{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex"},"task_id":"78ee7458-5db5-5e0f-96a7-580d4c29a0c2","task_key":"train--Challenge~5f42~5fmain","task_revision_id":"1","upstream_id":"Challenge_42_main","short_description":"Consider a graphene layer sitting on a 3D substrate. The substrate has charged…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the values of scaling exponents $\\\\alpha$ and $\\\\beta$.\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    alpha: float, scaling exponent in $\\\\xi\\\\propto n_i^{\\\\alpha}$.\\n    beta: float, scaling exponent in $\\\\Delta V_g\\\\propto n_i^{\\\\beta}$.\\n    have_plateau: bool, True if such plateau appear in 3D topological insulator.\\n    are_important: bool, True if charge impuritis are still important.\\n    is_long_range: bool, True if long-range scattering give longer mean free path, False if short-range scattering.\\n    is_longer: bool, True if long-range scattering in both graphene and 3D topological insulator give longer mean free path.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    alpha = ...\\n    beta = ...\\n    have_plateau = ...\\n    are_important = ...\\n    is_long_range = ...\\n    is_longer = ...\\n    # ---------------------------------------------------------------\\n\\n    return alpha, beta, have_plateau, are_important, is_long_range, is_longer\",\"problem_description\":\"\\n\\n# Problem setup:\\nConsider a graphene layer sitting on a 3D substrate. The substrate has charged impurities of which the density is $n_i$ $\\\\AA^{-3}$. Assume domains with characteristic linear size $\\\\xi$ will form,  in which the 3D substrate and the sample layer together are charge neutral and form a thermal equilibrium. Such domains lead to electron and hole puddles. As a result, there will be a plateau in conductivity when changing the gate voltage $V_g$.\\n\\n# Main problem:\\n\\nHow will $\\\\xi$ and the width of the plateau $\\\\Delta V_g$ scale with the impurity density $n_i$?  Assuming $\\\\xi\\\\propto n_i^{\\\\alpha}$, $\\\\Delta V_g\\\\propto n_i^{\\\\beta}$, find $\\\\alpha$ and $\\\\beta$. Will such a plateau appear in a 3D topological insulator? Are the charged impurities still important there? Do charged impurities give long-range or short-range scattering? Will the long-range scattering in both graphene and a 3D topological insulator give longer mean free path than short-range scattering?\\n\\n\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}