{"nodes": [{"id": "Potential", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "V", "formula": "V{\\left(\\{u\\} \\right)} = V_{\\mathrm{provided}}", "formulas": [{"op": "provide_potential", "latex": "V{\\left(\\{u\\} \\right)} = V_{\\mathrm{provided}}"}], "tier": "Sources", "uid": "80a4273845e09207860e06db18d1f1ee5670a9b9c8b1020a99928c872878f901"}, {"id": "Temperature", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "T", "formula": "T = T_{\\mathrm{provided}}", "formulas": [{"op": "provide_temperature", "latex": "T = T_{\\mathrm{provided}}"}], "tier": "Sources", "uid": "63f9bd32d94d77e12afbb96f5b1b7a870de6b1f5805213b7f3c3ff227da301f0"}, {"id": "ForceConstants[order=2]", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "\\Phi^{(2)}", "formula": "{\\Phi^{(2)}}_{i,j,\\mathbf{R}} = \\frac{d^{2}}{d u_{j(R)}d u_{i(0)}} V{\\left(\\{u\\} \\right)}", "formulas": [{"op": "compute_force_constants[order=2]", "latex": "{\\Phi^{(2)}}_{i,j,\\mathbf{R}} 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k_{B} \\log{\\left(1 - e^{- \\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}}} \\right)} + \\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{2}"}], "tier": "Thermodynamics", "uid": "38b082d42f5136a903752e5425c65c112c90bc564693ddb929ef3f175d9d96d8"}, {"id": "Entropy", "type": "observable", "layer": 7, "kind": "symbolic", "symbol": "S", "formula": "{s}_{\\mathbf{q},\\nu} = - k_{B} \\log{\\left(1 - e^{- \\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}}} \\right)} + \\frac{\\bar{\\h} n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}} \\right)} {\\omega}_{\\mathbf{q},\\nu}}{T}", "formulas": [{"op": "compute_entropy", "latex": "{s}_{\\mathbf{q},\\nu} = - k_{B} \\log{\\left(1 - e^{- \\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}}} \\right)} + \\frac{\\bar{\\h} n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}} \\right)} {\\omega}_{\\mathbf{q},\\nu}}{T}"}], "tier": "Thermodynamics", "uid": 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"contract_molar_free_energy", "latex": "F_{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {f}_{\\mathbf{q},\\nu}}{N_{q}}"}], "tier": "Thermodynamics", "uid": "419335bd5f3a1949576e608e4147d8f18bef980a69a9c6b99199f812c806b7be"}, {"id": "MolarEntropy", "type": "observable", "layer": 8, "kind": "symbolic", "symbol": "S_{\\mathrm{m}}", "formula": "S_{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {s}_{\\mathbf{q},\\nu}}{N_{q}}", "formulas": [{"op": "contract_molar_entropy", "latex": "S_{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {s}_{\\mathbf{q},\\nu}}{N_{q}}"}], "tier": "Thermodynamics", "uid": "0f08fcb37791dc7cc3e0e0ee5c611abbefc5075867719b275fcaa77ada4f1ac2"}, {"id": "MolarInternalEnergy", "type": "observable", "layer": 8, "kind": "symbolic", "symbol": "E_{\\mathrm{m}}", "formula": "E_{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {e}_{\\mathbf{q},\\nu}}{N_{q}}", "formulas": [{"op": "contract_molar_internal_energy", "latex": "E_{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {e}_{\\mathbf{q},\\nu}}{N_{q}}"}], "tier": "Thermodynamics", "uid": "220926f4974baec5c511f5acae2c9c4e09a8276d41c1314391b91dec537ef2fb"}, {"id": "Linewidth[channel=anharmonic_3ph]", "type": "hidden", "layer": 7, "kind": "symbolic", "symbol": "\\Gamma_{\\mathrm{3ph}}", "formula": "{\\Gamma}_{\\mathbf{q},\\nu} = \\frac{\\pi \\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N\\\\1 \\leq \\nu'' \\leq 3 N}} \\left(\\left(- 2 n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''}}{T k_{B}} \\right)} + 2 n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q'},\\nu'}}{T k_{B}} \\right)}\\right) \\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} + {\\omega}_{\\mathbf{q'},\\nu'} + 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{\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} + {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)} + \\left(n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''}}{T k_{B}} \\right)} + n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q'},\\nu'}}{T k_{B}} \\right)} + 1\\right) \\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} - {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)}\\right) \\operatorname{|V}^{2}_{3|}{\\left(\\mathbf{q},\\nu,\\mathbf{q'},\\nu',- \\mathbf{q'} + \\mathbf{q},\\nu'' \\right)}}{2 N_{q} \\bar{\\h}^{2}}"}], "tier": "Scattering", "uid": "f52d161ad0be284c121c6fd10fe403c922aaa59646746c12719622004da5fa3a"}, {"id": "Linewidth[channel=isotope]", "type": "hidden", "layer": 7, "kind": "symbolic", "symbol": "\\Gamma_{\\mathrm{iso}}", "formula": "{\\Gamma^{iso}}_{\\mathbf{q},\\nu} = \\frac{\\pi {\\omega}_{\\mathbf{q},\\nu}^{2} \\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N}} \\delta{\\left(- {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)} \\sum_{b=1}^{N} \\left(\\left|{\\sum_{\\alpha=1}^{3} \\overline{{e}_{\\alpha + 3 b - 3,\\mathbf{q},\\nu}} {e}_{\\alpha + 3 b - 3,\\mathbf{q'},\\nu'}}\\right|^{2}\\right) {g}_{b}}{4 N_{q}}", "formulas": [{"op": "compute_isotope_scattering", "latex": "{\\Gamma^{iso}}_{\\mathbf{q},\\nu} = \\frac{\\pi {\\omega}_{\\mathbf{q},\\nu}^{2} \\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N}} \\delta{\\left(- {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)} \\sum_{b=1}^{N} \\left(\\left|{\\sum_{\\alpha=1}^{3} \\overline{{e}_{\\alpha + 3 b - 3,\\mathbf{q},\\nu}} {e}_{\\alpha + 3 b - 3,\\mathbf{q'},\\nu'}}\\right|^{2}\\right) {g}_{b}}{4 N_{q}}"}], "tier": "Scattering", "uid": "6cbd3322c5b6eb0105f9ca22d9ddc88334206ec9c115a31e6fb87409cc374c45"}, {"id": "Linewidth[channel=boundary]", "type": "hidden", "layer": 8, "kind": "symbolic", "symbol": "\\Gamma_{\\mathrm{bnd}}", "formula": "{\\Gamma^{bnd}}_{\\mathbf{q},\\nu} = \\frac{\\sqrt{\\sum_{\\alpha=1}^{3} {v}_{\\alpha,\\mathbf{q},\\nu}^{2}}}{2 L}", "formulas": [{"op": "compute_boundary_scattering", "latex": "{\\Gamma^{bnd}}_{\\mathbf{q},\\nu} = \\frac{\\sqrt{\\sum_{\\alpha=1}^{3} {v}_{\\alpha,\\mathbf{q},\\nu}^{2}}}{2 L}"}], "tier": "Scattering", "uid": "81288248a2b40f0abfe78b058ffa4025a0312324e8af26b801017b9a0971961e"}, {"id": "Linewidth[channel=total]", "type": "hidden", "layer": 9, "kind": "symbolic", "symbol": "\\Gamma", "formula": "{\\Gamma^{tot}}_{\\mathbf{q},\\nu} = {\\Gamma^{anh}}_{\\mathbf{q},\\nu} + {\\Gamma^{bnd}}_{\\mathbf{q},\\nu} + {\\Gamma^{iso}}_{\\mathbf{q},\\nu}", "formulas": [{"op": "sum_linewidths", "latex": "{\\Gamma^{tot}}_{\\mathbf{q},\\nu} = {\\Gamma^{anh}}_{\\mathbf{q},\\nu} + {\\Gamma^{bnd}}_{\\mathbf{q},\\nu} + {\\Gamma^{iso}}_{\\mathbf{q},\\nu}"}], "tier": "Scattering", "uid": "50c33a1edd60bff6fb74124af1a8847a4c8a2c58d39554b4661e6a2cddb62f30"}, {"id": "IsotopeAbundances", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "g_{\\mathrm{iso}}", "formula": "{g}_{i} = g_{provided}", "formulas": [{"op": "provide_isotope_abundances", "latex": "{g}_{i} = g_{provided}"}], "tier": "Sources", "uid": "f8e4253dc1f218daff8e76af5b433dc230bf952f910ba163f87c6fa331aa033d"}, {"id": "PhononDOS", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "g(\\omega)", "formula": "g = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\delta{\\left(\\omega - {\\omega}_{\\mathbf{q},\\nu} \\right)}}{N_{q}}", "formulas": [{"op": "compute_dos", "latex": "g = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\delta{\\left(\\omega - {\\omega}_{\\mathbf{q},\\nu} \\right)}}{N_{q}}"}, {"op": "fourier_to_dos", "latex": "{g}_{\\omega} = \\frac{6 \\int\\limits_{0}^{\\infty} \\cos{\\left(\\omega \\tau \\right)} {Cv}_{\\tau}\\, d\\tau}{\\pi {Cv}_{0}}"}], "tier": "Harmonic", "uid": "e14b6cdd007ca3af971aa52880811138606efdb47e30d5a26c2335903ce796f3", "aliases": ["9b0af04f9828cfb02e19d5e41ab2c61fbe4759f049258b85c650baf22e43ebf1"]}, {"id": "Gruneisen", "type": "observable", "layer": 7, "kind": "symbolic", "symbol": "\\gamma", "formula": "{\\gamma_{G}}_{\\mathbf{q},\\nu} = - \\frac{\\sum_{\\substack{1 \\leq i \\leq 3 N\\\\1 \\leq j \\leq 3 N\\\\1 \\leq k \\leq 3 N\\\\-\\infty \\leq \\mathbf{R} \\leq \\infty\\\\-\\infty \\leq \\mathbf{R'} \\leq \\infty}} \\frac{e^{i \\mathbf{R} \\mathbf{q}} \\overline{{e}_{i,\\mathbf{q},\\nu}} {\\Phi^{(3)}}_{i,j,k,\\mathbf{R},\\mathbf{R'}} {e}_{j,\\mathbf{q},\\nu} {r}_{k,\\mathbf{R'}}}{\\sqrt{{m}_{i} {m}_{j}}}}{6 {\\omega}_{\\mathbf{q},\\nu}^{2}}", "formulas": [{"op": "compute_gruneisen", "latex": "{\\gamma_{G}}_{\\mathbf{q},\\nu} = - \\frac{\\sum_{\\substack{1 \\leq i \\leq 3 N\\\\1 \\leq j \\leq 3 N\\\\1 \\leq k \\leq 3 N\\\\-\\infty \\leq \\mathbf{R} \\leq \\infty\\\\-\\infty \\leq \\mathbf{R'} \\leq \\infty}} \\frac{e^{i \\mathbf{R} \\mathbf{q}} \\overline{{e}_{i,\\mathbf{q},\\nu}} {\\Phi^{(3)}}_{i,j,k,\\mathbf{R},\\mathbf{R'}} {e}_{j,\\mathbf{q},\\nu} {r}_{k,\\mathbf{R'}}}{\\sqrt{{m}_{i} {m}_{j}}}}{6 {\\omega}_{\\mathbf{q},\\nu}^{2}}"}], "tier": "Scattering", "uid": "a62ac1eae1e972a0b265a14f8a2af5610deb61eb0c883b7668ac4a64313b3b75"}, {"id": "PhaseSpace3Phonon", "type": "observable", "layer": 7, "kind": "symbolic", "symbol": "P_{3}", "formula": "{P_{3}}_{\\mathbf{q},\\nu} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N\\\\1 \\leq \\nu'' \\leq 3 N}} \\left(\\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} - {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)} + 2 \\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} + {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)}\\right)}{N_{q}}", "formulas": [{"op": "compute_phase_space_3phonon", "latex": "{P_{3}}_{\\mathbf{q},\\nu} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N\\\\1 \\leq \\nu'' \\leq 3 N}} \\left(\\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} - {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)} + 2 \\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} + {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)}\\right)}{N_{q}}"}], "tier": "Scattering", "uid": "ec483cecbdbfec9e13ce3b2033b03e4581edf7bc0ff935d09ab9c24a7e481080", "aliases": ["41a47eada8fee04f0c84f46e2f4e73d50b3a51967fab666636c047cb2ddce0be"]}, {"id": "MeanFreeDisplacement[bte_solver=rta]", "type": "hidden", "layer": 10, "kind": "symbolic", "symbol": "\\lambda_{\\mathrm{rta}}", "formula": "{F}_{\\alpha,\\mathbf{q},\\nu} = \\frac{{v}_{\\alpha,\\mathbf{q},\\nu}}{2 {\\Gamma}_{\\mathbf{q},\\nu}}", "formulas": [{"op": "solve_bte[bte_solver=rta]", "latex": "{F}_{\\alpha,\\mathbf{q},\\nu} = \\frac{{v}_{\\alpha,\\mathbf{q},\\nu}}{2 {\\Gamma}_{\\mathbf{q},\\nu}}"}], "tier": "Transport", "uid": "ad1373f31b9dbb8bd24ea16e1f90f1d92a39d309f2dc22583fad4fd82eca6f59"}, {"id": "MeanFreeDisplacement[bte_solver=direct_inverse]", "type": "observable", "layer": 10, "kind": "symbolic", "symbol": "\\lambda_{\\mathrm{dinv}}", "formula": "\\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N}} {F}_{\\alpha,\\mathbf{q'},\\nu'} {\\mathcal{M}}_{\\mathbf{q},\\nu,\\mathbf{q'},\\nu'} = {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}", "formulas": [{"op": "solve_bte[bte_solver=direct_inverse]", "latex": "\\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N}} {F}_{\\alpha,\\mathbf{q'},\\nu'} {\\mathcal{M}}_{\\mathbf{q},\\nu,\\mathbf{q'},\\nu'} = {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}"}], "tier": "Transport", "uid": "b24a26725f5641ffb16720514a7fdb23bf03304ce0d51c9f39501d1c8c009b6c"}, {"id": "ThermalConductivity[bte_solver=rta]", "type": "hidden", "layer": 11, "kind": "symbolic", "symbol": "\\kappa^{\\mathrm{rta}}", "formula": "{\\kappa}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "formulas": [{"op": "contract_kappa[bte_solver=rta]", "latex": "{\\kappa}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}"}], "tier": "Transport", "uid": "d79d88d53266c147ea8d806cdbd30df309199469351ee13bcc9040a98745ebc4"}, {"id": "ThermalConductivity", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "\\kappa", "formula": "\\text{True}", "formulas": [{"op": "resolve_thermal_conductivity", "latex": "\\text{True}"}, {"op": "resolve_effective_conductivity", "latex": "\\text{True}"}], "tier": "Transport", "uid": "a03324d4def54619f16ee80ccf176b95a59808141651debabbf1611b7e3a12d5"}, {"id": "ThermalConductivity[bte_solver=direct_inverse]", "type": "observable", "layer": 11, "kind": "symbolic", "symbol": "\\kappa^{\\mathrm{dinv}}", "formula": "{\\kappa}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "formulas": [{"op": "contract_kappa[bte_solver=direct_inverse]", "latex": "{\\kappa}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}"}], "tier": "Transport", "uid": "a369d33e9643b9f8766b111cde72c44fbf34a13ae46dfab31e1afab1c2129ca9"}, {"id": "ThermalConductivity[transport_model=wigner_populations]", "type": "observable", "layer": 11, "kind": "symbolic", "symbol": "\\kappa_{\\mathrm{pop}}", "formula": "{\\kappa^{W,pop}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "formulas": [{"op": "compute_kappa[transport_model=wigner_populations]", "latex": "{\\kappa^{W,pop}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}"}], "tier": "Transport", "uid": "3f837e061bf37d1ad104455abad3b9bd3d8bb16048a656977c2488fd3c4bd521"}, {"id": "ThermalConductivity[transport_model=wigner_coherences]", "type": "observable", "layer": 10, "kind": "symbolic", "symbol": "\\kappa_{\\mathrm{coh}}", "formula": "{\\kappa^{W,coh}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N\\\\1 \\leq \\nu' \\leq 3 N}} \\frac{\\left(\\frac{{c}_{\\mathbf{q},\\nu}}{{\\omega}_{\\mathbf{q},\\nu}} + \\frac{{c}_{\\mathbf{q},\\nu'}}{{\\omega}_{\\mathbf{q},\\nu'}}\\right) \\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right) \\left(\\frac{{\\omega}_{\\mathbf{q},\\nu'}}{2} + \\frac{{\\omega}_{\\mathbf{q},\\nu}}{2}\\right) {v}_{\\alpha,\\mathbf{q},\\nu} {v}_{\\beta,\\mathbf{q},\\nu'}}{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right)^{2} + \\left(- {\\omega}_{\\mathbf{q},\\nu'} + {\\omega}_{\\mathbf{q},\\nu}\\right)^{2}}}{N_{q} V_{cell}}", "formulas": [{"op": "compute_kappa[transport_model=wigner_coherences]", "latex": "{\\kappa^{W,coh}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N\\\\1 \\leq \\nu' \\leq 3 N}} \\frac{\\left(\\frac{{c}_{\\mathbf{q},\\nu}}{{\\omega}_{\\mathbf{q},\\nu}} + \\frac{{c}_{\\mathbf{q},\\nu'}}{{\\omega}_{\\mathbf{q},\\nu'}}\\right) \\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right) \\left(\\frac{{\\omega}_{\\mathbf{q},\\nu'}}{2} + \\frac{{\\omega}_{\\mathbf{q},\\nu}}{2}\\right) {v}_{\\alpha,\\mathbf{q},\\nu} {v}_{\\beta,\\mathbf{q},\\nu'}}{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right)^{2} + \\left(- {\\omega}_{\\mathbf{q},\\nu'} + {\\omega}_{\\mathbf{q},\\nu}\\right)^{2}}}{N_{q} V_{cell}}"}], "tier": "Transport", "uid": "7f8d407ef3f36290c7f8e385cddb4a06a3c1bf34563f1df35ded2bb532e69fa4"}, {"id": "ThermalConductivity[transport_model=wigner]", "type": "observable", "layer": 12, "kind": "symbolic", "symbol": "\\kappa^{\\mathrm{W}}", "formula": "{\\kappa^{W}}_{\\alpha,\\beta} = {\\kappa^{W,coh}}_{\\alpha,\\beta} + {\\kappa^{W,pop}}_{\\alpha,\\beta}", "formulas": [{"op": "combine_kappa_wigner", "latex": "{\\kappa^{W}}_{\\alpha,\\beta} = {\\kappa^{W,coh}}_{\\alpha,\\beta} + {\\kappa^{W,pop}}_{\\alpha,\\beta}"}], "tier": "Transport", "uid": "35e6000d8a0f2e4572e667befbd4cd5a8c6005fa8f6ef08791b6879c77baba64"}, {"id": "ThermalConductivity[transport_model=qhgk]", "type": "hidden", "layer": 10, "kind": "symbolic", "symbol": "\\kappa_{\\mathrm{qhgk}}", "formula": "{\\kappa^{QHGK}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N\\\\1 \\leq \\nu' \\leq 3 N}} \\frac{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right) {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu} {v}_{\\beta,\\mathbf{q},\\nu'}}{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right)^{2} + \\left(- {\\omega}_{\\mathbf{q},\\nu'} + {\\omega}_{\\mathbf{q},\\nu}\\right)^{2}}}{N_{q} V_{cell}}", "formulas": [{"op": "compute_kappa[transport_model=qhgk]", "latex": "{\\kappa^{QHGK}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N\\\\1 \\leq \\nu' \\leq 3 N}} \\frac{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right) {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu} {v}_{\\beta,\\mathbf{q},\\nu'}}{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right)^{2} + \\left(- {\\omega}_{\\mathbf{q},\\nu'} + {\\omega}_{\\mathbf{q},\\nu}\\right)^{2}}}{N_{q} V_{cell}}"}], "tier": "Transport", "uid": "8932e3cbadef74a82802b43fd1d25328cb40e6136c90df3b6a4a998e77f9567b"}, {"id": "CumulativeKappa[wrt=omega]", "type": "observable", "layer": 11, "kind": "symbolic", "symbol": "\\kappa^{\\mathrm{cum}}_{\\omega}", "formula": "{\\kappa^{cum}_\\omega}_{\\alpha,\\beta,\\omega_{c}} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\theta{\\left(\\omega_{c} - {\\omega}_{\\mathbf{q},\\nu} \\right)} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "formulas": [{"op": "contract_cumulative_kappa[wrt=omega]", "latex": "{\\kappa^{cum}_\\omega}_{\\alpha,\\beta,\\omega_{c}} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\theta{\\left(\\omega_{c} - {\\omega}_{\\mathbf{q},\\nu} \\right)} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}"}], "tier": "Transport", "uid": "55eaa32f15db94d55448456fb26f1c3ecaafcc9a3da2947f1615258c492542ef"}, {"id": "CumulativeKappa[wrt=mfp]", "type": "observable", "layer": 11, "kind": "symbolic", "symbol": "\\kappa^{\\mathrm{cum}}_{\\lambda}", "formula": "{\\kappa^{cum}_\\Lambda}_{\\alpha,\\beta,\\Lambda_{c}} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\left(\\theta{\\left(\\Lambda_{c} - \\sqrt{\\sum_{\\alpha=1}^{3} {F}_{\\alpha,\\mathbf{q},\\nu}^{2}} \\right)}\\right) {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "formulas": [{"op": "contract_cumulative_kappa[wrt=mfp]", "latex": "{\\kappa^{cum}_\\Lambda}_{\\alpha,\\beta,\\Lambda_{c}} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\left(\\theta{\\left(\\Lambda_{c} - \\sqrt{\\sum_{\\alpha=1}^{3} {F}_{\\alpha,\\mathbf{q},\\nu}^{2}} \\right)}\\right) {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}"}], "tier": "Transport", "uid": "b6a79d756cf45fc64a3a07feb9e46c7372f398b3ee54806bf1c9903d8a42039f"}, {"id": "Trajectory", "type": "hidden", "layer": 1, "kind": "symbolic", "symbol": "\\mathbf{r}(t)", "formula": "{r}_{i,\\alpha,t + 1} = \\frac{\\Delta t^{2} {F^{md}}_{i,\\alpha,t}}{2} + \\Delta t {v}_{i,\\alpha,t} + {r}_{i,\\alpha,t}", "formulas": [{"op": "run_md", "latex": "{r}_{i,\\alpha,t + 1} = \\frac{\\Delta t^{2} {F^{md}}_{i,\\alpha,t}}{2} + \\Delta t {v}_{i,\\alpha,t} + {r}_{i,\\alpha,t}"}], "tier": "Molecular dynamics", "uid": "12cde23e9a333fdd03313c4cdbd82d51033f9bb3bd505466a8db5360faf166b4"}, {"id": "HeatCurrent", "type": "hidden", "layer": 2, "kind": "symbolic", "symbol": "\\mathbf{J}", "formula": "{J}_{\\alpha,t} = \\frac{\\sum_{i=1}^{N_{atoms}} {E}_{i,t} {v}_{i,\\alpha,t} + \\sum_{\\substack{1 \\leq i \\leq N_{atoms}\\\\1 \\leq \\beta \\leq 3}} {W}_{i,\\alpha,\\beta,t} {v}_{i,\\beta,t}}{V_{cell}}", "formulas": [{"op": "compute_heat_current", "latex": "{J}_{\\alpha,t} = \\frac{\\sum_{i=1}^{N_{atoms}} {E}_{i,t} {v}_{i,\\alpha,t} + \\sum_{\\substack{1 \\leq i \\leq N_{atoms}\\\\1 \\leq \\beta \\leq 3}} {W}_{i,\\alpha,\\beta,t} {v}_{i,\\beta,t}}{V_{cell}}"}], "tier": "Molecular dynamics", "uid": "aab7019fbc014398b74ece01428527aa95e89da6f6d51efc0323b0acf59248c6"}, {"id": "HeatCurrentACF", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "\\langle JJ\\rangle", "formula": "{Jcorr}_{\\alpha,\\beta,\\tau} = \\frac{\\sum_{t=1}^{n_{lag}} {J}_{\\alpha,t} {J}_{\\beta,\\tau + t}}{n_{lag}}", "formulas": [{"op": "autocorrelate_heat_current", "latex": "{Jcorr}_{\\alpha,\\beta,\\tau} = \\frac{\\sum_{t=1}^{n_{lag}} {J}_{\\alpha,t} {J}_{\\beta,\\tau + t}}{n_{lag}}"}], "tier": "Molecular dynamics", "uid": "d8f44a4cce5e3dfea1ec9329189d49ac70e33d69cab222e2679f7791a30613ac"}, {"id": "VelocityAutocorrelation", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "\\langle v(0)v(t)\\rangle", "formula": "{Cv}_{\\tau} = \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq i \\leq N_{atoms}\\\\1 \\leq t \\leq n_{lag}}} {v}_{i,\\alpha,\\tau + t} {v}_{i,\\alpha,t}}{N_{atoms} n_{lag}}", "formulas": [{"op": "compute_velocity_autocorrelation", "latex": "{Cv}_{\\tau} = \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq i \\leq N_{atoms}\\\\1 \\leq t \\leq n_{lag}}} {v}_{i,\\alpha,\\tau + t} {v}_{i,\\alpha,t}}{N_{atoms} n_{lag}}"}], "tier": "Molecular dynamics", "uid": "f8b092359048e5130d56697ca9504bb4740b875aca685a8f760316bfb5e9a661"}, {"id": "MeanSquaredDisplacement", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "\\langle u^{2}\\rangle", "formula": "{M}_{\\tau} = \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq i \\leq N_{atoms}\\\\1 \\leq t \\leq n_{lag}}} \\left({r}_{i,\\alpha,\\tau + t} - {r}_{i,\\alpha,t}\\right)^{2}}{N_{atoms} n_{lag}}", "formulas": [{"op": "compute_msd", "latex": "{M}_{\\tau} = \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq i \\leq N_{atoms}\\\\1 \\leq t \\leq n_{lag}}} \\left({r}_{i,\\alpha,\\tau + t} - {r}_{i,\\alpha,t}\\right)^{2}}{N_{atoms} n_{lag}}"}], "tier": "Molecular dynamics", "uid": "28ebe96bc049a92e4318fd5242778f30e5647531d0a2b39c5891546329c8bca0"}, {"id": "ThermalConductivity[transport_model=green_kubo]", "type": "observable", "layer": 4, "kind": "symbolic", "symbol": "\\kappa_{\\mathrm{gk}}", "formula": "{\\kappa^{MD}}_{\\alpha,\\beta} = \\frac{V_{cell} \\int\\limits_{0}^{\\tau_{max}} {Jcorr}_{\\alpha,\\beta,\\tau}\\, d\\tau}{T^{2} k_{B}}", "formulas": [{"op": "contract_kappa[transport_model=green_kubo]", "latex": "{\\kappa^{MD}}_{\\alpha,\\beta} = \\frac{V_{cell} \\int\\limits_{0}^{\\tau_{max}} {Jcorr}_{\\alpha,\\beta,\\tau}\\, d\\tau}{T^{2} k_{B}}"}], "tier": "Molecular dynamics", "uid": "25dc9f7af8aec52efb6be16b7b7c76f8334b03056f8838c3ff432fdad8482fb6"}, {"id": "ThermalConductivity[transport_model=nemd]", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "\\kappa_{\\mathrm{nemd}}", "formula": "{\\kappa^{MD}}_{\\alpha,\\beta} = - \\frac{{J}_{\\alpha,t}}{{\\nabla T}_{\\beta}}", "formulas": [{"op": "contract_kappa[transport_model=nemd]", "latex": "{\\kappa^{MD}}_{\\alpha,\\beta} = - \\frac{{J}_{\\alpha,t}}{{\\nabla T}_{\\beta}}"}], "tier": "Molecular dynamics", "uid": "d3e529fd2e7fa0a009440c21bd64949d6326c04eaa109fbed4cafac798608118"}, {"id": "ThermalConductivity[transport_model=hnemd]", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "\\kappa_{\\mathrm{hnemd}}", "formula": "{\\kappa^{MD}}_{\\alpha,\\beta} = \\frac{{J}_{\\alpha,t}}{T V_{cell} {F_{e}}_{\\beta}}", "formulas": [{"op": "contract_kappa[transport_model=hnemd]", "latex": "{\\kappa^{MD}}_{\\alpha,\\beta} = \\frac{{J}_{\\alpha,t}}{T V_{cell} {F_{e}}_{\\beta}}"}], "tier": "Molecular dynamics", "uid": "e7157a52da5eb7d2ae79e57692acc30d7e63a3a0d1c731b14da4d6ab8fd5eb88"}, {"id": "ParticipationRatio", "type": "observable", "layer": 7, "kind": "symbolic", "symbol": "p", "formula": "{p}_{\\mathbf{q},\\nu} = \\frac{1}{N \\sum_{i=1}^{N} {a}_{i,\\mathbf{q},\\nu}^{2}}", "formulas": [{"op": "compute_participation_ratio", "latex": "{p}_{\\mathbf{q},\\nu} = \\frac{1}{N \\sum_{i=1}^{N} {a}_{i,\\mathbf{q},\\nu}^{2}}"}], "tier": "Harmonic", "uid": "09138ae29cee41ae5f1a1758ac65a479625ef4d3c6026bf7ad5b60f884d668a0"}, {"id": "ModalDiffusivity", "type": "observable", "layer": 10, "kind": "symbolic", "symbol": "D_{\\mathrm{mode}}", "formula": "D_{mode} = D^{QHGK}{\\left(\\omega,e,\\Gamma^{tot} \\right)}", "formulas": [{"op": "compute_modal_diffusivity", "latex": "D_{mode} = D^{QHGK}{\\left(\\omega,e,\\Gamma^{tot} \\right)}"}], "tier": "Transport", "uid": "6052fe7e6aa8d2116b369886bffe59b04859f21145640184d24dfd96fbb60513"}, {"id": "PhononTransmission", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "\\mathcal{T}(\\omega)", "formula": "{\\mathcal{T}}_{\\omega} = \\mathcal{T}_{provided}", "formulas": [{"op": "provide_transmission", "latex": "{\\mathcal{T}}_{\\omega} = \\mathcal{T}_{provided}"}], "tier": "Transport", "uid": "1e8b145f7ea4fc232aac842c985a46fee4a6d2b2a04118a38d67dce470ed1b50"}, {"id": "ThermalConductance[transport_model=landauer]", "type": "observable", "layer": 7, "kind": "symbolic", "symbol": "\\mathcal{G}_{\\mathrm{L}}", "formula": "\\mathcal{G} = \\frac{\\int\\limits_{0}^{\\infty} \\bar{\\h} \\omega \\frac{\\partial}{\\partial T} n_{BE}{\\left(\\frac{\\bar{\\h} \\omega}{T k_{B}} \\right)} {\\mathcal{T}}_{\\omega}\\, d\\omega}{2 \\pi}", "formulas": [{"op": "compute_conductance[transport_model=landauer]", "latex": "\\mathcal{G} = \\frac{\\int\\limits_{0}^{\\infty} \\bar{\\h} \\omega \\frac{\\partial}{\\partial T} n_{BE}{\\left(\\frac{\\bar{\\h} \\omega}{T k_{B}} \\right)} {\\mathcal{T}}_{\\omega}\\, d\\omega}{2 \\pi}"}], "tier": "Transport", "uid": "50084cef8405999bb14c2ff71b98f70f982d71254e3ec35bed543a10bbbbf542"}, {"id": "QuantumKineticEnergy", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "\\langle E_{K}\\rangle", "formula": "E_{K} = E_K^{cv}{\\left(r,T \\right)}", "formulas": [{"op": "sample_quantum_kinetic_energy", "latex": "E_{K} = E_K^{cv}{\\left(r,T \\right)}"}], "tier": "Molecular dynamics", "uid": "a9a8eae4014c21afe87f7332c4aa842b0fd9002a7ed33e0bdb1ec339402fc9c9"}, {"id": "HeatCapacity[method=pimd]", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "C_{V}^{\\mathrm{pimd}}", "formula": "C_{V} = C_V^{pimd}{\\left(r,T \\right)}", "formulas": [{"op": "sample_quantum_heat_capacity", "latex": "C_{V} = C_V^{pimd}{\\left(r,T \\right)}"}], "tier": "Molecular dynamics", "uid": 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U\\rangle/N", "formula": "\\bar{U} = \\frac{\\sum_{m=1}^{N_{s}} V{\\left({R}_{m} \\right)}}{N N_{s}}", "formulas": [{"op": "sample_mean_potential_energy", "latex": "\\bar{U} = \\frac{\\sum_{m=1}^{N_{s}} V{\\left({R}_{m} \\right)}}{N N_{s}}"}], "tier": "Molecular dynamics", "uid": "31af780e2daebb229b18642e20d5bb48a9ad42e73a0a7689e5ab148e31eaec6f"}, {"id": "Structure", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "\\mathcal{S}", "formula": null, "formulas": [], "tier": "Sources", "uid": "db1df71467f5a79440f849704ac20a33573ed4e270b5ebba59549272b2475239"}, {"id": "TotalEnergy", "type": "observable", "layer": 1, "kind": "symbolic", "symbol": "E_{tot}", "formula": "E_{tot} = E_{KS}{\\left(\\mathcal{S},V \\right)}", "formulas": [{"op": "solve_ground_state", "latex": "E_{tot} = E_{KS}{\\left(\\mathcal{S},V \\right)}"}], "tier": "Ground state", "uid": "cf5a36298543de2027b173b757d38d80b38b66a85d2d3a56f55097e3da0e9a5e"}, {"id": "Forces", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "F^{at}", "formula": "{F^{at}}_{i,\\alpha} = - \\frac{d}{d {R^{at}}_{i,\\alpha}} E_{tot}", "formulas": [{"op": "compute_forces_hf", "latex": "{F^{at}}_{i,\\alpha} = - \\frac{d}{d {R^{at}}_{i,\\alpha}} E_{tot}"}], "tier": "Ground state", "uid": "9c5a2fdedab6b668ccf5f2043fb4c29a982bcde5063fd6ac8633ee2c7858ecbf"}, {"id": "Stress", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "\\sigma", "formula": "{\\sigma}_{\\alpha,\\beta} = - \\frac{\\frac{d}{d {\\varepsilon^{str}}_{\\alpha,\\beta}} E_{tot}}{V_{cell}}", "formulas": [{"op": "compute_stress_cell", "latex": "{\\sigma}_{\\alpha,\\beta} = - \\frac{\\frac{d}{d {\\varepsilon^{str}}_{\\alpha,\\beta}} E_{tot}}{V_{cell}}"}], "tier": "Ground state", "uid": "183e66d4e443326d2c346a0327cdb125026bd8f6a91a9bb97b61265fcade7e85"}, {"id": "MagneticMoment", "type": "observable", "layer": 1, "kind": "symbolic", "symbol": "m^{spin}", "formula": "{m^{spin}}_{i} = m^{KS}{\\left(\\mathcal{S},V \\right)}", "formulas": [{"op": "compute_magnetic_moments", "latex": "{m^{spin}}_{i} = m^{KS}{\\left(\\mathcal{S},V \\right)}"}], "tier": "Ground state", "uid": "1423a9b16e18881e4388fee44f536891a6295aaa6e0fc3775bbbe89ee480847a"}, {"id": "BandGap", "type": "observable", "layer": 1, "kind": "symbolic", "symbol": "E_{gap}", "formula": "E_{gap} = E^{gap}_{KS}{\\left(\\mathcal{S},V \\right)}", "formulas": [{"op": "compute_band_gap", "latex": "E_{gap} = E^{gap}_{KS}{\\left(\\mathcal{S},V \\right)}"}], "tier": "Ground state", "uid": "7164425d1695208e46171b5030c938c3545c180b57018d51748d0507f9f08135"}, {"id": "PotentialEnergyFitError", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "\\epsilon_E", "formula": "\\epsilon_{E} = \\epsilon^{rmse}{\\left(V,E_{tot},N \\right)}", "formulas": [{"op": "compute_potential_energy_fit_error", "latex": "\\epsilon_{E} = \\epsilon^{rmse}{\\left(V,E_{tot},N \\right)}"}], "tier": "Ground state", "uid": 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\\nu^{H}{\\left(\\mathcal{S},V \\right)}", "formulas": [{"op": "compute_molecular_frequencies", "latex": "{\\nu_{mol}}_{m} = \\nu^{H}{\\left(\\mathcal{S},V \\right)}"}], "tier": "Molecular", "uid": "e1270ea9a293c9107226bf831f60985f5525a0c386dca70841743bdc5d082c26"}, {"id": "StaticDielectricTensor", "type": "observable", "layer": 7, "kind": "symbolic", "symbol": "\\varepsilon_0", "formula": "\\varepsilon_{0} = \\varepsilon_0^{stat}{\\left(\\varepsilon_{\\infty},Z^{*},\\omega,e \\right)}", "formulas": [{"op": "compute_static_dielectric", "latex": "\\varepsilon_{0} = \\varepsilon_0^{stat}{\\left(\\varepsilon_{\\infty},Z^{*},\\omega,e \\right)}"}], "tier": "Sources", "uid": "41f6c971e70bc42117a79b8ec51039f498280fdd9b77ef433bb602f90f966d93"}, {"id": "ElectronicDOS", "type": "observable", "layer": 1, "kind": "symbolic", "symbol": "ElectronicDOS", "formula": "g_{E} = g_{el}{\\left(E,\\mathcal{S},V \\right)}", "formulas": [{"op": "compute_electronic_dos", "latex": "g_{E} = g_{el}{\\left(E,\\mathcal{S},V \\right)}"}], "tier": "Electronic transport", "uid": "a32e08f3d5a147643986f961b701aea560791721df573f09188a557da369fb4f"}, {"id": "ElectricalConductivity[carrier=electronic]", "type": "observable", "layer": 8, "kind": "symbolic", "symbol": "\\sigma_{el}", "formula": "\\sigma_{el} = \\sigma^{bte}{\\left(\\mathcal{S},\\varepsilon_{0},C,\\omega \\right)}", "formulas": [{"op": "compute_electronic_conductivity", "latex": "\\sigma_{el} = \\sigma^{bte}{\\left(\\mathcal{S},\\varepsilon_{0},C,\\omega \\right)}"}], "tier": "Electronic transport", "uid": "c4a81f649b3964aed4d4b3c48dfec4f252958c9560332286e2105b175794144b"}, {"id": "SeebeckCoefficient", "type": "observable", "layer": 8, "kind": "symbolic", "symbol": "S", "formula": "S = S^{bte}{\\left(\\mathcal{S},\\varepsilon_{0},C,\\omega \\right)}", "formulas": [{"op": "compute_seebeck", "latex": "S = S^{bte}{\\left(\\mathcal{S},\\varepsilon_{0},C,\\omega \\right)}"}], "tier": "Electronic transport", "uid": 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{"id": "Diffusivity", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "D", "formula": "D = \\frac{\\mathrm{slope}_{MSD}}{2 d}", "formulas": [{"op": "contract_diffusivity", "latex": "D = \\frac{\\mathrm{slope}_{MSD}}{2 d}"}], "tier": "Diffusion", "uid": "37eb727e4cf5e46c3ca04ff4bea4fe9dbb3f7f3c3fad8a1681c7e9d04bafd34f"}, {"id": "ActivationEnergy", "type": "observable", "layer": 4, "kind": "symbolic", "symbol": "E_a", "formula": "D{\\left(T \\right)} = D_{0} e^{- \\frac{E_{a}}{T k_{B}}}", "formulas": [{"op": "fit_arrhenius", "latex": "D{\\left(T \\right)} = D_{0} e^{- \\frac{E_{a}}{T k_{B}}}"}], "tier": "Diffusion", "uid": "5d35f8f6975629291c51d432d3b137fb5bd92165706d37dbab71366a4714b71d"}, {"id": "MigrationBarrier", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "E_{m}", "formula": null, "formulas": [], "tier": "Diffusion", "uid": "c59c8d2bc372392cff55803e7b8b742878dac82f3ab7623bd267acd006d23d07"}, {"id": "ElectricalConductivity[carrier=ionic]", "type": "observable", "layer": 4, "kind": "symbolic", "symbol": "\\sigma_{ion}", "formula": "\\sigma_{ion} = \\frac{D e_{c}^{2} n_{c} z^{2}}{T k_{B}}", "formulas": [{"op": "compute_ionic_conductivity", "latex": "\\sigma_{ion} = \\frac{D e_{c}^{2} n_{c} z^{2}}{T k_{B}}"}], "tier": "Diffusion", "uid": "3f5753096d30cc6741fe9c6fd5e247e9afbb87b94f5844ec233f9243345d3712"}, {"id": "ConfigurationalEnergy", "type": "observable", "layer": 1, "kind": "symbolic", "symbol": "E_{\\mathrm{cfg}}", "formula": "E_{cfg} = E^{ce}{\\left(V,\\mathcal{S} \\right)}", "formulas": [{"op": "compute_configurational_energy", "latex": "E_{cfg} = E^{ce}{\\left(V,\\mathcal{S} \\right)}"}], "tier": "Diffusion", "uid": "12ec58944efc3849b267bf3017d9049179ed21717b7791d06fc17d69b7150c39"}, {"id": "CarrierDensity", "type": "observable", "layer": 1, "kind": "symbolic", "symbol": "n_c", "formula": "n_{c} = n^{c}{\\left(\\mathcal{S} \\right)}", "formulas": [{"op": "compute_carrier_density", "latex": "n_{c} = n^{c}{\\left(\\mathcal{S} 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S^{2} \\sigma_{el}"}], "tier": "Thermoelectric", "uid": "6c2382614be093f934d4f830419db9d60d32610041a8c213b677efc42f6a98ee"}, {"id": "ZT", "type": "observable", "layer": 13, "kind": "symbolic", "symbol": "ZT", "formula": "ZT = \\frac{PF T}{\\kappa_{tot}}", "formulas": [{"op": "contract_zt", "latex": "ZT = \\frac{PF T}{\\kappa_{tot}}"}], "tier": "Thermoelectric", "uid": "cf5879d2e17024fb396d83e19e71852f6cedf446fcf9d2feb81eb994e3e66e3a"}, {"id": "ThermalConductivity[role=matrix]", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "\\kappa_m", "formula": null, "formulas": [], "tier": "Composite", "uid": "b5cf0ca0f9b648341000b76e8839e45557973bb82a6e99460b05c066a448adf3"}, {"id": "ThermalConductivity[role=filler]", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "\\kappa_f", "formula": null, "formulas": [], "tier": "Composite", "uid": "1375fb6172571befa9a6551aa1d72bf541272a3f94ad6a762017ba3e4a44f78d"}, {"id": "InterfaceConductance", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "G", "formula": null, "formulas": [], "tier": "Composite", "uid": "2995dc3d963a6d387aa5b62d35a7248391ce3ce8047c0dc381cfc1cc99ef08f9"}, {"id": "FillerVolumeFraction", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "f", "formula": null, "formulas": [], "tier": "Composite", "uid": "093e6e3278c242894f90f40f3ef1d4b7fda19c58061ceb9e65ceddc2e0000c0d"}, {"id": "DepolarizationFactor", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "L", "formula": "L_{33} = \\begin{cases} \\frac{1}{3} & \\text{for}\\: p = 1 \\\\\\frac{- \\sqrt{1 - \\frac{1}{p^{2}}} + \\operatorname{atanh}{\\left(\\sqrt{1 - \\frac{1}{p^{2}}} \\right)}}{p^{2} \\left(1 - \\frac{1}{p^{2}}\\right)^{\\frac{3}{2}}} & \\text{for}\\: p > 1 \\\\\\frac{\\sqrt{-1 + \\frac{1}{p^{2}}} - \\operatorname{atan}{\\left(\\sqrt{-1 + \\frac{1}{p^{2}}} \\right)}}{p^{2} \\left(-1 + \\frac{1}{p^{2}}\\right)^{\\frac{3}{2}}} & \\text{otherwise} \\end{cases}", "formulas": [{"op": "depolarization_factors", "latex": "L_{33} = \\begin{cases} \\frac{1}{3} & \\text{for}\\: p = 1 \\\\\\frac{- \\sqrt{1 - \\frac{1}{p^{2}}} + \\operatorname{atanh}{\\left(\\sqrt{1 - \\frac{1}{p^{2}}} \\right)}}{p^{2} \\left(1 - \\frac{1}{p^{2}}\\right)^{\\frac{3}{2}}} & \\text{for}\\: p > 1 \\\\\\frac{\\sqrt{-1 + \\frac{1}{p^{2}}} - \\operatorname{atan}{\\left(\\sqrt{-1 + \\frac{1}{p^{2}}} \\right)}}{p^{2} \\left(-1 + \\frac{1}{p^{2}}\\right)^{\\frac{3}{2}}} & \\text{otherwise} \\end{cases}"}], "tier": "Composite", "uid": "297b3254a09037f017769b771e19ad849443911048f77078a804cbc334c38309"}, {"id": "ThermalConductivity[effective_medium=nan,orientation=random]", "type": "observable", "layer": 1, "kind": "symbolic", "symbol": "\\kappa_c", "formula": "\\kappa_{c} = \\frac{k_{m} \\left(f_{vol} \\left(\\frac{2 \\left(1 - L_{11}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right)}{L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right) + k_{m}} + \\frac{\\left(1 - L_{33}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}}\\right) + 3\\right)}{- f_{vol} \\left(\\frac{2 L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right)}{L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right) + k_{m}} + \\frac{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}}\\right) + 3}", "formulas": [{"op": "nan_effective_kappa", "latex": "\\kappa_{c} = \\frac{k_{m} \\left(f_{vol} \\left(\\frac{2 \\left(1 - L_{11}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right)}{L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right) + k_{m}} + \\frac{\\left(1 - L_{33}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}}\\right) + 3\\right)}{- f_{vol} \\left(\\frac{2 L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right)}{L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right) + k_{m}} + \\frac{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}}\\right) + 3}"}, {"op": "hasselman_johnson", "latex": "\\kappa_{c} = \\frac{k_{m} \\left(2 f_{vol} \\left(- k_{m} + \\left(1 - \\frac{k_{m}}{G_{int} a_{rad}}\\right) {k_{f}}_{3,3}\\right) + 2 k_{m} + \\left(1 + \\frac{2 k_{m}}{G_{int} a_{rad}}\\right) {k_{f}}_{3,3}\\right)}{- f_{vol} \\left(- k_{m} + \\left(1 - \\frac{k_{m}}{G_{int} a_{rad}}\\right) {k_{f}}_{3,3}\\right) + 2 k_{m} + \\left(1 + \\frac{2 k_{m}}{G_{int} a_{rad}}\\right) {k_{f}}_{3,3}}"}], "tier": "Composite", "uid": "8cee68745e003909cd8e65688628af53a35c0803ace3f330394bbf0edf338d7b"}, {"id": "ThermalConductivity[effective_medium=nan,orientation=aligned]", "type": "observable", "layer": 1, "kind": "symbolic", "symbol": "\\kappa_c^{\\parallel}", "formula": "\\kappa_{c} = \\frac{k_{m} \\left(\\frac{f_{vol} \\left(1 - L_{33}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}} + 1\\right)}{- \\frac{L_{33} f_{vol} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}} + 1}", "formulas": [{"op": "nan_effective_kappa_aligned", "latex": "\\kappa_{c} = \\frac{k_{m} \\left(\\frac{f_{vol} \\left(1 - L_{33}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}} + 1\\right)}{- \\frac{L_{33} f_{vol} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}} + 1}"}], "tier": "Composite", "uid": "fc292985123b709c76547dc02721e60797cf5693fbad40b874826291314d5dd3"}, {"id": "InterfaceConductance[transport_model=nemd]", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "G_{\\mathrm{nemd}}", "formula": "G_{int} = \\frac{q_{\\mathrm{int}}}{\\Delta T_{\\mathrm{int}}}", "formulas": [{"op": "contract_interface_conductance[transport_model=nemd]", "latex": "G_{int} = \\frac{q_{\\mathrm{int}}}{\\Delta T_{\\mathrm{int}}}"}], "tier": "Molecular dynamics", "uid": "9dabd5c3e66ac0b87479050482b623c9d72de947eb2aa74730dcb7ad7b4c8d81"}, {"id": "ConductanceState", "type": "observable", "layer": 2, "kind": "symbolic", "symbol": "G_c", "formula": "G_{c} = \\frac{A_{g} \\sigma_{el}}{L_{g}}", "formulas": [{"op": "contract_device_conductance", "latex": "G_{c} = \\frac{A_{g} \\sigma_{el}}{L_{g}}"}, {"op": "apply_conductance_drift", "latex": "G_{c} = G_{0} \\left(\\frac{t_{d}}{t_{d,0}}\\right)^{- \\nu_{drift}}"}, {"op": "emit_thermionic_conductance", "latex": "G_{c} = \\frac{P_{th} T q_{e} e^{- \\frac{\\Phi_{Bn}}{T k_{B}}}}{k_{B}}"}], "tier": "Analog device metrology", "uid": "325f09352a8872995fe6ab0a40b9df8780a778389fa0fe00ebcbd9ef7758f4a3"}, {"id": "ConductanceWindow", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "W_G", "formula": "W_{G} = \\frac{G_{c}}{G_{min}}", "formulas": [{"op": "contract_conductance_window", "latex": "W_{G} = \\frac{G_{c}}{G_{min}}"}], "tier": "Analog device metrology", "uid": "ec3515d40f800b3c58242b644285b045dcfaec8102799a44dea2a61a9783673c"}, {"id": "ConductanceDriftExponent", "type": "observable", "layer": 0, "kind": "symbolic", "symbol": "\\nu_{drift}", "formula": null, "formulas": [], "tier": "Analog device metrology", "uid": "d9c306ec2d77f10a80335d58a2f12a09124ef4a0061ddf1ae18c90c3ef597aef"}, {"id": "StateCoefficientOfVariation", "type": "observable", "layer": 3, "kind": "symbolic", "symbol": "c_G", "formula": "c_{G} = \\frac{\\sigma_{G}}{G_{c}}", "formulas": [{"op": "contract_state_coefficient_of_variation", "latex": "c_{G} = \\frac{\\sigma_{G}}{G_{c}}"}], "tier": "Analog device metrology", "uid": "165019489c9bfee532b8ff4c164d71f2a16516a933f3f40950554bdc8a0f84a5"}, {"id": "DotProductError", "type": "observable", "layer": 4, "kind": "symbolic", "symbol": "\\epsilon_{dot}", "formula": "\\epsilon_{dot} = \\frac{c_{G} 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"kind": "param", "uid": null}, {"source": "AtomCount", "target": "MolarInternalEnergy", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "PhononDOS", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "Gruneisen", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "PhaseSpace3Phonon", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "CumulativeKappa[wrt=omega]", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "CumulativeKappa[wrt=mfp]", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "ParticipationRatio", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "MeanPotentialEnergy", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "PotentialEnergyFitError", "op": "provide_AtomCount", "kind": "param", "uid": null}, {"source": "AtomCount", "target": "PotentialVirialFitError", "op": "provide_AtomCount", "kind": "param", "uid": null}], "tiers": [{"name": "Sources", "order": 0, "description": "Inputs a calculation is given: the potential, temperature, force constants, and polar-response tensors."}, {"name": "Harmonic", "order": 1, "description": "The harmonic phonon picture: dynamical matrix, frequencies, eigenvectors, group velocities, density of states."}, {"name": "Thermodynamics", "order": 2, "description": "Equilibrium thermodynamics of the phonon gas: heat capacity, entropy, free energy, and their molar forms."}, {"name": "Scattering", "order": 3, "description": "Phonon lifetimes: anharmonic, isotope, and boundary linewidth channels, phase space, and Gruneisen anharmonicity."}, {"name": "Transport", "order": 4, "description": "Thermal conductivity from the Boltzmann transport equation, the Wigner and QHGK models, and cumulative-kappa distributions."}, {"name": "Molecular dynamics", "order": 5, "description": "Direct MD: trajectories, heat current, correlation functions, and the Green-Kubo / NEMD / HNEMD conductivities."}, {"name": "Ground state", "order": 6, "description": "First-principles ground state: the total energy, forces, and stress a DFT engine computes for a structure under a chosen potential, and the error of a fitted potential against those labels."}, {"name": "Mechanics", "order": 7, "description": "Continuum mechanical response: the elastic tensor, its isotropic moduli, and the pressure."}, {"name": "Stability", "order": 8, "description": "Phase stability and electrochemistry: formation energies, distance to the convex hull, surface energetics, and intercalation voltages."}, {"name": "Thermochemistry", "order": 9, "description": "Assessed phase thermodynamics: Gibbs energies, chemical potentials, phase fractions, and transition temperatures from CALPHAD databases."}, {"name": "Quasi-harmonic", "order": 10, "description": "Finite-temperature thermodynamics from the quasi-harmonic approximation: Gibbs energy, thermal expansion, constant-pressure heat capacity, and the thermal Gruneisen parameter."}, {"name": "Molecular", "order": 11, "description": "Molecular quantum chemistry and reaction energetics: orbital gaps, reaction barriers, and bond dissociation."}, {"name": "Electronic transport", "order": 12, "description": "Carrier transport from ab-initio scattering: electronic conductivity, Seebeck, electronic thermal conductivity, and mobility."}, {"name": "Diffusion", "order": 13, "description": "Mass transport from MD: self-diffusivity via the Einstein relation and the Arrhenius activation energy."}, {"name": "Thermoelectric", "order": 14, "description": "Thermodynamic identities combining the map's own formulas: the thermal Gruneisen relation and C_P - C_V (second producers), the molar volume, the total thermal conductivity (lattice + electronic), the thermoelectric power factor, and the dimensionless figure of merit ZT."}, {"name": "Composite", "order": 15, "description": "Composite effective thermal conductivity of a filled matrix with interfacial (Kapitza) resistance: the matrix and filler conductivities, the interface conductance, the filler volume fraction and depolarization factors, and the Nan-type effective-medium conductivity (random and aligned) cross-checked against the Hasselman-Johnson spherical limit."}, {"name": "Analog device metrology", "order": 16, "description": "The conductance family of analog resistive devices (memristors, memtransistors): the programmed conductance, the conductance window (on/off ratio and LTP/LTD windows under their protocols), the drift exponent, the per-level spread, the error of an analog dot product on a crossbar of such cells, the Schottky barrier with the work function and electron affinity that set it, the energy of a programming pulse, and the set voltage and retention time of a device switched by a defect hop. Biased conductance relaxation differs from interface-trap occupation relaxation; benchmark-qualified classification accuracy requires a conductance response family and an externally evaluated readout protocol."}, {"name": "Cure kinetics", "order": 17, "description": "Cure of a thermosetting resin: the degree of cure (measured, or Kamal-Sourour kinetics over an isothermal hold) and the glass transition temperature it sets (DiBenedetto relation)."}]}