# 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 = ∂ ∂ =− ∴ ∂ ∂ =− =− =− ∂

• Published on
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