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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    20-May-2018

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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

    RSTUVW

    =

    +

    +

    RSTUVW =

    +

    =

    +

    FHG

    IKJ

    (10.1)

    6

  • 1 1 1 1 1 1

    1 1

    1 1 1 1

    2 2

    2

    2 2

    2

    2 2

    2

    2

    2

    2

    2 2

    2

    2

    2

    2

    2 2 2 2

    2

    2

    r r

    r

    r r r

    sinsin

    sin sincos sin

    sin

    cossin sin

    cossin sin

    FHG

    IKJ +

    LNM

    OQP=

    +

    FHG

    IKJ +

    LNM

    OQP

    =

    +

    +

    LNM

    OQP

    =

    +

    +

    (10.2)

    =

    FHG

    IKJ +

    FHG

    IKJ +

    LNM

    OQP

    22

    2 2 2

    2

    2

    1 1 1 1r r

    rr sin

    sinsin

    (11)

    +LNM

    OQP =

    +LNM

    OQP =

    LNMOQP =

    =

    + =

    FHG

    IKJ +

    FHG

    IKJ +

    L

    22

    22

    2 2

    22

    2

    2 2

    2 2

    2

    2 2 2

    2

    2

    2

    2 2

    2

    0

    1 1 1 1

    mV r E

    mV r k

    m

    mV r k

    U r k

    U r k

    r rr

    r

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) { ( ) } ( )

    ( )sin

    sinsin

    r r

    r r

    r r

    r r

    r r

    rNM

    OQP

    + = ( ) { ( ) } ( )r rU r k 2 0

    (12)

    E km

    U r mV r= =2 2

    222, ( ) ( ) (13)

    ( ) ( , , ) ( ) ( , )r = =r R r Y (14)

    7

  • 1 1 1 1 0

    1 1 1 1

    2

    2 2 2

    2

    22

    2

    2 2 2

    2

    22

    r rr R r Y

    rR r Y U r k R r Y

    Yr r

    rR r R rr

    Y U r k R r Y

    FHG

    IKJ +

    FHG

    IKJ +

    LNM

    OQP

    + =

    FHG

    IKJ +

    FHG

    IKJ +

    LNM

    OQP

    +

    ( ) ( , )sin

    sinsin

    ( ) ( , ) { ( ) } ( ) ( , )

    ( , ) ( ) ( )sin

    sinsin

    ( , ) { ( ) } ( )

    ( , )

    ( )( ) { ( ) }

    ( , ) sinsin

    sin( , )

    =

    FHG

    IKJ + +

    FHG

    IKJ +

    LNM

    OQP

    =

    0

    1 1 1 02

    22 2

    2

    2

    2r

    R r rrR r r U r k

    YY

    ( , , )r

    rR r r

    rR r r U r k

    YY

    ( )( ) { ( ) }

    ( , ) sinsin

    sin( , )

    FHG

    IKJ + =

    FHG

    IKJ +

    LNM

    OQP

    =

    2

    22 2

    2

    2

    21 1 1

    rR r r

    rR r r U r k

    rR r rrR r U r k

    r

    r rrR r U r k

    rR r

    ( )( ) { ( ) }

    ( )( ) { ( ) }

    ( ) ( ) ( )

    FHG

    IKJ + =

    FHG

    IKJ + =

    FHG

    IKJ +

    RSTUVW =

    2

    22 2

    2

    22

    2

    2

    22

    2

    1

    1 0

    (15)

    1 1 1

    1 1 0

    1 1 0

    2

    2

    2

    2

    2

    2

    2

    2

    2

    YY

    Y Y

    Y

    ( , ) sinsin

    sin( , )

    sinsin

    sin( , ) ( , )

    sinsin

    sin( , )

    FHG

    IKJ +

    LNM

    OQP

    =

    FHG

    IKJ +

    LNM

    OQP

    +

    FHG

    IKJ +

    +LNM

    OQP

    =

    = (16)

    Y F G( , ) ( ) ( ) = (17) (16)

    8

  • 1 1 0

    1 1 0

    1 1 0

    1 1

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    sinsin

    sin( , )

    sinsin

    sin( ) ( )

    sinsin ( ) ( )

    sin( ) ( ) ( ) ( )

    ( )sin

    sin ( ) ( )sin

    FHG

    IKJ +

    +LNM

    OQP

    =

    FHG

    IKJ +

    +LNM

    OQP

    =

    FHG

    IKJ +

    + =

    FHG

    IKJ +

    Y

    F G

    F G F G F G

    G F F

    + =

    FHG

    IKJ + +

    =

    2

    22

    2

    0

    1 1 0

    G F G

    FF

    GG

    ( ) ( ) ( )

    ( )sin sin ( ) sin

    ( )( )

    ( , )

    1

    1

    2 2

    2

    22

    FF

    GG

    ( )sin sin ( ) sin

    ( )( )

    FHG

    IKJ + =

    =

    (18)

    1

    0

    2

    22

    2

    22

    GG

    G G

    G A i

    ( )( )

    ( ) ( )

    ( ) exp( )

    =

    + =

    =

    G d A d A A( ) ,

    2

    0

    2 2

    0

    2 22 1 12z z= = = =

    A > 0 G( )

    G( ) exp( )

    = 12

    i (19)

    G G

    im

    ( ) ( ) exp( )

    exp( ), , , ,

    0 12

    2 12

    2

    2 10 1 2 3

    = = =

    == =

    i

    (20)

    9

  • (18) m

    1

    0

    1 0

    2 2 2

    2 2

    2

    2

    FF m

    F F m F

    F m F

    ( )sin sin ( ) sin

    sin sin ( ) ( ) sin ( )

    sinsin ( )

    sin( )

    FHG

    IKJ + = =

    FHG

    IKJ +

    FHG

    IKJ +

    FHG

    IKJ =

    = (21)

    cos , sin

    ( ) (cos )

    = = =

    =

    x dd

    dxd

    ddx

    ddx

    F f

    1 0

    1 0

    1 0

    1

    2

    2

    2

    2

    2

    2

    22

    2

    sinsin ( )

    sin( )

    sinsin ( )

    sin( )

    sinsin sin sin ( )

    sin( )

    sinsin sin ( )

    sin

    FHG

    IKJ +

    FHG

    IKJ =

    FHG

    IKJ +

    FHG

    IKJ =

    FHGIKJ

    RSTUVW +

    FHG

    IKJ =

    FHGIKJ

    RSTUVW +

    FHG

    IKJ

    F m F

    dd

    dd

    F m F

    ddx

    ddx

    F m F

    ddx

    ddx

    F m F( )

    sinsin sin cos sin ( )

    sin( )

    sin cossin

    sin ( )sin

    ( )

    cos sin ( )sin

    ( )

    ( )

    =

    FHG

    IKJ

    RSTUVW

    + FHG

    IKJ =

    FHGIKJ +

    RSTUVW +

    FHG

    IKJ =

    +FHG

    IKJ +

    FHG

    IKJ =

    +

    0

    1 2 0

    2 1 0

    2 0

    2 1

    22

    2

    2

    2

    22

    2

    2

    2

    22

    2

    2

    2

    2

    ddx

    ddx

    ddx

    F m F

    ddx

    ddx

    F m F

    ddx

    ddx

    F m F

    x ddx

    x d2

    2

    2

    2

    22

    2

    2

    2

    10

    1 21

    0

    dxF m

    xF

    x d f xdx

    x df xdx

    mx

    f x

    FHG

    IKJ +

    FHG

    IKJ =

    +

    FHG

    IKJ =

    ( ) ( )

    ( ) ( ) ( ) ( )

    (22)

    m = 0

    10

  • ( ) ( ) ( ) ( )1 222

    02

    00 +x

    d f xdx

    x df xdx

    f x 0= (23)

    f x0 ( )

    f x C x x

    df xdx

    k m C x

    d f xdx

    k m k m C x

    mk m

    m

    mk m

    m

    mk m

    m

    00

    00

    1

    02

    02 0 0

    2

    0

    0

    0

    0

    1 1

    1

    ( ) , ( )

    ( )

    ( ) ( )( )

    =

    = +

    = + +

    +

    =

    +

    =

    +

    =

    b g (24)

    x m(20) m(23)

    ( ) ( ) ( ) ( )

    ( ) ( )( ) ( )

    ( )( ) ( )( )

    (

    1 2 0

    1 1 2

    1 1

    2

    22

    02

    00

    20 0

    2

    00

    1

    0 0

    0 02

    00 0

    0

    0

    0 0

    0 0

    + =

    + + RST

    UVW +RST

    UVW +RST

    UVW =

    + + + +

    +

    =

    +

    =

    +

    =

    +

    =

    +

    =

    x d f xdx

    x df xdx

    f x

    x k m k m C x x k m C x C x

    k m k m C x k m k m C x

    k

    mk m

    mm

    k m

    mm

    k m

    m

    mk m

    mm

    k m

    m

    + + =

    + + + + +

    + + + + =

    + +

    +

    =

    +

    =

    +

    =

    +

    =

    +

    =

    +

    =

    m C x C x

    k k C x k k C x k m k m C x

    k m k m C x k m C x C x

    k k C x k k C x

    mk m

    mm

    k m

    m

    k km

    k m

    m

    mk m

    mm

    k m

    mm

    k m

    m

    k k

    )

    ( ) ( ) ( )( )

    ( )( ) ( )

    ( ) ( )

    0 0

    0 0 0

    0 0 0

    0 0

    0 0

    0 0 02

    0 0 11

    0 02

    2

    0 00

    00 0

    0 0 02

    0 0 1

    0

    1 1 1

    1 2 0

    1 1

    +

    +

    =

    +

    =

    +

    =

    +

    =

    ++

    + + + + +

    + + + + =

    + +

    + + + + + + + + +

    10 0 2

    0

    0 00

    00 0

    0 0 02

    0 0 11

    0 0 2 0 0 0

    2 1

    1 2 0

    1 1

    2 1 1 2

    0

    0 0 0

    0 0

    0

    ( )( )

    ( )( ) ( )

    ( ) ( )

    ( )( ) ( )( ) ( )

    k m k m C x

    k m k m C x k m C x C x

    k k C x k k C x

    k m k m C k m k m C k m C C x

    mk m

    m

    mk m

    mm

    k m

    mm

    k m

    mk k

    m m mk

    l q

    00

    mm

    mk k

    m mk m

    m

    k k C x k k C x

    k m k m C k m k m C x

    =

    ++

    =

    =

    + +

    + + + + + + + + + =

    0

    0 0 02

    0 0 11

    0 0 2 0 00

    0

    1 1

    2 1 1 0

    0 0

    0

    ( ) ( )

    ( )( ) { ( )( )}

    C k k0 0 k0 0 01 0 0( ) , 1, = = ( )k k0 01 0+ = k C C0 0= 1 1 0=

    k0 1=

    C1 0=

    11

  • ( )( ) { ( )( )}

    ( )( )( )( )

    k m k m C k m k m C

    C k m k mk m k m

    C

    m m

    m m

    0 0 2 0 0

    20 0

    0 0

    2 1 11

    2 1

    + + + + + 0+ + + =

    =+ + + + + + +

    +

    +

    (25)

    k0 0=

    C m mm m

    Cm+ =+

    + +21

    2 1( )

    ( )( ) m (26)

    C m mm m

    C m mm m

    C m mm m

    Cm m m+ =+

    + +=

    + + +

    =+ + +2

    2

    2

    2

    21

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