

    +

    =

    +

    +

    =

    +

    +

    =

    +

    +

    =

    xrx r x x r r r

    yry r y y r r r

    zrz r z z r r

    sin cos cos cos sinsin

    sin sin cos sin cossin

    cos sin

    (5)

    x, y, z(5)

    FHG

    IKJ =

    +

    FHG

    IKJ

    +

    FHG

    IKJ

    =

    +

    FHG

    IKJ

    +

    +

    FHG

    IKJ

    +

    x x r r r r r r

    r r r r

    r r r r

    r r

    sin cos cos cos sinsin

    sin cos cos cos sinsin

    sin cos sin cos cos cos sinsin

    cos cos sin cos cos cos sinsin

    sinsin

    sin cos cos cos

    r r

    r r r r r r r

    rr r r r

    r r

    rr

    FHG

    IKJ

    =

    FHG

    IKJ +

    FHG

    IKJ

    +

    +

    +

    F

    H

    GGGG

    I

    K

    JJJJ

    +

    sinsin

    sin cos sin cos cos sin cos

    cos coscos cos sin cos sin cos cos cos

    sin cossin

    sinsin

    sinsin

    sin sin

    2 22

    2

    2 2 2

    2 2

    2

    2

    2

    1 1

    sin cos cos sin cos cos

    cossin

    sinsin

    +

    +

    +

    +

    F

    H

    GGGG

    I

    K

    JJJJ

    2 2

    2

    2

    r r r

    r r

    2

  • =

    FHG

    IKJ +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    FHG

    IKJ

    +

    FHG

    IKJ

    +

    +

    FHG

    IKJ

    sin cos sin cos cos sin cos

    cos cos sin cos cos

    cos sin cossin

    cossin

    sinsin

    sin cos

    2 22

    2

    2 2

    2 2 2

    2

    2 2

    2

    2

    2

    2

    2

    2

    1 1

    1 1

    1

    r r r r r r r

    r r r r r r

    r

    r r r r

    2

    FHG

    IKJ

    +

    FHG

    IKJ

    2

    2

    2

    2

    1

    r r

    r

    sin

    cos sinsin

    sin cos

    (6)

    FHG

    IKJ =

    +

    +

    FHG

    IKJ

    +

    +

    FHG

    IKJ

    =

    +

    +

    FHG

    IKJ

    +

    +

    +

    FHG

    IKJ

    +

    +

    y y r r r r r r

    r r r r

    r r r r

    r r

    sin sin cos sin cossin

    sin sin cos sin cossin

    sin sin sin sin cos sin cossin

    cos sin sin sin cos sin cossin

    cossin

    sin sin cos sin

    r r

    r r r r r r r

    rr r r r

    r r

    r

    +

    FHG

    IKJ

    =

    +

    +

    FHG

    IKJ

    +

    +

    +

    +

    F

    H

    GGGG

    I

    K

    JJJJ

    +

    cossin

    sin sin sin sin cos sin cos sin cossin

    cossin

    cos sincos sin sin sin sin sin cos sin

    cos cossin

    cossin

    cossin

    sin cos

    2

    2 2

    2

    2

    2

    2 2

    2

    2

    2

    r r r r

    r r

    +

    +

    +

    +

    F

    H

    GGGG

    I

    K

    JJJJ

    sin sin cos cos cos sin

    sinsin

    cossin

    2 2

    2

    2

    =

    +

    +

    FHG

    IKJ +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    FHG

    IKJ

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    sin sin sin cos sin sin cos

    cos sin sin cos sin

    cos sin cossin

    cossin

    cossin

    sin cos

    2 22

    2

    2 2 2

    2 2 2

    2

    2 2

    2

    2

    2

    2

    2

    2

    1 1

    1 1

    1

    r r r r r r r

    r r r r r r

    r

    r r r r r r

    r

    FHG

    IKJ

    +

    +

    FHG

    IKJ

    2

    2

    2

    2

    1

    sin

    cos cossin

    cos sin

    (7)

    3

  • FHG

    IKJ =

    FHG

    IKJ

    FHG

    IKJ

    =

    FHG

    IKJ

    FHG

    IKJ

    =

    +

    FHG

    IKJ

    +

    FHG

    IKJ

    =

    z z r r r r

    r r r r r r

    r r r r

    r r r r r

    cos sin cos sin

    cos cos sin sin cos sin

    cos cos sin sin

    sin sin cos cos sin

    cos

    2

    2 2

    2

    2

    2

    2

    2

    2

    2

    2

    2 2

    2

    2

    1

    1 1

    +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    FHG

    IKJ

    r r r r

    r r r r r r

    sin cos

    sin sin cos

    (8)

    x, y, z

    4

  • +

    +

    =

    FHG

    IKJ +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    FHG

    IKJ

    +

    FHG

    IKJ +

    +

    2

    2

    2

    2

    2

    2

    2 22

    2

    2 2 2

    2 2 2

    2

    2 2

    2

    2 2

    2

    1 1

    1 1

    1

    x y z

    r r r r r r r

    r r r r r r

    r r r r

    sin cos sin cos cos sin cos

    cos cos sin cos cos

    cos sin cossin

    cossin

    sinsin

    2

    2

    2

    2

    2

    2

    2 22

    2

    2 2 2

    2 2 2

    2

    2 2

    1

    1 1

    1 1

    FHG

    IKJ

    FHG

    IKJ

    +

    FHG

    IKJ

    +

    +

    +

    FHG

    IKJ +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    FH

    sin cossin

    cos sinsin

    sin cos

    sin sin sin cos sin sin cos

    cos sin sin cos sin

    r r r

    r

    r r r r r r r

    r r r r r rGIKJ

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    FHG

    IKJ

    +

    +

    FHG

    IKJ

    +

    +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    cos sin cossin

    cossin

    cossin

    sin cossin

    cos cossin

    cos sin

    cos sin cos

    sin sin cos

    r

    r r r r r r

    r

    r r r r

    r r r r

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    22

    2

    2

    2 2

    2

    1 1

    1

    1 1 2

    r r

    FHG

    IKJ

    5

  • =

    +

    +

    +

    +

    FHG

    IKJ +

    +

    FHG

    IKJ

    +

    FHG

    IKJ +

    FHG

    IKJ

    +

    FHG

    IKJ +

    sin cos sin sin cos

    sin cos cos sin cos sin

    sin cos cos sin cos sin

    sin cos sin cos

    2 22

    22 2

    2

    22

    2

    2

    2 2 2 2

    2 2 2 2

    2

    1 1

    1 1

    1 1

    r r r

    r r r r r r

    r r r r r r

    r r r r r

    FHG

    IKJ

    +

    FHG

    IKJ +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    +

    FHG

    IKJ +

    +

    FHG

    IKJ

    +

    FHG

    IKJ +

    2

    2 2

    2 2 2

    2

    2 2 2

    2

    2 2

    2

    2

    2

    2

    1 1

    1 1

    r

    r r r r r r

    r r r r r r r r r

    r r

    sin cos sin cos

    cos cos cos sin sin

    cos sin cossin

    cossin

    cos sin cossin

    cos

    1

    sin

    sinsin

    cossin

    sin cossin

    sin cossin

    cos sinsin

    sin cos cos cossin

    cos sin

    +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    +

    FHG

    IKJ

    +

    +

    FHG

    IKJ +

    FHG

    IKJ

    +

    FHG

    IKJ +

    +

    FHG

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    1 1

    1 1

    r r r r r r

    r r r r r r

    r rIKJ

    =

    +

    +

    FHG

    IKJ +

    +

    FHG

    IKJ +

    =

    +

    +

    +

    +

    =

    FHG

    IKJ +

    FHG

    IKJ +

    LNM

    OQP

    2

    2

    2

    2 2

    2

    2 2

    2

    2 2

    2

    2 2 2 2

    2

    2

    2

    2 2 2

    2

    2

    1 1 1 1

    2 1 1

    1 1 1 1

    r r r r r r r r

    r r r r r r

    r rr

    r

    sincos

    sin

    cossin sin

    sinsin

    sin

    (9)

    1 1 1 1

    1 2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    r rr f

    r rr f

    r r rr f

    r rf r f

    r

    rfr

    fr

    r fr

    fr r

    fr r r r

    f

    2

    FHG

    IKJ =

    FHG

    IKJ =

    FHG

    IKJ

    RSTUVW=

    +

    FHG

    IKJ

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