The computer model

Computer code was written to model the scattering system so that the scattered spectrum could be predicted for a given set of parameters and to help in the design of an optimum diffraction system for a given application. As stated above, there are several parameters in the system that play a part in the final profile of the spectrum. The program uses diffraction data for a chosen material obtained from the powder diffraction files, (JCPDS 1961). The data used in the program are the scattering plane spacings, d, and the relative intensities of the scattering from those planes. The program calculates the scattered spectrum for a given incident x-ray spectrum up to 150 kV and any scatter angle from 3 to 10 degrees. The program also enables the geometry of the collimation to be changed. Three types of photon beam are modelled, a pencil beam, a ribbon beam, and a conical beam (obtained using annular collimation). Different shapes of target sample are also modelled, so that for a given material the parameters can be varied until the optimum spectrum for a given application is achieved.

Chapter Two Modelling Coherent Scattered Spectra

The format of the modelled data is either intensity as a function of energy or intensity as a function of momentum transfer. The energy or momentum transfer axis has 512 values to simulate the number of channels in the multi channel analyser being used in this study. In principle, this number of channels could be changed in the model to suit any number of channels used in the MCA.

2.1.1. The diffraction data line spectrum

The spectrum produced from the JCPDS diffraction data is one which gives a line of a certain intensity at each energy from the incident x-ray spectrum that satisfies the Bragg law for the material under investigation. The model first calculates this hue spectrum for a particular material up to the end point energy of the incident spectrum and a given scatter angle. The diffraction data is read in and the wavelengths that satisfy the Bragg law (see equation 1.12) are calculated and converted to energy values using equation 1.15. The model then places the calculated energy values into the appropriate energy channel that simulates the MCA.

Figure 2.1 shows the line spectra for calcium carbonate, polyethylene and hydroxylapatite and compares them to the measured spectra which was measured with an incident spectrum of 70 kV and the angle of scatter 6 degrees. Clearly there are factors that are spreading and blurring the spectrum and these factors have to be considered in the model. The following are considered and their effect on the line spectrum modelled.

Angular blurring

The probability of a coherent scattering event occurring as a function of energy.

The polyenergetic incident spectrum.

Attenuation of the incident and scattered beam in the sample material. The contribution to the spectrum from Compton scattering.

Detector resolution.

— C a C 0 3 m easured — M ixlelled line spectrum

— Polyethylene m easured — M odelled line spectrum

H ydroxylapatite m easured — M ixlelled line spectrum

0.6 0.6 0.4 0.4 0.2 70 30 40 5( E n e r g y k e V 60 70 30 40 5( E n e r g y k e V 60 E n e r g y k e V

Figure 2.1 : Comparison o f the line spectrum and m easured spectrum fo r calcium carbonate, polyethylene and hydroxylapatite. The incident spectrum used fo r the measurement was 70 kV and an angle o f scatter 6 degrees. A ll spectra have been normalised to the highest peak fo r comparison

purposes. 2.1.2 Angular blurring T h e l i n e s p e c t r a s h o w n i n f i g u r e 2 . 1 a r e f o r a f i x e d s c a t t e r i n g a n g l e o f 6 d e g r e e s , i . e . i t i s a s s u m e d t h a t o n l y p h o t o n s s c a t t e r e d a t t h i s a n g l e f r o m t h e o b j e c t r e a c h t h e d e t e c t o r a n d c o n t r i b u t e t o t h e s p e c t r u m . I n o r d e r t o t r y t o a c h i e v e t h i s i d e a l i n t h e m e a s u r e d s p e c t r u m , b o t h t h e i n c i d e n t a n d s c a t t e r e d b e a m s n e e d t o b e c o l l i m a t e d . T o o b t a i n g o o d a n g u l a r r e s o l u t i o n t h e p h o t o n b e a m w i d t h n e e d s t o b e a s s m a l l a s p o s s i b l e , w h i l e t h e d i s t a n c e s b e t w e e n t h e p r i m a r y c o l l i m a t o r a p e r t u r e s a n d t h e s c a t t e r e d c o l l i m a t o r a p e r t u r e s n e e d t o b e a s l a r g e a s p o s s i b l e . T h i s h a s d i s a d v a n t a g e s i n p r a c t i c e b e c a u s e a s t h e p h o t o n b e a m w i d t h i s r e d u c e d a n d t h e a p e r t u r e d i s t a n c e s i n c r e a s e d , t h e n u m b e r o f p h o t o n s t h a t p a s s t h r o u g h t h e s y s t e m d e c r e a s e s . T h i s i n t u r n l e a d s t o t h e n e e d f o r i n c r e a s e d m e a s u r i n g t i m e s t o o b t a i n a c c e p t a b l e c o u n t i n g s t a t i s t i c s . F i g u r e 2 . 2 s h o w s t h a t b e c a u s e o f t h e f i n i t e d i m e n s i o n s o f t h e c o l l i m a t o r g e o m e t r y , t h e r e a r e a r a n g e o f s c a t t e r i n g a n g l e s t h a t c a n r e a c h t h e d e t e c t o r . T h e r a n g e o f a n g l e s t h a t c a n r e a c h t h e d e t e c t o r w i l l d e p e n d o n t h e t y p e o f c o l l i m a t i o n u s e d , i . e . p e n c i l b e a m c o l l i m a t i o n , r i b b o n b e a m c o l l i m a t i o n o r a n n u l a r b e a m c o l l i m a t i o n . T h e r a n g e c a n b e c a l c u l a t e d f r o m t h e g e o m e t r y a n d f i g u r e 2 . 3 s h o w s t h e a n g u l a r d e v i a t i o n f o r p e n c i l b e a m g e o m e t r y e i t h e r s i d e o f t h e c h o s e n s c a t t e r a n g l e f o r

Chapter Two Modelling Coherent Scattered Spectra

four beam diameters, and various collimator aperture separation distances. The calculations assume that the distance between the primary beam collimator apertures and the scattered beam collimator apertures are the same.

S c a tte re d beam colllmatore

Figure 2.2 : The finite dimensions o f the collimation geometry give rise to angular blurring.

2.2