B The PISA profiling function

The PISA profiling function is made up of three functions, a Gaussian, an exponential and a Lorentzian. Inside a radius Rc fixed proportions of the Gaussian and Lorentzian functions are used, outside of Rc the exponential replaces the Gaussian and is added to the continuing Lorentzian. The exponential is joined smoothly to the Gaussian. Inside of Rc the function takes the form:-

1 πσ2(1 + ( τ2 ln ( 1 τ)))¯ Q (1 + r2 σ2 ln(2))¯ + (1 Q) exp(r2 σ2 )

and outside of Rc it takes the form:-

1 πσ2(1 + ( τ2 ln ( 1 τ)))¯ Q (1 + r2 σ2 ln(2))¯ + (1 Q) exp(2r σ ln (1 τ))̲ τ


τ =
the fraction of the peak intensity at which to change from the gaussian to an exponential function (CROSS/100),
σ =
the gaussian function sigma (GSIGM),
Q =
the fraction of the Lorentzian function to add to the gaussian or exponential function at each point (COMIX).

The radius at which the exponential replaces the gaussian is:-

Rc = σln ( 1 τ)

The actual function is that which when multiplied by the integrated intensity gives the intensity at the given radius. Basically the functional forms from which the above equations are derived are:-

Gaussian — exp(r2 σ2 )
Lorentzian — 1/(1 + r2 σ2 )
Exponential — exp((rRc σ )).