Content-Type: multipart/mixed; boundary="-------------0006050157110" This is a multi-part message in MIME format. ---------------0006050157110 Content-Type: text/plain; name="00-254.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-254.keywords" ionization, magnetic fields ---------------0006050157110 Content-Type: application/x-tex; name="ion_magn.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ion_magn.tex" %%Version 29/05/00 %%Robert Seiringer %%On the maximal ionization of atoms in strong magnetic fields \documentclass[12pt]{article} \usepackage{a4,amsthm,amsfonts,latexsym} \pagestyle{myheadings} \newtheorem{thm}{THEOREM}[] \newtheorem{cl}[thm]{COROLLARY} \newtheorem{lem}{LEMMA} \theoremstyle{definition} \newtheorem{rem}{Remark} %Macros \def\mfr#1/#2{\hbox{${{#1} \over {#2}}$}} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\bsigma}{\mathord{\hbox{\boldmath $\sigma$}}} \newcommand{\rmi}{i} \newcommand{\rme}{e} \newcommand{\rmd}{d} \newcommand{\OO}{O} \newcommand{\infspec}{{\rm inf\ spec\ }} \newcommand{\A}{{\bf A}} \newcommand{\B}{{\bf B}} \newcommand{\0}{{\bf 0}} \newcommand{\nab}{\bf nabla} \newcommand{\Aa}{{\bf A}} \newcommand{\x}{{\bf x}} \newcommand{\abf}{{\bf a}} \newcommand{\y}{{\bf y}} \newcommand{\X}{{\bf X}} \newcommand{\Tr}{{\rm Tr}} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\N}{{\mathbb N}} \newcommand{\half}{\mbox{$\frac{1}{2}$}} \newcommand{\D}{{\mathcal D}} \newcommand{\Ll}{{\mathcal L}} \newcommand{\Hh}{{\mathcal H}} \newcommand{\Cc}{{\mathcal C}} \newcommand{\eps}{\varepsilon} \newcommand{\dx}{\frac\partial{\partial x}} \newcommand{\dy}{\frac\partial{\partial y}} \newcommand{\HN}{H_{N,Z,\A}} \newcommand{\sig}{\bsigma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \date{\small June 5, 2000} \begin{document} %\markboth{\scriptsize{RS \version}}{\scriptsize{RS \version}} \title{\bf{On the maximal ionization of atoms \\ in strong magnetic fields}} \author{\vspace{5pt} Robert Seiringer\\ \vspace{-4pt}\small{Institut f\"ur Theoretische Physik, Universit\"at Wien}\\ \vspace{-4pt} \small{Boltzmanngasse 5, A-1090 Vienna, Austria}\\ \small{E-Mail: \texttt{rseiring@ap.univie.ac.at} }} \maketitle \begin{abstract} We give upper bounds for the number of spin $\half$ particles that can be bound to a nucleus of charge $Z$ in the presence of a magnetic field $\B$, including the spin-field coupling. We use Lieb's strategy, which is known to yield $N_c<2Z+1$ for magnetic fields that go to zero at infinity, ignoring the spin-field interaction. For particles with fermionic statistics in a homogeneous magnetic field our upper bound has an additional term of order $Z\times\min\left\{(B/Z^3)^{2/5},1+|\ln(B/Z^3)|^2\right\}$. \end{abstract} \section{Introduction and main result} Let $\HN$ be the Hamiltonian for $N$ identical particles with spin $\half$ in the Coulomb field of a nucleus of charge $Z$ and in a magnetic field $\B=\rm{curl\,} \A$, \beq\label{ham} \HN=\sum_{i=1}^N\left(H_\A^{(i)}-\frac Z{|\x_i|}\right)+\sum_{i1$. \end{thm} Note that since the ground state energy is superadditive in $N$, $E(N,Z,\B)/N$ is bounded by some some function independent of $N$. Moreover, the best bound in (\ref{arrive}) is achieved in the limit $k\searrow 1$, which exists by concavity of $E(N,Z,\B)$ in $Z$. To apply Theorem \ref{thm1} to the case of fermionic electrons in a homogeneous magnetic field $\B=(0,0,B)$, one needs upper and lower bounds to the ground state energy. These were derived in \cite{lsya} and \cite{lsyb} and are given in section \ref{sf}. The result is the following: \begin{thm}[Maximal ionization for fermions]\label{fermi} Let $\HN$ be the restriction of (\ref{ham}) to the fermionic subspace, and let $\B=(0,0,B)$. Then, for some constants $C_1$ and $C_2$, and for all values of $B\geq 0$ and $Z>0$, \beq\label{arferm} N_c< 2Z+1+C_1 Z^{1/3}+C_2 Z\min\left\{(B/Z^3)^{2/5},1+|\ln(B/Z^3)|^2\right\}. \eeq \end{thm} Of course we do not believe that these bounds are optimal. One might assume that Lieb's bound $N_c<2Z+1$ holds also in this case, at least for large $Z$ (compare with the lower bound in \cite{lsya} stated above), but it remains an open problem to show this. However, Theorem \ref{fermi} improves a result obtained in \cite{brummel}, which states that $N_c<2Z+1+cB^{1/2}$ for the Hamiltonian (\ref{ham}) restricted to some special wave functions in the lowest Landau band, which reduces the problem to an essentially one-dimensional one. One might ask how the Pauli principle affects the result in Theorem \ref{fermi}. It turns out that the analogue for bosonic particles is the following: \begin{thm}[Maximal ionization for bosons]\label{bose} Let $\HN$ be the restriction of (\ref{ham}) to the bosonic subspace, and let $\B=(0,0,B)$. Then for some constant $C_3$ and for all $B\geq 0$ and $Z>0$ \beq\label{26} N_c< 2Z +1 +\frac Z2\min\left\{\left(1+ \frac B{Z^2}\right),C_3\left(1+\left[\ln\left(\frac B{Z^2}\right)\right]^2 \right)\right\}. \eeq \end{thm} In the next section we will give the proof of Theorem \ref{thm1}. In section \ref{sg} several possible generalizations are stated, and in section \ref{se} the necessary energy bounds for the case of a homogeneous magnetic field are given, which will prove Theorems \ref{fermi} and \ref{bose}. \section{Proof of Theorem \ref{thm1}} Let $\Psi$ be a normalized ground state for $\HN$. Assume, for the moment, that $\langle\Psi||\x_N|\Psi\rangle$ is finite. Then \begin{eqnarray}\nonumber & &E(N,Z,\B)\langle|\x_N|\Psi|\Psi\rangle=\langle|\x_N|\Psi|\HN\Psi\rangle\\ \nonumber & &= \langle|\x_N|\Psi| \left(H_{N-1,Z,\A}+H_\A^{(N)}-\frac Z{|\x_N|}+\sum_{i=1}^{N-1}\frac 1{|\x_i-\x_N|}\right)\Psi\rangle.\\ \label{hn1} \end{eqnarray} Because $\Gamma=\int \Psi^*\Psi |\x_N|d\x_N$ is an acceptable trial density matrix, we can use the variational principle to conclude that \beq \langle|\x_N|\Psi|H_{N-1,Z,\A}|\Psi\rangle\geq E(N-1,Z,\B)\langle|\x_N|\Psi|\Psi\rangle. \eeq By assumption the energy is monotonously decreasing in $N$, so \begin{eqnarray}\nonumber & &\langle|\x_N|\Psi|\left(H_\A^{(N)}-\frac Z{|\x_N|}+\sum_{i=1}^{N-1}\frac 1{|\x_i-\x_N|}\right)\Psi\rangle\\ \label{eq1} & &\leq \left(E(N,Z,\B)-E(N-1,Z,\B)\right)\langle|\x_N|\Psi|\Psi\rangle\leq 0. \end{eqnarray} Now using the demanded symmetry of $\Psi$, we get \beq \langle\Psi| \frac{|\x_N|}{|\x_i-\x_N|}\Psi\rangle=\frac12 \langle\Psi| \frac{|\x_N|+|\x_i|}{|\x_i-\x_N|}\Psi\rangle > \frac12 \eeq (the strict inequality follows from the fact that $\left\{(\x,\y), |\x-\y|=|\x|+|\y|\right\}$ has measure zero), so (\ref{eq1}) gives \beq\label{eq2} Z> \frac12 (N-1)+\langle|\x_N|\Psi|H_\A^{(N)}|\Psi\rangle. \eeq As in \cite{brummel} we have for any positive function $\varphi(\x_N)$ \begin{eqnarray}\nonumber \langle\varphi\Psi|H_\A^{(N)}|\Psi\rangle &=&\langle\varphi^{1/2}\Psi|\left(H_\A^{(N)}-\left| \frac{\nabla\varphi}{2\varphi}\right|^2\right) \varphi^{1/2}\Psi\rangle\\ \label{brum} & &-\rmi{\rm Re\,} \langle\varphi^{1/2}\Psi| \frac{\nabla\varphi}{\varphi}\cdot(-\rmi\nabla+\A) \varphi^{1/2}\Psi\rangle. \end{eqnarray} Now $\langle|\x_N|\Psi|H_\A^{(N)}\Psi\rangle$ is certainly real, because all the other quantities in equation (\ref{hn1}) are real. So choosing $\varphi(\x_N)=|\x_N|$ in equation (\ref{brum}) we get \beq\label{real} \langle|\x_N|\Psi|H_\A^{(N)}\Psi\rangle= \langle|\x_N|^{1/2}\Psi|\left(H_\A^{(N)}-\frac 1{4|\x_N|^2} \right) |\x_N|^{1/2}\Psi\rangle. \eeq Using that $H_\A^{(N)}\geq 0$ equation (\ref{eq2}) reads \beq N< 2 Z+1+\frac 12\langle\Psi||\x_N|^{-1}\Psi\rangle. \eeq Moreover, \begin{eqnarray}\nonumber \langle\Psi||\x_N|^{-1}\Psi\rangle(k-1)&=&\frac 1{NZ}\langle \Psi|(\HN-H_{N,kZ,\A})\Psi\rangle\\ &\leq& \frac 1{NZ}\left(E(N,Z,\B)-E(N,kZ,\B)\right), \end{eqnarray} so we arrive at the desired bound for $N_c$. Throughout, we have assumed that $\langle|\x_N|\Psi|\Psi\rangle$ is finite. A priori, this need not be the case. However, one could arrive at the same conclusions using the bounded function $\varphi_\eps(\x_N)=|\x_N|(1+\eps|\x_N|)^{-1}$ instead of $|\x_N|$ in (\ref{hn1}), and letting $\eps\to 0$ at the end (see \cite{lieb}). Note that \beq \left|\frac{\nabla\varphi_\eps}{\varphi_\eps}\right|^2= \frac1{|\x|^2(1+\eps|\x|)^2}\leq\frac 1{|\x|\varphi_\eps(\x)}, \eeq so our conclusions remain valid. \begin{rem} Instead of ignoring the kinetic energy in (\ref{real}) one could use the operator inequality \beq (-\rmi\nabla+\A)^2-\frac 1{4|\x|^2}\geq 0 \eeq to conclude that \beq N_c< 2Z+1-2\langle\Psi||\x_N|\B(\x_N)\cdot\sig_N\Psi\rangle. \eeq This may especially be of interest if $|\B(\x)|\leq b|\x|^{-1}$ for some constant $b$. And for $\B=\0$ Lieb's bound $N_c<2Z+1$ is reproduced. \end{rem} \section{Generalizations of Theorem \ref{thm1}}\label{sg} As in \cite{lieb} several generalizations of Theorem \ref{thm1} are possible: \begin{itemize} \item One can allow different statistics than the bosonic or fermionic one, or even consider independent particles. Moreover, the particles could have different masses and charges. \item Hartree- and Hartree-Fock theories can be treated in the same manner. \item One can replace the Coulomb interaction (everywhere) by some positive $v(\x)=1/w(\x)$, with $w$ satisfying \beq w(\x-\y)\leq w(\x)+w(\y), \eeq and for some constant $C$ \beq \left|\nabla w\right|^2\leq C . \eeq Looking at the proof of Theorem \ref{thm1} we see that these two properties are really what we needed. \end{itemize} \section{Application to homogeneous fields}\label{se} We will now apply Theorem \ref{thm1} to the case of a homogeneous magnetic field $\B=(0,0,B)$, and prove Theorems \ref{fermi} and \ref{bose}. The magnetic potential, in the symmetric gauge, is given by $\A=\half \B\times\x$. The energy in this case will be denoted by $E(N,Z,B)\equiv E(N,Z,\B)$. To derive explicit bounds on $N_c$, we need upper and lower bounds to the ground state energy of (\ref{ham}). However, since we are not trying to give the optimal constants, the upper bound $E(N,Z,B)\leq 0$ will suffice for our purposes. So we will concentrate on the lower bounds. We will distinguish between the fermionic and the bosonic case. Throughout, every fixed constant will be denoted by $C$, although the various constants may be different. \subsection{The fermionic case}\label{sf} In \cite{lsya} and \cite{lsyb} the following lower bounds on the ground state energy of ({\ref{ham}) were derived: \begin{lem}[Lower bounds on the fermionic energy] Let $\lambda=N/Z$. The ground state energy of (\ref{ham}) restricted to the fermionic subspace satisfies: \begin{enumerate} \item[(a)] If $B\leq C Z^{4/3}$ then \beq E(N,Z,B)\geq -CZ^{7/3}\lambda^{1/3}\left(1+C\lambda^{2/3}\right). \eeq \item[(b)] If $B\geq C Z^{4/3}$ then \beq\label{useful} E(N,Z,B)\geq -CZ^{9/5}\lambda^{3/5}B^{2/5}\left(1+C\lambda^{-2/5}\right). \eeq \item[(c)] If $B\geq C Z^2$ then \beq\label{only} E(N,Z,B)\geq -CNZ^2\left(1+\left[\ln\left(\frac C{\lambda^{1/2}} \left(\frac B{Z^3}\right)^{1/2}+1\right)\right]^2\right). \eeq \end{enumerate} \end{lem} \begin{rem} Part (c) of Theorem \ref{fermi} follows from omitting the repulsion terms in Theorem 1.2 (\lq\lq Confinement to the lowest Landau band\rq\rq) in \cite{lsya} and then using the bound (4.11) there. Although this theorem is applicable for $B\geq CZ^{4/3}$, with an additional error term, the result is simpler for $B\geq CZ^2$. However, the bound (\ref{only}) is only of interest for $B\geq CZ^3$, because for smaller $B$ (\ref{useful}) is more useful. \end{rem} Using (\ref{arrive}) and the bounds in the preceding Lemma we find that $\lambda_c\equiv N_c/Z$ satisfies \begin{eqnarray}\nonumber \lambda_c&<& 2+Z^{-1}+C\lambda_c^{-2/3}Z^{-2/3}\left(1+C\lambda_c^{2/3}\right) \quad {\rm if}\quad B\leq C Z^{4/3},\\ \nonumber \lambda_c&<& 2+Z^{-1}+C\lambda_c^{-2/5}\left(\frac B{Z^3}\right)^{2/5}\left(1+C\lambda_c^{-2/5}\right)\quad {\rm if}\quad B\geq C Z^{4/3}, \\ \nonumber \lambda_c&<& 2+Z^{-1}+C\left(1+\left[\ln\left(\frac C{\lambda_c^{1/2}} \left(\frac B{Z^3}\right)^{1/2}+1\right)\right]^2\right)\quad {\rm if}\quad B\geq C Z^{2}.\\ \end{eqnarray} Putting together these three bounds, we obtain the result stated in Theorem \ref{fermi}. Note that these bounds imply in particular \beq \limsup_{Z\to\infty}\frac {N_c}Z\leq 2\qquad {\rm if}\quad B/Z^3\to 0. \eeq \subsection{The bosonic case}\label{sb} To get a lower bound the bosonic energy we will first omit the positive repulsion terms in (\ref{ham}). By scaling the variables $\x_i\to Z^{-1}\x_i$ we see that \beq E(N,Z,B)\geq N Z^2 e(B/Z^2), \eeq where $e(b)$ is the ground state energy of hydrogen in a homogeneous magnetic field of strength $b$. For small $b$, we will use the diamagnetic inequality, which implies \beq e(b)\geq -\frac14-b. \eeq A large $b$ expansion of $e(b)$ is given in \cite{ahs3}. From there we get the following lower bound: \begin{lem}[Lower bound for the hydrogen energy] For large enough $b$ the ground state energy of hydrogen, $e(b)$, satisfies \beq e(b)\geq -\frac14\left(\ln b\right)^2\left(1+\frac C{\ln b}\right). \eeq \end{lem} Note that the bosonic energy is at least of order $NZ^2$, even for small $B$. Therefore the contribution to (\ref{arrive}) is always at least $\OO(Z)$, in contrast to fermions, where the energy is of order $N^{1/3}Z^2$ for small $B$ (this is the reason for the additional factor $Z^{1/3}$ in (\ref{arferm})). Setting $k=2$ we arrive at the bound given in Theorem \ref{bose}. We remark that since the bosonic energy is always less than the fermionic energy, Theorem \ref{bose} holds also for fermions; but the bound stated there is certainly worse than the one given in Theorem \ref{fermi}. \begin{thebibliography}{99} \bibitem{lieb} E.H. Lieb, {\it Bound on the maximum negative ionization of atoms and molecules}, Phys. Rev. A {\bf 29}, 3018--3028 (1984) \bibitem{brummel} R. Brummelhuis and M.B. Ruskai, {\it A One-Dimensional Model for Many-Electron Atoms in Extremely Strong Magnetic Fields: Maximum Negative Ionization}, J. Phys. A {\bf 32}, 2567--82 (1999) \bibitem{lsya} E.H. Lieb, J.P. Solovej, and J. Yngvason: {\it Asymptotics of Heavy Atoms in High Magnetic Fields: I. Lowest Landau Band Regions}, Commun. Pure Appl. Math. {\bf 47}, 513--591 (1994) \bibitem{lsyb} E.H. Lieb, J.P. Solovej, and J. Yngvason: {\it Asymptotics of Heavy Atoms in High Magnetic Fields: II. Semiclassical Regions}, Commun. Math. Phys. {\bf 161}, 77--124 (1994) \bibitem{ahs3} J.E. Avron, I.W. Herbst, and B. Simon, {\it Schr\"odinger Operators with Magnetic Fields III. Atoms in Homogeneous Magnetic Field}, Commun. Math. Phys. {\bf 79}, 529--572 (1981) \bibitem{lsst} E.H. Lieb, I.M. Sigal, B. Simon, and W. Thirring, {\it Approximate Neutrality of Large-Z Ions}, Commun. Math. Phys. {\bf 116}, 635--644 (1988) \end{thebibliography} \end{document} ---------------0006050157110--