## Density of states → Helmholtz free energy density f(T)

Using statistical mechanics, it can be shown that the Helmholtz free energy density f(T) for bosons can be expressed as the following integral,

Here $\hbar\omega$ is the energy of phonons with frequency ω and D(ω) is the density of states. This result is discussed in Statistical Physics, Part 1 by Landau and Lifshitz and in the notes on the thermodynamic properties of non-interacting bosons.

The form below uses this formula to calculate the temperature dependence of the Helmholtz free energy density from tabulated data for the density of states. The density of states data is input as two columns in the textbox at the lower left. The first column is the angular-frequency ω in rad/s. The second column is the density of states in units of s rad-1m-3.

After the 'DoS → f(T)' button is pressed, the density of states is plotted on the left and f(T) is plotted from temperature Tmin to temperature Tmax on the right. The data for the f(T) plot also appears in tabular form in the lower right textbox. The first column is the temperature in Kelvin and the second column is the Helmholtz free energy density in units of J m-3.

 D(ω)[1015 s rad-1m-3] f(T)[108 J m-3] ω [1012 rad/s] T [K] Tmin:  K  Tmax:  K Input: ω [rad/s] D(ω) [s rad-1m-3] Output: T [K] f(T) [J m-3] 0.624275E12 0.001522E15 0.945885E12 0.003513E15 1.267495E12 0.005505E15 1.589105E12 0.008631E15 1.910715E12 0.012338E15 2.232325E12 0.016848E15 2.553935E12 0.021385E15 2.875545E12 0.027111E15 3.197155E12 0.033198E15 3.518765E12 0.038952E15 3.840375E12 0.046311E15 4.161985E12 0.054002E15 4.483595E12 0.06302E15 4.805204E12 0.072177E15 5.126814E12 0.083769E15 5.448424E12 0.094254E15 5.770034E12 0.106039E15 6.091644E12 0.118045E15 6.413254E12 0.132154E15 6.734864E12 0.144853E15 7.056474E12 0.159626E15 7.378084E12 0.176003E15 7.699694E12 0.192353E15 8.021304E12 0.208205E15 8.342914E12 0.226712E15 8.664524E12 0.245691E15 8.986133E12 0.267712E15 9.307743E12 0.288543E15 9.629353E12 0.309485E15 9.950963E12 0.334052E15 10.272573E12 0.36025E15 10.594183E12 0.382022E15 10.915793E12 0.408138E15 11.237403E12 0.439482E15 11.559013E12 0.466012E15 11.880623E12 0.494867E15 12.202233E12 0.526045E15 12.523843E12 0.553682E15 12.845453E12 0.585469E15 13.167063E12 0.619496E15 13.488672E12 0.653496E15 13.810282E12 0.689599E15 14.131892E12 0.732949E15 14.453502E12 0.773008E15 14.775112E12 0.815418E15 15.096722E12 0.859986E15 15.418332E12 0.907708E15 15.739942E12 0.955844E15 16.061552E12 1.006332E15 16.383162E12 1.061164E15 16.704772E12 1.116549E15 17.026382E12 1.175917E15 17.347992E12 1.234871E15 17.669601E12 1.297061E15 17.991211E12 1.361658E15 18.312821E12 1.43497E15 18.634431E12 1.510578E15 18.956041E12 1.59335E15 19.277651E12 1.675155E15 19.599261E12 1.765342E15 19.920871E12 1.853952E15 20.242481E12 1.951968E15 20.564091E12 2.05513E15 20.885701E12 2.168473E15 21.207311E12 2.291442E15 21.528921E12 2.427525E15 21.850531E12 2.568477E15 22.17214E12 2.720965E15 22.49375E12 2.897106E15 22.81536E12 3.088629E15 23.13697E12 3.305632E15 23.45858E12 3.63E15 23.78019E12 3.974841E15 24.1018E12 4.275308E15 24.42341E12 4.550794E15 24.74502E12 4.772554E15 25.06663E12 4.864014E15 25.38824E12 4.908194E15 25.70985E12 4.966041E15 26.03146E12 5.007594E15 26.35307E12 5.053379E15 26.674681E12 5.110036E15 26.996288E12 5.174218E15 27.317898E12 5.219091E15 27.639509E12 5.277767E15 27.961119E12 5.327232E15 28.282729E12 5.375203E15 28.604339E12 5.416893E15 28.925949E12 5.46301E15 29.247559E12 5.51195E15 29.56917E12 5.56833E15 29.89078E12 5.606839E15 30.21239E12 5.65943E15 30.533998E12 5.711384E15 30.855608E12 5.748538E15 31.177218E12 5.782621E15 31.498828E12 5.833884E15 31.820438E12 5.888826E15 32.142048E12 5.936769E15 32.463658E12 5.995778E15 32.785269E12 6.057083E15 33.106879E12 6.10818E15 33.428489E12 5.927142E15 33.750097E12 5.439164E15 34.071707E12 4.935804E15 34.393317E12 4.41922E15 34.714927E12 3.897242E15 35.036537E12 3.579402E15 35.358147E12 3.565071E15 35.679758E12 3.550962E15 36.001368E12 3.539177E15 36.322978E12 3.536023E15 36.644588E12 3.526258E15 36.966198E12 3.519314E15 37.287806E12 3.497791E15 37.609416E12 3.481662E15 37.931026E12 3.451065E15 38.252636E12 3.41333E15 38.574246E12 3.370727E15 38.895857E12 3.328566E15 39.217467E12 3.272185E15 39.539077E12 3.201916E15 39.860687E12 3.116792E15 40.182297E12 3.01861E15 40.503907E12 2.899458E15 40.825515E12 2.737591E15 41.147125E12 2.577164E15 41.468735E12 2.588977E15 41.790346E12 2.815302E15 42.111956E12 3.347544E15 42.433566E12 4.043422E15 42.755176E12 4.841578E15 43.076786E12 5.587336E15 43.398396E12 6.224925E15 43.720006E12 6.693095E15 44.041617E12 7.121401E15 44.363224E12 7.561962E15 44.684835E12 7.631179E15 45.006445E12 7.347366E15 45.328055E12 6.804529E15 45.649665E12 6.064304E15 45.971275E12 5.112305E15 46.292885E12 4.339048E15 46.614495E12 3.735403E15 46.936106E12 3.177239E15 47.257716E12 2.613736E15 47.579326E12 1.945827E15 47.900934E12 1.300842E15 48.222544E12 0.342074E15

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