1. . . , (h,k,l) [ h,k,l ] . . . (h,k,l) OA, OB, OC .
, , , :
OA = a / h, OB = b / k, OC = c / l
, ON [ hkl ],
ON = h a + k b+ l c
[ h k l ], , (h k l), ON , , AB, :
(ON × AB) º 0.
(ON × AB) = { ON (b / k - a / h)} = {(h a + k b + l c) (b / k - a / h)}= h/k (ab) + b 2 + l/ k (c b) - a 2 - k/h (ba) - l/ h (ca) º 0
, :
1) (a × b) = (c × b) = (b × a) = (c × a) = 0, , a=b=g=90
2) a= b.
AB (hkl): BC, AC , a=c b=c.
, , , .
( ) (1.4)
cos j= (1.11)
: , .
2. . , . I.
a, b, c. :
I = m a + n b + p c, (1.12)
m, n, p - , . ( m, n, p , , ).
I :
I = m 2a2 + n 2b2 + p 2c2 + 2 mn abcosg + 2 mp accosb + 2 np bccosa, (1.13)
I :
I =a (1.14)
3. . , , . a, b, c, S
|
|
S=|a| |b| sin(ab) S=|[ ab ]| (1.15)
V :
V=S×h=| a |×| b ×|sin(ab)×| c |×cos(cs)=| c |×| s |×cos(cs)=(c×s)=(c [ ab ])
V= (c [ ab ])
:
V = (a [ bc ]) = (b [ ca ]) =(c [ ab ]) (1.16)
:
V = abc (1.17)
, (1.17) :
V=abc (1.18)
:
V=a3 (1.19)
4. . d - . , , . d .
. (hkl) x, y, z OA, OB, OC, / h, b/ k, c/ l . ON , , |ON|= d OAN ON/OA= cos a d h /a=cosa.
: d k /b= cosb; d l /c= cosg.
, :
cos2 a + cos2 b + cos2 g = 1 (1.20)
(1.21)
(1.21) . , , , a¹b¹c, a=b=g=90. 1.21, :
(1.22)
a=b=c a=b=g=90:
(1.23)
, .
1.11.
. . . , (hkl) [[ hkl ]] . , :
(aa *)=(bb *)=(cc *) = 1 (1.24)
(ab *)=(ac *)=(ba *)=(bc *)=(ca *)=(cb *) = 0, (1.25)
.. 1, . a, b, c - , a *, b *, c * - x*, y*, z*.
, , , , .
, x*, y*, z* a *, b *, c *.
, (ba *)=0 (ca *)=0 , a * b , , , . , b * * ac ab , .
|
|
. a * bc, a *=j1[ bc ], j1 -
| a *|=
a, (aa *)=j1(a[ bc ])=j 1×V=1, j1=1/V.
: a *=[ bc ]/V
b *=[ ac ]/V c *=[ ab ]/V. (1.26)
(1.26) . , . , . , , .
(hkl) , [[ hkl ]].
H = h a * + k b *+ l c *. (1.27)
, H (hkl). x, y, z (.1.14). (hkl) , .
OA = a / h; OB = b / k; OC = c / l. (1.28)
, (hkl). , , , , [ AB × BC ]. (1.28) :
[ AB×BC ] = [(b / k - a / h) (c / l - b / k)] = [ bc ]/ kl - [ ac ]/ hl + [ ab ]/ hk = 1/ hkl ×{ h [ bc ] + k [ ca ] + l [ ab ].
, hkl /V,
[ AB×BC ] = h [ bc ]/ V + k [ ca ]/ V + l [ ab ]/ V = h a * + k b * + l c * = H,
.. [ AB×BC ] , H , .
, H, (hkl).
H. H , / h n N (.1.14). n = H /| H |. , | n | a / h N. a / h N , d, (n × a / h)=1×d
(n × a / h)=(a / h × H /| H |)= a / h ×(a* h + b* k + c* l)/| H |= 1/| H |=d. (1.28)
. 1.14. (hkl).
(1.28) H . , d, . .
. H
H2=1/d2= h 2a*2+ k 2b*2+ l 2c*2+2 hk a*b*cosg+2 hl a*c*cosb+2 kl b*c*cosa. (1.29)
: , Hhkl . , H, (hkl) . .
|
|
, , (.1.15). . , R mnp: R = m a + n b + p c.
R . R, . , .
. 1.15. .
(hkl) H , H = h a *+ k b *+ l c*. (hkl) R, H R - (HR)=0. H R,
(HR) = hm + kn + lp = 0. (1.30)
(h1k1l1) (h2k2l2). :
(1.31)
:
, , , (1.32)
D - , D=
R 1/D. . - , .
:
(111) () .
(1.30)
=1, =1 = 2
, [ mnp ]=[ ]. .1.11.