Fitting a Lorentzian curve to data

Question

Good morning everyone, regarding my research "high resolution laser spectroscopy" I would like to fit the data obtained from the experiment with a Lorentzian curve using Mathematica, so as to calculate the value of FWHM (full width at half maximum). To do this I have started to transcribe the data into "data", as you can see in the picture and also I written the basic instruction to do this:

data = {{1.098`, -1.064`}, {1.1`, -1.064`}, {1.102`, -1.06`}, 
{1.104`, -1.06`}, {1.106`, -1.056`}, {1.108`, -1.056`}, {1.11`, 
-1.056`}, {1.112`, -1.056`}, {1.114`, -1.056`}, {1.116`, -1.056`}, 
{1.118`, -1.052`}, {1.12`, -1.052`}, {1.122`, -1.048`}, {1.124`, 
-1.048`}, {1.126`, -1.044`}, {1.128`, -1.044`}, {1.13`, -1.044`}, 
{1.132`, -1.044`}, {1.134`, -1.044`}, {1.136`, -1.044`}, {1.138`, 
-1.04`}, {1.14`, -1.04`}, {1.142`, -1.04`}, {1.144`, -1.04`},
{1.146`, -1.036`}, {1.148`, -1.036`}, {1.15`, -1.036`}, {1.152`, 
-1.032`}, {1.154`, -1.032`}, {1.156`, -1.032`}, {1.158`, -1.028`}, 
{1.16`, -1.028`}, {1.162`, -1.028`}, {1.164`, -1.028`}, {1.166`, 
-1.024`}, {1.168`, -1.024`}, {1.17`, -1.024`}, {1.172`, -1.024`}, 
{1.174`, -1.02`}, {1.176`, -1.02`}, {1.178`, -1.024`}, {1.18`, 
-1.02`}, {1.182`, -1.016`}, {1.184`, -1.016`}, {1.186`, -1.016`},
{1.188`, -1.012`}, {1.19`, -1.012`}, {1.192`, -1.012`}, {1.194`, 
-1.008`}, {1.196`, -1.008`}, {1.198`, -1.008`}, {1.2`, -1.008`}, 
{1.202`, -1.008`}, {1.204`, -1.004`}, {1.206`, -1.004`}, {1.208`, 
-1.004`}, {1.21`, -1}, {1.212`, -1}, {1.214`, -1}, {1.216`, -1}, 
{1.218`, -1}, {1.22`, -0.996`}, {1.222`, -1}, {1.224`, -0.996`}, 
{1.226`, -0.992`}, {1.228`, -0.996`}, {1.23`, -0.992`}, {1.232`, 
-0.992`}, {1.234`, -0.992`}, {1.236`, -0.992`}, {1.238`, -0.988`}, 
{1.24`, -0.988`}, {1.242`, -0.988`}, {1.244`, -0.984`}, {1.246`, 
-0.984`}, {1.248`, -0.984`}, {1.25`, -0.98`}, {1.252`, -0.98`}, 
{1.254`, -0.98`}, {1.256`, -0.98`}, {1.258`, -0.976`}, {1.26`, 
-0.976`}, {1.262`, -0.976`}, {1.264`, -0.976`}, {1.266`, -0.976`}, 
{1.268`, -0.972`}, {1.27`, -0.972`}, {1.272`, -0.972`}, {1.274`, 
-0.972`}, {1.276`, -0.968`}, {1.278`, -0.968`}, {1.28`, -0.964`}, 
{1.282`, -0.964`}, {1.284`, -0.964`}, {1.286`, -0.964`}, {1.288`, 
-0.96`}, {1.29`, -0.96`}, {1.292`, -0.96`}, {1.294`, -0.956`}, 
{1.296`, -0.952`}, {1.298`, -0.952`}, {1.3`, -0.952`}, {1.302`, 
-0.952`}, {1.304`, -0.952`}, {1.306`, -0.952`}, {1.308`, -0.948`}, 
{1.31`, -0.952`}, {1.312`, -0.948`}, {1.314`, -0.948`}, {1.316`, 
-0.948`}, {1.318`, -0.944`}, {1.32`, -0.948`}, {1.322`, -0.944`}, 
{1.324`, -0.944`}, {1.326`, -0.944`}, {1.328`, -0.94`}, {1.33`, 
-0.944`}, {1.332`, -0.944`}, {1.334`, -0.94`}, {1.336`, -0.94`}, 
{1.338`, -0.94`}, {1.34`, -0.936`}, {1.342`, -0.936`}, {1.344`, 
-0.936`}, {1.346`, -0.936`}, {1.348`, -0.936`}, {1.35`, -0.932`}, 
{1.352`, -0.928`}, {1.354`, -0.928`}, {1.356`, -0.928`}, {1.358`, 
-0.928`}, {1.36`, -0.928`}, {1.362`, -0.928`}, {1.364`, -0.928`}, 
{1.366`, -0.924`}, {1.368`, -0.924`}, {1.37`, -0.924`}, {1.372`, 
-0.92`}, {1.374`, -0.924`}, {1.376`, -0.92`}, {1.378`, -0.92`}, 
{1.38`, -0.92`}, {1.382`, -0.92`}, {1.384`, -0.92`}, {1.386`, 
-0.92`}, {1.388`, -0.916`}, {1.39`, -0.916`}, {1.392`, -0.916`}, 
{1.394`, -0.92`}, {1.396`, -0.916`}, {1.398`, -0.916`}, {1.4`, 
-0.916`}, {1.402`, -0.916`}, {1.404`, -0.912`}, {1.406`, -0.912`}, 
{1.408`, -0.912`}, {1.41`, -0.912`}, {1.412`, -0.912`}, {1.414`, 
-0.908`}, {1.416`, -0.912`}, {1.418`, -0.912`}, {1.42`, -0.908`}, 
{1.422`, -0.908`}, {1.424`, -0.908`}, {1.426`, -0.908`}, {1.428`, 
-0.908`}, {1.43`, -0.908`}, {1.432`, -0.908`}, {1.434`, -0.908`}, 
{1.436`, -0.908`}, {1.438`, -0.908`}, {1.44`, -0.904`}, {1.442`,
-0.908`}, {1.444`, -0.904`}, {1.446`, -0.904`}, {1.448`, -0.904`}, 
{1.45`, -0.9`}, {1.452`, -0.9`}, {1.454`, -0.904`}, {1.456`, -0.9`}, 
{1.458`, -0.9`}, {1.46`, -0.9`}, {1.462`, -0.9`}, {1.464`, -0.9`}, 
{1.466`, -0.9`}, {1.468`, -0.9`}, {1.47`, -0.904`}, {1.472`, -0.9`}, 
{1.474`, -0.9`}, {1.476`, -0.9`}, {1.478`, -0.9`}, {1.48`, -0.896`}, 
{1.482`, -0.9`}, {1.484`, -0.9`}, {1.486`, -0.896`}, {1.488`, -0.9`},
{1.49`, -0.9`}, {1.492`, -0.896`}, {1.494`, -0.9`}, {1.496`, -0.9`}, 
{1.498`, -0.9`}, {1.5`, -0.896`}, {1.502`, -0.9`}, {1.504`, -0.9`}, 
{1.506`, -0.9`}, {1.508`, -0.9`}, {1.51`, -0.9`}, {1.512`, -0.9`}, 
{1.514`, -0.904`}, {1.516`, -0.9`}, {1.518`, -0.9`}, {1.52`, -0.9`}, 
{1.522`, -0.9`}, {1.524`, -0.9`}, {1.526`, -0.9`}, {1.528`, -0.9`}, 
{1.53`, -0.904`}, {1.532`, -0.904`}, {1.534`, -0.904`}, {1.536`, 
-0.904`}, {1.538`, -0.904`}, {1.54`, -0.904`}, {1.542`, -0.904`}, 
{1.544`, -0.904`}, {1.546`, -0.904`}, {1.548`, -0.908`}, {1.55`, 
-0.904`}, {1.552`, -0.904`}, {1.554`, -0.908`}, {1.556`, -0.908`}, 
{1.558`, -0.904`}, {1.56`, -0.916`}, {1.562`, -0.92`}, {1.564`, 
-0.92`}, {1.566`, -0.92`}, {1.568`, -0.924`}, {1.57`, -0.92`}, 
{1.572`, -0.924`}, {1.574`, -0.92`}, {1.576`, -0.92`}, {1.578`, 
-0.928`}, {1.58`, -0.924`}, {1.582`, -0.924`}, {1.584`, -0.928`}, 
{1.586`, -0.928`}, {1.588`, -0.924`}, {1.59`, -0.928`}, {1.592`, 
-0.928`}, {1.594`, -0.928`}, {1.596`, -0.932`}, {1.598`, -0.932`}, 
{1.6`, -0.932`}, {1.602`, -0.932`}, {1.604`, -0.932`}, {1.606`, 
-0.932`}, {1.608`, -0.932`}, {1.61`, -0.936`}, {1.612`, -0.936`}, 
{1.614`, -0.94`}, {1.616`, -0.94`}, {1.618`, -0.94`}, {1.62`, 
-0.94`}, {1.622`, -0.94`}, {1.624`, -0.94`}, {1.626`, -0.944`}, 
{1.628`, -0.944`}, {1.63`, -0.944`}, {1.632`, -0.944`}, {1.634`, 
-0.944`}, {1.636`, -0.944`}, {1.638`, -0.948`}, {1.64`, -0.944`}, 
{1.642`, -0.948`}, {1.644`, -0.948`}, {1.646`, -0.948`}, {1.648`, 
-0.952`}, {1.65`, -0.952`}, {1.652`, -0.952`}, {1.654`, -0.952`}, 
{1.656`, -0.952`}, {1.658`, -0.956`}, {1.66`, -0.96`}, {1.662`, 
-0.96`}, {1.664`, -0.96`}, {1.666`, -0.96`}, {1.668`, -0.96`}, 
{1.67`, -0.964`}, {1.672`, -0.968`}, {1.674`, -0.964`}, {1.676`, 
-0.968`}, {1.678`, -0.968`}, {1.68`, -0.968`}, {1.682`, -0.968`}, 
{1.684`, -0.968`}, {1.686`, -0.972`}, {1.688`, -0.972`}, {1.69`, 
-0.972`}, {1.692`, -0.976`}, {1.694`, -0.976`}, {1.696`, -0.976`}, 
{1.698`, -0.976`}, {1.7`, -0.976`}, {1.702`, -0.976`}, {1.704`, 
-0.976`}, {1.706`, -0.976`}, {1.708`, -0.98`}, {1.71`, -0.98`}, 
{1.712`, -0.984`}, {1.714`, -0.984`}, {1.716`, -0.984`}, {1.718`, 
-0.984`}, {1.72`, -0.984`}, {1.722`, -0.988`}, {1.724`, -0.988`}, 
{1.726`, -0.992`}, {1.728`, -0.988`}, {1.73`, -0.992`}, {1.732`, 
-0.992`}, {1.734`, -0.996`}, {1.736`, -0.996`}, {1.738`, -0.996`}, 
{1.74`, -0.996`}, {1.742`, -0.996`}, {1.744`, -1}, {1.746`, -1}, 
{1.748`, -1}, {1.75`, -1}, {1.752`, -1.004`}, {1.754`, -1.004`}, 
{1.756`, -1.004`}, {1.758`, -1.004`}, {1.76`, -1.008`}, {1.762`, 
-1.008`}, {1.764`, -1.008`}, {1.766`, -1.008`}, {1.768`, -1.008`}, 
{1.77`, -1.012`}, {1.772`, -1.012`}, {1.774`, -1.012`}, {1.776`, 
-1.016`}, {1.778`, -1.016`}, {1.78`, -1.016`}, {1.782`, -1.016`}, 
{1.784`, -1.02`}, {1.786`, -1.02`}, {1.788`, -1.02`}, {1.79`, 
-1.024`}, {1.792`, -1.024`}, {1.794`, -1.024`}, {1.796`, -1.024`}, 
{1.798`, -1.028`}, {1.8`, -1.024`}, {1.802`, -1.028`}, {1.804`, 
-1.028`}, {1.806`, -1.032`}, {1.808`, -1.032`}, {1.81`, -1.032`}, 
{1.812`, -1.036`}, {1.814`, -1.036`}, {1.816`, -1.036`}, {1.818`, 
-1.04`}, {1.82`, -1.036`}, {1.822`, -1.04`}, {1.824`, -1.04`}, 
{1.826`, -1.04`}, {1.828`, -1.04`}, {1.83`, -1.044`}, {1.832`, 
-1.044`}, {1.834`, -1.044`}, {1.836`, -1.048`}, {1.838`, -1.048`}, 
{1.84`, -1.048`}, {1.842`, -1.048`}, {1.844`, -1.048`}, {1.846`, 
-1.052`}, {1.848`, -1.052`}, {1.85`, -1.052`}, {1.852`, -1.056`}, 
{1.854`, -1.056`}, {1.856`, -1.056`}, {1.858`, -1.06`}, {1.86`, 
-1.06`}
ListPlot[data, PlotTheme -> "Scientific"]
model = a/((b - t)^2 + d) + c;
result = 
 NonlinearModelFit[data, {model}, {a, b, c, d}, t, 
  Method -> "NMinimize"]
Show[ListPlot[data], 
 Plot[result[t], {t, 1.098`, 1.86`}, PlotRange -> Full]]
max = NMaximize[result[t], t][[1]] 
min = NMinimize[result[t], t][[1]] 
minmax = data[[All, 1]] // MinMax
fwhm = t /. 
    NSolve[{result[t] == (max + min)/2, 1.098` < t < 1.86`}, t] // 
   Differences // First
max1 = FindMaximum[result[t], t] // First
min1 = FindMinimum[{result[t], 1.098` < t < 1.86`}, t] // First
middle = max1 + min1/2
t1t2 = NSolve[{result[t] == (max1 + min1)/2, 1.098` < t < 1.86`}, t]
fwhm1 = t /. 
    NSolve[{result[t] == (max1 + min1)/2, 1.098` < t < 1.86`}, t] // 
   Differences // First

The problem is that the Lorentziana does not come out as it should:

Can anyone help me? My goal afterwards will be to use the obtained Lorentzian and make it the sum of three Lorentzians.

Answer 1

Clear["Global`*"]
    
plt = ListPlot[data, PlotTheme -> "Scientific"];
model = a/((b - t)^2 + d) + c;

(result = NonlinearModelFit[data, {model}, {a, b, c, d}, t]) // Normal

(* -1.23954 + 0.0569007/(0.168084 + (1.48017 - t)^2) *)

minmax = data[[All, 1]] // MinMax

(* {1.098, 1.86} *)

Show[
 plt,
 Plot[result[t],
  {t, minmax[[1]], minmax[[-1]]}]]

{min, max} = #[{result[t], minmax[[1]] < t < minmax[[-1]]}, t] & /@ 
  {MinValue, MaxValue}

(* {-1.0584, -1.57806} *)

The value of max is clearly wrong (reported Mathematica Tech Support CASE: 4951661). (EDIT: This has been resolved in version 13.2)

The correct value is

max = result[
   SolveValues[
     {result'[t] == 0, result''[t] < 0}, t][[1]]] //
  Quiet

(* -0.901011 *)

fwhm = t /. 
     Solve[{result[t] == (max + min)/2, minmax[[1]] < t < minmax[[-1]]}, t] //
     Differences // First // Quiet

(* 0.451257 *)