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METROPOLIS MONTE CARLO X-RAY AND NEUTRON DIFFRACTION IN ICE Ih
P. Deutsch
To cite this version:
P. Deutsch. METROPOLIS MONTE CARLO X-RAY AND NEUTRON DIFFRACTION IN ICE
Ih. Journal de Physique Colloques, 1987, 48 (C1), pp.C1-9-C1-14. �10.1051/jphyscol:1987102�. �jpa-
00226232�
JOURNAL DE PHYSIQUE
Colloque C1, suppl6ment a u no 3, T o m e 48, m a r s 1 9 8 7
METROPOLIS MONTE CARL0 X-RAY AND NEUTRON DIFFRACTION I N ICE Ih
P.W. DEUTSCH
T h e P e n n s y l v a n i a S t a t e U n i v e r s i t y , B e a v e r C a m p u s , B r o d h e a d R o a d , Monaca, P A 15061, U.S.A.
~gsumg.--&ous prgsentons et discutons les extensions de nos s$mulations pour des angles plus <lev& de dispersion de rayons. Nous prgsentons seulement des exp/eri- ences qualitatives. Nous donnons les rQsultats de simulation de diffraction de neutrons et nous les comparons avec l'exp6rience disponible.
Abstract.--Extensions of our simulations to higher x-ray scattering angles are presented and discussed. Here only qualitative comparisons are available. Neutron diffraction simulation results are presented and compared with available experiment.
I. INTRODUCTION. Continuing previous studies [la, lb] we present the results gf Metropolis Monte Carlo [2] simulations on a 192 water molecule unit cell with periodic boundary conditions. These equilibrium calculations are directed toward structural properties of ice Ih--the discrete fourier transforms elucidated, if indirectly, by x-ray diffraction powder spectra. To date the work has been con- fined to the scope of the Copper k a diffraction powder spectrum of Dowel1 and Rinfret [3]. While we have probed a variety of model temperatures beside that cor- responding to the experiment, we have not expanded the diffraction angle beyond this range. Moreover, this is a test primarily of the oxygen structure, and we have not probed the structure of the hydrogens. In.this study we attempt both extensions. The primary means is to calculate more structure factors: oxygen structure factors with Miller indices corresponding to larger Bragg diffraction angles, and as a first for us, hydrogen structure factors. These hydrogen struc- ture factors are to make correspondence with the elastic scattering of coherent neutrons by the deuteriums of heavy ice. While we don't possess the "half hydrogen model" ice sample consistent with neutron diffraction we do have a sample which simulates the hydrogen disorder by having a low point dipole moment while closely obeying the Bernal Fowler rules [4].
11. PROCEDURE. Following methods outlined in earlier work [la, lb, 51 we calcul- ate average structure factors for an equilibrated unit cell of ice Ih consisting of 192 water molecules using Metropolis Monte Carlo [2] methods as adapted to molecular systems by Barker et a1 [6]. Runs of one million steps for each of the input temperatures T = ZOK, 60K, LOOK, ZOOK, and 260K were performed and stored after equilibration runs were executed separately for each temperature. As was discussed previously, the eqyilibration for each temperature absorbed at least one million Monte Carlo steps. This is also true of the new temperature 60K added-sb our study where 1.002 million steps were run based upon an initial equilibrated 20K unit cell. In all cases the step parameters were adjusted so that very close to half of the steps were accepted by the Monte Carlo procedure. The ice samples were employed with periodic bounda~y conditions in the Metropolis Monte Carlo cal- culations. A cutoff distance of 5A between oxygen atoms was employed for different water molecules units, for by this distance most of the noncoulombic or non-ionic potential is negligible [fb]. As in previous studiea, water molecules comprislGB
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987102
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the sample were allowed to rot~tc: dnu r;^ar~slate freely but they were conscraiiieci to a rigid uniform internal geometry. Only in the initial parent configuration for all these studies were the individual 192 water molecule units permitted to adjust internally to a uniform set-of intramolecular coordinates [la]. As in previous studies, the separable two body central force potential RLS2 [7] was employed for the interactions. While it may not capture all the details of water molecule interactions [ 8 ] , this potential has the virtues of comparative theoretical simpli- city and of close contact with the traditions and techniques of classical liquid theory [9].
A reconstruction procedure described elsewhere [ 5 ] is used to produce average structure factor values
where =
+
kB+
:;1- A, B, and C are the primitive three dimensional reciprocal lattice vectors of the hexagonal lattice which determine the fundamental periodi~ity of the wurzite oxygen sublattices, and they are generated from the-primitive wurzite lattice vectors by standard rules [lo]. The total wave vector G in the structure factor expression is the scattering vector for the associated Bragg reflection, and the Rj are the oxy- gen position vectors in the case of x-ray diffraction. For the neutron diffraction work the Rj are the oxygen and hydrogen positions. Ao and Aa are respectively the oxygen and hydrogen scattering lengths of Thiesson and Narten [ll]. In both types of diffraction the averages are performed by computing a complete structure factor at each 1000 step interval. This spacing of sampling helps to retain statistical independence among the configurations. In x-ray diffraction we are simulating a powder spectrum. Thus while Miller indices h, k, and 1 are picked for strength of scattering from preliminary 170,000 step runs at 20K, we also group all the 6's by
( E l .
As is appropriate for a powder simulation we then use all G's of the samelength to calculate a composite structure factor [5]:
This means that we are comprehensive including all possible Miller indices that yield a targeted value of G. We then correct this elastic scattering contribution by the standard geometrical and polarization factors shown below [12].
I =
z
S(G) 1+cosL29 f2 Sin B C ~ S Q 2Here I is the elastic scattering intensity in arbitrary units;
z
S(G) is the oxygen lattice structure factor sum described above; 9 is the Bragg diffraction angle, and f is the appropriate local scattering form factor. In a previous study [5] the work with a variety of form factors had not substantial impact upon the diffraction features so we employed the atomic structure factors listed by B. D. Cullity 1131.Since the neutron scattering experiments we compared our results with were from a single crystal heavy ice experiment [14], we confine ourselves to calcula- tions of structure factors. Accordingly, we interpret these directly as the diffraction intensities.
111. RESULTS AND DISCUSSION. The results are depicted in Tables I, 11, 111, and IV. The binding energies in Table I fall off with increasing temperatures.
Table I. Calculated binding energies and prominent (110) average structure factor for T = 20K, 60K, loOK, 200K, and 260K.
Tables 1I and iV show the x-ray diffraction results, T a b l e LL i l l t h e i o u OL btruc- ture factors and Table IV in the form of normalized Bragg powder intensities. The x-ray diffraction results are generally in fairly good agreement with the experi- ments of Bertie, Calvert, and Whalley [15]. Their results show weak scattering for extended angles except for the (203) and (212) reflections. This is consistent with the simulation work at small angle Bragg peaks where the peaks fall off generally with increasing scattering angle. We also note a marked decrease in the structure factor for fixed Miller indices with increasing temperature, consistent both with the lower Bragg angle results and with the textbook theory.
Table,II. Average structure factors for indicated temperatures multiplicity is Miller indices M ?OK 'JOK
- - -
100~" 2 0 0 ~ ~ 2 6 0 ~ ~ 300Ka 2ab(100) 6 43.5 42 4 1 3 9 31 2.9 22.6
a~esults for 20 152.8"
from refs. 5 and 16.
b~ngles in degrees
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Table 111. Calculated a and experimental hydrogen structure factors for similar . - temperatures for the first 20 reflections.
hkl Scal. (100K) Sexpt (123K) Scal. (200K) -
100 2.64 2.64 2.43 2.87
002 5.00 4.57 4.92 3.99
101 1.01 1.91 0.94 1.99
102 0.69 1.50 0.63 1.61
103 1.02 1.55 0.90 1.55
200 0.01 0.81 0.02 0.88
201 0.44 1.48 0.31 1.60
004 1.55 3.31 1.53 3.14
202 0.23 1.74 0.22 1.79
104 0.20 1.16 0.16 1.20
203 0.31 1.18 0.20 1.05
105 0.67 1.96 0.68 1.83
204 0.24 1.56 0.16 1.52
300 0.14 1.46 0.13 1.25
30 1 0.04 0.28 0.26 0.21
302 0.07 0.72 0.05 0.63
006 0.33 1.15 0.26 0.98
205 0.17 1.45 0.07 1.24
106 0.03 0.34 0.05 0.29
303 0.10 0.63 0.06 0.59
a ~ h e calculated structure factor values are normalized to 2.64 for the (100) reflection at 100K.
b~xperimental values from Peterson and Levy ref. 14.
Table IV. Powder spectrum values at T = 60K and lOOK including some extended angles. Intensities normalized to 100 for the (100) peaks. Ten lowest Bragg angle peaks from reference 5.
Miller indices - T=60K T=~OOK-
(100) 100 100
(002) 54 5 4
(101) 6 2 5 4
(102) 2 7 2 7
(110) 4 6 41
(103) 45 45
(200) 7 6
(112) 2 4 21
(201) 05 05
(202) 03 02
(203) 01 01
(210) 0 2 02
(005) 0 0
(105) 04 04
(212) 0 1 0 1
(300) 03 03
(213) 05 03
a~alues for the (first) ten peaks with the lowest Bragg angles are taken from refrence 5.
The neutron results depicted in Table 111 do not work out quantitatively.
Perhaps this is because we do not have a fully proton shared ice arrangement. We could also parallel the x-ray powder spectrum simulations described above. Here we would incorporate all the structure factors associated with a given value of G
into a structure factor sum.
P. W. Deutsch, B. N. Hale, R. C. Ward, and D. A. Reago, Jr., (a) J. Chem.
Phys. 78, (1983) 5103-5107; (b) J. Phys. Chem. 87, (1983) 4309-4311.
N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, J.
Chem. Phys. 21, (1953) 1087-1099.
L. G. Dowel1 and A. P. Rinfret, Nature 188, (1960) 1144-1148.
N. H. Fletcher, The Chemical Physics of Ice (Cambridge Univ. Press, London) 1970 D. 31.
P. w.-~eutsch and T. D. Stanik, J. Chem. Phys. To be published October 15, 1986.
J. A. Barker and R. 0. Watts, Chem. Phys. Lett., 3, (1969) 144-149.
F. H. Stillinger and A. Rahman, J. Chem. Phys., 68, (1978) 666-670.
See for example G. Jancso and P. Bopp, Z. for Naturforschung, 38A, (1983) 206-213.
A. Rahman, F. H. Stillinger, and H. L. Lemberg, J. Chem. Phys. 63, (1975) 5223-5229.
W. E. Thiessen and A. H. Narten, J. Chem. Phys., 77, (1982) 2656-2662.
C. Kittel, Introduction to Solid State Physics, 5th ed., (John Wiley, New York) 1976.
B. D. Cullity, Elements of x-ray Diffraction, 2nd ed., (Addison-Wesley, Reading, MA) 1978, p. 139.
B. D. Cullity, Loc. Cit., p. 520.
S. W. Peterson and H. A. Levy, Acta Cryst. 10, (1957) 70-76.
J. E. Bertie, L. D. Calvert, and E. Whalley, J. Chem. Phys. 38, (1963) 840- 846. These experimenters appear to employ a cylindrical sample whose dia- meter is not given. Thus we are not permitted a quantitative comparison with
their experiment.
P. W. Deutsch and E. S. Banas, Phys. Stat. Sol. (b), 136, (1986) K 1-3.
COMMENTS
W.B. HOLZAPFEL
Are you sure that the powder data are not affected by texture ? Answer :
I suppose they could be. My estimates of the experimental uncertainties, I should say, were based upon some telephone conversations with Prof. John Bertie of the University of Alberta at Edmonton. He certainly mentioned the possibility that mosaic structure could influence line widths. Perhaps texture would be a further uncertainty.
W.F. KUHS Remark :
Probably the most powerful test for our calculations is a comparison with our recent high-resolution neutron diffraction structure factors. These data are corrected few all systematic errors (Lorentz effect, extinction, absorption,thermal diffuse scattering) and are accurate to 1% for medium strong reflections at 15 K (3% at 123K).
Questions :
Could you please comment on the possible reasons (and possible remedies) for the relatively large discrepancies you find between your calculated and the observed
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neutron structure factors ? Could you quote numbers for the mean value and the scatter in the hydrogen bond (O...H-0) angle after the equilibration ?
Answer :
We fixed the H.. .O-H angle at 101° in the initial cell optimization as reported in J. Chem. Phys. 78, 5103 (1983).We held this angle fixed for all our simulations.
Plans are now being made to permit this internal bond angle to vary. This and other matters stated in my manuscript are under consideration. It would be interesting to see details of your data which are much more recent than the data I used for comparison with my calculations. I am curious to see also what form your basic data take. Perhaps there are additionai areas of contact with simulations such as mine besides structure factors.
J.S. TSE
The RSL2 model for water interactions may not give the correct ice Ih structure at zero pressure. In your "geometry optimization" process, did you allow the angles of the simulation box to vary ? Were the ensemble averaged pressure of the system at different MC temperatures close to zero ?
Answer :
First we answer about the geometry optimization.
The angles of the simulation box were not allowed to vary in the initial geometry optimization. It should be pointed out that we were not trying to produce transitions to other phases of ice where you might find torsional processes to be involved that might twist the geometry of the unit cell. In our runs, we found no inclination toward other phases other than the melting of the sample at higher temperatures in the range of 260 K to 300 K. For example, we looked for ice I, using ice Ic reciprocal lattice vectors, and we found virtually no contributions, very low structure factors. Plots of specific configurations show the hexagonal structural features to be preserved at temperatures below melting. There was input from Prof.
Hale's group here and in the following part of the question : To answer about the pressure, the pressure may build to two or three kilobars at T = 300 K or so. But at low temperatures, it doesn't appear to be too significant. We may well start varying the box dimensions to lower the pressure towards zero.
