# Peak shapes

Some various peak shapes that I've found useful in doing things like crystallite size distributions, solid solutions and the like.

Peak shape is defined by a FWHM, not a standard deviation. The lognormal peak is defined by a peak position (being the point of maximum intensity) and a FWHM, which is naturally asymmetric. All peak areas are normalised to 1 and allow for scaling. They will work with v5; some more work on them is required to back them in to v4.

Gaussian

```macro & Gau(& int,& posn,& fwhm,& x)	{
int (2 Sqrt(Ln(2)/Pi)/fwhm) Exp(-4 Ln(2) ((x-posn)/fwhm)^2)
}```

Lorenztian

```macro & Lor(& int,& posn,& fwhm,& x)	{
int (1/(2 Pi)) (fwhm / ((fwhm/2)^2 + (x-posn)^2))
}```

Pearson VII

```macro & PeaVII(& int,& posn,& fwhm,& m,& x)	{
int (2 Sqrt((2^(1/m)-1)/(fwhm^2)) Gamma_Approx(m)/(Sqrt(Pi) Gamma_Approx(m-1/2))) ((fwhm/2)^(2 m)/((fwhm/2)^2 + (2^(1/m) - 1)*(x-posn)^2)^m)
}```

Lognormal

```macro & lognormal(& int, & posn, & fwhm, & x)	{
int/(Sqrt(2 Pi) LN_sigma(posn,fwhm) x) Exp(-((Ln(x)-LN_mu(posn,fwhm))/(Sqrt(2) LN_sigma(posn,fwhm)))^2)
}

macro & LN_r(& p, & w) {
(w + Sqrt(4 p^2 + w^2)) / (2 p)
}
macro & LN_mu(& p, & w) {
LN_r(p,w)^2/(2 Ln(2)) + Ln(p)
}
macro & LN_sigma(& p, & w) {
LN_r(p,w)/Sqrt(2 Ln(2))
}```

pseudo-Voigt

```macro & PV(& int,& posn,& fwhm, & mix, & x)	{
mix Gau(int, posn, fwhm,x) + (1 - mix) Lor(int, posn, fwhm,x)
}```

Split functions

```'all are normalised to unit area'
macro & split_Gaussian(& int,& posn,& hwL,& hwR,& x)	{
If(x < posn, 2 Gau(int,posn,2 hwL,x) / (1 + hwR/hwL),
2 Gau(int,posn,2 hwR,x) (hwR/hwL) / (1 + hwR/hwL)
)
}

macro & split_Lorentzian(& int,& posn,& hwL,& hwR,& x)	{
If(x < posn, 2 Lor(int,posn,2 hwL,x) / (1 + hwR/hwL),
2 Lor(int,posn,2 hwR,x) (hwR/hwL) / (1 + hwR/hwL)
)
}

macro & split_PV(& int,& posn,& hwL,& hwR, & mixL, & mixR, & x)	{
If(x < posn, 2 PV(int,posn,2 hwL, mixL, x) / (1 + (PV(int,posn,2 hwL,mixL,posn)/PV(int,posn,2 hwR,mixR,posn))),
2 PV(int,posn,2 hwR, mixR, x) (PV(int,posn,2 hwL,mixL,posn)/PV(int,posn,2 hwR,mixR,posn)) / (1 + (PV(int,posn,2 hwL,mixL,posn)/PV(int,posn,2 hwR,mixR,posn)))
)
}

macro & split_PeaVII(& int,& posn,& hwL,& hwR, & mL, & mR, & x)	{
If(x < posn, 2 PeaVII(int,posn,2 hwL, mL, x) / (1 + (PeaVII(int,posn,2 hwL,mL,posn)/PeaVII(int,posn,2 hwR,mR,posn))),
2 PeaVII(int,posn,2 hwR, mR, x) (PeaVII(int,posn,2 hwL,mL,posn)/PeaVII(int,posn,2 hwR,mR,posn)) / (1 + (PeaVII(int,posn,2 hwL,mL,posn)/PeaVII(int,posn,2 hwR,mR,posn)))

)
}

macro & split_Lognormal(& int,& posn,& hwL,& hwR,& x)	{
If(x < posn, 2 lognormal(int,posn,2 hwL,x) / (1 + (lognormal(int,posn,2 hwL,posn)/lognormal(int,posn,2 hwR,posn))),
2 lognormal(int,posn,2 hwR,x) (lognormal(int,posn,2 hwL,posn)/lognormal(int,posn,2 hwR,posn)) / (1 + (lognormal(int,posn,2 hwL,posn)/lognormal(int,posn,2 hwR,posn)))
)
}```