viernes 2 de septiembre de 2011 - Bilbao Crystallographic Server

Transcripción

viernes 2 de septiembre de 2011
SPACE-GROUPS DATABASES
and
ACCESS TOOLS
PRACTICAL EXERCISES
Mois I. Aroyo
Universidad del Pais Vasco, Bilbao, Spain
PRACTICAL EXERCISES
On
IT :
Symmetry Database
http://158.227.0.68:8080/
Bilbao Crystallographic Server
http://www.cryst.ehu.es
ITOn: Symmetry Database
http://158.227.0.68:8080/
Main Menu
Links to the ITA and ITA1 data
http://www.cryst.ehu.es
MATRIX-COLUMN
PRESENTATION OF
SYMMETRY OPERATIONS
Description of isometries
coordinate system:
{O, a, b, c}
isometry:
point X −→ point X̃
linear/matrix
part
matrix-column
pair
translation
column part
Seitz symbol
Short-hand notation
isometry:
(W,w)
X
~
X
notation rules: -left-hand side: omitted
-coefficients 0, +1, -1
-different rows in one line
examples:
1/2
-1
0
1
-1
1/2
{
-x+1/2, y, -z+1/2
x+1/2, y, z+1/2
Symmetry Database
Problem: Matrix-column presentation
short-hand
notation
matrix-column
pair
GEOMETRIC MEANING
OF
MATRIX-COLUMN
PAIRS (W,w)
Crystallographic symmetry operations
characteristics:
fixed point P̃ = P
of isometries
Types of isometries preserve handedness
identity:
the whole space fixed
translation t:
no fixed point
rotation:
one line fixed
rotation axis
φ = k × 360 /N
no fixed point
screw axis
screw vector
screw rotation:
x̃ = x + t
◦
do not
Types of isometries
preserve handedness
roto-inversion:
inversion:
reflection:
glide reflection:
centre of roto-inversion fixed
roto-inversion axis
centre of inversion fixed
plane fixed
reflection/mirror plane
no fixed point
glide plane
glide vector
Problem: Matrix-column presentation
Geometrical interpretation
GENPOS
space group
15
Example GENPOS: Space group C2/c(15)
Matrix-column
presentation
of symmetry
operations
Geometric
interpretation
ITA
data
Problem: Geometric
Interpretation of (W,w)
SYMMETRY
OPERATION
EXERCISES
Problem 1.1
Construct the matrix-column pairs (W,w) (and the
corresponding (4x4) matrices) of the following
coordinate triplets:
(1) x,y,z
(2) -x,y+1/2,-z+1/2
(4) x,-y+1/2, z+1/2
(3) -x,-y,-z
Characterize geometrically these matrix-column
pairs taking into account that they refer to a
monoclinic basis with unique axis b,
Use the program SYMMETRY OPERATIONS for the
geometric interpretation of the matrix-column pairs of
the symmetry operations.
Problem 1.1
SOLUTION
(i)
(ii)
ITA description: under Symmetry operations
EXERCISES
Problem 1.2 (additional)
Problem 2.11
of ITA exercises
Consider the matrices
(ii) Applying the SYMMETRY OPERATIONS program determine
the geometrical meaning of (A, a), (B, b), (C, c) and (D, d) if they
refer to a cubic basis.
Problem 1.2
SOLUTION
Problem 1.2
SOLUTION
The matrix-column pair (A, a): translation part
ITA description:
(A, a): screw rotation 21
screw rotation axis x,x,1/4
2(1/2,1/2,0) x,x,1/4
(B, b): rotation 3
rotation axis x,x,x
3- x,x,x
Problem 1.2
SOLUTION
The matrix-column pair (C, c): translation part
(C, c): screw rotation 42
screw rotation axis x,0,1/2
ITA description:
4+(1/2,0,0) x,0,1/2
EXERCISES
Problem 1.3
1. Characterize geometrically the matrix-column pairs
listed under General position of the space group
P4mm in ITA.
2. Consider the diagram of the symmetry elements of
P4mm. Try to determine the matrix-column pairs of
the symmetry operations whose symmetry
elements are indicated on the unit-cell diagram.
3. Compare your results with the results of the program
SYMMETRY OPERATIONS
GENERAL and SPECIAL
WYCKOFF POSITIONS
SITE-SYMMETRY
Symmetry Database
and special
Problem: General
Wyckoff positions
Calculation of the Site-symmetry groups
Group P-1
S={(W,w), (W,w)Xo = Xo}
(
0
-1
0
-1
-1
0
)
1/2
0
1/2
=
-1/2
0
-1/2
Sf={(1,0), (-1,101)Xf = Xf}
Sf≃{1, -1}
isomorphic
Problem: Wyckoff positions
Site-symmetry groups
WYCKPOS
space group
Example WYCKPOS: Wyckoff Positions Ccce (68)
2 1/2,y,1/4
2 x,1/4,1/4
Example WYCKSPLIT: Wyckoff Positions Ccce (68)
2 1/2,y,1/4
2 x,1/4,1/4
EXERCISES
Problem 1.4
Consider the special Wyckoff positions of the
the space group P4mm.
Determine the site-symmetry groups of Wyckoff
positions 1a and 2b. Compare the results with the
listed ITA data
The coordinate triplets (x,1/2,z) and (1/2,x,z),
belong to Wyckoff position 4f. Compare their
site-symmetry groups.
Compare your results with the results of the
program WYCKPOS.
Problem 1.4
Space group P4mm
SOLUTION
CO-ORDINATE
TRANSFORMATIONS
IN
CRYSTALLOGRAPHY
&
& &
& &
& &
They are briefly reviewed in
r
"
x
a
(
y
b
(
z
c"
The same point X is given with respect to a new coordinate system,
&
&
&
&
i.e. the new basis
vectors
a
,
b
,
c
and
the
new
origin
O
(Fig.
5.1. TRANSFORMATIONS
OF THEthe
COORDINATE
SYSTEM
In this section,
relations between
the primed and unprimed
5.1.3.1), by the position vector quantities are treated.
P ! Q"1 :
notation
& &
transformation !3-dimensional
(affine
transformation)
of the
space
r& " x& a& ( yThe
b ( general
z& c& "
$
!
$
'
'#
coordinate
system
consists
of
two
parts,
a
linear
part and a shift
a
a
are written in the following
'&
'# &
# borigin
In this section, the relations between
primed
andmatrix
of origin.theThe
#3 ! 3$
of
the
linear
#3 ! y,
1$ z)
bpart
! P #O:
!and the
(a,unprimed
b,Pc),
point
X(x,
c'
c'# the shift vector p,
column matrix p, containing the components
of
quantities are treated.
the transformation
by the symbol
The general transformationdefine
(affine
transformation)uniquely.
of the It is represented
(P,p)
These part
transformation
rules apply also to the quantities covariant
(P, p).
coordinate system consists of two
parts, a linear
and a shift
' ' '
, b , c and contravariant! with! !
with
respect
to
the
basis
vectors
a
!
!
!
of
Thespace
#3 ! 3$ matrix P of(i)theThe
linear
part
and
the
#3
!
1$
orsorigin.
of direct
(a
,
b
,
c
),
origin
O’: point
X(xorare, ythe, z )
linear
partto implies
a change
of orientation
or length
respect
a, b, c, which
are written
as column
matrices.
They
column
components
of the
p,direct space, [uvw], which are transformed
dices
of amatrix
planep,(orcontaining
a set of theboth
indices
ofshift
a direction
of the basis
vectors
a,vector
b, c, ini.e.
neral
affine
consisting
of a shiftItofisorigin
define
thetransformation,
transformation
uniquely.
represented
by& & &by the symbol
ect
space
or
the
coordinates
by
a shift
vector pTransformation
with components p1 and p2 and amatrix-column
change #a , b , c $ " #a, b,
pair
c$P! (P,p)
(P,
p).
#
# space
$
! $
reciprocal
m a, b to a , b . This implies a change in the coordinates of
#
#
&
u
u
# #
y.
rom(i)
x, yThe
to x , linear
#11
part implies a change of orientation or length or
#P
& !PQ12# vP&13!
v
rices:
$ #
'
(i) linear
change
of orientation
orb, length:
both of the basis
vectorspart:
a, b, c,
i.e.
w21 P22 wP23 (
" #a,
c$% P
es
of matrices
a pointofinP& direct
space
verse
and
& p& are needed. They are
#a , bspace
, c $ " #a, b, c$P
P31
P32 above,
P33 the components of a
ors of reciprocal
In
contrast
to
all
quantities
mentioned
#
&
Q ! P"1
thePcoordinates
of a point X in direct space
P11 P12 position
P13 vector
" #Pr11ora (
f a direction in direct space
21 b ( P31 c,
x, y, z '
depend also on the shift of the origin in direct space. The
$
nts of a shift vector from
P32 c,by
P12 a ( P22 b is(given
" #a, b, c$% P21 P22 general
P23 ((affine) transformation
& "1
he new origin
q!O
"P p!
! #P$13 a (!P23$b ( P33 c$"
P
P
P
31
32
33
x
x
nts
of of
anthe
inverse
origin
onsists
components
of
the
negative
shift
vector
&
" #%
" %
gintheOcoordinate
to origin
O, with
P
b
(
P
c,
y
!
Q
q
#
&
# yshift
& $ vector
b# , #P
c# , 11
i.e.a (
to
system
a# , "
31
For 21
a pure linear
transformation, the
p is zero and the
(ii)
shift
by
a
shift
vector
p(p
#1,p2,p3):
#
# origin
#
symbol
is
(P,
o).
z
z
q ! q1 a $ q2 b $ q3 c ! P12 a ( P22 b ( P32 c,
$
n part of a symmetry
The determinant of P, det#P$,!should
be
positive.
If
det#P$ is
x
$
Q
y
$
Q
z
$
q
Q
11
12
13
1
nsformation
(Q, q) is the inverse P
transformation
of
the
origin
O’
has
a
(
P
b
(
P
c$"
13
23
33
in direct space
negative,
a
right-handed
coordinate
system is transformed
"
% into a
# O
# #+ p
O’
=
x ,p
$ 3Q)22in
y $ Q23 z $ q2 &!
!(p
ng (Q, q) to the basis vectors a , b , c and the origin
#1Q,p
212
coordinates
left-handed
one
(or
vice
versa).
If
det#P$
"
0,
the new basis vectors
d
#4
!
1$
column
matrix
of
sis
a, b,
c withtransformation,
origin O are obtained.
Forvectors
a pure
linear
the
shift
vector
p
is
zero
and
the
Q31
x $system
Qa32 ycomplete
$ Q33 z $ qcoordinate
3
the
old
coordinate
#
#
are
linearly
dependent
and
do
not
form
es
of
a
point
in
direct
space
-dimensional
transformation
of
a
and
b
,
some
symbol is (P, o).
Q 2 deareseptiembre
set de 2011
as follows: Qsystem.
and
viernes
33 ! 1
Co-ordinate transformation
Transformation of the coordinates of a point X(x,y,z):
(X’)=(P,p)-1(X)
=(P-1, -P-1p)(X)
x’
y’
z’
=
(
P11 P12 P13 p1
P21 P22 P23 p2
P31 P32 P33 p3
special cases
-origin shift (P=I):
-change of basis (p=o) :
Transformation of symmetry operations (W,w):
(W’,w’)=(P,p)-1(W,w)(P,p)
Transformation by (P,p) of the unit cell parameters:
metric tensor G:
G´=Pt G P
)
-1
x
y
z
Symmetry Database
Problem: Coordinate transformations
Generators and General positions
Coordinate
Transformations
ITA-settings
symmetry data
ITA-settings
symmetry data
ITA-settings
Symmetry Data of Pn21m
(standard setting Pmn21)
Coordinate
Transformations
(W’,w’)=(P,p)-1(W,w)(P,p)
Coordinate Transformations
(W’,w’)=(P,p)-1(W,w)(P,p)
det P>1
additional
symmetry
operations
t(1,0,0)
b
a
2a
Symmetry Database
transformations
Problem: Coordinate
Wyckoff positions
Coordinate
Transformations
ITA-settings
symmetry data
Transformed
Symmetry Data
X’=(P,p)-1X
Transformed Symmetry Data
New Parameter Presentation
Problem: Coordinate transformations
Generators
General positions
GENPOS
space group
Transformation
of the basis
ITA-settings
symmetry data
Example GENPOS:
default setting C12/c1
(W,w)A112/a=
(P,p)-1(W,w)C12/c1(P,p)
final setting A112/a
Example GENPOS: ITA settings of C2/c(15)
default setting
A112/a setting
Problem: Coordinate transformations WYCKPOS
Wyckoff positions
space group
Transformation
of the basis
ITA
settings
EXERCISES
Problem 1.6
Consider the space group P21/c (No. 14). Show that the
relation between the General and Special position data of
P1121/a (setting unique axis c ) can be obtained from the data
P121/c1(setting unique axis b ) applying the transformation
(a’,b’,c’)c = (a,b,c)bP, with P= c,a,b.
EXERCISES
Problem 1.7
Use the retrieval tools GENPOS or Generators and General
positions, WYCKPOS (or Wyckoff positions) for accessing
the space-group data on the Bilbao Crystallographic Server or
Symmetry Database server. Get the data on general and
special positions in different settings either by specifying
transformation matrices to new bases, or by selecting one
of the 530 settings of the monoclinic and orthorhombic
groups listed in ITA.
Consider the General position data of the space group Im-3m
(No. 229). Using the option Non-conventional setting obtain
the matrix-column pairs of the symmetry operations with
respect to a primitive basis, applying the transformation
(a’,b’,c’) = 1/2(-a+b+c,a-b+c,a+b-c)

viernes 2 de septiembre de 2011 - Bilbao Crystallographic Server

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