A photographic method for neutron small-angle scattering

Nuclear Instruments and Methods 186 (1981) 621-636 North-Holland Publishing Company

621

A PHOTOGRAPHIC METHOD FOR NEUTRON SMALL-ANGLE SCATTERING A.J. ALLEN * and D.K. ROSS

Department of Physics, Universityof Birmingham, P.O. Box 363, Birmingham B15 2TT, England Received 12 November 1980

A photographic method of measuring small angle neutron scattering (SANS) is described, and results obtained using a cold neutron beam and the 25 m flight tube on the DIDO reactor, AERE HarweU are presented. The experimental arrangement is similar to a conventional position-sensitive detector (p.s.d.) SANS instrument but with a gadolinium foil/film converter in place of the p.s.d. The high resolution of the converter allows all dimensions behind the secondary collimator to be scaled down by a factor 10. Analysis is given for two types of secondary collimation namely edge and point geometry. With the former the SANS can be measured by a single linear scan using a microdensitometer but for the latter a two dimensional scan is required. It is shown that use of the complete cold neutron spectrum need not affect the accuracy of results provided the form of the spectrum and scattering functions are known. A complete analysis is given for the case of Guinier behaviour and results using weighted least-squares fits to the data are presented. The photographic method used works at smaller Q values than do many p.s.d, systems and so looks for larger particle sizes. It is shown that the method is suited to treat the multiple refraction phenomenon found in some ferromagnetic alloys. While the technique is more restricted than are conventional p.s.d, systems where the shape of S(Q) is unknown or where the scattering is of very low intensity, it has the advantage of being much cheaper and of allowing the simultaneous measurement of a large number of samples. Further, by removing the secondary collimation a simple neutron radiograph of the sample arrangement is obtainable.

1. Introduction Neutron small angle scattering (SANS) is now widely used for the study of small inhomogeneities in materials o f technological and biological interest [ 1]. In many such studies however, a large number o f samples must be used or a large number of positions along a component (e.g. a turbine blade) must be studied. The disadvantage of using conventional position-sensitive detectors (p.s.d.) is that a given facility can carry out only one measurement at a time. The number of measurements and hence the extent o f the study may be seriously constrained by the available neutron time. A detector system allowing more rapid and simultaneous assessment of a number of small angle scattering samples is therefore of interest The experimental system used here is basically similar to a conventional p.s.d. SANS spectrometer but a gadolinium foil/film converter replaces the p.s.d. The high spatial resolution of the foil-film combination for neutron detection allows the geometry after the secondary collimation to be scaled down * Present address: Materials Physics Division AERE Harwell, Oxfordshixe OX11 ORA England. 0 0 2 9 - 5 5 4 X / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © North-Holland

considerably. The neutron scatter-profile is imaged on the radiograph which is then scanned using a microdensitometer. The relation between film density and neutron exposure, and the nature o f the separate contributions to the spatial resolution for various possible combinations are now well established [2]. It is also possible to make quite accurate physical measurements from a microdensitometer trace taken across an image on a neutron radiograph. In the present work a 25 #m thick gadolinium foil has been used in contact with photographic films (Ilford SP352-1ine and IIA) X-ray films (Kodak types C and R, Agfa D2) and electron-image film (Kodak). The neutrons pass through the film and are absorbed in gadolinium; some of the resulting low energy internal conversion electrons pass back into the film where they develop the grains. The inherent spatial resolution of the foil/film converter is largely determined by the range of the conversion electrons in the emulsion, and is less than 100 pxn. The optical density o f the exposed film is linear with respect to the neutron exposure [2]. In contrast with scintillation screen systems the total exposure does not depend on the exposure time (no reciprocity failure) which means that long exposures are possible.

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A.J. Allen, D.K. Ross/Neutron small-angle scattering

Obvious advantages of the Gd foil/film converter over p.s.d, systems conventionally used in SANS facilities are its relative cheapness and simplicity. Another major advantage lies in its good spatial resolution. The spatial resolutions for Gd foil/film and p.s.d, systems are typically ~100/~m and 0.01 m respectively, so there are ~106 resolution elements in a film area of 10 -2 m 2 compared with 4 × 103 in a 64 X 64 grid p.s.d. Given that the reactor beam emerges from a collimator ~0.01 m wide in the p.s.d, case, for optimum resolution, the sample has to be placed midway between reactor and detector. In contrast for the Gd: system the sample-detector distance may be as little as 0.01 of the total flight path. This means that for a given flight path and the same beam collimation from the reactor, the angular resolution will be better by a factor 2 in the film system. Further, the secondary collimation may be of millimeter dimensions and cut in a gadolinium foil mask with the sample placed behind and in contact with it. Thus many such point (or edge) geometry combinations may be set up simultaneously and recorded in the same radiograph. Against these advantages there is the disadvantage that to get a discernible density on the developed film it is necessary to achieve a neutron exposure of >108 neutrons cm -2. For many practical facilities using filters to produce the cold neutron beam this means it is not possible to monochromate the beam if exposure times are to be kept within reasonable limits. It is therefore necessary to convolute the scattering function with the cold neutron spectrum before fitting to experimental data. It is also necessary to use as low a cut-off wavelength as possible to maximise the neutron flux. Provided a given form of the scattering function S(Q) may be assumed it should be possible to gain much of the information from SANS that conventional p.s.d, systems can give-i.e, particle size distribution and shape, sometimes particle spacing, ordering and internal structure. However measurements are generally confined to smaller values of Q than in p.s.d, systems because: 1) the scattering from smaller particles at larger Q is frequently too small to show up on film above the background ; 2) the background cannot be as accurately subtracted out as in the conventional method as will be explained in a later section. In compensation the photographic method described here has sufficient collimation to study small Q values (0.006 A -1) not often accessible with p.s.d, systems•

An exception is of course the D11 instrument on the High Flux Reactor, ILL, Grenoble [3]. If only the particle size is required and the SANS intensity is sufficiently strong, the Gd foil/film converter has one more advantage in that an "edge" geometry may be used. In this case a sharp gadolinium edge which cuts the neutron beam at right angles is "viewed" through the sample which is in contact with it. Much shorter exposures will then suffice for particle size determinations alone, but the convolution of the SANS signal with the dominant straightthrough beam makes determination of further parameters extremely difficult. 2. Theoretical discussion

2.1. The scattering profile The experimental arrangement for SANS measurements using a photographic foil/film converter is shown in fig. 1. In the normal case of SANS the theoretical form of the density profile on the film may be formally obtained for an athitrary isotropic small angle scattering function (1/4~r)acohS(Q)per scattering atom where Q is the change in neutron wave vector k, i.e. Q = 2k sin ~0 = (47r/X) sin ~0, and 0 is the angle of scatter. The total coherent scattering cross-section per atom may be written 7r

°c°h f S ( Q ) 47r

•

27rsin0d0

x

= STOTOco

h ,

(1)

0

where the integration is over all possible directions of the scattered neutron. At a given neutron wavelength, X, the total removal cross-section, Orx may be written x

= Oax + Oi.c + S T o T O c o h ,

(2a)

where o ax= absorption cross section and Oinc = incoherent scattering cross section. If the sample is of thickness r and has mean atomic number density N, then the small angle scattering intensity per unit area of the second collimator A2 in unit solid angle is given, for incident flux qS(X), by

I(Q) = 4)(X) e -Ngxrr Noc°hr S(Q) 41r

(3)

provided that the probability of small angle scattering is small, i.e. NrSXoTacoh ~ 1 SO that multiple small

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A.J. Allen, D.K. Ross/Neutron small-angle scattering

Air- filled cavity (old rotor t a n k ) ~ x

'25m Blockhouse (air filled ) sample

Helium Filled Flight Tube

Cassette containing foil/film converter

Beam Switch

]

"x

(At Diaphragms at each end )

Reactor Face

I---'-.... ....4-j

GdmA2~k on sample plate

[ Be-

Bi filter

I

Reactor

Shell

I I I

Colli m at or-Sample Distance L

Fig. 1. Experimental arrangement for SANS studies using the photographic method and the 6H beam, DIDO reactor, AERE Harwell.

,.ngle scattering can be neglected. The direction of the scattered neutron here corresponds to a wavevector transfer of Q. It is convenient to work with the macroscopic cross-sections Zra =NoXr etc. so that, if EXAS is the total macroscopic SANS cross section at wavelength X, then;

We now assume that the scattering is confined to small angles. From eqs. (3) and (4) the contribution of a point ( x , y ) to the scattered intensity per unit area of film from an area element 6A2 of the second collimator is given, for wavelength X, by 6Is(x, y, X) where

]~rh = ~ak + ~inc + ~ A S

6Is(x,y, X)

(2b)

q~(X)i=aA2 e -xrxrd ~s~As (0) =

and the term Noconr/(47r) S(Q) in eq. (3) is replaced by dZXAS(0)/d~2, the macroscopic differential small angle scattering cross section. For widely separated identical particles randomly spaced and orientated throughout the sample it may be shown that [4] : ~AS =

npT(NpTbpT - Nobo) 2 VpT'X2C'd,

(4)

where N p T , N O =mean atomic number densities inside and outside the particles; bpT , bo = mean neutron scattering lengths inside and outside the particles; VpT=the particle volume; C o = m e a n chord length of each particle (for spheres of radius R, Ca = 3R/2); npT = number density of the scattering particles in the sample.

-

d,.Q,

r.

(5)

Here l is the sample-film distance and x, y are measured in the plane of the film relative to where the unscattered beam cuts this plane. Using the small angle approximation we may substitute

0 ~ / ( x 2 +y2)/l,

(6)

and integration of eq. (5) over all x and y should give the same ~ A S as in eq. (4), provided large angle contributions to the scattering are negligible. For non-monochromatic beams it is necessary to integrate the contribution 8Is(x,y, X) over the incident neutron wavelength distribution f(X) where q~(X) = ~bof(X), f o f(X) dX = 1 and ~0 is the total incident flux. This gives:

624

A.J. Allen, D.K. Ross /Neutron small-angle scattering

Ms(X, y ) = 8A2 12 " 7 " ~o ? 0

X (-0) f(X) dX. e-ErXr d ~-§AS da (7)

In eq. (7) the area element 8A2 was taken to be at x = y = 0. In general it is at (x',y') where we make the transformation x - x - x ' etc. in eq. (5), (6) and (7), and we must sum the contributions 6Is(X x', y - y ' ) over all points x', y ' within the area of the collimator A2 to get the resulting scattered intensity at the point (x,y) on the film:

I's(x'Y)

=f / 8Is(x -SA-Ex"Y- y ' ) dx' dy'.

(8)

Finally we convolute I's(X,y) with an unsharpness function c(x,y) to allow for the finite beam collimation, film resolution, scattering in the film base etc. The result for the observed final scattered intensity I's'(X,'y) is thus:

approximation at all Q however. From eqs. (4) and (10) we may write for eq. (5) for all particle shapes:

8Is(x,Y,

X) = q~(X)SA2 e ErXr ~ A S T ( a[~] \2n /

× exp[ -~a~,(x 2 +y2)] •

We now work through to the final scattered intensity as before but the Gaussian approximation allows a number of simplifications. For the wavelength integration we note in eq. (2b) that 12ax generally has a " l / v " dependence, i.e. )Ea~cx X while 22SXAS(x X2 from eq. (4). However, provided Z,aXr and Z~AST are ~1 we may ignore the X dependence of the term e -Erxr. Also the term Z~AS ' a~, is independent of wavelength, which leaves the Gaussian term, the argument of which may be expressed as ( K2/X2) where K is a constant. Hence the integrand of eq. (7) is effectively: f(X)exp(K2/X2)'-f(X).

/s(X,Z) : "

f

f

--oo

_ ~

¢i(x . . . . .,y ) c(.-, - x",y - y")

× dy".

(9)

Usually the integrals above must be determined numerically and many of the functions involved are numerically determined themselves. In much of the present work however a number of simplifying assumptions have been made. Most importantly the Guinier approximation has been assumed for the scattering function so that [4,5]:

(l 1)

K 2f(X) K4f(X) X2 ~- 2X4 ,

(12)

assuming that K2/X 2 is small for all X values where f(X) is significant, i.e. assuming the Guinier approximation holds for the whole wavelength distribution. On integrating over X this series becomes l-K2

~5

+)-

~-""-exp(-KE/XeE),

(13)

where Xe2 = 1/(1/X 2) and (x) denotes a mean of x averaged over the wavelength distribution. As might have been expected Xe is the root mean inverse square wavelength of the distribution. Hence:

d ~ A S (0)

d~2 X V~,T

- npT(NpTbpT -- N°b°)Z exp(-Q2R~/3)

8Is(X, y) - dPo"6A2 (1 0a)

= rtpT(NpTbp T __ nobo)2 V~T exp[gax( 1 2 x 2 +yE)] , (lOb)

where a [ = 87r2R~/3XEl2 and Rg = the radius of gyration of each particle. The Guinier approximation holds for widely separated, randomly oriented and spaced particles of any shape and mean dimension 2R at values of Q such that QR < 1 (or 2 for spherical particles) [4,5]. For spherical particles where R~ = 3R2/5 integration of eq. (10a) over 47r or (10b) over all x and y yields a value of Y'~AS which is (10/9) times that in eq. (4). This factor is simply a result of using the Guinier

" e -Err ZSAS T " ~-~,

× exp[-~a2(x z +yE)] ,

(14)

where a 2 = 8n2R~/3X~l2 and ~SAS, 52r are the crosssections at wavelength Xe. For point geometry measurements, the integration over A2 is over the pinhole area at the second collimator. Provided the pinhole radius "r" is such that the radial variance, o~ (=~r 2) is small with respect to that of the Guinier Gaussian we may use the central limit theorem in the integration [6]. We may again use this theorem to deal with the unsharpness function, provided that the radial variance o 2c of this function is also small with respect to that of the Guinier Gaus-

625

A.J. Allen, D.K. Ross/Neutron small-angle scattering Edge geometry."

sian. The final scattered intensity is then: I s (x, y ) = qSo • 7rr2 • e-ZCrr

]~SAS r • k=l\

" exp[-~-b2(x = +Y2)I ,

~,.

/

(16b)

(lSa) 2

where 1/b = = 1/a 2 + @ + o2 . Since the above expression is completely analytic it is easy to go a stage further and allow for multiple scattering*, dropping our original assumption that £SAS r < 1 to get [7]: Point geometry."

' e x p [ - ~ b ~ ( x= +YZ)I ,

(16a)

where 1

k

b~, - - a" +o~+Oc2 and k is the number o f scatters. For edge geometry measurements where we assume the gadolinium edge at the second collimator. A= to be the y ' = 0 axis, eq. (14) must be integrated for all x ' values (_oo to +oo) and for y ' values from _oo to zero : 0

oo

a 2

ff

2~- exp [ - l a 2 [(x - x ' ) 2 + (y - y')2 II dx'

Xdy' a

- X/(2n)

a

f

exp(-~a2 y2) dY,

where Y = y - y ' . Hence we get: Is(X, Y) = 4~0" e -:err • F--SASr " E R F ( a y ) ,

(17)

where E R F ( ) denotes an error function. The central limit theorem may be applied to i/lclude the unsharpness function as before to give ls'(X, Y) = 4~o • e -y-rr • ESAS r q E R F ( b y ) ,

(15b)

where 1/b~ = k/a = + % . The above expressions for the scattered intensity hold in the diffraction limit of SANS, i.e. where the difference in neutron scattering length densities within and outside the particles and the particle size itself are such that the phase difference between neutron De Broglie waves passing through and around a particle ¢ ~ 1. In the opposite limit o f 4~>> 1 the phenomenon o f multiple refraction occurs [8]; here refraction is identical to the phenomenon o f primary extinction encountered sometimes in Bragg scattering. Whereas in the diffraction limit of SANS the above expressions for the scattered intensities are small components superimposed on the unscattered beam, in the refraction limit (~>> 1) the incident beam suffers a series o f refractions which broaden it into a single Gaussian shape as it passes through the sample. The refractions occur at boundaries between regions o f different neutron refractive index nR in the sample medium but the statistical accumulation o f deviations results in a Gaussian profile if the direction of incidence at such a boundary is assumed to be random. In practice the multiple refraction limit occurs when the scattering particles have dimensions ~ 10/~m while the SANS diffraction limit arises from particles with dimensions <~1000 3, [9]. A typical example o f multiple refraction is that which occurs due to the fluctuation of magnetisation direction between different domains in an unmagnetised ferromagnet [5,9]. The neutron beam is small angle scattered as it encounters successive domain boundaries because the neutron magnetic scattering length and hence the refractive index o f the medium depends on the orientation o f the magnetic induction B in the domain relative to the neutron direction. Hence the change in magnetization direction in crossing from one domain to another causes a change in refractive index at the boundary between the two. The refractive index n~ at wavelength X is given b y

[5]

where now l i b 2 = 1/a z + 02 . On allowing for multiple scattering we get

X2 X2Nb m n B nR -- 1 = + , +r E.

* The following eqs. (16) only hold if the wavelength spectrum is well peaked as it is here (see fig. 3).

where N and b are the mean atomic number density and nuclear scattering length in the medium, mn and

(18)

A.J. Allen, D.K. Ross/Neutron small-angle scattering

626

En are the magnetic moment and kinetic energy (cq/X 2) of the neutron. The maximum angle of deviation 2x0 of the beam at one refraction is 2ffcx given by: sin ~eh = (2mnB/En) 1/2 ,

(19)

where ffcx is the critical glancing angle of the refraction when the change in refractive index between the two domains is a maximum, i.e. when the magnetizations are oppositely directed. In fact A0 only approaches t~c if the incident beam direction makes a very small angle with the domain boundary, and if random orientation of the domain boundaries relative to the incident beam direction is assumed then [5] : A0XR"~ *c

2.2. The response o f the film [2]

Consider a simple film emulsion of thickness x containing grains of constant cross-section a. After exposure the processed film is scanned with a microdensitometer which has an incident light beam intensity Bo. For a mean number n of developed grains per unit volume of emulsion the transmitted light intensity B T is given (regardless of how the developed grains are distributed through the emulsion) by BT = Bo exp(-ncoc) ,

and the conventional film density D is defined by the equation D = loglo(Bo/BT) = 0.43nax .

and A0XRcx AnXRcc X2 '

(22)

(23)

where A0~ is the mean angle of deviation of the beam at a single refraction. Since the whole incident beam is affected we expect ZSA s to be independent of X but it may be shown that the differential crosssection is given by [5] :

For previously unexposed Film with such a simple emulsion let v~ be the average number of grains rendered developable by the absorption of one neutron in the Gd foil. As the neutron exposure E increases fewer new grains are rendered developable because an increasing number have been made developable already. It may be shown that

d ]~AS (0)

D = 0.43Nax(1

da

A

[

,/30 z

- P(A0~) 2 e x P ~ 2 p ( ~ R ) 2 ] '

(20)

where A and t3 are constants and P is the mean number of refractions suffered by the beam in the sample. The width of this Gaussian is proportional to X2 whereas that of the Guinier Gaussian coX. The width does not explicitly depend on particle size but for a ferromagnet we may write P = (r/6),

(21 )

where 6 = the mean domain dimension. Hence the width or-6-1/2 as compared to 6-1 in the Guinier case. For unmagnetised ferromagnets 13is close to unity. The integration over the wavelength distribution proceeds as before but is complicated by the X'4 dependence of the term in front of the exponential [(A0XR)-2]. If the same assumptions are made the result is still a Gaussian but the effective wavelength in the argument does not correspond to (1/~,2) -1/2 but to ((1/Xs)/(1/X4))-va. The central limit theorem may be used for the spatial integrations subject to the same assumptions given previously, and an error function may be obtained for the edge geometry. The fit is to a single Gaussian profile and the broadening relative to the incident beam profile gives an estimate of the magnetic domain size.

-

e-~E/Nx),

(24)

where N is the grain number density in the film. Hence the density rises to a saturation value of 0.43Nax while for small E the density response is linear: D = 0.43NaE. In practice the film will contain a number density distribution of grain cross-sections N(c0 such that f o N(a) da = N and the mean number of grains of a certain size rendered developable in unexposed film by a neutron will also have a distribution ~ ( a ) such that f o ~ ( a ) dc~ = N. Eq. (24) becomes D = 0.43 /

N(o0 ax[1 - e -m(c')E/N(c')x] d a .

(25)

o

If we denote the mean cross-section as ~-then whatever the actual distribution functions m(a) and N ( a ) the density saturates at D = 0.43N-ax and is linear for small E where D = 0.43~--aE.

(26)

Otherwise we see from eq. (25) that eq. (24) is rather an over-simplification, and in practice the film manufacturers design their emulsions to extend the linear region of the response curve to as high an E-value as possible. The complete curve for ILFORD LINE film (formerly SP352), as determined experimentally on the DIDO 6H beam at HarweU is shown in fig. 2. This

A.J. Allen, D.K. Ross

/Neutron small-angle scattering

627

24

\

2.1

Orclin Sa|uration 1'8 1.5

c 0.9 o

~, Linear Response

0-6

o 3~,0

1

I

5

10

I

I

I

15

20

25

Neutron Exposure/(xlOaneutrons crn2) Fig. 2. Photographic density response curve of Ilford LINE film with respect to neutron exposure. film is linear in response to neutron exposure for densities up to 2.0 when used with a 25 /ira Gd foil in a foil/film converter.

3. Experimental discussion We return to fig. 1 to describe the layout of the 6H beam on DIDO reactor at AERE Harwell as set up to take SANS exposures. After the 50 cm diameter Cd collimator there is a 1 m air-filled cavity before the helium-filled flight tube. The flight tube passes out of the reactor building to a "25 m blockhouse" in which the aluminium sample holder plate and film cassette are mounted on an optical bench. The distance from collimator to sample plate is 24.5 m while the separation between sample plate and film may be usefully varied in the range 2 0 - 5 0 cm. The Gd mask and sample are attached to the sample plate while the 25 /xrn Gd converter foil is permanently attached to an aluminium sheet and is pressed in contact with the back of the film inside the cassette. Good contact between foil and film is essential to get a uniform response over the area of the film. The cold neutron spectrum of the incident flux used for the photographic SANS exposures is shown in fig. 3. It was measured using a cold neutron disc chopper and a aHe gas detector (which was black to

cold neutrons when placed end-on in the neutron beam), with the whole 6H flight tube (L in fig. 1) as the distance of flight. Neutrons of k < 4 A are removed by the liquid N2 cooled Be-Bi filters since Xo = 4 A is the beryllium Bragg edge. In theory the transmitted spectrum should reflect the low energy end o f the Maxwellian distribution of neutron velocities in the reactor and should therefore show a k -s dependence. In fact the spectrum is affected by the neutrons passing through components from the reactor core out to the filters themselves, and this results in: 1)peaks and dips in the spectrum-particularly due to the variation with k of the total neutron crosssection for the bismuth single crystal; 2) a stronger X dependence because of the presence of strong " l / v " absorbers (the X dependence here is closer to 3,-6). The root mean inverse square wavelength was found numerically for the actual spectrum and is 4.69 A. This value of 3,e was used in eq. (14) etc. for SANS diffraction while the effective wavelength used for the width of the Gaussian in eq. (20) for SANS refraction is 4.40 A. With the collimation above the cold neutron flux before the sample plate is ~ 2 X 10 s neutrons cm -2 s -1 . The 1/8" thick aluminium sample plate and the 1/16" aluminium front face on the cassette atten-

A.J. Allen, D.K. Ross/Neutron small-angle scattering

628

41

46

3.96

-6 U

b [3

r

r'h

b~ c" (9 cO 0

57

.-$ C ¢-

Backgr°Ind

C3 AZ C)

3.0 11.1

0"

I-

.3

4

I

h

i

I

I

I

5

6

7

8

9

10

Wavelength in A---,-. Fig. 3. Cold neutron wavelength spectrum used in photographic SANS measurements, 6H beam, DIDO reactor, AERE tlarwell.

uate the beam by ~6%. About 2% of the beam is isotropically scattered in the aluminium-here including the 1 mm thick diaphragms at the ends of the flight tube, and there is further attenuation and isotropic scattering both in the sample and in the air. Fortunately the effect of isotropic scattering is simply to add a virtually flat background to the scattering profile. However the gelatine base of the film itself also gives rise to scattering and this does contribute to the unsharpness function discussed earlier. For this reason thin-based films are of interest in these studies (e.g. Ilford IL4). A number of films have been found useful in SANS measurements. The main criteria for selecting a film, apart from linearity of response, are the need for a reasonable exposure time and maximum signalto-noise ratio in the density profile. The signal-tonoise ratio is increased by increasing the number of grains that appear in the microdensitometer scanning window at any one time; the digitised density values are then based on better statistics. This requirement is met by having a finer grain size, which unfortunately means a slower film. The most readily available film with the best compromise between these criteria was found to be Ilford LINE film (formerly SP352). For

the point geometry with a 1 mm diameter pinhole the exposure time for the main SANS exposures was initially about 18 h, although later work has shown that it is of value to run longer exposures of up to 72 h For the edge geometry the exposure times were about 70 min. With the point geometry the pinhole diameter must be small enough to satisfy the central limit theorem in eq. (15a) and (15b), and must be large enough to give a reasonable scattered intensity. The experiment set-up with this geometry, is compared in table 1 for two sample-film separations, with two conventional SANS instruments-the PLUTO SANS spectrometer [10] at AERE Harwell and D11 at ILL Grenoble [3]. The table shows that although a 50 cm sample-film distance was used for the earlier results, a 20 cm configuration, which just satisfies the central limit theorem for many samples gives a much better intensity at the film, and this was used for the later point geometry measurements. The resolution in wavevector, (AQ/Q) may be quoted for a detector system at a given Q when a monochromatic neutron beam is used. Since the whole cold neutron spectrum is used for the foil/film converter system, (AQ/Q) depends on the scattering

A.J. Allen, D.K. Ross /Neutron small-angle scattering

629

Table 1 A comparison between conventional and foil/film small angle scattering systems PLUTO SANS instrument [10l

DIDO 6tt foil/film SANS system (point geom)

D l l SANS instrument ILL Grenoble [31

Detector:

128 × 128 grid BF 3 LETI detector

25/xm Gd foil in contact with photo-emulsion 50 um microdensitometer window (or <100 ~lrl film spatial resolution)

64 X 64 grid BF3 detector

Detector Mesh size:

0.5 cm

Collimator: Collimator sample distance:

2 cm diameter 2m

5 cm diameter 24.5 m

Collimation and distances variable ( 2 - 4 0 m)

Flux at second collimator: Second collimator:

3.5 X 104 cm -2 s-1 at 6 A 2 cm diameter

2 × l0 s cm -2 s-1 1 mm diameter pinhole

4 X 106 ncm -2 s-1 at 6 A for same resolution as PLUTO

Sample-detector distance:

2.1m

50 cm

20 cm

1-40 m

Normalised intensity at detector:

1.0 at 6 A

0.22

1.36

ll4at6A (same resolution as PLUTO)

Angular resolution LxO a): First coUimator (rad): Second coUimator (tad): Detector resolution (rad): Resultant Lx0 (tad):

0.007 0.007 0.002 0.01

0.0014 0.0014 0.00008 0.002

0.0014 0.0035 0.0002 0.004

Total uncertainty in Q, LxQ:

0.01 A -1 at 6 A

0.0025 A -1 0.005 A -1 (if calculated at 4.7 A)

1.0 cm

Variable

0.01A -1 a t 6 A with above flux. For lower fluxes can go down to 0.001 A -1

a) The angular resolution has been taken as twice the r.m.s, angular deviation of the beam. With these wavelength assumptions the PLUTO instrument generally studies the Q-range 0.02-0.1 A -1 and the 6H system is frequently working in the range 0 . 0 0 6 0.02 A -1 . (The lower limit is somewhat less at 50 cm). D l l of course more than covers both these ranges. profile as well as Q. In table 1 h o w e v e r for c o m p a r i son p u r p o s e s an effective w a v e l e n g t h o f 4.7 A h a s b e e n a s s u m e d to give the u n c e r t a i n t y in w a v e v e c t o r , A Q for t h e foil/film s y s t e m f r o m the angular resolut i o n A0. The t a b l e s h o w s t h a t t h e l o n g 6 H flight t u b e leads to b e t t e r r e s o l u t i o n t h a n is possible w i t h t h e restricted p a t h l e n g t h o f t h e c o n v e n t i o n a l P L U T O s p e c t r o m e t e r . Because t h e b e t t e r r e s o l u t i o n m a k e s t h e l o w e r l i m i t o f t h e Q-range smaller w i t h t h e foilf'flm s y s t e m while t h e s c a t t e r i n g f r o m small particles is f r e q u e n t l y i n s u f f i c i e n t to show u p o n p h o t o g r a p h i c film, t h e foil-film t e c h n i q u e t e n d s to l o o k for scattering f r o m larger particles t h a n does t h e P L U T O instrument. The digitised d a t a f r o m a film-scan differs in a n u m b e r o f ways f r o m t h a t p r o d u c e d b y c o n v e n t i o n a l

p.s.d, s y s t e m s . First the a c t u a l d e n s i t y values are not b a s e d o n a real a c c u m u l a t e d n e u t r o n c o u n t b u t vary d e p e n d i n g o n w h a t is c o n s i d e r e d zero d e n s i t y i.e. t h e d e n s i t y o f clear film. The d e n s i t y o f clear film will differ slightly f r o m one i n d i v i d u a l e m u l s i o n to a n o t h e r , a n d so a zero c o r r e c t i o n or flat b a c k g r o u n d m u s t be s u b t r a c t e d o u t for all d e n s i t y p r o f i l e s ; - t h i s also takes a c c o u n t o f t h e flat i s o t r o p i c s c a t t e r i n g b a c k g r o u n d . S e c o n d l y t h e r e s p o n s e curves for t w o e m u l s i o n s o f n o m i n a l l y t h e same film, a l t h o u g h b o t h linear, will n o t b e a b s o l u t e l y identical. A p r o b l e m t h e r e f o r e arises w i t h t h e p o i n t g e o m e t r y w h e n a " n o s a m p l e " e x p o s u r e is to b e n o r m a l i s e d so t h a t it c o m pares d i r e c t l y w i t h a SANS e x p o s u r e , p a r t i c u l a r l y since it has p r o v e d difficult to get t w o d i f f e r e n t pinh o l e s sufficiently i d e n t i c a l to have c o r r e s p o n d i n g

630

A.J. Allen, D.K. Ross /Neutron small,angle scattering

SANS and "no-sample" profiles on the same exposure. As long as the SANS is not so strong that it affects the centre of the profile the general procedure in the current work has been to take both long and short exposures with and without the sample present. The long exposures blow up the region away from the straight-through peak to show up the SANS and provide the raw data for analysis. The short exposures do not exhibit saturation at the centre of the straightthrough beam and hence the four exposures may be used together to normalise the "no-sample" data with respect to the SANS data. Finally because the density data comes from a photographic film the "noise" is not based on normal statistics. It might be expected from the previous analysis that the noise in the density data for a simple emulsion, OD ccx/D where D is the density. This is not the case because of the complicated grain size distribution in real films and more important because of the so-called "quantum mottle" * effect [2] encountered in foil-film converters which both increases the noise observed and complicates its dependence on the density. For these measurements the noise has been deduced empirically for the individual data points by a moving average technique. The estimates for a~) so produced have then been used to weight the data in the fitting process. The observed data noise also depends on the area of the microdensitometer scanning window, and for the present point geometry scans a square 50 X 50 /ml 2 window has been used. After scanning the two-dimensional array of data is radially averaged. In the edge geometry scans the window measures 1/60 mm in the direction of the scan and 1 mm perpendicular to it (the scan direction being across the edge right angles). The computer codes for fitting the theoretical functions to the data were originally based on Hatwell subroutine VA05A [11] but for the most recent data they are based on a more modern subroutine from the N.P.L. Library [12]. These contour-fitting routines work towards optimum values of several function parameters ci by successive iterations from initial estimates using a quasi-Newton method. For the point geometry the first few data points in the straight through beam are masked out, and the normalised "no-sample" data NSG(r) is included in the fitting function DF(r) which physically should be given by • "Quantum mottle" arises because several adjacent grains in the film are rendered developable by the same electron produced in the foil.

DF(r)

+

= C 1

C2

+ NSG(r)

e - c 3 r ( ~ ( b ~ R 2 ] (c3r)k (k= 1\ ~ / ~ . r

J - b 2 2 )} exp(--U ,(27)

where r = radius from centre of the profile and 2 (1/b~) = (k/c~) + %. In fact for the majority of samples this function is ill-conditioned in that c2, c3, c4 cannot be easily discerned as 3 independent parameters by the fitting routines working with real data. It is necessary to rewrite the function as: DF(r) = c, + NSG(r) + c'2 exp t ~ - NS (Y3r)k~

)

.exp(_b~r2/2) l

where (1/b2a) = (1/c4z) + o2. Comparing eqs. (27) and (28) the factor e-C3r(c3r)'(b~RZ/2) has been absorbed into c2 in the second equation. In eq. (28) Ys [the same as c3 in eq. (27)] merely determines the relative importance of the multiple scattered components. Since the contribution from these is frequently small it is necessary to magnify the effects of changes in this parameter during the fitting. Hence Y3 is determined from the actual fitted parameter ca by the equation: r s = ca ' SC3 - (SC3 - 1),

(29)

where SC3 may be up to about 30 for weakly scattering samples. Changes in Ya are then magnified SC3 times the corresponding change in c3. For the edge geometry the experimental "no sample" NSG(r) is replaced by a theoretical convolution function of the circular collimator with the Gd edge, FC(y) which is normalised to unity. The width parameter for F c ( y ) may be determined accurately by fitting the appropriate function to the "no-sample" data. There is no problem of ill-conditioned function h e r e - d u e to the way in which the usually large straight-through component is included. The function DF(y) is given by NS

DF(y) = cl + c2 (FC(y)

+ ~ (c3~)~ k:l

k~-~--"ERF[bk(y -

Cs)] (30)

where Cs is the position of the centre of the error function profile.

}

A.J. Allen, D.K. Ross /Neutron small-angle scattering A single particle size has been assumed but the size given by the fit should be (R~)/(R~) [4] if there is in fact any size distribution. This is very heavily weighted in favour of large particles. The value of ESAS then quoted assumes the whole volume fraction o f scattering particles is at the size given. However in the point geometry function different parameters control the weight attached to the single-and multip l e - s c a t t e r e d components. It is necessary therefore to check the fits to ensure that the fitting routine has not selected sizes out of the distribution weighting them appropriately rather than genuinely allowing for multiple SANS using a single particle size. This problem can occur when the SANS is particularly weak and is best resolved by cutting out the multiple scattered components of eq. (28). For ferromagnetic alloys where the multiple refraction phenomenon occurs and the straight through beam itself is broadened into a Gaussian, there is no "no sample" contribution and only a flat background to consider. Here a single Gaussian width is assumed and the "multiple scattering" terms are ignored. The object in this case has been simply to measure the beam broadening caused by the refraction.

4. Some experimental results In previous sections we have outlined the advantages and problems in using a photographic detection system in SANS and how such studies differ in range, resolution etc. from conventional p.s.d. SANS work. In presenting results for a new method o f this kind it

63 l

is necessary to identify two clear objectives. These are : 1) to show the consistency and reproducibility o f the method ; 2) to show how studies using the new method compare with similar studies carried out by other means, and that the results obtained are based on the real physics o f the systems under study. Results are presented here (see table 2) for some nickel- and ffon-based a l l o y s - t h e latter showing multiple magnetic refraction. Included in the nickel based samples were 3 over-aged high pressure turbine blades (T1-T3). Some o f these samples were also studied using the conventional p.s.d. SANS spectrometer on PLUTO reactor, AERE Harwell.

4.1. S A N S diffraction results A number o f edge geometry exposures were made for each of the nimonic and NiCrMo alloys. Films used were Ilford LINE and IL4 films, A G F A D2, Kodak types C and R and Electron Image film. Exposure times were 15 min for Kodak Type C (a fast doublesided emulsion), 70 min for LINE, D2, Type R films, 2¼ h for I I A and ~ 1 2 h for electron image film. The D2 film really required much longer exposures while Kodak Type R gave results inconsistent with the others when the SANS intensity was small. This film also showed the best linear density response to neutron exposure when used in a Gd foil/film converter. For each o f the N75 and NiCrMo alloys the Type C, LINE, IL4 and Electron Image films gave results which agreed to within the uncertainties quoted by the fitting routine. The results for the radius of gyra-

Table 2 Results from SANS measurements of radius of gyration, Rg and cross-section ZSA S Sample and geometry

Film-foil converter method on 6H beam

Pluto SANS spectrometer

Thickness

Film-sample distance

Rg

ESA S

Thickness

Rg

ESAS

N75 Point N75 Edge

5 mm 10

20 cm 50

511 A 531

0.2 0.3 cm -1

3 mm

124 A

0.03 cm -1

N90 Point

5 mm

20 cm

(790 A)

Wide variations

3 mm

83A

0.03 cm -1

5 mm 10

20 cm 50

419 A 424

0.20 cm -1

2.5 2.5 2.5

20 cm 20 20

378 A 408 358

0.7-0.9 cm -1

NiCrMo Point NiCrMo Edge T 1 Point T2 Point T3 Point

3 mm

148

0.035 cm -1

632

A.J. Allen, D.K. Ross /Neutron small-angle scattering

FILM

DENSITY -

--0

35 •

• •

"NO SAS

It

AAIb

&

data

htfed Sample

,L •

- -0.3

SAMPLE" data

film

tP

..F

Function distance

- 50cm

0"1

•

I f & mm

1.0mm

•

•

•

A &

I

I

0,5mm

Distance

0

From

0~mm

I

10am

l'/aTl~

Centre of Profite

l:ig. 4. 10 mm thick N75 alloy sample, edge geometry: density profile for SANS data with normalised "no sample" data for comparison Rg = (531 + 29) A, VSAS = 0.25 cm -1 .

lion Rg for these two alloys were 530 A and 420 A for N75 and NiCrMo respectively, see fig. 4. In each case the quoted uncertainties were -+20 40 A while the actual consistency of individual results was slightly better than this. Tile N90 sample did not show any measurable SANS in the edge geometry exposures. Using a point geometry, a different sample film distance (20 cm instead of 50 c a ) and sample thickness (5 mm instead of 10 nrm), the N75 alloy yielded a mean Rg value (51 l A) in rough agreement with the edge geometry results and again within the quoted uncertainties of the fitting routine. The edge and point geonretry exposures gave the macroscopic scattering cross-section XZSAs values in the range 0.2 0.3 cm -1. The NiCrMo alloy had a smaller ESA s (~0.20 cm -1) but tile point geometry result for Rg (419 A) again agreed with values obtained using the edge geometry. Sonre discernible SANS was seen for tire N90 sample using point geometry but the width of tile Guinier function seemed dangerously narrow for tile various assumptions made earlier to held over the range of measurements. Still, statistically stable results were obtained for Rg = 800 A. Values given lot the mean particle size at the middle of the leading

edges of tile over-aged turbine blades T~, T2, T 3 using point geometry were certainly consistent to within the error margins quoted by the fits [i.e. ( 3 8 0 -+ 3 0 ) A ] . Here the SANS was considerably stronger than in the un-aged samples and ESA s ~ 0.7 0.9 cm -1, see fig. 5. It was hoped originally to present some results/'or graphite samples but this was not possible due to tile high Bragg edge (6.7 A) of graphite. Double Bragg scattering would have masked the true SANS intensity. However, it should be pointed out that when the scattering profiles front the same graphite sample using the same geometry but different films (Type C and LINE) and exposure times are normalised to allow for tile different exposures, fihn speeds, etc., tile curves are indistinguisable to within the scatter caused by fihn noise. This says much about the reproducibility of the scatter profiles using photographic fihns. Tile composition of the various alloys studied is given in table 3. In the nickel based alloys the scattering cross-section of nickel is ~18 b as compared to 2 4 b for m o s t o f tile other constituents [15]. Thus the neutron scattering length density may be different inside a precipitate phase from that in the sur-

A.J. Allen, D.K. Ross /Neutron small-angle scattering

A1

r125 1-0

• •

633

SAS data "No

sampl.e" background

0.75

,.,~-:[rnage of Pinhole 0.5 ¢,E3 "• • & •&

E 0.25

lb •

± _- ~ , . A , ~ A ~

iF_

0

0-5ram

I

1.0mm Distance from

I

1.5 mm Center of

I

2-0 turn

I

2.5ram

,_

Profile

Fig. 5. 2.5 mm thick leading edge of turbine blade T3, point geometry density profile for SANS data with normalised "no sampie" background for comparison. Rg = (358 _+31) A, 2SA S = 0.7 cm-1 .

rounding solid solution because: 1) there may be a different fraction of nickel in the precipitate ; 2) the atomic packing inside the precipitate may differ from that in the surrounding solid solution. The main hardening precipitate in the nimonic alloys [13,14] N75, N90 and in the turbine blades is an fcc Ni3X structure 7' where X is A1 to Ti. The 7' phase may or may not be coherent with the basic fcc structure solid solution 7. The initial size depends

on heat treatment and alloy composition but typically Rg <~ 100 ~ while the particles increase in size with ageing to typically Rg ~ 4 0 0 - 6 0 0 A. The volume fraction of 7' also increases with ageing as Ti and A1 come out of the basic solid solution [9,13]. The other major scattering precipitates which form in nickel based alloys are carbides. The primary precipitate is TiC which forms at grain boundaries. This reacts with the basic 7 phase to form a secondary carbide coated in 7'; the reaction is:

Table 3 Composition of alloy samples [ 13, 14]

6 TiC + 7 ~ 6 Ni3Ti + Cr23C 6 .

Ref. name

Composition by weight

N75 Nimonic N90 Nimonic NiCrMo

Ni-20 Cr-0.4 Ti-0.1 C Ni-20 Cr-17 Co-2.5 Ti-l.5 A1 50 Ni-40 Cr-10 Mo

T1, T2, T3

3 over-aged nickel alloy turbine blades similar in composition to N90 but with more Ti and A1 plus traces of C, B, Nb, Zr, etc.

RI and R5

Fe-12 Cr-2.5 Ni-2 Mo-0.7 mn-0.4 Si0.3 V-0.1 C plus traces of P, S, N

(31)

The primary carbides are of micrometer dimensions and hence too large to show up in SANS measurements. The secondary carbides however are much more likely to be of a size which will give SANS [13]. In the NiCrMo alloy the likely phases present are the o-phase which has atomic composition: 51.3 N i 18.3 C r - 3 0 . 4 M o and a phase Ni2Cr3 [16]. There must also be a residual solid solution of Ni, Cr, and Mo, but otherwise least was known about this sample. The surface of the N75, N90, NiCrMo and also the iron-based samples were polished and etched for

634

A.J. Allen, D.K. Ross /Neutron small-angle scattering

Table 4 Heat treatment and multiple refraction broadening for ferromagnetic alloy R1 and R5 Sample ref. name

R1

R5

Heat treatment: Softening: ttardening: Tempering: Stress relief:

Heat to 690°C for 6 h, air cool to room temperature Heat to 1050°C, air cool Heat to 1050°C, quench in oil Heat to 650°C, air cool Heat to 300-370°C for 3 h and air cool Heat to 280°C, air cool

Thickness: Broadening: (full widths at half height)

9.5 mm 0.0108 rad = 37'arc

-

examination in a scanning electron microscope to supplement the information gained from SANS measurements. It was not possible to SEM the turbine blades or to run them on the conventional PLUTO SANS spectrometer because they were only available for a short time. In N75 alloy there is insufficient titanium (and no aluminium) to form 3'' particles [13]. The 0.1% carbon by weight is sufficient to form carbides and indeed these were seen at grain boundaries in the SEM runs. Assuming half the carbon goes into tile secondary carbides [14] and these are all of size Rg 520 • a value ZSA s of 0.3 cm -1 is expected (using data from the Powder Diffraction File[16] on the phase Cr23C6). This compared well with the result given in table 2. If we assume a particle size distribution tailing off at smaller Rg values it is also possible to explain the results from the conventional PLUTO SANS spectrometer. The scattering profile in the latter case did not give a good computer fit and this may be explained [4] if this represents only about 20% of the size distribution at the small particle end as is suggested by the measured ESAS of 0.03 cm -1 . In N90 alloy there is <0.05% by weight [13] of carbon and hence rather less SANS from secondary carbides is likely, although some grain boundary carbides were seen in the SEM. Scattering from these carbides may have explained the 790 A size measurement using the film-foil converter method but this result implies that the Guinier and spectrum convolution approximations do not really hold. The PLUTO SANS spectrometer result gives a very good computer fit; so this is less likely to be scattering from the lower end of the carbide particle size distribution and is more likely to be due to SANS from fine 7' precipitates which would be too small to show up in the SEM run. The value o f ZSAs is determined not only by the amount o f Ti and AI that has formed 7' in the

16.0 mm 0.0134 rad = 46'arc

sample but also by tile difference in packing of the 3" and 7 phase structures. The value of ESA s of 0.03 cm -1 is about right if the 7' particles are of size Rg 1 0 0 A . * The SANS intensity is clearly sufficient (fig. 5) to make studies of precipitate growth in wellaged samples with reasonable precision. The SEM run showed the presence of two phases co-existing in the NiCrMo alloy. The high electron number density phase may reasonably be assumed to be the Mo-rich a-phase which has the atomic composition 51.3 N i - 1 8 . 3 C r - 3 0 . 4 Mo [16]. The other phase is probably the residual Ni, Cr and Mo but a finer precipitate phase was visible within this. This finer phase is likely to be of composition Ni2Cr3 and the scatter length density difference between this and the surrounding solid solution is probably what causes the SANS.

4.2. Multiple refraction results Table 4 indicates the heat treatment given to the ferromagnetic alloys R1 and R5 which otherwise had the same composition (table 3). Here the scattering completely broadens the straight through beam and the fit lbr the increased Gaussian width of the beam gives an error of less than 1% compared to the 5 10% uncertainties quoted for the samples above, see fig. 6. Ahhough in both the SEM and optical microscope the quenched and unquenched samples showed different fine structure, the multiple refraction seen obeyed eqs. (20) and (21) as if the magnetic domain size in tile two samples was very similar. Hence from eq. (20) it is expected that the refraction broadening * More recent results have shown that extended heat treatment of N90 alloy causes the 3'' to grow in size such that the SANS observed using the photographic method becomes much more prominent. The radius of gyration obtained is then 650 A.

A.J. Allen, D.K. Ross/Neutron small-angle scattering

635

0.5

:

...

•

•

~

•

Sample Data "No sample" Data

~Fltted 0.375--

- ' ~

•

function

Sample-film distance : 20cm

"~ 0 2 5 -

O.I O

I1 0.125 --

0

I

[

10'

20 t

Angle

from

I Centre

30 ~

of

I 40 '

1 4 5 ~A R C

Profile

Fig. 6. 16 mm thick Fe-based alloy sample RS, point geometry: density profile for sample showing multiple refraction with normalised "no-sample" profile for comparison. After correcting for width due to beam collimation, pinhole size etc. the full width at half maximum of the sample profile is 46' arc.

would be proportional to the square root of the thickness. Results obtained are shown in table 4 and the ratio of the two results is 1.24 while that of the square roots of thickness is 1.30. At ?, = 4.25 )k * (effective) these results are consistent with a grain size of < 3 0 ~m [5]. The multiple refraction broadening was found to completely disappear when the samples were placed in a high magnetic field (~1.3 T) as the domain magnetizations lined up to give a constant magnetic neutron refractive index through the sample. A soft iron sample showed narrower refraction broadening when unmagnetised **. Although a point geometry was used here, the multiple refraction phenomenon clearly lends itself very well to edge geometry measurements and hence such studies may be carried out comparatively rapidly. It should be • These results have been checked using a wavelength of 6.0 A on tire PLUTO SANS spectrometer. Good agreement is obtained if the ~.-4 dependence of eq. (20) is assumed. • * Since the domains in soft iron are known to have a mean dimension of 30 ~m, this result implies that the samples R1 and R5 have smaller domains, assuming the magnetic induction within tire domains is comparable to that in iron.

noted that to get data of the quality of fig. 6 using a conventional instrument it would be necessary to go to D11 at the ILL.

5. Conclusion We have shown that the photographic method for SANS using a Gd foil/film converter not only has the advantages over conventional p.s.d, systems of allowing multiple measurements simultaneously and having potentially better angular resolution and relative compactness and cheapness, but that it also gives reliable and reproducible SANS results over much of the particle size range of interest. Since the photographic method is more suited to looking at large particles while many conventional p.s.d, system resolutions are such that they are best suited to studying smaller particles the two techniques may frequently complement each other. The scope of the photographic method can be increased further by the use of multi-pinhole arrays particularly where a very large number of measurements are needed and work is currently in hand in this respect.

636

A.J. Allen, D.K. Ross /Neutron small-angle scattering

We wish to acknowledge AERE ltarwell for provision of neutron beam facilities; in particular we thank Mr. P. Schofield, Dr. M.R. Anderson, Mr. D.H.C. Harris and other members of staff for advice and assistance with the experimental side of the project. At the University of Birmingham we would like to thank Dr. M.R. Hawkesworth of the Department of Physics for advice on the foil/film converter technique, and Dr. I.R. Harris and Mr. C. King of the Department of Physical Metallurgy for help in characterising many of the alloy samples. We would also like to acknowledge Mr. G.E. Hickford who carried out a preliminary assessment of the method using edge geometry, and the SRC film digitization service at Daresbury laboratory who scanned the point geometry exposures. One of us (AJA) wishes to acknowledge the Science Research Council and AERE Harwell t'or financial support.

References [1 ] Proc. 4th Int. Conf. on Small angle scattering of X-rays and neutrons, J. Appl. Cryst. 11 (1978) 15. [2] M.R. ttawkesworth, At. En. Rev. 15 (1977) 2.

[3] Neutron beam facilities at the ttFR available tbr users, ILL Grenoble (1977). [4] Guinier and I:ournet, Small angle scattering of X-rays (J. Wiley, New York, 1955). [5] G. Kostorz, Treatise on materials science and technology, vol .15 (Academic Press, New York, 1979). [6] M.L. Boas, Mathematical methods for the physical sciences (J. Wiley, New York, 1966). [7] D.L. Dexter and W.W. Beeman, Phys. Rev. 76 (1949) 1782. [8] G.E. Bacon, Neutron diffraction (Oxford Univ. Press, 1962). [9] [t. Walther and A. Pizzi (FIAT-CECA), to be published. [10] AERE Harwell R-9278 (1978). [ 11 ] ftarwell Subroutine Library (VA05A by Powell), AERE Harwell R-7477 (1973). [12] Nottingham Algorithm Group Subroutine Library Mk. 8 (E04FCF), University of Nottingham, Crisps Computer Centre (1978). [13] W. Betteridge, Tire ninronic alloys (Arnold, London, 1959). [14] Private communication from Department of Physical Metallurgy, University of Birmingham (1979). [15] Neutron cross sections, Brookhaven National Laboratory and USAEC (1955). [16] Powder Diffraction File, JCPDS International Centre for Diffraction Data (1979).