Derivation of FWHM and Height for Peak Fitting Functions
29 Nov 2017 | Learning- 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
- Doniach 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 complementary 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 complementary 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:
\[\operatorname{FWHM}\approx2\sigma\sqrt{2\ln{2}}\approx 2.3548\sigma\]Doniach Sunjic (photo-emission)
A = amplitude, \(\mu\) = center, \(\sigma\) = sigma, and \(\gamma\) = gamma (see Casaxps for more info)
\[f(x; A, \mu, \sigma, \gamma) = \frac{A}{\sigma^{1-\gamma}}\frac{\cos\left[\pi\gamma/2 + (1-\gamma) \arctan{((x - \mu)/\sigma)}\right]} {\left[1 + (x-\mu)/\sigma\right]^{(1-\gamma)/2}}\]Doniach Sunjic Height
Max height occurs at x = \(\mu\)
\[f(\mu; A, \mu, \sigma, \gamma) = \frac{A}{\sigma^{1-\gamma}}\frac{\cos\left[\pi\gamma/2 + (1-\gamma) \arctan{((\mu - \mu)}/\sigma)\right]} {\left[1 + (\mu-\mu)/\sigma\right]^{(1-\gamma)/2}}\] \[=\frac{A}{\sigma^{1-\gamma}}\frac{\cos\left[\pi\gamma/2\right]} {\left[1\right]^{(1-\gamma)/2}}\] \[\operatorname{height}=\frac{A}{\sigma^{1-\gamma}}\cos(\pi\gamma/2)\]Doniach 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.
\[\operatorname{FWHM}=\]