{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "# Infinite matter, from the electron gas to nuclear matter\n", "\n", " **[Morten Hjorth-Jensen](http://computationalphysics.no), National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA & Department of Physics, University of Oslo, Oslo, Norway**\n", "\n", "Date: **July 2015**\n", "\n", "## Introduction to studies of infinite matter\n", "\n", "\n", "Studies of infinite nuclear matter play an important role in nuclear physics. The aim of this part of the lectures is to provide the necessary ingredients for perfoming studies of neutron star matter (or matter in $\\beta$-equilibrium) and symmetric nuclear matter. We start however with the electron gas in two and three dimensions for both historical and pedagogical reasons. Since there are several benchmark calculations for the electron gas, this small detour will allow us to establish the necessary formalism. Thereafter we will study infinite nuclear matter \n", "* at the Hartree-Fock with realistic nuclear forces and\n", "\n", "* using many-body methods like coupled-cluster theory or in-medium SRG as discussed in our previous sections.\n", "\n", "## The infinite electron gas\n", "\n", "The electron gas is perhaps the only realistic model of a \n", "system of many interacting particles that allows for a solution\n", "of the Hartree-Fock equations on a closed form. Furthermore, to first order in the interaction, one can also\n", "compute on a closed form the total energy and several other properties of a many-particle systems. \n", "The model gives a very good approximation to the properties of valence electrons in metals.\n", "The assumptions are\n", "\n", " * System of electrons that is not influenced by external forces except by an attraction provided by a uniform background of ions. These ions give rise to a uniform background charge. The ions are stationary.\n", "\n", " * The system as a whole is neutral.\n", "\n", " * We assume we have $N_e$ electrons in a cubic box of length $L$ and volume $\\Omega=L^3$. This volume contains also a uniform distribution of positive charge with density $N_ee/\\Omega$. \n", "\n", "The homogeneous electron gas is one of the few examples of a system of many\n", "interacting particles that allows for a solution of the mean-field\n", "Hartree-Fock equations on a closed form. To first order in the\n", "electron-electron interaction, this applies to ground state properties\n", "like the energy and its pertinent equation of state as well. The\n", "homogeneus electron gas is a system of electrons that is not\n", "influenced by external forces except by an attraction provided by a\n", "uniform background of ions. These ions give rise to a uniform\n", "background charge. The ions are stationary and the system as a whole\n", "is neutral.\n", "Irrespective of this simplicity, this system, in both two and\n", "three-dimensions, has eluded a proper description of correlations in\n", "terms of various first principle methods, except perhaps for quantum\n", "Monte Carlo methods. In particular, the diffusion Monte Carlo\n", "calculations of [Ceperley](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.45.566) \n", "and [Ceperley and Tanatar](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.39.5005) \n", "are presently still considered as the\n", "best possible benchmarks for the two- and three-dimensional electron\n", "gas. \n", "\n", "\n", "\n", "The electron gas, in \n", "two or three dimensions is thus interesting as a test-bed for \n", "electron-electron correlations. The three-dimensional \n", "electron gas is particularly important as a cornerstone \n", "of the local-density approximation in density-functional \n", "theory. In the physical world, systems \n", "similar to the three-dimensional electron gas can be \n", "found in, for example, alkali metals and doped \n", "semiconductors. Two-dimensional electron fluids are \n", "observed on metal and liquid-helium surfaces, as well as \n", "at metal-oxide-semiconductor interfaces. However, the Coulomb \n", "interaction has an infinite range, and therefore \n", "long-range correlations play an essential role in the\n", "electron gas. \n", "\n", "\n", "\n", "\n", "At low densities, the electrons become \n", "localized and form a lattice. This so-called Wigner \n", "crystallization is a direct consequence \n", "of the long-ranged repulsive interaction. At higher\n", "densities, the electron gas is better described as a\n", "liquid.\n", "When using, for example, Monte Carlo methods the electron gas must be approximated \n", "by a finite system. The long-range Coulomb interaction \n", "in the electron gas causes additional finite-size effects that are not\n", "present in other infinite systems like nuclear matter or neutron star matter.\n", "This poses additional challenges to many-body methods when applied \n", "to the electron gas.\n", "\n", "\n", "\n", "\n", "\n", "## The infinite electron gas as a homogenous system\n", "\n", "This is a homogeneous system and the one-particle wave functions are given by plane wave functions normalized to a volume $\\Omega$ \n", "for a box with length $L$ (the limit $L\\rightarrow \\infty$ is to be taken after we have computed various expectation values)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\psi_{\\mathbf{k}\\sigma}(\\mathbf{r})= \\frac{1}{\\sqrt{\\Omega}}\\exp{(i\\mathbf{kr})}\\xi_{\\sigma}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $\\mathbf{k}$ is the wave number and $\\xi_{\\sigma}$ is a spin function for either spin up or down" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\xi_{\\sigma=+1/2}=\\left(\\begin{array}{c} 1 \\\\ 0 \\end{array}\\right) \\hspace{0.5cm}\n", "\\xi_{\\sigma=-1/2}=\\left(\\begin{array}{c} 0 \\\\ 1 \\end{array}\\right).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Periodic boundary conditions\n", "\n", "\n", "We assume that we have periodic boundary conditions which limit the allowed wave numbers to" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "k_i=\\frac{2\\pi n_i}{L}\\hspace{0.5cm} i=x,y,z \\hspace{0.5cm} n_i=0,\\pm 1,\\pm 2, \\dots\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We assume first that the electrons interact via a central, symmetric and translationally invariant\n", "interaction $V(r_{12})$ with\n", "$r_{12}=|\\mathbf{r}_1-\\mathbf{r}_2|$. The interaction is spin independent.\n", "\n", "The total Hamiltonian consists then of kinetic and potential energy" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H} = \\hat{T}+\\hat{V}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The operator for the kinetic energy can be written as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{T}=\\sum_{\\mathbf{k}\\sigma}\\frac{\\hbar^2k^2}{2m}a_{\\mathbf{k}\\sigma}^{\\dagger}a_{\\mathbf{k}\\sigma}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Defining the Hamiltonian operator\n", "\n", "The Hamiltonian operator is given by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}=\\hat{H}_{el}+\\hat{H}_{b}+\\hat{H}_{el-b},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with the electronic part" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_{el}=\\sum_{i=1}^N\\frac{p_i^2}{2m}+\\frac{e^2}{2}\\sum_{i\\ne j}\\frac{e^{-\\mu |\\mathbf{r}_i-\\mathbf{r}_j|}}{|\\mathbf{r}_i-\\mathbf{r}_j|},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have introduced an explicit convergence factor\n", "(the limit $\\mu\\rightarrow 0$ is performed after having calculated the various integrals).\n", "Correspondingly, we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_{b}=\\frac{e^2}{2}\\int\\int d\\mathbf{r}d\\mathbf{r}'\\frac{n(\\mathbf{r})n(\\mathbf{r}')e^{-\\mu |\\mathbf{r}-\\mathbf{r}'|}}{|\\mathbf{r}-\\mathbf{r}'|},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which is the energy contribution from the positive background charge with density\n", "$n(\\mathbf{r})=N/\\Omega$. Finally," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_{el-b}=-\\frac{e^2}{2}\\sum_{i=1}^N\\int d\\mathbf{r}\\frac{n(\\mathbf{r})e^{-\\mu |\\mathbf{r}-\\mathbf{x}_i|}}{|\\mathbf{r}-\\mathbf{x}_i|},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "is the interaction between the electrons and the positive background.\n", "\n", "\n", "\n", "## Single-particle Hartree-Fock energy\n", "\n", "In the first exercise below we show that the Hartree-Fock energy can be written as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{HF}=\\frac{\\hbar^{2}k^{2}}{2m_e}-\\frac{e^{2}}\n", "{\\Omega^{2}}\\sum_{k'\\leq\n", "k_{F}}\\int d\\mathbf{r}e^{i(\\mathbf{k}'-\\mathbf{k})\\mathbf{r}}\\int\n", "d\\mathbf{r'}\\frac{e^{i(\\mathbf{k}-\\mathbf{k}')\\mathbf{r}'}}\n", "{\\vert\\mathbf{r}-\\mathbf{r}'\\vert}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "resulting in" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{HF}=\\frac{\\hbar^{2}k^{2}}{2m_e}-\\frac{e^{2}\n", "k_{F}}{2\\pi}\n", "\\left[\n", "2+\\frac{k_{F}^{2}-k^{2}}{kk_{F}}ln\\left\\vert\\frac{k+k_{F}}\n", "{k-k_{F}}\\right\\vert\n", "\\right]\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The previous result can be rewritten in terms of the density" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "n= \\frac{k_F^3}{3\\pi^2}=\\frac{3}{4\\pi r_s^3},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $n=N_e/\\Omega$, $N_e$ being the number of electrons, and $r_s$ is the radius of a sphere which represents the volum per conducting electron. \n", "It can be convenient to use the Bohr radius $a_0=\\hbar^2/e^2m_e$.\n", "For most metals we have a relation $r_s/a_0\\sim 2-6$. The quantity $r_s$ is dimensionless.\n", "\n", "\n", "In the second exercise below we find that\n", "the total energy\n", "$E_0/N_e=\\langle\\Phi_{0}|\\hat{H}|\\Phi_{0}\\rangle/N_e$ for\n", "for this system to first order in the interaction is given as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "E_0/N_e=\\frac{e^2}{2a_0}\\left[\\frac{2.21}{r_s^2}-\\frac{0.916}{r_s}\\right].\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## Exercise 1: Hartree-Fock single-particle solution for the electron gas\n", "\n", "The electron gas model allows closed form solutions for quantities like the \n", "single-particle Hartree-Fock energy. The latter quantity is given by the following expression" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{HF}=\\frac{\\hbar^{2}k^{2}}{2m}-\\frac{e^{2}}\n", "{V^{2}}\\sum_{k'\\leq\n", "k_{F}}\\int d\\mathbf{r}e^{i(\\mathbf{k'}-\\mathbf{k})\\mathbf{r}}\\int\n", "d\\mathbf{r}'\\frac{e^{i(\\mathbf{k}-\\mathbf{k'})\\mathbf{r}'}}\n", "{\\vert\\mathbf{r}-\\mathbf{r'}\\vert}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**a)**\n", "Show first that" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{HF}=\\frac{\\hbar^{2}k^{2}}{2m}-\\frac{e^{2}\n", "k_{F}}{2\\pi}\n", "\\left[\n", "2+\\frac{k_{F}^{2}-k^{2}}{kk_{F}}ln\\left\\vert\\frac{k+k_{F}}\n", "{k-k_{F}}\\right\\vert\n", "\\right]\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "**Hint.**\n", "Hint: Introduce the convergence factor \n", "$e^{-\\mu\\vert\\mathbf{r}-\\mathbf{r}'\\vert}$\n", "in the potential and use $\\sum_{\\mathbf{k}}\\rightarrow\n", "\\frac{V}{(2\\pi)^{3}}\\int d\\mathbf{k}$\n", "\n", "\n", "\n", "\n", "\n", "**Solution.**\n", "We want to show that, given the Hartree-Fock equation for the electron gas" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{HF}=\\frac{\\hbar^{2}k^{2}}{2m}-\\frac{e^{2}}\n", "{V^{2}}\\sum_{p\\leq\n", "k_{F}}\\int d\\mathbf{r}\\exp{(i(\\mathbf{p}-\\mathbf{k})\\mathbf{r})}\\int\n", "d\\mathbf{r}'\\frac{\\exp{(i(\\mathbf{k}-\\mathbf{p})\\mathbf{r}'})}\n", "{\\vert\\mathbf{r}-\\mathbf{r'}\\vert}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "the single-particle energy can be written as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{HF}=\\frac{\\hbar^{2}k^{2}}{2m}-\\frac{e^{2}\n", "k_{F}}{2\\pi}\n", "\\left[\n", "2+\\frac{k_{F}^{2}-k^{2}}{kk_{F}}ln\\left\\vert\\frac{k+k_{F}}\n", "{k-k_{F}}\\right\\vert\n", "\\right].\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We introduce the convergence factor \n", "$e^{-\\mu\\vert\\mathbf{r}-\\mathbf{r}'\\vert}$\n", "in the potential and use $\\sum_{\\mathbf{k}}\\rightarrow\n", "\\frac{V}{(2\\pi)^{3}}\\int d\\mathbf{k}$. We can then rewrite the integral as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", "\\frac{e^{2}}\n", "{V^{2}}\\sum_{k'\\leq\n", "k_{F}}\\int d\\mathbf{r}\\exp{(i(\\mathbf{k'}-\\mathbf{k})\\mathbf{r})}\\int\n", "d\\mathbf{r}'\\frac{\\exp{(i(\\mathbf{k}-\\mathbf{p})\\mathbf{r}'})}\n", "{\\vert\\mathbf{r}-\\mathbf{r'}\\vert}= \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", "\\frac{e^{2}}{V (2\\pi)^3} \\int d\\mathbf{r}\\int\n", "\\frac{d\\mathbf{r}'}{\\vert\\mathbf{r}-\\mathbf{r'}\\vert}\\exp{(-i\\mathbf{k}(\\mathbf{r}-\\mathbf{r}'))}\\int d\\mathbf{p}\\exp{(i\\mathbf{p}(\\mathbf{r}-\\mathbf{r}'))},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and introducing the abovementioned convergence factor we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", "\\lim_{\\mu \\to 0}\\frac{e^{2}}{V (2\\pi)^3} \\int d\\mathbf{r}\\int d\\mathbf{r}'\\frac{\\exp{(-\\mu\\vert\\mathbf{r}-\\mathbf{r}'\\vert})}{\\vert\\mathbf{r}-\\mathbf{r'}\\vert}\\int d\\mathbf{p}\\exp{(i(\\mathbf{p}-\\mathbf{k})(\\mathbf{r}-\\mathbf{r}'))}.\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With a change variables to $\\mathbf{x} = \\mathbf{r}-\\mathbf{r}'$ and $\\mathbf{y}=\\mathbf{r}'$ we rewrite the last integral as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\lim_{\\mu \\to 0}\\frac{e^{2}}{V (2\\pi)^3} \\int d\\mathbf{p}\\int d\\mathbf{y}\\int d\\mathbf{x}\\exp{(i(\\mathbf{p}-\\mathbf{k})\\mathbf{x})}\\frac{\\exp{(-\\mu\\vert\\mathbf{x}\\vert})}{\\vert\\mathbf{x}\\vert}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The integration over $\\mathbf{x}$ can be performed using spherical coordinates, resulting in (with $x=\\vert \\mathbf{x}\\vert$)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\int d\\mathbf{x}\\exp{(i(\\mathbf{p}-\\mathbf{k})\\mathbf{x})}\\frac{\\exp{(-\\mu\\vert\\mathbf{x}\\vert})}{\\vert\\mathbf{x}\\vert}=\\int x^2 dx d\\phi d\\cos{(\\theta)}\\exp{(i(\\mathbf{p}-\\mathbf{k})x\\cos{(\\theta))}}\\frac{\\exp{(-\\mu x)}}{x}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We obtain" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", "4\\pi \\int dx \\frac{ \\sin{(\\vert \\mathbf{p}-\\mathbf{k}\\vert)x} }{\\vert \\mathbf{p}-\\mathbf{k}\\vert}{\\exp{(-\\mu x)}}= \\frac{4\\pi}{\\mu^2+\\vert \\mathbf{p}-\\mathbf{k}\\vert^2}.\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This results gives us" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", "\\lim_{\\mu \\to 0}\\frac{e^{2}}{V (2\\pi)^3} \\int d\\mathbf{p}\\int d\\mathbf{y}\\frac{4\\pi}{\\mu^2+\\vert \\mathbf{p}-\\mathbf{k}\\vert^2}=\\lim_{\\mu \\to 0}\\frac{e^{2}}{ 2\\pi^2} \\int d\\mathbf{p}\\frac{1}{\\mu^2+\\vert \\mathbf{p}-\\mathbf{k}\\vert^2},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have used that the integrand on the left-hand side does not depend on $\\mathbf{y}$ and that $\\int d\\mathbf{y}=V$.\n", "\n", "Introducing spherical coordinates we can rewrite the integral as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", "\\lim_{\\mu \\to 0}\\frac{e^{2}}{ 2\\pi^2} \\int d\\mathbf{p}\\frac{1}{\\mu^2+\\vert \\mathbf{p}-\\mathbf{k}\\vert^2}=\\frac{e^{2}}{ 2\\pi^2} \\int d\\mathbf{p}\\frac{1}{\\vert \\mathbf{p}-\\mathbf{k}\\vert^2}= \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", "\\frac{e^{2}}{\\pi} \\int_0^{k_F} p^2dp\\int_0^{\\pi} d\\theta\\cos{(\\theta)}\\frac{1}{p^2+k^2-2pk\\cos{(\\theta)}},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and with the change of variables $\\cos{(\\theta)}=u$ we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{e^{2}}{\\pi} \\int_0^{k_F} p^2dp\\int_{0}^{\\pi} d\\theta\\cos{(\\theta)}\\frac{1}{p^2+k^2-2pk\\cos{(\\theta)}}=\\frac{e^{2}}{\\pi} \\int_0^{k_F} p^2dp\\int_{-1}^{1} du\\frac{1}{p^2+k^2-2pku},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which gives" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{e^{2}}{k\\pi} \\int_0^{k_F} pdp\\left\\{ln(\\vert p+k\\vert)-ln(\\vert p-k\\vert)\\right\\}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Introducing new variables $x=p+k$ and $y=p-k$, we obtain after some straightforward reordering of the integral" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{e^{2}}{k\\pi}\\left[\n", "kk_F+\\frac{k_{F}^{2}-k^{2}}{kk_{F}}ln\\left\\vert\\frac{k+k_{F}}\n", "{k-k_{F}}\\right\\vert\n", "\\right],\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which gives the abovementioned expression for the single-particle energy.\n", "\n", "\n", "\n", "**b)**\n", "Rewrite the above result as a function of the density" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "n= \\frac{k_F^3}{3\\pi^2}=\\frac{3}{4\\pi r_s^3},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $n=N/V$, $N$ being the number of particles, and $r_s$ is the radius of a sphere which represents the volum per conducting electron.\n", "\n", "\n", "\n", "**Solution.**\n", "Introducing the dimensionless quantity $x=k/k_F$ and the function" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "F(x) = \\frac{1}{2}+\\frac{1-x^2}{4x}\\ln{\\left\\vert \\frac{1+x}{1-x}\\right\\vert},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "we can rewrite the single-particle Hartree-Fock energy as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{HF}=\\frac{\\hbar^{2}k^{2}}{2m}-\\frac{2e^{2}\n", "k_{F}}{\\pi}F(k/k_F),\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and dividing by the non-interacting contribution at the Fermi level," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{0}^{F}=\\frac{\\hbar^{2}k_F^{2}}{2m},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\varepsilon_{k}^{HF} }{\\varepsilon_{0}^{F}}=x^2-\\frac{e^2m}{\\hbar^2 k_F\\pi}F(x)=x^2-\\frac{4}{\\pi k_Fa_0}F(x),\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $a_0=0.0529$ nm is the Bohr radius, setting thereby a natural length scale. \n", "\n", "\n", "By introducing the radius $r_s$ of a sphere whose volume is the volume occupied by each electron, we can rewrite the previous equation in terms of $r_s$ using that the electron density $n=N/V$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "n=\\frac{k_F^3}{3\\pi^2} = \\frac{3}{4\\pi r_s^3},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "we have (with $k_F=1.92/r_s$," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\varepsilon_{k}^{HF} }{\\varepsilon_{0}^{F}}=x^2-\\frac{e^2m}{\\hbar^2 k_F\\pi}F(x)=x^2-\\frac{r_s}{a_0}0.663F(x),\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with $r_s \\sim 2-6$ for most metals.\n", "\n", "\n", "\n", "It can be convenient to use the Bohr radius $a_0=\\hbar^2/e^2m$.\n", "For most metals we have a relation $r_s/a_0\\sim 2-6$.\n", "\n", "**c)**\n", "Make a plot of the free electron energy and the Hartree-Fock energy and discuss the behavior around the Fermi surface. Extract also the Hartree-Fock band width $\\Delta\\varepsilon^{HF}$ defined as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\Delta\\varepsilon^{HF}=\\varepsilon_{k_{F}}^{HF}-\n", "\\varepsilon_{0}^{HF}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Compare this results with the corresponding one for a free electron and comment your results. How large is the contribution due to the exchange term in the Hartree-Fock equation?\n", "\n", "\n", "\n", "**Solution.**\n", "We can now define the so-called band gap, that is the scatter between the maximal and the minimal value of the electrons in the conductance band of a metal (up to the Fermi level). \n", "For $x=1$ and $r_s/a_0=4$ we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\varepsilon_{k=k_F}^{HF} }{\\varepsilon_{0}^{F}} = -0.326,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and for $x=0$ we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{\\varepsilon_{k=0}^{HF} }{\\varepsilon_{0}^{F}} = -2.652,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which results in a gap at the Fermi level of" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\Delta \\varepsilon^{HF} = \\frac{\\varepsilon_{k=k_F}^{HF} }{\\varepsilon_{0}^{F}}-\\frac{\\varepsilon_{k=0}^{HF} }{\\varepsilon_{0}^{F}} = 2.326.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This quantity measures the deviation from the $k=0$ single-particle energy and the energy at the Fermi level.\n", "The general result is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\Delta \\varepsilon^{HF} = 1+\\frac{r_s}{a_0}0.663.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following python code produces a plot of the electron energy for a free electron (only kinetic energy) and \n", "for the Hartree-Fock solution. We have chosen here a ratio $r_s/a_0=4$ and the equations are plotted as funtions\n", "of $k/f_F$." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "\n", "import numpy as np\n", "from math import log\n", "from matplotlib import pyplot as plt\n", "from matplotlib import rc, rcParams\n", "import matplotlib.units as units\n", "import matplotlib.ticker as ticker\n", "rc('text',usetex=True)\n", "rc('font',**{'family':'serif','serif':['Hartree-Fock energy']})\n", "font = {'family' : 'serif',\n", " 'color' : 'darkred',\n", " 'weight' : 'normal',\n", " 'size' : 16,\n", " }\n", "\n", "N = 100\n", "x = np.linspace(0.0, 2.0,N)\n", "F = 0.5+np.log(abs((1.0+x)/(1.0-x)))*(1.0-x*x)*0.25/x\n", "y = x*x -4.0*0.663*F\n", "\n", "plt.plot(x, y, 'b-')\n", "plt.plot(x, x*x, 'r-')\n", "plt.title(r'{\\bf Hartree-Fock single-particle energy for electron gas}', fontsize=20) \n", "plt.text(3, -40, r'Parameters: $r_s/a_0=4$', fontdict=font)\n", "plt.xlabel(r'$k/k_F$',fontsize=20)\n", "plt.ylabel(r'$\\varepsilon_k^{HF}/\\varepsilon_0^F$',fontsize=20)\n", "# Tweak spacing to prevent clipping of ylabel\n", "plt.subplots_adjust(left=0.15)\n", "plt.savefig('hartreefockspelgas.pdf', format='pdf')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the plot we notice that the exchange term increases considerably the band gap\n", "compared with the non-interacting gas of electrons.\n", "\n", "\n", "We will now define a quantity called the effective mass.\n", "For $\\vert\\mathbf{k}\\vert$ near $k_{F}$, we can Taylor expand the Hartree-Fock energy as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{HF}=\\varepsilon_{k_{F}}^{HF}+\n", "\\left(\\frac{\\partial\\varepsilon_{k}^{HF}}{\\partial k}\\right)_{k_{F}}(k-k_{F})+\\dots\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we compare the latter with the corresponding expressiyon for the non-interacting system" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\varepsilon_{k}^{(0)}=\\frac{\\hbar^{2}k^{2}_{F}}{2m}+\n", "\\frac{\\hbar^{2}k_{F}}{m}\\left(k-k_{F}\\right)+\\dots ,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "we can define the so-called effective Hartree-Fock mass as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "m_{HF}^{*}\\equiv\\hbar^{2}k_{F}\\left(\n", "\\frac{\\partial\\varepsilon_{k}^{HF}}\n", "{\\partial k}\\right)_{k_{F}}^{-1}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**d)**\n", "Compute $m_{HF}^{*}$ and comment your results.\n", "\n", "**e)**\n", "Show that the level density (the number of single-electron states per unit energy) can be written as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "n(\\varepsilon)=\\frac{Vk^{2}}{2\\pi^{2}}\\left(\n", "\\frac{\\partial\\varepsilon}{\\partial k}\\right)^{-1}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Calculate $n(\\varepsilon_{F}^{HF})$ and comment the results.\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "## Exercise 2: Hartree-Fock ground state energy for the electron gas in three dimensions\n", "\n", "We consider a system of electrons in infinite matter, the so-called electron gas. This is a homogeneous system and the one-particle states are given by plane wave function normalized to a volume $\\Omega$ \n", "for a box with length $L$ (the limit $L\\rightarrow \\infty$ is to be taken after we have computed various expectation values)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\psi_{\\mathbf{k}\\sigma}(\\mathbf{r})= \\frac{1}{\\sqrt{\\Omega}}\\exp{(i\\mathbf{kr})}\\xi_{\\sigma}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $\\mathbf{k}$ is the wave number and $\\xi_{\\sigma}$ is a spin function for either spin up or down" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\xi_{\\sigma=+1/2}=\\left(\\begin{array}{c} 1 \\\\ 0 \\end{array}\\right) \\hspace{0.5cm}\n", "\\xi_{\\sigma=-1/2}=\\left(\\begin{array}{c} 0 \\\\ 1 \\end{array}\\right).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We assume that we have periodic boundary conditions which limit the allowed wave numbers to" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "k_i=\\frac{2\\pi n_i}{L}\\hspace{0.5cm} i=x,y,z \\hspace{0.5cm} n_i=0,\\pm 1,\\pm 2, \\dots\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We assume first that the particles interact via a central, symmetric and translationally invariant\n", "interaction $V(r_{12})$ with\n", "$r_{12}=|\\mathbf{r}_1-\\mathbf{r}_2|$. The interaction is spin independent.\n", "\n", "The total Hamiltonian consists then of kinetic and potential energy" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H} = \\hat{T}+\\hat{V}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The operator for the kinetic energy is given by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{T}=\\sum_{\\mathbf{k}\\sigma}\\frac{\\hbar^2k^2}{2m}a_{\\mathbf{k}\\sigma}^{\\dagger}a_{\\mathbf{k}\\sigma}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**a)**\n", "Find the expression for the interaction\n", "$\\hat{V}$ expressed with creation and annihilation operators. The expression for the interaction\n", "has to be written in $k$ space, even though $V$ depends only on the relative distance. It means that you need to set up the Fourier transform $\\langle \\mathbf{k}_i\\mathbf{k}_j| V | \\mathbf{k}_m\\mathbf{k}_n\\rangle$.\n", "\n", "\n", "\n", "**Solution.**\n", "A general two-body interaction element is given by (not using anti-symmetrized matrix elements)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{V} = \\frac{1}{2} \\sum_{pqrs} \\langle pq \\hat{v} \\vert rs\\rangle a_p^\\dagger a_q^\\dagger a_s a_r ,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $\\hat{v}$ is assumed to depend only on the relative distance between two interacting particles, that is\n", "$\\hat{v} = v(\\vec r_1, \\vec r_2) = v(|\\vec r_1 - \\vec r_2|) = v(r)$, with $r = |\\vec r_1 - \\vec r_2|$). \n", "In our case we have, writing out explicitely the spin degrees of freedom as well" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", "\\hat{V} = \\frac{1}{2} \\sum_{\\substack{\\sigma_p \\sigma_q \\\\ \\sigma_r \\sigma_s}}\n", "\\sum_{\\substack{\\mathbf{k}_p \\mathbf{k}_q \\\\ \\mathbf{k}_r \\mathbf{k}_s}}\n", "\\langle \\mathbf{k}_p \\sigma_p, \\mathbf{k}_q \\sigma_2\\vert v \\vert \\mathbf{k}_r \\sigma_3, \\mathbf{k}_s \\sigma_s\\rangle\n", "a_{\\mathbf{k}_p \\sigma_p}^\\dagger a_{\\mathbf{k}_q \\sigma_q}^\\dagger a_{\\mathbf{k}_s \\sigma_s} a_{\\mathbf{k}_r \\sigma_r} .\n", "\\label{eq:V_original} \\tag{1}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Inserting plane waves as eigenstates we can rewrite the matrix element as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\mathbf{k}_p \\sigma_p, \\mathbf{k}_q \\sigma_q\\vert \\hat{v} \\vert \\mathbf{k}_r \\sigma_r, \\mathbf{k}_s \\sigma_s\\rangle =\n", "\\frac{1}{\\Omega^2} \\delta_{\\sigma_p \\sigma_r} \\delta_{\\sigma_q \\sigma_s}\n", "\\int\\int \\exp{-i(\\mathbf{k}_p \\cdot \\mathbf{r}_p)} \\exp{-i( \\mathbf{k}_q \\cdot \\mathbf{r}_q)} \\hat{v}(r) \\exp{i(\\mathbf{k}_r \\cdot \\mathbf{r}_p)} \\exp{i( \\mathbf{k}_s \\cdot \\mathbf{r}_q)} d\\mathbf{r}_p d\\mathbf{r}_q ,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have used the orthogonality properties of the spin functions. We change now the variables of integration\n", "by defining $\\mathbf{r} = \\mathbf{r}_p - \\mathbf{r}_q$, which gives $\\mathbf{r}_p = \\mathbf{r} + \\mathbf{r}_q$ and $d^3 \\mathbf{r} = d^3 \\mathbf{r}_p$. \n", "The limits are not changed since they are from $-\\infty$ to $\\infty$ for all integrals. This results in" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\mathbf{k}_p \\sigma_p, \\mathbf{k}_q \\sigma_q\\vert \\hat{v} \\vert \\mathbf{k}_r \\sigma_r, \\mathbf{k}_s \\sigma_s\\rangle\n", "= \\frac{1}{\\Omega^2} \\delta_{\\sigma_p \\sigma_r} \\delta_{\\sigma_q \\sigma_s} \\int\\exp{i (\\mathbf{k}_s - \\mathbf{k}_q) \\cdot \\mathbf{r}_q} \\int v(r) \\exp{i(\\mathbf{k}_r - \\mathbf{k}_p) \\cdot ( \\mathbf{r} + \\mathbf{r}_q)} d\\mathbf{r} d\\mathbf{r}_q\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "= \\frac{1}{\\Omega^2} \\delta_{\\sigma_p \\sigma_r} \\delta_{\\sigma_q \\sigma_s} \\int v(r) \\exp{i\\left[(\\mathbf{k}_r - \\mathbf{k}_p) \\cdot \\mathbf{r}\\right]}\n", "\\int \\exp{i\\left[(\\mathbf{k}_s - \\mathbf{k}_q + \\mathbf{k}_r - \\mathbf{k}_p) \\cdot \\mathbf{r}_q\\right]} d\\mathbf{r}_q d\\mathbf{r} .\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We recognize the integral over $\\mathbf{r}_q$ as a $\\delta$-function, resulting in" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\mathbf{k}_p \\sigma_p, \\mathbf{k}_q \\sigma_q\\vert \\hat{v} \\vert \\mathbf{k}_r \\sigma_r, \\mathbf{k}_s \\sigma_s\\rangle =\n", "\\frac{1}{\\Omega} \\delta_{\\sigma_p \\sigma_r} \\delta_{\\sigma_q \\sigma_s} \\delta_{(\\mathbf{k}_p + \\mathbf{k}_q),(\\mathbf{k}_r + \\mathbf{k}_s)} \\int v(r) \\exp{i\\left[(\\mathbf{k}_r - \\mathbf{k}_p) \\cdot \\mathbf{r}\\right]} d^3r .\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For this equation to be different from zero, we must have conservation of momenta, we need to satisfy\n", "$\\mathbf{k}_p + \\mathbf{k}_q = \\mathbf{k}_r + \\mathbf{k}_s$. We can use the conservation of momenta to remove one of the summation variables in \n", "Eq. (ref{eq:V_original}, resulting in" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{V} =\n", "\\frac{1}{2\\Omega} \\sum_{\\sigma \\sigma'} \\sum_{\\mathbf{k}_p \\mathbf{k}_q \\mathbf{k}_r} \\left[ \\int v(r) \\exp{i\\left[(\\mathbf{k}_r - \\mathbf{k}_p) \\cdot \\mathbf{r}\\right]} d^3r \\right]\n", "a_{\\mathbf{k}_p \\sigma}^\\dagger a_{\\mathbf{k}_q \\sigma'}^\\dagger a_{\\mathbf{k}_p + \\mathbf{k}_q - \\mathbf{k}_r, \\sigma'} a_{\\mathbf{k}_r \\sigma},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which can be rewritten as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "\\hat{V} =\n", "\\frac{1}{2\\Omega} \\sum_{\\sigma \\sigma'} \\sum_{\\mathbf{k} \\mathbf{p} \\mathbf{q}} \\left[ \\int v(r) \\exp{-i( \\mathbf{q} \\cdot \\mathbf{r})} d\\mathbf{r} \\right]\n", "a_{\\mathbf{k} + \\mathbf{q}, \\sigma}^\\dagger a_{\\mathbf{p} - \\mathbf{q}, \\sigma'}^\\dagger a_{\\mathbf{p} \\sigma'} a_{\\mathbf{k} \\sigma},\n", "\\label{eq:V} \\tag{2}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This equation will be useful for our nuclear matter calculations as well. In the last equation we defined\n", "the quantities\n", "$\\mathbf{p} = \\mathbf{k}_p + \\mathbf{k}_q - \\mathbf{k}_r$, $\\mathbf{k} = \\mathbf{k}_r$ og $\\mathbf{q} = \\mathbf{k}_p - \\mathbf{k}_r$.\n", "\n", "\n", "\n", "**b)**\n", "Calculate thereafter the reference energy for the infinite electron gas in three dimensions using the above expressions for the kinetic energy and the potential energy.\n", "\n", "\n", "\n", "**Solution.**\n", "Let us now compute the expectation value of the reference energy using the expressions for the kinetic energy operator and the interaction.\n", "We need to compute $\\langle \\Phi_0\\vert \\hat{H} \\vert \\Phi_0\\rangle = \\langle \\Phi_0\\vert \\hat{T} \\vert \\Phi_0\\rangle + \\langle \\Phi_0\\vert \\hat{V} \\vert \\Phi_0\\rangle$, where $\\vert \\Phi_0\\rangle$ is our reference Slater determinant, constructed from filling all single-particle states up to the Fermi level.\n", "Let us start with the kinetic energy first" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\Phi_0\\vert \\hat{T} \\vert \\Phi_0\\rangle \n", "= \\langle \\Phi_0\\vert \\left( \\sum_{\\mathbf{p} \\sigma} \\frac{\\hbar^2 p^2}{2m} a_{\\mathbf{p} \\sigma}^\\dagger a_{\\mathbf{p} \\sigma} \\right) \\vert \\Phi_0\\rangle \\\\\n", "= \\sum_{\\mathbf{p} \\sigma} \\frac{\\hbar^2 p^2}{2m} \\langle \\Phi_0\\vert a_{\\mathbf{p} \\sigma}^\\dagger a_{\\mathbf{p} \\sigma} \\vert \\Phi_0\\rangle .\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the possible contractions using Wick's theorem, it is straightforward to convince oneself that the expression for the kinetic energy becomes" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\Phi_0\\vert \\hat{T} \\vert \\Phi_0\\rangle = \\sum_{\\mathbf{i} \\leq F} \\frac{\\hbar^2 k_i^2}{m} = \\frac{\\Omega}{(2\\pi)^3} \\frac{\\hbar^2}{m} \\int_0^{k_F} k^2 d\\mathbf{k}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The sum of the spin degrees of freedom results in a factor of two only if we deal with identical spin $1/2$ fermions. \n", "Changing to spherical coordinates, the integral over the momenta $k$ results in the final expression" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\Phi_0\\vert \\hat{T} \\vert \\Phi_0\\rangle = \\frac{\\Omega}{(2\\pi)^3} \\left( 4\\pi \\int_0^{k_F} k^4 d\\mathbf{k} \\right) = \\frac{4\\pi\\Omega}{(2\\pi)^3} \\frac{1}{5} k_F^5 = \\frac{4\\pi\\Omega}{5(2\\pi)^3} k_F^5 = \\frac{\\hbar^2 \\Omega}{10\\pi^2 m} k_F^5 .\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The density of states in momentum space is given by $2\\Omega/(2\\pi)^3$, where we have included the degeneracy due to the spin degrees of freedom.\n", "The volume is given by $4\\pi k_F^3/3$, and the number of particles becomes" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "N = \\frac{2\\Omega}{(2\\pi)^3} \\frac{4}{3} \\pi k_F^3 = \\frac{\\Omega}{3\\pi^2} k_F^3 \\quad \\Rightarrow \\quad\n", "k_F = \\left( \\frac{3\\pi^2 N}{\\Omega} \\right)^{1/3}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This gives us" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "\\langle \\Phi_0\\vert \\hat{T} \\vert \\Phi_0\\rangle =\n", "\\frac{\\hbar^2 \\Omega}{10\\pi^2 m} \\left( \\frac{3\\pi^2 N}{\\Omega} \\right)^{5/3} =\n", "\\frac{\\hbar^2 (3\\pi^2)^{5/3} N}{10\\pi^2 m} \\rho^{2/3} ,\n", "\\label{eq:T_forventning} \\tag{3}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We are now ready to calculate the expectation value of the potential energy" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\Phi_0\\vert \\hat{V} \\vert \\Phi_0\\rangle \n", "= \\langle \\Phi_0\\vert \\left( \\frac{1}{2\\Omega} \\sum_{\\sigma \\sigma'} \\sum_{\\mathbf{k} \\mathbf{p} \\mathbf{q} } \\left[ \\int v(r) \\exp{-i (\\mathbf{q} \\cdot \\mathbf{r})} d\\mathbf{r} \\right] a_{\\mathbf{k} + \\mathbf{q}, \\sigma}^\\dagger a_{\\mathbf{p} - \\mathbf{q}, \\sigma'}^\\dagger a_{\\mathbf{p} \\sigma'} a_{\\mathbf{k} \\sigma} \\right) \\vert \\Phi_0\\rangle\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "= \\frac{1}{2\\Omega} \\sum_{\\sigma \\sigma'} \\sum_{\\mathbf{k} \\mathbf{p} \\mathbf{q}} \\left[ \\int v(r) \\exp{-i (\\mathbf{q} \\cdot \\mathbf{r})} d\\mathbf{r} \\right]\\langle \\Phi_0\\vert a_{\\mathbf{k} + \\mathbf{q}, \\sigma}^\\dagger a_{\\mathbf{p} - \\mathbf{q}, \\sigma'}^\\dagger a_{\\mathbf{p} \\sigma'} a_{\\mathbf{k} \\sigma} \\vert \\Phi_0\\rangle .\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The only contractions which result in non-zero results are those that involve states below the Fermi level, that is \n", "$k \\leq k_F$, $p \\leq k_F$, $|\\mathbf{p} - \\mathbf{q}| < \\mathbf{k}_F$ and $|\\mathbf{k} + \\mathbf{q}| \\leq k_F$. Due to momentum conservation we must also have $\\mathbf{k} + \\mathbf{q} = \\mathbf{p}$, $\\mathbf{p} - \\mathbf{q} = \\mathbf{k}$ and $\\sigma = \\sigma'$ or $\\mathbf{k} + \\mathbf{q} = \\mathbf{k}$ and $\\mathbf{p} - \\mathbf{q} = \\mathbf{p}$. \n", "Summarizing, we must have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{k} + \\mathbf{q} = \\mathbf{p} \\quad \\text{and} \\quad \\sigma = \\sigma', \\qquad\n", "\\text{or} \\qquad\n", "\\mathbf{q} = \\mathbf{0} .\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We obtain then" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\Phi_0\\vert \\hat{V} \\vert \\Phi_0\\rangle =\n", "\\frac{1}{2\\Omega} \\left( \\sum_{\\sigma \\sigma'} \\sum_{\\mathbf{q} \\mathbf{p} \\leq F} \\left[ \\int v(r) d\\mathbf{r} \\right] - \\sum_{\\sigma}\n", "\\sum_{\\mathbf{q} \\mathbf{p} \\leq F} \\left[ \\int v(r) \\exp{-i (\\mathbf{q} \\cdot \\mathbf{r})} d\\mathbf{r} \\right] \\right).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The first term is the so-called direct term while the second term is the exchange term. \n", "We can rewrite this equation as (and this applies to any potential which depends only on the relative distance between particles)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "\\langle \\Phi_0\\vert \\hat{V} \\vert \\Phi_0\\rangle =\n", "\\frac{1}{2\\Omega} \\left( N^2 \\left[ \\int v(r) d\\mathbf{r} \\right] - N \\sum_{\\mathbf{q}} \\left[ \\int v(r) \\exp{-i (\\mathbf{q}\\cdot \\mathbf{r})} d\\mathbf{r} \\right] \\right),\n", "\\label{eq:V_b} \\tag{4}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have used the fact that a sum like $\\sum_{\\sigma}\\sum_{\\mathbf{k}}$ equals the number of particles. Using the fact that the density is given by\n", "$\\rho = N/\\Omega$, with $\\Omega$ being our volume, we can rewrite the last equation as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\Phi_0\\vert \\hat{V} \\vert \\Phi_0\\rangle =\n", "\\frac{1}{2} \\left( \\rho N \\left[ \\int v(r) d\\mathbf{r} \\right] - \\rho\\sum_{\\mathbf{q}} \\left[ \\int v(r) \\exp{-i (\\mathbf{q}\\cdot \\mathbf{r})} d\\mathbf{r} \\right] \\right).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the electron gas\n", "the interaction part of the Hamiltonian operator is given by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_I=\\hat{H}_{el}+\\hat{H}_{b}+\\hat{H}_{el-b},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with the electronic part" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_{el}=\\sum_{i=1}^N\\frac{p_i^2}{2m}+\\frac{e^2}{2}\\sum_{i\\ne j}\\frac{e^{-\\mu |\\mathbf{r}_i-\\mathbf{r}_j|}}{|\\mathbf{r}_i-\\mathbf{r}_j|},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have introduced an explicit convergence factor\n", "(the limit $\\mu\\rightarrow 0$ is performed after having calculated the various integrals).\n", "Correspondingly, we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_{b}=\\frac{e^2}{2}\\int\\int d\\mathbf{r}d\\mathbf{r}'\\frac{n(\\mathbf{r})n(\\mathbf{r}')e^{-\\mu |\\mathbf{r}-\\mathbf{r}'|}}{|\\mathbf{r}-\\mathbf{r}'|},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which is the energy contribution from the positive background charge with density\n", "$n(\\mathbf{r})=N/\\Omega$. Finally," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_{el-b}=-\\frac{e^2}{2}\\sum_{i=1}^N\\int d\\mathbf{r}\\frac{n(\\mathbf{r})e^{-\\mu |\\mathbf{r}-\\mathbf{x}_i|}}{|\\mathbf{r}-\\mathbf{x}_i|},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "is the interaction between the electrons and the positive background.\n", "We can show that" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_{b}=\\frac{e^2}{2}\\frac{N^2}{\\Omega}\\frac{4\\pi}{\\mu^2},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{H}_{el-b}=-e^2\\frac{N^2}{\\Omega}\\frac{4\\pi}{\\mu^2}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the electron gas and a Coulomb interaction, these two terms are cancelled (in the thermodynamic limit) by the contribution from the direct term arising\n", "from the repulsive electron-electron interaction. What remains then when computing the reference energy is only the kinetic energy contribution and the contribution from the exchange term. For other interactions, like nuclear forces with a short range part and no infinite range, we need to compute both the direct term and the exchange term.\n", "\n", "\n", "\n", "**c)**\n", "Show thereafter that the final Hamiltonian can be written as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "H=H_{0}+H_{I},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "H_{0}={\\displaystyle\\sum_{\\mathbf{k}\\sigma}}\n", "\\frac{\\hbar^{2}k^{2}}{2m}a_{\\mathbf{k}\\sigma}^{\\dagger}\n", "a_{\\mathbf{k}\\sigma},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "H_{I}=\\frac{e^{2}}{2\\Omega}{\\displaystyle\\sum_{\\sigma_{1}\\sigma_{2}}}{\\displaystyle\\sum_{\\mathbf{q}\\neq 0,\\mathbf{k},\\mathbf{p}}}\\frac{4\\pi}{q^{2}}\n", "a_{\\mathbf{k}+\\mathbf{q},\\sigma_{1}}^{\\dagger}\n", "a_{\\mathbf{p}-\\mathbf{q},\\sigma_{2}}^{\\dagger}\n", "a_{\\mathbf{p}\\sigma_{2}}a_{\\mathbf{k}\\sigma_{1}}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**d)**\n", "Calculate $E_0/N=\\langle \\Phi_{0}\\vert H\\vert \\Phi_{0}\\rangle/N$ for for this system to first order in the interaction. Show that, by using" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\rho= \\frac{k_F^3}{3\\pi^2}=\\frac{3}{4\\pi r_0^3},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with $\\rho=N/\\Omega$, $r_0$\n", "being the radius of a sphere representing the volume an electron occupies \n", "and the Bohr radius $a_0=\\hbar^2/e^2m$, \n", "that the energy per electron can be written as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "E_0/N=\\frac{e^2}{2a_0}\\left[\\frac{2.21}{r_s^2}-\\frac{0.916}{r_s}\\right].\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here we have defined\n", "$r_s=r_0/a_0$ to be a dimensionless quantity.\n", "\n", "**e)**\n", "Plot your results. Why is this system stable?\n", "Calculate thermodynamical quantities like the pressure, given by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "P=-\\left(\\frac{\\partial E}{\\partial \\Omega}\\right)_N,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and the bulk modulus" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "B=-\\Omega\\left(\\frac{\\partial P}{\\partial \\Omega}\\right)_N,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and comment your results.\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "## Preparing the ground for numerical calculations; kinetic energy and Ewald term\n", "\n", "The kinetic energy operator is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\hat{H}_{\\text{kin}} = -\\frac{\\hbar^{2}}{2m}\\sum_{i=1}^{N}\\nabla_{i}^{2},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the sum is taken over all particles in the finite\n", "box. The Ewald electron-electron interaction operator \n", "can be written as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\hat{H}_{ee} = \\sum_{i < j}^{N} v_{E}\\left( \\mathbf{r}_{i}-\\mathbf{r}_{j}\\right)\n", " + \\frac{1}{2}Nv_{0},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $v_{E}(\\mathbf{r})$ is the effective two-body \n", "interaction and $v_{0}$ is the self-interaction, defined \n", "as $v_{0} = \\lim_{\\mathbf{r} \\rightarrow 0} \\left\\{ v_{E}(\\mathbf{r}) - 1/r\\right\\} $. \n", "\n", "The negative \n", "electron charges are neutralized by a positive, homogeneous \n", "background charge. Fraser *et al.* explain how the\n", "electron-background and background-background terms, \n", "$\\hat{H}_{eb}$ and $\\hat{H}_{bb}$, vanish\n", "when using Ewald's interaction for the three-dimensional\n", "electron gas. Using the same arguments, one can show that\n", "these terms are also zero in the corresponding \n", "two-dimensional system. \n", "\n", "\n", "\n", "\n", "## Ewald correction term\n", "\n", "In the three-dimensional electron gas, the Ewald \n", "interaction is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " v_{E}(\\mathbf{r}) = \\sum_{\\mathbf{k} \\neq \\mathbf{0}}\n", " \\frac{4\\pi }{L^{3}k^{2}}e^{i\\mathbf{k}\\cdot \\mathbf{r}}\n", " e^{-\\eta^{2}k^{2}/4} \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " + \\sum_{\\mathbf{R}}\\frac{1}{\\left| \\mathbf{r}\n", " -\\mathbf{R}\\right| } \\mathrm{erfc} \\left( \\frac{\\left| \n", " \\mathbf{r}-\\mathbf{R}\\right|}{\\eta }\\right)\n", " - \\frac{\\pi \\eta^{2}}{L^{3}},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $L$ is the box side length, $\\mathrm{erfc}(x)$ is the \n", "complementary error function, and $\\eta $ is a free\n", "parameter that can take any value in the interval \n", "$(0, \\infty )$.\n", "\n", "\n", "\n", "## Interaction in momentum space\n", "\n", "The translational vector" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\mathbf{R} = L\\left(n_{x}\\mathbf{u}_{x} + n_{y}\n", " \\mathbf{u}_{y} + n_{z}\\mathbf{u}_{z}\\right) ,\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $\\mathbf{u}_{i}$ is the unit vector for dimension $i$,\n", "is defined for all integers $n_{x}$, $n_{y}$, and \n", "$n_{z}$. These vectors are used to obtain all image\n", "cells in the entire real space. \n", "The parameter $\\eta $ decides how \n", "the Coulomb interaction is divided into a short-ranged\n", "and long-ranged part, and does not alter the total\n", "function. However, the number of operations needed\n", "to calculate the Ewald interaction with a desired \n", "accuracy depends on $\\eta $, and $\\eta $ is therefore \n", "often chosen to optimize the convergence as a function\n", "of the simulation-cell size. In\n", "our calculations, we choose $\\eta $ to be an infinitesimally\n", "small positive number, similarly as was done by [Shepherd *et al.*](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.86.035111) and [Roggero *et al.*](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.115138).\n", "\n", "This gives an interaction that is evaluated only in\n", "Fourier space. \n", "\n", "When studying the two-dimensional electron gas, we\n", "use an Ewald interaction that is quasi two-dimensional.\n", "The interaction is derived in three dimensions, with \n", "Fourier discretization in only two dimensions. The Ewald effective\n", "interaction has the form" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " v_{E}(\\mathbf{r}) = \\sum_{\\mathbf{k} \\neq \\mathbf{0}} \n", " \\frac{\\pi }{L^{2}k}\\left\\{ e^{-kz} \\mathrm{erfc} \\left(\n", " \\frac{\\eta k}{2} - \\frac{z}{\\eta }\\right)+ \\right. \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " \\left. e^{kz}\\mathrm{erfc} \\left( \\frac{\\eta k}{2} + \\frac{z}{\\eta }\n", " \\right) \\right\\} e^{i\\mathbf{k}\\cdot \\mathbf{r}_{xy}} \n", " \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " + \\sum_{\\mathbf{R}}\\frac{1}{\\left| \\mathbf{r}-\\mathbf{R}\n", " \\right| } \\mathrm{erfc} \\left( \\frac{\\left| \\mathbf{r}-\\mathbf{R}\n", " \\right|}{\\eta }\\right) \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " - \\frac{2\\pi}{L^{2}}\\left\\{ z\\mathrm{erf} \\left( \\frac{z}{\\eta }\n", " \\right) + \\frac{\\eta }{\\sqrt{\\pi }}e^{-z^{2}/\\eta^{2}}\\right\\},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the Fourier vectors $\\mathbf{k}$ and the position vector\n", "$\\mathbf{r}_{xy}$ are defined in the $(x,y)$ plane. When\n", "applying the interaction $v_{E}(\\mathbf{r})$ to two-dimensional\n", "systems, we set $z$ to zero. \n", "\n", "\n", "Similarly as in the \n", "three-dimensional case, also here we \n", "choose $\\eta $ to approach zero from above. The resulting \n", "Fourier-transformed interaction is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " v_{E}^{\\eta = 0, z = 0}(\\mathbf{r}) = \\sum_{\\mathbf{k} \\neq \\mathbf{0}} \n", " \\frac{2\\pi }{L^{2}k}e^{i\\mathbf{k}\\cdot \\mathbf{r}_{xy}}. \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The self-interaction $v_{0}$ is a constant that can be \n", "included in the reference energy.\n", "\n", "\n", "\n", "\n", "## Antisymmetrized matrix elements in three dimensions\n", "\n", "In the three-dimensional electron gas, the antisymmetrized\n", "matrix elements are" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation} \\label{eq:vmat_3dheg} \\tag{5}\n", " \\langle \\mathbf{k}_{p}m_{s_{p}}\\mathbf{k}_{q}m_{s_{q}}\n", " |\\tilde{v}|\\mathbf{k}_{r}m_{s_{r}}\\mathbf{k}_{s}m_{s_{s}}\\rangle_{AS} \n", " \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " = \\frac{4\\pi }{L^{3}}\\delta_{\\mathbf{k}_{p}+\\mathbf{k}_{q},\n", " \\mathbf{k}_{r}+\\mathbf{k}_{s}}\\left\\{ \n", " \\delta_{m_{s_{p}}m_{s_{r}}}\\delta_{m_{s_{q}}m_{s_{s}}}\n", " \\left( 1 - \\delta_{\\mathbf{k}_{p}\\mathbf{k}_{r}}\\right) \n", " \\frac{1}{|\\mathbf{k}_{r}-\\mathbf{k}_{p}|^{2}}\n", " \\right. \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " \\left. - \\delta_{m_{s_{p}}m_{s_{s}}}\\delta_{m_{s_{q}}m_{s_{r}}}\n", " \\left( 1 - \\delta_{\\mathbf{k}_{p}\\mathbf{k}_{s}} \\right)\n", " \\frac{1}{|\\mathbf{k}_{s}-\\mathbf{k}_{p}|^{2}} \n", " \\right\\} ,\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the Kronecker delta functions \n", "$\\delta_{\\mathbf{k}_{p}\\mathbf{k}_{r}}$ and\n", "$\\delta_{\\mathbf{k}_{p}\\mathbf{k}_{s}}$ ensure that the \n", "contribution with zero momentum transfer vanishes.\n", "\n", "\n", "Similarly, the matrix elements for the two-dimensional\n", "electron gas are" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation} \\label{eq:vmat_2dheg} \\tag{6}\n", " \\langle \\mathbf{k}_{p}m_{s_{p}}\\mathbf{k}_{q}m_{s_{q}}\n", " |v|\\mathbf{k}_{r}m_{s_{r}}\\mathbf{k}_{s}m_{s_{s}}\\rangle_{AS} \n", " \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " = \\frac{2\\pi }{L^{2}}\n", " \\delta_{\\mathbf{k}_{p}+\\mathbf{k}_{q},\\mathbf{k}_{r}+\\mathbf{k}_{s}}\n", " \\left\\{ \\delta_{m_{s_{p}}m_{s_{r}}}\\delta_{m_{s_{q}}m_{s_{s}}} \n", " \\left( 1 - \\delta_{\\mathbf{k}_{p}\\mathbf{k}_{r}}\\right)\n", " \\frac{1}{\n", " |\\mathbf{k}_{r}-\\mathbf{k}_{p}|} \\right.\n", " \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " - \\left. \\delta_{m_{s_{p}}m_{s_{s}}}\\delta_{m_{s_{q}}m_{s_{r}}}\n", " \\left( 1 - \\delta_{\\mathbf{k}_{p}\\mathbf{k}_{s}}\\right)\n", " \\frac{1}{ \n", " |\\mathbf{k}_{s}-\\mathbf{k}_{p}|}\n", " \\right\\} ,\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the single-particle momentum vectors $\\mathbf{k}_{p,q,r,s}$\n", "are now defined in two dimensions.\n", "\n", "In actual calculations, the \n", "single-particle energies, defined by the operator $\\hat{f}$, are given by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " \\langle \\mathbf{k}_{p}|f|\\mathbf{k}_{q} \\rangle\n", " = \\frac{\\hbar^{2}k_{p}^{2}}{2m}\\delta_{\\mathbf{k}_{p},\n", " \\mathbf{k}_{q}} + \\sum_{\\mathbf{k}_{i}}\\langle \n", " \\mathbf{k}_{p}\\mathbf{k}_{i}|v|\\mathbf{k}_{q}\n", " \\mathbf{k}_{i}\\rangle_{AS}.\n", " \\label{eq:fock_heg} \\tag{7}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Periodic boundary conditions and single-particle states\n", "\n", "When using periodic boundary conditions, the \n", "discrete-momentum single-particle basis functions" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\phi_{\\mathbf{k}}(\\mathbf{r}) =\n", "e^{i\\mathbf{k}\\cdot \\mathbf{r}}/L^{d/2}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "are associated with \n", "the single-particle energy" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\varepsilon_{n_{x}, n_{y}} = \\frac{\\hbar^{2}}{2m} \\left( \\frac{2\\pi }{L}\\right)^{2}\\left( n_{x}^{2} + n_{y}^{2}\\right)\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "for two-dimensional sytems and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\varepsilon_{n_{x}, n_{y}, n_{z}} = \\frac{\\hbar^{2}}{2m}\n", " \\left( \\frac{2\\pi }{L}\\right)^{2}\n", " \\left( n_{x}^{2} + n_{y}^{2} + n_{z}^{2}\\right)\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "for three-dimensional systems.\n", "\n", "\n", "We choose the single-particle basis such that both the occupied and \n", "unoccupied single-particle spaces have a closed-shell \n", "structure. This means that all single-particle states \n", "corresponding to energies below a chosen cutoff are\n", "included in the basis. We study only the unpolarized spin\n", "phase, in which all orbitals are occupied with one spin-up \n", "and one spin-down electron. \n", "\n", "\n", "The table illustrates how single-particle energies\n", " fill energy shells in a two-dimensional electron box.\n", " Here $n_{x}$ and $n_{y}$ are the momentum quantum numbers,\n", " $n_{x}^{2} + n_{y}^{2}$ determines the single-particle \n", " energy level, $N_{\\uparrow \\downarrow }$ represents the \n", " cumulated number of spin-orbitals in an unpolarized spin\n", " phase, and $N_{\\uparrow \\uparrow }$ stands for the\n", " cumulated number of spin-orbitals in a spin-polarized\n", " system.\n", "\n", "\n", "\n", "\n", "## Magic numbers for the two-dimensional electron gas\n", "\n", "\n", "$n_{x}^{2}+n_{y}^{2}$ | $n_{x}$ | $n_{y}$ | $N_{\\uparrow \\downarrow }$ | $N_{\\uparrow \\uparrow }$ |
---|---|---|---|---|
0 | 0 | 0 | 2 | 1 |
1 | -1 | 0 | ||
1 | 0 | |||
0 | -1 | |||
0 | 1 | 10 | 5 | |
2 | -1 | -1 | ||
-1 | 1 | |||
1 | -1 | |||
1 | 1 | 18 | 9 | |
4 | -2 | 0 | ||
2 | 0 | |||
0 | -2 | |||
0 | 2 | 26 | 13 | |
5 | -2 | -1 | ||
2 | -1 | |||
-2 | 1 | |||
2 | 1 | |||
-1 | -2 | |||
-1 | 2 | |||
1 | -2 | |||
1 | 2 | 42 | 21 |
$n_{x}^{2}+n_{y}^{2}+n_{z}^{2}$ | $n_{x}$ | $n_{y}$ | $n_{z}$ | $N_{\\uparrow \\downarrow }$ |
---|---|---|---|---|
0 | 0 | 0 | 0 | 2 |
1 | -1 | 0 | 0 | |
1 | 1 | 0 | 0 | |
1 | 0 | -1 | 0 | |
1 | 0 | 1 | 0 | |
1 | 0 | 0 | -1 | |
1 | 0 | 0 | 1 | 14 |
2 | -1 | -1 | 0 | |
2 | -1 | 1 | 0 | |
2 | 1 | -1 | 0 | |
2 | 1 | 1 | 0 | |
2 | -1 | 0 | -1 | |
2 | -1 | 0 | 1 | |
2 | 1 | 0 | -1 | |
2 | 1 | 0 | 1 | |
2 | 0 | -1 | -1 | |
2 | 0 | -1 | 1 | |
2 | 0 | 1 | -1 | |
2 | 0 | 1 | 1 | 38 |
3 | -1 | -1 | -1 | |
3 | -1 | -1 | 1 | |
3 | -1 | 1 | -1 | |
3 | -1 | 1 | 1 | |
3 | 1 | -1 | -1 | |
3 | 1 | -1 | 1 | |
3 | 1 | 1 | -1 | |
3 | 1 | 1 | 1 | 54 |
Diagrams which enter the definition of the ground-state shift energy $\\Delta E_0$. Diagram (i) is first order in the interaction $\\hat{v}$, while diagrams (ii) and (iii) are examples of contributions to second and third order, respectively.
\n", "\n", "\n", "\n", "\n", "\n", "Using the standard diagram rules (see the discussion on coupled-cluster theory and many-body perturbation theory), the various\n", "diagrams contained in the above figure can be readily calculated (in an uncoupled scheme)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " (i)=\\frac{(-)^{n_h+n_l}}{2^{n_{ep}}}\\sum_{ij\\leq k_F}\n", " \\langle ij\\vert\\hat{v}\\vert ij\\rangle_{AS},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with $n_h=n_l=2$ and $n_{ep}=1$. As discussed in connection with the diagram rules in the many-body perturbation theory chapter, $n_h$\n", "denotes the number of hole lines, $n_l$ the number of closed\n", "fermion loops and $n_{ep}$ is the number of so-called\n", "equivalent pairs.\n", "The factor $1/2^{n_{ep}}$ is needed since we want to count a pair \n", "of particles only once. We will carry this factor $1/2$ with us\n", "in the equations below. \n", "The subscript $AS$ denotes the antisymmetrized and normalized matrix element" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\langle ij\\vert\\hat{v}\\vert ij\\rangle_{AS}=\\langle ij \\vert\\hat{v}\\vert ij\\rangle-\n", " \\langle ji \\vert\\hat{v}\\vert ij\\rangle.\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly, diagrams (ii) and (iii) read" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " (ii)=\\frac{(-)^{2+2}}{2^2}\\sum_{ij\\leq k_F}\\sum_{ab>k_F}\n", " \\frac{\\langle ij\\vert\\hat{v}\\vert ab\\rangle_{AS}\n", " \\langle ab\\vert\\hat{v}\\vert ij\\rangle_{AS}}\n", " {\\varepsilon_i+\\varepsilon_j-\\varepsilon_a-\\varepsilon_b},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " (iii)=\\frac{(-)^{2+2}}{2^3}\\sum_{k_i,k_j\\leq k_F}\\sum_{abcdk_F}\n", " \\frac{\\langle ij\\vert\\hat{v}\\vert ab\\rangle_{AS}\n", " \\langle ab\\vert\\hat{v}\\vert cd\\rangle_{AS}\n", " \\langle cd\\vert\\hat{v}\\vert ij\\rangle_{AS}}\n", " {(\\varepsilon_i+\\varepsilon_j-\\varepsilon_a-\\varepsilon_b)\n", " (\\varepsilon_i+\\varepsilon_j-\\varepsilon_c-\\varepsilon_d)}.\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the above, $\\varepsilon$ denotes the sp energies defined by\n", "$H_0$.\n", "The steps leading to the above expressions for the various\n", "diagrams are rather straightforward. Though, if we wish to compute the\n", "matrix elements for the interaction $v$, a serious problem\n", "arises. Typically, the matrix elements will contain a term\n", "(see the next section for the formal details) $V(|{\\mathbf r}|)$, which\n", "represents the interaction potential $V$ between two nucleons, where\n", "${\\mathbf r}$ is the internucleon distance.\n", "All modern models\n", "for $V$ have a strong short-range repulsive core. Hence,\n", "matrix elements involving $V(|{\\mathbf r}|)$, will result in large\n", "(or infinitely large for a potential with a hard core)\n", "and repulsive contributions to the ground-state energy. Thus, the\n", "diagrammatic expansion for the ground-state energy in terms of the\n", "potential $V(|{\\mathbf r}|)$ becomes meaningless.\n", "\n", "One possible solution to this problem is provided by the well-known\n", "Brueckner theory or the Brueckner $G$-matrix, or just the\n", "$G$-matrix. In fact, the $G$-matrix is an almost indispensable\n", "tool in almost every microscopic nuclear structure\n", "calculation. Its main idea may be paraphrased as follows.\n", "Suppose we want to calculate the function $f(x)=x/(1+x)$. If\n", "$x$ is small, we may expand the function $f(x)$ as a power series\n", "$x+x^2+x^3+\\dots$ and it may be adequate to just calculate the first\n", "few terms. In other words, $f(x)$ may be calculated using a low-order\n", "perturbation method. But if $x$ is large\n", "(or infinitely large), the above\n", "power series is obviously meaningless.\n", "However, the exact function\n", "$x/(1+x)$ is still well defined in the limit\n", "of $x$ becoming very large.\n", "\n", "These arguments suggest that one should sum up the diagrams\n", "(i), (ii), (iii) in fig. ref{fig:goldstone} and the similar ones\n", "to all orders, instead of computing them one by one. Denoting this\n", "all-order sum as $1/2\\tilde{G}_{ijij}$, where we have\n", "introduced the shorthand notation\n", "$\\tilde{G}_{ijij}=\\langle k_ik_j\\vert \\tilde{G}\\vert k_ik_j\\rangle_{AS}$\n", "(and similarly for $\\tilde{v}$),\n", "we have that" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\frac{1}{2}\\tilde{G}_{ijij}=\\frac{1}{2}\\hat{v}_{ijij}\n", " +\\sum_{ab>k_F}\\frac{1}{2}\\hat{v}_{ijab}\\frac{1}{\\varepsilon_i+\\varepsilon_j-\\varepsilon_a-\\varepsilon_b}\n", " \\nonumber \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " \\times\\left[\\frac{1}{2}\\hat{v}_{abij}+\\sum_{cd>k_F}\n", " \\frac{1}{2}\\hat{v}_{abcd}\\frac{1}\n", " {\\varepsilon_i+\\varepsilon_j-\\varepsilon_c-\\varepsilon_d}\n", " \\frac{1}{2}V_{cdij}+\\dots \\right].\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The factor $1/2$ is the same as that discussed above, namely we want \n", "to count a pair of particles only once.\n", "The quantity inside the brackets is just\n", "$1/2\\tilde{G}_{mnij}$ and the above equation can be\n", "rewritten as an integral equation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\tilde{G}_{ijij}=\\tilde{V}_{ijij}\n", " +\\sum_{ab>F}\\frac{1}{2}\\hat{v}_{ijab}\\frac{1}{\\varepsilon_i+\\varepsilon_j-\\varepsilon_a-\\varepsilon_b}\n", " \\tilde{G}_{abij}.\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that $\\tilde{G}$ is the antisymmetrized $G$-matrix since\n", "the potential $\\tilde{v}$ is also antisymmetrized. This means that\n", "$\\tilde{G}$ obeys" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\tilde{G}_{ijij}=-\\tilde{G}_{jiij}=-\\tilde{G}_{ijji}.\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The $\\tilde{G}$-matrix is defined as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\tilde{G}_{ijij}=G_{ijij}-G_{jiij},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and the equation for $G$ is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " G_{ijij}=V_{ijij}\n", " +\\sum_{ab>k_F}V_{ijab}\\frac{1}\n", " {\\varepsilon_i+\\varepsilon_j-\\varepsilon_a-\\varepsilon_b}\n", " G_{abij},\n", " \\label{eq:ggeneral} \\tag{25}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which is the familiar $G$-matrix equation. The above\n", "matrix is specifically designed to treat a class of diagrams\n", "contained in $\\Delta E_0$, of which typical contributions\n", "were shown in fig. ref{fig:goldstone}. In fact the sum of the diagrams\n", "in fig. ref{fig:goldstone} is equal to $1/2(G_{ijij}-G_{jiij})$.\n", "\n", "Let us now define a more general $G$-matrix as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " G_{ijij}=V_{ijij}\n", " +\\sum_{mn>0}V_{ijmn}\\frac{Q(mn)}\n", " {\\omega -\\varepsilon_m-\\varepsilon_n}\n", " G_{mnij},\n", " \\label{eq:gwithq} \\tag{26}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "which is an extension of Eq. [(25)](#eq:ggeneral). Note that \n", "Eq. [(25)](#eq:ggeneral) has\n", "$\\varepsilon_i+\\varepsilon_j$ in the energy denominator, whereas\n", "in the latter equation we have a general energy variable $\\omega$\n", "in the denominator. Furthermore, in Eq. [(25)](#eq:ggeneral)\n", "we have a restricted\n", "sum over $mn$, while in Eq. [(26)](#eq:gwithq)\n", "we sum over all $ab$ and we have\n", "introduced a weighting factor $Q(ab)$. In Eq. [(26)](#eq:gwithq) $Q(ab)$\n", "corresponds to the choice" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " Q(a , b ) =\n", " \\left\\{\\begin{array}{cc}1,&min(a ,b ) > k_F\\\\\n", " 0,&\\mathrm{else}.\\end{array}\\right. ,\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $Q(ab)$ is usually referred to as the $G$-matrix Pauli\n", "exclusion operator. The role of $Q$ is to enforce a selection\n", "of the intermediate states allowed in the $G$-matrix equation. The above\n", "$Q$ requires that the intermediate particles $a$ and $b$\n", "must be both above the Fermi surface defined by $F$. We may enforce\n", "a different requirement by using a summation over intermediate states\n", "different from that in Eq. [(26)](#eq:gwithq).\n", "An example is the Pauli operator\n", "for the model-space Brueckner-Hartree-Fock method discussed below.\n", "\n", "\n", "Before ending this section, let us rewrite the $G$-matrix equation\n", "in a more compact form.\n", "The sp energies $\\varepsilon$ and wave functions are defined\n", "by the unperturbed hamiltonian $H_0$ as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " H_0\\vert \\psi_a\\psi_b=(\\varepsilon_a+\\varepsilon_b)\n", " \\vert \\psi_a\\psi_b.\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The $G$-matrix equation can then be rewritten in the following\n", "compact form" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " G(\\omega )=V+V\\frac{\\hat{Q}}{\\omega -H_0}G(\\omega ),\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with\n", "$\\hat{Q}=\\sum_{ab}\\vert \\psi_a\\psi_b\\langle\\langle \\psi_a\\psi_b\\vert$.\n", "In terms of diagrams, $G$ corresponds to an all-order sum of the\n", "\"ladder-type\" interactions between two particles with the\n", "intermediate states restricted by $Q$.\n", "\n", "The $G$-matrix equation has a very simple form. But its\n", "calculation is rather complicated, particularly for finite\n", "nuclear systems such as the nucleus $^{18}$O. There are a\n", "number of complexities. To mention a few, the Pauli operator\n", "$Q$ may not commute with the unperturbed hamiltonian\n", "$H_0$ and we have to make the replacement" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\frac{Q}{\\omega -H_0}\\rightarrow Q\\frac{1}{\\omega -QH_0Q}Q.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The determination of the starting energy $\\omega$ is also another\n", "problem. \n", "\n", "\n", "In a medium such as nuclear \n", "matter we must account\n", "for the fact that certain states are not available as intermediate\n", "states in the calculation of the $G$-matrix.\n", "Following the discussion above\n", "this is achieved by introducing the medium\n", "dependent Pauli operator $Q$. Further, the\n", "energy $\\omega$ of the incoming particles, given by a pure kinetic\n", "term in a scattering problem between two unbound particles (for example two colliding protons), must be modified so as to allow\n", "for medium corrections.\n", "How to evaluate the Pauli operator for\n", "nuclear matter is, however, not straightforward.\n", "Before discussing how to evaluate the Pauli operator for nuclear matter,\n", "we note that the $G$-matrix\n", "is conventionally given in terms of partial waves and\n", "the coordinates of the relative and center-of-mass motion.\n", "If we assume that the $G$-matrix is diagonal in $\\alpha$ ($\\alpha$ is a shorthand\n", "notation for $J$, $S$, $L$ and $T$), we write the equation for the $G$-matrix as a \n", "coupled-channels equation in the relative and center-of-mass system" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " G_{ll'}^{\\alpha}(kk'K\\omega )=V_{ll'}^{\\alpha}(kk')\n", " +\\sum_{l''}\\int \\frac{d^3 q}{(2\\pi )^3}V_{ll''}^{\\alpha}(kq)\n", " \\frac{Q(q,K)}{\\omega -H_0}\n", " G_{l''l'}^{\\alpha}(qk'K\\omega).\n", " \\label{eq:gnonrel} \\tag{27}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This equation is similar in structure to the scattering\n", "equations discussed in connection with nuclear forces (see the chapter on models for nuclear forces), except that we now have\n", "introduced the Pauli operator $Q$ and a medium dependent two-particle\n", "energy $\\omega$. The notations in this equation follow those of the chapter on nuclear forces\n", "where we discuss the solution of the scattering\n", "matrix $T$.\n", "The numerical details on how to solve the above $G$-matrix\n", "equation through matrix inversion techniques are discussed below\n", "Note however that the $G$-matrix may not be diagonal in $\\alpha$.\n", "This is due to the fact that the\n", "Pauli operator $Q$ is not diagonal\n", "in the above representation in the relative and center-of-mass\n", "system. The Pauli operator depends on the\n", "angle between the relative momentum and the center of mass momentum.\n", "This angle dependence causes $Q$ to couple states with different\n", "relative angular\n", "momentua ${\\cal J}$, rendering a partial wave decomposition of the $G$-matrix equation \n", "rather difficult.\n", "The angle dependence of the Pauli operator\n", "can be eliminated by introducing the angle-average\n", "Pauli operator, where one replaces the exact Pauli operator $Q$\n", "by its average $\\bar{Q}$ over all angles for fixed relative and center-of-mass\n", "momenta.\n", "The choice of Pauli operator is decisive to the determination of the\n", "sp\n", "spectrum. Basically, to first order in the reaction matrix $G$,\n", "there are three commonly used sp spectra, all\n", "defined by the solution of the following equations" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " \\varepsilon_{m} = \\varepsilon (k_{m})= t_{m} + u_{m}=\\frac{k_{m}^2}{2M_N}+u_{m},\n", " \\label{eq:spnrel} \\tag{28}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " u_{m} = {\\displaystyle \\sum_{h \\leq k_F}}\\left\\langle m h \\right| G(\\omega = \\varepsilon_{m} + \\varepsilon_h )\n", " \\left| m h \\right\\rangle_{AS} \\hspace{3mm}k_m \\leq k_M, \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation} \n", " u_m=0, k_m > k_M.\n", " \\label{eq:selfcon} \\tag{29}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For notational economy, we set $|{\\bf k}_m|=k_m$.\n", "Here we employ antisymmetrized matrix elements (AS), and $k_M$ is a cutoff\n", "on the momentum. Further, $t_m$ is the sp kinetic\n", "energy and similarly $u_m$\n", "is the\n", "sp potential.\n", "The choice of cutoff $k_M$ is actually what determines the three\n", "commonly used sp spectra.\n", "In the conventional BHF approach one employs $k_M = k_F$,\n", "which leads\n", "to a Pauli operator $Q_{\\mathrm{BHF}}$ (in the laboratory system) given by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " Q_{\\mathrm{BHF}}(k_m , k_n ) =\n", " \\left\\{\\begin{array}{cc}1,&min(k_m ,k_n ) > k_F\\\\\n", " 0,&\\mathrm{else}.\\end{array}\\right.\n", " \\label{eq:bhf} \\tag{30},\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "or, since we will define an\n", "angle-average Pauli operator in the relative and center-of-mass\n", "system, we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " \\bar{Q}_{\\mathrm{BHF}}(k,K)=\\left\\{\\begin{array}{cc}\n", " 0,&k\\leq \\sqrt{k_{F}^{2}-K^2/4}\\\\\n", " 1,&k\\geq k_F + K/2\\\\\n", "\t\\frac{K^2/4+k^2 -k_{F}^2}{kK}&\\mathrm{else},\\end{array}\\right.\n", " \\label{eq:qbhf} \\tag{31}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with $k_F$ the momentum at the Fermi surface.\n", "\n", "The BHF choice sets $u_k = 0$ for $k > k_F$, which leads\n", "to an unphysical, large gap at the Fermi surface, typically\n", "of the order of $50-60$ MeV. \n", "To overcome the gap\n", "problem, Mahaux and collaborators \n", "introduced a continuous sp spectrum\n", "for all values of $k$. The divergencies\n", "which then may occur in Eq. [(27)](#eq:gnonrel) are taken care of by\n", "introducing\n", "a principal value integration in Eq. [(27)](#eq:gnonrel),\n", "to retain only the\n", "real part contribution to the $G$-matrix.\n", "\n", "\n", "To define the energy denominators we will also make use of the\n", "angle-average approximation.\n", "The angle dependence is handled by the\n", "so-called effective mass approximation. The single-particle energies\n", "in nuclear matter are assumed to have the simple quadratic form" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " \\begin{array}{ccc}\n", " \\varepsilon (k_m)=&\n", " {\\displaystyle\\frac{\\hbar^{2}k_m^2}\n", " {2M_{N}^{*}}}+\\Delta ,&\\hspace{3mm}k_m\\leq k_F\\\\\n", " &&\\\\\n", " =&{\\displaystyle\\frac{\\hbar^{2}\n", " k_m^2}{2M_{N}}},&\\hspace{3mm}k_m> k_F ,\\\\\n", " \\end{array}\n", " \\label{eq:spen} \\tag{32}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $M_{N}^{*}$ is the effective mass of the nucleon and $M_{N}$ is the\n", "bare nucleon mass. For particle states above the Fermi sea we choose\n", "a pure kinetic energy term, whereas for hole states,\n", "the terms $M_{N}^{*}$ and $\\Delta$, the latter being \n", "an effective single-particle\n", "potential related to the $G$-matrix, are obtained through the\n", "self-consistent Brueckner-Hartree-Fock procedure.\n", "The sp potential is obtained through the same angle-average approximation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " \\label{eq:Uav} \\tag{33}\n", " U(k_m) =\\sum_{l\\alpha} (2T+1)(2J+1)\n", " \\left \\{ \\frac{8}{\\pi}\\int_{0}^{(k_F-k_m)/2}\n", " k^2dk G_{ll}^{\\alpha}(k,\\bar{K}_1) \\right. \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation} \n", " \\left.\n", " + \\frac{1}{\\pi k_m}\\int_{(k_F-k_m)/2}^{(k_F+k_m)/2}\n", " kdk (k_F ^2-(k_m-2k)^2)\n", " G_{ll}^{\\alpha}(k,\\bar{K}_2) \\right \\} \\nonumber,\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have defined" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\bar{K}_1^2=4(k_m^2+k^2),\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\begin{equation}\n", " \\bar{K}_2^2=4(k_m^2+k^2)-(2k+k_m-k_F)(2k+k_1+k_F).\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This\n", "self-consistency scheme consists in choosing adequate initial values of the\n", "effective mass and $\\Delta$. The obtained $G$-matrix is in turn used to\n", "obtain new values for $M_{N}^{*}$ and $\\Delta$. This procedure\n", "continues until these parameters vary little.\n", "\n", "\n", "\n", "\n", "\n", "\n", "## Exercise 4: Magic numbers for infinite matter and the Minnesota interaction model\n", "\n", "\n", "**a)**\n", "Set up the quantum numbers for infinite nuclear matter and neutron matter using a given value \n", "of $n_{\\mathrm{max}}$.\n", "\n", "\n", "\n", "**Solution.**\n", "The following python code sets up the quantum numbers for both infinite nuclear matter and neutron matter meploying a cutoff in the value of $n$." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from numpy import *\n", "\n", "nmax =2\n", "nshell = 3*nmax*nmax\n", "count = 1\n", "tzmin = 1\n", "\n", "print \"Symmetric nuclear matter:\" \n", "print \"a, nx, ny, nz, sz, tz, nx^2 + ny^2 + nz^2\"\n", "for n in range(nshell): \n", " for nx in range(-nmax,nmax+1):\n", " for ny in range(-nmax,nmax+1):\n", " for nz in range(-nmax, nmax+1): \n", " for sz in range(-1,1+1):\n", " tz = 1\n", " for tz in range(-tzmin,tzmin+1):\n", " e = nx*nx + ny*ny + nz*nz\n", " if e == n:\n", " if sz != 0: \n", " if tz != 0: \n", " print count, \" \",nx,\" \",ny, \" \",nz,\" \",sz,\" \",tz,\" \",e\n", " count += 1\n", " \n", " \n", "nmax =1\n", "nshell = 3*nmax*nmax\n", "count = 1\n", "tzmin = 1\n", "print \"------------------------------------\"\n", "print \"Neutron matter:\" \n", "print \"a, nx, ny, nz, sz, nx^2 + ny^2 + nz^2\"\n", "for n in range(nshell): \n", " for nx in range(-nmax,nmax+1):\n", " for ny in range(-nmax,nmax+1):\n", " for nz in range(-nmax, nmax+1): \n", " for sz in range(-1,1+1):\n", " e = nx*nx + ny*ny + nz*nz\n", " if e == n:\n", " if sz != 0: \n", " print count, \" \",nx,\" \",ny, \" \",sz,\" \",tz,\" \",e\n", " count += 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "" ] } ], "metadata": {}, "nbformat": 4, "nbformat_minor": 0 }