Neutron diffraction study of carbontetrachloride, siliciumtetrachloride, titaniumtetrachloride, and tintetrachloride in the liquid state at 295 K

NEUTRON DIFFRACTION STUDY OF

CARBONTETRACHLORIDE,

SILICIUMTETRACHLORIDE,

TITANIUMTETRACHLORIDE, AND

TINTETRACHLORIDE

IN THE LIQUID STATE AT 295 K

titaniumtetrachloride, and tintetrachloride

in the liquid state at 295 K

n

OB M fNj O Vfl o - 4 ^ BIBLIOTHEEK TU Delft P 1117 6234 C 248252

carbon tetrachloride

silicium tetrachloride

titanium tetrachloride and

tin tetrachloride

in the Hquid state at 295 K

PROEFSCHRIFT ter verkrijging

van de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus

prof. ir L. Huisman,

voor een commissie aangewezen

door het college van dekanen

te verdedigen op

woensdag 18 mei 1977

te 16.00 uur door

JAN BERNARD VAN TRICHT

natuurkundig ingenieur

geboren te Doetinchem

/ / / / ^^j^

Dit proefschrift is goedgekeurd door de promotor

PROF. DR J. J. VAN LOEF

C O N T E N T S

C H A P T E R I . I N T R O D U C T I O N 11

R E F E R E N C E S T O C H A P T E R I

^h

C H A P T E R I I . T H E E X P E R I M E N T S 15

I N T R O D U C T I O N 15

T H E I N S T R U M E N T 15

T H E R E S O L U T I O N 15

T H E S A M P L E S A N D

T H E S A M P L E C O N T A I N M E N T 21

The samples 21

The sample containment 2k

T H E M E A S U R E M E N T S 25

The measuring procedure 25

The aaoeptation of the data 27

The acceptation of the measurements 27

The experimental results 29

C O R R E C T I O N S T O T H E D A T A 30

I N T R O D U C T I O N 30

T H E C O R R E C T I O N S 31

R E F E R E N C E S T O C H A P T E R II

\0

C H A P T E R I I I . T H E T H E O R E T I C A L F R A M E W O R K 42

I N T R O D U C T I O N 1^2

T H E M O N A T O M I C A N D

D I A T O M I C L I Q U I D S Y S T E M !+3

The monatomia liquid system k3

The diatomic liquid system k6

T H E M O L E C U L A R L I Q U I D S Y S T E M 50

I N T R O D U C T I O N 53

N O R M A L I Z A T I O N O F D I F F R A C T I O N D A T A . 53

(i) The method of high scattering angles 'yk

(ii) The Krogh-Moe method 5^ (Hi) The Rahman and Mountain criteria 55

(iv) Calibration against vanadium 55

(v) Interval normalization 56

F O U R I E R T R A N S F O R M A T I O N O F

T H E N O R M A L I Z E D I N T E N S I T I E S 60

T H E S C A T T E R I N G F U N C T I O N A N D

I T S F O U R I E R T R A N S F O R M B A S E D

O N T H E H A R D S P H E R E M O D E L 6 T

R E F E R E N C E S T O C H A P T E R IV Tl

C H A P T E R V . R E S U L T S , D I S C U S S I O N , A N D

C O N C L U S I O N S 73

I N T R O D U C T I O N 73

I N T R A M O L E C U L A R P A R A M E T E R S 7"+

Interatomic distances 7^+ Mean amplitudes of vibration 77 E X P E R I M E N T A N D H A R D S P E R E M O D E L . . . 78 I N T E R A T O M I C I N T E R M O L E C U L A R D I S T A N C E S 8 3 P A R T I A L S T R U C T U R E F A C T O R S A N D P A R T I A L R A D I A L D I S T R I B U T I O N F U N C T I O N S . . . . 89 Carbontetrachloride 90 Titanium- and tintetrachloride 95

Intercomparison of the tetrachlorides 100

C O N C L U S I O N S 102

R E F E R E N C E S T O C H A P T E R V IOU

A P P E N D I C E S 107 A P P E N D I X A 108 A P P E N D I X B 112 A P P E N D i x c , . ^^h S U M M A R Y 119 S A M E N V A T T I N G 121 K O R T E L E V E N S B E S C H R IJ V I N G 123 N A W O O R D 123

C H A P T E R I . I N T R O D U C T I O N

I perceive with much regret that instead of a

well-constructed and strictly mathematical poem glorifying

The One State I am turning out some sort of a romance

of fantastic adventure.

Yevgeny Zamyatin: We.

Neutron diffraction has been demonstrated as early as 1936 [1]. Neutron diffractometers were installed at the first generation of research reactors that came into operation after 19^5 [2]. Review articles on diffraction of liquids appearing in the early sixties [3,*+] contain only a modest number of references to neutron diffraction. Since then much work has been done.

The present study has as its subject the molecular and liquid structure of four tetrachlorides. The interest in the study of liquid carbontetra-chloride was initiated through a paper by Rao [5] showing that in this molecular liquid, composed of spherical top molecules effects could be discerned suggesting a non-random orientation of the molecules within the liquid. Hence it seemed interesting to investigate the effect of non-sphericity of molecules upon the structure of the liquid. A systematic study was undertaken of four tetrachlorides; carbontetrachloride,

siliciumtetrachloride, titanium-,and tintetrachloride. It should be menti-oned that, with the exception of carbontetrachloride, studies of the liquid structure of these substances are scarce.

A study of thermal neutron diffraction of molecular liquids that aims at solving the liquid structure rather than the molecular structure has to

-12-cope with the disadvantage that the scattered intensity due to the inter-molecular contributions is small compared to the intrainter-molecular contribu-tions which depend upon well defined intramolecular atom-atom distances. The study of the liquid structure, therefore,demands accurate data, such that after correction for the large contribution due to the molecular structure the remaining intermolecular data are still statistically sig-nificant. The accumulation of these data has been the object of this study.

To this end much attention has been given to three aspects of data col-lection and data correction. These are (i) a critical examination of the instrumental stability, (ii) the application of a resolution correction, which usually is neglected in thermal neutron diffraction of liquids, and

(iii) the introduction of a new normalization method, called interval normalization. This normalization method yields the intramolecular structure parameters; the interatomic distances and the mean amplitudes of vibration. The intermolecular scattering contribution obtained from the normalized intensity distribution has been analysed in two steps. In the first step a comparison is made-between the experimental intermole-cular scattering function and the intermoleintermole-cular scattering calculated for a model of randomly oriented molecules regarded as hard spheres. From the comparison, information can be gained to which extent positional and orientational correlations in these molecular liquids are responsible for the intermolecular scattering function. In the second step,which is considered a further refinement, partial structure factors and partial radial distribution functions have been obtained.

A number of techniques exists to determine partial structure factors in polyatomic liquids. Isotopic substitution can be applied in neutron scattering as illustrated in molten CuCl by Page & Mika [6]. However, the number of isotopes, useful to apply in the substitution technique is limited. Moreover they are neither readily available in the quantity needed, nor easily convertible into the required substance [7], and they are not cheap. In another technique advantageous use is made of the difference in scattering amplitude for a given system using different types of radiation, for instance X-rays, electrons and neutrons. This

approach has been followed by Narten [7] who used data on carbontetra-chloride, from X-ray and neutron diffraction experiments.

The suggestion by Ramesh & Ramaseshan [8] to make use of anomalous scat-tering as a means to determine partial structure factors in case of liquids and non-crystalline solids consisting of two atomic species has not yet found application. The advantage of these techniques is that the scattering system differs little or not in the various experiments.

In X-ray studies of hydratation in aqueous solutions use ha"s been made of the concept of isomorphous substitution [9]- Recently such studies have been made with neutrons, applying isotopic substitution instead of re-placing the cations by isomorphous substitutes [10].

In the present study the concept of isomorphous molecular liquids is in-troduced. It should be emphasized that isomorphism in molecular liquids should have a meaning different from that in solutions. For solutions the size and the charge distribution of the ions are of importance whereas in comparing two molecular liquids the relevant parameters are the size and the degree of non-sphericity.

Combining the neutron data for carbontetrachloride of this study with X-ray data taken from the literature [7] intermolecular partial radial dis-tribution functions are calculated. For titaniumtetrachloride and tintetra-chloride the concept of isomorphous substitution is used and the 'common' partial radial distribution functions obtained for these liquids justify the adopted concept. The partial radial distributions were interpreted keeping in mind Prins'[11] caveat concerning 'wishful analysing' of liquid structure data.

The description of the equipment used in the measurements and the data correction procedures is given in Chapter II. In Chapter III the theoretical background is introduced which is indispensable in the sub-sequent normalization method and the Fourier transformations reported in Chapter IV. The experimental results, their discussion and the con-clusions form the contents of Chapter V. The fully corrected and norma-lized total and intermolecular intensities are tabulated in the Appen-dices A & B respectively. The equivalence of the different models, in-troduced in Chapter III describing the molecular and liquid structure

on the one hand, and the calculation of the weighted sum of intermolecu-lar partial pair correlation functions based upon a model liquid consis-ting of randomly oriented spherical top molecules regarded as hard spheres on the other hand is the subject of Appendix C.

R E R E R E N C E S T O C H A P T E R I

[1] Elsasser W.M., C.R.Acad.Sci.Paris 202(1936)1029 [2] Bacon G.E., Neutron Diffraction, Oxford(l962) [3] Kruh R.F., Chem.Rev.62(I962)319

[it] Furukawa K., Rept .Progr .Phys •25( 1962)395 [5] Rao K.R., J.Chem.Phys.1^8(1968)2395

[6] Page D.I. & K. Mika, J. Phy s .Cl+( 1971) 303^4 [7] Narten A.H., J.Chem.Phys.65(1976)573

[8] Ramesh T.G. & S.Ramaseshan, J.Phys.C^t (1971)3029

[9] Bol W., G.J.A. Gerrits & C.L. van Panthaleon van Eck, J.Appl .Cryst. _3( 1970)1+86

[10] Enderby J.E., R.A. Howe, G.W. Neilson & A.K. Soper, Annual Report SRC-SB (1971+) 135

[11] Prins J.A., Selected Topics in X-ray Crystallography, J.Bouman, Editor, Amsterdam(1951)p.191•

C H A P T E R I I . T H E E X P E R I M E N T S

"And people on Earth still firmly believe to this

very day that our physicists working on the most

complicated problems

..."

Arkadi & Boris Strugatski: Hard to be a god.

I N T R O D U C T I O N

In this chapter an account is given of the experiments. A description of the diffractometer is given and the instrumental resolution is described at some length. The effect of the resolution upon liquid structure'measure-ments is calculated. The molecular liquids that have been investigated are introduced with their relevant properties. The measuring procedure is ex-plained and the corrections to the measurements are dealt with following the order in which they.are applied.

T H E I N S T R U M E N T

The diffractometer at the R beamhole of the 2 MW reactor HOR is of the conventional type. The instrument is characterized by the parameters which

are listed in Table II. 1. . The thermal neutron flux at the entrance of the R beam hole is about

13 —2 — 1

10 cm s . The primary beam is collimated to within h2 minutes of arc

by means of a vertical Soller slit collimator, the other divergences are the natural ones.

-16-TABLE II.1. Characteristics of the neutron diffractometer at R .

S 2-1

Neutron flux at the sample (f) = (3il)l0^cm s at 2 MW (estimate) Neutron beam wavelength \ = (O.878 + 0.002) 2

o (Monochromator, Zn (0002)!

Diffraction angle 0 = 10.25 degrees of arc Mosaic spread g = 20 min. of arc F.W.H.M. Primary beam divergence a = 1*2 min. of arc F.W.H.M. Monochromatic beam divergence a_ = 60 min. of arc F.W.H.M. Detection system divergence a = 90 min. of arc F.W.H.M.

The monochromatic beam divergence a and the detection system divergence a depend upon the sample size in case samples are used which are smaller than the profile and the area of the monochromatic beam scanned by the detector. In case of plane slab samples the F.W.H.M. of both the beam profile and the detector profile have to be determined. The F.W.H.M. of these quantities enter the definition of divergence and they play a role in the determination of the scattering volume [I]. In this study the beam profile and the detector profile have been determined experimentally. The detector arm was placed at a position 29 = 90 and a thin plastic rod used as a probe was placed at the sample position. The probe was moved perpendicularly to the beam to scan its profile and subsequently moved

in the direction of the beam to scan the detector profile. The beam profile measurements were corrected for the distance between the rod and the detector. The detector profile measurements were carried out using rods of three different diameter. The scans yielded F.W.H.M. values w =

(32 + 1) mm for the beam profile and b = (27 ± I) mm for the detector profile. The intensity in the scans normalized to the average value in the flat center part is shown in Fig. II. 1. From the beam profile measure-ments an impression can be gained of the beam homogeneity. The scan through the detector profile using probes of different diameter show that the detector profile hardly depends upon the size of the probe.

d«laclDr profil.

Fig.II.1. Beam- and detector profile at the sample position. Detector at 26 = 90 degrees, probe moving perpendicular to the beam gives the beam profile, probe moving parallel to the beam gives the detector profile.

A beam shutter is placed just in front of the monochromator crystal in the primary beam. This shutter can be turned into three positions: one

po-. po-. po-. po-. 2

sition resulting m a 3.5 x 3-5 cm beam with a divergence of 2k minutes

• • . 2

of arc, one position resulting m a 3.5 x 7-5 cm beam with a divergence of 1*2 minutes of arc and a closed position. The closing and opening of the shutter takes 3 seconds. The Zn-monochromator crystal 25 x 8 x 1 cm in size, is in a symmetrical reflection position with the c-axis perpen-dicular to the largest area and it provides a beam of monochromatic neutrons with an average wavelength of O.878 A scattered from the (0002)-plane.

The monochromator setting at 20 = 20.5 degrees of arc introduces a wave-length spread

^^F.W.H.M. = \-^°^^%- («1^ + S ^ ) ' (II.1)

This wavelength spread in the monochromatic beam is one of the main sour-ces of the resolution of the diffractometer. A low efficiency fission chamber is placed at the end of the monochromatic beam channel, just in front of the sample. The fission chamber is used as a beam monitor and it covers the whole neutron beam. In the course of the experiments, monitor I, type RSN-312-M3 was replaced by monitor II, type LND-3008 with

-1(

a 30 times lower efficiency at the wavelength used. This change improved the stability of the operations of the system for the following reason. The beam intensity drifts between 10 and 15 percent during the five days reactor operation period, which gives small changes in the count rate due to the effect of monitor dead time that is not corrected for. In Table II.2 typical count rates for the two monitors are shown, using the same sample.

TABLE II.2. Comparison of two fission chambers used as monitors.

Monitor I Monitor II Ratio

Counts Collection time [s] Detector counts k

m^

~ 520 1+8 000 1 10^ ~ 350 32 600 1+0 ~ 3/2 ~ 3/2

The detector arm covers diffraction angles 20 between 2.5 and ll+5° in the measurements. The angular step increment is about HO min. of arc so that the detector has 217 possible different positions. The wave number transfer parameter, K , defined

UTT . _

K = — s m 0^ , (II.2)

subtends a range between O.k S~^ and 13.35 8 ~ \

The detection system consists of three neutron proportional detectors, 1 inch wide and 7 inches long, filled with BF gas 967» enriched in the

10 ^ isotope B and at a pressure of I50 cm Hg. The three detectors, type

RSN-II6S-MII were placed in a row above each other. The detection area 2

obtained in this way is 1U.7 cm , whereas the horizontal divergence contribution from the width of the detectors amounted to about 1+1+ minutes of arc. The calculated detection efficiency for O.878 8 neutrons is 81.7 peraent, using a = 181+3 tarn [7] as a cross section for the ''°B (n,a)'''Li* reaction.

T H E R E S O L U T I O N

The resolution o f the instrument is a smoothly varying function of the

scattering angle. Test runs were made on several powder samples in order

to verify whether the calculated parameters of the system adequately

describe the peak width distribution obtained. The following expression

can be derived for the F.W.H.M., A i , of the diffraction peaks [ 2 , 3 ] . 2 Ai =1 2 2 «3 . I*a^a^^6,^+(l+a^-l+a+l)a^^a22+(l+a^-8a + l + ) 6 % ^ 2 2 «1 ^"^2 k^' , (II.3) with a = tg0 /tg0 1.5 15.0

^m —

Fig.II.2. Comparison of calculated and measured F.W.H.M.'s for powder diffraction peaks of KI and N i . The relative resolution is shown as A K / K = ctgö.Ai.

In Fig. II.2 the calculated and measured F.W.H.M. values are given for

KI (.a = 7.052 8 ) and Ni (a = 3.517 8 ) in terms of K . The diffraction o o

peaks of the alkali halide occur at slightly smaller K-values than those

of nickel while the latter enable a peak width determination up to

7.1+ A , about half way the interval of measurement subtended in the

experiments. It is seen that the actual measurements agree with the

curve calculated using the parameters as indicated in Fig. II.2. In the

same figure the relative K-resolution is shown as ctg 0. A i . 2

-20-The effect of the instrumental resolution on a continuous structure factor S(ic) is calculated by Fredrikze and van Tricht [1+]. The structure factor obtained in a measurement, S (K),is related to S ( K ) through

m o 2

S

(K)

= a^

S(K

) + a , ( 0

)K

M ^ i + (0 )^ 2 d ^ M __ ^^_^)

m o o 1 o o die, , 2 o o dK, , k=k k=k o o

where 26 i s t h e s e t v a l u e o f t h e d e t e c t o r a n g l e , and k = 2TT/A ,

o 0 0 1+TT . K = T— s m 0 , o X o ' o

^ = ^ - °1 '

2 ^ N '^2 , , „ . 1 and a ^ ( 0 ^ ) = c / 3 + 2 c o t g 0^) - - ^ - c 3 ( c o t g 0„ + ^ ^ ^ ) c o t g 0 ^ , Q a^iQ^) = c^ c o t g 0j^ + — c o t g 0^ - c ^ c o t g 0j^ c o t g 0^ , 2 o 2 _ , 2 2 ^ - 2 2 a^ g^ + « . ^ 0 2 + 6.| ttg c , " '^ l+(a^^ + a/ + k 6^2) 2 (a^2 + k&^^)a^^ c„ = a^ + ^ ^ l+(a^2 ., ^^2 ^ ^^^2^

(a^2 + 2e^2)a2^

c, " "3 i+(a^2 +0^2 + Ue^2)

A Taylor series expansion of S ( K ) to second order in the relevant angular deviations has been used. Higher order terms are estimated to be much smal-ler than these first two orders, though they may turn out not to be

negligible. An impression of the order of magnitude of the resolution effect can be gained from Fig. II.3. For the calculation use has been made of a model structure factor appropriate for the description of the measured intensities of the substances under study in the region of

2r'

values larger than covered m the measurements.

13 _ 1.2 «L 1.0 - 09 0.7 0.03 0.01 1 000 - ; -001 < -0.03

;-f, 1 0 7 1

-v

" 2 "

-^

<b> CARBON-TETRACHLORIDE

r\

/ \

/-\^-V /-\^-V "

0° O j ^ i s ' $1 = 03° - N / A - ^ Ky \y \

^ /^^^

\_/

^u'- '°' *"

A , /

^ \ /

<b'> " c m

_ . —

. 0 9 J

1

^ • 1

v

i-'<"i«> 5 6 7 12 13 14 15 F i e : . I I . 3. The r e s o l u t i o n e f f e c t , A F ^ ( K ) . f o r an assumed s t r u c t u r e f a c t o r o f c a r b o n t e t r a c h l o r i d e , 1 + F ( K ) . Note t h e d i f f e r e n t o r d i n a t e s c a l e s . T H E S A M P L E S A N D T H E S A M P L E C O N T A I N M E N T The samples

Tetrahedral molecules or molecular groups form a wide class of which the AB,-type with A and B being atoms rather than molecular groups, consti-tute the most simple category. The symmetrical non-polar molecules which form liquids at room temperature with the A-atom belonging to the IV or IV groups of the periodic system and the B-atom to group VII have been considered in the present study. Of these the chlorides with the exception of ZrCl, and HfCl, form a continuous series within a liquid range inclu-ding room temperature. From these series two members have been excluded, GeCl. and PbCl, . The former was not readily available, and the latter explodes spontaneously at 388 K which is unfavourable,the more so as

-22-measurements at elevated temperatures were planned originally. A number of physical properties of the substances being investigated are assembled in Table II.3.

a b

TABLE II.3. Properties of IV and IV group tetrachlorides in the liquid state some of which at room temperature and 1 atm.

S u b s t a n c e P r o p e r t y M [ a . m . u ] p [gcm"-^] T [K] As [ c a l Mol m ^b ^^^ As, [ c a l Mol b X ^ . 1 0 ^ 2 ^ m V ^K-^] ' K - ' ] 'l CClj^ 153.81 1.595 2 5 0 . 1 2.1+0 ' 3I+9.9 2 0 . 7 1032 S i C l ^ 169.90 1.1+83 2 0 3 . 3 9 . 0 8 329.9 2 1 . 2 1513 TiCl^ 189.71 1.726 2I+8.9 8.96 U09.7 2 1 . 0 7 809 SnCl^ 260.50 2.226 239.9 9.11 387.1 2 1 . 8 981* *) see [5]

Table II.3 includes the entropies of melting AS and evaporation AS . The

m b latter indicates that these non-polar liquids closely obey Trouton's rule

— 1 — 1

which states that AS-^»:20 cal Mol K . The entropies of melting are quite similar with the exception of CCl, which has a much lower AS than

*+ m the other tetrahalides indicating that in the solid phase these molecules

already have some degree of rotational freedom [5].

There is evidence that within the solid a phase transition with an entropy change of 7.1+ cal Mol~ K~ occurs at 222 K. In addition, the liquid range of CCl, , defined as the ratio T.,/T ,is about 10^ smaller than that of the

^ D m

other three tetrahalides. The isothermal compressibility, Xm, derived from Brzostowski [6] is listed in Table II.3, because it is used in the deter-mination of the forward scattering.

In a thermal neutron diffraction study reference should be given to the relevant neutron data of the nuclei involved. In Table II.1+ for the

elements with natural isotopic abundance the absorption cross section (d ) , a the scattering cross section (a ), and the cross section for coherent (o )

scattering are listed. Moreover the scattering amplitude b is also given; note the negative scattering amplitude for titanium.

TABLE II.lt. Relevant neutron data for the atomic nuclei. [7,8,9].

Atom C Si Ti Sn CI Parameter a . 10^ [cm^] 5.55 2 . 2 3 1+.23 5.0 l6 2 ?U ? a . 10 [cm ] 0.003I+ 0.16 6 . 1 0 . 6 3 3 2 . 8 5 ^ 1?" b , . 10 [cm ] 0.661+ 0.1+15 -0.3I+ 0 . 6 1 O.95I+6 ^°'^ 2k 2 a ^. 10 [cm ] 5.51+ 2 . 1 6 1.1+5 k.67 11.1+5 coh

The scattering cross sections given here are for rigidly bound atoms as usual. In the case of an incoming neutron energy, E , high relative to the thermal energy quantum k_T, this cross section tends to the free

atom cross section, defined as follows

bound

a

free , . ™^2 ^ M^

with m being the neutron mass and M the molecular mass. The effect is about sixteen percent in the case of carbon which is the atom with the lowest mass considered here. However if we assume that the molecules are essentially rigid and the neutron energy low in comparison to k T, then the A-atoms in each molecule have an apparent mass equal to the molecular mass. In that case approximate expressions can be used for the effective mass of the peripheral atoms in the molecule [10,11]. In this study the introduction of an effective mass of the last type has not been considered. For the absorption cross section the values at

V = 2200 ™s are given, corresponding to a neutron wavelength of about 1.8 A. However, in the present experiments a wavelength of O.878 A is used. As the thermal neutron absorption cross section depends upon

-21+-1/v, or is proportional to X, the effective absorption cross sections are about a factor two lower than the ones given in Table II.U.

In neutron diffraction experiments use can be made of a variety of sample geometries two of which are most commonly used, i.e. a plane slab- and a cylindrical geometry. Although in the present investigations both have been used, the experiments carried out in cylindrical geometry are dis-cussed here in detail only. The experiments performed in plane geometry have been used in order to establish the small K-value behaviour of the corrected intensities (see page 39 )•

The simple construction of the sample cells is shown in Fig. II.lt, and

m

£5.

s^MiÜ

₁₄

Fig.II.l+. Cylindrical sample cell(a) and plane slab sample cell(b),

the dimensions of the cells are summarized in Table II.5. In both designs the neutron beam is traversing the cell through a window made of vanadium. Vanadium was chosen for its low coherent scattering amplitude, so that there is a smooth scattered intensity contribution to the diffraction pattern. The cylindrical cell system is constructed of a Ni-Fe alloy having a coefficient of linear expansion corresponding to that of V. For the construction of the plane cell brass was chosen. In order to tighten the cells use was made of metal o-rings consisting of hollow silvered stainless steel in the case of the cylinders, and of indium in the case

[mm] [mm] [mm] r 3-, [cm J

5

O.k 80 21+ 20 O.k 70 22

of the plane slab. The cylindrical cell was helium tested up to 90 atm of pressure at room temperature as well as at -170 C and showed no leakage [12]. The plane slab sample cell has been used between -80 C and + 1 2 0 C withstanding pressures of about 1 to 2.5 atm. In Table II.5 the main measures have been included.

TABLE II.5. Main design characteristics of the plane slab and cylindrical sample cells.

Shape Plane Cylinder Dimension Thickness/diameter [mm] Wall thickness Diameter/height Volume T H E M E A S U R E M E N T S

An outline will be given of the measuring procedure. A measurement consists of assembling scattered neutron intensities during one non-interrupted sequence in the detector at positions which are defined by the initial value of the scattering angle and a fixed angular increment. Measurements were repeated a number of times. The diffraction experiments with

CCl, and SiCl, were carried out with monitor I. In the course of the 1+ 1+

diffraction experiment with TiCl, monitor I was replaced by monitor II and in the experiment with SnCl, the second monitor was used. All the experiments were carried out at room temperature.

(i) A measurement was made without a cell at the sample position. This measurement was used as a background.

(ii) A series of two measurements was made with the empty sample cell at the sample position, which in combination with the preceding one determines the scattering by the empty cell.

-26-The background and empty cell measurements were repeated at the end of the period of the measurements.

(iii) The number of measurements performed with each substance at room temperature together with an order of magnitude estimate of the average intensity per scattering angle and an indication of the monitor used is given in Table II.6.

TABLE II.6. Number of measurements N, the estimated average intensity I per scattering angle in the detector, and the type of monitor used. Monitor Substance SiClj^ TiClj^ SnCl, 3

k

3 7 5 5 . 5 5 3.5 10" 10*^ 10^ 10^ I I I I I I

Care was taken to place the empty and filled sample cell in such a way that its position with respect to the beam is reproduced. The background and empty cell measurements and the measurements with TiCl, served to

1+

In Table II.7 this factor obtained with the three different types of measurement is given.

TABLE II.7. Normalization factor for monitor I and II in three types of measurement.

Type of measurement Average Intensity Normalization I II factor Background 1200 800 1.372 Empty cell + background 8300 5500 1.1+21

1+ k

TiCl, in cell + backgr. 5 10 3.5 10 1.1+71

— The acae£tanoe_of the_data

The acceptance of the raw experimental data for further data treatment depends upon the repeatability of measurements as this ascertains the stability of the measuring system. Mainly, for this reason each measure-ment is repeated a number of times. As the measured intensities are subject to chance variations, statistical methods are called upon in order to check the repeatability of measurements. An account is given here which criteria have been used, and to which statistical inferences they lead. The measured data presented here are obtained by averaging the accepted ones, and henceforth they are called the averaged measure-ments .

If several measurements were taken in a given K-interval an averaging was performed at each detector position within the interval and the

resulting mean values and the residuals were printed. The residuals for each position were assumed to follow a normal distribution law. The fraction of outliers, defined as the residuals with an absolute value larger than three times the expected standard deviation, amounted to not more than one percent of the total. This is well above expectations.

-28-It turned out that in case of a positive residual, the large deviation usually occurred when the measuring times were exceptionally long because the reactor had been shut down for some length of time. In the other cases, large deviations from the mean are caused by transients in the electronic equipment or by changes made in the environment of the diffractometer. These large residuals were not included in the determination of the average intensity at that particular position.

Apart from the study of the residuals we may ask for the distribution of residuals of either sign in two ways [13]. At first, assuming a normal distribution of the residuals one would expect an equal number of resi-duals of either sign for any individual measurement. From the number of residuals of either sign being sensitive to small changes of a measure-ment with respect to the average measuremeasure-ment an indication is obtained for overall systematic changes in the instrumental set-up through the in-equality of the numbers of residuals of either sign. Secondly, having obtained numbers n, and n„ of residuals of either sign taken over the whole measurement one may ask how these residuals are distributed in a time sequence sign plot [ill]. In order to check the obtained pattern of signs, runs are defined as uninterrupted sequences of residuals of either sign.

For measurements in which n and n„ are both greater than 10, a condition

which is almost always fulfilled in neutron diffractometry, the number of runs obtained can be tested against the expected number of runs calcu-lated from the normal approximation of the distribution of runs, with

2n n

P = ^ ^ 1 , (II.5)

2 2^1" (2n^n -n -n )

a = 2 . (IT.6) (n^+Ug) (n.^+n2-l)

This method is particularly sensitive to systematic changes in the in-strumental set-up leading to an increase or decrease of intensities over the duration of a measurement.

The result of these checks were positive, which means that no indication is found for any deviation other than that expected by chance. Hence the corrections and the data shaping procedures discussed in the following paragraphs were applied.

The averaged measurements are shown in Fig. II.5 together with the averaged measurements of the empty sample cell plus background and the background without a sample cell. The background and empty cell measurements shown in Fig. II.5 are those appropriate for CCl, and SiCl,. For TiCl, and SnCl, these intensities should be divided by the normalization factor

1.1+72 given in Table II.7.

6 0 5.0 4.0 • T 6 0 o 50 4 . 0 . •, 4 0 5iO 4.0 3 0 2 0 1.0 0 —I 1 1 1 1 1 r - —, 1 1 1— CARBONTETRACHLORIDE SUCIUMTETRACHLORIDE TITANIUMTETRACHLORIDE TINTETRACHLORIDE

EMPTY CELL* BACKGROUND BACKGROUND

I 2 3 4 5 6 7 8 9 10 , II 12 13 14 IS

,[l']

Fig.II.5. Averaged measurements for the tetrachlorides at room temperature, the averaged measurements of the empty sample cell and the

background.

It can clearly be seen that the K-increment between two adjacent positions of the detector is decreasing with increasing angle. This effect is due to the constant angular increment of the detector arm.

-30-C O R R E -30-C T I O N S T O T H E D A T A

I.l!:Èïl2.éh:£.ïi2'!l

Although the raw experimental data show the essential features of the ultimate results, they should nevertheless be corrected for a number of unwanted contributions inherently present while collecting the data.

The corrections to be applied are approximate in several respects. In all corrections that involve a scattering cross section the latter has been assumed to be sufficiently accurate if an isotropic elastic'cross section is used. This applies to the corrections for scattering by the sample cell (ii), correction for inelastic scattering (iv) the multiple scattering correction (v), all of which will be discussed.

The scattering from the empty cell has been corrected for the presence of the sample before subtracting its contribution. The incident beam intensity has not been corrected for absorption and scattering in the wall of the sample cell which weakens the homogeneity of the beam. The background and forward scattering intensities from the empty sample cell have been subtracted without further treatment. This correction includes a systematic error at small scattering angles as the presence of the sample should diminish in some way the forward-scattering inten-sity.

The inelastic scattering in the sample has been taken into account applying a modern version of the so-called Placzek [I5] correction. The correction for inelastic scattering (iv) arises from the failure of the static approximation of thermal neutron scattering.

The resolution correction applied in section (vi) depends upon a Taylor series expansion of the structure factor to second order leaving aside detailed considerations with respect to the higher order contributions. An estimate is given of a correction for multiple scattering from the sample cell (vii) and the sm.all K-behaviour of the scattered intensity (viii). The K-scaling and smoothing of the experimental data (iii) is performed using a spline smooth procedure.

Z'^£_22ï!rf2*i2"s

The background c o r r e c t i o n .

The average background measurement I, ( i ) accurate t o within 2% was

b

subtracted from the average empty sample cell measurement I (i) and from the average measurement of sample and cell I (i) together. Subtraction of a smoothed version of the average background showed to be less satisfactory than the subtraction of It, (i). This

indi-cates that systematic trends are present in the background data not easily observed in a measurement.

The scattering by the sample cell.

The average intensities I (i) due to scattering by the empty sample cell were corrected for background and for the presence of the sample by a method due to Boutron and Meriel [16]. According to this method the cell is considered to be thin such that neutron scattering and absorption is negligibly small. The function C(Z R,i) is calculated which describes the fraction of the empty cell scattering that should be subtracted. Here E is the macroscopic total scattering cross section of the sample and R is the radius of the cylindrical cell. In Table II.8 the quantities E R are given for the

tetrachlorides. The calculated transmission Tr of the

sample is shown and the ratio of the values of C(E R,i) at the lar-gest scattering angle C(max) and at zero scattering angle C(0). This ratio shows how much more should be subtracted of the empty cell scattering at large angles than at small angles.

TABLE II.8. Quantities pertinent to the scattering process.

Q u a n t i t y S u b s t a n c e CClj^ SiClj^ TiCl^ SnCl, ^ t ^ 0 . 8 0 5 0 . 6 6 5 0 . 7 0 1 0 . 6 9 8 -Tr 0.299 0.369 0.331 0 . 3 5 7 C(max)/C(0) ^ 3 9 1.28 1.31 1.31

-32-The function C(E^R,i) is depicted in Fig. II.6 for CCl, and SiCl^ which have the highest and lowest total cross section. The resulting scattered intensity distribution I (i) can be written

s

I j i ) = I^(i)-I^(i)-(l^(i)-I^(i)). C(E^R,i) . (II.7)

SiCl,

CCl.

0 5 10 15 20

4J-'] —

Fig.II.6. Fraction, C(E R,i), of the empty cell scattering to be sub-tracted according to Boutron & Meriel [l6].

(iii) The K-scaling and smoothing of the experimental data.

It is practical to transform the experimental data into a set at equally spaced K-intervals. An increment A K = 0.05 A was chosen. In the experimental data a K-increment of 0.05 A~ was present at K = 11.00 A , being larger at smaller values of K and smaller at larger values of K. The interpolation was accomplished by making use of a cubic spline interpolation and -smoothing program due to Reinsch [17]. For the spline smooth a weighting function was chosen according to the expected variances of the averaged expe-rimental data. The interpolated and smoothed intensities, 260 in number, are called intensity points, I ( K ) . The standard devia-tion for the intensity points was estimated to be diminished by a factor of /5, five being the number of measured positions involved in the calculation of each intensity point.

[iv) Correction for inelastic scattering.

In recent years many authors have given much attention to the con-tribution of inelastic scattering in neutron diffraction studies [18, 19, 20, 21, 22]. The results reported by Yarnell and coworkers [21] can be adapted most easily to our work and therefore it has been used in the present study. Inelastic scattering corrections have been applied to the total scattered intensity. In a first approximation use is made of an isotropic ideal gas scattering law neglecting the possible effects of interference scattering. Then the expression P ( K ) used for the correction for inelastic scattering including effects of recoil to second order reads

P K ; M k ^ T B + 2E 0 K 1 k „ T m „ B - — + K . 2M 3 E 0 2 K , 2 k m M 1+ o ; i i . 8 )

where M is the molecular mass and m is the neutron

mass. The energy of the incoming neutrons, E = IO6 meV so that k^T/E = 0.239. The constants K,, K^ and K., listed in Table Il'.9

D O 1 ^ 3

are related to the energy dependence of the detector system

E[k) = 1-exp(- 1.716 -^ ) , (II.9)

i n t h e f o l l o w i n g way : 1 +—• k — 2 o e o ( I I . 1 0 a ) XT - 3 ^ 5 , ^ 1 1 , 2 ^ 2 2 8 8 o e 8 0 G o o ( I I , 1 0 b ) 3 , ^ 1 1 , 2 ^2 3 k o c 1+ o e o o ( I I . 1 0 c )

-3k-with c = e(k ) , o o = ^ '1 dk k = k ' o '^ ' dk2 k = k

TABLE II.9. Constants related to the detection system efficiency.

K^ K3

\

= 0.878 2

o

0.817 0.158 0.253

In case of CCl, the expression for P ( K ) becomes

P ( K ) = (-7.8 + 1.152 K^ + 3x10"^ K ) 10 ,11.11

(v) The multiple scattering correction.

The correction for multiple scattering is based upon the separation of first order scattering from higher order scattering. The first order scattering should be known in order to be able to calculate the second and higher orders of scattering, A calculation of the transmission and the angular distributions of first order and of higher order scattering can be obtained for a cylindrical sample

using, the scattering- and total cross section, a„ and o , and

the number density, N, of the system and the radius R and the height h of the cylindrical sample. The values for the parameters used in the calculations are given in Table 11.10.

TABLE 11.10. Parameters used in the calculation of the multiple scattering effect. Cross sections in units of

-2l+ 2 . . -3 10 cm , R and h m cm, N m cm . P a r a m e t e r S u b s t a n c e CCl^ SiClj^ TiCl), SnClj^ -21 N*10 6.2U 5 . 1 0 5.1+7 5.15 '^S 6 9 . 5 5 6 6 . 2 3 6 8 . 1 3 6 9 . 0 0 "T 133.75 1 3 0 . 3 3 135.23 133.50 R 1.022 1.022 1.022 1.022 h 7 . 1 5 7 . 1 5 7 . 1 5 7 . 1 5 abs 2200 ma"^ 131 .1+ 131.2 137.5 132.0 A study o f t h e m u l t i p l e s c a t t e r i n g e f f e c t s f o r i s o t r o p i c s c a t t e r i n g systems h a s been made by Warmelink [ 1 2 ] . T h i s work can be r e g a r d e d a s an e x t e n s i o n and a r e f i n e m e n t of e a r l i e r work by Blech and Averbach [ 2 3 ] .

For t h e c a l c u l a t i o n of t h e m u l t i p l e s c a t t e r i n g use i s made of an i s o t r o p i c s c a t t e r i n g a p p r o x i m a t i o n . For t h e i s o t r o p i c s c a t t e r i n g model i n t e n s i t i e s t h e u n d e r c a s e t y p e i i s u s e d . I n t h i s a p p r o x i -mation t h e f i r s t o r d e r s c a t t e r i n g i ( K ) i s r e l a t e d t o t h e f i r s t o r d e r s c a t t e r i n g a t K = 0 , i . ( 0 ) t h r o u g h

I / K ) = i ^ ( 0 ) D ^ ( K ) , ( 1 1 . 1 2 )

where D ( K ) is a monotonously increasing function of K. The function D ( K ) is shown in Fig. II.7 for the substances under investigation. The k-th order scattering is- calculated as a fraction of the (k-l)-th order scattering. The sum of all orders of scattering yields the function M ( K ) .

-36-Fig.II.7. Single scattering volume effect, D (K),for a cylindrical sample cell.

The function M ( K ) has been calculated from the second and third order scattering at l6 equidistant K-values. The assumption is made that for all orders beyond the third one the ratio between any two successive orders is the same as that between the third

and the second order [lli]. A polynomial of degree four in K was

fitted to the l6 points and used to calculate all required values of M ( K ) . The function M ( K ) is a monotonously decreasing function of K, as it is shown in Fig. II.8 for all four tetra-chlorides.

.[S-';

i^(K) = i^(K) + ij^(K) , (11.13)

i^(K) = i^{ 0) D.J(K) , (11.11+)

ij^(K) = i^(K) M ( K ) = i^(0)D^(K) M ( K ) , (11.15)

l^(ic) = i^(K) D ^ ( K ) (1 + M ( K ) ) , (11.16)

where i ( K ) is the total scattering

i.(K) = l i.(K) , (11.17)

* j = 1 ^

i (K) is the multiple scattering.

In order to apply a correction for multiple scattering to a series of intensity points I.(K) at K-values where the interference scat-tering gives rise to an undulating intensity distribution, a reference value I ^ ( K ) is chosen in a way such that

ref "^

I ^ ( K ) = i,(K)D.(K) (1 + M ( K ) ) . (II.18) ref 1 1

This reference intensity is identical with that used for the inelas-tic scattering correction.

The multiple scattering contribution I „ ( K ) is calculated

y^) = w ( ^ ) T T ^ • ("-^9)

The first order scattering I (K) corrected for the volume effect D (K) reads

I, ( K ) = ( I . ( K ) - 1 ( K ) ) / D . ( K ) . (11.20)

-38-(vi) The resolution correction.

The expression for the resolution effect (II.3) was transformed into an iterative resolution correction. Estimates of the first and second derivatives I '(K) and I " ( K ) with respect to K were obtained in two different ways. Numerical differentation was applied in that part of the K-interval where no analytical expres-sion for the scattered intensity was available, K < 8.30 A , in the following way. The first derivative, I (K) was obtained and smoothed utilizing the spline smooth procedure introduced in (iii) and the standard deviations were estimated as if the intensity points were independent. The second derivative, I (K) was obtained from the smoothed I (K) in the same way.

Beyond this region the intensity was described by an analytical expression (IV.7) obtained in the normalization procedure. The

derivatives were derived analytically. Then the resolution correction was calculated using the approximate expression (ll.3) and added to

I, ( K ) and the procedure iterated. This was done in this way because the derivatives used were approximations of the real values with respect to at least two aspects. At first the intensity points I., (K) are influenced by the resolution and therefore they are not the true intensity points. Secondly the differentiation and the subsequent smoothing procedure tend to flatten the resulting

derivatives. The resolution correction needed a fourfold iteration, and this was perfomed analytically over the ultimate K-interval in the experiments. The analytical expression for the scattered intensity was derived from the normalization procedure (see para-graph IV.a) in each iteration.

In Fig. II. 9 the numerically calculated resolution correction is shown after the first iteration at K-values larger than 5 A~ as a result of the procedure described before. It is compared with the resolution effect to be expected assuming that the measured intensity in this K-interval can fully be described by the intra-molecular form factor F ( K ) , introduced in Chapter III. From the

result it can be concluded that the resolution effect is quite well reproduced already after the first iterative step. The reso-lution corrected intensity points are denoted I ( K ) .

Fig.II.9. Numerical resolution correction for carbontetrachloride (o) and analytical resolution correction for carbontetrachloride ( ).

(vii) The correction for multiple scattering from the sample cell. The scattering by the sample cell is reduced when the sample is present. In (ii) the calculation of the reduced contribution is outlined. A fraction of the neutrons scattered by the sample cell and scattered again in the sample. A rough estimate is made of this contribution to the multiple scattering. It was assumed to be isotropic and proportional to the transmission Tr of the sample, and to a factor 1-C (E fi,i) where C(E R,i) is the average value of this function, introduced in (ii), over the K-interval. (viii) The scattered intensity at small K-values.

The intensity point distribution, I ( K ) , corrected for resolution and for multiple scattering from the sample cell was compared to the corresponding distribution from a measurement in plane geometry. Although the intensity distribution measured in the last geometry was of inferior statistical quality, the intensity at small angles extrapolated to zero angle agrees more favourably with the value based upon the isothermal compressibility S(o). In the range of overlap from K = O.UO 'A up to K - 5.0 A~ the distributions ob-tained in either geometry were matched over the last peak in

-1+0-the interval. The two intensity distributions were subsequently

interchanged from K = O.UO A~ up to a K-value where the

distri-butions intersected, this value was at about K = 0.95 A

This yielded the final corrected molecular intensities I ( K ) ,

m

shown in F i g . 11.10.

F i g . I I . 1 0 . Fully corrected neutron d i f f r a c t i o n i n t e n s i t i e s for four

t e t r a c h l o r i d e s .

R E F E R E N C E S T O C H A P T E R I I

[1] Silva M.J. d a , IRI-internal report( 197I+)

[2] Caglioti C., A. Paoletti & F.P. Ricci, Nucl.Instr._3(1958)223

C3] Lieshout J.J. v a n , IRI-internal report(l968)

[1+] Fredrikze H. & J.B. van Tricht, Proc.Neutron Diffraction Conf.(l975)

Petten, RON 23l+,295

[5] Silver L. & R. Rudman, J.Phys .Chem.7|+.( 1970 ) 31 3I+

[6] Brzostowski W., Bull.Acad.Sc.Pol.13(1965)501

[7] Evaluated Nuclear Data File (ENDF/B-II,1II)

[8] Bacon G.E., Acta Cryst.A28(1972)357

[9] Shull C.G., M.I.T.compilation(1972)

[12] Warmelink D. J.H. , IRI-internal report( 197*+) [13] Tricht J.B. van, IRI-report( 1971+) ,IRI 132-71+-05

[1I+] Draper N.R. & H.Smith, Applied Regression Analysis(1966)lSBN 01+71221708

[15] Placzek G., Phys.Rev.86(1952)377

[16] Boutron F, & P. Meriel, Bull.Soc.Miner.Crist.83(196o)l25 [17] Reinsch C.H.,Num.Math.10(I967)177

[18] Ascarelli P. & G. Caglioti, II Nuovo Cimento 1+3(1966)375

[19] Egelstaff P.A. & M.J. Poole, Experimental Neutron Thermalisation(1966) p.153

[20] Powles J.G., Adv.Phys.22(1973)1

[21] Yarnell J.L., M.J. Katz, R.G. Wenzel & S.H. Koenig, Phys.Rev.A7(1973) 2130

[22] Blum L. & A.H. Narten, Adv.Chem.Phys . 3l+( 1976)203 [23] Blech I.A. & B.L. Averbach, Phys.Rev.A137(1965)1113.

-1+2-C H A P T E R I I I . T H E T H E O R E T I -1+2-C A L F R A M E W O R K

"Do not confuse what I shall say under this head

with the theories so much in vogue among the

meta-physicians and physicists, those weavers of the

wind ..."

Nathanael West: The dream life of Balso Snell.

I N T R O D U C T I O N

In this chapter the expressions for the thermal neutron differential

scattering cross section which have been used in the analysis of the

experimental data are brought together. In this thesis the scattering

of neutrons by compounds consisting of two atomic species is studied.

On one hand the scattering system can be associated with a diatomic

mixture, on the other hand it is equally well described as a molecular

system, the molecules of tetrahedral symmetry being composed of two

atomic species. The scattering system is a molecular liquid. The

mole-cular and the liquid structure both are the subject of this chapter.

The liquid structure depends upon the positional and the orientational

correlations between the molecules within the liquid, that cannot

be separated in any straightforward way [ 7 ] .

In the first section of this chapter the thermal neutron differential

scattering cross sections for a monatomic and a diatomic liquid

sys-tem are introduced. In the second section the corresponding expression

The scattering systems are assumed to be isotropic which we find

re-flected in the use of a scalar radial distribution function, g ( r ) . The

effects of orientational correlations have been ignored. This means that

the molecules within the liquid are assumed to be free one with respect

to another.

T H E M O N A T O M I C A N D D I A T O M I C L I Q U I D S Y S T E M

Neutron diffraction arises as the cooperative result of the neutron waves

scattered by all nuclei in the scattering system. The relative phase of

these scattered neutron waves gives rise to the diffraction pattern. The

measured diffraction pattern is governed on one side by the ratio of the

coherent and total scattering and on the other side by the spatial

arrange-ment of the nuclei in the system.

The_jnonatomic liquid system

For a monatomic isotropic system consisting of N nuclei located at

positions r_. within a volume V, for which the scattering is measured by the scattering amplitudes b - , the thermal neutron differential

scat-tering cross section per unit solid angle and per nucleus can be written

^ = 1

dfi N

> , 1 K . r .

.^, b . e 1

1=1 1 ,111.1

with K. = k^ - k , the difference between the outgoing wave vector k and

the incoming wave vector k of the neutrons. The change in wave vector

D — o

from k into k is assumed to be directionally only.

Expansion of (lll.l) leads to an expression in which the relative phases between all nuclei in the system are summed , and where all products b. b . appear

da _ 1 V V -u ^ i ^-(r.-r.) (ill.2) 1 V V •,-, i i c . ( r . - r - ,

jn = T? Z ) ^-^^ e - - 1 -J'

1 + 1 +

-Performing an ensemble average over both the spatial arrangement of the nuclei and the distribution of scattering amplitudes and assuming that no correlation exists between nuclear position and scattering amplitude

(III.2) yields

f =

<.=>

.

<.>=

p

, ^ -1 K.r i r ) e dr , (III.3) with N

<b > = -7 y b.b.

, , N • •, , 1 1' ' i=i'=1 and 1 ^ 1=1

The radial distribution function g(r) describes the average of the rela-tive occurrence of the internuclear distance r in the system. The con-vergence of (ill.3) is obtained by including a term

-<b>^ p e~^ -•- dr = - ( 2 T T ) W ^ P 6(K),

where iS(j<) is the Dirac delta function. Now,

-do <b2> <b> <b > ;g(r)-1) e -•- dr (III.1+) I t i s c u s t o m a r y t o d e f i n e t h e s t r u c t u r e f a c t o r , S ( K ) , by S ( K ) = 1 + P ( g ( r ) - l ) e~^ - - i - d r _{( I I I . 5 )}

We may a l s o w r i t e

S ( K ) = 1 + l+TTp r ( s ( r ) - 1 ) s i n (K r ) d r . ( I I I . 5 b )

For the sake of the brevity the scattering function i(K) and the pair correlation function h(r) are introduced.

They are defined as

i(K) = S ( K ) - 1 (III.6)

and

h(r) = g(r) - 1. (III.7)

Now the simplest way of writing —^ is

^ - < . > | 1 ^ ^ 1 ( K ) (III.8)

From (ill.5a), (ill.6),and (III.7) we may deduce the relation

h(r)

(2^)3p

. I s 1 K.r ,

I ( K ) e di< , ,111.9a)

which is the inverse Fourier transform of the structure factor (III.5). We may write (ill.9a) in the form

h(r) 1

„ 2 2ïï pr

K. i(K)sin(Kr)dK. (III.9b)

The expressions (ill.5) and (ill,9) form the analogon for thermal neutron scattering of the well-known Zernike-Prins [1] relations.

-1+6-in the limit-1+6-ing cases K->-0 [2] and r-n» respectively

Lim i(K) = -1 + p kg T x^,

with k the Boltzmann constant, T the absolute temperature and Xm "the

B 1 isothermal compressibility, whereas

Lim g(r) = 1.

r-K»

A s a consequence the relation (ill.8) in the two limiting cases K->-0

and K-x» transforms into

da dfi

= Lim 4^ = <bS

K-^0 K-O ^^ <b> , 1 - - — (1 - p k ^ T x , <b > and da dQ Lim da dQ <b2>

: I I I . 1 0 )

The quantities S ( K ) , i ( K ) , g(r) and h(r) defined here for a monatomic

system will be replaced in the following sections by corresponding e x

-pressions for a diatomic and a molecular system.

The diatomic liquid si^stem

The general treatment of X-ray scattering by a diatomic liquid system is

given by Wagner & Haider [ 3 ] . Their results adapted for the case of

ther-mal neutron diffraction in the static approximation are summarized in

this paragraph. The diatomic mixture consists of N atoms which occupy

a volume V. There are N. A-type and N B-type atoms such that

We i n t r o d u c e f r a c t i o n s and number d e n s i t i e s t h r o u g h

^A ~ N ' ^B ~ M

^A V ' ^B V

fulfilling the conditions

^ = ^A " ^B

and

D = p + p .

^ ^A '^B

Now the thermal neutron differential scattering cross section per unit

solid angle and per nucleus can be written

(g(r)-l) e~^ ^-^dr

1

- ,

(III.11)

with

<b > = f^ <^A >

^ h

"^B "'

and

<b> = f <b > + f <b > ,

A A B B '

[HI.12)

_

-1+8-while g(r)-1 Introducing ^ K ^ ^ S A A ^ ^ ) - ^ )

'

2 f^f3<b^><b3>(g^g(r)-l)

.

f2<b|>(g3g(r)-i:

<b> (III.13) i(ic) = p (g(r)-l) e ^ - • - dr , ; i i i . i i + ;

expression (III.10) can be rewritten in a simple form similar to (ill.8)

||=..^>{,.5!ru,

; i i i . i 5 )

Expression (ill.11) closely resembles (III.1+), the difference being that

a weighted sum of three partial pair correlation functions denoted

(g(r)-l) is used instead.

The values of the coefficients of the partial pair correlation functions

in (ill.13) for the substances investigated in this study are reproduced

in Table III.1, in which use is made of the neutron data given in Table II. 1+.

In the two limiting cases K->-0 and K->^ (ill. 11) becom.es

da dO. ^. da ^ , 2 . = Lim -7: =<b > K^O K->0 '^^ <b> , —^ (1 - pk T XT, and da dQ : i i i . i 6 )

TABLE III.1. Relative contributions to the weighted pair distribution function (ill.13). Rel. Contr. 'I < V ' 2f^f^<b^><b3> fl<^/ 2 2 2 <b> <b> <b> Substance CClj^ 0.022 0.252 0.726 SiClj^ 0.010 0.177 0.813 TiClj^ 0.010 -0.215 1.205 SnClj^ 0.019 0.238 0.71+3

The diatomic mixture approach does not enable the separation of the par-tial correlation functions in one experiment. Keating [1+] has shown how in principle the partial pair correlation functions in (111.13) can be obtained on applying isotopic substitution. However in practice only for a restricted number of elements suitable isotopes exist.

The differential cross section for a diatomic mixture can be used irres-pective of the structure of the scattering system involved. The latter may equally well be a mixture of atoms or a molecular liquid. In the last case sharp peaks in the weighted pair correlation function appear which can be associated with the intramolecular distances. The ana-lysis presented here applied to a molecular liquid may serve as a first step in the determination of the molecular and liquid structure of the substance under investigation. Based upon the knowledge of the structure of the molecules constituting the liquid an alternative derivation of the thermal neutron differential cross section may be devised. This is done in the next section.

-50-T H E M O L E C U L A R L I Q U I D S Y S -50-T E M

In discussing the molecular liquid in this section the assumption is made that no correlation exists between the orientations of the mole-cules. Although this assumption is not necessarily correct, it has the advantage that the resulting description of the diffraction pattern is very simple.

Starting from (III.l) we now write for the thermal neutron differential scattering cross section per unit solid angle and per molecule

da

an

N n

I I

i=1 j=1 1 K.r.. b. . e - -ij (III.17)

where i sums over the molecules and j over the atoms in one molecule. The position vector r.. can be written as a sum of r., the position

vec--ij - i ' ^

tor for the center of the molecule defined in the way advocated by de Vries [5]> £..,the position vector of the constituent atoms relative to the center of the molecule,and u..,the momentary position of the atom with respect to the equilibrium position of the atom,

r. . r. + p.. + u.. _{-1 -^ij -ij} (III.18)

The molecular center is chosen at the A-atom. Expansion of (III.17), introducing the radial distribution function for the molecular centers g^(r) and performing an ensemble average including an averaging over the molecular orientations, the statistical distribution of isotopes in the molecule and the deviations from their equilibrium positions, gives

2 ? 2

<b > = <b„> + 1+ <b^>

and

g = < . ^ > l l ^ F / K ) . F y K ) p ^

(gj^(r)-l) e"^ ''-•^dr^ , (III.20)

where Pw is the molecular number density. The intramolecular form factor P ( K ) is given by P.,(K) = e AB <b > Kp _AB 12<b„> sin(Kp. <b2> BB' e - ^ ^ ' ^ ^ B ^ Kp. BB ; i i i . 2 i )

and the intermolecular form factor P „ ( K ) by

T^M

=

< b 2 > < b ^ > + k < b g > sin(Kp^g) ^2 KP _AB (III.22)

The quantities p._ and p are the intramolecular distances between A-B 2 2 and B-B atoms pairs respectively. The quantities <u > and <u > are

Aij BB the mean square amplitudes of vibration.

The equations (III.20), (III.21) and (ill.22), but for the inclusion of the effect of molecular vibrations were in a slightly different form derived by Rao [ 6 ] .

Now the scattering function for the molecular center i,,(K) is introduced,

5 2

-The e x p r e s s i o n ( i l l . 1 7 ) may be w r i t t e n i n a c o n c i s e form u s i n g ( i l l . 2 0 ) and ( I I I . 2 3 )

da

dfi

= < b S 1 +

P ^ ( K )

+ V^^M i^

( K )

.

( I I I . 2 1 + )

The r a d i a l distributions g,^,(r) introduced in this paragraph is t h e same as g,.(r) introduced in t h e preceding paragraph if t h e system considerec there had consisted o f tetrahedral AB, - m o l e c u l e s .

In t h e two l i m i t i n g cases K ^ o and <->•«' (ill.20) becomes

da dü

= Lim §:= <b2>

K-^0 K^O <b > ^ ^ < b 3 > ^ «b^>^l+<b > ) ^

'- 2. " — • P M ^ B ^

X T

<b >

<b >

and

da

dfl _K^«J d a K-K»

<bs

: i i i . 2 5 ) R E F E R E N C E S T O C H A P T E R III [ 1 ] Z e r n i k e F . & J . A . P r i n s , Z . P h y s . A l ( 1 9 2 7 ) 181+

[ 2 ] O r n s t e i n L . S . & F . Z e r n i k e , Proc .Kon.Ned.Acad.Wet .XVIl( I91I+) 193 [ 3 ] Wagner C . N . J . & N.C. H a i d e r , Adv.Phys . l6( 1967)21+1

[1+] K e a t i n g D . T . , J . A p p l . P h y s . 3 ^ ( 1963)923 [ 5 ] V r i e s A. de , J . C h e m . P h y s . 1+6( 1967) 161O [ 6 ] Rao K . R . , J.Chem.Phys .]+8( 1968)2395.

C H A F T E R I V - Ï S Ë N O R M A L I Z A T I O N A N D F O U R I E R T R A N S F O R M P R O C E D U R E S

. . . the meaningful transformations and findings of

sorcerers were always done in states of sobre

consciousness.

Carlos Castaneda: Tales of Power.

I N T R O D U C T I O N

In this chapter- the experimental data corrected in the way described in Chapter II are normalized and Fourier transformed. The currently applied normalization methods are briefly reviewed and their applicability in the present study is considered. In view of their limitations an alternative normalization method, named interval normalization, is proposed. According to this method part of the diffraction pattern is analysed in K-space as a result of which intramolecular distances and mean amplitudes of vibra-tion are obtained. The normalized intensity distribuvibra-tion is reported in an Appendix. The normalized intensity and the associated Fourier trans-form are compared to the corresponding quantities calculated for a hard sphere model.

N O R M A L I Z A T I O N O F D I F F R A C T I O N D A T A

The normalization of liquid diffraction data can be performed in five dif-ferent ways. Four of these normalization procedures have recently been

-5U-reviewed [1s2] and therefore they will only be mentioned. Another way to normalize diffraction data has been developed in the course of this study and it will be discussed in more detail. The normalization procedures for liquid diffraction data reviewed by Moscinski [1] are the following:

This method was originally used for normalizing X-ray diffraction data at large scattering angles. It was expected that with the available w a v e -length a vanishingly small interference contribution to t h e scattered in-tensity would exist at large scattering angles. For neutron diffraction in monatomic liquids, utilizing neutrons of about 1 A wavelength, t h e method remains valid. The intensity at the highest K-values is used as a means of normalization.

(ii) "[h^J^^OQhJÉ.^^ method_L3']

This method makes use of t h e fundamental r e l a t i o n of l i q u i d d i f f r a c t i o n a n a l y s i s

p ( g ( r ) - l ) =—^ 2Tr r

dK K i(K) sin(Kr) . ( i V . l )

As g ( r ) goes r a p i d l y to zero for r<r , where r i s some p o s i t i o n i n s i d e t h e r e p u l s i v e p a r t of the two p a r t i c l e p o t e n t i a l function of the f l u i d , one may w r i t e for r<r ,

1 -P = - ^ 2TT 0 2 . / \ 3 i n ( K r ) / _ , , „ s dK K I ( K ) . ( I V . 2 ) Kr

This expression can be used to check the overall normalization of the dif-fraction data. Equation (IV.2) is used to this end in the limit r-+0.

_ 1 2lT^

Calculations by Fredrikze [li] starting from an analytical continuation for i(K) as proposed b y Verlet [ 5 ] for a monatomic fluid show that the Krogh Moe method should b e handled with utmost care.

(iii) The Rahman and Mountain criteria \.6,7~\

Starting from (IV.2) Rahman calculates

L . L -1 y.r i dr = -1 y .r e 2.2 2 . / % sin(Kr) , /-TTT i \ dK K I ( K ) ^ dr . (IV.4) —1 u .r . . . .

The testfunction e with y being any constant vector is integrated over a sphere of radius L<r yielding an exact quantity on the left hand side (LHS) while the right hand side ( R H S ) integral gives a result depen-ding on the measured quantity i(K). Mountain studied the effect of several types of error and came to the conclusion that small values of pL most clearly show the discrepancies between RHS and LHS of (IV.1+). He proposes a test criterium for data with errors smaller than 1?, utilizing the fact that for r<r both g(r) and the derivatives of g(r) vanish. Differentiating

(IV.2) gives 2-n^Q 2 dK K i(K)cos(Kr) 2Tr pr dK K i(K)sin(Kr) = 0 , (IV.5)

where each term in (IV.5) should b e - 1 . Calculations o f these terms for a number of r values (r<r ) enable the determination o f a rms deviation

c

from - 1 , which can b e used as a figure of m e r i t .

(iv) Calibration against vanadium

Calibration against vanadium, which scatters almost entirely incoherently, w a s proposed by North et al [ 8 ] . The sample intensities are compared with the intensity o f a vanadium sample o f similar size and total scattering cross section. Hence in principle an absolute intensity distribution is obtained.