Transcript Pauling`s rules File

Radius ratio
In ionic solids, cations try to maximize the number of neighboring anions. The maximum
number of
anions which have simultaneous contact to the central cation depends on the ratio between
the two ion radii.
Example:
Lower limiting radius ratio for triangular coordination C.N. 3 :
minimum radius ratio for
triangular coordination:
RA
R
= 1 + A = cos 30°
RA + Rc
Rc
RC
RA
RC
= (cos30 )-1 -1
RA
RC
= 0.155
RA
RA + Rc
For
RC
RA
< 0.155 => “rattling” cation
Coordination polyhedra
The number of nearest neighbors of an ion is called coordination number C.N. Usually the
coordination numbers for cations are given. For example, the titanium cation in rutile
(TiO2) is bonded to six oxygens e.g. its coordination number is 6. The lines connecting
nearest oxygen atoms depicts an polyhedron, which has the shape of an octahedron. The
titanium is, therefore, coordinated octahedrally. In representation of crystal structures,
instead of showing each individual atom, coordination polyhedra are shown:
The rutile structure depicted atom by
atom. The cluster consisting of the 6
nearest oxygens to every titanium
atom
describes
an
octahedron.
Replacing all the clusters by such
octahedra (= polyhedral representation) facilitates the "reading" of
the structure (top)
The rutile structure
Juxtaposition of the rutile structure projected down the c-axis: atom by atom with
the the ion radii at right scale (left) and the polyhedral representation (right).
http://www.uwsp.edu/geo/projects/geoweb/participants/dutch/
Coordination polyhedra II
Coordination polyhedra
Pauling rules I
Pauling’s Rules make predicition about the
arrangement of anions and cations in a ionic
structure:
1.
Pauling
Polyhedra
Rule:
Coordination
A coordination polyhedron of anions is formed
around every
cation (and vice-versa) - a
cation will try to be in simultaneous contact
with
the maximum number of anions.The
maximum number = probable coordination
number (coordination polyhedron) is given by
the ratio rule:
Rc /Ra:
< 0.15
0.15 -0.22
0.22 - 0.414
0.414 - 0.723
0.723 - 1.0
> 1.0
: pair
: triangle
: tetrahedron
: octahedron
: cube
: cuboctahedron
Pauling rules II
2. Pauling Rule: Electrostatic Valence Principle : “ Bond Strength”
In a stable ionic structure the charge on an cation is balanced by the sum of the electrostatic
bond strengths to the anions in the coordination polyhedron, i.e. a stable ionic structure must
be arranged to preserve local electroneutrality.
Electrostatic bond strength (e.b.s) of a M (cat.) - X (an.) bond
Mm+ coordinated by n Xx-
=> e.b.s of M:
m
n
The second Pauling‘s rule is followed when
Xx- coordinated by p Mm+
=> p
x
m
n = x
Prewitt's addendum: Given that the chemical formula for a crystal is charge
balanced, then the sum of the coordination numbers of the cations must equal the
sum of the coordination numbers of the anions.
Pauling rules III
Example for the second Pauling rule: Rutile TiO2
4/6
4/6
4/6
4/6
Each titanium ion Ti4+ (charge = +4) is bonded Each oxygen ion O2- (charge = -2) is bonded to
to 6 oxygens ions (O2- ), cation coordination
3 titanium ion Ti4+ , anion coordination
number n = 6
number p = 3
bond strength of one Ti4+ - O2- bond: 4/6
Pauling second rule:
m
p * n
bond strength of one Ti4+ - O2- bond: 4/6
= x
4
3 * 6
= 2
fullfilled!
Pauling‘s second rules allows predictions of how individual polyhedra are linked together.
Bond valence
Although Pauling‘s second rule works well for rutile, it is not a general rule. Bond strength
does not only depend on CN and the charge of the ions but also on bond length. Brown
and Shannon (1973) have derived a semi-empirical expression for the bond strength that
does take into account the bond length:
n
 R 
S  S0 
R0 
S0, R0 and n are characteristic of each cation – anion pair. Universal bond valence curves
have been given by Brown (1981). The variables R0 and n are characteristic for
isoelectronic cations.For cation – oxygen pairs the parameters are:
Cations
No of el.
R1 (A)
n
H+
0
0.86
2.170
Li+,Be2+,B3+
2
1.378
4.065
Na+,Mg2+,Al3+,Si4+,P5+,S6+
10
1.622
4.290
K+,Ca2+,Sc3+,Ti4+,V5+,Cr6+
18
1.799
4.483
Mn2+,Fe3+,
23
1.760
5.117
Zn2+,Ga3+,Ge4+,As5+
28
1.746
6.050
Pauling rule IV
3. Pauling Rule: Polyhedral Linking
The stability of different polyhedral linkings is
corner-sharing > edge-sharing > face-sharing
- effect is largest for cations with high charge and low coordination number.
- especially large when r+ /r- approaches the lower limit of the polyhedral stability.
corner sharing
Why?
edge sharing
face sharing
- Sharing edges/faces brings ions at the centre of each polyhedron closer together,
hence increasing electrostatic repulsions.
- Sharing edges/faces lowers the screening of the negative charges
lower cation
not visible
lower cation
visible
lower cation
visible
Pauling rules V
4. Pauling Rule: Cation Evasion In a crystal containing
different cations those of high valency and small coordination
number tend to share the mininum number of polyhedral elements
with each other.
Example: Perovskite structure CaTiO3
Ti4+ cation in octahedral (6) coordination
=> corner shared
5. Pauling Rule: Environmental Homogeneity
Ca2+ cation in cuboctahedral (12) coordination
=> face shared
The number of essentially different kinds of constituent elements in a
crystal tend to be small.
The rocksalt structure
Stoichiometry:
Ion charge:
NaCl
Na: +1 (m)
Cl: -1 (x)
Each sodium cation
is coordinated by 6
(n) chlorine anions.
The octahedra
share edges.
Each chlorine anion
is coordinated by 6
(p) sodium atoms.
sodium cation
chlorine anion
Pauling rules:
1. rule fulfilled
Ion radius:
Na+(IV): 0.99Å
Na+(VI): 1.02Å
Na+(VIII): 1.18Å
Cl-: 1.81
Radius ratio: IV: 0.54
VI: 0.56
VIII:0.65
2. rule fulfilled
e.b.s of Na - Cl bond: 1/6
C.N. of anion: 6
6 * 1/6 = anion charge
3. rule not fulfilled: octahedra share edges
too large for four-fold coord.
in the range for six fold coord.
too small for eight-fold coord.
Minerals with rocksalt structure: uraninite KCl, MgO
The fluorite structure CaF2
Cation distribution in a
The fluorine anions are cube layer of the fluorite
Perspective view of Calcium cations are
coordinated by 8
coordinated by 4 calcium structure. The filled cubes
two unit-cells
share edges with each
fluorine anions.
cations.
Ca
other.
The fluorine cube adjacent
F
to the first one is empty.
Cations are coordinate by 8 anions forming a cube.
Anions are tetrahedrally coordinated.
Pauling’s rules:
1. rule: does predict cube coordination for calcium.
2. rule: e.b.s of Ca - F bond: 2/8 = 1/4, C.N. of anion: 4
3. rule: not fulfilled!
Minerals with fluorite structure: uraninite UO2
4* 1/4 = 1 fulfilled!
The perovskite structure I
B-cations are coordinated by
A-cation 12-fold coordinated
6 oxygen anions
B-cation, octahedrally coord.
Oxygen
A1+:
2+:
3+:
cations:
Na, K
Ca, Sr, Ba, Pb
La, Y
B- cations:
5+: Nb
4+: Ti, Zr, Sn, Ce, Th, Pr
3+: Al, Fe, Cr
A-B cation radii relationship:
RO  RA  t  2RO  RB 
t: tolerance factor, for ideal perovskite structure t = 1.0
for t 0.7 - 1.0 perovskite structure can be expected,
but slightly distorted.
A-cations are coordinated by
12 oxygen anions
The perovskite structure II
Pauling’s rules:
1.rule
Ion radius: Ca2+(X): 1.23Å
Ca2+(XII): 1.34Å
Radius ratio: Ca/O: X:0.87
XII:0.95 too small
but close
Ti4+(IV): 0.42Å
Ti4+(VI): 0.605Å
Ti/O:
VI: 0.43 in the range for six-fold coord.
VIII:0.528 too small for eight-fold coor.
2. rule: e.b.s of Ca - O bond: 2/12 = 1/6
e.b.s of Ti - O bond: 4/6
Coordination of oxygen: 2 Ti and 4 Ca cations => 2*4/6 + 4*1/6 = 2
3. and 4. rule: partially fulfilled
Highly charged, small Ti is in
corner sharing octahedra
Low charged, large Ca
is in face-sharing
cuboctahedra
Minerals with perovskite structure: perovskite CaTiO3, p MgSiO3 under mantel conditions
Structural mapping
Sorting of structures based on ionic radii and other parameters such as ionicity,
electron negativity etc.
Structural map as function
of radius ratios for AB
compounds (pm: picometer).
Structural map as function
of radius ratios for A2BO4
compounds.
Packing of spheres I
1. Dense sphere packings in 2-D
a0
a0
tetragonal dense packed
hexagonal dense packed
2. Dense sphere packings in 3-D
Stacking of tetragonal dense packed layers
A-layer
A-layer
B-layer
Projection
A-layer
2. layer on top of 1. layer 2. layer displaced by a /2
0
cubic primitive stacking
interstitial void: cube
Projection
A-layer
3. layer = 1.
cubic body centered lattice
interstitial voids: octahedra
Packing of spheres II
Stacking of hexagonal dense packed layers
Two sets
of interstitials:
blue and green
The spheres of the next
layer can either beplaced in
the green or in the blue
interstitials.
Hexagonal closest packed (hcp)
Cubic closest packed (ccp) array of spheres
1. layer A
1. layer A
2.layer B in
blue interstitials
2.layer B in blue
inter stitials
3. layer A on top
of first layer
3.layer C in green
interstitials
4. layer on top of 1.
layer
Packing of spheres III
Shape of interstitial voids
B-layer
C-layer
A-layer
tetrahedron
[111] direction is perpendicular to
closed packed chlorine layers
octahedron
The rocksalt structure: Chlorine forms a ccp
array and sodium (black circles) fills all octahedral
voids