## Density of states → energy spectral density for electrons in metals

Fermions obey Fermi-Dirac statistics. The probablility that a fermion state at energy E is occupied is given by the Fermi function,

Here μ is the chemical potential. The energy spectral density u(E) is the energy times the density of states D(E) times the Fermi function.

The form below calculates the energy spectral density numerically from the density of states. The density of states is input as two columns of text in the textbox below. The first column is the energy in eV. The second column is the density of states in unit of eV-1 m-d, where d is the dimensionality (1,2, or 3). The electron density can be calculated from the number of electrons per unit cell and the volume of the unit cell or it can be input directly. After the 'DoS → u(E)' button is pressed, the density of states is plotted on the left and u(E) is plotted on the right. The data for the u(E) plot also appear as six columns of text in the lower right textbox. The first column is the energy in eV. Columns 2-6 are the energy spectral density u(E) for five temperatures 0.2Tmax, 0.4Tmax...Tmax in units are m-d.

 D(E) [eV-1 m-3] u(E) [m-3] E [eV] E [eV] dimensionality: 1  2  3 number of electrons per primitive unit cell:  volume of a primitive unit cell:  m2 electron density =  m-5 Tmin:  K  Tmax:  K E [eV]  D(E) [eV-1 m-3] E [eV] u(E,0.2Tmax) [m-3] ... u(E,Tmax) [m-3] 0 0 0.007055332 5.72E+26 0.014110664 8.09E+26 0.021165996 9.91E+26 0.028221327 1.14E+27 0.035276659 1.28E+27 0.042331991 1.40E+27 0.049387323 1.51E+27 0.056442655 1.62E+27 0.063497987 1.72E+27 0.070553319 1.81E+27 0.077608651 1.90E+27 0.084663982 1.98E+27 0.091719314 2.06E+27 0.098774646 2.14E+27 0.105829978 2.22E+27 0.11288531 2.29E+27 0.119940642 2.36E+27 0.126995974 2.43E+27 0.134051306 2.49E+27 0.141106637 2.56E+27 0.148161969 2.62E+27 0.155217301 2.68E+27 0.162272633 2.74E+27 0.169327965 2.80E+27 0.176383297 2.86E+27 0.183438629 2.92E+27 0.190493961 2.97E+27 0.197549292 3.03E+27 0.204604624 3.08E+27 0.211659956 3.13E+27 0.218715288 3.19E+27 0.22577062 3.24E+27 0.232825952 3.29E+27 0.239881284 3.34E+27 0.246936616 3.39E+27 0.253991947 3.43E+27 0.261047279 3.48E+27 0.268102611 3.53E+27 0.275157943 3.57E+27 0.282213275 3.62E+27 0.289268607 3.66E+27 0.296323939 3.71E+27 0.303379271 3.75E+27 0.310434602 3.80E+27 0.317489934 3.84E+27 0.324545266 3.88E+27 0.331600598 3.92E+27 0.33865593 3.96E+27 0.345711262 4.01E+27 0.352766594 4.05E+27 0.359821926 4.09E+27 0.366877257 4.13E+27 0.373932589 4.17E+27 0.380987921 4.20E+27 0.388043253 4.24E+27 0.395098585 4.28E+27 0.402153917 4.32E+27 0.409209249 4.36E+27 0.416264581 4.40E+27 0.423319912 4.43E+27 0.430375244 4.47E+27 0.437430576 4.51E+27 0.444485908 4.54E+27 0.45154124 4.58E+27 0.458596572 4.61E+27 0.465651904 4.65E+27 0.472707236 4.68E+27 0.479762567 4.72E+27 0.486817899 4.75E+27 0.493873231 4.79E+27 0.500928563 4.82E+27 0.507983895 4.86E+27 0.515039227 4.89E+27 0.522094559 4.92E+27 0.529149891 4.96E+27 0.536205222 4.99E+27 0.543260554 5.02E+27 0.550315886 5.05E+27 0.557371218 5.09E+27 0.56442655 5.12E+27 0.571481882 5.15E+27 0.578537214 5.18E+27 0.585592545 5.21E+27 0.592647877 5.24E+27 0.599703209 5.28E+27 0.606758541 5.31E+27 0.613813873 5.34E+27 0.620869205 5.37E+27 0.627924537 5.40E+27 0.634979869 5.43E+27 0.6420352 5.46E+27 0.649090532 5.49E+27 0.656145864 5.52E+27 0.663201196 5.55E+27 0.670256528 5.58E+27 0.67731186 5.61E+27 0.684367192 5.64E+27 0.691422524 5.66E+27 0.698477855 5.69E+27 0.705533187 5.72E+27 0.712588519 5.75E+27 0.719643851 5.78E+27 0.726699183 5.81E+27 0.733754515 5.84E+27 0.740809847 5.86E+27 0.747865179 5.89E+27 0.75492051 5.92E+27 0.761975842 5.95E+27 0.769031174 5.97E+27 0.776086506 6.00E+27 0.783141838 6.03E+27 0.79019717 6.06E+27 0.797252502 6.08E+27 0.804307834 6.11E+27 0.811363165 6.14E+27 0.818418497 6.16E+27 0.825473829 6.19E+27 0.832529161 6.22E+27 0.839584493 6.24E+27 0.846639825 6.27E+27 0.853695157 6.29E+27 0.860750489 6.32E+27 0.86780582 6.35E+27 0.874861152 6.37E+27 0.881916484 6.40E+27 0.888971816 6.42E+27 0.896027148 6.45E+27 0.90308248 6.47E+27 0.910137812 6.50E+27 0.917193144 6.52E+27 0.924248475 6.55E+27 0.931303807 6.57E+27 0.938359139 6.60E+27 0.945414471 6.62E+27 0.952469803 6.65E+27 0.959525135 6.67E+27 0.966580467 6.70E+27 0.973635799 6.72E+27 0.98069113 6.75E+27 0.987746462 6.77E+27 0.994801794 6.79E+27 1.001857126 6.82E+27 1.008912458 6.84E+27 1.01596779 6.87E+27 1.023023122 6.89E+27 1.030078454 6.91E+27 1.037133785 6.94E+27 1.044189117 6.96E+27 1.051244449 6.98E+27 1.058299781 7.01E+27 1.065355113 7.03E+27 1.072410445 7.05E+27 1.079465777 7.08E+27 1.086521109 7.10E+27 1.09357644 7.12E+27 1.100631772 7.15E+27 1.107687104 7.17E+27 1.114742436 7.19E+27 1.121797768 7.22E+27 1.1288531 7.24E+27 1.135908432 7.26E+27 1.142963764 7.28E+27 1.150019095 7.31E+27 1.157074427 7.33E+27 1.164129759 7.35E+27 1.171185091 7.37E+27 1.178240423 7.39E+27 1.185295755 7.42E+27 1.192351087 7.44E+27 1.199406418 7.46E+27 1.20646175 7.48E+27 1.213517082 7.50E+27 1.220572414 7.53E+27 1.227627746 7.55E+27 1.234683078 7.57E+27 1.24173841 7.59E+27 1.248793742 7.61E+27 1.255849073 7.63E+27 1.262904405 7.66E+27 1.269959737 7.68E+27 1.277015069 7.70E+27 1.284070401 7.72E+27 1.291125733 7.74E+27 1.298181065 7.76E+27 1.305236397 7.78E+27 1.312291728 7.80E+27 1.31934706 7.82E+27 1.326402392 7.85E+27 1.333457724 7.87E+27 1.340513056 7.89E+27 1.347568388 7.91E+27 1.35462372 7.93E+27 1.361679052 7.95E+27 1.368734383 7.97E+27 1.375789715 7.99E+27 1.382845047 8.01E+27 1.389900379 8.03E+27 1.396955711 8.05E+27 1.404011043 8.07E+27 1.411066375 8.09E+27 1.418121707 8.11E+27 1.425177038 8.13E+27 1.43223237 8.15E+27 1.439287702 8.17E+27 1.446343034 8.19E+27 1.453398366 8.21E+27 1.460453698 8.23E+27 1.46750903 8.25E+27 1.474564362 8.27E+27 1.481619693 8.29E+27 1.488675025 8.31E+27 1.495730357 8.33E+27 1.502785689 8.35E+27 1.509841021 8.37E+27 1.516896353 8.39E+27 1.523951685 8.41E+27 1.531007017 8.43E+27 1.538062348 8.45E+27 1.54511768 8.47E+27 1.552173012 8.49E+27 1.559228344 8.51E+27 1.566283676 8.53E+27 1.573339008 8.54E+27 1.58039434 8.56E+27 1.587449672 8.58E+27 1.594505003 8.60E+27 1.601560335 8.62E+27 1.608615667 8.64E+27 1.615670999 8.66E+27 1.622726331 8.68E+27 1.629781663 8.70E+27 1.636836995 8.72E+27 1.643892327 8.73E+27 1.650947658 8.75E+27 1.65800299 8.77E+27 1.665058322 8.79E+27 1.672113654 8.81E+27 1.679168986 8.83E+27 1.686224318 8.85E+27 1.69327965 8.86E+27 1.700334982 8.88E+27 1.707390313 8.90E+27 1.714445645 8.92E+27 1.721500977 8.94E+27 1.728556309 8.96E+27 1.735611641 8.97E+27 1.742666973 8.99E+27 1.749722305 9.01E+27 1.756777636 9.03E+27 1.763832968 9.05E+27 1.7708883 9.07E+27 1.777943632 9.08E+27 1.784998964 9.10E+27 1.792054296 9.12E+27 1.799109628 9.14E+27 1.80616496 9.16E+27 1.813220291 9.17E+27 1.820275623 9.19E+27 1.827330955 9.21E+27 1.834386287 9.23E+27 1.841441619 9.24E+27 1.848496951 9.26E+27 1.855552283 9.28E+27 1.862607615 9.30E+27 1.869662946 9.31E+27 1.876718278 9.33E+27 1.88377361 9.35E+27 1.890828942 9.37E+27 1.897884274 9.38E+27 1.904939606 9.40E+27 1.911994938 9.42E+27 1.91905027 9.44E+27 1.926105601 9.45E+27 1.933160933 9.47E+27 1.940216265 9.49E+27 1.947271597 9.51E+27 1.954326929 9.52E+27 1.961382261 9.54E+27 1.968437593 9.56E+27 1.975492925 9.57E+27 1.982548256 9.59E+27 1.989603588 9.61E+27 1.99665892 9.63E+27 2.003714252 9.64E+27 2.010769584 9.66E+27 2.017824916 9.68E+27 2.024880248 9.69E+27 2.03193558 9.71E+27 2.038990911 9.73E+27 2.046046243 9.74E+27 2.053101575 9.76E+27 2.060156907 9.78E+27 2.067212239 9.79E+27 2.074267571 9.81E+27 2.081322903 9.83E+27 2.088378235 9.84E+27 2.095433566 9.86E+27 2.102488898 9.88E+27 2.10954423 9.89E+27 2.116599562 9.91E+27 2.123654894 9.93E+27 2.130710226 9.94E+27 2.137765558 9.96E+27 2.14482089 9.98E+27 2.151876221 9.99E+27 2.158931553 1.00E+28 2.165986885 1.00E+28 2.173042217 1.00E+28 2.180097549 1.01E+28 2.187152881 1.01E+28 2.194208213 1.01E+28 2.201263545 1.01E+28 2.208318876 1.01E+28 2.215374208 1.01E+28 2.22242954 1.02E+28 2.229484872 1.02E+28 2.236540204 1.02E+28 2.243595536 1.02E+28 2.250650868 1.02E+28 2.2577062 1.02E+28 2.264761531 1.03E+28 2.271816863 1.03E+28 2.278872195 1.03E+28 2.285927527 1.03E+28 2.292982859 1.03E+28 2.300038191 1.03E+28 2.307093523 1.03E+28 2.314148855 1.04E+28 2.321204186 1.04E+28 2.328259518 1.04E+28 2.33531485 1.04E+28 2.342370182 1.04E+28 2.349425514 1.04E+28 2.356480846 1.05E+28 2.363536178 1.05E+28 2.370591509 1.05E+28 2.377646841 1.05E+28 2.384702173 1.05E+28 2.391757505 1.05E+28 2.398812837 1.06E+28 2.405868169 1.06E+28 2.412923501 1.06E+28 2.419978833 1.06E+28 2.427034164 1.06E+28 2.434089496 1.06E+28 2.441144828 1.06E+28 2.44820016 1.07E+28 2.455255492 1.07E+28 2.462310824 1.07E+28 2.469366156 1.07E+28 2.476421488 1.07E+28 2.483476819 1.07E+28 2.490532151 1.08E+28 2.497587483 1.08E+28 2.504642815 1.08E+28 2.511698147 1.08E+28 2.518753479 1.08E+28 2.525808811 1.08E+28 2.532864143 1.08E+28 2.539919474 1.09E+28 2.546974806 1.09E+28 2.554030138 1.09E+28 2.56108547 1.09E+28 2.568140802 1.09E+28 2.575196134 1.09E+28 2.582251466 1.09E+28 2.589306798 1.10E+28 2.596362129 1.10E+28 2.603417461 1.10E+28 2.610472793 1.10E+28 2.617528125 1.10E+28 2.624583457 1.10E+28 2.631638789 1.11E+28 2.638694121 1.11E+28 2.645749453 1.11E+28 2.652804784 1.11E+28 2.659860116 1.11E+28 2.666915448 1.11E+28 2.67397078 1.11E+28 2.681026112 1.12E+28 2.688081444 1.12E+28 2.695136776 1.12E+28 2.702192108 1.12E+28 2.709247439 1.12E+28 2.716302771 1.12E+28 2.723358103 1.12E+28 2.730413435 1.13E+28 2.737468767 1.13E+28 2.744524099 1.13E+28 2.751579431 1.13E+28 2.758634763 1.13E+28 2.765690094 1.13E+28 2.772745426 1.13E+28 2.779800758 1.14E+28 2.78685609 1.14E+28 2.793911422 1.14E+28 2.800966754 1.14E+28 2.808022086 1.14E+28 2.815077418 1.14E+28 2.822132749 1.14E+28 2.829188081 1.15E+28 2.836243413 1.15E+28 2.843298745 1.15E+28 2.850354077 1.15E+28 2.857409409 1.15E+28 2.864464741 1.15E+28 2.871520073 1.15E+28 2.878575404 1.16E+28 2.885630736 1.16E+28 2.892686068 1.16E+28 2.8997414 1.16E+28 2.906796732 1.16E+28 2.913852064 1.16E+28 2.920907396 1.16E+28 2.927962727 1.17E+28 2.935018059 1.17E+28 2.942073391 1.17E+28 2.949128723 1.17E+28 2.956184055 1.17E+28 2.963239387 1.17E+28 2.970294719 1.17E+28 2.977350051 1.18E+28 2.984405382 1.18E+28 2.991460714 1.18E+28 2.998516046 1.18E+28 3.005571378 1.18E+28 3.01262671 1.18E+28 3.019682042 1.18E+28 3.026737374 1.19E+28 3.033792706 1.19E+28 3.040848037 1.19E+28 3.047903369 1.19E+28 3.054958701 1.19E+28 3.062014033 1.19E+28 3.069069365 1.19E+28 3.076124697 1.19E+28 3.083180029 1.20E+28 3.090235361 1.20E+28 3.097290692 1.20E+28 3.104346024 1.20E+28 3.111401356 1.20E+28 3.118456688 1.20E+28 3.12551202 1.20E+28 3.132567352 1.21E+28 3.139622684 1.21E+28 3.146678016 1.21E+28 3.153733347 1.21E+28 3.160788679 1.21E+28 3.167844011 1.21E+28 3.174899343 1.21E+28 3.181954675 1.22E+28 3.189010007 1.22E+28 3.196065339 1.22E+28 3.203120671 1.22E+28 3.210176002 1.22E+28 3.217231334 1.22E+28 3.224286666 1.22E+28 3.231341998 1.22E+28 3.23839733 1.23E+28 3.245452662 1.23E+28 3.252507994 1.23E+28 3.259563326 1.23E+28 3.266618657 1.23E+28 3.273673989 1.23E+28 3.280729321 1.23E+28 3.287784653 1.24E+28 3.294839985 1.24E+28 3.301895317 1.24E+28 3.308950649 1.24E+28 3.316005981 1.24E+28 3.323061312 1.24E+28 3.330116644 1.24E+28 3.337171976 1.24E+28 3.344227308 1.25E+28 3.35128264 1.25E+28 3.358337972 1.25E+28 3.365393304 1.25E+28 3.372448636 1.25E+28 3.379503967 1.25E+28 3.386559299 1.25E+28 3.393614631 1.25E+28 3.400669963 1.26E+28 3.407725295 1.26E+28 3.414780627 1.26E+28 3.421835959 1.26E+28 3.428891291 1.26E+28 3.435946622 1.26E+28 3.443001954 1.26E+28 3.450057286 1.27E+28 3.457112618 1.27E+28 3.46416795 1.27E+28 3.471223282 1.27E+28 3.478278614 1.27E+28 3.485333945 1.27E+28 3.492389277 1.27E+28 3.499444609 1.27E+28 3.506499941 1.28E+28 3.513555273 1.28E+28 3.520610605 1.28E+28 3.527665937 1.28E+28 3.534721269 1.28E+28 3.5417766 1.28E+28 3.548831932 1.28E+28 3.555887264 1.28E+28 3.562942596 1.29E+28 3.569997928 1.29E+28 3.57705326 1.29E+28 3.584108592 1.29E+28 3.591163924 1.29E+28 3.598219255 1.29E+28 3.605274587 1.29E+28 3.612329919 1.29E+28 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6.984778555 1.80E+28 6.991833887 1.80E+28 6.998889218 1.80E+28 7.00594455 1.80E+28 7.012999882 1.80E+28 7.020055214 1.80E+28 7.027110546 1.81E+28 7.034165878 1.81E+28 7.04122121 1.81E+28

Free electron model:
Tight binding: