instrumentation for neutron reflectivity

Physica B 173 (1991) 1-10 North-Holland

Invited paper

Instrumentation for neutron reflectivity

J. P e n f o l d

ISIS Science Division, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon 0 X l l OQX, UK

The basic requirements for neutron reflectivity experiments are discussed, and some current instrumentation is described. Examples from the reflectometer CRISP are used to demonstrate the present capabilities of the technique and analysis methods in the context of surface chemistry and solid films.

I. Introduction

A range of phenomena similar to those in classical optics are exhibited by slow neutrons, and include reflection, refraction and interference. One of the earliest demonstrations of the analogy with electromagnetic radiation was the observation of total reflection from a mirror surface by Fermi et al. [1]. Subsequently, total reflection has been used extensively in neutron guides [2] and neutron spin polarisers [3].

Following the pioneering work of Thomas et al. [4] in surface chemistry and Felcher [5] in surface magnetism, the specular reflection of neutrons has been applied to the study of surfaces and interfaces.

In recent years, the advent of dedicated spec- t rometers [6] has seen a rapid expansion in the scientific applications of the technique to problems in surface chemistry, surface magnetism and solid films, and specular neutron reflection is now becoming an established technique for the study of surfaces and interfaces over a wide range of inter-disciplinary problems [4].

In this presentation some current instrumentation is described in detail, and the factors affecting the data reduction and quality and range of the data are discussed. The scope of the technique is illustrated with some recent examples in surface chemistry and solid films, with particular emphasis on the application of recently de- veloped model fitting procedures to these problems. The main emphasis throughout will be

associated with experience gained on the CRISP reflectometer [6] at ISIS [7].

2. Instrumentation

The essence of a neutron reflection experiment is to measure the specular reflection as a function of the wave vector transfer, Q, perpendicular to the reflecting surface. This can be related to the neutron refractive index profile normal to the surface or interface, and is often simply related to the scattering length density, yielding information about the composition and density gradients at surfaces and interfaces.

A wide range of Q can be achieved either by using monochromatic beams and scanning a large number of angles, or by using the broad band neutron time-of-flight (TOF) method to determine A at fixed 0.

The white beam T O F method is ideally suited to pulsed neutron sources, whereas the monochromatic beam method is appropriate to reactor based sources, and spectrometers based on each approach have been constructed [4].

As the critical glancing angles are small, nar- row well collimated beams are required, and long wavelength neutrons are an advantage. The two basic methods offer a number of distinct advantages and disadvantages.

The fixed angle of incidence, 0, and hence sample geometry, of the white beam T O F method is ideal for the study of the air-liquid

0921-4526/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)

2 J. Penfold / Instrumentation for neutron reflectivity

and liquid-liquid interfaces, and ensures constant sample illumination, whereas for the monochromatic beam case the illumination and sample geometry varies with 0. The Q resolution AQ/Q has contributions from A0 (collimation) and AA (At/t for the white beam TOF method). In general for the monochromatic beam case, the contributions from A0 and AA are both significant and AQ/Q varies over the reflectivity profile. In the white beam TOF method, in contrast, the At/t contribution is small (the resolution is dominated by A0) and hence AQ/Q is constant: even when the At contribution is significant, the time channels can be chosen for constant At/t.

The monochromatic beam method measures a profile point by point, whereas the white beam TOF method measures the entire profile simulta- neously. In this case, the monochromatic beam method offers distinct advantages when a re- stricted region of a reflectivity profile is required, as in time dependent studies.

The reactor based monochromatic beam spectrometers for specular reflection monochromate long wavelength neutrons using velocity selectors or shorter wavelengths ( -4 .~ ) using a mono- chromating crystal such as graphite. The small angle scattering spectrometer (SANS) D17 [8] at the Institut Laue Langevin has been extensively used for reflection experiments and is a good example of the former type. The spectrometer is on a highly curved guide viewing a cold source. The neutrons are monochromated by a velocity selector to give a usable wavelength range of 8 -30A with a wavelength resolution of 5 or 10%. The longer wavelength neutrons can be of particular advantage when very low values of Q are important, and for some studies at the liquid-solid interface. The sample to detector distance is variable and the two-dimensional detector (ideal for non-specular studies) can be rotated through 20, enabling a wide range of momentum transfer to be covered for solid samples.

As most of the subsequent discussion is based on experience with the white beam TOF reflectometer CRISP [6] on the ISIS pulsed neutron source, we will describe CRISP in some detail.

The spectrometer views the 20K hydrogen

moderator, giving it an effective wavelength range of 0.5-6.5 ,A, at the source frequency, and a maximum wavelength of 13 A at 25 Hz. The beam is inclined at 1.5 ° to the horizontal (specifically for liquid surfaces). The incident beam is tailored by coarse and adjustable fine collimation to give variable beam size and divergence, with typical beam dimensions of 40mm width and between 0.25 and 5.0 mm height (see fig. 1). A single disc chopper defines the wavelength band (AA) and provides some frame overlap suppression. Additional suppression is provided by a series of frame overlap mirrors: nickel coated silicon wafers set to refect out of the main beam neutrons of wavelengths greater than 13 A. The 180 ° aperture disc chopper rotating at either 50 or 25 Hz provides a 6.5 or 13 A wavelength band (see fig. 2). Efficient operation at 25 Hz will be facilitated by the incorporation of a nimonic chopper which will alleviate contamination from the initial power pulse from the target. The detector (a well shielded single He 3 detector, or a one-dimensional multidetector with a position- al resolution - 1 m) is located from 1.75 m from the sample position.

The experimental arrangement is extremely flexible, and liquid surfaces can be studied at angles less than 1.5 ° (down to -0 .2 ° ) by the insertion of a supermirror. Furthermore, solid films can be studied with ease, and the spectrometer is also used extensively for polarised neutron studies on magnetic systems [9].

3. Data reduction

For the reflectometer, CRISP, the experimental data is reduced to reflectivity (on an absolute scale) as a function of Q using now well established procedures, such that

[Id(Ai) -- bd(i)]~m(Ai) R(Q(Ai, 0)) = f [im(Ai ) bm(i)]~d(Ai) , (1)

where the subscripts d, m refer to the detector and monitor, I~, m are measured intensities, ~d,m are the appropriate wavelength dependent ef-

J. Pen fold / Instrumentation for neutron reflectivity 3

D $4 $3 S

/ /

52

SF P

/

S]

C

Fig. 1. Schematic diagram of the CRISP reflectometer, showing the chopper, C, collimating slits, S1-$4, frame overlap mirrors, F, beam monitor, M, sample, S, and detector, D. Shown also for completeness is the neutron spin polariser, P, and spin flipper, SF, used in its polarised neutron mode.

ficiencies, bd, m is the background in the detector and monitor, and f is the scale factor.

The absolute scale (and hence scale factor, f ) is determined by normalisation to the straight through beam, by reference to the region of total reflection, or by normalisation to a standard surface such as D 2 0 , each of the procedures has been shown to be equivalent.

In the white beam TOF method, the resolution in Q has contributions from At/t and A0/0 such that

A2Q _ A2tch d- A2tm A20 + (2) Q2 t 2 0 2 '

where A0 is the geometrical resolution, Atch the TOF channel width, At m the moderator pulse width and t is the TOF.

In general for CRISP the At/t contribution is

small, and A Q / Q is dominated by A0/0, and so is constant over the whole Q range measured. Even when the At contribution is significant, the time channels can be chosen for constant At/t. In the subsequent calculated reflectivity profiles described, the instrumental resolution is included using a Gaussian smoothing function of width AQ.

The useful Q range is determined not only by instrument, but also by sample effects. The minimum Q is 3 x 10 -3 A-~, and the maximum is

0 .65~ 1 (for liquid surfaces). Ornaximurn may however be sample limited as the reflectivity falls off as at least Q 4 (Fresnel's law), and even faster for a non-perfect surface or for particular refractive index profiles. Qmaximum is therefore often determined by the sample dependent background. For CRISP, the instrumental background is less than 10-7-10 -8 , and typical sample

4 J. Penfold I Instrumentation for neutron reflectivity

(a)

13 65 . . / . 0 ..,:"26 , t / 195

D(in me te rs ) / t V J~" k~

i / ~//,

6 / f / /

0 20 40 60 80 100 Time (miUisecs)

(b)

12-

0 ~

/

/

2O

!

4O 6O 80 100

Fig. 2. Distance time diagram for 180 ° aperture disc chopper at 6 m on CRISP for (a) 50 Hz, (b) 25 Hz: band pulses are shown as solid lines, dashed lines indicate a possible frame overlap contamination.

backgrounds are - 2 x 10 -6 for D 2 0 , - 4 x 10 ~' for H 2 0 and - 5 x 10 7 for silicon (measured at 0 : 1.5°). Qmaximum essentially determines the min imum dimension scale to which the technique is sensitive, and is of the order of a few ring- stroms. For example , the roughness of a liquid surface can be measured down to - 2 A (see fig. 4).

4. Scope of application

It has been shown [10] that the intensity of the reflected and transmit ted neutrons follows the same laws as electromagnetic radiation, with the electric vector perpendicular to the plane of incidence. The refractive index for neutrons is then commonly written as

n = 1 - AZA + i h C , (3)

I

1"Ore

I I I I

104 I >

1-2 a~ lO

10-3 l

10-4 0 015 0020 0025 0 030

Momentum transfer, O (,~-T)

Fig. 3. Specular reflection from a A/10 optical flat measured at an angle of incidence, 0 = 0.3 °, • data points, - - least squares model fit (for parameters see text).

where A = N b / 2 v , C = Ng, /4w, N is the atomic number density, b is the bound coherent scattering length, o- a is the adsorption cross section, and h is the neutron wavelength.

In contrast to X-rays, neutron scattering am- plitudes vary randomly f rom element to element, and isotopic substitution can be used to produce large contrasts in the scattering densities. Of particular importance is the large difference in scattering powers of hydrogen and deuterium; this has already been used to great effect in small angle neutron scattering, and has been crucial in the study of surface chemistry.

In surface chemistry, the specular reflection of neutrons can be used to provide information about adsorption at interfaces, where not only can the amount adsorbed be determined, but also the structure of the adsorbed layer. Specifi- cally, hydrogen-deu te r ium exchange has been used not only to highlight particular parts of the surface, but even to eliminate altogether the reflection f rom anything other than the adsorbed layer. The initial scientific interest has been adsorption at the air- l iquid interface of surfactants, fatty acids and polymers. The penetrability of thermal neutrons has assisted in the recent ex- tensions of the technique to the study of adsorption at the solid-liquid liquid-liquid interfaces and to the study of in situ electrochemistry.

J. Penfold / Instrumentation for neutron reflectivity 5

For magnetic materials magnetised in the plane of the surface, the neutron spin dependent refractive index gives rise to a neutron spin dependent reflectivity R±, where the refractive index is now defined as

n_+ = n N ----- n M = 1 -- (NAZ/2,rr) (b -+ C / z ) , (4)

where N, M refer to the nuclear and magnetic contributions, ~ is the moment per atom, and C is the constant (0.265 × 10 -12/zB/cm ).

The neutron spin reflectivity ratio R + / R _ has been shown to be a sensitive probe of the surface magnetisation profile [11]. The main areas of application have been in the study of magnetisation and ultrathin ferromagnetic films, the nature of magnetism in multilayers, and the investigation of flux penetration in superconductors.

In solid films, the advantages of neutron reflection are less obvious because X-ray reflection and other surface probes may often be more appropriate. However, a number of recent experiments have illustrated that in this area of surface science, neutron reflection has some dis- tinctive features which make it more effective in certain circumstances: especially where the neutron contrast can be particularly favourable, such as in polymer films and Langmuir-Blodgett films. Table 1 gives a clear indication of the wide range of the scientific applications of specular neutron refection on the CRISP reflectometer, and many of these applications have been recently reviewed [4].

5. Model fitting

The analysis of reflection data can proceed by model fitting or inversion. From the comparison with electromagnetic radiation most of the standard formulisms in classical optics [12] can be used with only minor modification. In particular, the optical matrix methods provide a convenient form to calculate exactly the neutron reflectivity profile from structural models of interfaces [13], and has, so far, been the preferred method of data analysis.

A number of matrix methods exist for calculat-

Table 1 Range of applications of specular neutron reflection on the CRISP spectrometer.

Surface chemistry (i) Adsorption at the air-liquid interface

Fatty acids Surfactants Polymers Lipids, proteins

(ii) Adsorption at the liquid-solid interface Surfactants Polymers Electrochemistry

(iii) Adsorption at the liquid-liquid interface Fatty acids Polymers

Solid films Langmuir-Blodgett films Polymer films Amorphisation in multilayers Thin semiconductor films Multilayers

Magnetism Ultra thin ferromagnetic films Magnetic glasses Magneto-optical films Magnetic multilayers Flux penetration in superconductors

ing reflectivity profiles from some arbitrary interfacial profiles, and have been described in detail elsewhere [14]. A convenient method is that of Abeles [15] and it has been used as the basis of a standard package of analysis programs de- veloped for CRISP [16]. In the Abeles method, a characteristic matrix per layer is derived, in optical terms, from the relationships between electric vectors in successive layers in terms of Fresnel coefficients and phase factors, such that,

[ e i~j-' r~ e i~j , ] Cj=[r,e-'~'-' e "~,-' J ' (5)

rj is the Fresnel coefficient at the j - 1 , jth interface,

rj = (Pi , - P j ) / ( P ~ - i + P j ) ,

pj = n~ sin ~ ,

~j = ( 2 ~ / A ) n f l j sin O,

and here 0 is the angle of incidence.

6 J. Pen fold / Instrumentation for neutron reflectivity

For n layers the matrix elements MI], M21 of the resultant matrix M R = [ M 1 ] [ M 2 ] ' " [ M . + I ] gives the reflectivity

R = M21M211M, ,MII •

If the surface is not entirely smooth its local roughness will modify the specular reflectivity in a manner similar to that of a diffuse interface. A roughened or diffuse interface is included in the Abeles description by modifying the Fresnels coefficients to include a gaussian roughness factor [13] such that

(pj , - p j ) r j - ( p j _ , T p j ) e x p ( - O ' 5 q i - l q J ( ° - ) 2 ) ' (7)

of 0.36 x 10 -5 ,~ -2 a surface roughness, ~r, of 33,~ and a A0 of 4.7%. The quality of the fit indicates that the profile is well described by Fresnel's law, modified for surface roughness and instrumental resolution. For this measure- ment the intrinsic instrumental resolution B0 was 3.7%, and it is assumed that the additional contribution is from surface undulations.

To resolve much smaller values of surface roughness it is necessary to measure the reflectivity to much higher Q values and lower reflectivities. Figure 4(a) shows the reflectivity of a pure D20 surface, where the solid line is the calculated profile for a surface roughness of 2.8 A. The deviation from ideality is seen more clearly in fig. 4(b), where the Q-4 dependence

where (~r) is the root mean square roughness qj = 2k sin 0j, and k is the neutron wave vector.

Based on the Abeles matrix, a range of standard fitting programs for unconstrained and con- strained model fittings and tailored to a range of specific problems have been written for the CRISP reflectometer and are described in detail elsewhere [16] (see table 2).

The simplest approach is to impose no constraints upon the model fitting, and some simple examples are illustrated.

Figure 3 shows the application of such an approach to a model fit of the reflectivity of a bare A/10 optical flat. The solid line is a least squares model fit for a scattering length density

_e

e¢

10

i0 -I

1o-Z

1o .3

i0 -&

10-5

t66

10-v 001

(o) - , i i i i i i i i i i i i

002 004 007 02 03

Momentum Transfer, Q (A -t)

065

~ o - 6 (b)

i i i i i i 1 i i i i i 1

Table 2 Standard analysis programs for the reflectometer, CRISP.

1. L_MULFIT: General multilayer program for up to 50 layers

2. L RATIO: Version of L_MULFIT, specifically for data ratioed to the bare subphase reflectivity or as RQ 4

3. MULTI_LAYER: Version of L_MULFIT written for multiple bilayers

4. T O P G R A D : Version of L_MULFIT with a user specified graded upper interface

5. M_GRADFIT2: For a single bilayer, with a graded interface between the two layers

6. FL1PFIT: For polarised neutron reflection data, simulta- neously refines spin up and spin down reflectivities, and flipping ratios ( R , / R )

7. GFIT: Generalised model fitting program for user specified model

~o

10 -5

1 0 - / •

1 0 . 3

lO -2

1 0 -1

0014 i i , , A , , i l i ! ' '0.63~

0-03 005 006 02 03

Momentum Tronsfer. Q {A "~)

Fig. 4. (a) Specular reflection from a pure D20 surface measured at angles of incidence of 0 = 0.4, 0.8 and 1.5 °, • data points with background of 1.5 x 10 6 subtracted, - - least squares fit for a roughnes of 2.8 A.. (b) As (a) except plotted as RQ 4, --- calculation for an ideal interface (Fres- nel 's law).

J. Penfold / Instrumentation for neutron reflectivity 7

(arising from Fresnel's law) has been removed, and the dashed line is the calculated curve for an ideal interface.

The presence of a thin film on such a substrate will modify the simple Fresnel reflectivity profile from the bare substrate, and for a well defined layer a series of discrete interference fringes will be observed. The interference pattern from such a well defined layer is shown in fig. 5 for a 48 layer deuterated cadmium eicosanoate Lang- muir-Blodgett (L-B) film deposited onto a polished silicon wafer. In this case, interracial and surface roughness will modify the reflectivity profiles, and this has been discussed in detail elsewhere [13]. The solid line is a least squares fit, assuming that in this Q range the L-B film can be described as a single uniform layer of thickness 1186A and a scattering density of 0.74 × 10 -5 .&-2. The calculated reflectivity profile is modified by a 5e of 3.7% and an interfacial roughness of 20 .& at both the air-film and film- substrate interfaces.

For multiple layers the reflectivity profile is more complex, and for multiple bilayers in addi- tion to the interference near the region of total refection which arises from the whole layer, a series of Bragg peaks will be observed and between these peaks will be a series of higher order interference fringes. The factors affecting the

I I I I l I

I 0

,., 10 -1

~. 10 -2

10 -z

10 I i I ~ [ = I ~ I , l 0 ' 0 2 0 03 0 0 4 0 0 5 . 0 0 6 0 - 0 7 -1

Momentum trQnsfer, Q ( A )

Fig. 5. Specular reflection from a 48 layer deutera ted cad- m i u m eicosanoate L a n g m u i r - B l o d g e n film deposited onto a silicon water , measured at 0 = 0.5 °, • data points, - - least squares model fit (for parameters see text).

detailed nature of such reflectivity profiles are more involved, and have been described in detail elsewhere [17]. Figure 6 shows the first order Bragg peak and its neighbouring interference fringes from a 15 bilayer nickel/titanium multilayer [18]. The solid line is a model fit assuming ideal interfaces and scattering densities for the nickel and titanium, and layer thicknesses of 47.8 ,~ (nickel) and 55.7 ,~ (titanium).

The examples shown so far have been well described by interfaces that are ideal or that have a width that can be characterised by a Gaussian. This is not always the situation and recent measurements by Higgins et al. [19] on the nature of the interface between the partially miscible polymer pair of deuterated polymethyl- methacrylate (d-PMMA) and hydrogeneous solution chlorinated polyethylene (h-SCPE) are a good example.

Figure 7(a) shows the reflectivity profile after thermal annealing of the d-PMMA h-SCPE pair. The low frequency fringes arise from the upper d-PMMA layer (--1000,~), whereas the higher frequency oscillations arise from the lower h- SCPE layer ( -3000A) . The damping of the fringes results from the nature of the interface between the two polymer films. The detailed form of the interface which fits the data is shown in the inset.

1.0

10 - I

1o -2

10 -3 _

1 0 - 4 1

.÷

I I I I I

0 0 3 0 0 " 0 4 0 0 " 0 5 0 0 0 6 0 0 0 7 0 Momenturn t r ans fe r ,Q ( in ,~-1)

Fig. 6. Specular reflection from a 15 bilayer nickel t i tanium mult i layer measured at 0 = 0.8 °, + data points, - - least squares fit (for parameters see text).


i l i xlO-3

O-lO 0007 i ~ , , i i i

00, . :.°0°2 ~:~ 0"0• o 003

0 002 t I 004 o 500 tooo 1500 2 0 0 0 ?,500

0"0~

-00 I - - l -- I 0 016 0 02 0 03 0 04 0 0544

Fig. 7. Neutron reflectivity profile (plotted as RQ 4) for d -PMMA ( - 1000 ,~) h-SCPE ( - 3000 A) polymer bilayer in a glass optical flat. The solid line is the calculated profile for the refractive index profile shown in the inset.

In the investigation of problems in surface chemistry it is possible to impose constraints upon the model fitting which significant increase the power of the technique, and contribute greatly to the uniqueness of the model. Hydro- gen deuterium isotopic substitution can be used in chemical systems to manipulate the refractive index profile. This has been particularly important in recent work [20, 21] on the adsorption of surfactants at the air-solution interface, where isotopic substitution can be applied to both solute and solvent. It is possible to choose the hydrogen deuterium ratio such that the solvent is null reflecting (i.e. its refractive index is matched to air). If the surfactant is deuterated, then any reflection results entirely from the surface adsorption of the surfactant. It is also possible to eliminate the reflection from the solute and determine the surface profile of the solvent. Isotopic substitution can be further used to highlight particular parts of the solute molecule by selective deuteration.

The first application of this approach is simply for the determination of surface excesses. It is straightforward to show for a single component system that, in the case of a deuterated surfactant in null reflecting water, reflectivity arises solely from the adsorbed layer and that it is directly related to the surface excess, F, or area

per molecule, A H G [22]. From a simple analysis, if such a reflectivity curve is fitted by a single uniform layer, then the area per molecule is given by

Eb A H G = - - (8)

dr. Nbf '

where E b is the total scattering length of the deuterated surfactant molecule, and dr, Nbf are length density. The product d~.Nbf can be measured to good accuracy ( - 1 % ) .

Figure 8 shows a typical fit to a deuterated dodecanol layer on null reflecting water, where the solid line is a model fit for a layer of 20.4 * thickness and a scattering density of 0 .6x 10 5 ~ 2. The lower curve shows the difference between the measured and fitted curves.

For such adsorbed amounts it is possible in this Q range to determine the thickness and density of the adsorbed layer independently. This is shown in fig. 9(a) for a 3 x 10 -3 M deuterated tetradecyltrimethyl ammonium bromide layer in null reflecting water (dr = 20 ~ , Nbf = 0.46 x 10 -5/~-2), where in fig. 9(b) the X 2 from the fits to a single uniform layer is plotted as a function of film thickness, and shows a well defined minimum at the optimal film thickness.

10-3

o~- ~ ~ ~ 10 -510-4 _ ~ e

•_$

10" i l I 0 1 0 2 0 5 0 4 _

Momentum tronsfer, Q (in/~ -1)

°,°/\"

I I

05 06

Fig. 8. Measured (@) reflectivity of a deuterated dodecanol layer on null reflecting water. The solid line is a fit to a single uniform layer (for parameters see text). The lower curve is the difference between the measured and fitted curves.

J. Penfold Instrumentation for neutron reflectivity 9

10 .3

10 TM

10-~

1 0 "

i i I I I

+

0 I. 1 I I I I o 2 o + 0 ,

Wove vector transfer, 0 ( 0 6

X 2

9 0

8 0

7 ' 0

6 0

5 0

4 ' 0

5 ' 0

2 ' 0

1 0

i •

\ / 't,,./•

I 1 I 15"0 20"0 2 5 0

Thickness, d (in ~)

Fig. 9. (a) Measured (O) reflectivity for 3 × 1 0 -3 M deuterated tetradecyltrimethyl ammonium bromide in null reflecting water. The solid line is a fit for a single uniform layer (for parameters see text). (b) X 2 versus film thickness.

Even for lower surface adsorptions when df and Nbf are coupled, the product (which is what is important in determining the surface excess) can still be measured with good accuracy.

This approach has recently been applied to several cationic and anionic systems, and some mixed surfactant systems. Neutron reflectivity is particularly important for such systems as, by selective deuteration, it is possible to measure unambiguously the relative surface excesses of each component.

Using selective deuteration, it is possible to obtain reflectivities from the same interface but viewed with different refractive index profiles. We can then constrain a model of the interface

to predict each of the reflectivities without adjustment, and this approach has been applied to some advantage on a range of systems [23, 24].

Having measured the reflectivity profiles for as many different refractive index profiles as possible, the procedure adopted for analysing the data is to construct the model of the interface and to refine the model parameters for one of the refractive index profiles (say deuterated surfactant in null reflecting water). The same model is then used to generate the reflectivities for the other refractive index profiles (obtained from isotopic substitution) for the same system. The model is then only acceptable if it can be used to predict the reflectivity for each refractive index profile with no adjustment of the parameters between the different profiles.

The simplest model that is consistent with the reflectivity from a range of absorbed surfactant layers is a two layer model where the first layer adjacent to the vapour phase contains some fraction (1-FHG) of the hydrocarbon chains, and the second layer, adjacent to the aqueous subphase, contains the headgroups, some fraction, FHG, of the chains, solvent and bound counterions. The model is then characterised by three structural parameters, A H G , FHG and FC, the extent to which the chains are fully extended; and from known molecular volumes, sizes and scattering lengths it is possible to calculate the extent and scattering densities of the two layers.

In fig. 10 we show the application of this approach to 0.0067 M SDS, where we have measured the reflectivity profiles for deuterated SDS in null reflecting water, deuterated SDS in D20, and protonated SDS in D20: the model fits are for an A H G of 44,~, FHG of 0.2 and FC of 0.87, and show excellent agreement with the data.

Acknowledgements

I acknowledge several colleagues and col- laborators who have contributed to the develop- ment of the CRISP reflectometer, and who have allowed me to present their data, but especially R.K. Thomas, J.S. Higgins, W.G. Williams, R.


70 -3 n [ u I ' I ' t ,

]0-4

10- s +

,L

1 0 - f l z L z z I t [ * I i o.o o. ,o o.2o o.2 . .3o

Momentum tronsfer, Q (

] I

0-35 0-40

Fig. 10. Specular reflection from 0.0067 M SDS at 0 = 0.5 ° for (i) deuterated SDS in null reflecting water (+) , (ii) deuterated SDS in D20 (O) and (iii) protonated SDS in D20 (O). Solid lines are least square model fits (for parameters see text).

Felici, R.C. Ward, C. Shackleton and G.J. Herdman.

References

[1] E. Fermi and W. Zinn, Phys. Rev. 70 (1946) 103. [2] H. Maier-Leibnitz and T. Springer, Reactor Sci. Tech.:

J. Nucl. Energy 17 (1963) 217. [3] J.B. Hayter, J. Penfold and W.G. Williams, J. Phys. E

11 (1978) 454. [4] J. Penfold and R.K. Thomas, J. Phys. Condens. Mat. 2

(1980) 1369. [5] G.P. Felcher et al., Rev. Sci. Intrum. 58 (1981) 609.

[6] J. Penfold, R.C. Ward and W.G. Williams, J. Phys. E 20 (1987) 1411.

[7] See for example ISIS Annual Report 1990 (RAL-90- 050).

[8] Neutron beam facilities at the high flux reactor available to users (Institut Laue Langevin, Grenoble, 1985).

[9] R. Felici, J. Penfold, R.C. Ward and W.G. Williams, Appl. Phys. A 45 (1988) 169.

[10] M.L. Goldberger and F. Seitz, Phys. Rev. 71 (1947) 294.

[11] G.P. Felcher, K.E. Gray, R.T. Kampwirth and M.P. Brodsky, Physica B 136 (1986) 59.

[12] J. Leckner, Theory of Reflection (Martinus Nijhoff, Dordecht, 1987).

[13] J. Penfold, J. de Phys. 50 (1989) C7-99; J. Penfold, Rutherford Appleton Laboratory Report, RAL-88-088 (1988).

[14] M. Born and E. Wolf, Principes in Optics (Pergamon, Oxford, 1970).

[15] O.S. Heavens, Optical Properties of Thin Films (Butter- worth, London, 1955).

[16] G.J. Herdman, J. Penfold and C. Shackleton, Ruther- ford Appleton Laboratory report RAL-90-040 (1990).

[17] G.J. Herdman and J. Penfold, to be published. [18] I.A. Anderson et al., to be published. [19] M.L. Fernandez, J.S. Higgins, J. Penfold and C.

Shackleton, Polymer 31 (1990) 2017. M.L. Fernandez, J.S. Higgins, J. Penfold and C. Shackleton, ACS Polymer preprints 2 (1990) in press.

[20] E.M. Lee, R.K. Thomas, J. Penfold and R.C. Ward, J. Phys. Chem. 93 (1989) 381.

[21] J. Penfold, E.M. Lee and R.K. Thomas, Mol. Phys. 68 (1989) 33.

[22] J. Penfold, in: Proc. WONSDA II Workshop on Data Analysis Techniques (1990), IOP Conference Series No. 107.

[23] R.K. Thomas, A. Rennie, J. Penfold, E.M. Lee and E. Simister, to be published.

[24] J. Penfold, R.K. Thomas, E. Simister, E. Lee and A. Rennie, J. Condens. Matter 3 (1990) SA411.

instrumentation for neutron reflectivity

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