iGrafx Designer 1 - Wisla Morniroli.dsf

Transcription

S P A C E G R O U P ID E
F R O M
P R E C E S S IO N E L E C T R O N
A N D
C O N V E R G E N T -B E A M E L E C
(C B E D
N T IF IC A T IO N
D IF F R A C T IO N (P E D )
T R O N D IF F R A C T IO N
)
J e a n -P a u l M O R N IR O L I
L a b o r a to ir e d e M é ta llu r g ie P h y s iq u e e t G é n ie d e s M a té r ia u x , U M R C N R S 8 5 1 7
U S T L & E N S C L , C it é S c ie n t if iq u e , V ille n e u v e d 'A s c q , F r a n c e
J e a n - P a u l.M o r n ir o li@ u n iv - lille 1 .fr
w w w .E le c tr o n - D iffr a c tio n .fr
X IV In te r n a tio n a l C o n fe r e n c e o n E le c tr o n M ic r o s c o p y , E M
L A B O R A T O IR E D E M É T A L L U R G IE P H Y S IQ U E
E T G É N IE D E S M A T É R IA U X
U M R C N R S 8 5 1 7
2 0 1 1 , W is la
S y m m e tr y d e te r m in a tio n s
C r y s ta llo g r a p h ic fe a
- 7 c ry s ta l s y s te m
- 1 4 B r a v a is la ttic
- 1 1 L a u e c la s s e
- 3 2 p o in t g r o u p
- 2 3 0 s p a c e g ro u
- th e s tru c tu re fa c
tu re s
s
e s
s
s
p s
to r
A ls o p o s s ib le fr o m e le c tr o n d iffr a c tio n
( m ic r o s c o p ic a n d n a n o s c o p ic s c a le s
in c o r r e la tio n w ith th e im a g e o f th e d iffr a c te d a r e a )
- S e
- M
- C o n v
- L a r g e - A n g le
le
ic
e
C
c te d -A re
r o d iffr a c
rg e n t-B e
o n v e rg e
a E
tio n
a m
n t-B
le
a
E
e
c tro n
n d E
le c tr
a m E
D iffr a c
le c tr o n
o n D iffr
le c tr o n
tio n ( S
P re c e
a c tio n
D iffr a
A E
s s io
(C B
c tio
D )
n
E D )
n (L A C B E D )
- 2 3 0 s p a c e g ro u
- th e s tru c tu re fa c
- U s u a lly o b ta in e d fr o m
tu re s
s
e s
s
s
p s
to r
X - r a y a n d n e u tr o n d iffr a c tio n s
- 2 3 0 s p a c e g ro u
(- th e s tru c tu re fa c
tu re s
s
e s
s
s
p s
to r)
- m ic r o s c o p ic a n d n a n o s c o p ic s c a le s
- in c o r r e la tio n w ith th e im a g e o f th e d iffr a c te d a r e a
- S e
- M
- C o n v
- L a r g e - A n g le
le
ic
e
C
c te d -A re
r o d iffr a c
rg e n t-B e
o n v e rg e
a E
tio n
a m
n t-B
le
a
E
e
c tro n
n d E
le c tr
a m E
D iffr a c
le c tr o n
o n D iffr
le c tr o n
tio n ( S
P re c e
a c tio n
D iffr a
A E
s s io
(C B
c tio
D )
n
E D )
n (L A C B E D )
- 2 3 0 s p a c e g ro u
tu re s
s
e s
s
s
p s
to r)
- S e le
- M ic r o d iffr a c tio n
- C o n v e
- L a r g e - A n g le C
c te d - A r e a E le
in c o n v e n tio n
rg e n t-B e a m E
o n v e rg e n t-B e
c tro n
a l a n
le c tr o
a m E
D iffr a c tio
d in p r e c
n D iffr a c
le c tr o n D
n (S A E D
e s s io n m
tio n ( C B E
iffr a c tio n
)
o d e s (P E D )
D )
(L A C B E D )
F E A T U R E S A V A I L A B LS E y m O m N e E t r L y E d C e Tt e R r m O i N n a D t i Io F n F s R A C T I O N P A T T E R N S
- 2 3 0 s p a c e g ro u
tu re s
s
e s
s
s
p s
to r)
P e r tin e n t m e th o d
- S e le
- M ic r o d iffr a c tio n
- C o n v e
- L a r g e - A n g le C
c te
in
rg e
o n
d - A r e a E le
c o n v e n tio n
n t-B e a m E
v e rg e n t-B e
c tro n
a l a n
le c tr o
a m E
D iffr a c tio
d in p r e c
n D iffr a c
le c tr o n D
n (S A E D
e s s io n m
tio n ( C B E
iffr a c tio n
)
o d e s (P E D )
D )
(L A C B E D )
O U T L IN E
I - In tr o d u c tio n
II - R a p id
- C o n v e
- M
- M ic r
d e s c r ip tio n o
rg e n t-B e a m E
ic r o d iffr a c tio n
o d iffr a c tio n in
III - F e a tu r e s a v a ila
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th e
le c tr
in c o
p re c
b le
" id e
/ Z
p e r
d e n
o n
a l"
O L
io d
re
e x p e r im
o n D iffr a
n v e n tio n
e s s io n m
e n ta
c tio n
a l m
o d e
l m
(C
o d
(P
e th o d s
B E D )
e
E D )
d iffr a c tio n p a tte r n s
s y m m e tr ie s
Z s h if ts
ic ity d iffe r e n c e s
fle c t io n s
IV - Id e n tific a tio n o f th e c r y s ta l s y s te m
V - Id e n tific a tio n o f th e e x tin c tio n s y m b o l fr o m P E D p a tte r n s
V I - S e le c tio n o f th e a c tu a l s p a c e g r o u p fr o m C B E D p a tte r n s
V II - E x a m p le
V III - L im ita tio n s a n d d iffic u lt ie s
IX - C o n c lu s io n
O U T L IN E
II - R a p id
- C o n v e
- M
- M ic r
rg e n t-B e a m E
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th e
le c tr
in c o
p re c
b le
" id e
/ Z
p e r
d e n
o n
a l"
O L
io d
re
e x p e r im
o n D iffr a
n v e n tio n
e s s io n m
e n ta
c tio n
a l m
o d e
l m
(C
o d
(P
e th o d s
B E D )
e
E D )
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
O p tic a l a x is
T E M
c o lu m n
C o n d e n s o r le n s
O b je c t p la n e
O b je c tiv e le n s
D iffr a c tio n
p a tte rn
B a c k fo c a l
p la n e
Im a g e p la n e
C o n ju g a te p la n e s
in
d iffr a c tio n m o d e
D iffr a c tio n
p a tte rn
In te r m e d ia te
le n s
S c re e n
C B E D
M o lle n s te d ( 1 9 3 9 ) ,
J . S te e d s , M . T a n a k a ...
S ta
a n d fo
(c o n
(s p o t s
tio
c u
v e
iz e
n a
s e
rg
d
ry c o
d in c
e n c e
o w n
n v e rg e n
id e n t b e
a b o u t 1
to a fe w
t
a m
° )
n m )
T ra n s
(tra n s m
a s a
th e b e a
S p e c im e n
D IS K P A T T E R N
m itte d
itte d in
fu n c tio
m o r ie
d is k
te n s ity
n o f
n ta t io n )
B a c k fo c a l
p la n e
T r a n s m itte d
a n d d iffr a c te d
d is k s
h k l d iffr a c te d d
( d iffr a c t e d in te
a s a fu n c tio n
th e b e a m o r ie n
w ith r e s p e c t to th
la ttic e p la n e
S c re e n
is k s
n s it y
o f
ta t io n
e ( h k l)
s )
O U T L IN E
II - R a p id
- C o n v e
- M
- M ic r
rg e n t-B e a m E
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th e
le c tr
in c o
p re c
b le
" id e
/ Z
p e r
d e n
o n
a l"
O L
io d
re
e x p e r im
o n D iffr a
n v e n tio n
e s s io n m
e n ta
c tio n
a l m
o d e
l m
(C
o d
(P
e th o d s
B E D )
e
E D )
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
C B E D
B a c k fo c a l
p la n e
S c re e n
C O N V E N T IO N A L M IC R O D IF F R A C T IO N
B e a m
c o n v e rg e n c e a
a b o u t 0 .0 1 °
S P O T P A T T E R N
S ta tio n a r y in c id e n t b e a m
n e a r ly p a r a lle l a n d fo c u s e d
( fille d c o n e )
B a c k fo c a l
p la n e
S c re e n
O U T L IN E
II - R a p id
- C o n v e
- M
- M ic r
rg e n t-B e a m E
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th e
le c tr
in c o
p re c
b le
" id e
/ Z
p e r
d e n
o n
a l"
O L
io d
re
e x p e r im
o n D iffr a
n v e n tio n
e s s io n m
e n ta
c tio n
a l m
o d e
l m
(C
o d
(P
e th o d s
B E D )
e
E D )
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
P E D
P re c
E le c
R . V
(1 9 9
P r e - s p e c im e n d e fle c t io n c o ils
e s s io n
tr o n D iffr a c tio n
in c e n t & P . M id g le y
4 )
N e a r ly p a r a lle
b e a m r o ta tin
h
( p r e c e s s io
B a c k fo c a l
p la n e
l a n d
g o n
o llo w
n a n g
fo c
th e
c o n
le a
u s e d in c id e n t
s u rfa c e o f a
e
u p to 4 ° )
S y n c h r o n iz e d
a n d o p p o s ite
d e fle c tio n s
C IR C L E P A T T E R N
P o s t- s p e c im e n d e fle c tio n c o ils
h k l
c ir c le
h k l d iffr a c te
a s a fu n c tio n o f
( in te g r a te d
a lo n g th e
d in te n s ity
th e o r ie n ta tio n
in te n s ity
h k l c ir c le )
h k l
s p o t
S c re e n
M a in p r o p e r tie s o f th e P E D te c h n iq u e
- P E D p a tte r n s a r e le s s d y n a m ic a l
( th e in c id e n t b e a m is n e v e r a lo n g th e z o n e a x is )
- 'f e w b e a m ' o r 's y s t e m a t ic r o w ' c o n d it io n s d u r in g t h e p r e c e s s io n m o v e m e n t
d o u b le d iffr a c tio n p a th s s tr o n g ly r e d u c e d
( id e n tific a tio n o f th e k in e m a tic a l fo r b id d e n r e fle c tio n s )
- m o r e Z O L Z a n d H O L Z r e fle c tio n s th a n o n c o n v e n tio n a l p a tte r n s
- in te g r a te d in te n s ity ( p a tte r n w e ll a lig n e d )
O U T L IN E
II - R a p id
- C o n v e
- M
- M
d e
rg e
ic r o
ic r
s c r ip
n t-B e
d iffr a
o d iffr
tio n o
a m E
c tio n
a c tio n
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th
le c
in
in
b le
" id e
/ Z
p e r
d e n
e e x p
tro n D
c o n v e
p re c e
o n
a l"
O L
io d
re
e r im
iffr a
n tio n
s s io
e n ta l m e th o d s
c tio n ( C B E D )
a l m o d e
n m o d e
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
S Y M M E T R Y O F D IF F R A C T IO N
C B E D
( W h o le p a tte r n )
P A T T E R N S
C B E D
P A T T E R N S
M ic r o d iffr a c tio n
C B E D
P A T T E R N S
P r e c e s s io n E le c tr o n M ic r o d iffr a c t io n
(Z e ro O rd e r L a u e Z o n e : Z O L Z )
S y m m e tr y e le m e n ts ?
C B E D
P A T T E R N S
P A T T E R N S
m
m
m
m
m
C B E D
m
P A T T E R N S
m
2 D fin ite f ig u r e
1 0 2 D p o in t g r o u p s
P
m
P
P
2
1
P
P
P
P
m
P
P
P
P
P
m
P
P
P
P
C B E D
4
P
m
3
6
m
P
P
P
P
P
2 m m
m
3 m
P
P
P
P
P
P
P
P
P
P
4 m m
6 m m
S Y M M E T R Y O F D IF F R A C T IO N P A T T E R N S
F o u r d iffe r e n t ty p e s o f s y m m e tr y
m
P
m
P
P
2
1
P
P
P
P
m
P
P
P
P
P
m
P
P
P
P
C B E D
4
P
m
3
6
m
P
P
P
P
P
2 m m
m
3 m
P
P
P
P
P
P
P
P
P
P
4 m m
6 m m
1 - W h o l e P a t t e r n s( W y m P m ) s e y t r m y m e t r y
m
P
m
P
P
2
1
P
P
P
P
m
P
P
P
P
P
m
P
P
P
P
C B E D
4
P
m
3
m
P
P
2 m m
P
P
4 m m
6
P
2 m m
m
3 m
P
P
P
P
P
P
4 m m
P
P M r i e c c r o e d s is f fi o r a n c E t i o l e n c e t r l o e n c t M r o i n c r p o r d e i f c f r e a s c s t i i o o n n
P
P
P
2 m m
6 m m
2 - Z O L Z s y m m e try
P A T T E R N S
m
P
m
P
P
2
1
P
P
P
P
m
P
P
P
P
P
m
P
P
P
P
C B E D
4
P
m
3
m
P
P
(2 m m )
P
P
(4 m m )
6
P
2 m m
m
3 m
P
P
P
P
P
P
4 m m
P
P
P
P
(2 m m )
6 m m
3 - " N E T " s y m m e tr y ( p o s itio n o n ly )
P
m
P
P
2
1
P
P
P
P
m
P
P
P
P
P
m
P
P
P
P
C B E D
4
P
m
3
m
P
P
(2 m m )
2 m m
P
P
(4 m m )
4 m m
6
P
2 m m
m
3 m
P
P
P
P
P
P
4 m m
P
P
P
P
(2 m m )
6 m m
4 - " ID E A L " s y m m e tr y ( p o s itio n + in te n s it y )
P
m
P
P
2
1
P
P
P
P
m
P
P
P
P
P
m
P
P
P
P
C B E D
4
P
m
3
m
P
P
(2 m m )
2 m m
P
P
(4 m m )
4 m m
6
P
2 m m
m
3 m
P
P
P
P
P
P
4 m m
P
P
P
P
(2 m m )
6 m m
T H E 'N E T ' S Y M M E T R Y ( p o s it io n o f t h e r e f le c t io n s o n ly )
E a s ily a v a ila b le o n s p o t p a tte r n s
m
m
(2 m m )
2 m m
E a s y to o b ta in
( d o e s n o t r e q u ir e a
p e r fe c t c r y s ta l a lig m e n t)
m
(2 m m )
It is c o n n e c te d w ith th e C R Y S T A L S Y S T E M
H ig h e s t W P
"n e t" s y m m e try
o b s e rv e d
1
2
m
2 m m
(4 m m )
(4 m m )*
6 m m
3 m
o r
a n
a b s e
o
(6 m
T r ic l.
C ry s ta l
s y s te m
M o n o .
O rth o .
d
n c e
f
m )
T e tra .
a n d
p re s e n c e
o f
(6 m m )
C u b .
a n
a b s e
o
(4 m
d
n c e
f
m )
R h o m b .
la ttic e
H e x .
la ttic e
T r ig .
H e x .
* e x c e p t fo r P a 3
F ro m
J .P . M o r n ir o li a n d J .W . S te e d s , U ltr a m ic r o s c o p y , 1 9 9 2 , 4 5 , 2 1 9
In m o s t c a s e s th e W H O L E P A T T E R N s y m m e tr y is r e q u ir e d
M ic r o d iffr a c tio n - T h e F ir s t- O r d e r L a u e Z o n e ( F O L Z ) is n o t v is ib le
m
m
m
m
m
m
m
m
(4 m m )
m
m
m
Z O L Z
F O L Z
m
m
m
m
m
m
m
m
(4 m m )
m
m
m
Z O L Z
F O L Z
m
m
m
m
m
m
m
(4 m m )
m
m
m
m
m
m
m
m
m
m
m
(4 m m )
4 m m
m
m
m
m
T H E 'I D E A L ' S Y M M E T R Y ( p o s it io n a n d in t e n s it y o f t h e r e f le c t io n s )
C B E D
m
m
m
m
C B E D
(4 m m )
4 m mM ic r o d iffr a c tio n
- V e r y a c c u r a te d e te r m in a tio n
- N o t e a s y e x p e r im e n ta lly
- N o t a lw a y s p o s s ib le
m
m
(2 m m )
2 m m
- N o n a c c u r a te d e te r m in a tio n
- R e q u ir e s a p e r fe c t c r y s ta l o r ie n ta t io n
- N o t re c o m m a n d e d
m
m
m
m
m
m
- A c c u
- D o e s n o t r e q u ir e
- E
- R
(2 m m )
ra te
s a
a s y
e c o
d e te
p e rfe
to p e
m m a
rm
c t
rfo
n d
in a tio n
c r y s ta l o r ie n ta tio n
rm
e d
If th e F O L Z is n o t v is ib le
m
m
m
m
m
m
m
m
D iffic u lt a n d te d io u s
( p e r fe c t o r ie n ta tio n r e q u ir e d )
m
m
m
m
m
m
m
m
E a s y a n d a c c u ra te
( p e r fe c t o r ie n ta tio n n o t r e q u ir e d )
It is c o n n e c te d w ith th e 3 2 P O IN T G R O U P S
B u x to n e t a l 1 9 7 6
C ry s ta l
L a u e
P o in t
< 1 1 1 >
< 0 0 1 >
< 1 1 0 >

s y s te m
c la s s
C u b ic
g ro u p
m -3 m
-4 3 m
m -3 m
3
(6 m
3
(3
m
m )
4 3 2
m
m )
T a b le v a lid fo r C B E D
W h a t a b o u t P E D ?
m -3
m -3
2 3
6 /m m m
6 /m m m
-6 2 m
6 m m
H e x a g o n a l
6 /m
-6
6
-3 m
3 m
-3 m
T r ig o n a l
-3
3
4 /m m m
4 /m m m
-4 2 m
4 m m
T e tra g o n a l
4 2 2
4 /m
4 /m
-4
m )
4
(3 m )
3
(6 )
3
(3 )
(4 m m )
2 m m
(2 m m )
2
(2 m m )
[0 0
6 m
(6 m
3
(3
6 m
(6 m
0 1 ]
m
m )
m
m )
m
m )
6
(6 m m )
6
(6 )
3
(3 )
6
(6 )
[0 0
3
(6 m
3
(3
0 1 ]
m
m )
m
m )
3 2
-3
m
m )
m
3
6 2 2
6 /m
4 m
(4 m
2 m
(4 m
3
(3 m )
3
(6 )
3
(3 )
[0 0
4 m
(4 m
2 m
(4 m
4 m
(4 m
1 ]
m
m )
m
m )
m
m )
4
(4 m m )
4
(4 )
2
(4 )
< 1 1 -2 0 >
2 m m
(2 m m )
m
(m )
m
(m )
2
(2 m m )
< 1 1 2
(2
1
(1
2
(2
< 1 0
2 m
(2 m
2
(2 m
m
(m
2
(2 m
2 m m
(2 m m )
m
(m )
2
(2 m m )
< 1 -1
2 m
(2 m
2 m
(2 m
m
(m
0 0 >
m
m )
m
m )
)
2
(2 m m )

[u v w ]
m
(2 m m )
1
(m )
1
(m )
m
(2 m m )
1
(m )
m
(2 m m )
m
(m )
1
(m )
1
(1 )
1
(1 )
1
(1 )
1
(2 )
1
(1 )
[u v .0 ]
m
(2 m m )
m
(m )
1
(m )
1
(m )
m
(2 m m )
m
(m )
1
(m )
[u u .w ]
m
(2 m m )
1
(m )
m
(m )
1
(m )
2 0 >
)
)
)
0 >
m
m )
m )
)
m )
< 1 1
2 m
(2 m
m
(m
m
(m
2
(2 m
0 >
m
m )
)
)
m )
[u 0 w ]
m
(2 m m )
1
(m )
m
(m )
1
(m )
[u v
m
(2 m
1
(m
1
(m
1
(m
m
(2 m
1
(m
0 ]
m )
)
)
[u -u
m
(2 m
m
(m
m
(m
.w ]
m )
)
)
1
(m )
[u -u .w ]
m
(2 m m )
m
(m )
1
(m )
[u v tw ]
1
(2 )
1
(1 )
1
(1 )
1
(2 )
1
(1 )
[u u
m
(2 m
m
(m
m
(m
[u v w ]
1
(2 )
1
(1 )
1
(1 )
1
(1 )
1
(2 )
1
(1 )
w ]
m )
)
)
1
)
(m )
m )
)
[u v .w ]
1
(2 )
1
(1 )
1
(1 )
1
(1 )
1
(2 )
1
(1 )
1
(1 )
C a s e o f th e n o n - c e n tr o s y m m e tr ic a l p o in t g r o u p s
C B E D p a tte rn
c o n n e c te d w ith
th e 3 2 p o in t g r o u p s
P r e c e s s io n p a tte r n
th e 1 1 L a u e c la s s e s
C a s e o f th e n o n - c e n tr o s y m m e tr ic a l p o in t g r o u p s
C B E D p a tte rn
th e 3 2 p o in t g r o u p s
P E D p a tte rn
th e 1 1 L a u e c la s s e s
P r e c e s s io n E le c tr o n D iffr a c tio n
L a u e c la s s
Z o n e a x e s
C U B IC
T h
o f
c o
1 1
e 'I d
P E D
n n e c
L A U
e a
p
te
E
l' s y m
a tte rn
d w ith
C L A S
m e try
s is
th e
S E S
m -3 m
W P
Z O L Z
< 1 1 1 >
3 m
(6 m m )
< 0 0 1 >
4 m m
(4 m m )
m -3
W P
Z O L Z
< 1 1 1 >
3
(6 )
< 0 0 1 >
2 m m
(2 m m )
6 /m m m
< 0 0 0 1 >
W P
Z O L Z
6 m m
(6 m m )
2 m m
(2 m m )
6 /m
W P
Z O L Z
< 0 0 0 1 >
6
(6 )

m
(2 m m )
-3 m
W P
Z O L Z
< 0 0 0 1 >
3 m
(6 m m )
< 1 1 -2 0 >
2
(2 )
-3
W P
Z O L Z
< 0 0 0 1 >
3
6

1
2
4 /m m m
W P
Z O L Z
< 0 0 1 >
4 m m
(4 m m )
< 1 0 0 >
2 m m
(2 m m )
4 /m
W P
Z O L Z
< 0 0 1 >
4
(4 )
m m m
W P
Z O L Z
< 1 0 0 >
2 m m
(2 m m )

m
(2 m m )
O R T H O
< 0 1 0 >
2 m m
(2 m m )
M O N O C L IN IC
C R Y S T A L S Y S T E M
< 1 1 0 >
2 m m
(2 m m )

< 1 -1

M
(2 m m )

1
m )
(2 )
A L C R Y S T A L S Y S T E M
0 0 >


1
(2 )
0 >
2 m m
(2 m m )

m
(2 m m )



m
m
m
(2 m m )
(2 m m )
(2 m m )
1
(2 )

m
(2 m m )

m
(2 m m )

1
(2 )

m
)
(2 m m )
( u n iq u e a x is b )

m
(2 m m )

1
(2 )
>
C R Y S T A L S Y S T E M

1
)
(2 )
>
N A L C R Y S T A L S Y S T E M
0 >

m
m
m )
(2 m m )

B IC
C R Y S T A L
< 0 v w
m
)
(2 m m
S T A L S Y S T E M
>
S Y S T E M
>
Id e n tific a tio n o f th e L A U E C L A S S E S fr o m P E D p a t te r n s
H ig h e s t
" id e a l" s y m m e tr y
o b s e rv e d
L a u e
c la s s
C ry s ta l
s y s te m
1
2
(2 )
(2 )
1
2 /m
2 m m
(2 m m )
m m m
T r ic l.
O rth o .
M o n o .
4
(4 )
4 /m
3
4 m m
(4 m m )
3 m
(6 ) (6 m m )
a n d
a b s e n c e
o f
3 m
(6 m m )
a n
a b s e
o
4 m
(4 m
4 /m m m
T e tra .
3
3 m
T r ig .
6
(6 )
6 m m
(6 m m )
3
(6 )
3 m
(6 m m )
a n d
a n d
p re s e n c e p re s e n c e
o f
o f
2 m m
4 m m
(2 m m )
(4 m m )
d
n c e
f
m
m )
6 /m
6 /m m m
H e x .
m 3
m 3 m
C u b .
O U T L IN E
II - R a p id
- C o n v e
- M
- M
d e
rg e
ic r o
ic r
s c r ip
n t-B e
d iffr a
o d iffr
tio n o
a m E
c tio n
a c tio n
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th
le c
in
in
b le
" id e
/ Z
p e r
d e n
e e x p
tro n D
c o n v e
p re c e
o n
a l"
O L
io d
re
e r im
iffr a
n tio n
s s io
c tio n ( C B E D )
a l m o d e
n m o d e
s y m m e tr ie s
Z s h if ts
fle c t io n s
V I - S e le c tio n o f th e s p a c e g r o u p fr o m C B E D p a tte r n s
V II - E x a m p le
M ic r o d iffr a c tio n p a tte r n
( v e r y th in c r y s ta l)
m
m
P a tte r n A .R e d ja ïm ia
M ic r o d iffr a c tio n p a tte r n
T y p ic a l s
Z O L Z
a lo n
o n s o m
h ifts b e tw
a n d th e F
g th e m ir r
e s p e c ific
e e n th e
O L Z
o rs
Z A P s
m
m
C o n n e c te d
w ith th e
B r a v a is la ttic e
M ic r o d iffr a c tio n a n d P r e c e s s io n E le c tr o n M ic r o d iffr a c tio n
m
m
2
P r e c e s s io n E le c tr o n M ic r o d iffr a c tio n
( id e n tific a tio n m u c h e a s ie r a n d s u r e )
1
s h ift
m
m
m
s h ift
m
m
m
m
m
F O L Z / Z O L Z s h ift id e n tific a tio n
e a s y a n d v e ry s u re
N o s h ift
m
m
m
m
m
m
m
m
O U T L IN E
II - R a p id
- C o n v e
- M
- M ic r
rg e n t-B e a m E
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th e
le c tr
in c o
p re c
b le
" id e
/ Z
p e r
d e n
o n
a l"
O L
io d
re
e x p e r im
o n D iffr a
n v e n tio n
e s s io n m
e n ta
c tio n
a l m
o d e
l m
(C
o d
(P
e th o d s
B E D )
e
E D )
s y m m e tr ie s
Z s h if ts
fle c t io n s
V I - S e le c tio n o f th e s p a c e g r o u p fr o m C B E D p a tte r n s
V II - E x a m p le
P e r io d ic ity
b e tw e e n th e
in th e Z O
in th e
d iffe r e n c e
r e f le c tio n s
L Z a n d
F O L Z
C o n n e c te d
w ith
g lid e p la n e s
M ic r o d iffr a c tio n a n d P r e c e s s io n E le c tr o n M ic r o d iffr a c tio n
m
m
2
( id e n tific a tio n m u c h e a s ie r a n d s u r e )
1
Id e n tific a tio n o f a
F O L Z / Z O L Z p e r io d ic ity d iffe r e n c e
d iffic u lt
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
O U T L IN E
II - R a p id
- C o n v e
- M
- M
d e
rg e
ic r o
ic r
s c r ip
n t-B e
d iffr a
o d iffr
tio n o
a m E
c tio n
a c tio n
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th
le c
in
in
b le
" id e
/ Z
p e r
d e n
e e x p
tro n D
c o n v e
p re c e
o n
a l"
O L
io d
re
e r im
iffr a
n tio n
s s io
c tio n ( C B E D )
a l m o d e
n m o d e
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
F o r b id d e n
d u e to :
- s
- g
- (
D iffic u lt to
d u e to d o u
r e fle c tio n s
c re w a x e s
lid e p la n e s
W y c k o ff p o s itio n s )
id e n tify o n c o n v e n tio n n a l e le c tr o n d iffr a c tio n
b le d iffr a c tio n ( d y n a m ic a l b e h a v io r )
R o ta tio n
E x tr e m e ly th in c r is ta l
T h in c r y s ta l
m u ltip le d iffr a c tio n
S u p p r e s s io n
o f th e
m u ltip le d iffr a c tio n p a th s
F o r b id d e n
d u e to :
- s
- g
- (
D iffic u lt to
d u e to d o u
r e fle c tio n s
c re w a x e s
lid e p la n e s
R o ta tio n
T h in c r y s ta l
o f th e
F o r b id d e n
d u e to :
- s
- g
- (
D iffic u lt to
d u e to d o u
r e fle c tio n s
c re w a x e s
lid e p la n e s
R o ta tio n
T h in c r y s ta l
o f th e
F o r b id d e n r e fle c tio n s
C B E D p a tte r n s : o b s e r v a tio n o f G jo n n e s a n d M o o d ie lin e s
in th e Z O L Z a n d in th e H O L Z s
E le c tr o n b e a m
G lid e
p la n e // e
S c r e w a x is
e -
-
m
D . J a c o b
m
P e r fe c t c r y s ta l o r ie n ta tio n r e q u ir e d
P r e c e s s io n a n g le : 1 °
O U T L IN E
II - R a p id
- C o n v e
- M
- M
d e
rg e
ic r o
ic r
s c r ip
n t-B e
d iffr a
o d iffr
tio n o
a m E
c tio n
a c tio n
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th
le c
in
in
b le
" id e
/ Z
p e r
d e n
e e x p
tro n D
c o n v e
p re c e
o n
a l"
O L
io d
re
e r im
iffr a
n tio n
s s io
c tio n ( C B E D )
a l m o d e
n m o d e
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
Id e n tific a tio n o f th e C R Y S T A L S Y S T E M
C o n v e n tio n a l M ic r o d iffr a c tio n p a tte r n s
fro m
t h e 'n e t ' s y m m e t r ie s
H ig h e s t W P
o b s e rv e d
1
2
m
2 m m
(4 m m )
(4 m m )*
6 m m
3 m
o r
a n
a b s e
o
(6 m
T r ic l.
C ry s ta l
s y s te m
M o n o .
O rth o .
d
n c e
f
m )
T e tra .
a n d
p re s e n c e
o f
(6 m m )
C u b .
a n
a b s e
o
(4 m
d
n c e
f
m )
R h o m b .
la ttic e
H e x .
la ttic e
T r ig .
H e x .
* e x c e p t fo r P a 3
F ro m
J .P . M o r n ir o li a n d J .W . S te e d s , U ltr a m ic r o s c o p y , 1 9 9 2 , 4 5 , 2 1 9
Id e n tific a tio n o f th e C R Y S T A L S Y S T E M
s y m m e tr ie s
v ia th e L A U E C L A S S fr o m
t h e 'id e a l'
H ig h e s t
o b s e rv e d
L a u e
c la s s
C ry s ta l
s y s te m
1
2
(2 )
(2 )
1
2 /m
2 m m
(2 m m )
m m m
T r ic l.
O rth o .
M o n o .
4
(4 )
4 /m
3
4 m m
(4 m m )
3 m
(6 ) (6 m m )
a n d
a b s e n c e
o f
3 m
(6 m m )
a n
a b s e
o
4 m
(4 m
4 /m m m
T e tra .
3
3 m
T r ig .
6
(6 )
6 m m
(6 m m )
3
(6 )
3 m
(6 m m )
a n d
a n d
o f
o f
2 m m
4 m m
(2 m m )
(4 m m )
d
n c e
f
m
m )
6 /m
6 /m m m
H e x .
m 3
m 3 m
C u b .
O U T L IN E
II - R a p id
- C o n v e
- M
- M
d e
rg e
ic r o
ic r
s c r ip
n t-B e
d iffr a
o d iffr
tio n o
a m E
c tio n
a c tio n
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th
le c
in
in
b le
" id e
/ Z
p e r
d e n
e e x p
tro n D
c o n v e
p re c e
o n
a l"
O L
io d
re
e r im
iffr a
n tio n
s s io
c tio n ( C B E D )
a l m o d e
n m o d e
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
Id e n tific a tio n o f th e e x tin c tio n s y m b o l
fr o m X - r a y d iffr a c tio n
B u e rg e rs
E x a m p le o f th e m o n o c lin ic c r y s ta l s y s te m
U n iq u e a x is b
L a u e c la s s 1 2 /m
R e fle c tio n c o n d itio n s
P o in t g r o u p
h k l
0 k l h k 0
h 0 l
h 0 0 0 0 l
0 k 0
P 1 -1
P 1 2 11
P 1 a 1
P 1 2 1/a 1
k
h
h
k
l
h +
h +
k +
k +
h +
h +
l
h + l
h + l
h
k
k
k
k
k
k
l
l
k -l
k -l
h ,l
h ,l
h + l
h ,l
k
k
k
k
E x tin c tio n s y m b o l
P 1 c 1
P 1 2 1/c 1
P 1 n 1
P 1 2 1/n 1
C 1 -1
C 1 c 1
A 1 -1
A 1 n 1
I1 -1
I1 a 1
2
P 1 2 1 (3 )
P 1 2 11 (4 )
1
m
2 /m
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
P 1 2 1/m 1 (1 1 )
P 1 a 1 (7 )
P 1 2 /a 1 (1 3 )
P 1 2 1/a 1 (1 4 )
P 1 c 1 (7 )
P 1 2 /c 1 (1 3 )
P 1 2 1/c 1 (1 4 )
P 1 2 /n 1 (1 3 )
P 1 2 1/n 1 (1 4 )
C 1 2 /m 1 (1 2 )
C 1 2 /c 1 (1 5 )
A 1 2 /m 1 (1 2 )
A 1 2 /n 1 (1 5 )
I1 2 /m 1 (1 2 )
I1 2 /a 1 (1 5 )
P 1 n 1 (7 )
C 1 2 1 (5 )
A 1 2 1 (5 )
I1 2 1 (5 )
C 1
C 1
A 1
A 1
I1 m
I1 a
m 1 (8 )
c 1 (9 )
m 1 (8 )
n 1 (9 )
1 (8 )
1 (9 )
In te r n a tio n a l T a b le s fo r C r y s ta llo g r a p h y
B u e rg e rs
U n iq u e a x is b
P o in t g r o u p
h k l
0 k l h k 0
h 0 l
h 0 0 0 0 l
0 k 0
P 1 -1
P 1 2 11
P 1 a 1
P 1 2 1/a 1
k
h
h
k
l
h +
h +
k +
k +
h +
h +
l
h + l
h + l
h
k
k
k
k
k
k
l
l
k -l
k -l
h ,l
h ,l
h + l
h ,l
k
k
k
k
P 1 c 1
P 1 2 1/c 1
P 1 n 1
P 1 2 1/n 1
C 1 -1
C 1 c 1
A 1 -1
A 1 n 1
I1 -1
I1 a 1
2
P 1 2 1 (3 )
P 1 2 11 (4 )
1
m
2 /m
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
P 1 2 1/m 1 (1 1 )
P 1 a 1 (7 )
P 1 2 /a 1 (1 3 )
P 1 2 1/a 1 (1 4 )
P 1 c 1 (7 )
P 1 2 /c 1 (1 3 )
P 1 2 1/c 1 (1 4 )
P 1 2 /n 1 (1 3 )
P 1 2 1/n 1 (1 4 )
C 1 2 /m 1 (1 2 )
C 1 2 /c 1 (1 5 )
A 1 2 /m 1 (1 2 )
A 1 2 /n 1 (1 5 )
I1 2 /m 1 (1 2 )
I1 2 /a 1 (1 5 )
P 1 n 1 (7 )
C 1 2 1 (5 )
A 1 2 1 (5 )
I1 2 1 (5 )
C 1
C 1
A 1
A 1
I1 m
I1 a
m 1 (8 )
c 1 (9 )
m 1 (8 )
n 1 (9 )
1 (8 )
1 (9 )
B u e rg e rs
E x a m p le o f th e m o n o c lin ic c r y s ta l s y s te m : 1 4 e x tin c tio n s y m b o ls
U n iq u e a x is b
P o in t g r o u p
h k l
0 k l h k 0
h 0 l
h 0 0 0 0 l
0 k 0
P 1 -1
P 1 2 11
P 1 a 1
P 1 2 1/a 1
k
h
h
k
l
h +
h +
k +
k +
h +
h +
l
h + l
h + l
h
k
k
k
k
k
k
l
l
k -l
k -l
h ,l
h ,l
h + l
h ,l
k
k
k
k
P 1 c 1
P 1 2 1/c 1
P 1 n 1
P 1 2 1/n 1
C 1 -1
C 1 c 1
A 1 -1
A 1 n 1
I1 -1
I1 a 1
2
P 1 2 1 (3 )
P 1 2 11 (4 )
1
m
2 /m
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
P 1 2 1/m 1 (1 1 )
P 1 a 1 (7 )
P 1 2 /a 1 (1 3 )
P 1 2 1/a 1 (1 4 )
P 1 c 1 (7 )
P 1 2 /c 1 (1 3 )
P 1 2 1/c 1 (1 4 )
P 1 2 /n 1 (1 3 )
P 1 2 1/n 1 (1 4 )
C 1 2 /m 1 (1 2 )
C 1 2 /c 1 (1 5 )
A 1 2 /m 1 (1 2 )
A 1 2 /n 1 (1 5 )
I1 2 /m 1 (1 2 )
I1 2 /a 1 (1 5 )
P 1 n 1 (7 )
C 1 2 1 (5 )
A 1 2 1 (5 )
I1 2 1 (5 )
C 1
C 1
A 1
A 1
I1 m
I1 a
m 1 (8 )
c 1 (9 )
m 1 (8 )
n 1 (9 )
1 (8 )
1 (9 )
B u e rg e rs
U n iq u e a x is b
P o in t g r o u p
h k l
0 k l h k 0
h 0 l
h 0 0 0 0 l
0 k 0
P 1 -1
P 1 2 11
P 1 a 1
P 1 2 1/a 1
k
h
h
k
l
h +
h +
k +
k +
h +
h +
l
h + l
h + l
h
k
k
k
k
k
k
l
l
k -l
k -l
h ,l
h ,l
h + l
h ,l
k
k
k
k
P 1 c 1
P 1 2 1/c 1
P 1 n 1
P 1 2 1/n 1
C 1 -1
C 1 c 1
A 1 -1
A 1 n 1
I1 -1
I1 a 1
2
P 1 2 1 (3 )
P 1 2 11 (4 )
1
m
2 /m
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
P 1 2 1/m 1 (1 1 )
P 1 a 1 (7 )
P 1 2 /a 1 (1 3 )
P 1 2 1/a 1 (1 4 )
P 1 c 1 (7 )
P 1 2 /c 1 (1 3 )
P 1 2 1/c 1 (1 4 )
P 1 2 /n 1 (1 3 )
P 1 2 1/n 1 (1 4 )
C 1 2 /m 1 (1 2 )
C 1 2 /c 1 (1 5 )
A 1 2 /m 1 (1 2 )
A 1 2 /n 1 (1 5 )
I1 2 /m 1 (1 2 )
I1 2 /a 1 (1 5 )
P 1 n 1 (7 )
C 1 2 1 (5 )
A 1 2 1 (5 )
I1 2 1 (5 )
C 1
C 1
A 1
A 1
I1 m
I1 a
m 1 (8 )
c 1 (9 )
m 1 (8 )
n 1 (9 )
1 (8 )
1 (9 )
B u e rg e rs
U n iq u e a x is b
P o in t g r o u p
h k l
0 k l h k 0
h 0 l
h 0 0 0 0 l
0 k 0
P 1 -1
P 1 2 11
P 1 a 1
P 1 2 1/a 1
k
h
h
k
l
h +
h +
k +
k +
h +
h +
l
h + l
h + l
h
k
k
k
k
k
k
l
l
k -l
k -l
h ,l
h ,l
h + l
h ,l
k
k
k
k
P 1 c 1
P 1 2 1/c 1
P 1 n 1
P 1 2 1/n 1
C 1 -1
C 1 c 1
A 1 -1
A 1 n 1
I1 -1
I1 a 1
2
P 1 2 1 (3 )
P 1 2 11 (4 )
1
m
2 /m
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
P 1 2 1/m 1 (1 1 )
P 1 a 1 (7 )
P 1 2 /a 1 (1 3 )
P 1 2 1/a 1 (1 4 )
P 1 c 1 (7 )
P 1 2 /c 1 (1 3 )
P 1 2 1/c 1 (1 4 )
P 1 2 /n 1 (1 3 )
P 1 2 1/n 1 (1 4 )
C 1 2 /m 1 (1 2 )
C 1 2 /c 1 (1 5 )
A 1 2 /m 1 (1 2 )
A 1 2 /n 1 (1 5 )
I1 2 /m 1 (1 2 )
I1 2 /a 1 (1 5 )
P 1 n 1 (7 )
C 1 2 1 (5 )
A 1 2 1 (5 )
I1 2 1 (5 )
C 1
C 1
A 1
A 1
I1 m
I1 a
m 1 (8 )
c 1 (9 )
m 1 (8 )
n 1 (9 )
1 (8 )
1 (9 )
fr o m e le c tr o n d iffr a c tio n
3 p ie c e s o f in fo r m a tio n r e q u ir e d :
- th e B r a v a is la ttic e
- F O L Z / Z O L Z s h ift
- P E D v e r y w e ll a d a p te d
- th e g lid e p la
- F O L Z / Z
- F o r b id d e
- P E
- C B
n e s
O L Z p
n r e fle
D v e ry
E D w e
- th e s c re w
- fo r b id
-
e s
r e fle c tio n s w ith o r w ith o u t G M
D w e ll a d a p te d
E D w e ll a d a p te d fo r G M lin e s
a x
d e n
P E
C B
e r io d ic
c tio n s
w e ll a
ll a d a p
ity d iffe r e n c e
w ith o r w ith o u t G M
d a p te d
te d fo r G M lin e s
lin e s
lin e s
C ry s ta l
s y s te m
P e r tin e n t z o n e a x e s
B r a v a is
la ttic e
T R IC L IN IC
a P
M O N O C L IN IC
u n iq u e a x is b
m P m C
m A m I
c a n // (0 1 0 )
M O N O C L IN IC
u n iq u e a x is c
m P m A
m B m I
a b n // (0 0 1 )
O R T H O R H O M B IC
o P o C
o A o B o I
o F
b c n d // (1 0 0 )
c a n d // (0 1 0 )
a b n d // (0 0 1 )
b c
c a
a b
c n
c n
T E T R A G O N A L
T R IG O N A L
S c re w a x e s
G lid e p la n e s
n //
n //
n //
d //
d //
(1 0
(0 1
(0 0
(1 1
(1 1
// [0 1 0 ]
[0 1 0 ]b W P
o r
[0 1 0 ]b W P

Z O L Z
// [0 0 1 ]
[0 0 1 ]c W P
o r
[0 0 1 ]c W P

Z O L Z
2 1 // [1 0 0 ]
2 1 // [0 1 0 ]
2 1 // [0 0 1 ]
[1 0 0 ] W P
[0 1 0 ] W P
[0 0 1 ] W P
// [1 0 0 ]
// [0 1 0 ]
[0 0 1 ] W P
4 3 // [0 0 1 ]
[1 0 0 ] W P
o r
[0 1 0 ] W P
[1 1 0 ] W P
o r
[1 1 0 ] W P
2
1
2
1
2
0 )
1 )
0 )
0 )
C U B IC
c P c I c F
(1 0
(0 1
(0 0
0 ) (
1 ) (
1 ) (
1
4
1
1
3
2
6
1
6
2
6
3
6
4
6
5
2
3
h P
d t I/ /
d //
d //
(1 1
(0 1
(1 0
1
2
4
h P h R
b c t nP
c a n
a b n
c n d //
a n d //
b n d //
b
c
0 )
c // {1 1 2 0 }
c // {1 1 0 0 }
H E X A G O N A L
Z o n e a x e s u s e fu l fo r th e
id e n tific a tio n o f th e
e x tin c tio n s y m b o l
// [0
// [0
// [0
// [0
// [0
// [0
// [0
0 1 ]
0 1 ]
0 1 ]
0 1 ]
0 1 ]
0 1 ]
0 1 ]
< 1 1 2 0 >
W P
< 1 1 0 0 >
W P
0 )
0 )
1 )
1 1 0 )
0 1 1 )
1 0 1 )
2
2 4 1 4 3 / /
[1 0 0 ]
[0 1 0 ] [0 0 1 ]
1
4
< 0 0 1 > W P < 1 1 0 > W P
< 1 1 1 > W P
o p tio n a l
C ry s ta l
s y s te m
P e r tin e n t z o n e a x e s
- E x a m p le
M o n o c lin ic c r y s ta l s y s te m
( u n iq u e a x is b d e s c r ip tio n )
- [0 1 0 ]b o r [0 1 0 ]b W P
- b Z O L Z
r e q u ir e d
B r a v a is
la ttic e
T R IC L IN IC
a P
M O N O C L IN IC
u n iq u e a x is b
m P m C
m A m I
c a n // (0 1 0 )
M O N O C L IN IC
u n iq u e a x is c
m P m A
m B m I
a b n // (0 0 1 )
O R T H O R H O M B IC
o P o C
o A o B o I
o F
b c n d // (1 0 0 )
c a n d // (0 1 0 )
a b n d // (0 0 1 )
b c
c a
a b
c n
c n
T E T R A G O N A L
T R IG O N A L
S c re w a x e s
G lid e p la n e s
n //
n //
n //
d //
d //
(1 0
(0 1
(0 0
(1 1
(1 1
// [0 1 0 ]
[0 1 0 ]b W P
o r
[0 1 0 ]b W P

Z O L Z
// [0 0 1 ]
[0 0 1 ]c W P
o r
[0 0 1 ]c W P

Z O L Z
2 1 // [1 0 0 ]
2 1 // [0 1 0 ]
2 1 // [0 0 1 ]
[1 0 0 ] W P
[0 1 0 ] W P
[0 0 1 ] W P
// [1 0 0 ]
// [0 1 0 ]
[0 0 1 ] W P
4 3 // [0 0 1 ]
[1 0 0 ] W P
o r
[0 1 0 ] W P
[1 1 0 ] W P
o r
[1 1 0 ] W P
2
1
2
1
2
0 )
1 )
0 )
0 )
C U B IC
c P c I c F
(1 0
(0 1
(0 0
0 ) (
1 ) (
1 ) (
1
4
1
1
3
2
6
1
6
2
6
3
6
4
6
5
2
3
h P
d t I/ /
d //
d //
(1 1
(0 1
(1 0
1
2
4
h P h R
b c t nP
c a n
a b n
c n d //
a n d //
b n d //
b
c
0 )
c // {1 1 2 0 }
c // {1 1 0 0 }
H E X A G O N A L
Z o n e a x e s u s e fu l fo r th e
id e n tific a tio n o f th e
e x tin c tio n s y m b o l
// [0
// [0
// [0
// [0
// [0
// [0
// [0
0 1 ]
0 1 ]
0 1 ]
0 1 ]
0 1 ]
0 1 ]
0 1 ]
< 1 1 2 0 >
W P
< 1 1 0 0 >
W P
0 )
0 )
1 )
1 1 0 )
0 1 1 )
1 0 1 )
2
2 4 1 4 3 / /
[1 0 0 ]
[0 1 0 ] [0 0 1 ]
1
4
< 0 0 1 > W P < 1 1 0 > W P
< 1 1 1 > W P
o p tio n a l
T h e o r e tic a l [0 1 0 ], [0 1 0 ] a n d Z A P s
d r a w n fo r th e 1 4 m o n o c lin ic e x tin c tio n s y m b o ls
M O N O C L IN IC C R Y S T A L S Y S T E M
P B r a v a is la ttic e
m P 1
P 1 -1
[0 1 0 ]b Z O L Z
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
s p a c e g ro u p s
P 1 -1
P 1 2 1 (3 )
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
m P 1
[0 1 0 ]b Z O L Z
P 1 -1
[0 1 0 ]b F O L Z
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
P 1 -1
P 1 2 1 (3 )
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
- N o F O L Z / Z O L Z s h ift
- N o F O L Z / Z O L Z p e r io d ic ity d iffe r e n c e
- N o fo r b id d e n r e fle c tio n
m P 1
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
P 1 -1
1
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
P 1 -1
P 1 2 1 (3 )
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
m P 1
P 1 -1
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
1
[1 0 1 ]b Z O L Z
m
[0 0 1 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
1
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
1
P 1 -1
P 1 2 1 (3 )
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
[u 0 w ]b
- [u v 0 ]c
1
m P 2
P 1 2 11
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
1
m
[0 0 1 ]b Z O L Z
[u 0 w ]b
- [u v 0 ]c
2
//
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
//
- F o r b id d e n r e fle c tio n w ith G M lin e s
[1 0 1 ]b Z O L Z
m
2
P 1 2 11
P 1 2 11 (4 )
P 1 2 1/m 1 (1 1 )
[u 0 w ]b
- [u v 0 ]c
2
//
m P 3 a
P 1 a 1
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
- F O L Z / Z O L Z p e r io d ic ity d iffe r e n c e
- F o r b id d e n r e fle c tio n s w ith G M lin e s
2
[1 0 1 ]b Z O L Z
m
[0 0 1 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
2
p
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
1
P 1 a 1
P 1 a 1 (7 )
P 1 2 /a 1 (1 3 )
[u 0 w ]b
- [u v 0 ]c
2
p
m P 3 b
P 1 c 1
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
3
m
[0 0 1 ]b Z O L Z
[u 0 w ]b
- [u v 0 ]c
1
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
p
[1 0 1 ]b Z O L Z
m
2
P 1 a 1
P 1 c 1 (7 )
P 1 2 /c 1 (1 3 )
[u 0 w ]b
- [u v 0 ]c
2
p
m P 3 c
P 1 n 1
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
4
m
[0 0 1 ]b Z O L Z
[u 0 w ]b
- [u v 0 ]c
2
p
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
p
[1 0 1 ]b Z O L Z
m
2
P 1 a 1
P 1 a 1 (7 )
P 1 2 /a 1 (1 3 )
[u 0 w ]b
- [u v 0 ]c
1
m P 4 a
P 1 2 1/a 1
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
2
m
[0 0 1 ]b Z O L Z
[u 0 w ]b
- [u v 0 ]c
3
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
//
[1 0 1 ]b Z O L Z
m
2
P 1 2 1/a 1
P 1 2 1/a 1 (1 4 )
[u 0 w ]b
- [u v 0 ]c
3
m P 4 b
P 1 2 1/c 1
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
3
[1 0 1 ]b Z O L Z
m
[0 0 1 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
2
//
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
3
P 1 2 1/c 1
P 1 2 1/c 1 (1 4 )
[u 0 w ]b
- [u v 0 ]c
3
m P 4 c
P 1 2 1/n 1
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m P
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
4
[1 0 1 ]b Z O L Z
m
[0 0 1 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
3
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
3
P 1 2 1/n 1
P 1 2 1/n 1 (1 4 )
[u 0 w ]b
- [u v 0 ]c
2
//
m S 1 a
C 1 -1
[1 0 0 ]b Z O L Z
S h ift
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
[0 1 0 ]b
- [0 0 1 ]c
- [u v 0 ]c
1
S h ift
[1 0 1 ]b Z O L Z
m
[0 0 1 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
4
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
1
- N o fo r b id d e n r e fle c tio n s
S h ift
m S
[u 0 w ]b
C 1 -1
C 1 2 1 (5 )
C 1 m 1 (8 )
C 1 2 /m 1 (1 2 )
[u 0 w ]b
- [u v 0 ]c
4
m S 1 b
S h ift
A 1 -1
[1 0 0 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
4
S h ift
[0 1 0 ]b Z O L Z
m S
[0 1 0 ]b
- [0 0 1 ]c
[0 1 0 ]b F O L Z
2
[1 0 1 ]b Z O L Z
S h ift
m
m
[0 0 1 ]b Z O L Z
[u 0 w ]b
- [u v 0 ]c
1
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
A 1 -1
A 1 2 1 (5 )
A 1 m 1 (8 )
A 1 2 /m 1 (1 2 )
[u 0 w ]b
- [u v 0 ]c
4
m S 1 c
I1 -1
S h ift
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m S
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
3
[1 0 1 ]b Z O L Z
S h ift
m
[0 0 1 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
4
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
1
I1 -1
I1 2 1 (5 )
I1 m 1 (8 )
I1 2 /m 1 (1 2 )
[u 0 w ]b
- [u v 0 ]c
1
m S 2 a
C 1 c 1
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m S
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
4
S h ift
m
[0 0 1 ]b Z O L Z
[u 0 w ]b
- [u v 0 ]c
4
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
p
[1 0 1 ]b Z O L Z
m
2
C 1 c 1
C 1 c 1 (9 )
C 1 2 /c 1 (1 5 )
[u 0 w ]b
- [u v 0 ]c
4
m S 2 b
A 1 n 1
[1 0 0 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
4
S h ift
[0 1 0 ]b Z O L Z
m S
[0 1 0 ]b
- [0 0 1 ]c
[0 1 0 ]b F O L Z
5
[1 0 1 ]b Z O L Z
m
m
[0 0 1 ]b Z O L Z
[u 0 w ]b
- [u v 0 ]c
2
p
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
A 1 n 1
A 1 n 1 (9 )
A 1 2 /n 1 (1 5 )
[u 0 w ]b
- [u v 0 ]c
4
m S 2 c
I1 a 1
S h ift
[1 0 0 ]b Z O L Z
m
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m S
[0 1 0 ]b
- [0 0 1 ]c
[u 0 w ]b
- [u v 0 ]c
6
[1 0 1 ]b Z O L Z
S h ift
m
[0 0 1 ]b Z O L Z
m
[u 0 w ]b
- [u v 0 ]c
4
U n iq u e a x is b
E x tin c tio n
s y m b o l
P o s s ib le
4
I1 a 1
I1 a 1 (9 )
I1 2 /a 1 (1 5 )
[u 0 w ]b
- [u v 0 ]c
2
p
R e m a rk
T h e p a tte r n s fo r e a c h o f th e 1 4 e x tin c tio n s y m b o l
a r e d iffe r e n t a n d ty p ic a l
R e c a p itu la tio n
1 0 0
0 1 0
[0 1 0 ]b
0 0 1
1 0 0
c
b
c
[0 1 0 ]b F O L Z
[0 0 1 ]c F O L Z
0 0 1
1 0 0
1 0 0
0 1 0
b
c
1
- [0 0 1 ]c
m P
[0 1 0 ]b
2
- [0 0 1 ]c
m P
[0 1 0 ]b
0 0 1
1 0 0
b
c
1 0 0
0 1 0
m P
[0 1 0 ]b
2 0 0
0 2 0
m P
c
c
3
- [0 0 1 ]c
m P
c
c
[0 1 0 ]b
0 0 2
2 0 0
b
b
4
- [0 0 1 ]c
b
c
c
c
1
b
b
b
- [0 0 1 ]c
0 0 2
2 0 0
b
[0 1 0 ]b
0 0 2
2 0 0
0 0 1
1 0 0
c
c
0 0 2
2 0 0
[0 1 0 ]b
2 0 0
0 2 0
b
b
[0 1 0 ]b Z O L Z
[0 0 1 ]c Z O L Z
m P
m ir r o r r e la te d
p a tte rn s
2 0 0
0 2 0
b
[0 1 0 ]b
1 0 0
0 1 0
b
c
- [0 0 1 ]c
2
m P
[0 1 0 ]b
2 0 0
0 2 0
b
c
- [0 0 1 ]c
3
m P
[0 1 0 ]b
b
c
- [0 0 1 ]c
2
S B r a v a is la ttic e s
S h if t
[0 1 0 ]b
2 0 0
0 2 0
0 0 1
1 0 0
b
1 0 0
0 1 0
[0 1 0 ]b F O L Z
[0 0 1 ]c F O L Z
c
b
S h if t
S h if t
2 0 0
0 2 0
b
c
S h if t
c
0 0 2
2 0 0
[0 1 0 ]b Z O L Z
[0 0 1 ]c Z O L Z
b
0 0 2
2 0 0
c
S h if t
2 0 0
0 2 0
b
c
0 0 2
2 0 0
b
c
2 0 0
0 2 0
b
c
0 0 2
2 0 0
b
c
2 0 0
0 2 0
b
c
S h if t
b
0 0 2
2 0 0
c
b
c
b
c
S h if t
S h if t
S h if t
S h if t
m S
[0 1 0 ]b
m S
1
- [0 0 1 ]c
[0 1 0 ]b
- [0 0 1 ]c
2
m S
[0 1 0 ]b
- [0 0 1 ]c
S h if t
3
S h if t
S h if t
S h if t
m S
[0 1 0 ]b
- [0 0 1 ]c
m S
4
[0 1 0 ]b
- [0 0 1 ]c
5
m S
[0 1 0 ]b
- [0 0 1 ]c
S h if t
6
S h if t
S h if t
S h if t
S h if t
[0 1 0 ]b
[0 1 0 ]b F O L Z
[0 0 1 ]c F O L Z
2 0 0
0 2 0
1 0 0
0 1 0
b
c
2 0 0
0 2 0
b
c
2 0 0
0 2 0
b
c
2 0 0
0 2 0
b
c
2 0 0
0 2 0
b
c
b
c
S h if t
S h if t
[0 1 0 ]b Z O L Z
[0 0 1 ]c Z O L Z
S h if t
m S
[0 1 0 ]b
- [0 0 1 ]c
1
S h if t
m S
[0 1 0 ]b
- [0 0 1 ]c
2
S h if t
m S
[0 1 0 ]b
- [0 0 1 ]c
3
S h if t
S h if t
m S
[0 1 0 ]b
- [0 0 1 ]c
4
m S
[0 1 0 ]b
- [0 0 1 ]c
5
m S
[0 1 0 ]b
- [0 0 1 ]c
6
m
[u 0 w ]b
- [u v 0 ]c
1
m
[u 0 w ]b
- [u v 0 ]c
2
//
m
[u 0 w ]b
- [u v 0 ]c
[u 0 w ]b
2
m
[u 0 w ]b
- [u v 0 ]c
3
m
[u 0 w ]b
- [u v 0 ]c
4
L a u e c la s s 1 2 / m
1 1 2 /m
P o in t g r o u p
2
m P 4
m P 4
P 1 -1
P 1 1 P 1 2 11
P 1 1 2 1
P 1 a 1
P 1 1 b
P 1 c1
P 1 1 a
m P 5
m P 5
P 1 n 1
P 1 1 n
m P 1
m P 1
m P 2
m P 2
m P 3
m P 3
m P 6
m P 6
m P 7
m P 7
m P 8
m P 8
m S 1
m S 1
m S 2
m S 2
m S 3
m S 3
m S 4
m S 4
m S 5
m S 5
m S 6
m S 6
P 1
P 1
P 1
P 1
P 1
P 1
2 1/ a
1 2 1/
2 1/ c
1 2 1/
2 1/ n
1 2 1/
C 1 -1
A 1 1 A 1 -1
B 1 1 I1 -1
I1 1 C 1 c1
A 1 1 a
A 1 n 1
B 1 1 n
I1 a 1
I1 1 b
P 1 2 1 (3 )
P 1 1 2 (3 )
m
P 1 m 1 (6 )
P 1 1 m (6 )
2 /m
P a tte rn ty p e
Z o n e a x e s
[0 1 0 ]b - [ 0 0 1 ] c
o r
[0 0 ]b - [0 0 ]c
P 1 2 / m 1 (1 0 )
P 1 1 2 / m (1 0 )
m P
[0 0 1 ]b
-
[0 0 1 ]c
m P
[0 0
]b
-
[0 0
]c
P 1 2 1/ m 1 (1 1 )
P 1 1 2 1/m (1 1 )
m P
[0 0 1 ]b
-
[0 0 1
]c 1
m P
[0 0
]b
-
[0 0
]c
P 1 a 1 (7 )
P 1 1 b (7 )
P 1 2 /a 1 (1 3 )
P 1 1 2 /b (1 3 )
m P
[0 0 1 ]b
-
[0 0 1 ]c
m P
[0 0
]b
-
[0 0
P 1 c 1 (7 )
P 1 1 a (7 )
P 1 2 / c 1 (1 3 )
P 1 1 2 / a (1 3 )
m P
[0 0 1 ]b
-
[0 0 1 ]c
m P
[0 0
]b
-
[0 0
P 1 n 1 (7 )
P 1 1 n (7 )
P 1 2 /n 1 (1 3 )
P 1 1 2 /n (1 3 )
m P
[0 0 1 ]b
-
[0 0 1 ]c
m P
[0 0
]b
-
[0 0
P 1 2 1/ a 1 (1 4 )
P 1 1 2 1/ b (1 4 )
m P
[0 0 1 ]b
-
[0 0 1 ]c
m P
[0 0
]b
-
[0 0
P 1 2 1/ c 1 (1 4 )
P 1 1 2 1/ a (1 4 )
m P
[0 0 1 ]b
-
[0 0 1 ]c
m P
[0 0
]b
-
[0 0
]c
P 1 2 1/ n 1 ( 1 4 )
P 1 1 2 1/ n (1 4 )
m P
[0 0 1 ]b
-
[0 0 1
]c 4
[0 0
-
[0 0
]c
[ 1 0 0 ] b - [0 1 0 ]
P 1 2 11 (4 )
P 1 1 2 1 (4 )
1
b
1
a
1
n
C 1 2 1 (5 )
A 1 1 2 (5 )
C 1 m 1 (8 )
A 1 1 m (8 )
C 1 2 / m 1 (1 2 )
A 1 1 2 /m (1 2 )
A 1 2 1 (5 )
B 1 1 2 (5 )
A 1 m 1 (8 )
B 1 1 m (8 )
A 1 2 /m 1 (1 2 )
B 1 1 2 /m (1 2 )
I1 2 1 (5 )
I1 1 2 (5 )
I1 m 1 (8 )
I1 1 m (8 )
I1 2 / m 1 (1 2 )
I1 1 2 / m (1 2 )
C 1 c 1 (9 )
A 1 1 a (9 )
C 1 2 / c 1 (1 5 )
A 1 1 2 /a (1 5 )
A 1 n 1 (9 )
B 1 1 n (9 )
A 1 2 /n 1 (1 5 )
B 1 1 2 /n (1 5 )
I1 a 1 (9 )
I1 1 b (9 )
I1 2 / a 1 (1 5 )
I1 1 2 / b ( 1 5 )
m P
m S
m S
m S
m S
m S
m S
m S
m S
m S
m S
m S
m S
]b
1
m
m
-
[u 0 w ]b
2
m
m
3
]c
-
2
[u v 0 ]c
m
4
]c
1
m
[u v 0 ]c
-
[u 0 w ]b
1
[u v 0 ]c
-
[u 0 w ]b
2
[u v 0 ]c
m
p a r
-
[u 0 w ]b
2
[u v 0 ]c
p a r
1
m
[u 0 w ]b
-
[u v 0 ]c
2
p e rp
m
[u 0 w ]b
-
[u v 0 ]c
2
p e rp
m
-
[u 0 w ]b
2
[u v 0 ]c
p e rp
m
2
-
-
[u 0 w ]b
2
-
[u 0 w ]b
1
[u v 0 ]c
m
[u 0 w ]b
-
[u v 0 ]c
1
m
[u 0 w ]b
-
[u v 0 ]c
3
m
[u 0 w ]b
-
[u v 0 ]c
3
m
-
[u 0 w ]b
2
[u v 0 ]c
p e rp
p e rp
-
[u 0 w ]b
m
p e rp
[u v 0 ]c
[u v 0 ]c
2
p a r
m
-
[u 0 w ]b
3
[u v 0 ]c
2
]c
3
m
[u 0 w ]b
-
[u v 0 ]c
3
m
[u 0 w ]b
-
[u v 0 ]c
3
m
[u 0 w ]b
-
[u v 0 ]c
1
m
[u 0 w ]b
-
[u v 0 ]c
4
m
[u 0 w ]b
-
[u v 0 ]c
4
m
[u 0 w ]b
-
[u v 0 ]c
[u 0 w ]b
-
[u v 0 ]c
4
m
[u 0 w ]b
-
m
[u 0 w ]b
m
[u 0 w ]b
m
-
[u 0 w ]b
2
[u v 0 ]c
p a r
3
m
-
[u 0 w ]b
2
[u 0 w ]b
-
[u v 0 ]c
3
m
[u 0 w ]b
-
[u v 0 ]c
4
4
m
[u 0 w ]b
-
[u v 0 ]c
1
[u v 0 ]c
1
m
[u 0 w ]b
-
[u v 0 ]c
4
-
[u v 0 ]c
4
m
[u 0 w ]b
-
[u v 0 ]c
4
-
[u v 0 ]c
4
[u v 0 ]c
m
p a r
1
]c
1
[0 0 1 ]c
2
4
[0 0
]b
-
[0 0
]c
2
[0 0 1 ]b
-
[0 0 1 ]c
3
[0 0
]b
-
[0 0
]c
3
[0 0 1 ]b
-
[0 0 1 ]c
4
m
[0 0
]b
-
[0 0
]c
4
[u v 0 ]c
[0 0 1 ]b
-
[0 0 1 ]c
5
[0 0
]b
-
[0 0
]c
5
[0 0 1 ]b
-
[0 0 1 ]c
6
[0 0
-
[0 0
6
]c
m
p a r
[u v 0 ]c
[u 0 w ]b
4
[0 0 1 ]c
]b
2
[u v 0 ]c
[u 0 w ]b
3
-
[0 0
-
[u 0 w ]b
2
]c
-
m
c
1
[0 0
[0 0 1 ]b
1
[u v 0 ]c
[0 0 1 ]b - [1 0 0 ]
1
[0 0 1 ]b
]b
-
[u 0 w ]b
[ 1 0 1 ] b - [ 1 1 0 ]c
c
m
[u 0 w ]b
2
-
p e rp
m
[u 0 w ]b
-
[u v 0 ]c
4
m
[u 0 w ]b
-
[u v 0 ]c
4
m
[u 0 w ]b
-
[u v 0 ]c
O n e p a tte r n r e q u ir e d
2
p e rp
m
m
[u 0 w ]b
-
[u 0 w ]b
[u v 0 ]c
-
2
[u v 0 ]c
p e rp
4
O T H E R C R Y S T A L S Y S T E M S
M O N O C L IN IC : 1 4
O R T H O R H O M B IC : 1 1 1
T E T R A G O N A L : 3 1
T R IG O N A L : 6
H E X A G O N A L : 7
C U B IC : 1 7
T o ta l: 1 8 6 e x
'A t la s o f E le c t r o n
A
p d f file s a v a ila b le : je a n -
tin c tio n s y m b o ls
Z o n e - A x is P a tte r n s '
T k q
p a u l.m o r n ir o li@ u n iv - lille 1 .fr
O U T L IN E
II - R a p id
- C o n v e
- M
- M
d e
rg e
ic r o
ic r
s c r ip
n t-B e
d iffr a
o d iffr
tio n o
a m E
c tio n
a c tio n
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th
le c
in
in
b le
" id e
/ Z
p e r
d e n
e e x p
tro n D
c o n v e
p re c e
o n
a l"
O L
io d
re
e r im
iffr a
n tio n
s s io
c tio n ( C B E D )
a l m o d e
n m o d e
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
m P 1
P 1 -1
[1 0 0 ]b Z O L Z
0 1 0
0 0 1
0 0 1
1 0 0
1 0 0
0 1 0
[0 1 0 ]b Z O L Z
0 0 1
1 0 0
b
c
b
c
b
b
c
c
[0 1 0 ]b F O L Z
0 1 0
0 0 1
1 0 1
1 1 0
c
b
c
b
[1 0 1 ]b Z O L Z
[0 0 1 ]b Z O L Z
1 0 0
0 1 0
0 1 0
0 0 1
b
c
b
c
U n iq u e a x is b
E x tin c tio n
s y m b o l
1 2 1
1 m 1
1 2 /m 1
P o in t g r o u p s
P o s s ib le
P 1 -1
P 1 2 1 (3 )
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
S p a c e g ro u p s
E a c h e x tin c tio n s y m b o l is in a g r e e m e n t
w ith a fe w p o s s ib le s p a c e g r o u p s
b e lo n g in g to d iffe r e n t p o in t g r o u p s
m P 1
P 1 -1
[0 1 0 ]b s y m m e tr ie s
0 1 0
0 0 1
0 0 1
1 0 0
1 0 0
0 1 0
[0 1 0 ]b Z O L Z
0 0 1
1 0 0
b
b
c
b
c
[0 1 0 ]b F O L Z
P 1 2 1 (3 )
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
(2 ),2
(1 ),1
(2 ),2
C o n n
w ith
3
P O IN T G
0 1 0
0 0 1
1 0 1
1 1 0
1 0 0
0 1 0
0 1 0
0 0 1
e c te d
th e
1
L A S S E S
c
[u 0 w ]b s y m m e tr ie s
c
P o s s ib le
C B E D
(Z O L Z )
P r e c e s s io n
(Z O L Z )
b
c
b
c
E x tin c tio n
s y m b o l
P o in t g r o u p s
C o n n
w ith
1
L A U E C
b
U n iq u e a x is b
1 2 1
1 m 1
1 2 /m 1
e c te d
th e
2
R O U P S
(2 ),2
(2 ),2
(2 ),2
b
[1 0 1 ]b Z O L Z
[0 0 1 ]b Z O L Z
P r e c e s s io n
(Z O L Z ),W P
c
b
c
C B E D
(Z O L Z ),W P
P o s s ib le
P o s s ib le
P 1 -1
P 1 2 1 (3 )
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
S p a c e g ro u p s
P 1 2 1 (3 )
P 1 m 1 (6 )
P 1 2 /m 1 (1 0 )
(m )
(m )
(2 m m )
(2 m m )
(2 m m )
(2 m m )
S e le c tio n o f th e a c tu a l s p a c e g r o u p
fr o m id e n tific a tio n o f th e p o in t g r o u p
fr o m C B E D ( id e a l s y m m e tr y )
O U T L IN E
II - R a p id
- C o n v e
- M
- M
d e
rg e
ic r o
ic r
s c r ip
n t-B e
d iffr a
o d iffr
tio n o
a m E
c tio n
a c tio n
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th
le c
in
in
b le
" id e
/ Z
p e r
d e n
e e x p
tro n D
c o n v e
p re c e
o n
a l"
O L
io d
re
e r im
iffr a
n tio n
s s io
c tio n ( C B E D )
a l m o d e
n m o d e
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
E x a m p le : C o e s ite
F ir s t s te p
C ry s ta l s y s te m
?
H ig h e s t 'n e t ' a n d
'id e a l' s y m m e t r ie s ?
P a tte rn : D . J a c o b
F ir s t s te p
(6 m m )
m
m
m
m
m
m
F ir s t s te p
" Id e a l" s y m m e tr y
(2 )
m
m
m
m
m
m
F ir s t s te p
" Id e a l" s y m m e tr y
(2 ), 2
- F ir s t s te p : id e n tific a tio n o f th e c r y s ta l s y s te m
H ig h e s t
o b s e rv e d
L a u e
c la s s
C ry s ta l
s y s te m
1
2
(2 )
(2 )
1
2 /m
2 m m
(2 m m )
m m m
T r ic l.
O rth o .
M o n o .
4
(4 )
4 /m
3
4 m m
(4 m m )
3 m
(6 ) (6 m m )
a n d
a b s e n c e
o f
3 m
(6 m m )
a n
a b s e
o
4 m
(4 m
4 /m m m
T e tra .
3
3 m
T r ig .
6
(6 )
6 m m
(6 m m )
3
(6 )
3 m
(6 m m )
a n d
a n d
o f
o f
2 m m
4 m m
(2 m m )
(4 m m )
d
n c e
f
m
m )
6 /m
6 /m m m
H e x .
m 3
m 3 m
C u b .
S e c o n d s te p
E x tin c tio n s y m b o l ? :
P e r io d ic ity d iffe r e n c e
S h ift a lo n g o n e d ir e c tio n
S h ift
1 0 0
0 1 0
[0 1 0 ]b
0 0 1
1 0 0
2 0 0
0 2 0
b
c
b
c
[0 1 0 ]b F O L Z
[0 0 1 ]c F O L Z
0 0 1
1 0 0
1 0 0
0 1 0
b
c
1
- [0 0 1 ]c
m P
[0 1 0 ]b
2
- [0 0 1 ]c
m P
[0 1 0 ]b
m ir r o r r e la te d
p a tte rn s
0 0 1
1 0 0
b
c
1 0 0
0 1 0
m P
[0 1 0 ]b
2 0 0
0 2 0
m P
c
3
- [0 0 1 ]c
m P
c
c
[0 1 0 ]b
0 0 2
2 0 0
b
b
4
- [0 0 1 ]c
b
c
c
c
1
c
b
b
- [0 0 1 ]c
0 0 2
2 0 0
b
[0 1 0 ]b
0 0 2
2 0 0
0 0 1
1 0 0
b
c
0 0 2
2 0 0
[0 1 0 ]b
c
b
[0 1 0 ]b Z O L Z
[0 0 1 ]c Z O L Z
m P
2 0 0
0 2 0
b
[0 1 0 ]b
1 0 0
0 1 0
b
c
- [0 0 1 ]c
2
m P
[0 1 0 ]b
2 0 0
0 2 0
b
c
- [0 0 1 ]c
3
m P
[0 1 0 ]b
b
c
- [0 0 1 ]c
2
S B r a v a is la ttic e s
S h if t
[0 1 0 ]b
2 0 0
0 2 0
0 0 1
1 0 0
b
1 0 0
0 1 0
[0 1 0 ]b F O L Z
[0 0 1 ]c F O L Z
c
b
S h if t
S h if t
2 0 0
0 2 0
b
c
S h if t
c
0 0 2
2 0 0
[0 1 0 ]b Z O L Z
[0 0 1 ]c Z O L Z
b
0 0 2
2 0 0
c
S h if t
2 0 0
0 2 0
b
c
0 0 2
2 0 0
b
c
2 0 0
0 2 0
b
c
0 0 2
2 0 0
b
c
2 0 0
0 2 0
b
c
S h if t
b
0 0 2
2 0 0
c
b
c
b
c
S h if t
S h if t
m S
[0 1 0 ]b
S h if t
S h if t
m S
1
- [0 0 1 ]c
[0 1 0 ]b
- [0 0 1 ]c
2
m S
[0 1 0 ]b
- [0 0 1 ]c
S h if t
3
S h if t
S h if t
S h if t
m S
[0 1 0 ]b
- [0 0 1 ]c
m S
4
[0 1 0 ]b
- [0 0 1 ]c
5
m S
[0 1 0 ]b
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S h if t
6
S h if t
S h if t
S h if t
S h if t
[0 1 0 ]b
[0 1 0 ]b F O L Z
[0 0 1 ]c F O L Z
2 0 0
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1 0 0
0 1 0
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c
2 0 0
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c
2 0 0
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b
c
2 0 0
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2 0 0
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b
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S h if t
S h if t
[0 1 0 ]b Z O L Z
[0 0 1 ]c Z O L Z
S h if t
m S
[0 1 0 ]b
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1
S h if t
m S
[0 1 0 ]b
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2
S h if t
m S
[0 1 0 ]b
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3
m S
m S
S h if t
S h if t
[0 1 0 ]b
[0 1 0 ]b
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4
- [0 0 1 ]c
4
m S
[0 1 0 ]b
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5
m S
[0 1 0 ]b
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6
L a u e c la s s 1 2 / m
1 1 2 /m
P o in t g r o u p
2
m P 4
m P 4
P 1 -1
P 1 1 P 1 2 11
P 1 1 2 1
P 1 a 1
P 1 1 b
P 1 c1
P 1 1 a
m P 5
m P 5
P 1 n 1
P 1 1 n
m P 1
m P 1
m P 2
m P 2
m P 3
m P 3
m P 6
m P 6
m P 7
m P 7
m P 8
m P 8
m S 1
m S 1
m S 2
m S 2
m S 3
m S 3
m S 4
m S 4
m S 5
m S 5
m S 6
m S 6
P 1
P 1
P 1
P 1
P 1
P 1
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1 2 1/
2 1/ c
1 2 1/
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A 1 1 A 1 -1
B 1 1 I1 -1
I1 1 C 1 c1
A 1 1 a
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I1 a 1
I1 1 b
P 1 2 1 (3 )
P 1 1 2 (3 )
m
P 1 m 1 (6 )
P 1 1 m (6 )
P 1 2 11 (4 )
P 1 1 2 1 (4 )
2 /m
P 1 2 / m 1 (1 0 )
P 1 1 2 / m (1 0 )
P 1 2 1/ m 1 (1 1 )
P 1 1 2 1/m (1 1 )
P a tte rn ty p e
Z o n e a x e s
m P
m P
m P
m P
m P
m P
m P
m P
P 1 a 1 (7 )
P 1 1 b (7 )
P 1 2 /a 1 (1 3 )
P 1 1 2 /b (1 3 )
P 1 c 1 (7 )
P 1 1 a (7 )
P 1 2 / c 1 (1 3 )
P 1 1 2 / a (1 3 )
P 1 n 1 (7 )
P 1 1 n (7 )
P 1 2 /n 1 (1 3 )
P 1 1 2 /n (1 3 )
m P
m P
P 1 2 1/ a 1 (1 4 )
P 1 1 2 1/ b (1 4 )
m P
m P
m P
m P
m P
m P
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m S
m S
m S
m S
m S
m S
m S
m S
m S
m S
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P 1 1 2 1/ a (1 4 )
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C 1 m 1 (8 )
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C 1 2 / m 1 (1 2 )
A 1 1 2 /m (1 2 )
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B 1 1 2 (5 )
A 1 m 1 (8 )
B 1 1 m (8 )
A 1 2 /m 1 (1 2 )
B 1 1 2 /m (1 2 )
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I1 1 2 (5 )
I1 m 1 (8 )
I1 1 m (8 )
I1 2 / m 1 (1 2 )
I1 1 2 / m (1 2 )
C 1 c 1 (9 )
A 1 1 a (9 )
C 1 2 / c 1 (1 5 )
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A 1 n 1 (9 )
B 1 1 n (9 )
A 1 2 /n 1 (1 5 )
B 1 1 2 /n (1 5 )
I1 a 1 (9 )
I1 1 b (9 )
I1 2 / a 1 (1 5 )
I1 1 2 / b ( 1 5 )
[ 1 0 0 ] b - [0 1 0 ]
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m
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O n e p a tte r n r e q u ir e d
[u v 0 ]c
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[0 1 0 ]b Z O L Z
m S
[0 1 0 ]b
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E x tin c tio n
s y m b o l
P o s s ib le
C 1 c 1
C 1 c 1 (9 )
C 1 2 /c 1 (1 5 )
m S 2 a
C 1 c 1
[0 1 0 ] - [0 1 0 ] s y m m e tr ie s
P o s s ib le
S h if t
C 1 c 1 (9 )
C 1 2 /c 1 (1 5 )
[0 1 0 ]b F O L Z
[0 1 0 ]b Z O L Z
m S
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4
U n iq u e a x is b
E x tin c tio n
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C 1 c 1
C 1 c 1 (9 )
C 1 2 /c 1 (1 5 )
C B E D
(Z O L Z ),W P
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P r e c e s s io n
(Z O L Z ),W P
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(2 ),2
C B E D e x p e r im e n t
[0 1 0 ] Z A P - Z O L Z
[0 1 0 ] - [0 1 0 ] s y m m e tr ie s
P o s s ib le
C 1 c 1 (9 )
C 1 2 /c 1 (1 5 )
C B E D
(Z O L Z ),W P
(1 ),1
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P r e c e s s io n
(Z O L Z ),W P
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C B E D e x p e r im e n t
[0 1 0 ] Z A P - Z O L Z
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C B E D
(Z O L Z ),W P
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P r e c e s s io n
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O U T L IN E
II - R a p id
- C o n v e
- M
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d e
rg e
ic r o
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s c r ip
n t-B e
d iffr a
o d iffr
tio n o
a m E
c tio n
a c tio n
- "N e t" a n d
- F O L Z
- F O L Z / Z O L Z
- F o r b id
f th
le c
in
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/ Z
p e r
d e n
e e x p
tro n D
c o n v e
p re c e
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O L
io d
re
e r im
iffr a
n tio n
s s io
c tio n ( C B E D )
a l m o d e
n m o d e
s y m m e tr ie s
Z s h if ts
fle c t io n s
V II - E x a m p le
L im ita tio n s a n d d iffic u ltie s
N e t s y m m e tr y : v e r y e a s y w ith a n y s p o t p a tte r n s
Id e a l s y m m e tr y : e a s y w ith P E D
v e r y a c c u r a te w ith C B E D
- P r o b le m s w ith la r g e la ttic e p a r a m e te r s
- P r o b le m s w ith n a n o c r y s ta ls ( < 2 0 n m )
- P r o b le m s w ith p s e u d o - s y m m e tr ie s
T h e p r o p o s e d m e th o d a s s u m e s th a t a n y z o n e a x is is a v a ila b le
C O N C L U S IO N
Id e n tific a tio n o f th e s p a c e g r o u p
- a t m ic r o s c o p ic a n d n a n o s c o p ic s c a le s
P E D a n d C B E D v e r y v a lu a b le
E a s y m e th o d : c o m p a r is o n w ith ta b le s
( n o in te n s ity m e a s u r e m e n ts )
1 , 2 o r 3 Z A P s a r e r e q u ir e d
- C r y s ta l id e n tific a tio n
- S ta r tin g p o in t fo r s tr u c tu r e d e te r m in a tio n
A C K N O W L E D G E M E N T S
D a m ie n J A C O B
G a n g J I
S ta v ro s N IC O L O P O U L O S
A b d e lk r im R E D J A IM IA
J o h n S T E E D S
P ie r r e S T A D E L M A N N
A tla s o f z o n e a x is p a tte r n s a v a ila b le
h ttp ://w w w .E le c tr o n - D iffr a c tio n .fr

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