The Fitting Program MOSFIT

The general fitting program MOSFIT is based on work by Longworth at U.K.A.E.A., Harwell [1]. It takes as input a set of folded data and a file containing the initial guess to the parameters which describe the spectrum. The fitting parameters are adjusted until the best fit to the data is achieved. Output from the program consists of the generated fit to the data and the associated final parameter values. MOSFIT can be used to fit both

experimental data and calibration spectra.

Calibration spectra are fitted to independent Lorentzian line shapes with no constraints placed on their relative position, width or depth. No scaling of the velocity range is undertaken, instead the fit is generated in

terms of channels. This enables a calibration constant CAL (channels/mms’1) to be determined and allows for an assessment of transducer drive linearity

($2.2.6).

The calibration constant enables the velocity scale for successive spectra of ’unknown’ materials to be defined. The fit to these ’unknown’ materials can then be generated in terms of Doppler velocity (mms'1) rather than channels. The initial guess for the Mossbauer parameters which describe the fit are made in terms of logical Mossbauer ’phases’. A phase is either a singlet, doublet or sextet. This system leads to the imposition of certain logical constraints on the relative position, width and depth of the lines of each phase.

The program operates so as to minimise the sum of the squares of the residuals R(i).

R(i) _ FIT(i) - DATA® (DATA®)*

3.1

hence, X2 ~ —— --- 5 (R(0)2 3.2

NCHAN-N <

NCHAN = number of (folded) data points

N = number of independent fitting parameters FIT(i) = calculated value of the spectrum in channel i DATA(i) = experimental value of the spectrum in channel i

For this data capture process the errors in DATA® will be due to counting statistics and will have a standard deviation of (DATA(i))V2. Hence the denominator in Eqn. 3.1 is the statistical weighting factor.

The values of FIT(i) are composed of a single background count value BGND and the sum of contributions from each phase present in the spectrum:

F 3.3

FIT(i)=BGND + > B(j)COUNT(i j) j-i

NP = number of phases B(j) = depth of phase j

COUNT(ij) = calculated value of phase j at channel i The program proceeds as follows:

i) Read the initial guesses to the Mossbauer parameters PRMTR(k) which describe the spectrum DATA®.

iii) Calculate the ’best’ values for BGND and B(j) using a linear least squares regression algorithm (LISFT)

iv) Calculate FIT(i) using Eqn. 3.3 v) Calculate chi-squared using Eqn. 3.2

vi) Adjust the parameters PRMTR(k) using the routine VA05A.

vii) Repeat steps ii), iii), iv), v), vi) until a minimum chi-squared value is reached.

viii) Write FIT(i) and COUNT(ij) to an output file and the final values of the parameters PRMTR(k) to a listing file.

The routine VA05A contains a non-linear least squares regression algo­ rithm which uses aspects from Newton-Raphson, steepest descent and Leven- berg-Marquardt methods [2]. For a simple spectrum, for instance two quad­ rupole doublets, the minimum chi-squared value is found within approximately twenty calls to VA05A. This assumes that reasonable guesses are made for the initial parameter values PRMTR(k).

Each phase will be described by a number of fitting parameters. For each singlet there are two parameters:

PRMTR(l) = 5 (Chemical isomer shift) PRMTR(2) = T/2 (half width at half height)

hence the contributions to the COUNT array will describe a Lorentzian line (Eqn. 2.19) with a half width T/2 centred at 5 :

COUNT(i,singlet)=

2 3.4

For each doublet there are four parameters :

PRMTR(3) = 5 (chemical isomer shift) PRMTR(4) = Q (half quadrupole splitting) PRMTR(5) = I L

2 (half width of lower velocity line) PRMTR(6) = r R (half width of higher velocity line)

2

hence contributions to the COUNT array is the sum of two Lorentzian lines 1 COUNT(i,doublet) = 2 ^ (1 + P l2) + t ^ ( 1 +Pr2)) 2 2 3.5 Pl = i - 6+ Q_{_£} 2 p _ » - 5 - Q r rA_d

The situation for a sextet is even more complex with seven independent fitting parameters.

PRMTR(7) = 5 (chemical isomer shift) PRMTR(8) = H (magnetic field strength PRMTR(9) = e (quadrupole perturbation) PRMTR(IO) = T (half width of lines) PRMTR(ll) = dr (incremental half width)

PRMTR(12) = I25 (relative intensity of lines 2 and 5) PRMTR(13) = I16 (relative intensity of lines 1 and 6)

The magnetic field strength H is expressed as the half splitting (in mm/s) of lines 1 and 6 of the sextet. The incremental width dT is designed to cope with slight distributions in field strength which will result in the 3 pairs of lines having different widths. The half widths of the lines are given by:

wk =

+ (5T

Hk = half splitting of each pair of lines

The relative intensity parameters 1^ and I16 allow for any orientation and magnetisation effects in the absorber which may alter the transition probabilities. For a polycrystalline absorber I25 = 0.67 and I16 = 1.0. The contribution to the COUNT array from a sextet is given by :

/ 3.7

COUNT(i^extet) = ^ }

k-J

RI(k) is the relative intensity of each pair of lines : RI(1) = RI(6) - I16 / (I16 + I* + 0.33)

RI(2) = RI(5) = I v I (I16 + 125 + 0.33) RI(3) = RI(4) = 0.33 / (I16 + I25 + 0.33) Starting from the most negative velocity line :

Pj « (i - 6 - e + H) / (T + 0O P2 = (i - 5 + e + ZH) I CT+Z0T) P3 = (i - 5 ♦ e - 2ZH) / (T + 2ZdT) P4 = (i - 5 + e + 2ZH) / (T + 2ZdT) P5 = (i - 5 + e - ZH) I (T + ZdT) P6 = (i - 5 - 8 - H) I (T + 0D

Where Z = separation of lines 5 and 6 H

= separation of lines 4 and 5 H

Various constraints may be placed on the otherwise independent varia­ tion of the fitting parameters PRMTR(k) by the routine VA05A. The parame­ ters may be fixed at their starting values, for instance it may be desirable to force the incremental line width 5T to remain at O.Omm/s. It is also possible to link parameters so that they vary, yet maintain the same starting ratio. This is often of use for forcing the line widths of a quadrupole doublet to be equal. The facility exists for forcing different phases of a fit to account for a set percentage of the total absorption area. This constraint could conceivably be employed if the ratio of two (or more) chemical sites had been determined by an independent analytical method. It must be assumed that the chemical sites yield the same recoil-free fraction in order to apply this constraint.

The regression routine VA05A operates so as to minimise the chi- squared value of the fit. A final chi-squared value close to or below unity can be regarded as satisfactory. However, it is difficult to use the chi-squared test to compare the adequacy of fits of different spectra unless they have been recorded under identical conditions. Better chi-squared values automatically follow from poorer quality data. For this reason a second goodness-of-fit parameter is also calculated which is independent of the quality of the data. This parameter is known as MISFIT [3]:

3.8 _ Discrepancy = D_

The discrepancy D is defined in a similar manner to chi-squared

NCHAN _ „ ^

3.9

p ■ i t rosfti

" nchan (x2~i}

i-i

The signal S is a measure of the difference between the data DATA® and the background count BGND. Its value is dependent upon the count rate and duration of the experiment, hence a higher signal S implies higher quality data: NCHAN