CHAPTER 2 - THE SCALING FUNCTION

2.1 INTRODUCTION

foverall = fL * fP * fOb * fPlate * fPack * fReference * fLambda * fAbs

where

fL is the Lorentz factor
fP is the Polarisation factor
fOb is the Obliquity correction
fPlate is the Plate scaling factor
fPack is the Pack scaling factor
fReference is the Reference scaling factor
fLambda is the Wavelength normalisation curve
fAbs is the Absorption correction

List of sections:

The Lorentz Factor
The Polarisation factor
The Obliquity Correction
The Plate Scaling Factor
The Pack Scaling Factor
The Reference Scaling Factor
The Wavelength Normalisation Curve
The Absorption Correction

2.2 THE LORENTZ FACTOR

The Lorentz factor is given by the expression:

fL = sin**2(theta)

LSM parameters: none

2.3 THE POLARISATION FACTOR

The polarisation scaling factor may be one of the following:

1. For X-rays (synchrotron)

   fP = 2/(1+cos**2(2*theta) - tau*cos (2*psi)*sin**2(2*theta)

   where 'psi' is the polar angle i.e. psi = acrtan(xf/yf) and 'tau' is the degree of polarisation.

2. For neutrons

   fP = 1.0

2.4 THE OBLIQUITY CORRECTION

This is a correction made to allow for a covering on the front of the detector. The obliquity scaling factor is:

fOb = exp(ut/cos(2*theta))

where 'ut' is the product of the linear absorption coefficient and thickness of the cover material.

LSM parameters: COVER_FAC (not refineable)

2.5 THE PLATE SCALING FACTOR

The plate to plate scaling factor which scales up the intensities on a lower plate to those on the top plate is given by:

fPlate = kappa * exp(ups/cos(2*theta))

where 'kappa' and 'ups' are the coefficients determined for the plate in question and the plate to plate scaling wavelength range appropriate to the wavelength of the reflection being scaled.

The scaling factor for the top plate is fixed at 1.0.

LSM parameters: PLATE_SCALE_r[][] almin almax kappa ups

(plate based parameters with multiple wavelength ranges - 'kappa' and 'ups' are refineable except for the top plate for which any values are, in any case, ignored)

2.6 THE PACK SCALING FACTOR

The pack scaling factor scales the intensities to those of the first pack and is calculated from the expression:

fPack = Spack * exp(-2.0*Bpack*sin**(theta)/lambda**2)

where 'Spack' is the linear scale factor and 'Bpack' is a temperature factor.

The scaling factor for the first pack is fixed at 1.0.

LSM parameters: PACK_SCALE[], PACK_BFAC[]

(pack based parameters - refineable except for the first pack)

2.7 THE REFERENCE SCALING FACTOR

This is a scaling factor to scale the Laue data to a reference set of data and is calculated from the expression:

fReference = Sreference * exp(-2.0*Breference*sin**(theta)/lambda**2)

where 'Sreference' is a linear scale factor and 'Breference' is a temperature factor.

LSM parameters: REFSET_SCALE, REFSET_BFAC (refineable)

2.8 THE WAVELENGTH NORMALISATION CURVE

The wavelength normalisation curve is modelled in one or more wavelength ranges using Chebyshev polynomials. A reference wavelength 'lamref' (which must lie within one of the ranges) is defined and the wavelength normalisation scaling factor is taken to be 1.0 at that wavelength.

For a wavelength range 'lmin' to 'lmax', the normalisation curve is modelled using the following function:

     nord
    Sigma Pi*Ti(lam')
      i=0

The normalised wavelength coordinate, lam', is calculated from the wavelength 'lam' as follows:

lam' = (2*lam - lmax - lmin)/(lmax - lmin)

The normalisation factor for a given value of 'lam' is calculated using the expression:

           nord                nord
fLambda = Sigma Pi*Ti(lam') / Sigma Prefi*Ti(lam'ref)
            i=0                 i=0

Note that the LSM refinement routines will scale any refined polynomial coefficients, for all the ranges, such that the ploynomial expression for the range containing the reference wavelength will evaluate to 1.0 at that reference wavelength; input polynomial coefficients need not, however, already be normalised in such a manner.

LSM parameters: NORM_LAMREF, NORM_COEFF_r nord almin almax Pi{...}

(dataset based but in wavelength ranges - 'Pi' are refineable)

2.9 THE ABSORPTION CORRECTION

The path dependent component of the absorption scaling factor fAbs(lambda) (not taken care of by the wavelength normalisation factor fLambda) is a (lambda, psi, nu) dependent function based on polar coordinates as shown in the following diagram:

Figure 2.1 Diagram of Absorption Correction Coordinates

The function may be expressed as:

fAbs = exp (mu(lam)*Pg)

where 'mu(lam)' is the wavelength dependent linear absorption coefficient and where 'Pg' is a general path length through the crystal (a function of 'psi', 'nu').

Int the Laue Scaling Module mu(lam) may be calculated for input coefficients using the expression:

A + C*lam**3

or interpolated from a table of input values.

In the LSM there are options to carry out a global correction for the whole dataset or a local absorption correction for each individual pack. For the global correction, all 'psi' values are relative to the pattern centre for the first pack. For a local correction, the 'psi'values are relative to the pattern centre for the pack in question.

Two options for the actual function to be modelled are allowed. The first of these models the (normalised) general path length 'Pgnorm' as a function of 'psi' and 'nu'. This will have a value of 1.0 at psi=0.0, nu=0.0. The second option models the actual (normalised) absorption scaling factor itself 'Nlamref'with respect to a reference wavelength. This will also have a value of 1.0 at psi=0.0, nu=0.0. This latter option should have the advantages that it relates directly to the scaling and should be potentially more stable in a refinement as it does not need to be related to the applied scaling factor through an exponential.

For a global absorption correction, the selected function may be modelled using either Chebyshev polynomials, within angular ranges of 'psimin' to 'psimax' and 'numin' to 'numax' matched to the range of angles actually present, or using Fourier coefficients which model the whole of angular space.

For a local absorption correction, the selected function will be modelled using Chebyshev polynomials within an angular range of 'psimin' to 'psimax' and 'numin' to 'numax' matched to each individual pack.

If the normalised general path length 'Pgnorm' is modelled, the scaling factor required is calculated from the expression:

fAbs(lam) = exp (mu(lam)*t0*Pgnorm)

where 't0' is the (general) path length through the crystal at psi=0.0, nu=0.0.

If the normalised absorption scaling factor 'Nlamref' is modelled, the scaling factor required is calculated from the expression:

fAbs(lam) = (knorm*Nlamref)**(mu(lam)/mu(lamref))

where knorm = exp(mu(lamref)*t0) and where 't0' is the (general) path length through the crystal at psi=0.0, nu=0.0.

Options are available to model these functions using either Chebyshev or Fourier coefficients.

1. Chebyshev Polynomials

   This option may be used with a global or local absorption correction. The functions are modelled as follows:

                     nord  mord
   Pgnorm(psi,nu) = Sigma Sigma Pij*Ti(psi')*Tj(nu')
                      i=0   j=0
   
   or
   
                      nord  mord
   Nlamref(psi,nu) = Sigma Sigma Pij*Ti(psi')*Tj(nu')
                       i=0   j=0
   

   where 'Ti()', 'Tj()' are the Chebyshev polynomials and 'Pij' are the polynomial coefficients. Note that the first terms for each half of the sum are divided by two following the common convention for Chebyshev polynomials and hence the first term of the overall summation is divided by four.

   The function is calculated within bounds 'psimin' to 'psimax' and 'numin' to 'numax'. The normalised coordinates psi', nu' are calulated from psi, nu as follows:

   psi' = (2*psi - psimax - psimin)/(psimax - psimin)

   nu' = (2*nu - numax - numin)/(numax - numin)

   To ensure that the value at psi=0.0, nu=0.0 is 1.0, the first coefficient is calculated from the values of the remaining coefficients as follows:

                        nord  mord
   P(0,0) = 4.0*(1.0 - Sigma Sigma Pij*Ti(psi'(0,0))*Tj(nu'(0,0)) )
                         i=0   j=0
   

   calculated with P(0,0) set to 0.0. P(0,0) is not refined or input/output as an LSM parameter.

2. Fourier Coefficients

   This option may only be used with a global absorption correction. The functions are modelled as follows:

                     nord  mord
   Pgnorm(psi,nu) = Sigma Sigma [Pij*sin(i*psi+j*nu) + Qij*cos(i*psi+j*nu)]
                      i=0   j=0
   

   or

                      nord  mord
   Nlamref(psi,nu) = Sigma Sigma [Pij*sin(i*psi+j*nu) + Qij*cos(i*psi+j*nu)]
                       i=0   j=0
   

   To ensure that the value at psi=0.0,nu=0.0 is 1.0, the coefficient Q(0,0) is calculated from the values of the remaining coefficients as follows:

                   nord  mord
   Q(0,0) = 1.0 - Sigma Sigma [Pij*sin(i*psi+j*nu) + Qij*cos(i*psi+j*nu)]
                    i=0   j=0
   

   calculated with Q(0,0) set to 0.0. Note also that P(0,0) can make no contribution to the function. Thus these two parameters are not refined or input/output as LSM parameters.

LSM parameters:

ABSCOR_MU C [A] or ABSCOR_MU alam1 mu1 alam2 mu2 ... (fixed)

ABSCOR_THICK[] (pack based - fixed)

plus

1. For a global absorption correction

   Chebyshev modelling

   ABSCOR_GLOBAL 1 nord mord psimin_glob psimax_glob numin numax p{...}

   P{...} refineable except for P(0,0) (omitted), others all fixed.

   Fourier modelling

   ABSCOR_GLOBAL 2 nord mord P{...} Q{...}

   P{...} refineable except for P(0,0) (omitted), Q{...} refineable except for Q(0,0) (omitted), others all fixed.

2. For a local absorption correction

   ABSCOR_LOCAL[] nord mord psimin_loc psimax_loc numin numax P{...}

   P{...} refineable except for P(0,0) (omitted), others all fixed.