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Chemical Physics 107 (1986) 47-60

North-Holland, Amsterdam

47

A NEW DETERMINATION OF THE STRUCTURE OF WATER AT 25°C

AK SOPER and M.G. PHILLIPS

Guelph - Waterl oo Program for Graduate Work in Physics, Deparr ment of Physics, Unioersity of Guelph.

Guelph, Ontario, Canada NI G 2 WI

Received 13 November 1985; in final form 10 April 1986

The results of a new neutron diffraction experiment to measure the structure of water are presented. The data, measured at

the McMaster Nuclear Reactor, are of a high quality and are analysed to yield the hydrogen-hydrogen pair correlation

function using a subtraction procedure which has been used in previous experiments of this kind. This procedure circumvents

the necessity of applying inelasticity corrections. The results are in good agreement with earlier work and serve to establish the

general correctness of the subtraction procedure when used to determine hydrogen correlations. The data are further analysed

to yield separate oxygen-hydrogen and oxygen-oxygen partial structure factors for liquid water. For the second part of the

analysis an effective mass model of the dynamic scattering law is used, with the model parameter, the effective mass of the

scattering particle, chosen by a least-squares fit to the measured differential cross sections. The final pair correlation functions

are obtained using a maximum entropy analysis of the structure functions.

1. Introduction

There have been three recent measurements of

the structure of water [l-3] which were performed

almost simultaneously, and which attempted to

extract partial structure factors from neutron dif-

fraction data. These experiments showed sufficient

discrepancies between them to warrant a reap-

praisal of the experimental techniques being used

[4]. It was concluded that in neutron experiments

on water the unique properties of hydrogen as a

scatterer of neutrons meant that some of the ex-

perimental arrangement and data analysis ap-

proximations which are conventionally used in

diffraction studies of most other liquids were not

applicable to studies of water and, by implication,

of other hydrogen containing liquids. In parallel

with these experiments there have been a number

of new simulations of water via computer modell-

ing. In particular quantum effects have been intro-

duced [5,6] and modern calculations include the

effects of intramolecular vibrational motions [7,8].

Therefore in a situation in which simulations are

predicting effects at the limit of measurability, it is

timely to present the results of a new neutron

diffraction experiment on water. Because the pair

correlation functions calculated by computer

simulation depend rather sensitively on the details

of the intermolecular potential, which for water is

anisotropic, the experimental correlation functions

provide an important test of the accuracy of the

simulations.

The present measurements were taken at the

McMaster Nuclear Reactor, Hamilton, Canada

using the University of Guelph liquids diffractom-

eter. Although McMaster is a low-flux facility (2

MW) the availability of a substantial length of

beam time plus low neutron background meant

that the runs could be repeated to check for

consistency. It also meant that excellent statistical

precision was obtained. A reactor experiment is

performed at constant incident neutron energy,

with the momentum transfer, Q, varied by chang-

ing the scattering angle. This is in contrast to the

previous time-of-flight (TOF) experiment [l] in

which the scattering angle was held fixed while the

incident energy was varied. Therefore the dynamic

effects on the diffraction data, which arise from

the nuclear recoil, have a quite different depen-

dence on Q for the two types of experiments.

0301-0104/86/ 03.50 Q Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

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48

A.K. Soper. M.G. Phillips / The structure

ofwater

t 25 OC

Because the correction is strongly angle depen-

dent, the reactor data show a substantial fall-off

with increasing Q which is not present in the

time-of-flight data.

The dynamic correction appears most dramati-

cally in the incoherent or “self” scattering from

individual atoms. To remove this term in the past

an empirical procedure [1,9], which makes no ref-

erence to a dynamical model of the liquid, has

been used. The result is the hydrogen-hydrogen

structure factor and corresponding pair correla-

tion function. This scheme is shown here to work

for the reactor data as well. In an effort to obtain

the remaining structure factors (O-O and O-H)

which are required to characterize water structure

we have adopted a simple effective mass model for

S(Q, w), the dynamic scattering law [lo] with the

effective mass an adjustable parameter chosen to

give a least-squares fit to the measured data. As

will be seen this model removes most of the self-

scattering. In addition a special procedure, based

on the principle of maximizing the entropy of the

pair correlation function, was employed for per-

forming the necessary Fourier transforms. This

method [ll] inhibits the transfer of systematic

errors which vary slowly with Q to the calculated

distribution function. In this way it is shown that

the subtraction procedure used for the H-H corre-

lation apparently removes all the single-atom-

scattering background for the present experiment

within measuring errors. The effective mass ap-

proach removes most of this background, but the

maximum entropy analysis indicates that there is

still a residual background which has not been

subtracted.

2. Theory

The underlying theory to the experiment has

already been described in detail [4] and so will be

reviewed only briefly here. The measured differen-

tial cross section in a reactor experiment is a

quantity summed over the atomic components

and integrated over energy transfers:

where Q, = 4~ sin 8/X, with 28 the scattering

angle and h the neutron wavelength,

b,

is the

neutron scattering length of atom (Y, k, and k,

refer to the incident and final neutron wave vec-

tors respectively, and E(k,) is the detector ef-

ficiency for the scattered neutron. Q =

1 i - k, 1

is

the momentum transfer and c =

E, - E,

is the

energy transfer, with Ei and E, the incident and

final neutron energies. &,(Q, E) is the van Hove

dynamic scattering law between atoms (Y and /3

[

The angular brackets in (1) represent an aver-

age over the spin and isotope states of the compo-

nent nuclei. Since these states are normally uncor-

related with the atomic positions the result is

different depending on whether (Y, /3 refer to the

same atom, the so-called self- or single-atom-

scattering terms, or whether (Y, p refer to different

atoms, the distinct or interference terms. The self-

terms have traditionally presented a major ob-

stacle to neutron diffraction work on hydrogen

because the large spin incoherent scattering cross

section for hydrogen, coupled with the broad

spread of energy transfers due to the small mass

of the proton, superimposes the desired inter-

ference scattering on a large, Q-dependent back-

ground.

The hydrogen/deuterium subtraction proce-

dure in principle removes this incoherent back-

ground and yields the H-H correlation function

directly from the data. To do so it has to be

assumed that the single-atom scattering from a

proton in light water, when integrated over energy

transfers according to (l), is unchanged for mix-

tures of heavy and light water. A similar ap-

proximation is made for the deuterons in heavy

water. Such approximations are widely utilized in

reactor physics calculations, and imply that any

residual single-atom scattering is small. The ap-

proximation works well for a quantum-mechanical

rigid rotor model of the dynamic scattering law

[4]: this model includes a calculation for HDO

molecules and so it can be claimed that some of

the quantum-mechanical effects in the real liquid,

which might lead to a residual single-atom compo-

nent in the HH structure factor, have been in-

cluded. The present reactor data, in which the

relative accuracy between samples is probably bet-

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A.K. Soper M .G. Phil li ps / The structure of w ater at 25 OC

49

ter than the earlier time-of-flight results, indicate

that the residual single-atom scattering is even

smaller than for the time-of-flight data.

In the previous experiment [4] it was possible to

obtain further information, a weighted sum of

O-H and O-O correlations, because the self-

scattering from deuterium is flat at small scatter-

ing angles in a TOF experiment. This is not true

for reactor data. Therefore in order to extract any

more information from the experiment it is neces-

sary to set up a model for the dynamics of the

self-scattering. The freely rotating rigid molecule

model of Blum et al. [13] demonstrated the quali-

tative trends of the reactor data, but did not give

quantitative agreement. Recently Granada et al.

[14] have had considerable success at predicting

the angular dependence of neutron-scattering data

from water both for the present experiment, and

for the previous time-of-flight data. Granada’s

method, which is a generalization of an earlier

effective mass model of Rrieger and Nelkin [lo],

uses a “synthetic” dynamic scattering law [15],

and instead of attempting to reproduce all the

details of the scattering law for water, the model

satisfies a few well-known integral properties.

There are no adjustable parameters in this model,

but because it appeared to diverge somewhat from

the measured data for light water at small Q

values, probably due to the assumption in the

model of free translational motion,. and because

other available models gave worse fits over the

whole Q range, the simpler effective mass model

of Kreiger and Nelkin [lo] was used directly. In

principle the effective mass can be derived from

the Sachs-Teller inverse mass tensor [16], but in

practice the rigid molecule values are inap-

propriate for a real molecule in the liquid. There-

fore we treated the effective mass as an adjustable

parameter, to be determined by least-squares anal-

ysis, along exactly the same lines as used by

Bertagnolli et al. [17]. The aim of this model was

simply to subtract a realistic background from the

data. The background was to lie as close as possi-

ble to the measured data but with a sufficiently

smooth variation with Q that none of the inter-

ference scattering would be removed. In fact the

absolute scale of the measured data also has some

uncertainty, as described below, and so the neu-

tron cross section used in the model was also

varied in the least-squares analysis. Therefore the

model scattering law used for the self-scattering is

a perfect gas-scattering law with arbitrary mass

for each atom:

X exp[ - (e - En)*/4E,k,T],

(2)

where En = h2Q2/2M,,,, and M,,, is the effective

mass. Each nucleus in a molecule will have its own

effective mass. For water there will be three terms

like (2) one for hydrogen, one for deuterium and

one for oxygen. These three terms are combined in

the appropriate proportions and then substituted

into (1) which has to be integrated numerically for

each scattering angle of interest.

3. Experimental

The data were recorded on the University of

Guelph liquids diffractometer at McMaster nu-

clear reactor, Hamilton, Canada. This instrument

has an incident beam at the sample position of

flux 7

x

lo4 n/cm*/s and wavelength of 0.966 A.

The rectangular beam used was 6.4 cm high by 1.9

cm wide and was filtered inside the reactor wall by

a set of single crystals of sapphire. Scattered neu-

trons were detected by four 10 atm ‘He detectors,

2.5 cm diameter and 15.2 cm high. The sapphire

filter reduced the fast neutron background to a

low level, and combined with the long running

times which are possible at McMaster the quality

of the data was comparable with that at many

high-flux institutions.

The samples in this experiment were held in

thin, disk-shaped, flat plate sample containers

which were angled so that the normal to the plates

made an angle of 50” to the neutron beam. This

ensured that the scattered neutrons could be ob-

served by transmission over the full range of

0-120°. The cans were made of anodized alu-

minum and no corrosion was observed even though

the water samples were held in the cans for several

weeks at a time. Arguments for the use of flat

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A. K. Soper, M.G. Phillips / The structure of water at 25 OC

plate sample cans for these experiments over the

more conventional cylindrical containers are

strong. In particular the absorption corrections are

analytic [18], the multiple scattering can be calcu-

lated easily to all orders [19], and the results are

insensitive to sample positioning errors [20], pro-

vided the sample area is substantially larger than

the beam area. In the present experiment the area

of the samples was 79 cm’ compared to the beam

area of 12 cm*. In fact, since most of the multiple

scattering that reaches the detector requires at

least one scattering near 90” scattering angle in a

thin flat plate, while the single scattering from

hydrogen is concentrated in the forward angles,

the level of multiple scattering is smaller than

would be expected on the basis of isotropic

scattering. The

rel tive

accuracy of one sample to

another is controlled only by the accuracy to

which the thickness of each container is known. In

the present case this was on the order of = 4%

because of the difficulty of machining thin

aluminum precisely. Each wall of the sample cans

was 0.056 cm thick.

The sample thicknesses were also measured by

continuously monitoring the neutron transmission

of the samples. This was achieved by means of

fision monitors placed before and after the sam-

ple. The sample thicknesses were obtained from

these transmission readings by using the liquid

density and the known total cross sections of

heavy and light water at this energy [21], that is

12.5 &-0.2 barn and 75 k 1 barn respectively. In

fact the analysis to obtain the HH structure factor

described below revealed a small but consistent

trend for the sample containing 66% light water,

and for the subsequent data analysis an increase

of 3% over the originally measured thickness was

assumed for this sample. This alteration, which

represents a change of 0.002 cm is within the

machining error. Table 1 compares the thicknesses

used in the data analysis with the original speci-

fied thicknesses. In all cases the agreement is

good.

Data for each sample were collected over a

period of = 6 days per sample with 2 or 3 angle

scans in that time. When the detectors were com-

bined and binned into Q-bins of width 0.1 A-’

there were typically = 5 X lo5 counts in a bin. The

Table 1

Measured sample thicknesses from neutron transmission mea-

surements compared to original machine shop specifications. x

is the mole fraction of light water in the mixtures

x Measured Specified % difference

thickness

thickness

(cm) (cm)

0 0.282 0.272 4

l/3 0.092 0.089 3

2/3 0.064 0.061 5

1 0.043 0.041 5

chosen sample thicknesses correspond to = 15%

scattering in all cases, so the statistical accuracy of

the measured differential cross sections was the

same for all samples. Empty can, background and

vanadium calibration data were also taken with

the same precision.

The measured data were corrected for back-

ground, attenuation, can scattering and multiple

scattering, and then normalized to the vanadium

scattering. In this experiment a vanadium slab of

adequate size was not available. (The slab has to

be substantially larger than the beam if edge cor-

rections in the multiple scattering calculation are

to be insignificant [19], as just desribed.) Instead a

rod of vanadium of diameter 0.635 cm and length

15 cm was used. Because the width of the incident

beam was greater than the width of this vanadium

sample, the absolute normalization of the data

was subject to some error due to vanadium posi-

tioning uncertainty. The final absolute normaliza-

tion was chosen by comparison with other pub-

lished data on liquid water [1,2]. However, the

relative accuracy of one water sample to another

was better than this absolute normalization be-

cause it depended only on how well relative thick-

nesses were known.

Data were recorded for four mixtures of light

and heavy water. If x is the mole fraction of light

water in the mixtures, then x took the values 1,

0.667, 0.333, and 0. The four measured differential

cross sections which resulted are shown in fig. 1.

The data appear to be in reasonable agreement

with earlier reactor experiments [2,3]. As in the

earlier work the differential cross sections show a

dramatic fall with increasing scattering angle due

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A. K. Soper. hf . G. Phil li ps / The structur e of water at 25OC

51

Fig. 1. Measured differential scattering cross sections for mix-

tures of light and heavy water, for neutron wavelength of 0.966

A. From the top the mole fractions of light water in the

mixtures are 1, 0.67, 0.33 and 0.0 respectively.

recoil effects, particularly in light water. The inter-

ference contribution, which is the oscillatory term

on top of the incoherent background also appears

to have generally the same Q dependence as was

seen in refs. [2,3]. However, the light water data of

Thiessen and Narten [2] showed an oscillatory

feature which appeared to proceed beyond the

range of the data (Q,,, = 13 A-‘). This oscilla-

tory pattern was not seen in the present experi-

ment, nor was it seen in either the time-of-flight

data [l] or in the data of Dore [3]. The cause of

the discrepancy between the Oak Ridge data and

these other measurements is not known at present.

The subsequent data analysis was divided into

two parts which will be described separately. In

the first the HH structure factor and correlation

function were extracted directly from the data

without reference to the model dynamic scattering

law. In the second, the effective mass law was

applied to extract interference functions, which in

turn were analysed into separate O-O and O-H

correlation functions.

4.

Results

4.1. H-H correlations

Previously it was shown [4] that if differential

cross sections for three mixtures of heavy and

light water are measured, Ci, C, and Cj, and if f

is the mole fraction of sample 1 in sample 3, then

the HH structure factor is obtained from the

combination

f Q + (1 -f)&(Q -z,(Q

= 4f(l -f)(W - W)2 Q +A,

(3)

where (b,), (b2)

are isotope averaged scattering

lengths for hydrogen in samples 1 and 2 respec-

tively, and M&Q) is the H-H interference func-

tion. A is a correction which is assumed small [4].

The fraction, f, is determined as follows. Sup-

pose mixture 1 has f, moles of H,O (and there-

fore 1 - fi moles of D,O). Similarly mixture 2 has

f2 moles of H,O, and mixture 3 has f3 moles of

H,O, with

i.e. mixture 3 is intermediate in light water content

between mixtures 1 and 2, then f is defined by the

ratio

f = f3 - f2Mf , - f2, ).

Since four differential cross sections were mea-

sured, the combination (3) was obtained in four

ways, to yield four versions of M,,(Q). These are

shown in fig. 2. General agreement between the

four results is seen: positions and magnitudes of

peaks are generally comparable, although the stat-

istical precision varies quite widely between the

data sets. Fig. 2 shows that all four structure

factors appear to oscillate about a flat line close to

zero. This confirms that within the present mea-

suring accuracies there is no fundamental flaw

with the subtraction procedure (eq. (3)): if as little

as 1% of the single-atom scattering were left in the

results it would appear as a marked slope in one

or more of them because of the sharp fall in

differential cross section with increasing Q (fig. 1).

The four results should be different because

they each can be regarded as a weighted sum of

HH structure factors, Zliu and Z&o, in pure light

and heavy waters respectively [4], which are differ-

ent on account of quantum effects. In effect the

measured data, M”“(Q), are represented by

M,,(Q =x Q + (I- + Qt (4)

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A. K. Soper, M.G. Phillips / The structure of w ter t25 OC

I I

0 2 4 6 8

10

12

0 (A-‘)

Fig. 2. Measured H-H structure factors obtained from the four

possible combinations of the data in fig. 1. The results are

presented in order of increasing contribution from the light

water H-H structure factor. From the top the values of i (eq.

(4)) are -0.28, 0.052, 0.39 and 0.72 respectively.

where x is the proportion of light water structure

factor in the data. Each dataset in fig. 2 contains a

different proportion of the term .Z&, and the data

are plotted in order of increasing light water struc-

ture factor. Using the four results of fig. 2 the two

terms, Z u and Z&,,

were selected by least-

squares fit. They are shown in fig. 3. Clearly the

latter term has superior statistical accuracy. Al-

though there do appear to be some difterences,

particularly in the region of Q = 6-9 A-‘, the

relative accuracies between samples as shown in

table 1 and the poor statistical precision of &.,

make any statements about quantum effects unre-

liable. To be able to make a quantitative estimate,

relative and statistical errors would have to be

substantially better than at present.

The intermolecular pair correlation function

obtained by Fourier transform of &,, g,,(r), is

shown in fig. 4a. In this and the next section the

Fourier transforms were obtained using an al-

gorithm called MAXENTS, described in detail

elsewhere [ll], which attempts to minimize the

transfer of errors in the measured data to the

I

I

0

2 4

8

10 12

I I

I

- Gi ‘1. . ”

O.O-

, , :’

; ,; -

‘5.:

.

‘ .

:

‘a

.’ .

,\*’

*. ‘L

+

‘. .

,- 8

.

’ .

: ,.

. .

a’

- ,. .*

Cd- ’

‘? .

.

-0.3 - :’ ’

I

I

0 2 4

12

Fig. 3. Separated H-H structure factor for heavy water (a) and

light water (b). Note the poor statistics on the second result

which prevent any estimation of quantum effects from these

data.

correlation function. In effect a pair correlation

function is modelled which

(i) is as consistent as possible with the structure

factor data,

(ii) satisfies known limits, i.e. compressibility

limit, g(0) = 0, and g(cc) = 1,

(iii) has the least amount of structure compared

to all the distributions which satisfy (i) and (ii).

In essence the method attempts to maximize

the entropy of the radial distribution function

subject to the constraints (i) and (ii). The extent

that a particular distribution is consistent with the

original data is determined by plotting the dif-

ference

where M(Q) are the measured data, and &se”(Q)

is the structure factor obtained from the model

g(r). If structure-like features are obviously pre-

sent in D(Q) then the algorithm must be iterated

until those features are smaller than other obvious

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A.K. Soper, M .G. Phil li ps / The structur e of water at 2S°

53

2 4

6 6 10

r A)

-1

I I

I I

I

0

4

6

Q A_,p ‘ 6 20

Fig. 4. Maximum entropy analysis of the H-H intermolecular

structure factor for heavy water (fig. 3a). The intramolecular

interference scattering was subtracted from the original data

prior to the analysis. (a) shows the calculated pair correlation

function. (b) shows the regenerated structure factor (continu-

ous line) and the difference function, eq. (5) (dots).

errors such as statistical noise and slowly varying

systematic errors. In addition the model distribu-

tion is subject to a repulsive potential energy

constraint that attempts to force atoms away from

the origin. The role of the potential is to reduce

the transfer of systematic errors in the measured

data, which cause unphysical behaviour at prim-

arily small

r

values, to the calculated correlation

function. The range over which the potential acts

is governed by the requirement that the model

distribution satisfy the compressibility limit. How-

ever, the “hardness” of the potential is an input

parameter. If the potential is too soft then par-

ticles in the distribution are allowed unphysically

close to the origin and so systematic errors in the

data are partly reproduced in the model distribu-

tion. On the other hand if the edge of the potential

is near a real peak in g(r) then particles in that

peak can become unduly structured, if the poten-

tial is too hard, in order to fit the measured data.

Generally it is found that there is a range of

hardness values over which the model distribution

is invariant.

The main advantages of this algorithm over the

traditional method of direct Fourier transform of

the measured data are that statistical noise is not

reproduced in +,(Q) and the data are extended

smoothly into regions of Q where no measure-

ments are available. This can be seen in fig. 4b

where both D(Q) and S,._~ for g,,(r) are shown.

For the results shown in fig. 4 a function corre-

sponding to the intramolecular interference was

subtracted from the data prior to the MAXENTS

analysis. As discussed elsewhere [ll] a well-de-

fined distance in the pair correlation function can

introduce serious truncation effects unless the data

extend to large Q values. The function used for

this subtraction was

s, Q) = C sin Qd/Qd)exp - b2Q2),

6)

where

d

is the distance in question and y is the

standard deviation of atoms about this distance,

determined from the data. The factor C is either

defined from prior knowledge of the scattering

system, or is chosen to minimize the variance of

this function from the data. For the H-H struc-

ture factor in liquid water C = 0.5 exactly for the

intramolecular distance. The function (6) has an

analytic Fourier transform in r-space to a pair of

gaussian-like functions located at r = f

d,

and so

can be added back if necessary into the final pair

correlation function without introducing trans-

form errors. If there is any mismatch between the

shape of this function and the data then the

MAXENTS algorithm automatically introduces

additional intensity in

g r)

around the r value in

question to compensate for the mismatch. Since

the maximum entropy formalism requires g(r) to

be positive for all r values the compensation

mechanism can effect only a broadening of the

gaussian: it is unlikely to make the gaussian any

narrower. For the H-H function and for the O-H

function of section 4.2 the gaussians were not

reintroduced into the correlation functions plotted

in figs. 4 and 7 respectively. The results plotted

therefore represent the intermolecular functions.

For the case of H-H an intramolecular distance

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A. K. Soper, M .G. Phil li ps / The structur e of waterat 25 OC

of 1.56 A, with standard deviation of 0.07 A, was

used. These values were determined from the time-

of-flight data and were felt to be more reliable

than using values from the present data, because

they were chosen by fitting the intramolecular

scattering function over the much larger Q range

of the latter experiment. The bond length is also in

agreement with the value determined by Powles

[22] from both reactor and time-of-flight data.

The HH pair correlation function obtained has

the same general features as that obtained in the

TOF experiment [4]. Comparison of the two ex-

periments is given elsewhere [23]. The maximum

differences, mostly quantitative, are = 0.05 in g(r)

on an absolute scale. The present data indicate an

intermolecular near-neighbour coordination num-

ber of = 6 atoms out to 3.05 A as found previously

[4]. However, the residue, D(Q). from the

MAXENTS analysis shown in fig. 4b is much

smaller than for the time-of-flight data, which

indicates the present data may be more reliable

than before. The same analysis was also per-

formed on the _ZiH data of fig. 3. The results (not

shown here) were in good agreement with fig. 4.

The principal differences occurred at

r

values

greater than 4 .& resulting from the much larger

statistical errors in this second structure factor.

4.2. O-O

and O-H correlations

In order to obtain further correlation functions

it is necessary to subtract the single-atom scatter-

ing from the diffraction data, fig. 1. This was

achieved in the present case by integrating the

effective mass model (eq. (2)), numerically over

the appropriate energy transfers (eq. (1)). An ap-

propriate grid of Q and M,,, values was chosen,

intermediate values being obtained by interpola-

tion. The object was to find that mass which

minimized the variance

done on the data, using IV,,, and F(M,,,) as

parameters. The data were fitted over the entire

measured Q range of 1 to 11 A-‘. The model

function was actually a weighted sum of contribu-

tions from oxygen and hydrogen and/or deu-

terium, depending on the composition of the sam-

ple. A mass of 16 amu for oxygen was assumed for

all samples. The effective masses of hydrogen and

deuterium were determined from the data for the

pure liquids initially. For the mixture samples,

because the single-atom scattering from these sam-

ples is dominated by hydrogen scattering, the ef-

fective mass for deuterium for the mixtures was

held constant at its pure liquid value and the

variance minimized with respect to the hydrogen

mass. The effective masses obtained in this way

are shown in table 2, and the resulting interference

functions shown in fig. 5. It should be noted that

the hydrogen effective mass is almost independent

of the sample composition, confirming our hy-

pothesis that the hydrogen single-atom scattering

is largely independent of the mixture composition.

Also both hydrogen and deuterium effective

masses are less than their Sachs-Teller rigid mole-

cule values of 1.9 and 3.6 respectively.

Also shown in table 2 are the factors F( ,)

for each dataset, along with the ratios of the

expected factor (from the known scattering lengths

of the nuclei) to F(M,,,). It is seen that the ratios

differ appreciably from unity. In addition the

ratio of the factors is not the same for all samples,

but changes by = 5% with increasing hydrogen

content. If this ratio had been the same for all

samples, or showed no obvious trend with hydro-

gen content, this would suggest a simple absolute

Table 2

Effective masses and factors determined by least-squares fits to

the differential cross section data, fig. 1. x is the atomic

fraction of light water in the mixtures

x

Kff (amu)

F( Me,, )

Ratio

deuterium hydrogen

expected factor/

F( f )

0 2.64 -

1.39 1.12

The factor F(M,, ) was introduced because of

l/3 2.64 1.65 4.85 1.13

uncertainty in the absolute scale of the data.

2/3 2.64 1.61 8.16 1.15

Therefore a two-parameter least-squares fit was

1 - 1.61 11.37 1.17

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A.K. Soper, M .G. Phil li ps / The structur e of water at 25°

55

I I

I I

I

I

1

1.0

I

I

I

I

0 2 4

08w 8

10 12

,

Fig. 5. Interference scattering functions for mixtures of heavy

and light water as obtained from the data of fig. 1 after

subtracting the effective mass model for the singleatom

scattering. The results are presented in the same order as in fig.

1. The neutron weighting of the partial structure factors in

each of these data sets is given in table 3.

normalization correction would be needed. How-

ever, the relative normalization of one sample to

another is probably good to 3% and so together

with the trend seen in table 2, implies that the

disparity between actual factors and expected fac-

tors has more to do with the assumption of a

simple effective mass model for the single-atom

scattering and with the least-squares analysis used

to subtract it from the data, than it has to do with

the absolute normalization. This is in accord with

the calculations of Granada et al. [14] who were

successful at fitting these same data with a model

scattering law, particularly at large Q values,

without any fitted parameters. An alternative

scheme for subtracting the model was also tried

which used the small Q data to determine the

factor F(&) based on the compressibility limit

for the structure factors. However, because the

present data extend only down to Q = 1 A- ‘, the

results were less reliable than for the least-squares

analysis. Therefore the least-squares analysis is the

one shown here. The factors F(M,,,) were not

used to modify the data in any way: our intention

was simply to subtract a realistic curve for the

single-atom scattering. Of course there is no

guarantee that

ll

the single-atom scattering can

be subtracted in this way. However, by using the

MAXENTS routine after the partial structure fac-

tor analysis we assumed that transfer of any resid-

ual errors to the distribution functions would be

minimized.

The four interference functions were used to

obtain the H-H, O-H and O-O partial structure

factors, by the usual least-squares analysis of lin-

ear equations, i.e. the structure factors were cho-

sen to minimize the variance between the predic-

ted interference functions and the data in fig. 5.

This least-squares analysis also assumes quantum

effects are negligible. Table 3 shows the contribu-

tion each partial structure factor makes to the

interference functions of fig. 5. The results are

shown in fig. 6. In presenting these results it is

useful to discuss the relative accuracy with which

each structure factor can be determined, given the

neutron weightings of table 3. This is complicated

by the different backgrounds on which each of the

four interference functions are measured. How-

ever, since the relative statistical accuracies of the

four measured differential cross sections of fig. 1

are the same, we can use the statistical fluctua-

tions in the partial structure factors to estimate

their accuracies. It can be concluded from fig. 6

that all three structure factors have nearly equal

uncertainties. On the other hand the result for

%H

in fig. 3b was seen to have less accuracy than

for Ed

o by virtue of poorer statistical precision.

It will be seen that the HH partial structure

factor, fig. 6a, is equivalent to the HH results in

fig. 2. Therefore no further analysis was per-

formed on this function. The O-H and O-O data

Table 3

Neutron weighted contribution of O-O, O-H and H-H par-

tial structure factors to the interference functions of fig. 5

x

o-o O-H

H-H

0

0.336

1.547 1.780

l/3

0.336

0.742

0.410

2/3

0.336

- 0.063

0.003

1

0.336

- 0.868

0.560

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56 A.K. Soper, M .G. Phil li ps / The structur e o water at 2S°

’

I

I

I I

-0.6-

I

I

I I

I

0 2 4 6 6 10 12

a i-‘)

Fig. 6. H-H (a), O-H (b) and O-O (c) partial structure factors

for liquid water, obtained from the data of fig. 5 by least-squares

analysis, using the factors of table 3.

gOH

a

I

l-

2

4

r CA?

6 10

4

6

Q _p l6 2o

Fig. 7. Maximum entropy analysis of the O-H intermolecular

structure factor for liquid water. The intramolecular inter-

ference function was subtracted from the data of fig. 6 prior to

the analysis. The notation is the same as for fig. 4.

were subjected to the MAXENTS analysis as de-

scribed above, with the results shown in figs. 7

and 8 respectively. For the O-H data an intramo-

lecular scattering function corresponding to the

O-H bond was subtracted prior to analysis. The

distance used was 0.98 A with a standard devia-

tion of 0.07 A, which were the values found from

the time-of-flight data for heavy water [ll]. The

values are close to the accepted values for the

O-H bond in liquid water [22]. Initial results from

the MAXENTS program indicated the first peak

in the O-H function to be much sharper than that

shown here. Subsequent analysis indicated this

sharp peak was a consequence of the repulsive

potential rising too steeply at small r values. The

potential was therefore made softer as described

in the section on the H-H correlation function.

For the O-O correlation function the data indi-

cate an oscillatory function which proceeds well

beyond the measured range. Also a droop appears

for Q > 10 A-‘. Precisely why this droop appears

is unclear. Most likely it is related to the effective

4

I 4 4

goo

a

3-

2-

l- i.-

0’

’

I”

’ ’ ’ ’ ’ ’

’

0 2 4

r

(rJ6 8 10

b

0 4

8

12

16 20

Q (A-')

Fig. 8. Maximum entropy analysis of the O-O structure factor

for liquid water. The notation is the same as for fig. 4.

8/17/2019 1-s2.0-0301010486850583-main

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A. K. Soper, M G Phihips / The structur e of water at 25 OC. 57

mass procedure used to subtract the single-atom

scattering from the total differential cross section.

As is shown in the paragraphs below which de-

scribe the pair correlation functions, there may be

a slowly varying systematic error in the data of

maximum value 0.45. The droop, which is un-

physical in appearance, is well within that range.

Because unphysical features at large Q can

have a relatively strong bias on the pair corr$a-

tion function, the data were truncated at 10 A-’

before further analysis. In addition the function

S,,,(Q), eq. (6), was subtracted from the data prior

to the MAXENTS analysis, and later reintroduced

into the correlation function. The values of C and

d

were 2.54 and 2.875 A respectively and were

chosen by least-squares fitting, The value of y

used to plot fig. 8a was 0.12 A, although least-

squares analysis indicated the narrower value of

0.07 A. Inspection of the difference D(Q) (eq. (5)

and fig. 8b) showed there was little visual im-

provement for y < 0.12 A. In keeping with the

notion of maximum entropy the most reasonable

value would imply the least amount of structure in

goo( r), i.e. y should be as large as possible without

compromising the fit to the structure data. In

contrast yalues for y significantly larger than 0.12

A (0.15 A for example) gave a notably worse fit. A

more precise estimate of y would be obtained if

the data extended to larger Q values.

In figs. 7b and 8b it can be seen that the

Table 4

Measured intermolecular partial pair correlations functions for liquid water - HH and OH correlations

r (4

gHH gOH

r 6

gHH gOH

r (‘4

gHH gOH

0.05 0.000 0.000 3.35 0.878 1.596 6.65 1.007 1.003

0.15 0.000 0.000 3.45 0.955 1.489 6.75 1.006 1.008

0.25 0.000

0.000

3.55 1.035 1.347 6.85 1.006 1.009

0.35 0.000

0.000

3.65 1.103 1.226 6.95 1.007 1.008

0.45 0.000

0.000

3.75 1.150 1.137 7.05 1.007 1.010

0.55 0.000 0.000 3.85 1.168 1.072 7.15

1.008

1.014

0.65 0.000 0.000

3.95 1.163 1.020

7.25 1.008 1.019

0.75 0.000 0.000 4.05 1.148 0.979 7.35

1.008

1.021

0.85 0.000 0.004

4.15 1.117 0.955

7.45 1.008 1.019

0.95 0.000

0.000

4.25 1.082 0.952 7.55 1.008 1.014

1.05

0.000

0.000

4.35 1.049

0.967 7.65 1.007 1.010

1.15 0.000 0.000 4.45 1.024

0.983 7.75 1.005 1.007

1.25

0.000 0.002 4.55 1.005 0.983

7.85 1.003

1.006

1.35 0.000 0.006 4.65 0.992

0.963 7.95

1.001 1.006

1.45 0.000 0.017 4.75 0.992 0.933

8.05 1.000

1.005

1.55 0.000 0.076 4.85 0.994

0.914 8.15

0.999 1.002

1.65 0.000 0.357 4.95 0.991

0.920 8.25 0.999 0.997

1.75 0.018 0.994 5.05

0.984 0.950 8.35

0.999 0.991

1.85 0.150 1.385 5.15 0.979

0.991 8.45

0.999 0.986

1.95 0.395

1.117

5.25 0.969 1.022

8.55 0.998 0.984

2.05

0.662 0.680 5.35 0.962 1.030

8.65

0.998 0.986

2.15

0.923 0.414 5.45 0.961 1.017

8.75 0.997 0.989

2.25 1.123 0.300 5.55 0.963 0.999 8.85 0.996 0.993

2.35 1.239 0.265 5.65 0.967 0.988

8.95 0.996

0.996

2.45 1.256 0.269 5.75 0.973

0.989

9.05 0.996 0.998

2.55 1.183 0.296 5.85 0.976

0.997 9.15

0.996 0.999

2.65

1.067 0.349

5.95 0.984 1.001

9.25 0.997 1.001

2.75 0.939 0.441

6.05 0.992 0.995

9.35 0.998 1.004

2.85 0.835 0.596

6.15 0.997 0.984

9.45 0.999 1.006

2.95 0.779 0.832 6.25

1.004 0.975

9.55 1.000 1.007

3.05 0.756 1.132

6.35 1.007

0.973

9.65 1.001

1.005

3.15 0.773 1.418

6.45 1.009 0.981

9.75

1.002

1.001

3.25 0.815 1.587 6.55 1.008

0.993

9.85 1.002 0.997

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58

A. K. Soper, M.G. Phillips / The structure of water at 25 OC

difference functions for both O-H and O-O

The maximum value of D(Q) is = 0.45 for the

structure functions show a fairly substantial back- O-O function, and so is within the fitting errors.

ground in the partial structure factors of fig. 6.

The correlation functions obtained by the

However, such a background is not unreasonable

MAXENTS analysis are as consistent as possible

given the small signal-to-noise ratio for three of with the measured structure factors but without

the samples. For example if the model misfits the

reproducing unphysical features at small r values.

single-atom scattering by 1% for the second sam-

It is assumed that such features arise from the

ple in fig. 1 (x = 2/3), then the O-O structure failure of the model scattering law to subtract the

factor would have a corresponding error of = 0.3.

single-atom scattering completely. If it is de-

Table 5

Measured intermolecular partial pair correlation functions for liquid water - 00 distribution

r ‘Q

ho

r A)

ho

r A

goo

r @I

go0

0.025 0.000 2.025 0.001 4.025 1.003 6.025 0.930 8.025

0.977

0.075

0.000 2.075 0.002 4.075

1.027

6.075 0.944 8.075

0.978

0.125 0.000 2.125 0.008 4.125 1.050 6.125 0.958 8.125 0.978

0.175 0.000 2.175

0.019 4.175 1.071

6.175 0.972 8.175 0.979

0.225

0.000

2.225 0.040 4.225 1.089 6.225 0.987 8.225 0.980

0.275 0 000 2.275 0.072 4.275

1.104

6.275 1 OOl 8.275 0.981

0.325 0.000 2.325

0.116 4.325 1.116 6.325 1.015 8.325 0.983

0.375 0.000 2.375 0.170 4.375

1.126 6.375 1.027

8.375 0.984

0.425 0.000

2.425 0.233 4.425 1.133 6.425 1.039 8.425 0.986

0.475

0.000

2.475 0.306 4.475 1.136

6.475

1.049

8.475

0.987

0.525 0 000 2.525

0.399

4.525 1.136 6.525 1.057 8.525 0.989

0.575

0.000

2.575 0.541 4.575 1.135 6.575 1.064 8.575 0.990

0.625

0.000

2.625

0.782 4.625 1.130 6.625 1.068 8.625 0.991

0.675 0.000 2.675

1.179

4.675

1.122 6.675 1.071 8.675 0.993

0.725 0.000 2.725 1.746

4.725 1.113 6.725 1.072 8.725 0.994

0.775

0 000

2.775 2.388

4.775

1.102

6.775 1.072 8.775 0.995

0.825

0.000

2.825 2.907 4.825

1.088

6.825

1.071 8.825 0.996

0.875

0.000

2.875 3.092 4.875 1.074 6.875 1.068 8.875 0.997

0.925

0.000

2.925 2.869 4.925 1.058

6.925

1.065

8.925

0.997

0.975

0.000

2.975 2.351 4.975

1.041 6.975 1.060 8.975 0.998

1.025 0 000

3.025 1.758 5.025 1.024 7.025

1.055 9.025 0.999

1.075 0.000 3.075 1.273

5.075 1.007 7.075 1.050 9.075 0.999

1.125

0.000

3.125 0.966 5.125 0.990

7.125 1.045 9.125 1.000

1.175

0.000

3.175 0.813 5.175 0.973 7.175 1.039

9.175 1 OOl

1.225 0.000 3.225 0.752 5.225 0.957

7.225 1.033 9.225

1.001

1.275 0.000

3.275

0.735

5.275 0.942

7.275 1.027 9.275

1.002

1.325 0.000 3.325 0.734 5.325 0.928

7.325 1.022 9.325

1.002

1.375 0.000 3.375 0.741

5.375 0.916 7.375 1.016

9.375 1.003

1.425

0.000

3.425 0.749 5.425 0.906

7.425 1.011 9.425

1.003

1.475

0.000

3.475 0.760 5.475 0.897 7.475 1.006 9.475 1.004

1.525

0.000

3.525 0.773 5.525 0.890 7.525 1 OOl

9.525 1.004

1.575

0.000

3,575 0.790 5.575 0.885 7.575 0.996

9.575 1.005

1.625 0 000

3.625

0.807

5.625 0.883 7.625 0.992 9.625

1.005

1.675

0.000

3.675 0.827 5.675 0.882 7.675 0.988

9.675

1.005

1.725

0.000

3.725 0.850 5.725 0.883

7.725

0.985

9.725 1.006

1.775 0.000 3.775 0.873

5.775

0.886

7.775 0.983

9.775 1.006

1.825 0.000 3.825 0.898

5.825 0.892 7.825 0.980

9.825 1.006

1.875

0.000

3.875 0.924 5.875 0.899

7.875

0.979

9.875 1.006

1.925

0.000

3.925

0.950

5.925 0.908 7.925 0.978

9.925 1.005

1.975

0 000

3.975

0.978

5.975 0.919 7.975 0.978

9.975 1.005

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A.K. Soper, M .G. Phill ips / The siructur e of water at 25OC

59

termined that the single-atom scattering has an

oscillatory character of its own then these general

comments will be invalidated. All existing models

for single-atom scattering, which by definition in-

volve interferences of a neutron scattered by the

same atom, show only a monotonic behaviour

with Q or scattering angle.

The pair correlation functions shown in figs. 7

and 8 show general qualitative agreement with

computer simulations of water. The O-H distribu-

tion has a well-defined peak at 1.85 A consisting

of = 1.8 hydrogen atoms out to a distance of 2.35

A from an oxygen atom at the origin. The 0-G

distribution has a well-defined peak at 2.975 A

consisting of = 4.5 oxygen atoms out to a distance

of 3.3 A from an oxygen atom at the origin. These

numbers and distances suggest that the near

neighbour coordination of water molecules is well

defined and roughly tetrahedral at any instant in

time but that a substantial number (on the order

of 10%) of molecules are to be found in other

configurations. A fuller understanding of these

results will require comparison with molecular dy-

namics studies.

In the previous time-of-flight experiment [4]

although separate O-H and O-O functions were

not obtained, a composite function GOHOO,

where

GOHOO = 0.178 go,( r ) + 0.822 go,(r),

(7)

was extracted, and so to compare the present

results with the time-of-flight data, this function

was constructed from the present O-O and O-H

results. The two experiments show fair agreement

in peak positions and heights, despite the widely

different inelasticity corrections, and lend further

support to our contention that the neutron diffrac-

ton experiment on water can yield quite accurate

and useful partial structure factors. Values for the

measured pair correlation functions for water as

determined in this experiment are given in tables 4

and 5.

5. Conclusion

A new diffraction experiment on the structure

of liquid water has been performed at a reactor

neutron source. The results show generally good

agreement with an earlier time-of-flight experi-

ment. In this case a full set partial pair correlation

functions was obtained by using an effective mass

dynamic scattering law to subtract the single-atom

scattering. However, the results are not yet good

enough to measure the predicted small differences

in structure between heavy and light water due to

quantum effects [6]. These data when compared

with the time-of-flight data provide a useful esti-

mate of the margin of error associated with pre-

sent day experiments on water. The two datasets,

which were taken independently, agree more

closely with each other than they do with the other

neutron experiments on water [2,3]. A detailed

comparison of the previous time-of-flight H-H

pair correlation function with the other reactor

results is given in ref. [4]. Comparison of the

present results with the time-of-flight data is made

in ref. [23].

Acknowledgement

We like to thank G. Willis and T. Riddolls for

the manufacture of the sample vessels, and the

staff of the McMaster Nuclear Reactor for much

help in the course of this experiment. The work

was performed under a grant from the Natural

Science and Engineering Research Council of

Canada.

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A.K. Soper and R.N. Silver, Phys. Rev. Letters 49 (1982)

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A.K. Soper, M.G. Phil li ps / The structur e of water at 25OC

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