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General advantages of neutron over X-ray diffraction are gained due to differences in the scattering mechanism. Neutrons are scattered as interaction with the nucleus, and although highly dependant on particular isotopes, scattering amplitudes show only relatively small changes with increasing atomic number. In contrast. X-ray radiation is scattered by interaction with the electrons and shows a steady increase in scattering amplitude with atomic number, making the scattering from light atoms difficult to detect in the presence of heavy ones. A further consequence of scattering from electrons is a

large scattering area across the atom, causing a significant decrease on the form factor with increasing Bragg angle. This work has made particular use of the precision gained by neutron diffraction in locating oxygen atom positions, extracting information about subtle distortions not easily detectable in the X-ray data due to the presence of heavy metal atoms. Cu 220 C u3 Si 531 Neutron Beam Stop Beam Rotating shutter and collimator Cryostat and Sample holder 32 collimators and detectors

Figure 3.3 BTl constant wavelength powder diffractom eter at NIST, with 32

3 .5 Cu (220) 14’ 2 .5 PQ Cu (220) T Cu (311) 14' 0 .5 160 120 140 60 80 100 40 Two-Theta (Degrees)

Figure 3.4 Experimentally determined beta integral, as a function of two-theta for the Cu (220) at 14’ and 7’ collimation and the Cu (311) monochromators at 14’ collimation.

3.2.5

Rietveld Refinement

In a powder sample formed by randomly oriented crystallites, information is only gained about the lattice spacing. Consequently, one of the main problems is the extraction of accurate structure factors for each reflection, due to either degeneracy or insufficient resolution. In 1967 Rietveld^’ ^ introduced a procedure for fitting whole profile, based on a least squares minimisation of the residual, Sy, between the observed and calculated intensity for each individual step, i, in the profile, where,

S y = X " i ( y i - y c i f i

yi = observed intensity for the ith step and yd = calculated intensity in the ith step The weighting of the observables is calculated as.

1

W i = ---

' yi(obs)

It differs from other methods of powder diffraction analysis in that there is no advanced attempt to allocate observed intensity to particular Bragg reflections. Consequently, a reasonable knowledge of the starting parameters is required. Calculated intensity for each hkl is estimated as follows:

y ci -2 0 k )P h k iT + y|5i hkl

s is the scale factor

Lk is a contribution from the Lorentz, polarisation and multiplicity factors <]) is the peak shape function

Phki is the preferred orientation function, T is due to absorption l^hkil is the stmcture factor for the particular hkl reflection yt,i is the background contribution at the ith step

Ai is the asymmetry parameter

Model parameters refined during a typical fit originate from both the sample and instrument. Cell parameters, atomic co-ordinates and overall temperature factors, isotropic or anisotropic if a non-cubic structure is found to have directional effects, aU contribute directly to the structure factor. The background contribution originates from both the instrument and incoherent scattering from the sample. Reactor sources for neutron diffraction generally give low, featureless backgrounds compared with pulsed sources and are well described by a simple 5th order polynomial function originating from a user defined value, BKPOS, such that

Ybi = m

i BKPOS

m

A distortion to the peak shape is caused by the cutting of the curved surface (except at 90°) of the Debye-Scherrer cone, by a finite detector sht height. This is modelled by a single parameter, P, where

A ^ ^ l - P s ( 2 e ,- 2 9 ,)

where s is the sign of (2 0j - 2 0i^) and 0^ is the defined angle below which this

asymmetry is applied. A good description of the peak shape is essential in modelling any diffraction data. Traditionally, Bragg reflections from neutron diffraction have been modelled by a Gaussian peak shape whose breadth, H, has the following theta dependence,^

H = um«^ 0 + vm«0 +w where u, v and w are refined parameters.

It has been successful in modelling low to medium resolution data, where instrumental broadening swamps complicated aberrations in the peak profile, caused by defects such as microstrain and small crystallite size. Throughout this work, profiles collected on the high resolution diffractometer, BTl, at the National Institute of Standards and Technology have required the convolution of Gaussian and Lorentzian components in a pseudo-Voigt function, to fully describe the peak shape. The mixing of the Gaussian, G, and Lorentzian, L, components is defined by a parameter T |,

pseudo - Voigt('pV') = T|L -i- (1 - T|)G

The mixing parameters, t|, varies linearly with two-theta, through two refined parameters, T)o and X.

ri=riQ+X.26

Where extraction of specific information about particle size and microstrain strain is required a modified Thompson-Cox-Hasting pseudo-Voigt (TCHZ) function has been employed. The theta dependence of the full width at half maximum (FWHM) is described as follows,

FWHM^(Gaussian) = utari^ 6 + vtan6 + 'w + cos 0

In this function u and z describe the isotropic broadening to the Gaussian function caused by strain and size, respectively, x and y represent the strain and size contribution to the Lorentzian component. This differs from other pseudo-Voigt functions in that p, is not a refined parameter, but dependent on theta.

T1 = 136603q - 0.44719q^ + 0.1116q^ where.

r = ( r à + + B F^F^+C F ^F^ + DFqF^, +

A=2.69269 B=2.242843 C=4.47163 D=0.0784

The convergence of the residual to a minimum is not a satisfactory criterion for a good fit, since either a false 'local' minimum could have been achieved or the starting structural model might have been incorrect. A number of agreement factors are defined to determine the validity of the fit. Two are based on the observed and calculated intensity for each Bragg reflection, hkl, the Bragg factor being statistically the closest to the conventional single crystal R-factor. In regions of peak overlap, the contribution to the observed integrated intensity is taken to be proportional to the calculated contribution for the same points.

„ _

SlVlhkl(obs) - VihklMc)

Rf -

SVlhkl(obs) Structural Factor

Rb = Z |lh k ;^b s)y h k ,(calc)| Factor 2 .Ihkl(obs)

The pattern and weighted pattern R-factors are related to the intensity of each step, i, rather than the overall calculated intensity for the Bragg reflections.

R p = ^|y i(o b s)-y j(calc)|

Z y i(o b s) Pattern Factor

^wp -

2^Wi(yi(obs)-yi(calc))‘

]£wi(yi(obs))' Weighted Pattern Factor

A 'goodness of fit' factor, %, is defined as,

R_wp

R.

where Re is the expected R-factor,

Re - N -P -h C ^WiYi (obs)

N is the number of data points, P is the number of refined parameters, C is the number of constraints giving a value for N-P+C, which defines the number of degrees of freedom. Two sets of R-factors are given in this work. The conventional R-factors are corrected for background intensity, the ones in the second set are not.

3.3 Vibrational Spectroscopy