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I want to find fit to this data and the equation:

enter image description here

Clear["Global`*"];
data = {{1, 1.2926876`}, {2, 2.4296988`}, {3, 3.3392012`}, {4, 
    4.1653912`}, {5, 4.9024912`}, {6, 5.5450292`}, {7, 6.108284`}, {8,
     6.6103844`}, {9, 7.0690112`}, {10, 7.4914848`}, {11, 
    7.8963848`}, {12, 8.2873168`}, {13, 8.663396`}, {14, 
    9.0243192`}, {15, 9.3678864`}, {16, 9.6954168`}, {17, 
    10.0114072`}, {18, 10.316488`}, {19, 10.6176864`}, {20, 
    10.9121928`}, {21, 11.1988224`}, {22, 11.4784872`}, {23, 
    11.750596`}, {24, 12.015308`}, {25, 12.2748`}, {26, 
    12.5285192`}, {27, 12.7795192`}, {28, 13.0259888`}, {29, 
    13.2677224`}, {30, 13.5053088`}, {31, 13.738288`}, {32, 
    13.966112`}, {33, 14.1911264`}, {34, 14.4120976`}, {35, 
    14.6317952`}, {36, 14.8479104`}, {37, 15.060792`}, {38, 
    15.271192`}, {39, 15.4784384`}, {40, 15.6821936`}, {41, 
    15.8837296`}, {42, 16.0824256`}, {43, 16.2807056`}, {44, 
    16.47644`}, {45, 16.6694944`}, {46, 16.8607952`}, {47, 
    17.04932`}, {48, 17.23492`}, {49, 17.4187984`}, {50, 
    17.6000976`}, {51, 17.7814976`}, {52, 17.961104`}, {53, 
    18.1384144`}, {54, 18.3144992`}, {55, 18.4882992`}, {56, 
    18.6596016`}, {57, 18.829616`}, {58, 18.9977008`}, {59, 
    19.1663024`}, {60, 19.3337312`}, {61, 19.4992272`}, {62, 
    19.6638032`}, {63, 19.826504`}, {64, 19.9871216`}, {65, 
    20.1467344`}, {66, 20.304648`}, {67, 20.4630064`}, {68, 
    20.6204512`}, {69, 20.7763472`}, {70, 20.9313552`}, {71, 
    21.0848432`}, {72, 21.2365424`}, {73, 21.3874256`}, {74, 
    21.5367104`}, {75, 21.6865104`}};
n0 = 8;
FindFit[data, (-\[Omega]0 + Sqrt[4 n \[Pi] \[Alpha] + n0 \[Omega]0^2]/
  Sqrt[n0])/\[Alpha], {\[Omega]0, \[Alpha]}, n]

enter image description here

I have no idea, why it does not work. I get those errors and the output is also wrong. How to fix it?

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  • 1
    $\begingroup$ You don't have usable initial estimates for your parameters? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Dec 22, 2021 at 19:22
  • $\begingroup$ How can I set initial estimates for the parameters? $\endgroup$
    – Gal Zajc
    Commented Dec 22, 2021 at 19:24
  • 1
    $\begingroup$ You should look at the "Details and Options" section of the docs for FindFit[] for more details, but in brief, you do something like FindFit[data, model[x], {{param1, param1est}, ...}, x]. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Dec 22, 2021 at 19:48

1 Answer 1

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Ensure that w0 and alpha are positive by e.g.:

sol=FindFit[data, {(-ω0 + Sqrt[4 n π α + n0 ω0^2]/ Sqrt[n0])/α, ω0 > 0, α> 0}, {ω0, α}, n]

To check the solution we may plot it:

Plot[f[n], {n, 1, 75}, Epilog -> Point[data]]

enter image description here

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