Ionic radii for use in geochemistry

Geoohlmica etCosmochimica Acta,1970,Vol.34,pp.945to956. Persamon PIBE.Printed inNorthern Irelsnd

E. 5. W. WHITTAKER and R. MUNTUS Department of Geology and Mineralogy, Oxford (Received 18 March 1970; acceptedin revisedform

7 May

1970)

Ab&t&-!Pb ionic radii proposed by &ANNON and PRE’(VITT as the effective values in oxides and fluoridesh&v8 been analysed for their conformity with the radius ratio principlesof crystal chemistry. It is shown that a set of vaIues intermediate between the “IR” and “CR” values of SEANNONand PREWITTprovides the best conformity in this respect, and a table of the values adopted is provided in periodicform. These values correspondto a VI co-ordinate oxygen radius of 1.32 d and a VI co-ordinatefluorineradius of 1.25 A, and the radii of the cations are therefore appreciably larger than those previously accepted, although not as large ss SHANNONand PREWITT’S“CR” values. It is suggested that the radii obtained are the most suitable for use in silicate geochemistry, and constitute a major improvement over previous values for this particularpurpose. Although it is ~po~ible to maintain the accuracy with which radius sums represent~~ratomic distances and at the same time to extend the range of the radii to other anions, it seems desirableto provide compatible (although only very approximate) values of radii for Cl, Rr, I, S and Se anions. Use of the new cation radii with these anion radii cannot be expected to be an improvement over the use of older radii, but should be about as satisfactory. The changes of cation radii relative to one another, as compared with AHRENS’radii, are tabulated in order to draw attention to their possible geochemioal significance,and the implications are discussedin the case of scandium. 1. I~TR~D~C~O~ THE MOST extensive

compilation of ionic radii, and the most satisfactory for use in geochemistry, has until recently been that of AHRENS (1952). This was an advance over previous lists, because it not only took account of more recent structure determinations but also carefully evaluated some of the causes of discrepancy between earlier lists, and in particular eliminated those due to GOLDSCHMIDT’S choice of mutually incompatible radii for 02- and F-. While making substantial use of Paulmg’s radii, Ahrens also eliminated their excessive reliance on theory by introducing various empirical ~justm%nts to bring them better into line with observed interionic distances. He also derived additional ionic radii by interpolation methods involving comparisons with ionisation potentials. In spite of these improvements, Ahrens’ radii suffer from one defect in oommon with those of PAULIN@ (1927) and GOLDSCHMIDT(1926), namely that they are strictly applicable only to structturesin which the cations are in octahedral oo-ordination. PAULINE (1960, and earlier editions of this work) has given factors for the correction of interionic distanoes (derived from radius sums) for co-ordination numbers other than VI. This does not however give any indication of how the individual cation and anion radii will change with co-ordination number, so that it has to be assumed that they change in equal proportion, and therefore that the radius ratio is independent of co-ordination numbers. This leads to a considerable amount of confusion in the case of elements which are not normally VI co-ordinated; thus, in the case of an important element like silicon it is extremely difficult to remember that the normally quoted radius of 0.42 A is not appropriate to its normal IV co-ordination, but that

946

E. J. W. WEIY~AKERand R. MUNTUS

its preference for IV co-ordination is to be deduoed from the ratio of its VI coordinate radius to the VI co-ordinate radius of oxygen. Moreover, the Pauling correction factors consider only the co-ordination number of the cation, since in Pauling’s derivation of ionic radii it was assumed that cation and anion were of equal charge and therefore of equal co-ordination number, an assumption which is obviously very far from the truth in many circumstances. Furthermore, many elements occur with more than one co-ordination number, and the assumption of constant radius ratio then leads to the hypothesis of “a rattling cation” even in cases where there is no physical evidence for this, as for example in the important case of aluminium VI co-ordinated by oxygen. The accuracy with which any of the usual tables of ionic radii predict interatomic distances is well known to be rather limited, and mean discrepancies between calculated and observed interatomic distances over a wide variety of compounds are commonly of the order of 0.06 A. This is an inevitable consequence of the inadequacy of the ionic approximation to the chemical bonding in many of the compounds concerned, and some of these discrepancies were disoussed by AHRENS (1952). The difficulty is essentially that a single value of a cation radius cannot allow for the variations in ionicity of the bond from that cation to a variety of anions of widely varying electronegativity. However, if the range of anions considered is very limited, for example 02- and F- only, then the ionicity of the bond formed by a given oation with these anions will vary but little from one structure to another, and a single value for the cation radius will add to a single value for the anion radius to give an accurate prediction of the bond length, regardless of the degree of ionicity of the bonds formed by that cation with these particular anions. SHANNON and PREWITT (1969) have taken advantage of this fact in deriving a very comprehensive and accurate set of effective ionic radii for cations in a great variety of oxidation states and co-ordination numbers, and also in high and low spin states where appropriate, which are applicable specifically and only in deriving interionic distances to 02- and F- anions. They have in addition defined the variation in the radii of these anions with anionic co-ordination number. Both in deriving and in applying their radii, SHANNONand PREWITT assume that where co-ordination is not entirely regular the mean cation to anion distance within a co-ordination polyhedron is a more meaningful parameter than the minimum interionic distance. That is, they assume a deformable sphere model rather than a hard sphere model for the ions. With this proviso, and within the limitation to oxygen and fluorine anions which they impose, SHANNONand PREWITT’S radii give an average disorepancy between observed distances and radius sums for 128 compounds of only O-016 A compared with 0.064 A when AHRENS radii are applied Although from a general chemical point of view the limitato the same compounds. tion to oompounds involving bonding from cations only to oxygen and fluorine involves a substantial limitation, from the standpoint of silicate geochemistry SHANNONand PRE~ITT’S radii clearly constitute a major advance. The SHAXNON and PREWITT radii, however, do suffer from one important disadvantage. Although the relative values of the cation radii are based entirely on empirical data, the absolute values necessarily depend (like those in all other compilations of ionic radii) on an assumed value for some specific anion radius. SHANNON

Ionic radii for we in geochemistry

947

and PREWITT make two alternative assumptions about this: one is that the oxygen radius in VI co-ordination is 1.40 8, the same value as was used by PAULINEand -ENS, and on this basis they derive what they call their “ionic radii” (IR) which are therefore dire&ly comparable with &mENS’ radii; their other assumption is effectively that the oxygen radius in VI co-ordination is 1.26 8, a value that is based on the empirical diBerence between the radii of 02- and F- and the theoretical calculations of Fum and TOSI (1964) for the radius of the F- ion. The alternative value of the radius of 02- leads to an alternative set of radii which SHCANNON and PREWITT describe as “crystal radii” (CR). They mention *evidence from various sources that anionic radii are smaller and cationic radii correspon~ngly larger than has previously been assumed, but they clearly do not feel that the evidence is sufficiently strong for one set of radii to be totally preferred to the other. 2. CRYSTAL CHEMICAL CRITEI~U FOR ABSOLUTEIonrc RADII The GOLDSCHMIDT and PAULINEradii, in spite of their shortcomings, were sufficiently satisfactory to lead to the establishment of the well known crystal chemical principles relating co-ordination number to the radius ratio r+,ir-, and the converse argument that cations with borderline values for radius ratio to a particular anion often exhibit alternative co-ordination numbers, has been used to provide an approximate justification for the way in which the traditional radii divide up the observed interatomic distances between cation and anion. With the very much larger amount of data now available from SHANNON and PBEWITT’Spaper, and the large number of radii which they present for a great variety of co-or~nation numbers, it is possible to apply this criterion in a much more quantitative manner. SHANNON and PREW~TT(1969) have themselves given some indication of this approach in their Table 6 (p. 943) by testing the radius ratios for oxides of a number of common cations for conformity (either exactly or within a certain small latitude) with the appropriate radius ratio requirements of the given co-ordination numbers. The results of these tests are listed both for the IR and CR values for the elements concerned, and it is evident from this table that the CR values conform more frequently with the radius ratio principle than do the IR values. It is notable that of the most important cations Si*+ does not conform with the principle for either IV or VI co-ordination on the basis of the IR values, but does conform on the basis of the CR values. On the other hand, aluminium in IV co-ordination oonforms with the principle on the basis of the IR values, but on the basis of the CR values only by the admission of a small degree of latitude, and in VI co-ordination it only conforms with the principle on the basis of the CR values. Fe3f in IV co-ordination and Mn2f in VI co-ordination, however, only accord with the principle on the basis of the IR values and not of the CR values. Therefore, it seems that both of SHANNON and PREWXTT'Ssets of radii are unduly extreme, though in opposite directions, and that some intermediate assumption about the oxygen radius would lead to a more satisfactory conformity with the radius ratio principle. In order to find such an optimum value the following procedure was adopted. Of the 279 cation radii listed by SHANNONand PREW~TT(1969) the four relating to carbon, hydrogen and nitrogen are clearly unsuitable for treatment in this way because they take negative values in the IR series. Three more, boron (III and IV)

948

E. J. W.

WHITTAKER and R. MUNTUS

and S6+ give radius ratios below the appropriate minima, even using the CR values. These, therefore, also have to be excluded from consideration, and are obviously poor candidates for treatment by the ionic approximation. Cr5+ VIII also gives a radius ratio below the appropriate minimum for the CR values, but this can be ignored because the only occurrence of this ion with this co-ordination number is in the ion CrOs3- in which the ligands are peroxide groups and not single 02- ions. Of the remaining 270 cation radii, 26 fall below the appropriate minima for their coordination numbers if the IR values are taken, and for each of these the necessary reduction of the oxygen radius (Ar,,) to bring the radius ratio into the appropriate range was calculated. The radius of IV co-ordinate oxygen was used throughout the calculation as a suitable mean value, except in the case of a few elements which exhibit the stated co-ordination number only in combination with F, for which a similarly appropriate radius was used. The results are shown in Table 1. This table shows at once that an appreciable value of ArO is required, but in order to decide on the optimum value it is necessary to consider the opposite case of those ions whose radius ratio to oxygen exceeds the minimum for a higher co-ordination number. This is a much less olear-cut criterion however, because many oations often exhibit coordination numbers well below the maximum value for the number of ligands that will fit round them, regardless of any practicable datum level for the anion radii. This arises for various reasons: (a) Effects of covalent bonding (e.g. tetrahedral Fe2+, Zn2+, Cd2+); (b) Co-ordination imposed by the rigidity of the rest of a crystal structure; (c) Co-ordination imposed by a limiting value of r-/r+ when the cation is larger than the anion, which is rare but accounts for VI co-ordination in CsF ; (d) When the cation has a low charge the reduction in energy resulting from surrounding it by more anions is largely or entirely offset by the repulsions between the anions. This phenomenon is familiar in the alkali halides where it inhibits the switch from NaCl-structure to CsCl-structure even when the radius ratio favours it. The effect is likely to exaggerate (a) and (b) for lower charge cations. Because of these effects IV-co-ordination appears for Li+, Na+, Mg2+, Fe2+, Zn2+, Cd2+, Hg2+ (i.e. some mono- and di-valent ions) which are large enough for VI co-ordination at all values of Ar,. Similarly, VI co-ordination occurs where VIII would be expected for many alkali metals, alkaline earths and lanthanides, and it would not be profitable to adjust Ar, to accommodate a few of these when it is clear that all of them cannot be accommodated and there are good reasons for the apparent anomalies. However, none of these anomalous cases involves a charge above 2 for IV co-ordination or above 3 for VI co-ordination, and therefore it seems possible that such anomalies will not affect the result if consideration is restricted to ions with charges higher than 2 for IV co-ordination and higher than 3 for VI co-ordination. There are no anomalously large ions of higher charge than 2 which exhibit VIII coordination, but those with a charge of 2 occur frequently and are therefore excluded in the same way. V and VII oo-ordination are excluded from oonsideration altogether because their rarity is itself an indication that their occurrence is influenced by special features outside the scope of the ionic approximation. Subject to these

Ionic radii for use in geochemistry %bbfe 1. Reduction (A**} in 0”

049

radius below the IX values required to mske specified ioILg=SXCCXX% B minimum radius r&o

restrictions there are 17 radii which give an anomalously large radius ratio if the CR v&es are used, and none if the IR values are used. Table 2 shows these in a similar way to that adopted in Table I as a function of ArO. Table 3 shows khctcurn~l~tiv~ numbers of ~~~~~~~~~mm from both Table i a& Table 2 as one proceeds from left to right and right to left respectively, again as a function of Ar,. The total number of anomalies, as might be expected, falls to & minimum at an intermediate level of AT, between 0 and O-14 which lies at about O-10 8. However, it does not seem altogether desirable to regard this minimum as

X&X

C.N.

ratio

IV

0.414

Ar,

0.14

0.13

OS12

AI*+

Re7+

0.11

0.10

0.09

0.08

0.07

MO”+

Cd+

w*

Is*+

0.06 F&

a.05

OQO.40.03

0.02

0.01

E.

950

J.

W.

WEITTAKER

and R. MUNTUS

‘Ihblo 3. Numbers of anom&ous radius ratios as a fun&cm of Ar,,

A+-,

0.14 .---

0.13 -

0.12

0.11

0.10

0.09

0.08

0.07

O-05

0.04

0.03 0.02

0.01

6

14

19

21

26

26

0.06

No. below min.

radius ratio

(cumulative) No above next

0

0

2

higher radius ratio (cumulative)

17

16

13

Total

17

16

15

No. xvith cliscrepaney in Ar, 0.02: for minimum

0

0

0

Above next higher

9

6

Total

9

6

5

6

G

1

1

1

0

0

12

I5

20

22

26

26

0

2

8

Q

11

14

19

6

6

3

0

0

0

0

0

6

6

5

8

9

11

14

19

II

the optimum value of Are, because it takes no account of the extent of the remaining anomalies. If one only considers those anomalies for which the discrepancy in Ar,, exceeds 902 then the minimum number of such discrepancies occurs at ArO = O-08il, and none of these 3 extreme anomalies seems very important. P5fIV and F-VI are neither of them very susprising exceptions in a discussion of ionic radii. VI coordinated silicon occurs in the silica-fluoride anion, but here the radius ratio is a little more favourable than in the case of radius ratios to oxygen considered in the table, and would correspond to a discrepancy of only 0-02in Ar,. It also occurs in stishovite, but this phase is formed only under high pressure and clearly does not represent the normal behaviour of silicon. It is therefore proposed that the optimum value of AY, is O*OS11, and a table of absolute ionic radii has been drawn up by adding O-OSA to all the IR values of SHANNON and PREWITT for eations and subtracting 0.0s i% from the corresponding figures for anions. The results, which oorrespond to a VI coordinate oxygen radius of 1.32b, are shown in Table 4. The slight amendments and additions to their results subsequently published by SHANNON and PREWITT (1970) do not introduce any additional elements into Table 1 or Table 2, and only lead to a shift of one element (W) one place to the right from its original position in Table 2. This modification has been included in that Table, and the amendments and additions from SHANNON and PREWITT (1970) have been incorporated in Table 4. 3. DISCW~SIONOF THE ABSOLUTE

RADII

The adoption of 0.08I%as the optimum value of Ar, means of course that the cation radii derived are on average appreciably larger than those which have been used traditionally, although not so large as the CR series of SHANNON and PEEWITT. The value of 1.32A for the radius of VI co-ordinate oxygen is quite fortuitously equal to the value adopted by GOLDSCHMIDT, but this does not make the values of the cation radii in general particularly similar to those of GOLDSCHMIDT. It in no sense implies a return to the GOLDSCHMIDT radii, since the fluorine radii adopted are substantially smaller than the oxygen radii, just as they were in PAULINQ’S and AHREITS’ lists but

Ionic radii for we in geochemistry

961

not in GOL~S~HMIDT’S.The increase in the c&ion radii and co~es~n~ng decrease in the anion radii is in line with a number of pieces of evidence from recent work, in particular wave mechanical calculations on electron distributions in co-ordinated ions rather than the calculations relating to free ions used by Pamma, and arguments based on BORN repulsive parameters together with data on compressibility and thermal expansion. References to such work are given by SHANNON and PREWTT (1969). While the increases in cation radii adopted ZtFenot so large as some of these lines of work have suggested, they do at least receive qualitative support from them. While the absolute sizes of ions are of interest in geochemists in connection with the a.ppropri&eness of different co-ordincttion numbers, the usual considerations of geochemioal similarity between elements depend primarily on relative radii. The effect on such considerations of the present radii is therefore independent of the datum level ohosen for the radius of oxygen, and can be based equally well on the values given in Table 4 or on either of the sets of radii given by SHANNON and PREWITT. A full discussion of all the changes in relative radii compared with previous lists such as Ahrens’ is complicated by the present &v~il~bi~ty of different radii for different ho-or~nation numbers, and future geochemic&l considerations of specific elements will undoubtedly have to take account of this. For a general survey however, it is convenient simply to consider relative changes in the radii for VI ooordination, and to eliminate sJ1 considerations of the changes in absolute radii by expressing these relative changes with respect to the radii of one or two common elements. For this purpose the cations have been divided into two groups, those with VI co-ordinate radii in the range 0.5~0.85 L%and those with VI co-ordinate radii greater than 045 8. The first group have been compared with m~esium. If a, constant is added to all the radii in this group so as to make the magnesium radius equal to the figure given by AHRENSthen the changes in the radii of the other eIements as compared with their AHRENSradii indicate the extent of the changes relative to magnesium. These differences are given in Table 5 and it is notable that there are no positive changes but a substantial number of reductions in radius relative to magnesium. For radii greater than 0.85 ir, Table 5 shows comparison with calcium, and in this case both positive and negative changes occur. At~ntion is drawn to two specific features of these changes. The changes in the lanthanides in oxidation state 3, with respect to calcium, are large and negative for lanthanum, and to a Iesser extent for the other early lanthanides, and fall to zero for the later lanthanides. The curve for the change in radius along the lanthanide series is somewhat smoother and much less steep according to Shannon and Prewitt’s data than it is according to AHRENS’figures. This is shown in Fig. 1. Some of the changes in the radii of the smaller ions with respect to magnesium must undoub~dly affect geo~hemical considerations regarding the substitution of these elements in common minerals. Attention has already been drawn by FROXDEL (1968) to the fact that the previously accepted radius of scandium was substantially too large. This is confirmed by SHANNONand PREWITT’Sdata, but a detailed consideration of FRONDEL’S geochemical conclusions shows the dangers of considering some ionic radii out of context with others. FRONDEL(1968) uses a reference value of 1.35 A as the radius of 02-, and with respect to this assigns a value of 0.73 i%to the radius of scandium in VI co-ordination.

-I-

Li 1 IV VI

0.68 0.82

2

IV

--

-

mg

0.686 0.7‘5 V VI 0*8.O YIII es 1

1.07 1.08 1.10 I.21 lr24 1.40

a IV

--

K

1VI 1.46 VII VIII IX x XII

1.54 I.59 1.63 1.67 I+38

_I%

2

_-

I

vf

1.78 VIII lez IX 1.86 x I.R9 XII I*!%

1.21 YII I.29 VIII 1.33 x I.4 0 XII 1.4

3 VI 0.98 VIII 1.10 IX 1.18

4 VI 0.80 VII o@J VIII 0.92

BB

VI VII VIII IX s XII

_-

1‘1:I 1.111 1.21, 1.2[I 1.31i I.40,

ce

---

:s VI

1.09 VIII 1% IX 1.23 XII 1.37

o+w 0.52 0.63 0.43 0.38

Tc 0.72

!

W 4 VI 0.73 6 TV 0.50 VT O68

0.75 0.74 0.72 0.77

/ O.?l : 0.69 i 086 0.57 0.63 0.73 j

I ,

Rn

0.75 4 VI 0.73 0.71 a*50 ! 0*58 / 0*68 / O.TQ ! _

3 VI 4 VI ii VI VIII

Fe

I __

3 VI 4 VI 5 VI 61V V VI VII

TS

VI 5 VI 6 VI 7 IV VI 4

0.71 0.60 0.60 0.48 0.65

3 VI. 0.x 4 VI 0.70

Ij‘

/

0s

&VI

0.71’ I

L

_-

La

3 VI VII VIII IX s XII

0.79 3 VI 0.78 4 VI 0.77 5IV @40 VI 0.72 VIE 0.74

AeLw

VIII 1,s , I2XII 1.71 _i

H O*QU 3% 4 IV VI 5 IV BIV

~~

2 VIL 0.75 2 IVH H 0.91 VIL VIII I.01 H 3V 0.66 3 IVH VIL o+s VIL H 0.73 H 4 VI 0.62 6 iv 0.35 7 IV 0.34

MO

2 VI

4 VI 0.79 VIII 0.91

--

Ba

2 VI L 0‘81

___I.

Iii

1.4‘ 1.4 1.51 1.5: I.& 1.6!

O-87 0.78 0.67 0.44 0.54 0.62

ran

Cr

Nb

-I&.-LO

2 VI 3vr 4 VI 5 IV v VI

--

’

--

zr

8i --

v

a+34 0.75 0.61 O.F9

_-

--

-

i

2 VI 3 VI 4V VI

Y

_>I

F1

/

3VI 0~33 VIII 0.95

8r

1 VT

1.67 I.64 I*68 I.74 1.81

cs

81 1.15 0 I.2 1.26 1.36 1.43 I.0

T

Ti

SC

YI VII VIII IX x XII

Rb 1 VI VII VIII x XI1

r

0.5!5 0.: I5

III

la I IV V VI VII VIII IX

Be

Table 4. Ionic radii based

Nd

Pr 3 VI VIII 4VI VIII

1.08 1.22 O+% 1~07

3 VI I.06 VIII 1.20 IX 1.17

T

Pm

Sm

3 VJ. 1.04

3 VI ItO4 VIII 1.17

I1VI 0.88

Eu 2 VI VIII 3 VI VII VIII

1% 1.33 1.03 I 1.11 I 1.15

VIII I.05

AC

Th 1.08 VIII 1.12 IX 1.17

L VI

u

Pa 4 VIII l.OQ 5 VIII 699 IX 1.03

/ L

3 VI 4 VII VIII IX 5 VI VII F II IV VI VJI --

1.12 I.06 1.08 1.13 0.84 1.04 0.53 0.58 0.81 0.96

NP

PU

1.18 3 VI 1.10 4 VIII I.06

3 VI 1.09 4VI 0.88 VIII I.04

2

vx

Am I.08 1 4 VIII 1.03 ! 3 VI

on radius ratio criteria

-

-

T

B 3 III IV

C -

0.10 wzo

I -

1.27 III 1.28 IV 1.30 VI I.32 VIII l-34

Al

Q+ 4 :i IV 0.413

4IV VI

--

Ni

cu

0.77 1VIL 0.64 H 0.68

111 0.54 2 IVsq 0.70 v 0.73 VI 0.81

co 2‘ IVII VIL Jl 3 VI&

P

St 0.47 0.56 0.61

$VI

0.65 0.73 0.83 0.61

6

3I IV v VI

8 6 3 8

0.55 0,63 0.70

4 IV VI

(1.56) (1.72) (l-78) oG?o

iIv

p3: VI (I*?: VIII (1+31 i III 0% I IV 0.21

.-

a0

se

A8

Br [VI (Ml VIII (1.81 I IV 0,3!

(1W 0.42 E VI 0.58 v1n (l+o: IV 0.37 .6

0.413 *5 IV OSB! 3 VI

H 0.69

-

--

--

4

Pd Rh 0.67 3 VI 0.75 1 II ,4 VI 0.71 2 IVsq 0.72 VI 3 VI 4VI

0.94 0.84 070

AU

Pt 3 4

2 IVsq oG% 4 VI 0.71

1

w

3 IVsq O-76 1 III 2 II IV VI VIII

l,( IS 0.717 I.(14 1.1IO : 1.:!2

1.02 1’12 VIII 1.14

Pb

Bi

2 IVpp VI VIII IX XI XII 4 VI VIII

VII VIII 4 Vl VIII

1.10 I.12 0‘84 0.96

3 VI

E?

HO

l)Y

$VI I.00

3 VI

3 IVpy 0.86 v 088 5 VI 0.63

o-99

VIII 1.11

Bk

VI I.08 VIII 1.03

3 Vl I.04 1 VIII 1.01

Cf

-

3 VI

r VI (2.1: VIII (1.9 5 VI 1.0:

0.60

At

PO

1.02 3v I.07 1.26 VI 1.10 ‘~711 I.19 I*37 I.4 1 1.47. 1.671 O-8 1.0;/ -

3 VI 0.98 VIII 1.10

Tm

4vIn

l*lf

0.%7 I VI 0,96 VIII 1.08 VI11 1.07

im

LU

Yb

fI VI

036

VIII 1.06

-Cm

-

I

Te III

Tb

i VII

Sb

2 VIII 1.30 4 VI 0.77

--

Hg

-

-

I-

18 : ,6 13 18 51 IQ

1 II OS76 rvaq I.10 V 1.20 VI 1.23 WI 1.32 VIII I*38 3 IVsq 0.73

3n

-1

3VI o-94 VIII 1.05

-Pm

ES

Md

No

LW

1.03

-

-

Of the elements disowned by Shannon and Pewitt H, C end N are ignored. Of the remainderonly the radii of B’+(IlI), B*+(IV), S*+(IV), Pe+(IV) and SiC+(VI)are more than &02A below the minimum approptite to their ooedination.

963

I

I.21 1.22 I-23 1.26

Cl

B 0.26 3 IV VI VIII 6 IV

-_

bla

I II III IV VI

B II

3 IV v vr

F

0

954

E.

Table 5. Relative

0

-0.02 -0.03 - 0.04 - 0.05 - 0.06 -0.07 -0.08 -0.10 -0.11 -0.12 -0.13 -0.14 -0.15 -0.16

J. W.

WHITTAKER

and R. MUNTUS

changes in VI co-ordinate cationic radii compared AHRENS’ radii, AT = T - r,,,,,,

Pd2+, Li+ l?f52+

+ o-05 + 0.04 + 0.02 +0.01 0 - 0.01 -0~02 -0.03 - 0.04 -0.05 -0.06 -0.07 --0.09 -0.10 -0.12 -0.15

P&, cu”+, Zrl”’ NW, G& C$‘+, Mna+, Itha+, Ina+, Sb5+ Cc?+, Moe+, Sn4+, W+ V”+, TaS+ vs”, Nb*+, Nb5+, Rx++, I++ w-t, MI+‘, os*+, TF+, Pb4’ w+, Hf*+ Ti’+ Ti3+ v+

0 Ahrens’

Ce

Pr

Nd

Pm Sm Eu Gd

HI”’ &Tat,

K”

Cs’, TV Xb+, 13aa+ SrZ+, TmS+, LUG+ 1w+, Y bs+ Tb3+, Dy3+, No3’Cdzt, Ya+, Pb2+, Th”+ Eu”+, Gd9+ SIX?+ P@’

data

0 Shannon & Prewitt’s

La

with

Tb

Dy

Ho

data

Er

Tm Yb

Lu

Fig. 1. Ionic radii of the lanthanides (oxidation state 3 -t; VI co-ordination) relative to VI co-ordinated calcium, according to Ahrens’ data and SHANNON and PREWITT’S data.

Ionic radii for use in geochemistry

955

Since the compounds on which his derivation is based involve III or IV co-ordination of oxygen the appropriate radius for this from Table 4 is about 1.29 A. Thus our value for scandium of O-83 A is approximately equivalent to a value of 0.77 A relative to an oxygen radius of 1.35 A, in qualitative agreement with FRONDEL who is therefore correct in observing that the radius of scandium is much closer than has previously been supposed to the radii of Fe a+, Cra+ and AP+. He is therefore correct in his conclusion, which in any case he supports with experimental evidence, that there will be much less of a barrier than has previously been supposed to scandium substituting for these elements in these oxidation states. He did not however compare his new radius for scandium with that of magnesium. Using AHRENS’ radii adjusted to a basis of oxygen = 1.35 A the radius of magnesium would be 0.71 A, so that FRONDEL’S value of O-73 A for the scandium radius makes it very much closer to that of magnesium ; the same is true using the radii from Table 4, which gives rso = 0.83 A and rMg = 0.80 A (both in VI co-ordination). It would seem therefore that there is no good reason to discount the possibility that a charge balancing substitution of SC, Al for Mg, Si may account for a substantial portion of the scandium in ferro-magnesian minerals. 4. APPROXIMATE RADII OF OTHER ANIONS As has been pointed out in Section 1, it was an essential feature of SHANNONand PREWITT’S work that they oonsidered only bonds from cations to oxygen or fluorine. Bonds from cations to a wider range of anions will involve different covalent contributions and cannot be expected to conform with anything like the same degree of accuracy to simple radius sums. But it is inevitable that people using the newly proposed radii will at some stage wish to make comparisons, however approximately, with structures involving other anions. If this is done it is of course essential that the radii used for the other anions should be compatible with the present set of radii, and because of the change in datum level (the assumed radius of oxygen), this will not be so if anion radii are taken from earlier lists and used in conjunction with the present cation radii. It is therefore felt desirable to include in Table 4 approximate, but compatible, values for the radii of Cl, Br, I, S and Se. In order to draw attention to the approximate nature of these values they have been put in parenthesis in Table 4. They have been derived by subtracting the appropriate cationic radii from the interionic distances in halides and chalcogenides in the more readily accessible data given by WYCKOFF (1963, 1964). It is noteworthy that whereas at all other places in the table radius increases with co-ordination number for any given oxidation state, the values obtained in this way for Cl-, Br- and I- exhibit a fall in radius from VI co-ordination to VIII co-ordination. This is presumably due to the fact that changes in radius of the cations as a function of co-ordination number may well be different when they combine with different anions. Whatever is the true change in radius of the halide ions with co-ordination number, it will here be obscured by the fact that we have derived these values by the use of cation radii appropriate only to bonding with oxygen and fluorine. It follows that the radii given in parenthesis will give radius sums, with cation radii in the table, that will represent interionic distances only to about the same degree of accuracy as have been available from previous tables of radii, and there is no reason to ascribe any high degree of validity to radius ratios involving them.

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E. J. W.

WH~TTAEER and R. MUNT~S

Acknowledgement-One of us (R.M.) gratefully acknowledges a N.E.R.C. Studentship during the tenure of which part of this work was done.

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REFERENCES AHRENS L. H. (1952) The use of ionization potentials-l. Ionic radii of the elements. Qeochim. Cosmochim. Acta 2, 1555169. FRONDEL C. (1968) Crystal chemistry of scandium as a trace element in minerals. 2. Kristallyr. 127, 121-138. FUMI F. G. and TOSI M. P. (1964) Ionic size and Born repulsive parameters in the NaCl-type alkali halides. J. Phys. Chem. Solids 25, 31-43. GOLDSCHMIDTV. M. (1926) Geochemische Verteilungsgesetze der Elemento VII. Die Gesetze der Krystallochemie. Skr. Norske. Vid.-Akad. Oslo. 1, Mat.-Nat. Kl. No. 2. PAULINE L. (1927) The sizes of ions and the structure of ionic crystals. J. Amer. Chem. Sot. 49, 765-790. PAULINE L. (1900) The Nature of the Chemisal Bond, 3rd edition. Cornell University Press. SHANNONR. D. and PREWITT C. T. (1969) Effective crystal radii in oxides and fluorides. Acta Cry3talEogr. B!& 925-946. SHANNONR. D. and PREWITT C. T. (1970) Revised values of effective ionic radii. Acta Crystallogr. in press. WYCKOFF R. W. G. (1963, 1964) Crystal Structures, Vol. 1 and 2, 2nd edition. Interscience.