Derivation of FWHM and Height for Peak Fitting Functions

Gaussian (Normal distribution)
Lorentzian
Voigt
Pseudo Voigt
Moffat
Pearson 7
Students T
Breit-Wigner-Fano
Lognormal
Damped Oscillator
Damped Harmonic Oscillator
Exponential Gaussian
Skewed Gaussian
Donaich Sunjic (photo-emission)

Gaussian (Normal distribution)

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

$f(x;A,\mu,\sigma)=\frac{A}{\sigma\sqrt{2\pi}}\exp\left[\frac{-(x-\mu)^2}{2\sigma^2}\right]$

Gaussian Height

Max height occurs at x = $\mu$

$f(\mu;A,\mu,\sigma)=\frac{A}{\sigma\sqrt{2\pi}}\exp\left[\frac{-(\mu-\mu)^2}{2\sigma^2}\right]$ $\operatorname{height} = \frac{A}{\sigma\sqrt{2\pi}}$

Gaussian FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\exp\left[\frac{-x_0^2}{2\sigma^2}\right] = 2^{-1}$ $x_0 = \pm\sigma\sqrt{2\ln 2}$ $\operatorname{FWHM}=2\sigma\sqrt{2\ln{2}}\approx 2.3548\sigma$

Lorentzian

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

$f(x;A,\mu,\sigma)=\frac{A}{\pi}\left[\frac{\sigma}{(x-\mu)^2+\sigma^2}\right]$

Lorentzian Height

Max height occurs at x = $\mu$

$f(\mu;A,\mu,\sigma)=\frac{A}{\pi}\left[\frac{\sigma}{(\mu-\mu)^2+\sigma^2}\right]$ $\operatorname{height} = \frac{A}{\sigma\pi}$

Lorentzian FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{\sigma}{x_0^2+\sigma^2}=\frac{1}{2\sigma}$ $2\sigma^2=x_0^2+\sigma^2$ $x_0=\pm\sigma$ $\operatorname{FWHM}=2\sigma$

Voigt

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see Wikipedia for more info)

$f(x;A,\mu,\sigma,\gamma)=\frac{A\textrm{Re}[w(z)]}{\sigma\sqrt{2\pi}}$

where

$z=\frac{x-\mu +i\gamma}{\sigma\sqrt{2}}$ $w(z)=e^{-z^2}{\operatorname{erfc}}(-iz)$

erfc is the complimentary error function. As above,

Voigt Height

Max height occurs at x = $\mu$

$z(\mu;\mu,\sigma,\gamma)=\frac{i\gamma}{\sigma\sqrt{2}}$ $f(\mu;A,\mu,\sigma,\gamma)=\frac{A\textrm{Re}[w(z)]}{\sigma\sqrt{2\pi}}$ $\operatorname{height}=\frac{A}{\sigma\sqrt{2\pi}}\textrm{Re}\left[w\left(\frac{i\gamma}{\sigma\sqrt{2}}\right)\right]$

Voigt FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{A}{\sigma\sqrt{2\pi}}\textrm{Re}\left[w\left(\frac{x_0+i\gamma}{\sigma\sqrt{2}}\right)\right]= \frac{1}{2}\frac{A}{\sigma\sqrt{2\pi}}\textrm{Re}\left[w\left(\frac{i\gamma}{\sigma\sqrt{2}}\right)\right]$ $e^{-\left(\frac{x_0+i\gamma}{\sigma\sqrt{2}}\right)^2}{\operatorname{erfc}}(-i\left(\frac{x_0+i\gamma}{\sigma\sqrt{2}}\right))= e^{-\left(\frac{i\gamma}{\sigma\sqrt{2}}\right)^2}{\operatorname{erfc}}(-i\left(\frac{i\gamma}{\sigma\sqrt{2}}\right))$ $e^{-\left(\frac{x_0+i\gamma}{\sigma\sqrt{2}}\right)^2}{\operatorname{erfc}}(-i\left(\frac{x_0+i\gamma}{\sigma\sqrt{2}}\right))= e^{-\left(\frac{i\gamma}{\sigma\sqrt{2}}\right)^2}{\operatorname{erfc}}(-i\left(\frac{i\gamma}{\sigma\sqrt{2}}\right))$

Hmmm IDK where to go from here.

$\operatorname{FWHM}\approx 3.6013\sigma$

Pseudo Voigt

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\alpha$ = fraction (see Wikipedia for more info)

$f(x;A,\mu,\sigma,\alpha)= \frac{(1-\alpha)A}{\sigma_g\sqrt{2\pi}} e^{[{-{(x-\mu)^2}/{2\sigma_g^2}}]} +\frac{\alpha A}{\pi} \big[\frac{\sigma}{(x - \mu)^2 + \sigma^2}\big]$

where $\sigma_g = {\sigma}/{\sqrt{2\ln{2}}}$ so that the full width at half maximum of each component and of the sum is $2\sigma$

Pseudo Voigt Height

Max height occurs at x = $\mu$

$f(\mu;A,\mu,\sigma,\alpha)= \frac{(1-\alpha)A}{\sigma_g\sqrt{2\pi}} +\frac{\alpha A}{\pi\sigma}$ $\operatorname{height}=\frac{(1-\alpha)A}{\sigma\sqrt{\pi/\ln(2)}} +\frac{\alpha A}{\pi\sigma}$

Pseudo Voigt FWHM

The definition of the function depends on FWHM=2 $\sigma$ .

Please see derivations of Gaussian and Lorentzian.

$\operatorname{FWHM}=2\sigma$

Moffat

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\beta$ = beta (see Wikipedia for more info)

$f(x; A, \mu, \sigma, \beta) = A \big[\left(\frac{x-\mu}{\sigma}\right)^2+1\big]^{-\beta}$

Moffat Height

Max height occurs at x = $\mu$

$f(\mu; A, \mu, \sigma, \beta) = A \big[1\big]^{-\beta}$ $\operatorname{height}=A$

Moffat FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\big[\left(\frac{x}{\sigma}\right)^2+1\big]^{-\beta}=2^{-1}$ $\left(\frac{x}{\sigma}\right)^2=2^{1/\beta}-1$ $x=\pm\sigma\sqrt{2^{1/\beta}-1}$ $\operatorname{FWHM}=2\sigma\sqrt{2^{1/\beta}-1}$

Pearson 7

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $m$ = exponent (see wikipedia for more info)

$f(x;A,\mu,\sigma,m)=\frac{A}{\sigma{\beta(m-\frac{1}{2},\frac{1}{2})}}\left[1+\frac{(x-\mu)^2}{\sigma^2}\right]^{-m}$

where $\beta$ is the beta function

Pearson 7 Height

Max height occurs at x = $\mu$

$f(\mu;A,\mu,\sigma,m)=\frac{A}{\sigma{\beta(m-\frac{1}{2},\frac{1}{2})}}\left[1+\frac{(\mu-\mu)^2}{\sigma^2}\right]^{-m}$ $\operatorname{height}=\frac{A}{\sigma{\beta(m-\frac{1}{2},\frac{1}{2})}}$

Pearson 7 FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\left[1+\frac{(x_0)^2}{\sigma^2}\right]^{-m}=\frac{1}{2}$ $x_0=\pm\sigma\sqrt{\left(\frac{1}{2}\right)^{-1/m}-1}$ $\operatorname{FWHM}=2\sigma\sqrt{2^{1/m}-1}$

Students T

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

$f(x;A,\mu,\sigma)=\frac{A\Gamma(\frac{\sigma+1}{2})}{\sqrt{\sigma\pi}\,\Gamma(\frac{\sigma}{2})} \Bigl[1+\frac{(x-\mu)^2}{\sigma}\Bigr]^{-\frac{\sigma+1}{2}}$

where $\Gamma(x)$ is the gamma function.

Students T Height

Max height occurs at x = $\mu$

$f(\mu;A,\mu,\sigma,m)= \frac{A\Gamma(\frac{\sigma+1}{2})}{\sqrt{\sigma\pi}\,\Gamma(\frac{\sigma}{2})} \Bigl[1+\frac{(\mu-\mu)^2}{\sigma}\Bigr]^{-\frac{\sigma+1}{2}}$ $\operatorname{height}= \frac{A\Gamma(\frac{\sigma+1}{2})}{\sqrt{\sigma\pi}\,\Gamma(\frac{\sigma}{2})}$

Students T FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{A\Gamma(\frac{\sigma+1}{2})}{\sqrt{\sigma\pi}\,\Gamma(\frac{\sigma}{2})} \Bigl[1+\frac{(x_0)^2}{\sigma}\Bigr]^{-\frac{\sigma+1}{2}}= \frac{1}{2}\frac{A\Gamma(\frac{\sigma+1}{2})}{\sqrt{\sigma\pi}\,\Gamma(\frac{\sigma}{2})}$ $\Bigl[1+\frac{(x_0)^2}{\sigma}\Bigr]^{-\frac{\sigma+1}{2}}=2^{-1}$ $x_0^2=\sigma(2^{\frac{2}{\sigma+1}}-1)$ $x_0=\pm\sqrt{2^{\frac{2}{\sigma+1}}\sigma-\sigma}$ $\operatorname{FWHM}=2\sqrt{2^{\frac{2}{\sigma+1}}\sigma-\sigma}$

Breit-Wigner-Fano

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $q$ = exponent (see Wikipedia for more info)

$f(x;A,\mu,\sigma,q)=\frac{A(q\sigma/2+x-\mu)^2}{(\sigma/2)^2+(x-\mu)^2}$

Breit-Wigner-Fano Height

Max height occurs at x = $\mu$

$f(\mu;A,\mu,\sigma,m)=\frac{A(q\sigma/2+\mu-\mu)^2}{(\sigma/2)^2+(\mu-\mu)^2}$ $=\frac{A(q\sigma/2)^2}{(\sigma/2)^2}$ $\operatorname{height}=A(q)^2$

Breit-Wigner-Fano FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{A(q\sigma/2+x_0)^2}{(\sigma/2)^2+(x_0)^2}=\frac{1}{2}A(q)^2$ $2(q\sigma/2+x_0)^2=q^2\left((\sigma/2)^2+(x_0)^2\right)$ $\frac{q^2\sigma^2}{2}+2q\sigma x_0+2x_0^2=\frac{q^2\sigma^2}{4}+x_0^2q^2$ $x_0^2(2-q^2)+2q\sigma x_0+\frac{q^2\sigma^2}{4}=0$ $x_0^2(q^2-2)-2q\sigma x_0-\frac{q^2\sigma^2}{4}=0$ $x = 2q\sigma \pm \frac{\sqrt{4q^2\sigma^2 + (q^2-2)(q^2\sigma^2)}}{2(q^2-2)}=0$ $\operatorname{FWHM}=2\frac{\sqrt{q^2\sigma^2(q^2+2)}}{2(q^2-2)}$

Lognormal

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

$f(x; A, \mu, \sigma) = \frac{A}{\sigma\sqrt{2\pi}}\frac{e^{-(\ln(x) - \mu)^2/ 2\sigma^2}}{x}$

Lognormal Height

Max height occurs at x = $e^{\mu-\sigma^2}$

$f(e^{\mu-\sigma^2};A,\mu,\sigma)=\frac{A}{\sigma\sqrt{2\pi}} \frac{e^{-(\ln(e^{\mu-\sigma^2})-\mu)^2/(2\sigma^2)}}{e^{\mu-\sigma^2}}$ $=\frac{A}{\sigma\sqrt{2\pi}}\frac{e^{\sigma^2/2}}{e^{\mu-\sigma^2}}$ $\operatorname{height}=\frac{A}{\sigma\sqrt{2\pi}}e^{\frac{\sigma^2}{2}-\mu}$

Lognormal FWHM

FWHM is found by finding the values of x at 1/2 the max height.

$f(x_0;A,\mu,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{A}{\sigma\sqrt{2\pi}}\frac{e^{-(\ln(x_0) - \mu)^2/ 2\sigma^2}}{x_0}= \frac{A}{2\sigma\sqrt{2\pi}}e^{\frac{\sigma^2}{2}-\mu}$ $\frac{e^{-(\ln(x_0) - \mu)^2/ 2\sigma^2}}{x_0}= \frac{1}{2}e^{\frac{\sigma^2}{2}-\mu}$ $\frac{x_0}{2}=e^{\frac{-(\ln(x_0) - \mu)^2}{2\sigma^2}-(\frac{\sigma^2}{2}-\mu)}$ $2\sigma^2\ln{x_0}+(\ln(x_0) - \mu)^2=2\sigma^2\left(\mu-\frac{\sigma^2}{2}+\ln{2}\right)$ $2\sigma^2\ln{x_0}+\ln(x_0)^2 - 2\mu\ln(x_0)+\mu^2=2\sigma^2\ln{2}-\sigma^4+2\sigma^2\mu-\mu^2$ $\ln(x_0)^2 + 2(\sigma^2-\mu)\ln(x_0)+(\sigma^2-\mu)^2=2\sigma^2\ln{2}-(\sigma^2-\mu)^2+(\sigma^2-\mu)^2$ $(\ln(x_0)+(\sigma^2-\mu))^2=2\sigma^2\ln{2}$ $\ln(x_0)=(\mu-\sigma^2)\pm\sqrt{2\sigma^2\ln{2}}$ $\operatorname{FWHM}=e^{(\mu-\sigma^2)+\sqrt{2\sigma^2\ln{2}}}-e^{(\mu-\sigma^2)-\sqrt{2\sigma^2\ln{2}}}$

Damped Oscillator

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

$f(x;A,\mu,\sigma)=\frac{A}{\sqrt{[1-(x/\mu)^2]^2+(2\sigma x/\mu)^2}}$

Damped Oscillator Height

Max height occurs at x = $\mu$

$f(\mu;A,\mu,\sigma)=\frac{A}{\sqrt{[1-(\mu/\mu)^2]^2+(2\sigma \mu/\mu)^2}}$ $\operatorname{height}=\frac{A}{2\sigma}$

Damped Oscillator FWHM

FWHM is found by finding the values of x at 1/2 the max height.

$f(x_0;A,\mu,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{A}{\sqrt{[1-(x_0/\mu)^2]^2+(2\sigma x_0/\mu)^2}}=\frac{1}{2}\frac{A}{2\sigma}$ $(4\sigma)^2 = [1-(x_0/\mu)^2]^2+(2\sigma x_0/\mu)^2$ $1-2\frac{x_0^2}{\mu^2}+\frac{x_0^4}{\mu^4}+\frac{4\sigma^2 x_0^2}{\mu^2}-16\sigma^2=0$ $\frac{x_0^4}{\mu^4}+x_0^2\left(\frac{4\sigma^2-2}{\mu^2}\right)+1-16\sigma^2=0$ $x_0^4+x_0^2\mu^2(4\sigma^2-2)+\mu^2-16\mu^2\sigma^2=0$

Substitute $y=x_0^2$

$y^2+y\mu^2(4\sigma^2-2)+\mu^2-16\mu^2\sigma^2=0$ $y=\frac{-\mu^2(4\sigma^2-2)\pm\sqrt{(2\mu^2(2\sigma^2-1))^2-4(\mu^2-16\mu^2\sigma^2)}}{2}$ $y=\frac{\mu^2(2-4\sigma^2)\pm \sqrt{4\mu^4((4\sigma^4-4\sigma^2+1)-(1-16\sigma^2))}}{2}$ $y=\frac{\mu^2(2-4\sigma^2)\pm \sqrt{4\mu^4(4\sigma^4+12\sigma^2)}}{2}$ $y=\mu^2(1-2\sigma^2)\pm 2\sqrt{\mu^4(\sigma^4+3\sigma^2)}$

Substitute back $y=x_0^2$

$x_0=\pm\sqrt{\mu^2(1-2\sigma^2)\pm 2\sqrt{\mu^4\sigma^2(\sigma^2+3)}}$

Two peaks each with a FWHM of:

$\operatorname{FWHM}=\sqrt{\left|\mu^2(1-2\sigma^2)+ 2\sqrt{\mu^4\sigma^2(\sigma^2+3)}\right|} -\sqrt{\left|\mu^2(1-2\sigma^2)- 2\sqrt{\mu^4\sigma^2(\sigma^2+3)}\right|}$

Damped Harmonic Oscillator

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see Wikipedia and DAVE/PAN for more info)

$f(x;A,\mu,\sigma,\gamma)=\frac{A\sigma}{\pi [1 - \exp(-x/\gamma)]} \Big[ \frac{1}{(x-\mu)^2 + \sigma^2} - \frac{1}{(x+\mu)^2 + \sigma^2} \Big]$

DHO Height

Max height occurs at x = $\mu$ .

$f(\mu;A,\mu,\sigma,\gamma)=\frac{A\sigma}{\pi [1 - \exp(-\mu/\gamma)]} \Big[ \frac{1}{(\mu-\mu)^2 + \sigma^2} - \frac{1}{(\mu+\mu)^2 + \sigma^2} \Big]$ $\operatorname{height}=\frac{A\sigma}{\pi [1 - \exp(-\mu/\gamma)]} \Big[\frac{1}{\sigma^2} - \frac{1}{4\mu^2 + \sigma^2} \Big]$

DHO FWHM

FWHM is found by finding the values of x at 1/2 the max height.

$f(x_0;A,\mu,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{1}{1 - \exp(-x_0/\gamma)} \Big[ \frac{1}{(x_0-\mu)^2 + \sigma^2} - \frac{1}{(x_0+\mu)^2 + \sigma^2} \Big] =\frac{1}{1 - \exp(-\mu/\gamma)} \Big[\frac{1}{\sigma^2} - \frac{1}{4\mu^2 + \sigma^2} \Big]$ $(1 - \exp(-\mu/\gamma)) \Big[ \frac{1}{(x_0-\mu)^2 + \sigma^2} - \frac{1}{(x_0+\mu)^2 + \sigma^2} \Big] =(1 - \exp(-x_0/\gamma)) \Big[\frac{1}{\sigma^2} - \frac{1}{4\mu^2 + \sigma^2} \Big]$

Hmmm IDK where to go from here.

But as defined in DAVE/PAN:

$\operatorname{FWHM}\approx 2\sigma$

Exponential Gaussian

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see [Wikipedia(http://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution)

$f(x; A, \mu, \sigma, \gamma) = \frac{A\gamma}{2} \exp\left[\gamma(\mu - x + \gamma\sigma^2/2)\right] \operatorname{erfc}\left(\frac{\mu + \gamma\sigma^2 - x}{\sqrt{2}\sigma}\right)$

where :func:erfc is the complimentary error function.

Exponential Gaussian Height

Max height occurs at x = $\mu$

$f(\mu; A, \mu, \sigma, \gamma) = \frac{A\gamma}{2} \exp\left[\gamma(\mu - \mu + \gamma\sigma^2/2)\right] \operatorname{erfc}\left(\frac{\mu + \gamma\sigma^2 - \mu}{\sqrt{2}\sigma}\right)$ $\operatorname{height}=\frac{A\gamma}{2} \exp\left[\frac{\gamma^2\sigma^2}{2}\right] \operatorname{erfc}\left(\frac{\gamma\sigma}{\sqrt{2}}\right)$

Exponential Gaussian FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{A\gamma}{2} \exp\left[\gamma(x_0 + \gamma\sigma^2/2)\right] \operatorname{erfc}\left(\frac{\gamma\sigma^2 - x_0}{\sqrt{2}\sigma}\right) =\frac{A\gamma}{2} \exp\left[\frac{\gamma^2\sigma^2}{2}\right] \operatorname{erfc}\left(\frac{\gamma\sigma}{\sqrt{2}}\right)$ $\exp\left[\gamma(x_0 + \gamma\sigma^2/2)\right] \operatorname{erfc}\left(\frac{\gamma\sigma^2 - x_0}{\sqrt{2}\sigma}\right) =\exp\left[\frac{\gamma^2\sigma^2}{2}\right] \operatorname{erfc}\left(\frac{\gamma\sigma}{\sqrt{2}}\right)$

Hmmm IDK where to go from here.

But by observation:

$\operatorname{FWHM}\approx2\sigma\sqrt{2\ln{2}}\approx 2.3548\sigma$

Skewed Gaussian

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see Wikipedia for more info)

$f(x; A, \mu, \sigma, \gamma) = \frac{A}{\sigma\sqrt{2\pi}} e^{[{-{(x-\mu)^2}/{2\sigma^2}}]} \Bigl\{ 1 + {\operatorname{erf}}\bigl[ \frac{\gamma(x-\mu)}{\sigma\sqrt{2}} \bigr] \Bigr\}$

where :func:erf is the error function.

Skewed Gaussian Height

Max height occurs at x = $\mu$

$f(\mu; A, \mu, \sigma, \gamma) = \frac{A}{\sigma\sqrt{2\pi}} e^{[{-{(\mu-\mu)^2}/{2\sigma^2}}]} \Bigl\{ 1 + {\operatorname{erf}}\bigl[ \frac{\gamma(\mu-\mu)}{\sigma\sqrt{2}} \bigr] \Bigr\}$ $\operatorname{height}=\frac{A}{\sigma\sqrt{2\pi}}$

Skewed Gaussian FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{A}{\sigma\sqrt{2\pi}} e^{[{-{(x_0)^2}/{2\sigma^2}}]} \bigl\{ 1 + {\operatorname{erf}}\bigl[ \frac{\gamma(x_0)}{\sigma\sqrt{2}} \bigr] \bigr\} =\frac{A}{\sigma\sqrt{2\pi}}$ $e^{[{-{(x_0)^2}/{2\sigma^2}}]} \bigl\{ 1 + {\operatorname{erf}}\bigl[ \frac{\gamma(x_0)}{\sigma\sqrt{2}} \bigr] \bigr\}=0$

Hmmm IDK where to go from here.

But by observation:

$\operatorname{FWHM}\approx2\sigma\sqrt{2\ln{2}}\approx 2.3548\sigma$

Donaich Sunjic (photo-emission)

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see Casaxps for more info)

$f(x; A, \mu, \sigma, \gamma) = \frac{A}{\sigma^{1-\gamma}}\frac{\cos\left[\pi\gamma/2 + (1-\gamma) \arctan{((x - \mu)/\sigma)}\right]} {\left[1 + (x-\mu)/\sigma\right]^{(1-\gamma)/2}}$

Donaich Sunjic Height

Max height occurs at x = $\mu$

$f(\mu; A, \mu, \sigma, \gamma) = \frac{A}{\sigma^{1-\gamma}}\frac{\cos\left[\pi\gamma/2 + (1-\gamma) \arctan{((\mu - \mu)}/\sigma)\right]} {\left[1 + (\mu-\mu)/\sigma\right]^{(1-\gamma)/2}}$ $=\frac{A}{\sigma^{1-\gamma}}\frac{\cos\left[\pi\gamma/2\right]} {\left[1\right]^{(1-\gamma)/2}}$ $\operatorname{height}=\frac{A}{\sigma^{1-\gamma}}\cos(\pi\gamma/2)$

Donaich Sunjic FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

$f(x_0;A,0,\sigma,m)=\frac{1}{2}f(\mu;A,\mu,\sigma,m)$ $\frac{\cos\left[\pi\gamma/2 + (1-\gamma) \arctan{((x_0)}/\sigma)\right]} {\left[1 + (x_0)/\sigma\right]^{(1-\gamma)/2}} =\cos(\pi\gamma/2)$

Hmmm IDK where to go from here.

$\operatorname{FWHM}=$

Austin Fox's personal website.

Derivation of FWHM and Height for Peak Fitting Functions

Gaussian (Normal distribution)

Gaussian Height

Gaussian FWHM

Lorentzian

Lorentzian Height

Lorentzian FWHM

Voigt

Voigt Height

Voigt FWHM

Pseudo Voigt

Pseudo Voigt Height

Pseudo Voigt FWHM

Moffat

Moffat Height

Moffat FWHM

Pearson 7

Pearson 7 Height

Pearson 7 FWHM

Students T

Students T Height

Students T FWHM

Breit-Wigner-Fano

Breit-Wigner-Fano Height

Breit-Wigner-Fano FWHM

Lognormal

Lognormal Height

Lognormal FWHM

Damped Oscillator

Damped Oscillator Height

Damped Oscillator FWHM

Damped Harmonic Oscillator

DHO Height

DHO FWHM

Exponential Gaussian

Exponential Gaussian Height

Exponential Gaussian FWHM

Skewed Gaussian

Skewed Gaussian Height

Skewed Gaussian FWHM

Donaich Sunjic (photo-emission)

Donaich Sunjic Height

Donaich Sunjic FWHM

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