1437

Acta Cryst. (1964). 17, 1437 II: A Proton-Ordered Form of Ice*

BY BAl~CLAY I~AMB Division of the Geological Sciences, California Institute of Technology, Pasadena, California, U. S. A.

(Received 5 August 1963)

Ice II, one of the high-pressure polymorphs of ice, of density 1.17 g.cm -3, has 12H~O molecules in a rhombohedral unit cell with a = 7-78 A, a = 113.1 °. According to the Laue symmetry shown in single- photographs, the space group should be either R3m, R3m or R32. Centrosymmetry is indicated by a statistical test with essentially three-dimensional intensity data. However, the Patterson function indicates a pseudo-structure in space group R3c. The true structure must be appreciably distorted from this because of moderately strong violation of the c-glide extinction condition. The distortion could degrade the space group to /~32, but the resulting structure has unsatisfactory features, and does not conform well with the experimental data. Structural reason- ing indicates space group R3. The observed R3m Laue s:~nnmetry must therefore be the result of intimate twinning of Rg individuals. The structure in R3 can be refined, under the assumption of twinning, and with the inclusion of hydrogen atoms, to R = 0.08. The basic R-3c structure contains ice-I-like units built out of puckered 'six-rings' of mole- cules. These traits are linked together in a more compact way than in ice I. Each oxygen atom is bonded to four nearest neighbors at 2.80 + 0.04 A, and has in addition a next-nearest neighbor at 3.24 _~. The distortion of the true structure ~from the pseudo-structure leaves the bond lengths essentially unaltered but changes the bond angles appreciably. It constitutes evidence that the hydrogen atoms are in ordered arrangement in the crystal, rather than disordered, as in ice I. The particular ordered arrangement actually realized can be deduced from detailed features of the structure. This arrangement gives better agreement with the X-ray data than any other, although the X-ray distinction among various arrangements is necessarily marginal. The ordering energy in ice II, formulated in terms of the H-bond strain, must involve signifi- cantly the effects of both donor- and acceptor-misorientation. The accepter strain energy contribu- tion is formal to depend primarily not on deviation from tetrahedral bond orientation relative to the accepting H20 but on deviation from the 'accepting plane' that bisects the H-O-H angle of the molecule. The H20 environment in ice II deviates greatly from an ideal tetrahedral one, yet the H-bond strain energy, although greater than in ice I, is small enough to be offset by the extra van der Waals energy, so that the energy of ice II is only 0.01 kcal.mole -1 greater than that of ice I. The measured entropy difference between I and II is a direct reflection of the entropy of proton-disorder in ice I.

Introduction ice II structure. It is part of a systematic study of Ever since Pauling's fruitful proposal (1935) that the crystal structures of the high-pressure fo~ms of ice, the protons in ice are in a disordered arrangement, of which the results for ice III have been reported the idea of proton disorder has played an important previously (Kamb & Datta, 1960a, b). part in interpreting the structure and physical The six high-pressure forms of ice are produced at properties of ice and related substances. Careful pressures upwards of 2000 atmospheres; their various studies of the ice I structure (Peterson & Levy, 1957; fields of stability were determined by Bridgman see also Lonsdale, 1958, and Owston, 1958) have (1912, 1935, 1937). Ice II, the form of interest here, failed to reveal any sign of ordered proton arrange- occupies the lowest-temperature portion of the P, T ments such as the ones proposed originally by Bernal field investigated by Bridgman, as shown in Fig. 1. & Fowler (1933), and a theoretical attempt to demon- The structure of ice II was studied by McFarlan strate that proton ordering is favored energetically (1936a, b) using X-ray powder diffraction methods; (Bjerrum, 1952) has proved faulty (Pitzer & Polissar, however, as reported previously (Kamb & Datta, 1956). 1960a, b), the structure found is not a plausible one It now appears, however, that there is a high- and the powder data on which it is based are not pressure ice polymorph, ice II, in which the protons those proper to ice II. are ordered. The present paper presents the evidence for this conclusion, based on a determination of the Experimental * Contribution No. 1197 from Division of the Geological The experimental technique used for preparing the Sciences. high-pressure ice samples and investigating them by k C 17 -- 93 1438 ICE II: A PROTON-ORDERED FORM OF ICE / for III over the range 2 to 3.5 kbar by Bridgman ICE (1912, p. 535), and by assuming that they have the ...v.. J same coefficient of thermal expansion as that of ice I, 20 / which has been evaluated by Lonsdale (1958). The / inaccuracies involved in these assumptions are not serious, because the volume difference is small com- pared with the specific volumes themselves. The "315 density of ice II obtained is 1.170 g.cm-a. .,Q ICE "EI :2 X-ray data Because ice II forms only by transformation from r, other solid phases (Fig. 1), it had been expected that the product would be a fine-crystalline powder un- / suited for single-crystal study. This was suggested also by McFarlan's early observations (1936b, pp. 84-85). However, single or coarse aggregates of a few crystals formed with fair frequency in the glass capillaries used. The crystals proved to be easier to study than crystals of ice III, because ice II does not show the spontaneous inversion under -80 -60 -40 -20 0 20 40 60 80 1(30 X-irradiation that was found for ice III (Kamb & Temperature (°C) Datta, 1960c). Fig. 1. diagram of the NfO system, after Bridgman The unit cell of ice II is rhombohedral, with an= (1912, 1937). Solid phases are numbered from I to VII (IV omitted). 7"78_+0"01 _~, a = 113.1_+0.2°; the corresponding triply-primitive hexagonal cell has all= 12-97_+0.02, CH = 6"25 _+ 0"01 /~. These values were determined from X-ray diffraction has been described in connection precession photographs with Cu Ka radiation of with the results for ice III (Kamb & Datta, 1960c); assumed wavelength 2=1.542 A.* The Laue sym- it was used without essential modification here. metry shown is 3m, and there are no systematic Preparation of ice II is accomplished by cooling a extinctions. The space group indicated should, there- water sample at low pressure to -75 °C, and then fore, be one of the three R3m (D~d), R32 (D~), R3m (C~.). Two single crystals suitable for collecting quantita- At pressures below about 3000 atmospheres the rate tive diffraction dat~ were lounc~ in the course oi t'h~ of transformation is too slow to get a consistent yield work, and from these a nearly complete three- of ice II, and at pressures above 3500 atmospheres dimensional set of data was obtained by precession (at -75 °C) a different phase is formed, which has and rotation methods. The set of photographs avail- not yet been identified. At temperatures much above able for intensity measurement is shown in Table 1. about -70 °C but still within the ice II stability The rotation photographs were obtained with the range, ice III forms metastably instead of ice II. precession camera by setting the precession angle to Identification of ice II as the phase formed in the zero and turning the spindle axis at constant rate. way described was discussed by Kamb & Datta (1960c) ; this identification has been confirmed recently Table 1. X-ray data for ice II by Bertie, Calvert & Whalley (1963). The t.at)le lists tile number of photographs of each type u.aed The density of ice II was not measured directly in the intensity measurements but was determined by comparison with ice III, Radiation in the following way. Needed is the density under Method Reflection type* CuK~ MoK~ the conditions of X-ray observation: atmospheric I hkO 1 2 pressure and -150 °C. From the known structure hkl 1 2 of ice III its density under the pertinent conditions hk2 0 ] is calculated to be 1.141 g.cm -3. At the Precession hhl 1 0 I hk(21c-h) 1 0 ice I -- ice II -- ice III, the difference in specific hk(2k -- h + l ) 1 0 volume between III and II (Bridgman, 1912, p. 524) hk(2k - h + 2) 1 0 is V(III)-V(II)=0.021 cm3.g-L The alteration of l~otation (CH axis) hlcl 2 3 this volume difference in going from the triple point, * RhombohedraI indices. at P=2.1 kbar, T=-22 °C, to the conditions P=I bar, T=-150 °C is estimated by assuming that * Bertie et al. (1963) have recently reported a unit cell both II and III have a compressibility over the determination for ice II, obtained by methods very similar pressure range 0 to 2.1 kbar equal to that determined to those used here, and in agreement with the above results. BARCLAY KAMB 1439

The number of reciprocal planes photographed by rotation photographs and the Mo K0¢ rotation photo- the precession method was limited by accidental graphs have almost the same slope if plotted against loss of the crystals. The precession photographs sin 0, but not against sin 0/~t. intercept 252 out of the approximately 600 non- On this basis, the rotation data were scaled to the equivalent reciprocal lattice points out. to sin0/2 = 0.7. precession data by applying corrections obtained 142 reflections were strong enough to be observed. from the linear regression line. The sets of data from The rotation photographs record all points out to all photographs on a common scale were then aver- sin 0/~=0.57, except for a few points close to the aged, and the intensities of reflections involved in CH axis. There are approximately 320 non-equivalent superimposed pairs in the rotation photographs were points within this limit. 86 reflections were actually obtained by difference where possible. The complete observed, of which 22 are subject to ambiguity in set of data contains 152 observed reflections, 3 un- indexing because they can represent superimposed resolved pairs, and 126 reflections weaker than a pairs (rarely, triplets or quadruplets) of non- known limit of observation. equivalent reflections. With the help of the precession data it was possible to resolve all but 3 of these ambiguities. The rotation data contain only 13 Structure determination observed reflections that are not present in the preces- The density and cell dimensions indicate that the sion set of data; 3 of these 13 are unresolvable pairs. unit cell of ice II contains 12 water molecules. The It is thus seen that the intensity data are essentially density calculated on this basis is 1.17 g.cm -3, in three-dimensional as far as strong and moderate complete agreement with the density value estimated reflections are concerned. from experimental measurements. Intensities were estimated visually in the usual way, From the cell dimensions and content, two possible from the photographs listed in Table 1, and were structures can be constructed that give plausible corrected for the Lorentz and polarization effects. interatomic distances and coordination relationships The precession data were then reduced to a common among the water molecules. One structure is an scale, as were also, separately, the rotation data. analog of the silicate framework in the mineral No absorption corrections for the Cu Kc¢ data were sodalite, distorted from cubic to rhombohedral sym- applied, as none appeared to be indicated by com- metry. An H20 framework of the sodalite type occurs parisons with the Mo Ko~ data. Scaling of the preces- in the hydrate HPF6.6H20 (Bode & Teufer, 1955). sion planes to one another is facilitated by the The other structure has no analog among silica or rhombohedral symmetry, l~otation intensities were silicate structures, but may nevertheless be described corrected by the appropriate multiplicities. as a framework structure involving tetrahedral Comparison between the precession data and the linkages. However, neither of these structures is in rotation data indicated a considerably larger un- a space group proper to ice II, the first because of certainty in the estimate of the relative scaling factor body centering corresponding to a smaller primitive than had been found for the within-group scaling cell in space group R3m, and the second because of comparisons. When the logarithmic intensity dif- a c glide plane, the space group being R3c. ferences for reflections common to the two sets of Because these discrepancies appeared unavoidable data were plotted against sin 0, a well-defined linear for the a priori structures, a direct structure determina- regression resulted, about which the scatter was tion was undertaken. The intensities were put on an comparable to that shown in the within-group scaling absolute scale by means of a Wilson plot. A zero- comparisons. This effect is thought to be due to an moment test (International Tables for X-ray Crystallo- incipient splitting of the individual reflections that graphy, 1959, p. 357) showed clear centrosymme_try is observable in the rotation photographs. The for the structure, indicating the space group R3m. splitting is attributed to a coarse mosaic containing The three-dimensional Patterson function was cal- rather widely diverging crystal orientations, since it culated, using a Burroughs 220 program written by could not be eliminated by careful alignment of the K. Hoogsteen. crystals. The splitting causes a spreading out of the The Patterson function indicates that each oxygen reflection spots which increases progressively with atom has four nearest neighbors, at distances of sin 0; it is most severe on the higher layer lines 2.81_+ 0-03 ~. In spite of the indicated space group (lhex=4 and 5). The subjective technique of visual R3m, there are no vectors of the type (X, X, 0) intensity estimation tends more to measure peak required by the operation of the mirror plane. Instead, photographic density than integrated density, ap- there are three non-equivalent vectors of the type parently, and hence the splitting leads to a progressive (X + ½, X + ½, ½), suggesting a c glide plane. Because decrease in the estimated rotation intensities as a of the contradictory indications, a method was sought function of sin 0, relative to the precession inten- that would define the structure independently of sities for which the splitting effect is less strong. the assumed space group. This can be done by utilizing This interpretation is supported by the fact that the the threefold symmetry axis. Only one vector of the regression lines obtained separately for the Cu Ka type (X, Y,-X-Y) required by the operation of 1440 ICE II: A PROTON-ORDERED FORM OF ICE this axis is present, and it defines the positions of all cell so that they are unequally bonded to one another of the oxygen atoms in CH-aXiS projection. A nearest on opposite sides. The nearest-neighbor distances on neighbor vector (length 2.84 /~) is related to the one side of the ring become 3.01 J~, and on the other (X, Y, -X-Y) vector in such a way as to generate side 2.57 A. The distance 2.57 A is almost certainly a puckered hexagonal ring (symmetry 3) of six too short to be acceptable as a hydrogen-bond distance oxygen atoms; there must be two such rings in the in ice. structure. Application of the vectors of type Strong experimental evidence against the R32 (X + ½, X + ½, ½) leads to a placing of the rings relative structure is provided by the hhl reflections having to one another in the cell, and the resulting structure 1 odd, which are sensitive to the way in which the accounts for all of the remaining Patterson vectors. perturbation vector is applied. There are 28 reflections The structure so determined is in space group R3c, of this type in the data list, of which 11 were observed. and is equivalent to the second a priori structure An immediate test of the R32 structure is not possible, mentioned above. It involves a single 12-fold oxygen because experience shows that, with the exception position (12f in R-3c). With coordinates derived from of 001, all of these reflections are sufficiently sensitive the Patterson vectors, and an isotropic temperature to the oxygen positions that meaningful conclusions parameter from the Wilson plot, a structure factor can be drawn from the structure amplitude agreement calculation for this structure gives a good overall for refined structures only. In a least-squares refine- pattern of agreement (R=0.30) with the observed ment of the R32 structure accordingly carried out, structure amplitudes, except for the c-glide extinctions for which the overall structure amplitude agreement that are predicted. converged to R=0.17, the pattern of agreement for The absence of these extinctions requires that the the hhl reflections with 1 odd remained poor. It actual structure differ b_y some perturbation from the corresponds to a residual/~ = 0.46 figured by including pseudo-structure in R3c. The perturbation must in addition to the 11 observed reflections the 4 affect the oxygen positions and must be appreciable, unobserved reflections for which IFc] is greater than since it can be shown that no arrangement of the the limits of observation, which were assigned as hydrogen atoms could account for more than a small the IFol values in these cases. This residual is to be fraction of the amplitude of reflection 111, which compared with 0.17 for the correct structure. is one-third of the largest observed for all reflections. Another type of structural argument, given later, A measure of the required perturbation can be indicates that the actual space group must lack the obtained by carrying out a Fourier synthesis in which twofold axis contained in R-3c. There is, however, the structure is treated as though it were in R3c, no way to drop both the c glide and the twofold using the signs of the structure factors calculated axis in R-3c while remaining in Laue group 3m. This previously and omitting the hhl reflections with l odd. leads to the conlucion that the true Laue group must This calculation gives in some aense an 'average' be 3, and that the apparent Laue symmetry 3m structure obtained by superposing upon the actual shown in the X-ray photographs is produced by a structure the structure generated from it by carrying twinning of R3 individuals. The scale of this twinning out a c-glide operation. It is found that the peak must be fine enough to give essential equality of representing the single oxygen atom in the R3c intensities of 'hkl' and 'khl' reflections obtained by structure is elongated in a certain direction in the superposition of hkl and lchl intensities from equal crystal. This elongation can be interpreted as a volumes of untwinned and twilmed domains, and 'splitting' of the single oxygen atom into two un- yet it must be coarse enough that no diffuseness of resolved half-atoms. From the shape of the elongated the hhl reflections with 1 odd is observed. peak the magnitude of the splitting between the half If the perturbation vector is accordingly applied atoms can be estimated, and thus the perturbation in R3, a structure results in which the two puckered vector determined. The splitting can be controlled six-rings are non-equivalent but in which all nearest- also by the amplitude of the 111 reflection. It amounts neighbor distances are close to 2.80 .~. This structure to 0.25/~. can be tested against the observed data, under the The way in which the perturbation vector is to be assumption of twinning, by comparing O~- applied to the R-3c structure to get the actual struc- [½F~(hkl)+½F'~(hkl)]'~ against Go(hkl). The agreement ture depends on the space group of the latter. The residual is R=0.14 before any further refinement of most obvious choice for this is R32, which is the oifly the structure. space group in Laue group 3m obtainable from R3c The space group R3 is also possible, but the single by dropping the c glide plane. The resulting structure perturbation vector derived in the way described has unsatisfactory features, however. The puckered provides insufficient information to generate a struc- six-rings of oxygen atoms become distorted so that ture in this space group. The zero-moment-test the 0-0 distances within them are not all equal. indication of centros_ymmetry ca~mot be applied to The two rings remain equivalent, related by the distinguish between R3 and R3, because it is dominated twofold axis, but their relative position shifts in the by the pseudo-space-group R-3c. Structural reasoning, BARCLAY KAMB 1441

given later, does not support a structure in R3, To test the sensitivity of the data to contributions and no attempt was made to derive a set of oxygen from the hydrogen atoms, which had not been included positions for refinement in this space group. up to this point., structure factor calculations were made for various proton arrangements: (1) the eight ordered arrangements in space group R3 that involve Refinement of the structure only one proton perO • • • 0 bond; (2) several ordered Least-squares refinement of the R3 structure was arrangements in R3 that violate this requirement; carried out with a weighted sum of residuals of the (3) the proton-disordered arrangement, also in R3;

form Xw[2Go(hkl)-F~(hkl) -F~.(khl)].' The program (4) several ordered arrangements in B3. In these for doing this is a modification of the triclinic least- calculations the oxygen parameters were held fixed, squares program written by g.E. Marsh for the the scale factor and oxygen temperature factor having Burroughs 220 computer. The modified program been first adjusted to fit the inclusion of hydrogen calculates and solves the full 6 x 6 matrix of least- atoms. For simplicity the protons were placed on the squares coefficients for the positional coordinate shifts center lines between oxygen atoms, distant 1.0 _~ of the two non-equivMent oxygen atoms in the R3 from the appropriate centers; this is not exactly structure. Inclusion of all off-diagonal terms in the correct, because the oxygen coordination polyhedra matrix is necessary because the two atoms interact a~e distorted from ideal tetrahedral shape, but it strongly in their contributions to most of the reflec- probably introduces no significant error. tions, owing to the pseudo-symmetry and the twin- ning.* The refinement reduced the structure amplitude function was used to avoid overemphasis on intensities near t, hc limit of observation, where the a(;curacy of intensity residual to R=0.11 for the 152 observed reflections. estimation is inherently poorer than at higher levels. The (:r7~ fun(.tion was used instead of Go-~ to avoid instabilities * The weighting function chosen varies inversely as Go" that experience shows (.an arise with the latter weighting for (4o > 1O'0, and assigns unit weight to all residuals for which function (R. E. Marsh, personal communi(:ation). Unobserved Go lies between 10.0 and the limit of observation, which ranges reflections were included with unit weight when Gc wa,~ from about 2.0 to about 7.5 over the range of observational greater than the limit of observation, whi('h was assigned conditions involved. The constant-weight portion of the as the GO value in such eases.

Table 2. Observed and calculated structure amplitudes* G~= [½r~(hkl) + ½F~(lchl)]½

hk~ ~ O h kt G G h k~ G G hk4 O O .....•-.c - ~.~ ..-c ...... c ___ -c -c

o ~ 9 < 3.8 2.8 0 0 1 < 2,0 0,5 i ~ I h.2 5,1 I 2 2 5,2 5,2 2 h 2 3,0 3,1 3 ~ 3 1].', 1].v 8 < b.t, o.h 10.2 ii.~ 2 h.3 h.l 3 " 3.1 1.6 3 o.~ 6.L t+ < 3.1 2,3 _ 9 < h.O i.o 3 3.5 3.9 Z 3.0 2.3 h 7.7 7.5 h 0.5 b.~ < 7.! t..3 7 7 9.3 9.7 4 10.2 n.~ %7 o.1 ~, 2.:~ 1.7 ; 5.6 c.; 6 z 7.3 I.? 8 h.9 4,5 5 < 3,5 h,l 5 4,7 3,:; 3 3 3,3 2,{ 6 h,5 2,5 "! < 'J,5 1.3 9 4,2 2.7 6 < 4.4 3.1 6 < 3.1 1. L; L < 2.~ 1.5 7 ~ 7.1 h*3 1.9 3 5.1 3.~ 1~ 1.2 7 < 3.9 2.1 7 3.2 2. 2~ 3 < 7.1 I.I 8 ( 7.2 2.U , 5.9 5.1 6.6 6.6 I I 32.9 33.1 " i..7 L9 h < 7.2 2.2 "~ 2 7.2 c.9 c 3.; 3.3 h.l 2,2 2 1.1,6 12,3 _ 9 3,3 :',~ '5 ~ 7.) 2,5 3 12,7 12,0 _ c < 2.2 2.5 %5 5.2 8.1 8.9 5 -..9 L... 7.3 2.2 i. <, 2.7 3.5 r.2 o.:: 1 t w h ~, ~ ",.2 ,.! 190 &. 3 h . 9 2.9 3.2 2 ~,.b t.." ! ~ :'.? C,.'~ 5 5,7 C..~ l,h 2.7 5 h.6 h.3 3 IL8 ll,.l : ~ 7.1 3.1 6 ~ 4.5 ~.7 ": ¢,.L ,J.? 5 ~.s ?.~ 6 3.1 2.1 h t.? l.a "9 2 ". 7.3 3.2 ? ~ 7.3 .,.9 "~ < u.(. 2.5 6 h.6 h.2 7 2.5 2.0 2 15.7 li.C. 3,* 7.2 2.1 ~ < 7.0 2.7 _ 5 < 7.8 l.h 7 < h.3 I.I 2 2 9,~ 11.7 7.6 6.2 5 " 7,I o.3 ~ 20.5 2O.'J < ';,(, I.', < 4.5 3.1, - 7.L~ ' .,. 9• < 4.2 I.~ < L.h 1.9 5 3.2 3.7 b ~ 7.I 1.5 9.q 10.2 ,. !; II.i I'..7 < 2,5 2,9 5 < u.2 2.0 -9 < 2.5 1.2 7 7.2 3.7 5 5.2 :;.4 .: ~.o 2..o 5 lb.5 In.b 6 3.6 3.6 i, ] 23.5 2',.t ; < 7.3 2.3 0 <+ 7.I 1.5 " ' : L.~, 2.1 6 7.2 6.~t 3 ~ o.') 5.9 2 17.5 17.2 - ~ ~ "., 2.t, 7 ~ 7.3 2. ; "~ < 7. .... 7 < 4.3 1.9 4.5 3.b 3 9.6 "~.7 ,~ < 7.'Q 1.'~ 82 < 6.2 2.4 -, < 7." 2.' 8 < 4,1, 1.5 5 < 3.~ 2.0 ii.', n.> 3 < 5.C3 6.1 ~ 2.G I.U < ¢,.8 2.3 _ 9 4.1 4.3 ._4.1, < 3.6 1.3 7; -~,7 ;.% ," :,.5 1.9 3 2.'} 3.2 :. %6 ~.'; 3 3 1~.3 13.9 1 113 l; 3.3 3.5 ,C 3.9 3.3 ~ i(~.2 9.0 ia ," 2.5 t.3 5 < 2.1, I.', 1,.~ 5.9 5 < 2.7 I.i < 3.3 I.;. 6 < 6.~ 1.9 c 3.3 2.3 5 8.8 8.5 6 2.7 3.0 2~ *.O 3.'J 7 < 7.C 2.5 o 3,9 3.7 2 5 < o.8 h,l 6 15.;+ 15.4 7 < 2,6 i.I o 2.1 2.4 ~, < 7.3 2.6 2 2 < b.J h.h 7 < h.3 3,a J < 2.6 2.2 -3 i !..2 i~,h - 9 < 7.1 2.5 3 "~ 7.3 ,..h 8 4o5 h.3 2 " 2.5 2.h 2 ) ;.2 32.5 7 2 ~" ::.3 3.9 ;, ~ 7.3 2.2 9 < 3.2 1.7 < 3.2 1.7 3 1;:.5 13.7 3 < o.1 1.h ~ '" 7.h 2.1 2 24,1 23,3 3.1 2.7 ~ 7.-; 7.1 , 13.o 13.1 3 < 7..9 ].3 3 20,5 20,2 5 5.1 h.9 2.5 2.9 5 3,h 3.3 h < 5,5 2.2 ~ 5.5 6.0 6 3.2 2.9 6 3.0 2.6 (, ~ o.6 2.9 _ 5 < L.6 2.9 5 < 2.7 2.1 7 < 3.2 2.h " < 3.I 1.7 7 < 7.0 1.1 ? 3 < c.? 5,1 6 3,9 5,1 8 < 3.1 22+ - 8 ":.2 ,.'5 5 < 7.2 I.~, 1, < h.9 I., 3.9 3.9 ," 2.0 1.1 2~ 1~.~ ~.2 ~ < ~.~ 1., ~ -- ~,.2 ~." _ < 1,.I 2.h < 3.2 1.7 ~ 2,;.7 2~.~, ~ < L.7 0.2 5 3 d.3 5.3 i i 13.7 13.7 2 7.3 6.2 3 %0 6.:, 35 Iu.7 10.3 h ll.l 12.~ 2 6.5 7.5 3 2.9 3.0 . 11.5 13.h ~.I 7.2 5 6.6 6.I 3 9.7 I0.0 4 < 3.3 I.~ 9.~ I~., 5 u.7 4.7 6 < 5.6 2.5 h 15,~ 16.3 6 3.~ 3.6 o 0.5 u.~ ~ 9.1 5.9 ~ 3 < 7.h 1.~ 5 L~.2 4.7 4.1 3.q "/ 3.9 3., 7 < ~,2 L.6 h < 3.7 1.6 6 3.8 3.8 7 3.7 2.'3 "3 < 2.h 1.7 :~ < 7.2 2.1 5 < t,.9 u., 1,.6 h,4 < 8 < 3.3 1.9 ~ ] <. 2.7 0.!, O <+ 7.9 l.b 6 5.5 ~.7 2.7 0,5 9 < 3,0 2,3 2 29.3 29.9 ~2 < L.7 3.5 ~ 3 2h.9 23.9 3,2 3,0 3 12,; IL,i+ 3 12,2 11,9 1, L, i 5,7 2 3.6 2.7 . 2.d 2.7 i. '~.9 L,9 q < 4.2 L.7 12.5 11.3 ¢ 152, II.! 5 6.5 6.0 _ 6 < 5.3 3.2 < 3.1 0.8 o 5.9 5.L 6 < 5.9 5.~ 3 3 < 3.7 0.3 5 < 3.3 1.3 "! 1+.9 2.. :' ~ 9.7 9.3 L 7.0 8.3 6 3.7 3.7 I~ 1,.3 11.5 8 .e 7.2 2.5 5 + 1,.0 4.1 7 6,3 5,8 12,5 13.5 9 "" b,8 2,I O 7.3 6.5 3 2.9 3.0 9 < 3.0 O.9 i~ 2,.2 3.5 5 5.0 4,3 6 < 2.h 1.8

* Note.-- For the unresolved pairs (283, 465), (276, 384), and (355, 454) the Go value given would be the observed amplitude if the other member of tile pair were vanishingly weak. 1442 ICE II: A PROTON-ORDERED FORM OF ICE

The agreement residuals for the various proton Table 4 arrangements range from R=0.087 to 0.112, the Varianc~-covariance matrix for el and OI1 coordhmtes corresponding range in the weighted sum of squares xI Yl zI XlI YII zII being 0.99x105 to 2-76x10 a. The improvement obtainable by introducing the more-suitable proton 0.75 0.27 --0.12 --0-01 0-03 arrangements is probably meaningful, but the data II-I z1 I o.1.14o 0.01 0.06 --0-59 1x 10 -~ do not distinguish among the arrangements giving the xH 0-91 0.53 0-51 best agreement, namely: one ordered arrangement YlI I 1-23 0-41 zii L 1.31 _ in R3 (designated 2-I-4, i-II-3 in a notation defined later), one in R3, and the proton-disordered arrange- Inverse variance-eovariance matrix for marginM density of merit. The lack of distinction among these is under- Ol or On coordinates ,done standable, because all three look identical in the x 5' z pseudo-structure in R-$c, which contributes most of x [ 1"08 --0.50 --0.10] the diffracted X-ray intensity. The reflections hhl y 1"00 --0"39 x 10 ~* with 1 odd provide here again a more sensitive basis z 0"97 for discrimination. It is found that the R3 (2-1-4, Principal standard deviations and principal axes 1-II-3) structure gives an agreement residual R = 0.09 i ai xi yi z~ for these reflections, whereas both the R3 structure 1 0.0011 0.55 0.06 -0.65 and the disordered structure give R=0.18. Since the 2 0.0008 0.66 - 0.42 0.52 (2-I4, i-II-3) ordered structure is strongly indicated 3 0.0006 1.14 1.36 1-15 by structural considerations, as discussed later, it was accepted as giving the correct proton arrangement and a final refinement of the oxygen parameters was Table 5. Oxygen coordination data for the actual carried out for this structure. The final agreement structure (R-3) and the pseudo-structure (R-3c) residual is R=0.080 for the 152 observed reflections. R-3 R'3c ^ Of the 126 unobserved reflections, only 2 calculate i "1 II 0 significantly greater than the limit of observation. Distances (A) The observed and calculated structure amplitudes for i-1 2.81 2.75 2-78 __+0.012 the final structure are listed in Table 2, and the atomic i-2 2.81 2.75 2-78 --+0-012 parameters are given in Table 3. i-3 2.84 2-84 _+0.013 i-4 2-80 2-80 --+0-007 i-5 3.24 3.24 Table 3. Atomic parameters in ice II Angles Atom x y z B (A ~) 1-i-2 116* 120 ° 117.5 ° _+ 0-3 ° OI 0.273 0.024 -0.146 1"93+_0"16 1-i-3 85 99* 92 OII 0.478 0.750 0.332 1"78 1-i-4 115.5" 107.5" 111.5 H(I2) 0.165 0.069 --0-190 2-11 2-i-3 126 125 125.5 H(I4) 0.413 0.205 -- 0-003 2.1-~ 2-i-4 88* 80 84 H(II1) 0-719 0.401 0.395 2.1¢ 3-i-4 129 127 128 H(II3) 0.741 0.203 0.360 2.11 * Possible donor angles.

Pertm'bation vector: p 0.012 0-023 ().011 changing XH and yn, these matrices are found to be On* 0.250 --0"022 --0"168 quite similar, as expected on account of the pseudo- t Assumed. symmetry. The average between them is given in Table 3. Also listed are the principal axes for the quadratic form of this matrix, found by taking into Accuracy account the skew coordinate system, and also the The standard deviations of the oxygen coordinates corresponding principal values expressed as principal in Table 3 are all very close to 0.0010, as derived from standard deviations in units of the rhombohedral cell the full 6 x 6 least-squares matrix, which takes into edge. The directions are appropriate to the position account the considerable interaction between the of O~ as given in Table 3. The smallest principal oxygen atoms. The magnitude of the interaction ean standard deviation corresponds to the principal axis be judged from the complete variance-covariance most nearly parallel to the perturbation vector matrix, given in Table 4. To evaluate the anisotropy (Table 3) by which the 'splitting' of the oxygen atoms of positional accuracy caused by the interaction, in the pseudostructure is generated. This was surprising inverse variance-covariance matrices for the marginal at first sight, but reflects the fact that in tlfis direction densities of the 0i and On coordinates separately the resolution of the atoms in the pseudostructure is were obtained from the matrix in Table 4. On inter- actually the greatest, the 'split' atoms being corn- BARCLAY KAMB 1443 pletely superimposed as viewed in projection onto above the other. If these columns are detached from a plane perpendicular to the perturbation vector. one another, moved relatively up and down parallel It is seen that in spite of the interaction, O~ is well to the ell-axis SO as to give a rhombohedral stacking resolved from the position of its companion 'half atom' sequence for equivalent six-rings, and then rotated in the R3c structure, for which the coordinates are ca. 30 ° around their long (CH) axes, they can be given, under O~*i, in Table 3. relinked together in a more tightly-fitting way than Standard deviations for the O-0 bond distances in ice I. This is the structure of ice II (Fig. 2). The have been computed from the complete variance- scheme of cross-linking between the hexagonal columns covariance matrix, taking into account the correlation causes the six-rings in each column to twist relative between the atomic positions. They are given in to one another through an angle of about 15 ° . This twist, plus a considerable flattening of the puckered Table 5. six-rings, causes the CH-aXiS tO decrease from the value 7-36 A in ice I to 6.25 A in ice II. The coordina- Interpretation of the structure tion of the oxygen atoms is markedly distorted from Ice II can be described first in terms of a basic struc- the ideal tetrahedral one, as shown by the bond ture that is related in a certain way to Ice I, and then angles in Table 5. Each oxygen atom has four nearest a relatively small but significant perturbation'upon neighbors at distances of 2.80 _+ 0.04/~. The distortion this basic structure. enables a next nearest neighbor at about 3-3 -~ The basic structure (or pseudo-structure) in R-3c distance to be accommodated; this distance is the is related to ice I as follows. In the ice I structure one between the atoms labelled I and II in Fig. 2. (see e.g. the illustration in Pauling, 1960, p. 465) The basic pseudo-structure is perturbed by 0.13 /~ one can recognize hexagonal columns consisting of displacements of the oxygen atoms, eliminating the puckered six-rings of water molecules stacked one c glide plane of the pseudo-structure and degrading the space group to R3. The effect, though slight, is clearly visible in Fig. 2. It makes the six-ring involving Oi become more strongly puckered, while the On six-ring becomes almost flat. It is contended that this perturbation is caused by an ordered arrangement of the protons in the structure. If the protons were in a disordered arrange- ment, as they are in ice I, the environments of all of the oxygen atoms would be on the average_ equiv- alent and the pseudo-structure in R3c would be retained. Any tendency for the bond distances or angles to assume values other than those imposed by the basic structure could not express itself in an ordered way; the only structural effect that could be detected would have the appearance of anisotropic thermal motion, representing the necessary short- range order. Such a situation is probably present in ice I (Peterson & Levy, 1957). It is assumed that there is here no effect on an intermolecular scale, analogous to the Jahn-Teller effect, that could lead -19 ~ I / to a spontaneous distortion of a disorder-averaged structure. The cause of the observed distortion must, then, be long-range ordering of the protons. In the proton-ordered structure, the oxygen posi- tions must conform to the symmetry of the proton Fig. 2. The structure of ice II, viewed along the hexagonal arrangement. In the R-3c structure two nearest neigh- cg axis. Heights of the atoms above a hexagonal (0001) bor oxygen atoms are related by a twofold axis. plane are given in hundredths of the CH axis (CH -= 6-25 A). The rhombohedral unit cell is outlined. One six-ring of Since the O-O distance is long enough to require each of the two types, involving respectively Oi and OII, that the bond be asymmetric, the twofold axis must is emphasized, and the ordered water-molecule orientations be dropped in the actual structure. This leads, as in these rings are shown. Representative atoms Oi and OII discussed previously, to assignment of space group R3. are labelled I and II respectively, and their nearest neigh- bors are numbered 1 through 4 in a scheme used in discussing The strong experimental confirmation of this space the proton ordering. Hydrogen bonds are shown with a group vis.h-vis the possibility of space group R32 dashed line; the bonds linking up and down between the constitutes, conversely, further support for the inter- six-rings, to form the ice-I-like hexagonal columns, are pret~tion of proton ordering. omitted to avoid confusion. 1444 ICE II: A PI%OTON-ORDERED FORM OF ICE

The postulated twinning is a natural consequence 1-II-4) but this must be ruled out because it makes of this interpretation. Within a basic R-3c framework the two oxygen atoms equivalent and also violates of oxygen positions, the two possible ways of ordering the restriction that there be only one proton per the protons (corresponding to opposite signs for the H-bond. The pair (1-I.4, 1-II-3) avoids this difficulty, perturbation vector) are twinned counterparts. Since but presents another: the angle 1-]I-3 is improved the proton ordering is energetically doubtless a small by the distortion from the pseudo-structure, but the effect, it is to be expected that stacking faults in the angle 1-I4 is worsened. For (2-I.4, 1-II-3), on the proton ordering will arise, subdividing the basic other hand, both angles are improved by the distortion. single-crystal framework into twinned domains. The angle 1-I-2 cannot be used for a structure in space group R3, but it can be used in R3; however, this then requires use of 3-I.4, which is one of the Assi~,nment of proton positions worst angles of the group. From the detailed features of oxygen coordination It might appear from the bond angles that the in the structure (Table 5) it is possible to infer the proton arrangement (1-I-4, 1-II-3) would have lower positions of the protons. The bond distances are of total energy (higher H-bond energy) than (2-I.4, no direct help, because every H-bond must involve 1-II-3) and should therefore be preferred. However, a proton at one end or the other. It is the bond angles three features of the structure argue in favor of that provide the necessary clues, since, of the six (2-I--4, 1-II-3). bond angles at each oxygen atom, only one can be Consider first the perturbation by which the R3c occupied by the H-O-H angle of the water molecule pseudo-structure distorts into the actual structure. at that site. The data in Table 5 show that a considerable distortion The tendency for the water molecule to form bonds from the pseudo-structure is possible without essential at tetrahedral angles, as shown in ice I and in crys- change in any of the O • -. O bond distances. In fact talline hydrates (Wells, 1962 ; Baur, 1962 ; Clark, 1963), it can be shown, and also visualized from Fig. 2, indicates that the hydrogen bond energy has that the structure is free to distort in ways quite significant dependence on bond angle. The energy different from the perturbation actually observed, dependence results from bending of the H-bonds: still without sensibly changing the 0 • • • 0 distances. it is known that the water molecule in crystalline Consider the perturbation vector applied at the atom hydrates is little distorted from its configuration in labelled II in Fig. 2. Any perturbation vector lying the vapor (McGrath & Silvidi, 1961; Chidambaram, approximately in a plane parallel to the CH axis and 1962). In most hydrates, the 0... 0"." 0 angles containing the II.4 bond direction has the above presented to the water molecules are prevented by property. What then determines the direction of the other steric requirements from adjusting to the angle perturbation vector in this plane ? {avored by the water molecules, so that deviations Let the direction of the perturbation vector be of donor O--H by as much as 15 ° from the 0-.. 0 defined by the angle ~ that it makes with the CH aXiS, bond occur. being measured positive in the direction from In ice II, there is freedom for the bond angles to +cH toward the 0-O vector from 4 to ]I (Fig. 2). adjust without essential alteration of the bond dis- The changes produced in the various bond angles tances, as indicated by comparison of bond distances by the distortion can be calculated in terms of in the pseudo-structure and the actual structure. and the length p of the perturbation vector. To first We therefore assume that the structure distorts in order in p they are as follows such a way as to bring the 0...0...0 angles presented to the H-O-H groups (donor angles) ,:1(1-II-3) = (22 cos $ + 51 sin •)p closer to the angle for which H-bond energy is a ,1(2-I.4) = (38.4 cos ~ +2.7 sin $)p maximum. The favored angle might reasonably be l (1-I-4) = - (38.4 cos ~ + 2.7 sin ~)p taken either as 104.6 °, the H-O-H angle of the isolated water molecule (Darling & Dennison, 1940), where p is taken in A and the angle changes are or 109.5 °, the tetrahedral coordination angle in ice I. obtained in degrees. Since ~ngles reasonably near these values are present, If now the protons of the water molecules went the angles that deviate greatly may be eliminated, into the arrangement (1-I-4, l-II-3), one could leaving for consideration as donor angles only the expect the R3e structure to distort in such a way angles marked with an asterisk in Table 5. The that the original angles 1-0-4 and 1-0-3 become seemingly most favorable pair of angles* is (1-I.4, modified so as to minimize the quantity

* The angles are labelled in accordance with the scheme according to this same scheme; thus (1-I-4) designates a indicated in Fig. 2. The nearest-neighbor oxygen atoms of water molecule at Oi oriented so as to donate its protons to Oi and OII are numbered from 1 to 4 in a way that would neighbors 1 and 4 of 01. The angles in the undistorted pseudo- be symmetry-equivalent for Oi and OiI if a c-glide plane were structure are design~ated by replacing the I or II (which are actually operative. The proton positions are designated here equivalent) by 0. BARCLAY KAMB 1445

[(1-0-4) +/1(1-1-4)- 104.6] 2 molecule orientation, the angle/" between the acceptor + [(1-0-3) + A (1-II-3)- 104.6] 2 plane (0 ... 0 ... 0 plane defined by oxygen atoms from which H-bonds are accepted at the given water whereas for arrangement (2-1-4, 1-I1-3) the distortion molecule) and the H-O-I{ bisecting plane, it being should try to minimize assumed that the protons orient themselves sym- metrically with respect to the 0".. 0... 0 angle [(2-0-4) + A (2-I-4) - 104.6]" to which they donate H-bonds (donor angle). Acceptor- + [(l-0-3)+/] (1-1I-3)- 104.6J". plane misorientation angles I ~ are given in Table 6, first for the R3c structure, and then in more detail By minimizing these quantities with respect to ~, for the more favorable angles in the actual structure. for fixed p (assumed small), we can predict what the The (1-I-4) orientation, considered as possible in perturbation direction ~ should be for the two proton terms of the size of the donor angle, is seen to be arrangements. The predicted angles arc ~=90 ° for markedly unfavorable in relation to its aceeptor plane. (1-I-4, 1-I1-3) and $=39 ° for (2-I-4, 1-II-3). The observed angle is ~ = 42.6 _+ 2.7 °. The distortion is therefore appropriate to the (2-I-4, 1-II-3) structure Table 6. Acceptor-plane misorientation a~wle 1' only. St ruet ure Donor angle Aeeeptor angle /' In more general terms, the (1-1-4, 1-II-3) structure y,~,~ 2-0-3 I-0 4 25 '~ would seem to be unfavorable to the existence of R~c I 0-4 2-0-3 24 long-range proton ordering, regardless of whether n~, 3-0.4 I o -2 23 or not the 'most appropriate' distortion of the struc- n,~,. l 0-2 3 o 4 2(~ R3c I. () 3 204 12 ture were achieved. If the angles 1-I-4 and 1-II-4 to.% 2 ~) 4 I () 3 (; were available for occupancy by H-O-H, it appears R3 I 1-4 2 [--3 24.5 that (1-II-4) would be at least as favored if not more /¢,~ 1-11 4 2-lI 3 24 so than (1-I-4) ; this would tend to upset the ordering n~i 1-1-3 2-1=4 I I in the six-ring involving OII, and thus cause a break- Rg 1-]1-3 2-II -4 12.5 down of long range order in the crystal. It seems that R~ 2--]-4 l-I-3 2 21I 4 1 II 3 9.5 the existence of an ordered arrangement of energet- ically well-favored angles for H-O-H occupancy is essential to achieving long-range proton ordering. A complete picture of acceptor-bond o,'ientation in Its absence is probably responsible for the failure of relation to bisecting plane is obtained in stereographic ice I to achieve proton ordering at low temperature projection (Fig. 3). The acceptor-bond misorientation (Pauling, 1960, pp. 464-468). can be specified in terms of two angles o) and (y, We have so far considered the proton ordering in o) measuring the angular departure from the bisecting terms of the donor hydrogen bonds made by each plane, and q) the azimuthal orientation relative to the water molecule. The acceptor relationships appear to 'negative pole' of the water molecule (the axis bisecting be important also. In terms of the usually postulated the H-O-H angle, on the side away from the protons). tetrahedral character of the water molecule, the In Table 7 the misorientation parameters for the acceptor bond orientations indicated by the angles in H-bonds in structures (2-I-4, 1-II-3) and (1-I-4, Table 5 look rather unfavorable. For structure 1-II-3) are given. Donor misorientation is specified (2-II-4, l-I-3), the acceptor-acceptor bond angle is in terms of the angle .(2= ~(0D--104"6°), OD being the the smallest O -- • O • ". O angle at both OI and Oxi, donor angle. The large increases in 09 for bonds I-2 being much below the tetrahedral value. The distor- and II-3 in structure (1-I-4, 1-II-3), in comparison tion of the coordination polyhedra from ideal tetra- to the smaller increases in g2 for (2-I-4, 1-II-3), hedral configuration is so great that there is no tend to favor selection of the latter structure. possibility of achieving a tetrahedral acceptor orienta- tion in relation to donor protons ordered in a reason- ably favorable way. It is therefore necessary to seek Table 7. Bond-strain parameters a less restrictive criterion for favorable acceptor-bond (2-I-4, 1-II-3) (1-I-4, 1-II--3) orientation. ,., A necessary but not sufficient condition for tetra- Bond d ( A ) ~ o)

L marginal statistical significance. It is hoped to carry out a neutron diffraction study of ice II to test experimentally the arguments presented.

Thermodynamic considerations The ordering of water molecule orientations in ice II is reasonable in terms of the stability field of this phase in relation to other forms of ice (Fig. 1). The impossibility of superheating II into the fields of stability of III or V (Bridgman, 1912, p. 530) which contrasts with the readily achieved supercooling of III and V into the field of II, suggests a 'melting' of the proton ordering analogous to melting of the oxygen framework that occurs at the solid-liquid transition, which shows this same type of supercooling versus superheating relationship for all of the solid phases (except II). The proton 'melting' cannot take place within the oxygen framework of ice II because of the energetic unfavorability of most of the possible water a/(projected) molecule orientations in the structure; hence the oxygen framework changes to that of ice III (in which Triangles: neighbors to OI; circles: neighbors to OII. Filled the protons are probably disordered). symbols: upper hemisphere; open symbols: lower hemisphere. It seems that the proton-ordering transformations Fig. 3. Stereographic projection of the coordination polyhedra are particularly difficult to achieve. Bridgman (1912, around Oi (triangles) and OII (circles). The neighbors are p. 494) found that once II is completely transformed numbered in conformity with Fig. 2; 5 is the next-nearest to III, it is very difficult to retransform directly to II, neighbor. The orientation of the diagram conforms to O~r an exception to the typical 'memory' phenomenon in Fig. 2; for Oi, the orientation corresponds to the twinned counterpart of Fig. 2. The donor-bisecting planes for donor exhibited in the interconversion of ice polymorphs angles 2-I-4 and 1-II-3 are shown, solid on the upper (Bridgman, 1912, p. 532). Likewise, II cannot be hemisphere, dashed below. The points I and II are the formed initially from I without cooling to about 'negative poles' of the respective water molecules. Repre- -70 °C, far below the actual upper limit of ice II sentative accepter misorientation angles o) and ~0 are labelled. stability. As determined by Bridgman's measurements (1912, p. 492), the entropy of ice II is 0-77 cal. mole-l.deg -1 of the ice II structure: (1) The protons are ordered less than that of ice I, over the temperature range in the scheme (2-I-4, 1-II-3). (2) Both donor and from -34 °C to -75 °C. This value is close to the accepter relationships are important in achieving the residual entropy of ice I, 0.82 cal. mole-l.deg -1 as proton ordering. (3) The criterion for favorable measured by Giauque & Ashley (1933), or 0.81 as accepter bonds is that the bonds lie in or near the calculated by Pauling (1935) on the basis of proton H-O-It bisecting plane (w__0), without regard to disorder. The measured AS value can thus represent their precise orientation within this plane (~ value). directly the proton ordering in ice II, provided that This criterion corresponds to the electronic structure the difference in lattice-vibrational entropy between of the water molecule presented by Lennard-Jones I and II is small. There is no direct evidence for this, & Pople (1951, p. 157), in which a 'ridge' of high but it seems likely, on the basis of comparison of electron density follows the negative side of the relative entropies of the various ice polymorphs H-O-It bisecting plane, the maximum density being (Table 8). The entropy of ice II is less than that of at the 'negative pole' of the molecule. The presence the other ice pha~e~ by amount, s an order of magnitude in ice II of accepter-accepter angles rather smaller larger than tbe entropy differences among the other than tetrahedral is in accord with this, and also with phases themselves, it being therefore also likely that arguments of the type given by Pauling (1958) for these other forms are all proton-disordered. the plausibility of linear N-H---O=C bonds in Considering the rather large distortions from an polypeptides, and with the observation by Clark 'ideal' tetrahedral environment for the water molecules (1963) of the frequent occurrence of threefold coor- in ice II, it is remarkable that the internal energy of dination of water molecules in organic hydrates. this phase is so nearly equal to that of ice I, especially The proton-ordering scheme derived by structural by comparison with the other high-pressure forms of arguments is supported by the X-ray data, although ice (Table 9). Because in ice II the average donor the support provided can necessarily have only angle is only 94.5 ° , the idea presents itself that the BARCLAY KAMB 1447

Table 8. Entropy changes for ice-ice transitions From II II II I III V IV VI To I III V III V VI VI VII AS + 0.8 + 1.0 + 1.0 + 0.08 -- 0.06 + 0.05 - 0.13 + 0.05 Units of cal.mole-l.deg-1. Values are averages over the respective equilibrium curves, from determinations by Bridgman (1912, 1935, 1937). Value involving ice IV is for D20. energetically most-favored H-bond donor angle of the where ~ and u are constants, u defining the relative water molecule may for some reason be smaller than energy contributions of acceptor misorientation and the H-O-H angle of the free water molecule. However, donor misorientation. Dependence of this type on ~Q a case cannot be made for this idea, because of the has been introduced by several previous authors considerable stabilization provided by the increased (Table 10). The proposed dependence on ~o is based van der Waals interaction in ice II. on the structural indication in ice II that the acceptor orientation is sensitive to co but not to ~. Pople (1950) used a relation like (1), but took for the acceptor Table 9. Energy of ice polymorphs relative to ice I misorientation the departure angle from tetrahedral orientation relative to the water molecule; he also Ice II III IV V VI VII assumed in effect ~ = 1. Quoted values of ~ are listed E-E(I) 0.01 0.25 0-38 0.34 0.42 0.95 in Table 10; the value g = 6 cal. mole -1. deg -2, which Units of kcal.mole-1. Values are approximate, based on AE will be used here, is not likely to be in error by more determinations by Bridgman (1912, 1935, 1937); OE/OP and OE/OT for the individual phases have been neglected. than a factor of 2. The van der Waals energy of non-nearest neighbors can be estimated by the method used by Pauling & The energy difference between ice I and ice II can Simonetta (1952, p. 31; see also Pauling, 1961), which be estimated by considering (1) H-bond strain energy, for a pair of water molecules at distance r gives an (2) van der Waals energy of non-nearest neighbors, energy and (3) electrostatic interaction of non-nearest neigh- Evw = - l'67(do/r) 6 kcal.mo le-1 (2) bors. Of these, (2) is the most reliably estimated, (3) requires extensive calculations beyond the scope where d0=2.76 ~. We obtain the van der Waals of this paper, and will not be considered here. The energy of ice I by summing the individual contribu- H-bond strain energy (1) involves bond-length strain tions for O-O distances out to 6-7 ~, and carrying (which includes the energy of repulsive forces and out an integration beyond; for ice II the integration van der Waals attractive forces for nearest neighbors) is started at 6.0 J~. and also bond-angle strain, represented by the angular The estimated energy contributions are given in misorientation parameters in Table 7. Table 11. The calculated energy of ice II is too low, To the extent that bond-length and bond-angle in spite ol the significant amount of bond-strain strains are independent, the bond-length strain energy energy. This is because the denser packing of ice II, can be estimated in the way given by Pauling (1960, which involves eight O-O distances in the range 3-24 p. 453). The bond-angle strain energy will be assumed to 3.92 A and five more between 4-24 and 4-47 A, to have the form gives a much higher van der Waals energy than the open structure of ice I, with twelve next-nearest ~'= z (f22 + ~,o)") (1) neighbors at 4.50/~ and one at 4"60/~. The calculated

Table 10. Values of ~ in equation (1) Source ~ (cal.mole-l.deg-e) Method Chidambaram (1962) 8.8 Theoretical* after Lippincott & Schroeder (1955) Cross et al. (1937) 4-2 Raman spectrum of water Pauling & Corey (1954) 7.3 Empirical Pimentel & McClellan (1960) 4.9 Libration frequency vR in ice Pimentel & McClellan (1960) 1.1 Theoretical (electrostatic model) Pople (1950) 8.2 Radial distribution curve for water Pople (1950) 8.2 Theoretical (electrostatic model) Present worker 2.2 Bending frequency v.2 in ice * The bending energy was calculated for ~Q = 2.5 ° only, the dependence on ~Q not being given explicitly. t Calculated from spectroscopic data from Ockman (1958), by assuming that the aceeptor oxygen is rigidly fixed during the bending vibration. This improbable assumption is probably responsible for the discrepancy between g values obtained from VR and v2, which has been noted by Pimentel & McClellan (1960, p. 127). An interpretation that does not require such a low value of g is given by Zimmerman & Pimentel (1962, p. 735). 1448 ICE II: A PROTON-0RDERED FORM OF ICE energy difference could be made to agree with ex- Note added in pro@ ~ l%ecently Bertie & Whalley periment by taking ~ = 9 cal. mole-1, deg-2. (1964) have published spectroscopic data which they interpret as indicating proton ordering in ice II and ice 1II, although it is not clear whether long-range Table 11. Calculated energy contributions Jbr ice I and II or short-range order is contemplated. The evidence 11 discussed here casts doubt on interpretation in terms (keal..~ole -1) of long-range proton order because crystallographic Bond-lenglAl strain 0 0.04 and thermodynamic data indicate a sharp distinction Bond-angle strain between ice III (disordered) and ice II (ordered). Donor rnisorientation o.07 0.42 Aeeeptor misorientation (v = 1) 0 o'3.~J The crystallographic results for ice II imply that the Van der Waals energy -- 0-96 - 2.17 acceptor relationship must be taken into account in Totals - 0.9 -- 1.3 interpreting the spectral bands that involve H-bond bending. The bending anisotropy should lead, for Calculated E(I I)-E(1) = -- 0.4 kcal.mole- 1 Measured (Bridgman, 1912) -- + 0' 01 example, to a splitting of the libration modes, which are usually assigned a single frequency. This may help to explain the two libration frequencies of about To compute the acceptor energy contribution on 800 cm-~ and 500 cm -~ found for HDO by Bertie & the basis of deviation from tetrahedral orientation, WhMley (1964) and it may contribute to the breadth as done by Pople (1950), we must add to equation (1) of the ~,~,. band in the pure H..,O and D20 . If a term zv(qg-54"7°) 2. From the data in Table 7, these interpretations are correct, the spectroscopic the resulting additional contribution (for v=l) to data constitute additional evidence for an appreciable the energy of ice II is +3.1 kcal.mole -1, much to() value for the parameter v in equation (1) above. large to be compatible with the observed energy. In another recent paper Bertie, Calvert & Whalley This argues against the concept of the tetrahedral (1964, p. 1377) report that ice II has a low dielectric water molecule in its elementary form, and in favor constant and no detectable dielectric relaxation, in of the interpretation presented here, in which the contrast to ices I, III, V, and VI. This provides acceptor energy is independent of 9~. striking confirmation of the interpretations of proton From Table 7 it is found, on the basis of equation (1), order versus disorder presented in the present paper. that water-molecule orientation (2-14) will be favored energetically over (1-I4) provided that v>0.1. On the same basis, the energy of a hypothetical, It is a pleasure to acknowledge the important completely proton-disordered form of ice II can be contribution to this work of Carolyn K_nobler, who estimated. From dat~ of the kind in Table 7, and read and reduced the X-ray intensities and helped assuming v= 1, the disordering energy is found to be in various other ways. R. E. Marsh kindly provided +3.0 kcal.mole-% so large as to prevent a transition his triclinic least-squares program, and gave helpful to the disordered form. discussion and advice. I am indebted to Linus Pauling The bond lengths in ice II do not conform in a for many discussions and valuable ideas, and for simple way to the bond strength criterion given by suggesting the investigation originally. The work was equation (1), as shown by the data in Table 7. The supported by grants G-4537 and G-19769 from the longest bond, 2.84 +_ 0"013 A, and the shortest, National Science Foundation. 2-75_0.01~ A, ought to correspond to almost the same strength in terms of bond-bending strain. Evidently the bond lengths are affected by other References factors. The long bond is the one between the ice-I-like BAUR, W. H. (1962). Acta Cryst. 15, 815. hexagonal columns of six-rings, and it could shorten BERNAL, J. D. & FO~'LER, ]~. H. (1933). J. Chem. Phys. without altering any of the other bond lengths if the 1, 515. columns were to come closer together by contraction BERTIE, J. E., CALVERT, L. D. & WI=[ALLEY, E. (1963). of the ~H axis. However, the n0n-bonded distance of J. Chem. Phys. 38, 840. 3.24/~ would be shortened by the full amount of the BERTIE. J-. E., CALVERT, L.D. & WttALLEY, E. (1964). contraction. Apparently a repulsion becoming effective Canad. J. Chem. 42, 1373. at this distance prevents shortening of the 2.84 A BERTIE, J. E. & WI~ALLEY, E. (1964). J. Chem. Phys. bond length. 40, 1646. It is worth noting that the favorable energetic BJERRUM, N. (1952). Science, llS, 385. BODE, H. & TEUFER, G. (1955). Acta Cryst. 8, 611. relations involving the proton ordering peculiar to BRIDGMAN, P. W. (1912). Proc. Amer. Acad. 47, 441. ice II are probably responsible fo~ the fact that this BRIDGMAN, P. ~V. (1935). J. Chem. Phys. 3, 597. structure, unlike the other known ice structures, BRIDGMAN, P. W. (1937). J. Chem. Phys. 5, 964. has no analog among the polymorphs of silica so far CHIDAMBARAIVI, 1~. (1962). J. Chem. Phys. 36, 2361. discovered. CLARX, J. R. (1963). Aust. J. Pure Appl. Phys. 13, 50. BARCLAY KAMB 1449

CRoss, P. C., BURNHAm, J. & LEIO~TON, P. A. (1937). PAULING, L. (1935). J. Amer. Chem. Soc. 57, 2680. J. Amer. Chem. Soc. 59, 1134. PAULING, L. (1958). In Symposium on Protein Structure, DARLING, B. W. & DENNISON, D. M. (1940). Phys. Rev. Ed. NEUBERGER, A. p. 17. London: Methuen. 57, 218. PAULING, L. (1960). The Nature of the Chemical Bond. GIAUQUE, W. F. & ASHLEY, M. (1933). Phys. Rev. 43, 81. Ithaca: Cornell Univ. Press. KAMB, B. & DATTA, S. K. (1960a). Nature, Lond. 187, PAULIne, L. (1961). Science, 134, 15. 140. PAULING, L. & COREY, R. B. (1954). Fortsehr. Chem. org. KAMB, B. & DATTA, S. K. (1960b). Acta Cryst. 13, 1029. Naturst. 11, 180. KAMB, B. & DATTA, S. K. (1960c). Unpublishedmanu- PAULING, L. & SIMONETTA, M. (1952). J. Chem. Phys. script. 20, 29. International Tables for X-ray Crystallography (1959). PETERSON, S. W. & LEVY, H. A. (1957). Acta Cryst. Vol. II. Birmingham: Kynoch Press. 10, 7O. LENNARD-JONES, J. & POPLE, J. A. (1951). Proc. Roy. PIMENTEL, G.C. & MCCLELLAN, A. L. (1960). The Hydro- Soc. A, 205, 155. gen Bond. San Francisco: Freeman. LIPPINCOTT, E. R. & SCHROEDER, R. (1955). J. Chem. PITZER, K. S. & POLISSAR, J. (1956). J. Phys. Chem. Phys. 23, 1099. 60, 1140. LO~SDALE, K. (1958). Proc. Roy. Soc. A, 247, 424. POPLE, J. A. (1950). 1)roe. Roy. Soe. A, 205, 163. MeFARLAN, R. L. (1936a). J. Chem. Phys. 4, 60. WELLS, A. F. (1962). Structural McF~_aLAN, R. L. (1936b). Rev. Sci. Instrum. 7, 82. (3d ed.) : Oxford. MCGRATH, J. ~V. & SILVIDI, A. A. (1961). J. Chem. Phys. ZIMIE[ERMAN, n. & PIMENTEL, (.~. C. (1962). In Advances 34, 322. in Molecular Spectroscopy. Ed. MANGINI, A. p. 726. 0CKMAN, N. (1958). Advanc. Phys. 7, 199. New York: Pergamon Press. 0WSTON, P. G. (1958). Advane. Phys. 7, 171.

Acta Cryst. (1964). 17, 1449 The Structures of Some Inorganic Cyanamides. I. t)reparation of Single Crystals and Preliminary Studies

BY K. M. ADXMS*, M. J. CooP~,~* t AND M. J. SOLES§ Cavendish Laboratory, Cambridge, England

(Received 26 November 1963)

The preparation of single crystals of sodium, thallous, silver and lead cyanamides is described. Preliminary crystallographic data on these compounds are reported.

Introduction as polycrystalline or amorphous powders and methods A detailed study has recently been made of the for their preparation in this form have been described physical properties, molecular structure and thermal previously (Shushunov & Pavlov, 1955; Deb & Yoffe, and radiation stability of a number of inorganic 1959; Bolis-Caunella & Costa, 1953). However the cyanamides (Sole, 1963; Sole & Yoffe, 1964). Crystallo- single crystals required for the above investigations graphic studies of these compounds have therefore are difficult to prepare and the methods used are been undertaken in order to correlate the physical therefore described in some detail in this paper. properties with the detailed crystal structures. The Preliminary crystallographic data are also reported. compounds investigated, namely sodium, thallous, silver and lead cyanamides, are fairly readily available Sodium cyanamide * Crystallographic Laboratory. An attempt to prepare sodium cyanamide by the t Formerly Mullard Research Fellow, Downing College, method of Shushunov & Pavlov (1955) was un- Cambridge. Present address: Brookhaven National Labora- successful and from infrared and X-ray analyses the tory, Long Island, New York, U.S.A. :~ Laboratory of Physics and Chemistry of Solids. product appeared to be mainly sodium acid cyanamide. § Present address: Atomic Energy Board, Pretoria, South Accordingly sodium cyanamide in the form of fused Africa. ingots was obtained from Bios Laboratories Incor-