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\begin{document} |
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\title[Real space electrostatics for multipoles. III. Dielectric Properties] |
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{Supplemental Material for: Real space electrostatics for multipoles. III. Dielectric Properties} |
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University |
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of Notre Dame, Notre Dame, IN 46556} |
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\author{Thomas Parsons} |
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\affiliation{Department of Chemistry and Biochemistry, University |
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of Notre Dame, Notre Dame, IN 46556} |
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\author{Kathie E. Newman} |
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\affiliation{Department of Physics, University |
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of Notre Dame, Notre Dame, IN 46556} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu.} |
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\affiliation{Department of Chemistry and Biochemistry, University |
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of Notre Dame, Notre Dame, IN 46556} |
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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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\maketitle |
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\newpage |
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\section{Boltzmann averages for orientational polarization} |
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The dielectric properties of the system is mainly arise from two |
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different ways: i) the applied field distort the charge distributions |
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so it produces an induced multipolar moment in each molecule; and ii) |
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the applied field tends to line up originally randomly oriented |
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molecular moment towards the direction of the applied field. In this |
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study, we basically focus on the orientational contribution in the |
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dielectric properties. If we consider a system of molecules in the |
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presence of external field perturbation, the perturbation experienced |
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by any molecule will not be only due to external field or field |
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gradient but also due to the field or field gradient produced by the |
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all other molecules in the system. In the following subsections |
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\ref{subsec:boltzAverage-Dipole} and \ref{subsec:boltzAverage-Quad}, |
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we will discuss about the molecular polarization only due to external |
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field perturbation. The contribution of the field or field gradient |
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due to all other molecules will be taken into account while |
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calculating correction factor in the section \ref{sec:corrFactor}. |
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\subsection{Dipoles} |
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\label{subsec:boltzAverage-Dipole} |
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Consider a system of molecules, each with permanent dipole moment |
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$p_o$. In the absense of external field, thermal agitation orients the |
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dipoles randomly, reducing the system moment to zero. External fields |
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will tend to line up the dipoles in the direction of applied field. |
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Here we have considered net field from all other molecules is |
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considered to be zero. Therefore the total Hamiltonian of each |
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molecule is,\cite{Jackson98} |
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\begin{equation} |
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H = H_o - \bf{p_o} .\bf{E}, |
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\end{equation} |
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where $H_o$ is a function of the internal coordinates of the molecule. |
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The Boltzmann average of the dipole moment is given by, |
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\begin{equation} |
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\braket{p_{mol}} = \frac{\displaystyle\int d\Omega\; p_o\; cos\theta\; e^{\frac{p_oE\; cos\theta}{k_B T}}}{\displaystyle\int d\Omega\; e^{\frac{p_oE\;cos\theta}{k_B T}}}, |
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\end{equation} |
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where $\bf{E}$ is selected along z-axis. If we consider that the |
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applied field is small, \textit{i.e.} $\frac{p_oE\; cos\theta}{k_B T} << 1$, |
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\begin{equation} |
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\braket{p_{mol}} \approx \frac{1}{3}\frac{{p_o}^2}{k_B T}E, |
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\end{equation} |
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where $ \alpha_p = \frac{1}{3}\frac{{p_o}^2}{k_B T}$ is a molecular |
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polarizability. The orientational polarization depends inversely on |
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the temperature and applied field must overcome the thermal agitation. |
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\subsection{Quadrupoles} |
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\label{subsec:boltzAverage-Quad} |
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Consider a system of molecules with permanent quadrupole moment |
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$q_{\alpha\beta} $. The average quadrupole moment at temperature T in |
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the presence of uniform applied field gradient is given |
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by,\cite{AduGyamfi78, AduGyamfi81} |
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\begin{equation} |
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\braket{q_{\alpha\beta}} \;=\; \frac{\displaystyle\int d\Omega\; e^{-\frac{H}{k_B T}}q_{\alpha\beta}}{\displaystyle\int d\Omega\; e^{-\frac{H}{k_B T}}} \;=\; \frac{\displaystyle\int d\Omega\; e^{\frac{q_{\mu\nu}\;\partial_\nu E_\mu}{k_B T}}q_{\alpha\beta}}{\displaystyle\int d\Omega\; e^{\frac{q_{\mu\nu}\;\partial_\nu E_\mu}{k_B T}}}, |
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\label{boltzQuad} |
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\end{equation} |
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where $\int d\Omega = \int_0^{2\pi} \int_0^\pi \int_0^{2\pi} |
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sin\theta\; d\theta\ d\phi\ d\psi$ is the integration over Euler |
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angles, $ H = H_o -q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of |
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a quadrupole in the gradient of the |
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applied field and $ H_o$ is a function of internal coordinates of the molecule. The energy and quadrupole moment can be transformed into body frame using following relation, |
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\begin{equation} |
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\begin{split} |
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&q_{\alpha\beta} = \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
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&H = H_o - q:\vec{\nabla}\vec{E} = H_o - q_{\mu\nu}\;\partial_\nu E_\mu = H_o -\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu. |
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\end{split} |
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\label{energyQuad} |
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\end{equation} |
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Here the starred tensors are the components in the body fixed |
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frame. Substituting equation (\ref{energyQuad}) in the equation (\ref{boltzQuad}) |
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and taking linear terms in the expansion we get, |
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\begin{equation} |
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\braket{q_{\alpha\beta}} = \frac{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)q_{\alpha\beta}}{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)}, |
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\end{equation} |
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where $\eta_{\alpha\alpha'}$ is the inverse of the rotation matrix that transforms |
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the body fixed co-ordinates to the space co-ordinates, |
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\[\eta_{\alpha\alpha'} |
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= \left(\begin{array}{ccc} |
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cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
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cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
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sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
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\end{array} \right).\] |
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Integration of 1st and 2nd terms in the denominator gives $8 \pi^2$ |
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and $8 \pi^2 /3\;\vec{\nabla}.\vec{E}\; Tr(q^*) $ respectively. The |
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second term vanishes for charge free space |
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(i.e. $\vec{\nabla}.\vec{E} \; = \; 0)$. Similarly integration of the |
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1st term in the numerator produces |
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$8 \pi^2 /3\; Tr(q^*)\delta_{\alpha\beta}$ and the 2nd term produces |
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$8 \pi^2 /15k_B T (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
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{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta$, |
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if $\vec{\nabla}.\vec{E} \; = \; 0$, |
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$ \partial_\alpha E_\beta = \partial_\beta E_\alpha$ and |
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${q}^*_{\alpha'\beta'}= {q}^*_{\beta'\alpha'}$. Therefore the |
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Boltzmann average of a quadrupole moment can be written as, |
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\begin{equation} |
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\braket{q_{\alpha\beta}}\; = \; \frac{1}{3} Tr(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
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\end{equation} |
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where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupolarizablity and ${\bar{q_o}}^2= |
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3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. |
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\section{External application of a uniform field gradient} |
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\label{Ap:fieldOrGradient} |
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To satisfy the condition $ \nabla . E = 0 $, within the box of molecules we have taken electrostatic potential in the following form |
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\begin{equation} |
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\begin{split} |
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\phi(x, y, z) =\; &-g_o \left(\frac{1}{2}(a_1\;b_1 - \frac{cos\psi}{3})\;x^2+\frac{1}{2}(a_2\;b_2 - \frac{cos\psi}{3})\;y^2 + \frac{1}{2}(a_3\;b_3 - \frac{cos\psi}{3})\;z^2 \right. \\ |
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& \left. + \frac{(a_1\;b_2 + a_2\;b_1)}{2} x\;y + \frac{(a_1\;b_3 + a_3\;b_1)}{2} x\;z + \frac{(a_2\;b_3 + a_3\;b_2)}{2} y\;z \right), |
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\end{split} |
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\label{eq:appliedPotential} |
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\end{equation} |
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where $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ are basis vectors determine coefficients in x, y, and z direction. And $g_o$ and $\psi$ are overall strength of the potential and angle between basis vectors respectively. The electric field derived from the above potential is, |
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\[\bf{E} |
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=\frac{g_o}{2} \left(\begin{array}{ccc} |
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2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+ (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
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(a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y + (a_2\; b_3 \;+ a_3\; b_3)\;z \\ |
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(a_3\; b_1 \;+ a_3\; b_2)\;x + (a_3\; b_2 \;+ a_2\; b_3)y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
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\end{array} \right).\] |
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The gradient of the applied field derived from the potential can be written in the following form, |
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\[\nabla\bf{E} |
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= \frac{g_o}{2}\left(\begin{array}{ccc} |
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2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
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(a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_3)\;z \\ |
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(a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
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\end{array} \right).\] |
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\section{Point-multipolar interactions with a spatially-varying electric field} |
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We can treat objects $a$, $b$, and $c$ containing embedded collections |
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of charges. When we define the primitive moments, we sum over that |
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collections of charges using a local coordinate system within each |
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object. The point charge, dipole, and quadrupole for object $a$ are |
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given by $C_a$, $\mathbf{D}_a$, and $\mathsf{Q}_a$, respectively. |
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These are the primitive multipoles which can be expressed as a |
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distribution of charges, |
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\begin{align} |
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C_a =&\sum_{k \, \text{in }a} q_k , \label{eq:charge} \\ |
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D_{a\alpha} =&\sum_{k \, \text{in }a} q_k r_{k\alpha}, \label{eq:dipole}\\ |
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Q_{a\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in } a} q_k |
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r_{k\alpha} r_{k\beta} . \label{eq:quadrupole} |
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\end{align} |
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Note that the definition of the primitive quadrupole here differs from |
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the standard traceless form, and contains an additional Taylor-series |
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based factor of $1/2$. In Paper 1, we derived the forces and torques |
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each object exerts on the others. |
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Here we must also consider an external electric field that varies in |
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space: $\mathbf E(\mathbf r)$. Each of the local charges $q_k$ in |
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object $a$ will then experience a slightly different field. This |
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electric field can be expanded in a Taylor series around the local |
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origin of each object. A different Taylor series expansion is carried |
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out for each object. |
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For a particular charge $q_k$, the electric field at that site's |
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position is given by: |
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\begin{equation} |
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E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
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+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
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r_{k \varepsilon} + ... |
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\end{equation} |
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Note that the electric field is always evaluated at the origin of the |
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objects, and treating each object using point multipoles simplifies |
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this greatly. |
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To find the force exerted on object $a$ by the electric field, one |
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takes the electric field expression, and multiplies it by $q_k$, and |
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then sum over all charges in $a$: |
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\begin{align} |
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F_\gamma &= \sum_{k \textrm{~in~} a} q_k \lbrace E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
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+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
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r_{k \varepsilon} + ... \rbrace \\ |
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&= C_a E_\gamma + D_{a \delta} \nabla_\delta E_\gamma |
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+ Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma + |
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... |
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\end{align} |
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Similarly, the torque exerted by the field on $a$ can be expressed as |
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\begin{align} |
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\tau_\alpha &= \sum_{k \textrm{~in~} a} (\mathbf r_k \times q_k \mathbf E)_\alpha \\ |
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& = \sum_{k \textrm{~in~} a} \epsilon_{\alpha \beta \gamma} q_k |
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r_{k\beta} E_\gamma(\mathbf r_k) \\ |
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& = \epsilon_{\alpha \beta \gamma} D_\beta E_\gamma |
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+ 2 \epsilon_{\alpha \beta \gamma} Q_{\beta \delta} \nabla_\delta |
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E_\gamma + ... |
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\end{align} |
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The last term is essentially identical with form derived by Torres del |
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Castillo and M\'{e}ndez Garrido,\cite{Torres-del-Castillo:2006uo} although their derivation |
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utilized a traceless form of the quadrupole that is different than the |
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primitive definition in use here. We note that the Levi-Civita symbol |
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can be eliminated by utilizing the matrix cross product in an |
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identical form as in Ref. \onlinecite{Smith98}: |
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\begin{equation} |
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\left[\mathsf{A} \times \mathsf{B}\right]_\alpha = \sum_\beta |
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\left[\mathsf{A}_{\alpha+1,\beta} \mathsf{B}_{\alpha+2,\beta} |
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-\mathsf{A}_{\alpha+2,\beta} \mathsf{B}_{\alpha+1,\beta} |
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\right] |
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\label{eq:matrixCross} |
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\end{equation} |
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where $\alpha+1$ and $\alpha+2$ are regarded as cyclic permuations of |
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the matrix indices. In table \ref{tab:UFT} we give compact |
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expressions for how the multipole sites interact with an external |
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field that has exhibits spatial variations. |
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\begin{table} |
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\caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque |
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$(\mathbf{\tau})$ expressions for a multipolar site embedded in an |
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electric field with spatial variations, $\mathbf{E}(\mathbf{r})$. |
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\label{tab:UFT}} |
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\begin{tabular}{r|ccc} |
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& Charge & Dipole & Quadrupole \\ \hline |
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$U$ & $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\ |
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$\mathbf{F}$ & $C \mathbf{E}(\mathbf{r})$ & $+\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ & $+\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\ |
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$\mathbf{\tau}$ & & $\mathbf{D} \times \mathbf{E}(\mathbf{r})$ & $+2 \mathsf{Q} \times \nabla \mathbf{E}(\mathbf{r})$ |
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\end{tabular} |
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\end{table} |
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289 |
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\section{Gradient of the field due to quadrupolar polarization} |
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\label{singularQuad} |
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In this section, we will discuss the gradient of the field produced by |
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quadrupolar polarization. For this purpose, we consider a distribution |
293 |
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of charge ${\rho}(r)$ which gives rise to an electric field |
294 |
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$\vec{E}(r)$ and gradient of the field $\vec{\nabla} \vec{E}(r)$ |
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throughout space. The total gradient of the electric field over volume |
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due to the all charges within the sphere of radius $R$ is given by |
297 |
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(cf. Jackson equation 4.14): |
298 |
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\begin{equation} |
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\int_{r<R} \vec{\nabla}\vec{E}\;d^3r = -\int_{r=R} R^2 \vec{E}\;\hat{n}\; d\Omega |
300 |
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\label{eq:8} |
301 |
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\end{equation} |
302 |
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where $d\Omega$ is the solid angle and $\hat{n}$ is the normal vector |
303 |
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of the surface of the sphere which is equal to |
304 |
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$sin[\theta]cos[\phi]\hat{x} + sin[\theta]sin[\phi]\hat{y} + |
305 |
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cos[\theta]\hat{z}$ |
306 |
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in spherical coordinates. For the charge density ${\rho}(r')$, the |
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total gradient of the electric field can be written as (cf. Jackson |
308 |
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equation 4.16), |
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\begin{equation} |
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\int_{r<R} \vec{\nabla}\vec{E}\; d^3r=-\int_{r=R} R^2\; \vec{\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; \vec{\nabla}\;\left(\int \frac{\rho(r')}{|\vec{r}-\vec{r'}|}\;d^3r'\right) \hat{n}\; d\Omega |
311 |
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\label{eq:9} |
312 |
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\end{equation} |
313 |
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The radial function in the equation (\ref{eq:9}) can be expressed in |
314 |
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terms of spherical harmonics as (cf. Jackson equation 3.70), |
315 |
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\begin{equation} |
316 |
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\frac{1}{|\vec{r} - \vec{r'}|} = 4\pi \sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\frac{{r^l_<}}{{r^{l+1}_>}}\;{Y^*}_{lm}(\theta', \phi')\;Y_{lm}(\theta, \phi) |
317 |
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\label{eq:10} |
318 |
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\end{equation} |
319 |
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If the sphere completely encloses the charge density then $ r_< = r'$ and $r_> = R$. Substituting equation (\ref{eq:10}) into (\ref{eq:9}) we get, |
320 |
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\begin{equation} |
321 |
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\begin{split} |
322 |
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\int_{r<R} \vec{\nabla}\vec{E}\;d^3r &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; \vec{\nabla}\;\left(\int \rho(r')\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\frac{{r'^l}}{{R^{l+1}}}\;{Y^*}_{lm}(\theta', \phi')\;Y_{lm}(\theta, \phi)\;d^3r'\right) \hat{n}\; d\Omega \\ |
323 |
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&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(r')\;{r'^l}\;{Y^*}_{lm}(\theta', \phi')\left(\int_{r=R}\vec{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right)\hat{n}\; d\Omega \right)d^3r |
324 |
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' |
325 |
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\end{split} |
326 |
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\label{eq:11} |
327 |
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\end{equation} |
328 |
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The gradient of the product of radial function and spherical harmonics |
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is given by (cf. Arfken, p.811 eq. 16.94): |
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\begin{equation} |
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\begin{split} |
332 |
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\vec{\nabla}\left[ f(r)\;Y_{lm}(\theta, \phi)\right] = &-\left(\frac{l+1}{2l+1}\right)^{1/2}\; \left[\frac{\partial}{\partial r}-\frac{l}{r} \right]f(r)\; Y_{l, l+1, m}(\theta, \phi)\\ &+ \left(\frac{l}{2l+1}\right)^{1/2}\left[\frac |
333 |
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{\partial}{\partial r}+\frac{l}{r} \right]f(r)\; Y_{l, l-1, m}(\theta, \phi). |
334 |
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\end{split} |
335 |
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\label{eq:12} |
336 |
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\end{equation} |
337 |
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Using equation (\ref{eq:12}) we get, |
338 |
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\begin{equation} |
339 |
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\vec{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right) = [(l+1)(2l+1)]^{1/2}\; Y_{l,l+1,m}(\theta, \phi) \; \frac{1}{R^{l+2}}, |
340 |
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\label{eq:13} |
341 |
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\end{equation} |
342 |
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where $ Y_{l,l+1,m}(\theta, \phi)$ is the vector spherical harmonics |
343 |
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which can be expressed in terms of spherical harmonics as shown in |
344 |
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below (cf. Arfkan p.811), |
345 |
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\begin{equation} |
346 |
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Y_{l,l+1,m}(\theta, \phi) = \sum_{m_1, m_2} C(l+1,1,l|m_1,m_2,m)\; {Y_{l+1}}^{m_1}(\theta,\phi)\; \hat{e}_{m_2}, |
347 |
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\label{eq:14} |
348 |
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\end{equation} |
349 |
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where $C(l+1,1,l|m_1,m_2,m)$ is a Clebsch-Gordan coefficient and |
350 |
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$\hat{e}_{m_2}$ is a spherical tensor of rank 1 which can be expressed |
351 |
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in terms of Cartesian coordinates, |
352 |
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\begin{equation} |
353 |
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{\hat{e}}_{+1} = - \frac{\hat{x}+i\hat{y}}{\sqrt{2}},\quad {\hat{e}}_{0} = \hat{z},\quad and \quad {\hat{e}}_{-1} = \frac{\hat{x}-i\hat{y}}{\sqrt{2}} |
354 |
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\label{eq:15} |
355 |
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\end{equation} |
356 |
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The normal vector $\hat{n} $ can be expressed in terms of spherical tensor of rank 1 as shown in below, |
357 |
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\begin{equation} |
358 |
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\hat{n} = \sqrt{\frac{4\pi}{3}}\left(-{Y_1}^{-1}{\hat{e}}_1 -{Y_1}^{1}{\hat{e}}_{-1} + {Y_1}^{0}{\hat{e}}_0 \right) |
359 |
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\label{eq:16} |
360 |
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\end{equation} |
361 |
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The surface integral of the product of $\hat{n}$ and |
362 |
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${Y_{l+1}}^{m_1}(\theta, \phi)$ gives, |
363 |
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\begin{equation} |
364 |
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\begin{split} |
365 |
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\int \hat{n}\;{Y_{l+1}}^{m_1}\;d\Omega &= \int \sqrt{\frac{4\pi}{3}}\left(-{Y_1}^{-1}{\hat{e}}_1 -{Y_1}^{1}{\hat{e}}_{-1} + {Y_1}^{0}{\hat{e}}_0 \right)\;{Y_{l+1}}^{m_1}\; d\Omega \\ |
366 |
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&= \int \sqrt{\frac{4\pi}{3}}\left({{Y_1}^{1}}^* {\hat{e}}_1 +{{Y_1}^{-1}}^* {\hat{e}}_{-1} + {{Y_1}^{0}}^* {\hat{e}}_0 \right)\;{Y_{l+1}}^{m_1}\; d\Omega \\ |
367 |
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&= \sqrt{\frac{4\pi}{3}}\left({\delta}_{l+1, 1}\;{\delta}_{1, m_1}\;{\hat{e}}_1 + {\delta}_{l+1, 1}\;{\delta}_{-1, m_1}\;{\hat{e}}_{-1}+ {\delta}_{l+1, 1}\;{\delta}_{0, m_1} \;{\hat{e}}_0\right), |
368 |
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\end{split} |
369 |
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\label{eq:17} |
370 |
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\end{equation} |
371 |
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where ${Y_{l}}^{-m} = (-1)^m\;{{Y_{l}}^{m}}^* $ and |
372 |
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$ \int {{Y_{l}}^{m}}^*\;{Y_{l'}}^{m'}\;d\Omega = |
373 |
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|
\delta_{ll'}\delta_{mm'} $. |
374 |
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Non-vanishing values of equation \ref{eq:17} require $l = 0$, |
375 |
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|
therefore the value of $ m = 0 $. Since the values of $ m_1$ are -1, |
376 |
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1, and 0 then $m_2$ takes the values 1, -1, and 0, respectively |
377 |
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provided that $m = m_1 + m_2$. Equation \ref{eq:11} can therefore be |
378 |
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|
modified, |
379 |
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\begin{equation} |
380 |
|
|
\begin{split} |
381 |
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|
\int_{r<R} \vec{\nabla}\vec{E}\;d^3r = &- \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;{Y^*}_{00}(\theta', \phi')[ C(1, 1, 0|-1,1,0)\;{\hat{e}_{-1}}{\hat{e}_{1}}\\ &+ C(1, 1, 0|-1,1,0)\;{\hat{e}_{1}}{\hat{e}_{-1}}+C( |
382 |
|
|
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d^3r'. |
383 |
|
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\end{split} |
384 |
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\label{eq:18} |
385 |
|
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\end{equation} |
386 |
|
|
After substituting ${Y^*}_{00} = \frac{1}{\sqrt{4\pi}} $ and using the |
387 |
|
|
values of the Clebsch-Gorden coefficients: $ C(1, 1, 0|-1,1,0) = |
388 |
|
|
\frac{1}{\sqrt{3}}, \; C(1, 1, 0|-1,1,0)= \frac{1}{\sqrt{3}}$ and $ |
389 |
|
|
C(1, 1, 0|0,0,0) = -\frac{1}{\sqrt{3}}$ in equation \ref{eq:18} we |
390 |
|
|
obtain, |
391 |
|
|
\begin{equation} |
392 |
|
|
\begin{split} |
393 |
|
|
\int_{r<R} \vec{\nabla}\vec{E}\;d^3r &= -\sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;d^3r'\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right)\\ |
394 |
|
|
&= - \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\;C_{total}\;\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right). |
395 |
|
|
\end{split} |
396 |
|
|
\label{eq:19} |
397 |
|
|
\end{equation} |
398 |
|
|
Equation (\ref{eq:19}) gives the total gradient of the field over a |
399 |
|
|
sphere due to the distribution of the charges. For quadrupolar fluids |
400 |
|
|
the total charge within a sphere is zero, therefore |
401 |
|
|
$ \int_{r<R} \vec{\nabla}\vec{E}\;d^3r = 0 $. Hence the quadrupolar |
402 |
|
|
polarization produces zero net gradient of the field inside the |
403 |
|
|
sphere. |
404 |
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|
405 |
|
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|
406 |
gezelter |
4399 |
\bibliography{dielectric_new} |
407 |
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\end{document} |
408 |
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% |
409 |
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% ****** End of file multipole.tex ****** |