Useful links for further learning:
(1) Allen Hunter's YSU X-Ray Structure Solution Guide: A Beginner's Introduction
http://www.as.ysu.edu/~adhunter/YSUSC/index.html
Pdf file with hints + simple examples to work with
(2) IUCr teaching pamphlets (www.iucr.org => Education)
(3) SHELX manual (http://shelx.uni-ac.gwdg.de/SHELX/index.html)
(4) Acta Crystallographica journals (www.iucr.org)
(5) Disorder - tutorial:
http://shelx.uni-ac.gwdg.de/~peterm/tutorial/disord.htm
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RES/INS file
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Model evaluation=> important values:
R-factor
wR2
S (goodness of fit)
temperature factors
difference Fourier map (highest peak/deepest hole)
Rint
Reasonable bond lengths/bond angles, special attention to short contacts
Examine most disagreeable reflections and bad equivalents
Check for any signs of pseudosymmetry (use PLATON software!)
Check largest correlation matrix elements
Make sure your refinement converged
Always use updated (IUCr website) CHECKCIF software!
Always check all numerical values! Always bear in mind that change in one value changes
also the other values!
Rint
R=
∑F
=
− Fo2 ( average)
2
o
∑
∑ (F
2
o
)
Fo − Fc
∑F
o
[(
) ]
[ ( ) ] 
 ∑ [w(F − F ) ]
S =

 ∑ w F 2 − F 2
o
c
wR 2 = 
2 2
 ∑ w Fo
2
o

(n − p )
2 2
c
2
1/ 2
1/ 2

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751
Acta Cryst. (1976). A32, 751
Revised Effective Ionic Radii and Systematic Studies of Interatomie Distances
in Halides and Chaleogenides
BY R. D. SHANNON
Central Research and Development Department, Experimental Station, E. L Du Pont de Nemours
and Company, Wilmington, Delaware 19898, U.S.A.
(Received 30 October 1975; accepted 9 March 1976)
The effective ionic radii of Shannon & Prewitt [Acta Cryst. (1969), B25, 925-945] are revised to include
more unusual oxidation states and coordinations. Revisions are based on new structural data, empirical
bond strength-bond length relationships, and plots of (1) radii vs volume, (2) radii vs coordination
number, and (3) radii vs oxidation state. Factors which affect radii additivity are polyhedral distortion,
partial occupancy of cation sites, covalence, and metallic character. Mean NbS+-O and Mo6+-O
octahedral distances are linearly dependent on distortion. A decrease in cation occupancy increases
mean Li+-O, Na+-O, and Ag+-O distances in a predictable manner. Covalence strongly shortens
Fe2+-X, Co2+-X, Ni2+-X, Mn2+-X, Cu+-X, Ag+-X, and M - H - bonds as the electronegativity of X
or M decreases. Smaller effects are seen for Zn2+-X, Cd2+-X, In3+-X, pb2+-X, and TI+-X. Bonds with
delocalized electrons and therefore metallic character, e.g. Sm-S, V-S, and Re-O, are significantly
shorter than similar bonds with localized electrons.
Introduction
Procedure
A thorough and systematic knowledge of the relative
sizes of ions in halides and chalcogenides is rapidly
being developed by crystal chemists as a result of (1)
extensive synthesis within certain structure types, e.g.
rocksalt, spinel, perovskite and pyrochlore; (2) preparation of new compounds with unusual oxidation states
and coordination numbers; and (3) the abundance of
accurate crystal structure refinements of halides, chalcogenides, and molecular inorganic compounds. A set
of effective ionic radii which showed a number of
systematic trends with valence, electronic spin state,
and coordination was recently developed (Shannon &
Prewitt, 1969, hereafter referred to as SP 69). This work
has since been supplemented and improved by studies
of certain groups of ions: rare earth and actinide ions
(Peterson & Cunningham, 1967, 1968); tetrahedral
oxyanions (K~ilm~in, 1971); tetravalent ions in perovskites (Fukunaga & Fujita, 1973); rare earth ions
(Greis & Petzel, 1974); and tetravalent cations (Knop
& Carlow, 1974).
Further, the relative sizes of certain ions or ion pairs
were studied by Khan & Baur (1972)" N H + ; Ribbe &
Gibbs (1971): O H - ; Wolfe & Newnham (1969):
Bi3+-La 3+; McCarthy (1971): Eu2+-Sr2+; Silva,
McDowell, Keller & Tarrant (1974): No 2+. These
authors' results have been incorporated here into a
comprehensive modification of the Shannon-Prewitt
radii.
In this paper the revised list of effective ionic radii,
along with the relations between radii, coordination
number, and valence is presented. The factors responsible for the deviation of radii sums from additivity
such as polyhedral distortion, partial occupancy of
cation sites, covalence, and metallic behavior (electron
delocalization) will be discussed.
The same basic methods used in SP 69 were employed
in preparing the revised list of effective ionic radii
(Table 1). Some of the same assumptions were made:
(1) Additivity of both cation and anion radii to reproduce interatomic distances is valid if one considers
coordination number (CN), electronic spin, covalency,
repulsive forces, and polyhedral distortion.*
(2) With these limitations, radii are independent of
structure type.
(3) Both cation and anion radii vary with coordination number.
(4) With a constant anion, unit-cell volumes of isostructural series are proportional (but not necessarily
linearly) to the cation volumes.
Other assumptions made in SP 69 have been modified:
(1) The effects of covalency on the shortening of
M - F and M - O bonds are not comparable.
(2) Average interatomic distances in similar polyhedra in one structure are not constant but vary in a
predictable way with the degree of polyhedral distortion (and anion CN). Both of these modified assumptions will be discussed in detail later.
The anion radii used in SP 69 were subtracted from
available average distances. Approximately 900 distances from oxide and fluoride structures were used,
and Table 2 lists their references according to CN and
spin. These references generally cover from 1969 to
1975. The cation radii were derived to a first approximation from these distances, and then adjusted to be
consistent with both the experimental interatomic distances and radii-unit cell volume (r 3 vs V) plots, as in
* Polyhedral
distortion
was not considered
A C 32A - 1
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in SP 69.
752
REVISED
EFFECTIVE
IONIC
RADII
IN
SP 69. Although such r a v s V plots are not always
linear (Shannon, 1975), their regular curvilinear nature
still allows prediction of radii. This system is particularly accurate for radii in the middle of a series, and
least reliable for large polarizable cations like Cs +,
Ba z +, and T13 +. Radii-volume plots were used by Knop
& Carlow (1974) and Fukunaga & Fujita (1973) to
derive radii of tetravalent cations. These radii were
used along with experimental interatomic distances in
deriving the final radii. Greis & Petzel (1974) derived
rare earth radii in eight- and nine-coordination using
accurate cell dimensions for rare earth trifluorides and
distances calculated using the structural parameters
of YF3 and LaF3. These radii were used in Table 1
after applying small corrections ( + 0.030 ,~ to lXLa3+,
IXCe3+, 'Xpr3+, and ~XNd3+; +0.025 A to all other
Greis & Petzel ~XRE3+ radii, and 0.015 A to all
HALIDES
AND
CHALCOGENIDES
VI"RE3+ radii) for consistency with experimental interatomic distances and radii-CN plots.
Where structural data were not available or not accurate, plots of (1) radii v s unit cell volumes, (2) radii
v s CN and (3) radii v s oxidation state, or combinations
of these were used to obtain estimated values. Fig. 1
shows examples of radii-valence plots used to provide
consistency between experimental radii and those anticipated from the regular nature of these plots. Cations
whose final radii values were derived from both
estimated values and experimental interatomic distances a r e : V l O s S + ,
VIOs6+,
VIOs
7+,
VlRe4+, VIRES+,
VlRe6+, VIReT+, VIRh4+,
vI1U4+,
VIIUS+,
and
VIIU6+.
Fig. 2(a)-(e) shows plots of radii v s C N . Generally,
it was assumed that radii-CN plots for two different
ions do not cross. Radii for 'VCu+, V'Cu+, IXRb+,
VNi2+, VIIEr3+' vIIyb3+ ' WITb3+ ' X.Nd3+ ' IVCr4+'
Table 1. Effective ionic radii
CR crystal radius, IR effective ionic radius, R from P vs V plots, C calculated, E estimated, ? doubtful, * most reliable, M from
metallic oxides.
ION
EC
CN
AC÷3 6P 6 V l
A G ÷ I 4D10 11
IV
VSQ
~
AGed 60 9
AGe3 60 8
AL*3
2P 6
AM*2 5F 7
AM÷3 5F 6
AM÷~ 5F 5
AS+3 4S 2
AS+$ 3D10
AT÷7 5DIO
AU÷I 5 0 1 0
AUe3 50 8
AU÷5 50 6
d +J 1S 2
8A÷~ 5P 6
BEe2 15 2
Ble3
65 2
vI
Vll
Vlll
IVSO
Vl
IVSQ
Vl
IV
v
Vl
Vll
Vlll
IX
Vl
VIII
Vl
Viii
Vl
lV
vl
Vl
Vl
IVSQ
Vl
Vl
Ill
I¢
Vl
Vl
Vii
Viil
IX
X
Xl
XI!
Ill
IV
l
~
Vl
Vlll
81÷5 5 0 1 0 V l
8K+3 5F 8 Vl
b K ÷ 6 5F 7 V l
VlIl
8 R - I 6P 6 V l
8 R e 3 6P 2 IVSO
8R+5 45 2 I I I P Y
8R÷7 3010 IV
Vl
c e 4 15 2 l l I
IV
VI
CA+2 3P 6 Vl
•
Vll
Vlli
x
~
Xll
CDeZ 6 0 1 0 I V
v
Vl
Vli
Vlll
Xll
CE+3 65 1 V l
Vll
Vlll
IX
II
CE÷4 5P 6 Vl
VIll
II
C F * 3 bD I VI
C F * 6 5F 8 V l
5P
Ck
1.26
.81
1.14
1.16
1.23
1.29
1.36
1.42
.93
1.08
.81
.89
• 53
.62
.675
1.35
1.40
1.65
1.115
1.23
.g9
l.og
.72
.675
.60
.76
1.51
.82
.g9
.71
.15
.25
.41
1.49
1.52
1.56
1.61
1.66
1.71
h75
.30
.41
.59
1.10
1.17
1.31
.go
1.1o
.97
1.07
1.82
.73
.65
.39
.53
.06
.29
.30
1.14
1.20
1.26
1.32
1.37
1.68
.92
1.01
1.09
1.17
1.24
1.45
1.15
1.21
1.283
1.336
1.39
1.48
1.Ol
1.11
1.21
1.28
l.og
.961
*IR*
1.12
R
.67
1.00
C
1.02
1.09
C
1.15
c
1.22
1.28
.79
.94
.67
.75
R
.39
.68
.535
*
1.21
1.26
1.31
.g75 R
1.09
.85
R
.g5
.58
A
. 3 3 5 Re
.46
C*
.62
A
1.37
A
.68
.85
A
.57
.01 *
.11
*
C
.2T
1.35
1.38
C
1.42
"
1.47
1.52
1.57
1.61
C'
.16
.27
*
. 6 5 'C
.96
C
1.03
Re
1.17
R
.76
E
.96
R
.83
R
.93
R
p
l.Vb
.59
.31
.25
.39
A
-.08
.15
P
.16
A
1.oo
1.06
*
1.12
•
1.18
1.23
1.34
.78
.87
.95
1.03
c
¢
1.10
1.31
1.01
R
1.07
E
1.163 R
1.196 R
1.25
1.36
C
.87
R
.97
R
1.07
1.16
.95
R
.821R
ION
EC
CN
C L - I 3P 6 V l
CL+b 35 2 l l l P Y
CL÷7 2P 6 I V
Vl
C~÷3 5F 7 V l
GHe6 5F 6 V l
VIll
CO÷2 3D 7 I V
V
Vl
Vlll
C0÷3 30 6 V l
GO÷4 3D 5 IV
Vl
CR+2.3D 4 VI
SP
HS
L5
HS
kS
HS
HS
L$
HS
CR+3 3D 3 V l
CR÷4 30 2 IV
Vl
Cg÷5 30 1 I V
Vl
VIll
eRe& 3P 6 IV
Vl
c $ + 1 5p 6 V I
Vlll
lx
X
Xl
Xll
CUe1 3010 11
IV
VI
CUe2 30 9 IV
IVSQ
1
CU*3 30 8 Vl
0 +1 15 0 I I
0Y÷2 6 F I O Vl
vI I
Vl I
UYe3 6F 9 Vl
VII
Vlll
IX
E&e3 4 F 1 1 V l
Vl 1
Vl I
IX
EU÷2 4F 7 V l
Vll
VIII
IX
x
EUe3 4F 6 V i
Vll
Vlll
IX
F - 1 2P 6 i l
Ill
IV
Vl
F e7 I S 2 V l
FEe2 30 6 IV
IVSQ
Vl
VIII
FE+3 30 5 I V
v
VI
Vlll
Vl
IV
Vl
IV
V
Vl
GD÷3 ~F ? V l
FEe4
FEe6
FR*I
GAe3
3D 4
30 2
6P 6
3010
kS
HS
H$
LS
HS
HS
H$
LS
HS
HS
CR
1.67
.26
.22
,61
1.11
.99
1.09
.72
.81
.79
.885
1.06
.685
.75
.56
.67
.87
.94
.755
.55
.69
.485
.63
.71
.40
.58
1.81
1.88
1.92
1.95
1.g9
2.02
.60
.76
.91
.71
.71
.79
.87
.68
.04
1.21
1.27
1.33
1.052
1.11
1.167
1,223
1.030
1.085
1.144
1.202
1.31
1.34
1.39
1.44
1.49
1,087
1.15
1.206
1.260
1.145
1 .16
1.17
1,19
.22
.77
.78
.75
.920
1.06
.63
.72
.69
aT05
.92
.725
.39
1.9~
.61
.69
.TbO
1.078
ION
fIR'
1.81
.12
*08
,27
.97
.85
.95
.58
.67
.65
.765
.90
.565
.61
.40
.53
.73
.80
.615
.41
.55
.365
.69
.$7
.26
.44
1,67
1.76
1.78
1.81
1.85
1.88
.46
.60
.77
.57
.57
.65
.73
.56
-.10
1.07
1.13
1.19
.g12
.97
1.027
1.083
.ago
.9~5
1.004
1.062
1.17
1.20
1.25
1.30
1.35
.g6~
1.01
1.066
1.120
1.285
1.30
1.31
1.33
.08
.63
.66
.61
.780
.g2
.49
.58
.55
.665
.78
.585
.25
1.80
.47
.55
.620
.938
P
•
A
R
R
R
C
R
Re
R•
R
E
R
R•
R
R
ER
c
EG
CN
SP
G 0 " 3 4F 7 V l l
VIII
IX
GE÷2 45 2 V I
GE*q 3010 I V
Yi
H e l 1S 0 I
II
H F * 4 "6F16 I V
VI
VII
VIII
HG÷I 6S 1 I l l
VI
HGe2 5010 I |
IV
VI
rill
HOe3 6 F 1 0 VI
VIll
IX
X
I :~ 8. )
v1
2.0.
5S
IIIPY
VI
1 ÷7 4D10 1V
Vl
IN÷3 6 0 1 0 I V
.58
l.Og
.56
.67
.76
Vl
E
E
IR+J
IR*4
IR÷5
K "1
50
50
50
3P
6
5
4
b
*
*
,6o
Vlll
Vl
Vl
Vl
IV
Vl
VII
VIII
x
1.06
.82
.765
.71
1.51
1.52
1.60
1.65
1.69
1.73
1.78
1.172
1.24
1.300
1.356
1.41
1.50
.730
.90
1.06
1.ooi
1.117
1.172
.71
~
Xll
LAe3 4 0 1 0 V l
Vl!
vllI
X
I
R
R
g
R
R
R
xli
IS 2 IV
Vl
VIII
LU*3 4F14 Vl
Vlll
1X
MG+2 2P 6 IV
Ll÷l
il
R
• 860
ll
HN÷2 30 5 I V
V
Vl
R
R
Vll
lll
MN÷3 30 6
~
Vl
A
E
Re
c
*
R
R•
R
R
A
*
R*
R
HN÷6 30 3 l V
Vl
MN÷5 30 2 I V
MNe6 30 l IV
MNe7 3P b I V
Vl
MO+3 40 3 Vl
MO*~ 40 2 Vl
~ o e 5 6D I IV
Vl
MO÷b 4P 6 l v
•
v
VI
Vll
"
:l
2,
) iv
2S
Vl
N
N ÷5 I S 2 I l l
Vl
CR
1,14
1,193
1,247
,87
.530
*670
--,26
-.06
.72
.85
.90
.97
1.11
1.33
*83
1.10
1.16
1.28
1.041
1,155
1.212
1.26
HS
H5
LS
H$
HS
L$
HS
1.03
......
.80
.89
.81
.970
1.04
1.10
.72
.72
.785
.53
.670
.67
.395
.3g
.60
.83
.790
.60
.75
.55
.64
.73
.87
*IR*
1,00
1,053
10107
.73
,390
*530
-,38
-.18
.58
.71
.76
.83
R
R¢
A
•
R•
:
,97
1.19
.69
.96
1.02
1.16
.901
1,015
1.072
1,12
220
.66
*g5
.62
k
R
R
R
,.
.53
.62
8oo.
Re
*g2
.68
E
.625 R
.57
E
1.37
1.38
1.66
1.51
1.55
1.$9
1.66
1.032 R
I.IO
l.lbO R
1.216 R
1.27
1.36
G
.590
.7&
•
.92
C
.8bl R
.977 R
1.032 R
.57
.720
.89
.66
.75
.67
.830
.90
.9b
.58
.58
.665
.3g
.530
.33
.255
.25
.66
.69
.650
.46
.61
.61
.50
.59
.73
1.32
.30
....
.044
.27
-.106
.16
.13
*
G
C
E
Re
C
R
R
Re
R
Re
R
A
RH
R
R
Re
R*
A
A
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753
R. D. S H A N N O N
Table 1 (cont.)
EC
ION
NA*I
2P 6 I V
V
VI
VlI
VIII
IX
XIl
NB+J 4 0 2 V l
N 8 , 4 4 0 1 Vl
VIII
N 8 , 5 4P 6 I V
Vl
VII
VIII.
ND+Z 6F 6 VIII
IX
N O + ~ 6F 3 V l
VIII
IX
XlI
N I ÷ 2 3 0 8 IV
IVSQ
v
VI
N I * J 3 0 7 Vl
NI÷4
NO+2
NP÷2
NP*3
NP+6
NP+5
NPeb
NP+7
o -2
OH-I
0S+6
0S+5
0S.0
05"7
0S,8
p .3
p *5
PA+3
PAt4
PA~5
P8+Z
PB*6
p0÷l
PD÷2
PO•J
POe6
P~*3
PO÷4
POeb
30 6
5F16
5F 5
5F 6
5F 3
Vl
VI
VI
VI
V[
Viil
5F 2 VI
5F 1 V I
6P 6 V I
2P b I f
Ill
IV
VI
VIII
11
Ill
IV
Vl
5D 6 VI
5 0 3 Vl
50 2 V
Vl
50 1Vl
5P 6 I V
3S 2 vl
2P 6 I V
v
Vl
5F 2 Vl
6D 1 V l
• viii
bP 6 Vl
VIII
IX
6 5 2 IVPY
VI
VII
VIII
IX
x
XI
XIl
5 0 1 0 IV
V
Vf
VIII
6 0 9 II
4 0 0 IVSO
Vl
60 7 VI
40 6 Vl
6F 4 V l
Viii
IX
65 2 VI
rill
5010 VI
LS
HS
LS
1.13
1.14
1.16
1.26
1.32
1.38
1.53
.86
.82
.93
.62
.78
.83
.88
1.63
1.49
1.123
1.249
1.303
1.61
.69
.63
.77
.830
.70
.74
.62
1.26
1.26
1.13
1.oi
1.12
.89
.86
.85
1.21
1.22
1.26
1.26
1.28
1.18
1.20
1.21
1.23
.770
.715
.63
.685
.665
.53
.58
.31
.43
.52
1.18
1.06
1.15
.92
1.05
1.09
1.12
1.33
1.37
1.63
1.69
1.54
1.59
1.63
.79
.87
.915
1.08
.73
.70
1.oo
.90
.755
1.11
1.233
1.286
1.08
1.22
.81
.99
1.00
1.02
I.IZ
1.18
1.26
1.39
.72
.68
.79
.68
.66
.69
.76
1.29
1.35
.983
1.109
1.163
1.27
.55
.69
.63
.690
.56
.60
.68
1.I
1.10
1.01
.87
.98
.75
.72
.71
1.35
1.36
1.38
1.40
1.62
1.32
1.36
1.35
1.37
.630
.575
.49
.545
.525
.39
.46
.17
.29
.38
1.0¢
.90
1.01
.70
.91
.95
.98
1.19
1.23
1.29
1.35
1.40
1.45
1.49
.65
.73
.775
.96
.59
.66
.86
.76
.615
.97
1.093
1.146
.96
1.08
.07
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RE
C
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R•
E
R
E
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R
R
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~o.05~vi
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Viii
IX
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PT+2 5 0 8 IVSQ
VI
PT+4 5D 6 V [
PT+5 5 0 3 V l
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vl
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v
vl
$8÷5 4010 Vl
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VIII
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vl
$H+2 6F 6 VIl
Vl I I
IX
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Vll
VIII
IX
Ill
SN*4 4010 IV
58+3
E
E
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E
E
E
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C
e
g
c
C
C
c
C
C
E
E
R
R
VII
VIII
SR*2 4P 6 V l
VII
VIII
IX
~lI
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TA*4
TA*5
R
R
k
g
R
R
A
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50 1 Vl
5P 6 V l
VII
VIII
T 8 + 3 4F 8 VI
VII
VIII
IX
T 0 . 6 4F 7 Vl
Vill
vIIIV4+, IVpb4+, and XTh4+ obtained from these plots
were used to help determine the values in Table 1. The
first estimate of vIHV4+ was made from distances in
C32H28SsV (Bonamico, Dessy, Fares & Scaramuzza,
19741.
Another method used to estimate radii was based on
the empirical relationship between interatomic distances and bond strengths. Brown & Shannon (1973)
derived these relationships for the cations in the first
three rows of the periodic table from a large number of
experimental interatomic distances. These curves can
be used to calculate hypothetical distances for cations
in any coordination (Brown & Shannon, 19731 Shannon, 19751 Brown, 1975). Examples of cations whose
radii were calculated in this way are: lVMn2+, V[Be2+,
VtB3+ ' wps+, v l S 6 + ' V m M g 2 + ' and VmFe 2+. These are
marked with a C in Table l. In certain cases, these
values were combined with known structural data (see
Table 2) to obtain the radii in Table 1. Although the
CR
ION
eIRt
1.13
1.266
1.319
.99
I.IO
.74
.99
.765
,71
1.14
1.oo
I,IO
.88
.99
1.126
1.179
.85
.96
.60
.80
.625
.57
1.00
.86
.96
.76
.85
.71
1,62
1.86
1.66
1o70
1.75
1.77
1.80
1.83
1.86
1.97
.77
.72
.69
.52
.67
.805
.74
.69
,82
.760
.705
.52
.50
1.48
1.70
1.52
1.56
1.61
1.63
1.66
1.69
1.72
1.83
.63
.58
.55
.38
.53
.665
.60
.55
.68
.620
.565
.38
.36
.51
.26
.43
.90
.96
.90
.76
.885
1.OLO
1.84
.64
.62
.56
.40
.540
1.36
1.41
1.46
1.098
1~16
1.219
1.272
1.38
.69
.37
.12
.29
.76
.80
.76
.60
.765
.870
1.98
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.28
.62
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1.22
1.27
1.32
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1.079
1.132
1.24
.55
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.09
.95
1.32
1.35
1.40
1.65
1.50
1.58
.86
,82
.78
.83
.88
1.063
1.12
1.180
1.235
.90
1.02
.690
.75
.81
1.18
1.21
1.26
1.31
1.36
1.46
.72
.68
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.69
,76
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.98
1.040
1.095
.76
.88
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TC+4 4 0 3 V I
TG*5 40 2 Vl
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Vl
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Vl
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Vl
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r l + ~ 3P 6 I V
v
Vl
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1 L * 1 6S 2 v l
VIII
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T L + 3 5D10 I V
Vl
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TM+2 4 F 1 3 V I
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TM+) k F l 2 V I
viii
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viii
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vii
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v +4 3 0 I v
Vl
VIII
V +5 3P 6 I V
v
Vl
w +~ 50 2 V I
w +5 50 1Vl
w +6 5P 6 IV
v
vl
XE+8 4D10 IV
vl
Y +J 4P 6 v I
VII
Vill
IX
YB+2 4 F 1 4 VI
Vll
VIII
Y8+3 4F13 Vl
Vll
Vlll
IX
I N * 2 3 0 1 0 IV
v
vI
VIII
Z R + ~ 6P 6 I V
V
Vl
Vll
VIII
IX
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CR
fiR*
.783
.74
.51
,70
2.07
066
.80
1.11
.57
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1.08
1+19
1.23
1,27
1,32
1,35
1.00
.810
.$6
.65
.745
.88
1.64
1.73
1.84
.89
1.025
1,12
1.17
1.23
1.020
1.13~
1.192
1,165
1.03
1.09
1.14
1.19
1,31
.90
.98
.59
.66
.87
.95
1.00
.93
.780,
.67
.72
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.68
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.76
.56
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.54
.62
1.040
1.10
1.159
1.215
1.16
1.22
1.28
1.008
1.065
1.125
1.182
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.82
.880
1.04
.73
.80
.86
.92
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1.03
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.37
.56
2.21
.52
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097
.43
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,94
1.05
1.09
1,13
1.18
1.21
.86
.670
.42
.51
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.74
1.50
1.59
1,70
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.98
1,03
l.Oq
.880
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1.052
1,025
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1.00
1.05
1,17
.76
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.45
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.40
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1.019
1.075
1.02
1.08
1.14
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AN
ER
A
e
G
C
RC
•
E
C
C
E
R•
G
C
R•
C
a
R
RE
R
¢
R
R
R
R
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Re
E
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E
Re
R*
E
R*
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RH
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*
*,
majority of radii were derived from oxides and fluorides,* some were taken from chlorides, bromides,
iodides, and sulfides. For large electropositive cations
with highly ionic bonds, very little covalent shortening
is believed to occur and radii derived from these other
compounds should differ only slightly from those
derived from fluorides and Oxides. Examples are divalent rare earths such as Yb 2+ , Tm 2+, Dy 2+ , Sm 2+, Nd 2+
and the ions Am 2+, Ac 3÷, Np a+, and U 4÷.
Another useful scheme for estimation of radii is the
comparison of unit-cell volumes of compounds containing cations of similar size. McCarthy (1971) prepared a number 6f isotypic Sr 2÷ and Eu 2÷ ternary
oxides and generally found the unit cells of the Sr 2÷
* Because of covalency differences in M - O and M - F bonds,
oxide distances were emphasized. Therefore the radii in Table 1
are m o r e applicable to oxides than fluorides. This subject is
treated further in the discussion Effects of covalence.
A C 32A - I*
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754
REVISED
EFFECTIVE
IONIC
Table
RADII
2.
IN
HALIDES
AND
CHALCOGENIDES
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47 3409 co| pz 07
70 ZXK~A
132
1 3 2 C03 A S | 0 8
C0*Z vx
09 CJCHA
4 7 3 4 0 9 CO2 P2 07
r o ZRKK& 13Z
332 C03 AS2 0 8
6 6 SPHOA
11
11 c o w 0 4
6 7 HCACA
SO 2 0 2 3 C0Z HN3 0 8
14 J C S I A 1 9 7 4
6 1 4 C0 C4 H6 0 6
74 ACSC&
JO 188o C02 G4 H12 0 1 2
C0.2 VII
74 4CS6&
2#A
119 0 o D i e H2 c o 0 ) 2 . 3
l l Z HZ 0
C0 DeC H2 c o u ) z . 3
Ha O
7~ J C S l A 197~ t q Z z c o c ~ H 5 0 S
co*z vlxx
.
98
9 0 3 CO2 NSZ 0 7
AC8CA
25 1806 00 (N 03)2.4
D2 u
6 7 6 CO C4 H6 0 6
t 4 J C S I ~ 1976
C E * 3 vlll
74 ZADC6 4 0 3
I R) v $ v ICE F31
74 J C S I 6 1974 1 1 6 5 C41 H 2 4 CE F I 2 N De 84
CE*3 IX
67 SPH~A
lZ
2 1 4 CE e s t O5
I R3 V$ V ICE F $ )
T4 ZADCA 4 0 3
cE*3 x
6O A N N I A
45
I CE~ ME ME2 7 1 2 5 1 4 0 2 2
CE,4 vl
72 ACOCA
28
9 5 6 8R CE o )
v
131R
)CE4,)
73 JS$C8
CE.4 VIII
8
$3 1~ N 4 1 2 CE F6
JC$1A 1174 2021Ni6
CE u l O 0 3 6 H 2 . 3 0 H 2 o
74 J$TCA
l~
39t CEIS 04)2
F4 ACSAA
2 8 1079 A - CE t A C A C ) 4
C t * 4 XXX
68 JSCSA
90 3 5 8 9 I N H 4 ) 2 H6 ICE NOt2 0 4 2 1 . 1 2 H2 0
C F * 3 VX
F4 J I N C A
36 2 0 2 3 R3 VS V ICF2 15 0 4 ) 3 )
C L * 5 111
T3 NASUA
8
791 8 8 CL 0 3
CL*/ Iv
r z ACDCA
28
039 TMPO CL U4
12 ZK~XA
0~
65 X CC 0 4
6O ACL~A
13
e s 5 N o z CL 0 4 , n CL OA,
Hct
0 4 . N 2 O , C l CL 0 4 . 3 H2 O
s q JPCNA
6)
ZTe H CL O 4 . H 2 0
~8 JRCSA
0 0 8 0 / 5 C6 HD.AG CL 0 4
$7 P l S A A
56
134 N H4 CL 04
ST P l S A A
~6
143 K CL 0 4
6 2 ACCJ~R
15 1 2 0 1 N N4 CL O4
71 J C S I A 1971 1 3 t l c u ) c l o H9 N ) ) 2 COL 0 4 ) 2
7O AC~CA
2 6 1928 Ha . s CL 0 4
11 AC8C6
zt
8 9 8 H CL 0 4 . 2 112 H2 o
13 I C " D A
1 4 1 7 ¢G N I 2 - T R I E N - C U CL 0 4
11A¢DCA
ZT 8 9 8 x CL 0 4 , Z I l l
,2 0
12
7t
62
bQ
13
NRSUA
7 1281CL(*7)
-0
J C S I A 1971 1 8 5 7 C L ( * 7 )
-o
ACCRA
15
I s N3 o CL 0 4 1 - 8 0 C I
ACDCA
25 ~875 N H3 o H CL 0 4
ACSAA
2 7 2 3 0 9 )PaR 1o H 1 4 1 3 C 0 3 (CL 0 4 1 1 0
. o H2 o
73 ACSAA
273523
CU i t 3 H4 N I ) *
(CL O412
CN,~ vi
67 )NUCA
)
)27 R ICE*4)
cu*z I V
69 ZAACA
369
306 C0 V2 U4
C0,2 v
12 ACBCA
2 8 2 8 0 3 C02 PZ 0 7 ALPHA
C0,2 v!
68 ZAACA 3 5 8
125 CO SE 0 6
08 ZKKKA
120
2 9 9 CO GE 0 3
70 CJCHA
48
881 C03 A$2 0 8
70 JCPSA
5 ) 3 2 / 9 BA C0 F4
10 P E P l A
3
| 6 1 C02 SI OA
7 J AC~CA
2 9 2 ) 0 4 CO3 V2 O8
T l HCACA
54 1621 CO3 (O N ) 2 ( S 0 4 | 2 ° 2
H2 0
REF I
C02 $ ! 0 4
72 ACECA
28 2083 COl pz 07
70 INOCA
9
l S I CO ( O N P A ) 3 (CL 0 4 1 2
73 ACBCA
2 9 2 7 4 1 CO S I F 6 . b H2 0
74 A N N I A
59
4 7 5 C02 $ | 0 4
T4 JCHL8
4
55 C | 6 H 1 8 CO 0 6
CO*Z V ) I I
6 6 INOCA
S 1208 (A$(C6 HSIRI2(CO(N O3J~)
co,) Vl
LS.
6 8 CCJDA 1 9 6 8
871 c0 (N 0313
O80JCHA
4 6 3 4 1 2 C03 0 4
6 6 JACSA
8 8 2 9 5 1 C0 (CS H7 O213
74 ACUCA
30
8 2 2 C0 1C5 H7 0 2 3 3
69 JACSA
e l 6 8 0 1 I N H 4 1 6 ( H 4 C02 N 0 1 0 O381
•
, 7 H2 o
74 ZAACA 4 0 8
97 K C02 0 4
C0"4 IV
71ZAACA
306
I 8A2 CO 0 4
73 ZAAOA
398
54 L l 8 G0 0 6
7 4 ZAACA 4 0 8
75 C52 CO 0 3
74 ZAACA 4 0 9
lsz 86 co2 o7
C0.4 v)
HS
6 7 STGBA
3
I 8 3 VS V ( F L U O R I D E S )
14 ZAACA 4 0 8
97 K C02 0 4
CR*2 vl
LS
7L ANCPA
6
411A2
CR 0 6
6 9 ACDGA
25
9 2 5 8 VS o ELECTRONS
CA*3 vl
6 9 MDUUA
4
6 2 | NA3 CR F6
7O INOCA
4• a Z Z 8 HA3 )CR NO 006HOZ4 M 6 3 . 8 , 2 0
1o ACSAA
2
3 6 2 7 ~A2 CA3 0 8
73 NROUA
8
5 9 3 CA CR FS
6 5 ACCRA
19
1 3 1 C R | C ~ H7 0 2 1 3
CR*4 IV
14 ZAACA 4 0 7
129 BA2 CR 0 4
CR*4 Vl
72 NRBUA
?
1 5 7 CR o z
CR*5 Vl
67 SEGOA
3
I A3 v s v ) F L O O R ) D E S )
CR*6 IV
6 8 CJCHA
96
9 3 5 K2 CR2 O7
70 ACGCA
26
2 2 2 CR 0 3
6 9 JC$1A 7 9 8 9 1 8 5 7 ( N H R ) 2 CR 0 4
h 9 ACAC8
2 5 $|16 AG~ CR2 0 7
70 5PHDA
15
5 3 0 K2 CR4 0 1 3
to ANNIA
55
7 8 4 P82 CR2 0 5
7 0 ACSAA
2 4 3 6 2 7 N2 CR3 0 8 0 H
71 $FHCO
15
8 2 0 NA2 CR2 0 7 . 2 H2 0
71SPHCA
15
8 2 6 L I 2 CR2 0 7 . 2 H2 o
73 AC8CA
29
BqO NA2 CR2 0 7 ALPHA
71ACSAA
2S
4 4 RB2 CR2 0 7
7 0 CJCHA
48
5 3 7 8 8 2 CR2 0 7
71ACSAA
2S
35 R82 CRZ 0 7
71JSSC8
3
3 6 4 A02 CR 0 4
72 ACeCA
Z8 2 8 4 5 82 CR 0 4
73 ACSAA
Z7
1 7 7 ZR4 CO H | 6 CCR O ~ ) S . H Z O
73 ACSCA
2 e 21~l R82 CR4 013
•73 6COCA
29 2 9 6 3 NA2 CA 0 4 . 4 H2 0
71JCSIA
letl
1 8 5 7 ( N H R I 2 CR 0 4
73 NRDUA
8
271 K2 CA2 O l
C R * 6 vl
74 AMMIA
59 1 1 0 0 P86 CR CL6 X6 V2
C$*l vlll
0 9 SPHCA
13 9 3 0 CS2 BE F4
CS*l x
6 9 INOCA
8 1 6 6 5 CS4 M03 F I O
6 9 $PHCA
13 9 3 0 C$2 BE F4
cs*l
xl
6 9 INOCA
8 1 6 6 5 C$4 NG3 F I O
C$.1 x11
67 ACCRA
23
8 6 5 C$ U F 6
6 8 ACSAA
2 2 2 7 9 3 CS CO CL3
71AC8CA
27
2~S C$ U6 F 2 5
CU*| II
b 9 ZKKKA
129
2 5 9 CU LA 0 2
70 ZAACA 3 7 9
1 1 3 SR CU2 O2
CU*| Iv
49 ACCRA
2
158
c u CL3
Cu*l vl
70 NRBUA
S 2 0 7 CU TA 0 3
cu*a Iv
57 ACCRA
10
5 5 4 c u CR2 0 4
71ACIEA
1o 4 1 3 SR CU 8 4 , CA CU F4
CU*2 IV $0
O7 ZK~KA
124
9 1 Z N 2 CU 6 8 2 O8
68 ACBCA
24
888 cuz IN2 05
t l ACBGA
27
6 7 7 CU 02 0 4
6 5 JCPSA
4 3 3 9 5 9 CU ( 0 6 H S I C H 3 ) 2 C ) O212
6 6 INOCA
$
5 1 7 CU 1 0 | 0 H9 0 2 1 2
61JCS~*
1967
3 0 9 CU ( 0 4 C l 2 H I 8 )
6 6 PRLAA 2 8 9
161 C14 HLO 0 4 CU
70 ACSCA
26
8 cu o
cu.z v
6 9 ACSAA
23
221CU) W 0 6
~8 C a ) H A
46
9 1 7 CU3 AS2 O8
6 8 JCPSA
40 2 6 1 9 CU NO 0 4
cu*z vI
:I .ACCDA
. . . . . . .|.6. . . |.z.4. .c.u.5
'68
;o
68
68
JCPSA
AC8CA
CJCHA
J&CSA
70 )NOCA
t 3 ACBCA
CU*3 Vl
12 MRBUA
DV*Z v )
UNPUI
DY*2 V l l
UNPUI
OY*Z V I I I
uNPul
or,3 vl
6 3 PHSSA
0Y'3 VII
I t JCNLB
oy.1 viii
I ~ .$$COA
......
ov.J ix
) p o 4 3 2 ~o , 1 4
4 8 2 6 1 9 CU NO 0 4
2 6 10Z0 CU i o ~
~6
60S CU2 P2 O7
90 5621 CU(((C H3)2 N)2 I P l 0 1 2 O111 ; e l 0 4 1 2
9
is| cu )O~FA)3 (eL 04)2
2 9 1 7 4 3 CU V 2 0 b
7
9 1 3 LA GU 0 3
OY 12
Of Ct2*
0Y 8R2
ov eL2
3 K446 0 ¥ 2 o )
|
83 OYITHO)3oH2 0
1 | 7 1 ~ .o.r. ). . . . .FE3
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R. D. SHANNON
Table
74 2AACA
ER*3 Vl
7~ ACbCA
ER*3 VII
TO SPHCA
72 JCMLB
ER+~ V I I I
6 8 CHPL8
?0 [NOCA
7O SSCOA
7L ACSAA
403
I
26
484
13
Z
36 ER2 GE2 0 7
1 9 7 ER8 U i T H D ) I O
~4 ZAACA
4O3
2
ER2 S12 OT
10 H ) 1 2
2
4 7 ER ~ 0 4 . ER V 0 4
9 2AO0 ER I C 2 0 4 ) I H C2 0 4 ) . 3 H 2
8 1 7 4 5 ER3 F E 5 0 L 2
28
372 E~ | H U C H2 G 0 O 8 3 . 2 H 2
3 6 2 EK I C 2 H3 S 0 4 ) 3 . 9 H 2
I R3 V$ V IER F31
201
oA+~ I V
16 ACRC~
T5 AC~CA
GA.) V l
14 A T R I A
GD*3 V I I
70 &CBCA
?2 ACSCA
T2 SPHC~
O9 [VNM&
T2 JSSCB
GO*J V I I I
IA SPHCA
l Z SPHCA
74 ZAACA
~ 0 ~ 3 ZX
72 SPHE&
6 9 IVNNA
7~ ZAACA
OE*4 I V
68 ZKKKA
69 $ C I E A
69 Z~KKA
7O JS$CB
Zl sP-c*
70 ACSAA
6 7 ACSAA
7~ NOCNe
I1 " 0 C ~ 6
72 SPHCA
12 ROCk5
GE*4 v i
10 SSCO~
I0 J$$C8
TA ,OCM8
TA AC~CA
12 ANN1&
12 2KKKA
TZ ~OCM8
.,A
I
............
L!
0
EU3 O4
EU F Z ,
EU 8RZ
7~ EU CL2
EU F2
2 1 8 EU2 SI
6,,cc.
EU4 AL2 o g
L 1 2 EU5 8 8
l o g 4 EU3 FE2 GA3 0 1 2
I R3 VS V (EU ~38
2 5 2 7 EU2 I C 3 H2 0 4 1 3 . e H 2
0
20
(cont.)
8 6 9 HG NO 0 4
2049
1745
437
1
| R3 V$ V I H U F J )
2 6 1 3 HOICZ H5 S 0 6 ) 3 . 9 H 2
409
IS
24
24
llbE
BA FE S14 0 1 0
139g
3O6
452
|29&
( N A , K ) 2 FE4 S [ A 2
FE V2 O4
FE2 TI 0 4
FE2 MO 04
0
030.H2
0
V25 R VS A I F E S21
4 3 0 FE AL~ IP O412 IO H J 2 l 0
.2H2 0
tgO ~E S O4
g 9 9 FE3 BE S I 3 0 9 [ F I O H I 2
7 7 5 FE ( N H 4 ) 2 ( $ 0 4 ) 2 , 6 H 2
0
290 LIFE
P O4
4 8 & F~2 SA 0 4
79A GARNE~$
333 FE3 A L 2 S I 3
2 6 6 CALCULATEO
1469
1745
A263
T2S
36A6
832
I
Z469
1745
335
I
6315
3616
239
440
0L2
CA2 FE2 OS
M3 FE5 0 1 2
BA FE2 0 4
BA CA FE4 0 8
CA2 FE2 05
6A FE2 0 4
FE V 0 4
CAZ FEZ O5
~3 FE5 0 1 2
K FE F4
FE v 0 4
B I FE O3
CA2 F E 2 0 S
FE lOS H7 0213
FE I C 7 H5 0 2 1 3
33L ESTIMATED
3 3 1 R3 VS V I P E R O V S K I T E S I
R3 VS V I S ~ Fe O3)
*3
K2 FE 0 4
R3 VS v (K2 FE 0 4 )
6t6 L | 5 GA 0 4
56O S~ OAZ S l Z
08
3O 1364 ItS HAl Ob O&
26
4 8 4 GDZ S I 2 0 7
28
6 0 GO2 NO3 O12
16 7 9 0 G O 2 0 E Z 07
5 1823 G O g . 3 3 S I 6 0 2 6
S 266 0 0 9 . 3 3 $ 1 b 0 2 6
1~
16
4O3
9 2 6 NA GO S ] 0 4
79O ~ 0 2 GE2 O?
A R3 V$ V IO0 F ) |
16
7 9 0 GO2 GEZ QT
5 IR23 GD9.33 S]6 026
4O3
A R3 VS V IO0 F 3 )
t26
165
129
Z
18
24
2A
A02
IOZ
t7
103
299
586
427
612
~a~
1287
12B!
964
I245
244
1560
CO 0E 0 8
AN2 GE 0 4
~N3 FEZ GE3 OAZ
~ 0 2 8 G E| O 0 4 8
~
sm GE o~
NA4 $N2 GE4 0 1 2 I 0 H I 4
NAB SN4 GEIO 0 3 0 (O H I 4
NAZ 0£ O3
K 2 0 E ~ O9
CO GE O3
GE5 0 ( P 0 4 1 6
1 $ 5 7 CA2 GE O4
2
662 . G 2 8 GEtO O48
Ao2 I Z 4 S K 2 0 E 4 0 9
2T 2 1 3 3 OE 02
~1
62 MN2 GE 0 4 OELT&
186
38T Ge I O H I P 04
103 AS60 GES 0 I P 0 4 ) 6
+
13
2 3 5 0 N H~
750 L| |
841LI
1
1015 CEII
794 C i I I
7 2 9 N H4
I 03
O]
03
0314
0314.H2
I 03
ZT~ ~
~
o2
~ ~C$AA
ZJ 3341
I0 ,42
14 AC>AA
2T 3467 . P ~ ( O , 1 8
.F*4 VIII
TJ &CSAA
27 Z * 5 5 NF ( 0 H I 2
.0,1 vl
? l CCJO& | 9 1 |
4 6 6 NO2 F2
2 5 5 6 N H4 I 03
1782 NA I 0 4
308 A I 04
1 8 5 7 II*TI-U
7o~.s.
,'/6
INOCA
~9 ZKKKA
S 04
0
0
97 ~82 IN4 07
2 8 0 SR2 I N Z 0 3
9 7 R82 I N 4 0 7
t437 IN O H S 04.lHZ
3 e 8 CUE I N Z OS
1662 IN 0 0 H
3583 , G 2 NAZ $ ] 6
2667 K~ NG 84
OAR
A966 ~G) P2 0 8
A42 MG 84 0 7
NG3 P2 O8
RG2 P2 07
A l S q . G 2 P2 07
36A ~G2 A$2 OT
1419 , G ,
H4 P O*
o41
S 04*~Z 0
IC~ 0 4 1 4 . , Z
0
012
AC6CA
30 1882 NA I N S l Z 0 6
74 SPHOA
18
7 6 t I N Z GE2 07
Vl
7A J 5 $ C 8
3
1 7 4 SR ZR 0 3
IR+S V I
74 NROUA
9 1177 R3 VS V ICO2 I R 2 071
K*I
IV
68 ZAACA
3SR 2 4 1 K
AG O
RE~ 2
K2 0
K*Z
VI
SA ZAACA 2 6 4
144 K 58 F6
6 8 SPHOA
12 1095 K Y N02 0 8
09 CCJDA
II
6 0 6 K2 ZR2 O5
b 9 ACUGA
2 5 1919 ~ U2 F9
K*L
vii
~e CJCHA
46
9 3 5 ~2 ¢R2 0 7
69 JCSIA 19~9
8 4 q KZ NO 0 4
TA S~COA
S 3 3 8 ~ FE F4
K*I
Vlll
~0 ZKKKA
74
3O6 K H2 P 0 4
62 ZKKKA
|AT
4A1K2
TI6 013
37 ZKKKA
98
2 6 6 K H2 I H 3 O) 85 0 1 0
7A INUCA
7
8 7 3 K H C2 0 4
6 8 CJCHA
46
9 3 5 K2 CR2 O7
70 J C S I A 1 9 7 0 3 0 9 2 K AU I N 0 3 ) 4
6 5 ACCRA
19 6 2 9 K4 H2 12 0 1 0 . 8 H 2 0
K*|
IX
70 ZKKKA
132
27 K A . 6 N A S . 5 CAO.3 A L T . 3
L18*5 032
6 9 ~CBCA
25
6 0 0 K CE F4
6 9 ACSCA
25 1 9 1 9 K U2 F9
K*A
X
73 CJCHA
$1 2 6 1 3 K AL P2 0 7
K +I
XII
6 8 SPHCA
13 4 2 0 K Y ~2 08
? l INOCA
10 1264 K2 P8 CO I N 0 2 1 6
67 I~0CA
5
5 1 4 K2 BA CO I N U 2 8 6
74 IACSA
9 6 6 6 0 6 K2 CA CO I N 0 2 ) 6
75 ACOEA
3A
$ 9 6 K2 8A CU I N 0 2 1 6
57 PASAA
56
643 K CL 0 ~
LA*3 Vl
6 9 ZKKKA
129
2 ~ 9 CU LA 02
T3 NRUUA
8 1 2 6 9 ~ 3 V$ V I ~ E 2 ~ 3 0 A 2 1
LA+3 V I I !
?4 A N N I A
59 1277 LA4 ~G2 T I 3 S [ 4 0 2 2
73 ACRCA
2 9 2 0 7 4 LA2 H03 U I 2
6 8 INOCk
7 2 2 9 5 LA ( C 5 H7 0 2 ) 3 I H Z 0 4 2
74 ZAACA +03
A R3 VS V ( L A F31
7~ SPHCA
L8 67S LA2 SR3 (B 0 3 1 4
LA*3 IX
71NRBUA
6
2 3 LA FE 0 3
?~ Z ~ C A
~O~
A ~ ) VS V ( L ~ ~31
74 A~IA
59 12T7 LA~ NG2 T13 S I 4 0 2 2
LI+A IV
3V ZKKKA
A02 119 L I O HoHZ O
TO 2AACA 3 7 9
| 5 7 L I Z CU 02
70 INOCA
9 1 0 9 6 Y8 L [ F4
71AMNIA
56
18 NA3 AL2 L l 3 F 1 2
71ACSCA
27
0 6 6 L 1 5 OA O4
T3 JSSCB
6
538 L I 3 V O4
73 ACRCA
2 9 2&Z$ L I {N~ HSI 8E F4
T ) ACBCA
2 9 ~ 6 2 8 L [ N H3 0 H $ 04
6~ ACCRA
17 7 8 3 L I 2 C2 0 4
14 ACSCA
30 2 4 h 6 L I Z 8E Sl 0 4
LI+L V|
&8 ACBCA
24
2 2 3 L | 3 AL F6
6 9 2AACA
)TA
3 0 6 L 1 2 ZR O3
70 ZKKKA
132
I 1 8 L I 2 AL2 S [ 3 0 l O
TA ~RRUA
6 ~
U Z ~0 F6
6~ A¢C~A
19 ~ 0 1 L I C6 07 H7
74 A¢IEA
66 819 LI N8 P
0204
68 CZ~YA
66
29O Ll Fe
?l A¢SA6
2~ 3387 LI N ~ O8
73 IJCHA
~Z 26~ LI V OJ
T3 ACBCA
29 2 2 9 4 L I 2 ZR F6
CU+3 V l
7O ZAACA 3T7
70 C~ LU2 O4
?z J*CC*
4
2 ~ ~ u e o~
LU*~ VIA!
74 ZAACA *03
| R3 VS V ( L U F31
LU+3 IX
7~ ZAACA ~ 0 3
A R~ VS v ( L U F31
~G*2 I v
72 AC§C~
28
2 6 7 KZ NGS $ I 1 2 0 3 0
AC~CA
2
~4 ACUC6
30
~0+2 V
6~ ~CSAA
ZZ
&6 , J N N A 1966
UNPU$
UNPU3
NO*2 V l
65 CJCH&
~)
68 UAPCA
11
TO ACeCA
26
,3 7
2 7 5 6 3 V5 V 1~4 . F
H236
0
2S35"HOIN2 014 IH C 0313.2H2
83L IN OIZIHOIN 03)51
19 6 2 9 K4
2 0t0.8H2
59 2 0 3 6 I N H 4 1 Z H ) [ 0 6
409
393
74
T|
73
74
K HO BE F6
H 0 3 FE5 0 1 2
H0 Pb 0 1 4
R3 VS V IHO F 3 I
IR'4
A R3 VS V I E U F 3 |
2 8 2 7 EU2 ( C 3 H2 0 4 ) 3 . 8 H 2
33~7 EU TRISGLYCOLATE
2T
36
2
73 ACBCA
29
H0*J VIII
T~ ACBCA
30
TO SSCOA
8
72 8UFCA
9S
74 ZAACA 4 0 3
HO+3 I x
14 ZAACA 4 0 3
7 4 ACOCA
30
H0+3 X
T4 INOCA
13
75 CJEHA
S3
1.5
Ill
71JCPSA
54
6 6 ACCRA
20
6 6 ACCRA
21
58 ACGRA
9
58 ACCRA
It
4 3 RTCPB
6Z
I+5
Vl
7 t JCPSA
54
1+7
IV
?0 RCBC~
26
2 6 ZEPYA
3g
71 J C S I A A971
I*7
VI
6 5 ACCRA
37 JACSA
IN.3 IV
T4 ZAACA
73 ZAACA
IN*3 VI
74 ZAACA
~ t ACSAA
6 0 ACOCA
70 ACSAA
04
EU4 AL2 0 9
L | EU3 0 4
L I 2 EU5 08
201
8
.......
0
201CI
EU3 O4
It04 EU 12
L I 2 EUS 0 8
7O SUFCA
93
71SPHCA
|5
6 ? ACCRA
22
6 8 GIWYA
68
14 A ~ N [ A
59
FE*2 rill
TA AMMIA
$6
71ZKK~A
134
T3 ACAC8
29
FE+J AV
H$
TO ACECA
26
70 SSCOA
8
?1ACRCA
27
I I MREUA
6
71AESAA
2S
73 ACBCA
29
FE+3 v
TA JS$CB
~
FE*3 vI
H$
TO ACSCA
26
70 SSCOA
6
71SSCOA
9
?L JS$CB
~
TL JPCSA
32
11ACSAA
25
67 ACCRA
23
6 9 CCJO~ 1 9 6 9
FE*) VIII
T3 JSSCB
8
FE*4 Vl
73 JSSCB
8
FE+o I V
z~ JSSCB
0
I R3 VS V (eR F31
I97 ER8 0 I T H D ) I O I 0 N ) 1 2
72 JGHL8
ER.3 IX
5 9 ZKKKA
112
?4 ZAACA 4O3
EU+2 V l
7O ZAACA
374
EU+2 V I I
70 2AACA 3 7 4
6 9 ACRCA
25
7 3 REF 3
EU*2 VIII
UNPUt
EU+2 I x
73 RVCMA
|0
UNPUl
EU+Z x
? l NATMA
S8
EU+3 V l
68 REF 4
7 0 ZAACA
374
73 REF 3
EU÷3 V I I
68 REF 4
13 REF 3
EU+S V I I I
65 JCP$A
48
~ ZAACA 4O3
3 ACSAA
27
EU+3 [ X
74 ZAAC~ 4O3
73 ACSAA
27
71ACSAA
25
F E * Z I V SQ H$
74 AMHIA
59
FE*Z IV
HS
69 $C]EA
L&6
6 9 ZAACA 3 6 9
TI JUPSA
31
12 JUPSA
33
FE*Z VI
LS
69 ACRCA
28
FE+2 V I
HS
&9 NJMMA 1 9 6 9
• 6 JCPSA
~P.~ Iv
TS J 5 5 C 8
NF.~ Vll
R3 VS V |OY F 3 )
2
755
8 A665 E$4 H03 FAO
IZ9
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698P.C,
lI
93,,..
10 J 5 5 C 8
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70 BSC++ 1970 + 2 4 3 "G $ 0 4 . . Z 0
71 l c e c l
z?
813 .~
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bB &CSAA
22 A466 MG$ P2 08
ro 6 k F A
C~ MO Sl O4
0
AGSCA
ANM|A
ANMIA
CJCNR
10 INOCA
72 C J C H i
7~ ACBCA
NG*Z V I I I
73 AGRC8
MN*Z I V
HS
70 ANNIA
6 9 ZAA¢A
F I AGBCA
6 9 PMSSA
13 ACACB
NN*~ V
HS
6e ANNIA
74 MPMTA
~N*Z VI
LS
0 9 ACBCA
MN*~ Vl
HS
6 9 SGIEA
6 q JCPSA
70 ZKK~A
6 g ANMIA
70 N J ~ I A
6 5 ACCRA
72 A N N I A
6 7 PRLAA
67 HCACA
MN*2 V I I
72 AMNIA
MN*Z V I A l
6 9 ZKKKA
71AMM|A
73 SSCOA
74 JCPS6
NN*J V!
HS
0 7 ACSAA
b 7 ZKKKA
b8 ACSCA
6 9 JCPSA
6 3 PHSSA
6 8 BUFCA
7A JS$C8
7J JSSC8
7~ A N N I A
6 8 ACBCA
* INOCA
4 INOCA
NN*~ IV
7S JSSC8
NN+~ V l
73 JS$CB
6 9 INOCA
6 3 CZYPA
6 7 HCACA
.N*6 IV
12 ACOCA
M~*7 I v
6 8 ACOCA
MO+3 V I
6 9 ACBCA
6 9 INOCA
.0.~ vl
7A MRUUA
MO+~ IV
7 4 INOCA
~O.S Vl
7 t INOCA
~
68 JCPSA
6 8 SPHDA
69 JCS|A
72 ACUCA
6 9 JCPSA
71 SPH~A
?A SPHCA
71JCPSA
73 ACBCA
71JCSIA
NO*6 V
67 CCJOA
08 J C S I A
NO+6 VI
68 JCSIA
70 JSSCB
70 INOCA
r o ACSAA
6 6 ACSAA
TO CCJOA
6~ INOCA
73 ACRCA
74 ACECA
Iv
REF 6
N*5
Ill
REF 6
JO
86
$8
$2
q
SO
29
29
2 4 9 1 M G 2 V2 0 1
| 5 8 3 NG 1 8 6 O f I 0 H I 6 1 * Z H |
1 0 Z 9 MG C OS
11S5 CA L l ~G2 HZ I P 0 4 1 1 6
I S I ~G I O N P ~ 1 3 (CA 0 4 1 2
3 6 1 9 NG Y2 U6
2 6 1 1 M G 3 AS2 0 8
266 CALGULAIIO
NN? S | AS U | 2
Iq~ VZ 0 4
NN CO CA 0 4
MN ¢R2 0 4
GALGULATEO
55 1 4 8 9
369
306
2T 1 0 4 6
J2
Kql
29
266
53
21
28
U
1 8 6 1 M N 2 0 H AS 04
2 6 6 NN2 AS OA OH
9 2 8 R VS 0 £LECTMONS
168
SR6 ~ Z
GE U6
51 4 9 2 8 BA MN F 4
l~Z
INNS
tO N l z $ 1 2 0 8
5 4 | 3 6 2 NN eE2 | P 0 4 l Z I O H I 2 . S H 2 0
113
1MN? ~A|2 IS 0~)1$.18H2
0
19
8 8 ~ MN S O~
ST
6 2 1 M N 2 GE 04
9~
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229
427
56
791
12
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GARNETS
NN3 AL2 GE3 0 1 2
MN U4 0 7
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124
24
50
3
9~
3
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NNZ 0 3
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I N H 4 ) 2 RN F5
f'~Z 03
TO NN 0 8 , PR n N O ) , N u x N O )
LA JCN 03, RN3 04,
L A . 9 S CA.o5 #N 0 3
NA MN? 0 1 2
NGZ ~N e o s
NA4 MN~ TLS OAR
~N I t 7 H5 0 Z 1 3 , 1 1 4 C6 HS CH$
~N IACACi3
ZO?|
428
1233
1066
K.46
339
238
6
16
39
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2 4 11A4
13 IRSA
13 1 8 6 4
L3
2 7 5 R3 V$ V IM4 NN O ~ l
8
23k
8
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39e
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8 6 NN O3
NIL2 NN NBL2 0 3 8 . S O N 2
NAT H4 MN I I 0 6 I S * I T H 2
NN5 GO, c o 2 MN) 0 5
0
2 ~ 1 0 5 3 AG NN 0 4
25
4 0 0 KJ NO CL~
8 2 6 9 4 K8 NO F6
6
5 5 5 L 1 2 MO F 6
13 2 7 1 5
t0
R3 VS V ¢AE N 0 0 * l
9 2 2 8A2 NO NO 0 6
48
12
1969
28
50
18
15
55
29
1971
2 6 t 9 CU N0 O~
LOgS K Y N 0 2 0 8
8 4 9 K2 1~3 0 4
6 0 GU2 NO) OA2
8 6 N02 NO3 0 1 2
611LI3
FE NO3 0 1 2
8 2 9 K AL N02 0 8 . K FE ROE 0 8
1093 CA N0 0 ~ , SR NO 0 *
2 0 7 4 LA2 NO) O12
1 8 5 7 NO(*63 - O
1967
1968
3 7 4 ~2 NO3 0 1 0
1 3 9 8 KZ X0$ O10
19&8
1
9
24
20
1970
13V8 K2 M03 0ZO
kS& AG6 M010 0 3 3
2 2 2 8 NA3 ICR MO 0 6 0 2 ~ H b I * R H 2
3711Cl
~ 0 0 2 AS O~
2 6 9 8 NO F 6 IGASI
SO NO 0 3 ( H 2 0 ) 2
1603 K
I R 0 O2 C2 O4I H2 0 1 2 O
29
8 6 9 HG NO 0 4
30 11V5 NO 03VHZ O
N-J
NA+I I V
74 ZAACA
REF 2
NA+A
V
68 ACeCA
6 8 SPHOA
6~ ZAACA
NA+I Vl
70 ACSAA
6 5 ACCeA
63
60
58
~6
,ccA,
ZKKRA
ZKKKA
ACCRA
0
2 8 2 8 4 S K2 MN O~
NG3 N 2 , S I 3
N~.8
NITI
N
NHk N O3t~L~ N 0 3 t K N OAt
RAin 0312,TIIN
O314
409
6 9 N A 6 ZN 0 4
NA2 O
2 4 1077 ~A2 S I 2 OS
A2 9 8 7 NA2 ZNZ S l 2
329
110 NA2 HO O2
07
2 4 1287 N&~ SN2 GE4 0 1 2
19
5 6 1 N A C6 0 7 H7
i~
115
111
~
s g ACCRA
1
74 ACRCA
30
78 ACBCA
31
NA*I VII
71 SPHCA
IS
;0 NJNlA
113
T ) ACBCA
29
NA*A V I I I
6 8 AGBGA
24
6 8 SPHOA
A2
71ANMIA
56
NA+I Xll
[ l J$SC8
3
32 ZKKRA
81
Ne.3 vl
7. ACIEA
U6
,8.~ Viii
75 JACSA
9?
NR+~ v l
6 8 JCPSA
*a
70 JS~C~
l
70 JSSC8
1
70 ~ l A
ss
$5 PRVAA
98
71J5$CB
J
71Z*~c~
3co
14 J I N C A
36
It JCSI& lVll
1233
6)0
Z4I
811
I0 H)4
NA o I O . I * . Z , 2 0
NA2 AL2 S I S O [ O * 2 H Z 0
NA CL 0 3
INAsAS RAIN
5 2 6 NA U ACETATE
1872 NA2 u O4
8 9 0 NA2 C O 3 . H Z O
9 2 6 NA GO S l 0 4
I NN7 NAA2 I $ O 4 1 3 . 1 5 N 2 0
RqO NA2 OR2 07 ALPHA
1 7 0 3 NA 8 F4
V8? NA~ Z . Z $12 0 7
18 NA3 AL2 L I 3 FAZ
89 NA|3 N833 094
13S NA A~ S I 0 4
RA9 L I
NU OZ
2713 NBIOPNI~
5048
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90
go3
09
119
1965
AZIO
8 A 2 7 S R 7 . 5 NB2 0 5 . 7 8
~ - , 8 2 OS
NA2 . 8 4 OAA
c a NSZ 0 6
C~2 NEZ 07
N ~ l $ NeS~ O94
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0
756
REVISED
EFFECTIVE
IONIC
RADII
IN
Table 2
70 ACOC&
2b
7 t ACSAA
25
59 IPHCA
4
73 JSSC8
e
66 ACSA6
20
74 8UFC&
97
N8"5 VII
t o JSSC8
1
TL J$$Co
3
? l ACSCA
27
; 5 ACBCA
31
N0,2 VIII
UNPUI
N0+2 i x
v~NPUt.
ND*3
?1 INOCA
1o
74 RRSUA
9
ND*3 V I I I
b9 JCPSA
50
71 JSSC6
3
lO SPHC&
14
?u 4C8CA
26
70 ACSAA
24
71 SPHDA
15
71SPHCA
15
74 MR6UA
9
74 Z4AC4 4O3
14 ACOCA
30
ND+J IX
TO ACSAA
24
71SPHCA
15
73 6C$AA
27
74 ZAAC& 403
73 ACSA&
ZI
73 ~ c s ~
27
74 AHH|A
5q
NO~$ X l l
7Z JSSCO
4
N[*2 Iv
bL J A P [ A
32
65 8SCFA 1965
NZ+2 t v SQ
b6 [NOCA
5
NI+2 v
67 8APCA
15
NI*2 Vl
?* &HNIA
5e
74 ACBCA
30
68 ZAACA 358
67 8APC&
15
70 ACOCA
28
70 ZAACA 3T8
70 JSSC5
2
7 t PHSSA 438
70 REF I
b4 ACCRA
t?
13 ACOCA
29
63 Z K K ~
11o
?4 JCPS&
61
73 JCRL8
3
?3 &C8C6
29
NI*~ vi
LS
?4 ZSAC4
405
71CH0C~
2tZ
NI*3 V]
NS
54 JACS&
76
Nle4 Vl
L$
&? STGBA
~
74 J[NCA
3
No+2 V l
?4 [NOCA
73
NP*J v [
68 J I N C ~ "
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NP*~ Vi
6? iNUC4
J
14 CJCH4
52
NPt6 Vl
105
354?
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045 $14 NSb O26
LI N03 00
I V . YS) NO O~
8X ~8 O4
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NA3 N8 04
454
89
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&?3
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NAI3 N095 09~
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NB2 05
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NO [2
NO e L 2 ,
NO ~R2
922 0A2 NO RO 06
1661 NO AL3 04 012
86
450
518
484
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129
I
4&8
N02 HO3 012
NO V 04
K NO U2 O8
NO2 T I 2 o?
N04 RE2 Oil
NO2 H 0b
N04 ~3 015
NO P5 014
~3 VS V IN0 F31
ND P3 09
2969
g91
2441
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2813
29?3
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0
N04 M3 015
NO2 IC3 H2 O 4 ) 3 . 6 H 2 o
R3 V$ v (NO F31
N02 (C3 H2 0 4 1 3 . 6 H 2 o
NO 0~ C O3
N04 HG2 T I 3 $14 022
IL NO A~ o ~
685 NI CR2 04
1085 SPINEL$
1200 N]
[OPN)2
47 N I 2 P2 07
N i 2 $I 04
NI IPV N 0 J 6 (B F 4 i 2
N[ SE 04
H I 2 P2 Q7
R8 NI F3
$R2 N[ TE 06
RO N[ F3
NI (O H)Z
NI2 51 04
1461 N| IC5 H? 0 2 1 2 . 2 H 2 o
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$1F6.6H2 0
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IH c o O ) 2 * Z H 2 O
852 NI C4 0 4 . Z H 2 0
1 8 1 N I IC5 H7 O 2 ) 2 . ( C 2 N 5 0
2304 H I 3 VZ 08
485
1686
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1464
12g
416
125
H)2
167 ~2 N* NI F6
2163 H0 N[ O3
14q9 N& NI 02
l R3 VS ~61FLOORIOE$1
L561 K2 N]
2233 EIT[NA?EO
823 NP CL3
32? ESTiNATEO
2175 R~ VS v
a3 vs v 186Z s~ NP O6)
OH-I II
71 6~X14
OH-I I[[
~ I 6H~16
oH-I
56 t155 R 0 6 . 6 Feo4 513 012 F 0 H,
R[OH-IIIRIF*Z)~.04
56 I I S 5
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F E . 4 S13 012 F 0 H.
R(OH-I)=8(F-I|+.04
Iv
OH-I Vl
RIOH-I)IRIF*11*.04
05.4 Vl
6q JC0HA
I T 45q 05 02
?0 ACSAA
24 125 05 02
OS+5 V l
71 ~CSI4 1 9 7 [ 2760 os F5
?4 SSCOA
14 357 R3 V$ V ICU2 O52 071
56 JINCA
2
79 K 05 F6
0S'6 Vl
~ VS V IPE~OVSKI?~Sl
o$+? v l
R3 V$ v IPEROV$~iTE$1
05+6 I v
66 ACSA&
20
395 o$ 04
7 ) ACOCA
29 1703 os o4
65 ~ c c ~
19 157 os o4
7 1 J C S I & 1971 1857 0 5 1 8 . ) - o
P*5
Iv
t z ACaC&
20 z083 co2 P2 07
60 CJCH6
46 6O5 CU2 PZ o?
65 C~¢H4
43 113~ nGz p z o7
60 IN0C~
7 1345 C42 P2 o?
71 8SCF~ I v 7 1 426 ZR P2 o?
?o ~C0CA
26 16Z6 H~ p 0 4 . 1 / Z HZ O
?1 ~C0CA
27 2 9 t N4Z H2 PZ 0 7 . 6 H 2 0
?~ NJNN& [ V ? I
241SR AL3 {P 0412 I0 H ) 5 * H 2 o
69 z x ~ x ~
130
148 K z ~ z I p O4)3
?1AC0C6
27 2124 N43 P 0 4 . 1 2 H 2 o
60 ~C$A6
22 18ZZ NA LR2 P~ 012
b8 2KKKA 127
2 1 A L 3 PZ 0 8 . $ H 2 o
68 ClWYA
68 290 Li FE P 04
lO &C6CA
26 1826 H3 p 04
?2 ANNIA
$?
45 NN,65 FE,35 P O~
tZ AG6GA
Z§ l g ~ $ | q H ~ | ~ M P U4
73 ACS¢A
Z~ 14t LU P O4
I I ACSCA
~ Z~4T CA {H2 p 0 4 ) 2 . H 2 U
?3 AC§¢A
ZZgZ A L P 0 4 . 2 H 2 o
?L ACSAA
25 512 K H5 [P 0412
pe$?OvJ$$CB
1 120 ZNZ P2 07
6T JACSA
09 2268
6? Jac$a
o~ 227o
P*5
Vl
71ZA~C~
380
5[
?2 CCJO& l g T 2
676
7J ACAC3
29 266
~A*~ vl
67 INUCA
3 3Z?
74 CJCH4
52 Z [ 7 5
P~+$ V l
z [ ac8c8
27 731
P~*5 I x
67 JC$1A l g 6 T 1429
PBe~ IV
PY
63 ZXKK~
126 98
PI*2 Vl
C23 H29 US p
CzJ HZ9 05 P
p eL5
ET3 N H IC6 H4 0 2 ) 3 P
C6LCULAtEO
R IP&+41
R3 v$ v
~ P6 O3
K2 PA F7
p$ $1 oJ
1o 4CAC8
PB*2 V I I
9 ZKKK&
4 &CCRA
PB*2 v i i i
t l SPHCA
64 ACGRA
?3 CJCH4
?2 HROUA
p o * z IX
67 ACCRA
73 GJGHA
?~ ZKKKA
?~ CJGHA
P8*Z x
70 ZKKKA
P8.2 Xl[
51 6CCR~
26
HALIDES
501 p e z 03
09
15 728 P8 w 04
17 1539 PB P2 06
51
?o P02 V2 O?
7 1025 BI t l T & N A T E S
22
744 PO F2
51
TO P02 V 2 0 t
139 215 P8 c 03
S2 2 7 0 1 P U V2 06
228 P03 P2 08
1o3 p8 IN 0312
R3 VS v IOA $ 041
70 ZKKKA 132 220 P83 P2 00
71 INOC~
10 1264 K2 P8 CU IN 0 2 1 6
P 6 " 4 IV
72 JCSIA 1972 2448 R3 VS V INA4 P8 041
po** v
7o ZAaC6 375 255 RS2 po 03
PB$4 v I
70 6CAC8
26 501 P02 03
65 JINCA
27 150g PB3 04
74 CJCHA
52 2175 ~3 V5 v
po.~ viii
60 N~ou~
3
153 P8 02
PO*2 I v SQ
67 INOCA
6 730 P0 lob H5 OH3 CHIC 01212
60 J $ [ C A
g
166 po I I C 6 H 5 ) 2 CH C2 0212
po*~ vI
68 NRBUA
3 699 R3 V$ V INZ P02 071
73 INOCA
12 1726 XE PO F I [
6 1 J C $ I A l g 6 1 3?ze K2 po F6
PH*~ vI
PH*$ V I I I
74 2AACA 403
I R3 VS v IPX F31
PH*3 I x
74 ZA4C4 403
I 03 VS V IPH F3i
PO+4 v l
74 OJCHA
52 2175 R3 v$ v
POt4 v i i i
R3 VS v ( F L U g R [ T E I
PR*J V l
t l MRBUA
6 545 R3 V$ v (PR2 N03 0121
P a * 3 VIII
70 SPHCA
1~
28 PR2 U2 09
74 ZAACA 403
I R3 V$ v (PR F 3 |
P 8 . 3 IX
7o SPHCA
15
28 PRZ U2 og
5g ZKKKA 112
362 PR IC2 MS S 0 4 1 3 . g H 2 0
74 ZAAC4 403
I R) v$ v IPR F ) i
e~*4 Vl
72 ACBCA
20 956 OA PR 03
?~ ACBC6
31 971 ~ R t 0Z2
73 JSSC8
U 3 3 1 R IPR941
74 CJCHA
52 2175 R3 VS V
PT*2 I V SQ
72 REF 5
PC3 C0 06
Pt*4 vI
89 JINCA
31 3803 PY 02
R3 v$ v 1N2 p l 2 o71
?4 CJCHA
52 2175 R3 V5 v
Pt*5 Vl
67 STBGA
3
I R3 v$ v (FLUORIDES)
67 J C $ I A 1967 478 XE PT F I I
pu*~ vt
61 [NUCA
3 327 R I P U * J )
75 JINCA
37 743 R (PU~31
PU*4 VI
b7 INUCA
J 32? R IPU*41
73 JS5C6
8 331R (PU*4)
t ~ CJCHA
52 2175 R3 VS v
PU+6 V l
R3 VS v tSA2 SR PU O61
ao+1Vl
lO 244c~
3/5
255 R02 P6 O3
~ o * z ix
74 ACSCA
30 L640 R02 s 04
RS+I x l
7~ ACSCA
30 1640 R02 s o*
RO+I x x i
70 ACSCA
26 1464 R8 N[ F3
?o J$$C6
2 416 R9 N I F3
70 JS$CB
2 562 86 NI F3
R O - I XXV
65 ACC86
19 2O5 ~8 u 02 IN 0312
aE+4 v i
4 CJCH~
RE+5 v |
70 ACSA~
uNPu2
O8 ACSCA
REe6 V l
t 5 JS$C6
1o
3
2175 R t vs v
24 3406 N04 RE2 011
co2 aE2 o?
24 6T4 Re CL5
13
77 BA2 HN RE 05
R3 V$ V IPEROVIRITE$1
R V$ VALENCE
~E*? IV
b8 ACIEA
? 295 RE2 o? IO H212
7 1 J C $ 1 6 1971 1857 R E I * 7 ) - o
70 CJCH6
40 219 IRE2 I N - C 4 H7 02121 (RE 0 4 | 2
kE÷Z v l
68 ACiEA
7 295 RE2 07 {H2 012
l o 4C8CA
RH*~ V l
?3 INOC6
Ru+3 v i
CHALCOGENIDES
(cont.)
120 213 P8 CAZ $ | 3
17 1539 Pe P2 06
132
AND
26 1876 R,2 o )
12 2640 ~N F5
R3 vs v I L 6 Ru 03i
RU*4 V l
?o ac$64
24
l l 6 RU o z
74 4 c o c ,
30 143~ N413-X~ ~U4 0~
7~ CJCH4
5~ 2175 R3 vs v
RU*5 v i
?I J C $ [ A IVTI 2789 RU F$
REF 7
k3 VS V IC02 RU2 o ? )
I ) INOC~
. I Z l / I T X~ RU F I I
R U * t IV
S4 J6CS~
76 3317 ~ RU O4
au+u Iv
67 A~$A6
21
T37 RU 04
S+6 Iv
GO 6C0CA
24 508 C0 3 04.3 H2 U
7o ZKKK&
132
99 PB2 S 05
?0 8UFC6
q3 LgO FE $ 04 ALPHA
?o BUFC4
~3
1e5 FE S 04 0 H
70 6$CFA lg?O 4Z43 n G $ 04 N2 0
71 6C0C6
27 ZTZ N H4 N $ o4
70 NJNIA
113
I NN? NA~2 IS 0 4 ) | 3 . 1 5 H 2
6~ ~CCaA
[9
664 NN $ O4
,i 6c566
~s 3 2 1 3 N A s H 0 5 4 ° 4 N z °
?z 6cuc6
28 864 SN
12 N~IU6
Z36
g5 C4 $ 0 4 . 2 H 2 o
72 6C6C4
26 284S x z s O4
T3 JSTCA
14 499 TL2 $ 04
14 ACUC~
30 g21 C6 $ O4.2N2 o
74 NJHIA
121 208 FE2 IS 0413
S*6 v t
13 ACA¢~
~9 265 ¢4LCU~6tEO
o
$0.J Iv
?o A¢$AA
24
320 38 P 04
$0.5 Vl
?o 4 n n t 4
5s 1480 NN? $8 AS 012
71JCSl&
I V T l 942 AS $8 f 8
1 1 J G S I A 1911 2318 8R2 $83 F16
74 JC$$8
q 345 Na 58 03
s~*J Vl
68 CJCHA
46 | 4 4 6 S C 2 0 J
60 ARKE4
29
)43 scz oJ
U~PU4
5CZ ~ I Z 01
73 5PHCA
17 749 SG2 $12 o?
t 3 INOC~
12 927 SC IC5 H? 0 2 ) 3
73 ACBCA
2V 2615 NA SG 512 06
14 INOCA
13 IS8 SC IC? N5 0 2 ) 3
73 ACSAA
2? 2 8 4 1 S C U H IC3 H2 0 4 1 . 2 H 2 o
69 5PHDA
14
9 NA3 SC 512 O7
sc*J rill
14 INUCA
tO
1 3 t SC H ICY H5 0 2 i 4
T2 ACSA4
25 1337 SG2 (G2 0 6 1 3 . b ~ I 0
74 INOCA
13 1a06 H sc I £ 1 N3 0214
74 :NOCA
13 ] e e o .
sc I c ? . s o z i 4
$E*O I V
be ZAACA 358
125 NN S£ 0 4 , CO $Z 0 4 * N i $ [ 04
b l JGSIA
2]?
968 H2 SE 04
70 ACOCA
26 436 NA2 SE 04
?o ACeCA
26 1451KZ SE 04
t o ZAACA 37V 20~ N I SE 0 4 . 6 H 2 0
?2 4C8CA
28 204~ K~ 5E 04
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as
[ 9 C U IN . 3 1 4 S~ 04
?1J$cIA
1971 1857 S e l * 6 1 - o
Sl*4 IV
63 NAT~A
50
9 1 F E 2 $1 04
;~ ZKaK4
137
86 MG2 SI 04
6 I 6CCR4
14 835 NG3 AL2 $13 012
1o ZKKKA 132
I RN5 10 H i 2 S12 00
$8 ACCRA
II
437 CA3 AL2 S i 3 012 I G R O S S U L A R I T ( I
71 4 N N I 6
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?1 5PHEA
15 926 ~A Gb $1 13~
71SPHCA
15 806 V2 5Z OS
71NATWA
58 218 EU2 51 o~
7o P E P l 6
3
l b l C02 $ I 04
70 ACOCA
25
105 043 S [ 4 Nab 026
71 4 c 8 c 4
27
747 CA2 $ i 0 4 . C A CL2
7| AGBCA
27
848 C42 $I 04
71 &NNIA
56 1222 N A . | 6 K . 8 4 CA4 1518 0 2 0 1 F . 8 H 2 0
71ANNIA
56 1155 HGS*b F E . 4 $13 0 1 2 . ~
F 0 N
69 NS4PA
2
31 L ; N $12 0 6 , H A M S I 2 ~6
CA NG $12 06
69 ~S4PA
2
95 F E 6 * I H N * I NG*8 C A * I $10
022.!
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2
101 L 1 2 . 4 N A . I H G I Z . 9 5 1 1 5 . 7
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132 288 C&5 $ I 2 0 t
IC 0312
71ACBC&
27 2269 NA2 $ l U J . 6 N 2 0
?z SPHC4
14 z o z l 8 e z s i o~
72 4CBC&
28 1899 AL2 BE3 S i 6 018
; 4 ACOCA
30 2434 112 BE 51 Ok.
IX*4 V[
62 NA[WA
49 34S $1 02
69 CJCH6
47 3859 CU $ l F b * 4 N 2 D
?o ACBCA
26 233 SI P2 o?
71ACOCA
27 2133 SI 02
? l ACSCA
27
594 CA3 S 1 1 0 H I 6 . | Z H 2 D . $ 0 4 . C 0 3
73 4CBCA
2g 2 7 4 1 N $i F b . b H 2 o
73 ACSCA
29 2748 C0 $ 1 F b * b H 2 0
14 CJCHA
52 2115 63 V5 V
SH*2 v l I
UNPUl
SN 12
SH*2 V i i i
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SH*Z i x
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IS 924 NA SN GE 04
$N.3 Vll
70 SPHC&
15 214 SH2 SI2 07
14 SPHCA
1o 575 K2 SN F5
SN*3 V I i i
74 ZAAGA 403
! R3 V$ V ($N F J I
74 4CaCA
30 Z ? S l SN P5 014
S~*3 ix
69 ACAC8
25 6 2 | SN ISR 0 3 1 3 . e H 2 0
?o SPHC4
15 214 $M2 $12 o?
14 ZNOC4
13 2eO N H4 SN IS 0 4 1 2 ° 4 H 2 0
7G ZAACA 403
I R3 VS V |SN F 3 |
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72 JSSC8
4
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SH*~ I V
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31 511 K4 SN 04
72 J C $ I A 1972 2448 R3 V$ V iNA4 SN 041
7J 4C4CB
29 2 b b C4LGUL4TEO
SN*~ V
70 6HHI4
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70 asses
2 410 x2 SN 03
SN*4 v I
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248 L18 I N 06
70 4C$44
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74 CJCHA
52 217~ R3 V$ v
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70 244C4
379
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72 ZAAC& 393 266 SR 4Gb 04
SR*Z V l l
72 6CBCA
28 3668 SRIO IP 0416 I 0 H ) 2
SR*2 v i i i
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71AKMIA
71ACSCA
74 SPHCA
SR*2 IX
69 4C,C4
to ZKKX4
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72 ACSCA
14 SP,C6
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74 SPHC4
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70 Z&AC4
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70 JSSC8
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R. D. S H A N N O N
Table 2
72 ACBCA
TC*~ VI
67 STBGA
I C ÷ 7 IV
6 g MILER
71ZRACA
7E*4 IV
b 9 ACBCA
7 l ACBGA
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72 ACBCA
13 ACBCA
VII
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6 5 ACCRA
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Vl
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74 MRBUA
7 0 JPCSA
V*4
V
b5 ACGRA
b l JCPSA
73 ACBCA
73 ACBCA
V*4
Vi
72 JSSC8
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72 PRVBA
74 ACBCA
71ACSAA
7 0 ACSAA
74 PRVBA
V*5
IV
be RCBCA
6O CHPLB
6 7 ACSAA
70 Z K • K •
71 JESTS
71ACBCA
70 INOCA
1L CJCHA
73 ACBCA
72 JSSCB
13 CJCHA
71CJCHA
73 JSSCB
12 CJCHA
73 CJCHA
73 ACBCA
73 ACBCA
73 CJCHA
757
(cont.)
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32
73 ACSAA
27
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3
ZN*Z V
70 JSSCB
1
73 CJCHA
51
7 l AMNIA
56
ZN*Z Vl
b5 CJEHA
43
6 8 SPNCA
13
70 JSSCB
1
71CJCHA
49
71AHMIA
56
73 ACUCA
29
73 ACSAA
27
ZR*4 IV
75 JSSCB
13
ZR*4 V
b 9 CCJOA 1 9 6 9
70 JSSCe
2
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Z5
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371
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71ACBCA
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52
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6 e ACBCA
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26
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53
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27
73 ACSAA
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27
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E3
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71ACBCA
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56
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2
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27
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71ACBCA
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27
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24
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129 SR2 NI TE Ob
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24 3 1 7 8 75 l 0 H I 6
66 ACSAA
2 0 153S TE F6
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6 5 ZAACA 3 3 4
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6 8 CHDBA 2 6 7 1433 CUE TE Ob
b 9 NOCMB 100 1809 AGE TE 0 2 l 0 H I 4
71 B U F C •
94
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73 ACBCA
29
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29
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40
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73 ACBCA
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70 ACBCA
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71ACBCA
27 2 2 T q I N H 4 1 3 TH FT
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20
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20
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30
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71,4 VI
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131
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136
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139
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11
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11
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30
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52
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66 JC$1A 196B
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75 • C B C •
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71ZAACA
73 ZAACA
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74 ZAACA
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72 ZAACA
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74 1 A R I A
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74 ZAACA
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R.M.G.MVC•0FF,CRYSTAL STRUCTURES,WILEY,
N.Y.eI96B
H.BARNIGHAUSEN ET A L . , P R O C . 1 0 7 H R . E .
RES.CONF.CAREEREE,AR|ZII973)P.490
C.BRANOLEeH.STEiNFIN•.PROC.7TH
R.E.
RES.CONF.eCORONAOO,CAL.OCT 2 8 , 1 9 6 §
R.D.SHANNON,U.S.PAT.3&bSIBLtNAYIb.1972
W.H.BAUR,NITROGEN,HANOBOOK OF GEUCHEM.
SPRINGER-VERLAG,N.Y.1974
A.M.SLEIGHTtU.S.PAT.38&9544,NOV
19,1974
H.BARNIGHAUSEN,PERSONAL COMMUNICATION
A.M.SLEIGMTePERSONAL COMMUNICATION
C.C•LVOePERSONAL COMMUNICATION
C . T . P R E M I T T , P E R S O N A L COMMUNICATION
M . H . B A U R , P E R S O N A L COMMUNICATION
REF 7
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REF S
REF b
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2 1 8 4 K3 V OZ C2 O 4 . 3 H Z
1743 GU VZ 0 6
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4 1119 vz 05
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270
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1 2 0 ZN2 P2 0 7
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REF 2
REF 3
3~ 16~BN, v 0 3
V*5
Y*~
B23 R l U * 3 !
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758
REVISED EFFECTIVE
I O N I C R A D I I IN H A L I D E S
compounds to be slightly larger than those of the E u 2 +
compounds. This difference was assumed to exist for
all Sr z+ and Eu 2+ coordinations. Because compounds
of Am 2+ and Sr 2+ have similar cell volumes, the radius
of Am 2+ was made equal to that of Sr 2+
Wolfe & Newnham (1969) studied Bi4_xRExTi3012
and concluded that Bi3÷ and La 3+ have nearly equal
radii. From a study of BiTaO4 Sleight & Jones (1975)
have concluded that although Bia + and La 3÷ have essentially equal radii, the size of Bi3÷ depends on the
degree of the 6s 2 lone-pair character. When BiTaO4
transforms from a structure where the lone-pair character is dominant to the LaTaO4 structure, it undergoes
a volume reduction. Table 3 shows a comparison of
isotypic Bia+ and La a+ compounds where the lone-pair
character of Bi3+ is (1) constrained and (2) dominant.
Bi pyrochlores such as Bi2Ru207, Bi2Ir207 and Bi2Pt207
were omitted from the table because no corresponding
La pyrochlore exists, but they have unit-cell volumes
close to those of the Sm or Nd pyrochlores and thus
have smaller volumes than those of La. When Bi3÷ is
forced into high symmetry, a Bi 3+ compound has a
smaller volume than that of La 3+, but when the lonepair character is dominant, the Bi a+ compound is distorted and Bi 3÷ and La a+ compounds have approximately equal volumes. This behavior was also noted
in the highly symmetric garnet structure where the
hypothetical BiaFesOlz was estimated to have cell
dimensions between those of the hypothetical
NdaFesO12 and Pr3FesO12 (Geller, Williams, Espinosa,
Sherwood & Gilleo, 1963). For practical purposes,
Bi a+ is listed as slightly smaller than La 3+ but this
dependence on lone-pair character must be kept in
mind when comparing the volumes of Bi a + and La a+
compounds. Similar behavior may also exist for Pb 2+
and Sr 2+, but this relationship was not investigated.
Table 3. Cell volumes of isotypic Bi 3+ and La 3+
compounds
(a) Lone pair character of Bi a+ constrained
Compound
BiLi(MoO4)2
LaLi(MoOa)z
BiNa(MoO4)z
LaNa(MoO4)2
BiOF
LaOF
Cell volume
314.7
328.7
320-5
332.1
87.6
97.7
Ratio
0.96
110.7
116.8
123.8
126.4
293-0
304.7
0.95
BiOCI
LaOC1
BiOBr
LaOBr
BiPO4
LaPO4
(b) Lone pair character of Bi a +
BizMoO6
LazMoO6
BiFeO3
LaFeO3
Bi2Sn207
La2SnzO7
dominant
268.5 ( × 8)
267.3
62.49 ( x 6)
60.77 ( x 4)
1219.9 ( x 8)
1225.3
AND CHALCOGENIDES
A similar study of relative cell volumes of isotypic
compounds involving the pairs Cu÷-Li +, Ag+-Na +,
TI+-Rb ÷, and pb2+-Sr 2÷ was used to obtain more
reliable estimates of the radii of Cu +, Ag +, TI ÷, and
Pb 2÷ (Shannon & Gumerman, 1975).
The nature of Sn 2÷, NH~-, and H - made it impossible to define their ionic radii. The coordination of
Sn 2÷ by oxygen or fluorine is always extremely irregular,* leading to average distances which depend
on the degree of distortion. Since this distortion varies
widely from one compound to another, it is not meaningful to define an ionic radius.
Khan & Baur (1972) derived an apparent radius of
the NH4+ ion by analyzing the N - O distances in a large
number of ammonium salts. They concluded that
NH + has an octahedral radius of 1.61 A, between that
of Rb ÷ (1.52 A) and Cs ÷ (1.67 A). Alternatively, cell
volumes of NH~ and Rb ÷ fluorides, chlorides,
bromides, iodides and oxides may be compared. This
leads to the conclusion that N H ~ is not significantly
different in size from Rb ÷. No explanation is offered
for this inconsistency and therefore the radius of NH~
is not included.
The radius of the hydride ion, H - , has been the
subject of some controversy. A number of different
radii have been proposed: 2.08 (Pauling, 1960); 1.40
(Gibb, 1962); and 1.53 A (Morris & Reed, 1965). Gibb
studied interatomic distances in many hydrides and
concluded that good agreement between observed and
calculated distances could be obtained using r(V~H -) =
1.40 A if corrected for cation and anion coordination.
The value of r(IVH -) was taken to be 1.22 A.
Morris & Reed (1965) concluded that differences in
observed distances in hydrides were caused by the
large H - polarizability. Because of such wide variations in the apparent H - radius, it was omitted. However, an explanation for the variations based on covalence differences will be discussed later.
* Although cell dimensions of Sn2M207 pyrochlores were
used in SP 69 to derive r(VmSn2+), Stewart, Knop, Meads &
Parker (1973) and Birchall & Sleight (1975) recently found that
the pyrochlore A site in Sn2Ta207 is not fully occupied. Thus,
even this example of apparently regular Sn 2÷ polyhedra is not
valid.
0"97
I
.90
0.90
.80
I
I
I
I
V
0.98
0.96
.70
1-00
.50
1"03
,40
1"00
~---------"~ To "91'
-
I
+2
r
I
I
4"5
+4
+5
OXIDATION STATE
Fig. 1. Effective ionic radius (,~)
vs
I
.+6
+7
oxidation state.
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•
R.
D.
SHANNON
Results and discussion
In Table 1 two sets of radii are included. The first is a
set of traditional radii based on r(V'Oz-) = 1.40 A. The
2.oo1---~--~
. . . .
I
r -
I
--
I
I
1
I
1.901
759
other set is based on r(WO z-) = 1.26 and r(V~F-)= 1.19
A,, and corresponds to crystal radii as defined by Fumi
& Tosi (1964). As pointed out in SP 69, crystal radii
differ from traditional radii only by a constant factor
1.70
I
J
. 1.60
1.50
1.40
1.30
,
. . . . .
1.50
Rbd
1.70
,
1.60
Cs -
~iaoi
,
1.40
1.30
1.20
1.10
r
1.00
.90
1.20
.80
~
1.10
Y0
1.00
.60
•
.90
Li +
Ni2-~
.50
80
Be +
.4O
.70
.30
.60
.20
.50~
I
4
I
I
I
I
5
6
7
8
CN
I
I
9
10
ill
12
I
.10
13
2
CN
(a)
I
I
1
•
1.20
(b)
I
.
I
I
.
- ~Bi3+
. . . .
,
1.10
J
1.00
•
j .
I
i
I
I
la3F+~]
~.~TL3÷
1.30
.90
.80
p 3+
.70
.60
.50
A0
.30
_ B
°/ ~
~
3
Go3+
1.10
~_---~-Tb 3+
AI 3+
~Tm
1.00
-
Er
..
~""~'~"Lu-'yb3÷
-.
.90
+
.20
Zi
.10
=
.7
0
CN
I
7
I
8
9
CN
(c)
F i g . 2.
I
6
.
(d)
(a)-(e)
Effective ionic radius ( • )
vs C N
for some common cations.
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1
10
!i
]
12
760
REVISED EFFECTIVE
I
I
t
I
I
I O N I C R A D I I IN H A L I D E S
p.4+
VXOs6+, VlRe6+, and V1Os7+. The symbol A means that
Ahrens (1952) ionic radius was used whereas P means
Pauling's (1960) crystal radius was used. The symbol
M means that the radius was derived from a comp o u n d having metallic conductivity. Distances calculated from these radii may be too small for use in
compounds having localized electrons. (See discussion
Bk 4+
Effects of electron delocalization.)
I
I
I
'1
Th 4÷
1.20
i.I0
!.00
~
.90
:Hf4 +
.80
Ti 4+
~/4+
.70
.60
/
.50
.40
.30
y
AND CHALCOGENIDES
Ge4+
Si +
.20
In addition, the sources of the radii are indicated
in Table 2.
Fig. 2(a)-(e) shows that r - C N plots are reasonably
regular. Notable exceptions are IVNa+, VNa+, and
~VK+. It is apparent that N a - O and K - O distances do
not decrease as much as anticipated from the r - C N
curve* when the CN falls below six. Typical distances
and corresponding radii in Table 4 show that N a - O
distances in four-coordination are only slightly less
than in six-coordination. The reduction in interatomic
distances is caused primarily by the decreased repulsive
forces due to fewer ligands according to the expression
of Pauling (1960):
Rc.<c~_[ANacl Bc,c~_]~I('-~)
.10
RNaCl
CN
(e)
Fig. 2. (cont.)
of 0.14 ~ . Although their inclusion in Table 1 may seem
superfluous, it is felt that crystal radii correspond more
closely to the physical size of ions in a solid. They
should be used, for example, in discussions of closest
packing of spheres, structure field maps (Muller &
Roy, 1974), and diffusion in solids (Flygare & Huggins,
1973). Traditional radii have been retained because of
their familiarity to crystal chemists and physicists.
They will probably continue to be used for comparison
of unit-cell volumes and interatomic distances. In the
table, the ion is followed by electron configuration
(EC), coordination number (CN), spin state (SP),
crystal radius (CR), and effective ionic radius (IR),
and in the last column, a symbol indicating the derivation of the radii and their reliability. Those with a
question mark are doubtful because of: uncertainty in
CN, or deviation from radii vs CN, or radii v s valence
plots. Where at least five structural determinations
resulted in radii differing by no more than + 0.01 A,
the values are marked with an asterisk.
When the choice of a radius was influenced by any
of the various correlations described earlier, it is indicated by the following: R - from r 3 vs unit cell volume
plots; C - calculated from bond length-bond strength
equations; E - estimated from one or more plots of r
v s valence, r v s CN, and r vs cell volume. E implies poor
or nonexistent structural data. Radii in this category
include VIFeZ+LS, WMn2+LS, vlcIa+LS, vIV2+,
VINo2+ ' VINia+HS, wit3+, WMo 3+, VITa3+, Wpaa+,
VlTa4+ ' IVpb4+ ' V~IrS+' WOsS+ ' WReS+, WpuS+, WBiS+,
L Acscl BNaCl
where R = i n t e r a t o m i c distance, A = M a d e l u n g constant, B = t h e cation CN and n = Born repulsion coefficient. It appears that this equation is not valid for
four-coordinated Na ÷ or K +.
There are a few small irregularities in r - C N plots
probably caused by poor or insufficient data, e.g.
curves for TI 3+ vs y3+. The differences in slopes of
Ti 4+ VS C r 4+ and V s + vs As 5÷ are probably caused by
Ti4+-O and V5+-O octahedra being generally more
distorted, which leads to greater average interatomic
distances.
It is also interesting to compare distances in square
planar coordination v e r s u s tetrahedral coordination.
Radii of square planar Cu 2÷ and Ag ÷ are equal to or
slightly greater than corresponding tetrahedral radii,
consistent with the trend anticipated from anion
* Extrapolation of the Na curve gives r(~VNa÷)=0-90 A.
\
Ul S b . . . . . .
"
-' .'!ll,
[ \ , %.
i\",
(.9
Z
"'
n,,
.~%
~
........
I- - -t~ --~,,
',
ci
I
I
!
i
7
!I
ii
I",,:,
I
I
I
i
I
_1
I
°
"' s,
.......
~
~
',
"__'__i
Ti- - Ti ~ I i
- ]l , ~
I
Rb R! R2
R,
AVERAGE INTERATOMIC DISTANCE , R
Fig. 3. Typical bond length vs bond strength plot.
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R. D. S H A N N O N
repulsion effects. A similar comparison with Fe 2+ and
Ni 2+ cannot be made because of electron distribution
changes from tetrahedral to square planar coordination.
I
i
2.05
,
-v'r NbS+
2.03
2.02
2.01
~¢~)
2.00
1.99
1.97
1.96
1.95
ii
1.94
i
•002
[
i
.004 .006 .008
i
i
I
.010 .012 .014 .016
Fig. 4. Mean NbS+-O b o n d length v s distortion. Vertical bars
represent average e.s.d.'s quoted by the authors. Solid
circles represent m o r e accurate data.
r
2.o0
i
i
"v'r M06 +
1:98
1.97
1.95
1.94
J
1.911
1.90
R (/~,)
2.40
2.37
2.36
2.39
Mean 2-38
2.38
2-37
2"39
2.48
2.44
2.41
2-415
2.45
2-460
2.375
2-406
2.42
r (/~)
Reference
1-02
0.99
1.00
0.99
1.00
60 A C C R A 13
57 A C C R A 10
69 Z A A C A 4 0 9
1.00
1.01
1-02
1.10
1.06
1.04
1.05
1.10
1-09
1.025
1.046
1.05
74
65
70
64
65
67
61
67
63
59
75
A C B C A 30
A C C R A 19
A C S A A 24
A C C R A 17
A C C R A 18
S C I E A 154
A C C R A 14
A C C R A 22
A C C R A 16
A C C R A 12
A C B C A 31
263
462
69
1872
561
1287
672
818
1453
555
182
1233
526
19
Additivity of radii to give mean interatomic distances
is not so important to the synthetic chemist who is
primarily interested in ionic radii for predicting substitution in crystal structures. Crystallographers and
physicists, however, are concerned with comparing
calculated and experimental interatomic distances and
predicting distances, e.g. for distance least-squares
(DLS) structure refinements (Baur, 1972; Tillmanns,
Gebert & Baur, 1973; Dempsey & Strens, 1975). The
effective ionic radii in Table 1 can be used to reproduce
moderately well most average interatomic distances in
oxides and fluorides. However, certain deviations do
occur. Some of these are unexplained but others can
be attributed to (1) polyhedral distortion, (2) covalence,
(3) partial occupancy of cation sites, or (4) electron
delocalization.
R=1.920 + 3 . 7 3 A
1.99
1.93
Compound
(a) lVNa+
Na20
NasP3010
NaOH.H20
Na6ZnO4
Factors affecting mean interatomic distances
1.9e
2.01
Table 4. lnteratomic distances in some compounds
containing tetrahedral and octahedral Na +
(b) VlNa+
Na2WO4
NaC6OTH7
Na4Sn2GeaO12(OH)a
Na2P2OT. 10H20
NaHCO3
Na2B406(OH)2.3H20
Na4P4OI2.4H20
NaAI(SO4)2.12H20
NaB(OH)4.2H20
N a U acetate
CloHI3NsNaO6P.6H20
Mean
R= 1.976 + 6.45 A
2.04
761
~
.002
,
.004
F i g . 5. M e a n
~
,
.006 .O08
M06+-O
v
.010
Z~
bond
,
.012
~
.014
,
.016
.018
l e n g t h vs d i s t o r t i o n .
1. Polyhedral distortion
To see the effects of polyhedral distortion consider
the relationship between bond length (R) and Pauling
bond strength (s) (Brown & Shannon, 1973). The analytical expression S=So(R/Ro) -N, where So is an ideal
bond strength associated with R0, and R0 and N are
fitted parameters, was evaluated for cation-oxygen
pairs for the first three rows of the periodic table. Using
these relationships, the sums of bond strengths about
cations and anions were found to equal the valences
with a mean deviation of about 5 %. Accepting the
approximate validity of Pauling's second rule, p = ~s
where p = valence, it is possible to derive the effects of
distortion of various polyhedra on their mean bond
distances. Fig. 3 shows a typical R-s curve. An undistorted octahedron results in an average bond
strength g and a mean distance/~a. A distorted octahedron with three bonds of length R= and three of
length Rb results in the same average bond strength, g,
but a mean distance Rz >/~1.
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762
REVISED
EFFECTIVE
IONIC
RADII
The effects of distortion on m e a n bond lengths in
numerous polyhedra have been determined. Although
distortions in tetrahedra are not as important as in
octahedra, they can contribute to variations in m e a n
tetrahedral distances (Baur, 1974; Hawthorne, 1973).
Strongly distorted octahedra like those containing V 5+,
Cu 2+, and M n 3+ show a significant variation in m e a n
distance with distortion, A* (Brown & Shannon, 1973;
S h a n n o n & Calvo, 1973a; Shannon, G u m e r m a n &
Chenavas, 1975). Octahedra containing Mg 2+, Zn z+,
Co 2+, and Li + are generally less distorted than those
of V 5+, Cu 2+, and M n 3+ and show a less pronounced
dependence on m e a n bond length (Brown & Shannon,
1973).
The effects of distortion on m e a n bond lengths in
N b S + , O and M o 6 + - O octahedra are illustrated in
Figs. 4 and 5. Tables 5 and 6 list the data used to derive
the figures.
Table 7 lists the results of linear regression analyses
of m e a n bond length on distortion for all octahedra
studied. It is clear from Fig. 4 that undistorted N b 5+
octahedra in pyrochlores have a distinctly smaller mean
value than in compounds like NbOPO4, CaNb206, and
NaaNbO4. Most of the accurately refined molybdates
have relatively distorted octahedra. However, certain
ordered perovskites with no octahedral distortion such
as BazCaMoO6 would be expected to have much
smaller m e a n Mo6+-O distances than a typical molyb* Octahedral distortion is defined by A=~Y(R~-R/R) z
where R=average bond length and Rl=an individual bond
length.
IN HALIDES
AND
CHALCOGENIDES
date. In fact, the M o 6 + - O octahedra in Mo2(O2C6C14)6
with a very small distortion have the short mean distance of 1.919 A.
Table 7 also lists the results of regression analyses
for TaS+-O and W 6 + - O octahedra but they are only
approximate because of the scarcity of accurate structural data. Analysis of Ti4+-O octahedra was unsuccessful because of scatter in the data. Distances in
Ba6Ti17040 (Tillmanns & Baur, 1970) and BaTiO3
(Evans, 1951) deviated significantly from a linear
relation.
Relations between m e a n distance and distortion
should be particularly useful to help determine oxidation states in mixed valence compounds with such
combinations as MoS+-Mo 6+, W s + - W 6+, v a + - v 5+,
Nb4+-Nb 5+ and Mn3+-Mn 4+. Such considerations
helped rationalize M n - O distances in NaMnvO12 and
the mineral pinakiolite (Shannon, G u m e r m a n &
Chenavas, 1975).
The radii in Table 1 are generally derived for an
average degree of distortion. Thus, interatomic distances calculated from these radii m a y be inaccurate
if the distortion in a particular c o m p o u n d is much less
or greater than usual. This applies particularly to cations whose polyhedra frequently show a large distortion, e.g. MO 6+, N b s+, V 5+, Ba 2+, and the alkali ions.
2. Effects of partial occupancy of cation sites on mean
cation-anion distances
In compounds with partially occupied sites, abnormally large c a t i o n - a n i o n distances are usually
found, as expected if the anions surrounding unoc-
Table 5. Compar&on of mean octahedral N b s + - O distances with distortion
Only structures with e.s.d.'s for Nb-O distances of < 0.025 ~ were used.
Compound
HgzNb207
Cd2Nb207
NazNb4Ol i
Ba0.27Sr0.75NbzOs.Ts
NalaNbasO94
Ba3Si4Nb6026
Na13Nb3sO94
Nax3Nb35094
Na13NbasO94
NaNbO3
NalaNb35094
Na13Nb35094
Na13Nb35Og4
Na13Nb35094
LiNb308
LiNbO3
Ca2Nb207
Ca2Nb207
SbNbO4
KNbO3
NaaNbO4
CazNb207
Ca2Nb207
Na3NbOa
CaNb206
GaNbOa
/~ (/~)
1.999
1.957
1.977
1.967
1"965
1"989
1"967
1.959
1"964
1.985
1"947
1"991
1"987
1"978
1-993
2"000
1.997
2.005
2.003
2.011
2.013
2.010
.2.015
2.021
2.021
2.031
Distortion
/t = ((AR/R) 2) × 104
0
0
1
6
7
9
11
12
12
16
18
22
22
24
28
31
31
34
37
42
52
53
58
60
76
83
Reference
68 INOCA
7
72 CJCHA
50
70 JSSCB
1
61 JCPSA
48
71 JSSCB
3
70 ACBCA
26
71 JSSCB
3
71 JSSCB
3
71 JSSCB
3
69 ACBCA
25
71 JSSCB
3
71 JSSCB
3
71 JSSCB
3
71 JSSCB
3
71 ACSAA
25
66 JPCSA
27
74 JINCA
36
74 JINCA
36
65 CCJDA
1965
67 ACACA
22
74 BUFCA
97
74 JINCA
36
74 JINCA
36
74 BUFCA
97
70 AMMIA
55
65 ACACA
18
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1704
3648
454
5048
89
102
89
89
89
851
89
89
89
89
3337
997
1965
1965
611
639
3
1965
1965
3
90
874
R. D. SHANNON
763
T a b l e 6. Comparison o f mean octahedral M o 6 + - O distances with distortion
Only structures with e.s.d.'s for M o - O distances of < 0.025 A were used.
Compound
Mo2(O2C6C14)6
Mo4On
Mo4On
Mo4On
Mo4On
Mo4Ot~
Mo4On
orthorhombic
monoclinic
monoclinic
orthorhombic
orthorhombic
monoclinic
(ClsHnO2)2MoO2
(NH4)6[M07024].4H20
(NH4)6[MoTOz4]. 4 H 2 0
(NH4)6[M07024].4H20
LiMoO2AsO4
(NH4)6MoaO27.4H20
HgMoO4
(NH4)6[MoTO24]. 4H20
(NH4)6[Mo7024]. 4H20 •
(NH4)6[MoTO24]. 4 H 2 0
MoOa .2H20
MoOa. 2H20
MOO3.2H20
MOO3.2H20
(NHg)5[MoOa)5(PO4) (HPO4)]. 3H20
Naa(CrMo6024H~). 8HzO
(NH4)6MoaO27.4H20
Na3CrMo6Oz4H6.8H20
(NH4)5[(MoO3)5(PO4) ( H P O 4 ) ] . 3 H 2 0
(NH4)dTeMo6024]. Te(OH)6.7HzO
CoMoO4
(NHg)6MoaO27.4H20
MoO3
(NH4)6[Mo7024].4H20
K2{[MoO2(C204) (H20)]20}
(NH4)6MoaO27.4H20
(NH4)s[(MoOz)s(PO4) (HPO4)]. 3H20
Na3CrMo6Oz4H6.8H20
(NH4)5[(MoOa)s(PO4) (HPO4)]. 3HzO
MOO3.H20
(NH4)~[(MoO3)5(PO4) (HPO4)]. 3H20
(NH4)6[ Mo7024]. 4 H 2 0
(NH4)6[Mo7Oz4]. 4 H 2 0
"
R (A)
1"919
1.944
1.946
1.937
1"951
1.911
1.945
1"952
1"962
1"972
1"960
1"967
1"960
1"965
1 "955
1"962
1.974
1.966
] "961
1"957
1"953
1"970
1.976
1.976
1.976
1.974
1.981
1.991
1.972
1.981
1"976
1"976
1"974
1"982
1.986
1 "977
1-984
1 "991
1"991
2"008
Distortion
zi = ((AR/R) z) x 10'
5
9
10
56
67
96
96
99
99
101
104
104
106
111
113
115
118
121
123
126
134
140
141
141
143
145
147
150
151
151
152
152
152
159
163
167
167
186
189
197
Reference
75 J A C S A
97
63 A R K E A
21
63 A R K E A
21
63 A R K E A
21
63 A R K E A
21
63 A R K E A
21
63 A R K E A
21
74 A C B C A
30
75 JCSIA
1975
75 JCSIA
1975
75 J C S I A
1975
70 A C S A A
24
74 A C B C A
30
73 A C B C A
29
75 JCSIA
1975
75 JCSIA
1975
68 J A C S A
90
72 ACBCA
28 .
72 A C B C A
28
72 A C B C A
28
72 A C B C A
28
74 JCSIA
1974
70 I N O C A
9
74 A C B C A
30
70 I N O C A
9
74 JCSIA
1974
74 A C B C A
30
65 A C A C A
19
74 A C B C A
30
63 A R K E A
21
68 J A C S A
90
64 I N O C A
3
74 A C B C A
30
74 J C S I A
1974
70 I N O C A
9
74 J C S I A
1974
74 A C B C A
30
74 J C S I A
1974
75 JCSIA
1975
75 JCSIA
1975
2123
365
365
365
365
365
365
300
505
505
505
' 3711
48
869
505
505
3275
2222
2222
2222
2222
941
2228
48
2228
941 2095
269
48
357
3275
1603
48
941
2228
941
1795
941
505
505
T a b l e 7. Variation o f mean M - O distance and effective ionic radius in octahedral environments as a function o f distortion
Maximum
Correlation G o o d n e s s
Ion
A x 104
N*
Rot
r0:l:
m
coefficient of fit ( x 103)
M o ~+
212
38
1.920
3-73
0.74
67
0.572
3.01
0-63
70
W e+
122
7
1-925
3-30
0.75
19
0"565
3.28
0.66
24
V 5+
576
16
1.887
2.62
0.98
8
N b s÷
83
29
1.976
6.45
0.69
71
0.599
6.83
0.44
99
Ta 5+
79
6
1.984
6.70
0.81
18
0.617
3.79
0-15
46
M n 3+
71
15
1"994
7"08
0-82
30
0-624
6.15
0"54
50
Cu 2+
316
26
2.085
3"99
0"82
77
Mg 2÷
156
28
2.094
8.31
0.72
21
'
0"728
8"86
0"77
• 18
Co 2+
46
15
2"106
7"38
0"42
19
0"734
11 "70
0-70
16
Zn 2 +
71
16
2"099
7"70
0-64
21
0"736
8"20
0.74
16
Li ÷
148
il
2"159
8"42
0"81
30
0"784
9.02
0-79
35
* N = n u m b e r of independent octahedra
t R=Ro+mA.
r = ro+ mzl.
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764
R E V I S E D E F F E C T I V E I O N I C R A D I I IN H A L I D E S A N D C H A L C O G E N I D E S
cupied sites relax toward their bonded cation neighbors. Therefore average distances should increase as
the occupancy factor decreases. In general, partial occupancy seems to be more prevalent for cations which
are weakly bonded to oxygen like Cu ÷, Ag ÷, alkali
ions, and large alkaline earths. The most prominent
examples are Li and Na compounds. Table 8 summarizes the existent data on some structures with partial cation occupancy. Fig. 6 shows the dependence of
mean Li-O bond length on the degree of occupancy.
Although the data are not extensive, it is apparent that
mean distance increases as occupancy factor decreases.
Extrapolation of the Li curve in Fig. 6 to zero occupancy, i.e. a tetrahedral Li vacancy, gives 2.10-2.15 A,
which is close to the 2.11 A found for c~-LisGaO4 by
Stewner & Hoppe (1971) and for fl eucryptite by
Tscherry, Schulz & Laves (1972).
Another example of the effects of partial occupancy
can be found in the non-stoichiometric feldspar
Sr0.a4Na0.03vq0.13All.69Si2.2908 reported by Grundy &
Ito (1974). The mean Sr-O distance in this compound
is 0.03 A greater than in the stoichiometric SrA1ESi2Oa
(Chiari, Calleri, Bruno & Ribbe, 1975).
The relation between mean distance and occupancy
probably cannot be quantified precisely because the
relaxation of oxygen ions will depend on the nature
and number of other cation neighbors.
3. Effects of covalence
Changes in interatomic distances due to covalence
effects are anticipated in compounds with (1) anions
less electronegative than fluorine or oxygen, i.e. chlor-
ides, bromides, sulfides, selenides, etc. and (2) tetrahedral oxyanions such as the VO]- and AsO ]- groups.
The effects of covalence show up as a lack of additivity
of the radii and are generally referred to as 'covalent
shortening'.
(a) Halides and chalcogenides. Covalence effects can
be observed by comparing the relative contraction of
cation-anion distances in two different isotypic compounds as the anion becomes less electronegative, e.g.
Fe 2+ in Fe2GeO4 and Fe2GeS4 vs Mg 2+ in Mg2GeO4
and Mg2GeS4. Covalence shortens both Fe-S and
Mg-S bonds relative to Fe-O and Mg-O bonds, but
because of the greater electronegativity of Fe 2+ (1.8)
compared to Mg 2+ (1.2), the Fe-S bonds are shortened
to a greater extent. Thus a 'covalency contraction'
parameter (Shannon & Vincent, 1974) can be defined:
d(Fe-X) a
Ra= d(Mg_X)3
where d ( F e - X ) = m e a n Fe-X distance.
A similar parameter
Rv=
V(Fer, Xn)
V(MgmXn)
compares the volume of an Fe 2+ compound with that
of an isotypic Mg 2+ compound. To see the effects of
covalence on the Fe-X distance relative to the Mg-X
distance, the ratio Rv or Rd may be plotted against the
difference in electronegativity of the Fe-X bond,
AZFe-x. Such schematic Rv-A Z plots are shown in
Fig. 7. The reference ions for Cd 2÷ and In 3+ are Ca 2+
and Sc3+ respectively. Such plots usually show a strong
Table 8. Mean distances in structures with partially occupied cation sites
Compound
(a) IVLi+
Typical
LiAISiO4 (fl eucryptite)
Occupancy
factor
Reference
Table 1
73 AMMIA
72 ZKKKA
68 ZKKKA.
69 ZKKKA
72 ZKKKA
70 ZKKKA
68 ZKKKA
71 ACBCA
72 ZKKKA
58
135
126
130
135
132
127
27
135
681
175
46
420
161
118
327
616
175
59
9
59
25
127
9
27
280
345
280
1503
94
345
1826
LiA1Si206 II (fl spodumene)
0"50
LiA1SiO4 (fl eucryptite)
Li2Al2Si3Olo
LiAISi206 III
~-LisGaO4
LiAISiO4
0-50
0-40
0.33
0.00
0.00
1.97
2.020 (4)
2.025 (7)
2-08 (4)
2-085 (9)
2.056 (2)
2.064 (4)
2.068 (5)
2.11
2.11
1.00
0.91
0.82
0.70
0.50
0"35
0.29
0.25
2.42
2.533 (6)
2.74
2.723 (6)
2.600 (9)
2"839 (I)
2.65
2.88
Table 1
74 AMMIA
74 JSSCB
74 AMMIA
69 ACBCA
68 ZKKKA
74 JSSCB
71 ACBCA
1.00
0"44
0.33
0.22
2.50
2"64
2.75
2.83
Table 1
74 JSSCB
74 JSSCB
72 JSSCB
(b) VINa+
Typical
Na2Fe2AI(PO4)3 (wyllieite)
NaSbO3
Na2Fe2AI(PO4)3 (wyllieite)
NaA1Si3Oa (high albite)
NaAlnOt7 (fl-A1203)
NaSbO3
Na2.58A121.81034 (fl-A1203)
(c) ViAg÷
Typical
AgSbO3
AgSbO3
Agz.4A122034.2.
1"00
1.00
R
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345
345
60
R.
z.15 /
,
,
,
i
|
2'10I
2.05~-
IL~
D. S H A N N O N
i
i
VACANCY
IN ,, UsC.-aO
4
AND .8 EUCRYPTITE
e
LiALSi206 T
(~ SPODUMENE)
l
LiAISi04~ ,,~
~LiAISiz06]I[
(~ EUCRYPTITE)
= UzAIzSisO~o
~UAtSi04
2"00i (,8 EUCRYPTITE)
e~TYPICAL
i .95F-Li+-O DISTANCE
I'S°/~-
i
I
t
1.0
.9
.8
.7
I
I
5
I
,
I
.2
I
,
0
OCCUPANCY
Fig. 6. Mean Li÷-O bond length us partial occupancy.
1.20
|
I
I
I
UNFILLED "d" SHELL
2+,Mn2+, C02+,N j2+
1.10
1.00
FI LLED "d" SHELL
Zn2+ Cd2+ in3+
~ - ~
Rv
or
Rd
0.90
0.80
FLUORIDES OXIDES
2.5
CHLORIDES SULFIDES/SELENIDESJ BROMIDES |
/IODIDES
1.5
2.0
1.0
0.5
0.0
L~
Fig. 7. Covalency contraction parameter, Ro or Ra, vs ,dZ for
filled and unfilled d shell cations.
1.30
t
i
i
i
i
,2o-C'-.": ~_~"
eNa-H
LIO
eLi-H
eLiBaH3
1.00
• Mg-H
0.90
Rv
aSi-H
f ~ 0.80
aAt-H
0.70
B-H
o
D P-H
a As-H
0.60
aC-H
O.5C
I
0.,~ l~
,Io
&
oo'
-~.s
N-Ho
-o9
z~x
Fig. 8. Covalency contraction parameter, R, or Rd, vs AX for
hydrides. Solid circles represent ratios of cell volumes of
isotypic compounds. Squares represent ratios of the cubed
M-H distances to the cubed M-F distances.
765
dependence of Rv on ZIZ. For Fe2+-Mg 2+ the Fe 2+
fluoride volumes are ~ 1 1 0 % of the corresponding*
Mg 2+ fluoride volumes whereas the Fe 2÷ sulfide volumes are ~ 9 6 % of the corresponding Mg z+ sulfide
volumes. Plots for the cations with filled 'd' shells show
a markedly smaller dependence on zlg. This appears
to be due to the difference in covalence of hybrid orbitals formed from metal 'd' orbitals vs metal 's-p' orbitals.
These relations show that effective ionic radii derived
primarily from oxides are not strictly applicable to
fluorides - note the change in Rv for Fe 2+, Co 2+, Ni 2+,
and Mn 2+ from fluorides to oxides. This effect is particularly noticeable in R~-ZIX plots for the pairs
Cu+-Li ÷ and Ag+-Na ÷ (Shannon & Gumerman,
1975). The Cu+-Li + and Ag+-Na ÷ plots are very steep,
e.g. the volume of AgF is 120 % of the volume of NaF,
whereas the volume of Ag2Se is only 72 % of the volume
of Na2Se. Although most of this change arises from
covalency, double repulsion effects present in the Li
and Na halides described by Pauling (1960) may also
play a role.
Covalence effects are useful in explaining certain
differences between the effective ionic radii of Table 1
and the ionic radii of Pauling (1927) and Ahrens (1952).
Pauling's radii for Cu ÷ (0.96 A) and Ag ÷ (1.26/~) are
considerably larger than those in Table 1 (0.77 and
1.15 A respectively). Since these radii were derived
from comparison of alkali halide distances, using an
equation relating effective nuclear charge and screening
constants (Pauling, 1927), they are valid in primarily
ionic crystals. The smaller radii in Table 1 are applicable in the more covalent oxides. Extrapolation of R vs
ZIX curves such as in Fig. 7 leads to values of 0.91 /~
and 1.23 A for fluorides, which are close to Pauling's
ionic values.
A final example of covalence effects concerns
M + - H - distances. According to Gibb (1962), the radius of the hydride ion is slightly larger than the radius
of the fluoride ion. To rationalize the behavior of the
hydride ion, the M - H bond has been treated as covalent. Therefore, it is useful to make R~ vs AZ plots similar
to those just discussed for Fe 2+, Cu +, etc. In this case,
the reference ion is F - and volumes of certain hydrides
are compared to those of isotypic fluorides. The results
of this analysis are shown in Fig. 8. The solid circles
represent volume ratios, R~ = V ( M = H , ) / V ( M m F , ) ; open
squares represent ratios of typical distances Rd=
d ( M - H ) 3 / d ( M - F ) 3. In the more ionic hydrides of Cs,
Rb, K, and Na, hydride volumes are considerably
larger than those of the fluorides. For the Li and Mg
compounds, hydride and fluoride volumes are approximately equal, whereas the more covalent hydrides have
increasingly smaller relative volumes than the corresponding fluorides. Fig. 8 partly explains the differences in reported radii. The Morris & Reed (1965)
value of 1.53 A was derived essentially from the large
alkali halides, while Gibb's value of 1.40 A was derived
primarily from hydrides of the more electronegative
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766
REVISED EFFECTIVE
I O N I C R A D I I IN H A L I D E S A N D C H A L C O G E N I D E S
metals such as: Sc, Ti, Y, Zr, HI', Nb, Ta, and Th.
Because of this strong dependence of M - H distances
on cation electronegativity, it does not seem very useful
to quote a unique radius for H - .
(b) Tetrahedral oxyanions. Lack of additivity also
appears in most small tetrahedral groups and is particularly noticeable for the ions lVB3+, ~VFe3+, IVGe4+,
tVA:+, IvVS+, IVS6+, XVSe6+, and IVflT+. The deviations in vanadates have been studied in detail (Shannon
& Calvo, 1973b). Assuming that the V-O bond is
strongly covalent, and that relatively electronegative
cations such as Cu 2+, Ni 2+, and Co 2+ tend to remove
electron density from the V-O bond, a V-O bond
length increase in Cu, Ni, and Co vanadates is anticipated. Plots of mean radii (~) vs mean cation electronegativity (:~) show a marked slope with a gradual increase in ~(~vvs+) from vanadates of the alkali and
alkaline earth ions to those of Cu, Ni, and Co. Similar
plots for other ions, ps+, AsS+ (Shannon & Calvo,
1973b), B 3+, Si 4+, Se 6+ (Shannon, 1975), showed the
same behavior. The statistical data on the tetrahedra
of B3+, SIS+, Ge4+, p5+, ASS+, 56+, Se6+, Cr6+, M06+,
W 6+, and C17+ have been summarized by Shannon
(1975). The slopes of the i vs '2 plots were greatest for
V 5+, Se 6+, and C17+, and least for Si4+. Although the
evidence for covalence as the origin of these effects in
the above systems is only indirect, this behavior is
consistent with accepted ideas of 'covalent shortening'
of bonds.
The evidence for covalent shortening of lVFe3+-O
bonds is more direct. Jeitschko, Sleight, McClellan &
Weiher (1976) have found a good correlation between
(1) the Fe M6ssbauer isomer shift and mean Fe-O
distance and (2) ~ and mean Fe-O distance (R). Thus,
in fl-NaFeO2/~= 1.86/~ and 3=0.18 mm s -1 relative
to ~ Fe whereas in Bia(FeO4) (MOO4)2/~ = 1.909 A and
= 0.282 mm s- 1.
4. Effects o f electron delocalization
At a pressure of 6.5 kbar SmS (NaCI structure)
undergoes a semiconductor to metal transition and a
reduction in cell edge from 5.97 to 5.70 A. (Jayaraman,
Narayanamurti, Bucher & Maines, 1970). The reduction in cell volume was attributed to a partial conversion of Sm 2+ to Sm 3+ ; some of the electrons presumably
go into a conduction band.
Electron delocalization effects can also be seen by
comparing the volumes of the conducting V sulfides
VS, VTSs, V3S4 and VsSs with the corresponding Cr
sulfides which have localized 'd' electrons (de Vries &
Jellinek, 1974). The V compounds have volumes ~ 5 %
smaller than the corresponding chromium compounds.
This does not agree with the relative sizes of V and Cr
in oxides and fluorides, e.g. r(WV3+)=0"64 and
r(WCr3+)=0.615 A. For the sulfides, this unit-cell volume anomaly is not simply attributable to metallic vs
semiconducting behavior. While Cr3S4, Cr556, and
Cr7Sa show a positive temperature dependence of resistivity typical of a metal, magnetic susceptibility
measurements indicate Curie-Weiss behavior and
therefore nearly localized electrons (van Bruggen,
1969). This is in contrast to the Pauli paramagnetic
behavior of the corresponding V sulfides (de Vries &
Haas, 1973) characteristic of delocalized electrons.
Thus, in SmS and the sulfides of V metallic character
accompanied by electron delocalization appears to be
associated with reduced bond distances.
A further example of delocalization effects occurs in
the compound NaVS2 (Weigers, van der Meer, van
Heinigen, Kloosterboer & Alberink, 1974). The molecular volume of Pauli paramagnetic NaVS2 1 (67.9 .&3)
is significantly less than that of NaVS2 II (72.7 .&3).
NaVS2 II is characterized by localized electrons (Jellinek, 1975) and its molecular volume is consistent with
that of isotypic NaCrS2 (71.1 N3).
If electron delocalization in oxides results in reduced
metal-oxygen distances and thereby an effective increase in valence, radii derived for the ions Mo 4+,
TC4+, Ru 4+, Rh 4+, W 4+, Re 4+, Os 4+, and Ir s+ from
metallic oxides may not be reliable when applied to
insulating oxides. Thus, radii obtained from distances
in the metallic phases, e.g. RhOz, ReOz, and Cd2IrzO7,
will be smaller than radii obtained from semiconducting or insulating compounds.* When both types of
compounds have been studied, a significant difference
in distances is generally found. The mean octahedral
Re4+-O distance in insulating K4[Re202(C204)4]. 3H20
(Lis, 1975) of 2.021 (10)/~ (r=0.671 A) is greater than
the estimated mean distance in metallic ReO2 of 1-99/~
(r=0.63/k). Knop & Carlow's (1974) value o f r = 0 . 6 6 2
A derived from cell volumes of the insulating Cs2ReF6
phases is consistent with the radius of Re 4+ from
K4[Re202(CEO4)4].3H20. The ReS+-O distance in
Nd4Re2Olt (Wilhelmi, Lagervall & Muller, 1970) of
1"987 (12) ,~ (r=0.607 A) is significantly greater than
the distance in metallic Cd2Re207 (Sleight, 1975) of
1.93 (2)/k (r=0.55 A). The radii of 0"58 A derived from
XeFsRuF6 and 0.60 A from XeFRuF6 (Bartlett, Gennis, Gibler, Morrell & Zalkin, 1973) are greater than
the radius of 0.565 A derived from the r a - V plot for
metallic Cd2Ru207. In contrast, however, the Mo 4+
radius of 0.64 /~ derived from insulating Li2MoF6
(Brunton, 1971) is not greatly different from the radius
of 0.65 A derived from metallic MoO2 (Brandt &
Skapski, 1967).
Although there appears to be ample evidence to
show that M - O bond distances in compounds with
localized electrons are greater than M - O distances in
compounds with delocalized electrons, the data are not
yet sufficient to derive a reliable set of radii for semicon2ucting compounds4containing+Mo4+, Tc 4+ , Ru 4+ ,
Rh , W , Re , Os , and Ir . This will become
possible as additional accurate structure refinements of
fluorides, molecular inorganic compounds, and semiconducting oxides containing these ions become available.
* This assumes that metallic character can be equated with
delocalized electron behavior in these compounds.
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R. D. S H A N N O N
I would like to acknowledge the help of F. Jellinek
for providing unpublished data on NaVS2, F. C. Hawthorne for pointing out numerous structures containing
partially occupied cation sites, O. Muller for several
sources of radii of unusual ions, M. Fouassier for unpublished data on K4MO4 compounds, I. D. Brown for
unpublished bond length-bond strength curves, and P. S.
Gumerman for assistance with data collection. Structure data on rare earth halides and an analysis of the
radii of divalent rare earths provided by H. Bg.rnighausen were especially valuable. I am particularly indebted to Ruth Shannon for the tabulation of data and
proof reading. Finally, I would like to thank R. J.
Bouchard, W. H. Baur, and H. B/irnighausen for
critically reviewing the manuscript prior to publication.
References
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A C 32A - 2
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Hints for the future...
Preparation for X-ray measurement
Write down cell constants for your ligand / possible by-products (use CCDC database
for organic compounds / ICSD database for inorganic compounds).
Check if different crystal forms are present in your sample.
Consider an EDX measurement for your crystals, especially if different forms are
present in your sample.
Things for noting down in one place
Date (reaction & X-ray measurement)
Crystal code
Reaction number
Crystal shape and colour
Crystal size
Experimental details: if the crystal was cut, if it decomposed quickly, if twinning was
detectable under microscope etc.
Diffractometer type (IPDS2? IPDS2t? IPDS1?
Temperature (note: e.g. on IPDS1 it is not 100 K, phase transitions require sometimes
different temperatures etc.)
Cell constants (after "Postrefine") - please pay attention to correct estimated standard
deviations
Number of reflections used to determine cell constants (with theta min, theta max - all
taken from SUM file)
Absorption correction type, absorption coefficient (make sure that it is calculated for
the right formula), Tmin, Tmax (note: always denote if absorption correction was
introduced in file names: e.g. use "fileabs" name. Introducing absorption correction to
an already corrected dataset is physically meaningless!).
If data reduction was carried out on full measurement data, A, B, EMS parameters
(useful when optimization of integration is needed).
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Files for saving in your structure directory (extensions)
X, SUM, HKL (not overwritten by XPREP - to retain direction cosines; different
names for files corrected/not corrected for absorption), CRS, LXR (after absorption
correction).
It is recommended to keep your own copy of the whole measurement folder.
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443
Acta Cryst. (1998). B54, 443±449
Re®nement of Twinned Structures with SHELXL97
Regine Herbst-Irmer* and George M. Sheldrick
Institut fuÈr Anorganische Chemie, UniversitaÈt GoÈttingen, Tammannstrasse 4, D-37077 GoÈttingen, Germany.
E-mail: rherbst@shelx.uni-ac.gwdg.de
(Received 1 October 1997; accepted 1 December 1997 )
Abstract
Four examples of the re®nement of twinned structures
are discussed. These structures illustrate three different
types of twins: twinning by merohedry, pseudo-merohedry and non-merohedral twins. How the twinning was
detected, how the structures were solved and how they
were re®ned are shown in this paper. The difference
between a re®nement as a disordered model and a
twinned model is illustrated. Sometimes the twin law
was necessary to solve the structure, while in the other
examples the twinning was ®rst recognized during
re®nement of the structures. Finally, a list of `characteristic warning signs' is presented which are indicative of
possible twinning.
1. Introduction
A typical de®nition of a twinned crystal is the following:
`Twins are regular aggregates consisting of crystals of the
same species joined together in some de®nite mutual
orientation' (Giacovazzo, 1992). Therefore, for the
description of a twin two things are necessary: a
description of the orientation of the different species
relative to each other (the twin law) and the fractional
contribution of each component. The twin law can be
expressed as a matrix which transforms the hkl indices of
one species into the other.
In SHELXL (Sheldrick, 1998) the twin re®nement
method of Pratt et al. (1971) and Jameson (1982) has
been implemented. F2c values are calculated by
Fc2 † ˆ …o:s:f:†2
n
X
mˆ1
km Fc2m
1†
where o.s.f. is the overall scale factor, km is the fractional
contribution of twin domain m, Fcm is the calculated
structure factor of twin domain m and n is the number of
twin domains. The sum of the fractional contributions km
must be unity, so (n ÿ 1) of them can be re®ned and k1 is
calculated by
k1 ˆ 1 ÿ
n
X
mˆ2
km :
Four types of twins may be distinguished:
# 1998 International Union of Crystallography
Printed in Great Britain ± all rights reserved
2†
(a) Twins by merohedry: The twin law is a symmetry
operator of the crystal system, but not of the point group
of the crystal. This means that the reciprocal lattices of
the different twin domains superimpose exactly and the
twinning is not directly detectable from the re¯ection
pattern. Two types are possible:
(i) The twin operator belongs to the Laue group, but
not to the point group of the crystal. These crystals are
racemic twins. There are no special problems in solving
and re®ning such structures. The only question to be
resolved is the determination of the absolute structure.
Even if the determination of the absolute con®guration
is not one of the aims of the structure determination, it is
important to re®ne any non-centrosymmetric structure
as the correct absolute structure in order to avoid
introducing systematic errors into the bond lengths etc.
(Cruickshank & McDonald, 1967). In some cases the
absolute structure will be known with certainty (e.g.
proteins), but in others it has to be deduced from the Xray data. Generally speaking, a single phosphorus or
heavier atom suf®ces to determine an absolute structure
using Cu K radiation and with accurate high-resolution
low-temperature data, including Friedel opposites, such
an atom may even suf®ce for Mo K.
The de®nition of the Flack parameter (Flack, 1983;
Bernardinelli & Flack, 1985) is a special case of (1)
Fc2 † ˆ …1 ÿ x†Fc2 …hkl† ‡ xFc2 …hkl†
3†
where x is the fractional contribution of the inverted
component of a `racemic twin'; it should be zero if the
absolute structure is correct, unity if it has to be inverted
and somewhere in-between if racemic twinning is really
present. Thus, the above formulae apply with n = 2 and
the twin law R = (1Å00, 01Å0, 001Å).
(ii) The twin operator is part of the symmetry of a
lattice point (holohedry). It does not belong to the Laue
group of the crystal. This type is possible in the trigonal,
tetragonal, hexagonal and cubic crystal systems. If the
different twin domains have similar contributions, the
re¯ection intensities appear to possess a higher
symmetry than the true structure. The determination of
the correct space group and solving the structure can be
dif®cult. Patterson or molecular replacement methods
may solve the problem. This type of merohedral twinActa Crystallographica Section B
ISSN 0108-7681 # 1998
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444
REFINEMENT OF TWINNED STRUCTURES
ning may, at least in theory, occur simultaneously with
racemic twinning (i).
(b) Twinning by pseudo-merohedry: The twin
operator belongs to a higher crystal system than the
structure. This may happen if the metric symmetry is
higher than the symmetry of the structure; typical
examples are monoclinic structures with either very
close to 90 or a and c almost equal. The problems are
the same as in the case of type (ii). Depending on how
well the higher metric symmetry is ful®lled it may
happen that the reciprocal lattices overlap exactly and
the twinning is not detectable from the diffraction
pattern. The structure appears to have a higher
symmetry than it in fact possesses. Solving and re®ning
such twins requires essentially the same procedures as
for merohedral twins.
In contrast to the two ®rst types of twinning, in the
remaining two types not every re¯ection is affected by
the twinning. This means that the twinning may be
detectable from the diffraction pattern. Structure solution may be possible by identifying and using only those
re¯ections that are contributed to by a single twin
domain alone.
(c) Twinning by reticular merohedry: A typical
example is obverse/reverse twinning of a rhombohedral
structure. This means that the reciprocal lattices of the
different twin domains again superimpose exactly, but
that systematic absent re¯ections for one domain are
present for the others and vice versa. If enough re¯ections are only present for one twin domain, structure
solution should be possible using these re¯ection
intensities alone, possibly augmented by the intensities
of the common re¯ections divided by the number of
contributors.
(d) Non-merohedral twins: The twin operator is an
arbitrary operator. The reciprocal lattices do not exactly
overlap. There are some re¯ections which overlap or
cannot be distinguished from each other, while the
majority of the re¯ections are not affected by the
twinning. The diffraction pattern should normally reveal
the twinning. Cell determination using automated
diffractometer software may be a problem. Space-group
determination may be a little complicated, because some
systematically absent re¯ections may be affected by the
other twin domains. With the correct unit cell and a
subset of the data, structure solution should normally be
possible, but the re®nement may well be unsatisfactory
unless overlap of some re¯ections caused by the twinning is taken into account.
If the reciprocal lattices of the twin domains do not
exactly overlap, even a re®nement taking the twinning
into account often still remains unsatisfactory. Different
reciprocal lattices have different orientation matrices. If
the data are measured on a normal four-circle diffractometer without any kind of area detector and only the
orientation matrix of one domain is used, the re¯ections
of the other domains are of poor quality. The re¯ections
that have contributions from several domains may have
bad shapes and macroscopic domains may well not be
centred on the goniometer. Sometimes it is easier to
look for better crystals than to collect such data and try a
re®nement taking twinning into account (see Example
4). Otherwise, special programs for data collection
should be used (Henke, 1995). Area detectors are much
better suited than point detectors for the collection of
data from twinned crystals, but special software is
required to analyse the data (Hahn, 1997). Sparks (1997)
has presented programs for area detectors and discussed
examples of their use.
2. Examples
Twinned crystals tend to have a poor effective data-toparameter ratio, so they often require restraints
to obtain a satisfactory re®nement (Watkin, 1994). In
these examples, except where otherwise stated, the
following restraints were employed: distance restraints
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REGINE HERBST-IRMER AND GEORGE M. SHELDRICK
for chemically equivalent 1,2- and 1,3-distances,
planarity restraints for groups such as phenyl rings,
rigid-bond anisotropic displacement parameter (ADP)
restraints (Hirshfeld, 1976; Rollett, 1970; Trueblood &
Dunitz, 1983) and `similar ADP restraints' (Sheldrick,
1998). Even when restraints are employed the distribution of the displacement parameters (ORTEP plot;
Johnson, 1965) and residual features in a difference
electron density map are often less satisfactory than for
a normal structure determination.
3. Example 1
Structure (1) is an example of twinning by merohedry. It
could not be solved by routine methods. The composition of the compound was not known with certainty, but
some triphenylphosphine ligands were anticipated. The
®rst problem was to determine the space group. The
crystals appear to be trigonal with a = b = 12.623 (2) and
Ê . There were systematic absences for a 31
c = 26.325 (5) A
or a 32 axis. The Rint value (Rint = |Fo2 ÿ Fo2 (mean)|/
[Fo2 ]) for the Laue group 3Å was quite acceptable
(0.067), but the value for the Laue group 3Åm was only
slightly higher (0.120). It was possible to obtain the
coordinates of the osmium and four P atoms from the
Patterson (Sheldrick, 1992) in the space group P32, but
the resulting difference electron density map was not
very satisfactory. In spite of the relatively low R values
[wR2(all data) = 0.57, R1 [I > 2(I)] = 0.24, wR2 =
{[w(Fo2 ÿ Fc2 )2]/[w(Fo2 )2]}1/2, R1 = ||Fo| ÿ |Fc||/|Fo|]
only a small part of the structure could be identi®ed.
However, there were some typical warning signs of
twinning. The mean value of |E2 ÿ 1| was very low
Fig. 1. Structure of (1), hydridochlorocarbonyltris(triphenylphosphine)osmium(II).
445
(0.510) and the Rint value for the higher symmetric Laue
group was signi®cantly, but only slightly, higher than for
the lower symmetric one. This could mean that the
twofold axis was not a true crystallographic one, but the
twin law. This axis interchanges h and k and reverses l, so
the matrix is 010, 100, 001Å. Taking this twinning into
account substantially improved the re®nement. The R
values dropped to 0.13 (R1) and 0.35 (wR2). Several
phenyl rings could now be located and after a few cycles
of re®nement the whole structure could be found. The
re®nement of k2 converged to 0.404 (5). Fig. 1 shows the
®nal structure. The structure will be discussed more fully
elsewhere.
In non-centrosymmetric space groups and with heavy
atoms such as osmium it should be possible to determine
the absolute structure. The Flack parameter x (Flack,
1983) re®ned to 0.61 (6). This could mean that the
absolute structure is wrong and the space group P31 is
correct rather than P32 and/or that there is some additional racemic twinning. Changing to the space group
P31 and re®ning as a four-component twin so as to take
racemic twinning into account showed that P31 is indeed
the correct space group and that there is no racemic
twinning, but that the twin law should be changed to 01Å0,
1Å00, 001. This could be recognized by the re®ned k
values: k2 = 0.064 (13) for matrix 010, 100, 001Å, k3 =
0.038 (17) for matrix 1Å00, 01Å0, 001Å and k4 = 0.329 (13) for
matrix 01Å0, 1Å00, 001. It was necessary to introduce some
restraints. There are nine chemically equivalent phenyl
rings, so all chemically equivalent 1,2- and 1,3-distances
in the nine rings were restrained to be the same. For
every phenyl ring a planarity restraint was employed.
For the anisotropic displacement parameters of the C
atoms it was necessary to use the rigid bond and simi-
Fig. 2. Structure of (2), (8-cyclooctatetraenyl)[hydrotris(pyrazol-1yl)borato]titan(III).
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446
REFINEMENT OF TWINNED STRUCTURES
Table 1. Comparison of the re®nements of (2) in Cm and
R3m
R1 [I > 2(I)]
wR2 (all data)
k2, k3
Flack x
Number of data
Parameters
Restraints
Data/parameter ratio
S.u. (CÐC) in C8H8
S.u. (CÐC) in pyrazole
Cm
R3m
0.025
0.069
0.341 (5), 0.327 (5)
ÿ0.06 (3)
1064
131
165
8.1
0.004±0.006
0.005±0.008
0.020
0.050
Ð
ÿ0.01 (3)
437
111
168
3.9
0.012±0.014
0.004
larity restraints. There is also one disordered ethanol
molecule in the cell; distance and ADP restraints were
employed to re®ne it. The re®nement of 606 parameters
with 4384 data and 1354 restraints converges to R1 =
0.046 [for I > 2(I)], wR2 = 0.102 (all data) and an
absolute structure parameter of 0.026 (13).²
4. Example 2
The second structure (Kilimann et al., 1994) is an
example of twinning by pseudo-merohedry. Fig. 2 shows
the ®nal structure. The structure appeared to be rhomÊ ]. Again the
bohedral [a = b = 11.083 (2), c = 11.953 (2) A
mean value of |E2 ÿ 1| (0.647) was lower than expected.
There were no additional systematic absences. The Rint
values for the Laue groups R3Å and R3Åm were identical
(0.042) and the structure was initially solved by direct
methods (Sheldrick, 1990) in the space group R3m. The
pyrazole ring system was quite normal, but rather than
the eight-membered cyclooctatetraene ring a sixmembered ring with very long carbon±carbon bonds
appeared in the electron density map. This was caused
by the assumption of a threefold axis; there was only one
sixth of a molecule in the asymmetric unit, so the eightmembered ring should be disordered. With distance
restraints for the equivalent 1,2- and 1,3-CÐC distances,
a planarity restraint for the ring, rigid-bond restraints
and hard similarity restraints (standard uncertainty, s.u.,
of 0.005), even an anisotropic re®nement of the disordered model was possible, but the data-to-parameter
ratio was awful.
Since part of the structure does not ful®l the threefold
symmetry, the correct space group may be Cm rather
than R3m, with the apparent threefold axis as the twin
law. The low value for Rint in R3m would then be caused
by twinning. With the matrix ÿ0.3333 0.3333 ÿ0.6667,
110, 0.3333 ÿ0.3333 ÿ0.3333, the hexagonal cell is
transformed into a monoclinic C-centred cell with a =
² Lists of atomic coordinates, anisotropic displacement parameters
and structure factors have been deposited with the IUCr (Reference:
JZ0004). Copies may be obtained through The Managing Editor,
International Union of Crystallography, 5 Abbey Square, Chester CH1
2HU, England.
Table 2. Systematic absence exceptions for example (3)
Number of re¯ections
Number of re¯ections with I > 3
Mean intensity
Mean intensity-to-sigma ratio
21
a
c
n
22
0
0.6
0.5
225
84
32.6
8.7
225
83
27.6
8.3
232
167
57.3
16
Ê , = 96.85 (3) .
10.220 (2), b = 11.083 (3), c = 7.538 (3) A
The structure was solved again by direct methods in the
new space group. Now there is one half of the molecule
in the asymmetric unit. The structure ful®ls the mirror
symmetry and there is no longer any disorder. To derive
the matrix notation of the threefold axis in the monoclinic cell it is necessary to multiply three matrices. The
®rst matrix is the one above that transforms the hexagonal cell into the monoclinic one, the second describes
the threefold axis in the hexagonal cell (010, 1Å1Å0, 001)
and the last is the transformation from monoclinic to
hexagonal, which is the inverse of the ®rst matrix. The
result is the matrix 0.5 ÿ0.5 1, 0.5 ÿ0.5 ÿ1, 0.5 0.5 0. For
the re®nement of this model again several restraints
were helpful. All chemically equivalent 1,2- and 1,3distances were restrained to be equal. For the anisotropic displacement parameters of the N and C atoms
relatively hard rigid-bond (s.u. 0.002) and softer similarity restraints (s.u. 0.01) were used. For the eightmembered ring a planarity restraint was imposed. It was
necessary to employ damping, otherwise the re®nement
did not converge. Since this leads to slightly lower s.u.'s
than the correct ones, the ®nal re®nement cycle was
performed with zero `shift multipliers' and without
damping in order to obtain unbiased standard uncertainties. Table 1 compares the two re®nements. Primarily
because of the better data-to-parameter ratio, we
preferred the twinned re®nement to the disordered
re®nement and the structure was published in Cm. The
almost identical k values show that the crystal was an
ideal `drilling' (a German word), which explains the low
Rint value in the rhombohedral space group.
5. Example 3
The third example is the structure of the thallium
compound C19H33LiN3Tl (Armstrong et al., 1993). It is
again an example of twinning by pseudo-merohedry.
The data were collected with the following monoclinic
primitive cell: a = 13.390 (3), b = 25.604 (6), c =
Ê , = 112.39 (3) . a and c have equal lengths
13.390 (3) A
and therefore the matrix 101, 1Å01, 01Å0 transforms this
cell into an orthorhombic C-centred one with a = 14.900,
Ê (LePage, 1982). The Rint values
b = 22.252, c = 25.604 A
are similar: 0.097 for the orthorhombic case and 0.081
for monoclinic, so the orthorhombic system was tried
®rst. The mean value for |E2 ÿ 1| was 1.074, but the
systematic absences were strange and the Siemens
SHELXTL-Plus program XPREP (Sheldrick, 1991)
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REGINE HERBST-IRMER AND GEORGE M. SHELDRICK
447
Table 3. Comparison of different re®nements of (4)
Unique data
R1 [I > 2(I)]
wR2 (all data)
K²
Ê ±3)
max (e A
k2
S.u. (PÐO,PÐC)
(A) original data
(B) without
F2/s.u. > 6
(C) without |h| =
0,1,5,6
(D) |h| = 0 or 6
split
(E) |h| = 0,1,5 or 6
split
(F) E less F2/s.u.
>6
1667
0.101
0.300
13.225
1.97
Ð
0.006±0.008
1654
0.091
0.251
8.490
1.70
Ð
0.005±0.007
991
0.040
0.104
0.023
0.33
Ð
0.002±0.004
2977
0.083
0.252
4.436
1.47
0.206 (6)
0.004±0.006
2992
0.057
0.221
0.388
0.80
0.170 (3)
0.003
2974
0.050
0.146
0.413
0.56
0.205 (2)
0.002±0.003
² K = mean (Fo2 )/mean (Fc2 ) for 0 < Fc/Fcmax < 0.016.
could not ®nd any acceptable space group. In the
monoclinic crystal system the mean value of |E2 ÿ 1| was
1.119. Again the systematic absences were strange, as
shown in the following reciprocal space plot of the h0l
layer (Fig. 3) and in Table 2.
The distributions for an a or a c glide plane were quite
similar. In the reciprocal space plane h0l there are many
very strong and also many very weak re¯ections (this is
the reason for the anomalously high value for the mean
of |E2 ÿ 1|). All re¯ections with h and l even are very
strong and all re¯ections with both h and l odd are
nearly absent. Taking into account that a and c have
equal lengths, there is a simple explanation. The crystal
is twinned and the twin law is 001, 01Å0, 100. Therefore, a
and c are interchanged and the correct space group is
P21/c. The absences are caused by the overlap of the two
twin domains with effective space groups P21/c and
P21/a. Again the relatively low Rint value for the higher
symmetric (i.e. orthorhombic) description is caused by
twinning. It was possible to obtain the coordinates of
two Tl atoms from a Patterson map in P21/c. With these
coordinates the whole structure could be found in a
difference electron density map. There are two independent and almost identical molecules in the asymmetric unit. The structure re®ned to R1 [I > 2(I)] =
Fig. 3. h0l layer of reciprocal space for example (3).
0.043, wR2 (all data) = 0.112, k2 = 0.474 (1). Fig. 4 shows
one of the cation±anion pairs.
6. Example 4
The last structure is methylene diphosphonic acid,
CH6O6P2 (DeLaMatter et al., 1973; Peterson et al.,
1977), and is an example of a non-merohedral twin. The
space group is P21/c with a = 7.820 (4), b = 5.468 (3), c =
Ê , = 103.57 (4) . Direct methods solved the
13.693 (6) A
structure with no problems. There were several re¯ections that violate the systematic absences. For most of
them |h| was 6 or 1. There were also no problems with
the re®nement, but the R values were very high (see
re®nement A in Table 3). Many re¯ections disagreed
substantially with the model. For all of them |h| was 0, 1,
5 or 6, and Fo was greater than Fc. The factor K =
mean(Fo2 )/mean(Fc2 ) was very high for the re¯ections
with low intensity. There were also many high residual
Ê ÿ3. Omission of
peaks with densities of more than 1 e A
the `most disagreeable' re¯ections with F2/s.u. > 6
(re®nement B) lowers the R values and also the residual
density, but the result was not yet satisfactory. Disorder
or solvent molecules were not detectable.
Again interpretation of the twinning solved the
problem. To derive the twin law we had to ®nd a matrix
that transforms the cell into an equivalent cell, see Fig. 5.
We knew that a factor of 1/6 or 5/6 in a was involved.
After several trials we found the matrix 100, 01Å0,
ÿ0.8217 (5/6)01Å. The reciprocal lattices coincide
Fig. 4. Structure of (3), showing one of the two independent cation±
anion pairs [PMDETA.Li]+[CpTlICp]ÿ.
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448
REFINEMENT OF TWINNED STRUCTURES
almost exactly when |h| = 0 or 6. When |h| = 1 or 5, the
re¯ections are so close that in most cases they cannot be
resolved from one another.
In test C all re¯ections with |h| = 0, 1, 5 or 6 were
omitted. The R values dropped substantially and the
residual density was now in the normal range. The K
value [mean(Fo2 )/mean(Fc2 )] for the re¯ections with 0 <
Fc/Fcmax< 0.016 is now very low. The most disagreeable
re¯ection has F2/s.u. = 6.96. In a second step we took
twinning for the re¯ections with |h| = 0 or 6 into account.
The R values dropped, but there were still many
re¯ections with |h| = 1 or 5 which were inconsistent.
Therefore, we also split the re¯ections with |h| = 1 and 5
(re®nement E). This re®nement was better, but now the
intensities of some re¯ections with |h| = 1 were underestimated. For these re¯ections, both reciprocal lattices
were distinguishable and only one re¯ection had been
measured. Therefore, for our ®nal re®nement (F) we
omitted all the re¯ections with F2/s.u. > 6.
This structure is known and in the literature there is
no indication of twinning. Re®nement F results in
similar s.u.'s to the published untwinned structure,
although the R value is higher, probably because of the
problem handling partial overlap; we also suspect that
one twin component was better centred in the beam
than the other.
7. Conclusions
Twinning usually arises for good structural reasons.
When the heavy atom positions correspond to a higher
symmetry space group it may be dif®cult or impossible
to distinguish between the twinning and disorder of light
atoms (Hoenle & von Schnering, 1988). Since re®nement as a twin usually requires only two extra instructions and one extra parameter, in such cases it should be
attempted ®rst before investing many hours in a detailed
interpretation of the `disorder'! Re®nement of twinned
crystals often requires the full arsenal of constraints and
restraints, since the re®nements tend to be less stable,
and the effective data-to-parameter ratio may well be
low. In the last analysis chemical and crystallographic
intuition may be required to distinguish between the
various twinned and disordered models, and it is not
easy to be sure that all possible interpretations of the
data have been considered.
Experience shows that there are a number of characteristic warning signs for twinning given in the
following list. Of course not all of them can be present in
any particular example, but if one ®nds several of them,
the possibility of twinning should be given serious
consideration.
(a) The metric symmetry is higher than the Laue
symmetry.
(b) The Rint value for the higher-symmetry Laue
group is only slightly higher than for the lowersymmetry Laue group.
(c) The mean value for |E2 ÿ 1| is much lower than the
expected value of 0.736 for the non-centrosymmetric
case. If we have two twin domains and every re¯ection
has contributions from both, it is unlikely that both
contributions will have very high or very low intensities,
so the intensities are distributed so that there are fewer
extreme values. Other tests based on intensity statistics
have been proposed (Rees, 1980). However, the use of
the mean |E2 ÿ 1| value is particularly simple, because it
is a single number and often calculated by data reduction and direct methods programs.
(d) The space group appears to be trigonal or hexagonal.
(e) The apparent systematic absences are not consistent with any known space group.
(f) Although the data appear to be in order, the
structure cannot be solved. This may also happen if the
cell is incorrect, e.g. with a halved axis.
(g) The Patterson function is physically impossible.
The following points are typical for non-merohedral
twins, where the reciprocal lattices do not overlap
exactly and only some of the re¯ections are affected by
the twinning.
(h) There appear to be one or more unusually long
axis.
(i) There are problems with the cell re®nement.
(j) Some re¯ections are sharp, others split.
(k) K = mean(Fo2 )/mean(Fc2 ) is systematically high for
the re¯ections with low intensity. This may also indicate
a wrong choice of space group in the absence of twinning.
(l) For all of the `most disagreeable' re¯ections, Fo is
much greater than Fc.
We thank N. W. Alcock for kindly donating the data
of structure (1) and P. G. Jones for the data of (4).
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