| 71 |
|
\begin{abstract} |
| 72 |
|
This document includes useful relationships for computing the |
| 73 |
|
interactions between fields and field gradients and point multipolar |
| 74 |
< |
representations of molecular electrostatics. We also provide |
| 74 |
> |
representations of molecular electrostatics. We also provide |
| 75 |
|
explanatory derivations of a number of relationships used in the |
| 76 |
|
main text. This includes the Boltzmann averages of quadrupole |
| 77 |
< |
orientations, and the interaction of a quadrupole with the |
| 77 |
> |
orientations, and the interaction of a quadrupole density with the |
| 78 |
|
self-generated field gradient. This last relationship is assumed to |
| 79 |
|
be zero in the main text but is explicitly shown to be zero here. |
| 80 |
|
\end{abstract} |
| 82 |
|
\maketitle |
| 83 |
|
|
| 84 |
|
\section{Generating Uniform Field Gradients} |
| 85 |
< |
One important task in performing out the simulations mentioned in the |
| 86 |
< |
main text was to generate uniform electric field gradients. We rely |
| 87 |
< |
heavily on both the notation and results from Torres del Castillo and |
| 88 |
< |
Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} In this work, |
| 89 |
< |
tensors were expressed in Cartesian components, using at times a |
| 90 |
< |
dyadic notation. This proves quite useful for computer simulations |
| 91 |
< |
that make use of toroidal boundary conditions. |
| 85 |
> |
One important task in carrying out the simulations mentioned in the |
| 86 |
> |
main text was to generate uniform electric field gradients. To do |
| 87 |
> |
this, we relied heavily on both the notation and results from Torres |
| 88 |
> |
del Castillo and Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} |
| 89 |
> |
In this work, tensors were expressed in Cartesian components, using at |
| 90 |
> |
times a dyadic notation. This proves quite useful for computer |
| 91 |
> |
simulations that make use of toroidal boundary conditions. |
| 92 |
|
|
| 93 |
|
An alternative formalism uses the theory of angular momentum and |
| 94 |
|
spherical harmonics and is common in standard physics texts such as |
| 313 |
|
${E}_\gamma$ terms are all evaluated at the center of the object (now |
| 314 |
|
a point). Thus later the ${E}_\gamma$ terms can be written using the |
| 315 |
|
same (molecular) origin for all point charges in the object. The force |
| 316 |
< |
exerted on object $a$ by the electric field is given by, |
| 316 |
> |
exerted on object $a$ by the electric field is given by,\cite{Raab:2004ve} |
| 317 |
|
\begin{align} |
| 318 |
|
F^a_\gamma = \sum_{k \textrm{~in~} a} E_\gamma(\mathbf{r}_k) &= \sum_{k \textrm{~in~} a} q_k \lbrace E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
| 319 |
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
| 370 |
|
\end{table} |
| 371 |
|
|
| 372 |
|
\section{Boltzmann averages for orientational polarization} |
| 373 |
< |
The dielectric properties of the system mainly arise from two |
| 374 |
< |
different ways: i) the applied field distort the charge distributions |
| 375 |
< |
so it produces an induced multipolar moment in each molecule; and ii) |
| 376 |
< |
the applied field tends to line up originally randomly oriented |
| 377 |
< |
molecular moment towards the direction of the applied field. In this |
| 378 |
< |
study, we basically focus on the orientational contribution in the |
| 379 |
< |
dielectric properties. If we consider a system of molecules in the |
| 380 |
< |
presence of external field perturbation, the perturbation experienced |
| 381 |
< |
by any molecule will not be only due to external field or field |
| 382 |
< |
gradient but also due to the field or field gradient produced by the |
| 383 |
< |
all other molecules in the system. In the following subsections |
| 373 |
> |
If we consider a collection of molecules in the presence of external |
| 374 |
> |
field, the perturbation experienced by any one molecule will include |
| 375 |
> |
contributions to the field or field gradient produced by the all other |
| 376 |
> |
molecules in the system. In subsections |
| 377 |
|
\ref{subsec:boltzAverage-Dipole} and \ref{subsec:boltzAverage-Quad}, |
| 378 |
< |
we will discuss about the molecular polarization only due to external |
| 379 |
< |
field perturbation. The contribution of the field or field gradient |
| 380 |
< |
due to all other molecules will be taken into account while |
| 388 |
< |
calculating correction factor in the paper. |
| 378 |
> |
we discuss the molecular polarization due solely to external field |
| 379 |
> |
perturbations. This illustrates the origins of the polarizability |
| 380 |
> |
equations (Eqs. 6, 20, and 21) in the main text. |
| 381 |
|
|
| 382 |
|
\subsection{Dipoles} |
| 383 |
|
\label{subsec:boltzAverage-Dipole} |
| 384 |
|
Consider a system of molecules, each with permanent dipole moment |
| 385 |
< |
$p_o$. In the absense of external field, thermal agitation orients the |
| 386 |
< |
dipoles randomly, reducing the system moment to zero. External fields |
| 387 |
< |
will tend to line up the dipoles in the direction of applied field. |
| 388 |
< |
Here we have considered net field from all other molecules is |
| 389 |
< |
considered to be zero. Therefore the total Hamiltonian of each |
| 390 |
< |
molecule is,\cite{Jackson98} |
| 385 |
> |
$p_o$. In the absense of an external field, thermal agitation orients |
| 386 |
> |
the dipoles randomly, and the system moment, $\mathbf{P}$, is zero. |
| 387 |
> |
External fields will line up the dipoles in the direction of applied |
| 388 |
> |
field. Here we consider the net field from all other molecules to be |
| 389 |
> |
zero. Therefore the total Hamiltonian acting on each molecule |
| 390 |
> |
is,\cite{Jackson98} |
| 391 |
|
\begin{equation} |
| 392 |
< |
H = H_o - \mathbf{p_o}\cdot \mathbf{E}, |
| 392 |
> |
H = H_o - \mathbf{p}_o \cdot \mathbf{E}, |
| 393 |
|
\end{equation} |
| 394 |
|
where $H_o$ is a function of the internal coordinates of the molecule. |
| 395 |
< |
The Boltzmann average of the dipole moment is given by, |
| 395 |
> |
The Boltzmann average of the dipole moment in the direction of the |
| 396 |
> |
field is given by, |
| 397 |
|
\begin{equation} |
| 398 |
< |
\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}}}, |
| 398 |
> |
\langle p_{mol} \rangle = \frac{\displaystyle\int p_o \cos\theta |
| 399 |
> |
e^{~p_o E \cos\theta /k_B T}\; d\Omega}{\displaystyle\int e^{~p_o E \cos\theta/k_B |
| 400 |
> |
T}\; d\Omega}, |
| 401 |
|
\end{equation} |
| 402 |
< |
where $\bf{E}$ is selected along z-axis. If we consider that the |
| 403 |
< |
applied field is small, \textit{i.e.} $\frac{p_oE\; cos\theta}{k_B T} << 1$, |
| 402 |
> |
where the $z$-axis is taken in the direction of the applied field, |
| 403 |
> |
$\bf{E}$ and |
| 404 |
> |
$\int d\Omega = \int_0^\pi \sin\theta\; d\theta \int_0^{2\pi} d\phi |
| 405 |
> |
\int_0^{2\pi} d\psi$ |
| 406 |
> |
is an integration over Euler angles describing the orientation of the |
| 407 |
> |
molecule. |
| 408 |
> |
|
| 409 |
> |
If the external fields are small, \textit{i.e.} |
| 410 |
> |
$p_oE \cos\theta / k_B T << 1$, |
| 411 |
|
\begin{equation} |
| 412 |
< |
\braket{p_{mol}} \approx \frac{1}{3}\frac{{p_o}^2}{k_B T}E, |
| 412 |
> |
\langle p_{mol} \rangle \approx \frac{{p_o}^2}{3 k_B T}E, |
| 413 |
|
\end{equation} |
| 414 |
< |
where $ \alpha_p = \frac{1}{3}\frac{{p_o}^2}{k_B T}$ is a molecular |
| 414 |
> |
where $ \alpha_p = \frac{{p_o}^2}{3 k_B T}$ is the molecular |
| 415 |
|
polarizability. The orientational polarization depends inversely on |
| 416 |
< |
the temperature and applied field must overcome the thermal agitation. |
| 416 |
> |
the temperature as the applied field must overcome thermal agitation |
| 417 |
> |
to orient the dipoles. |
| 418 |
|
|
| 419 |
|
\subsection{Quadrupoles} |
| 420 |
|
\label{subsec:boltzAverage-Quad} |
| 421 |
< |
Consider a system of molecules with permanent quadrupole moment |
| 422 |
< |
$q_{\alpha\beta}$. The average quadrupole moment at temperature T in |
| 423 |
< |
the presence of uniform applied field gradient is given |
| 424 |
< |
by,\cite{AduGyamfi78, AduGyamfi81} |
| 421 |
> |
If instead, our system consists of molecules with permanent |
| 422 |
> |
\textit{quadrupole} tensor $q_{\alpha\beta}$. The average quadrupole |
| 423 |
> |
at temperature $T$ in the presence of uniform applied field gradient |
| 424 |
> |
is given by,\cite{AduGyamfi78, AduGyamfi81} |
| 425 |
|
\begin{equation} |
| 426 |
< |
\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}}}, |
| 426 |
> |
\langle q_{\alpha\beta} \rangle \;=\; \frac{\displaystyle\int |
| 427 |
> |
q_{\alpha\beta}\; e^{-H/k_B T}\; d\Omega}{\displaystyle\int |
| 428 |
> |
e^{-H/k_B T}\; d\Omega} \;=\; \frac{\displaystyle\int |
| 429 |
> |
q_{\alpha\beta}\; e^{~q_{\mu\nu}\;\partial_\nu E_\mu /k_B T}\; |
| 430 |
> |
d\Omega}{\displaystyle\int e^{~q_{\mu\nu}\;\partial_\nu E_\mu /k_B |
| 431 |
> |
T}\; d\Omega }, |
| 432 |
|
\label{boltzQuad} |
| 433 |
|
\end{equation} |
| 434 |
< |
where $\int d\Omega = \int_0^{2\pi} \int_0^\pi \int_0^{2\pi} |
| 435 |
< |
sin\theta\; d\theta\ d\phi\ d\psi$ is the integration over Euler |
| 436 |
< |
angles, $ H = H_o -q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of |
| 437 |
< |
a quadrupole in the gradient of the |
| 438 |
< |
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, |
| 439 |
< |
\begin{equation} |
| 440 |
< |
\begin{split} |
| 441 |
< |
&q_{\alpha\beta} = \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
| 442 |
< |
&H = H_o - q:{\nabla}\mathbf{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. |
| 443 |
< |
\end{split} |
| 444 |
< |
\label{energyQuad} |
| 445 |
< |
\end{equation} |
| 434 |
> |
where $H = H_o - q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of a |
| 435 |
> |
quadrupole in the gradient of the applied field and $H_o$ is a |
| 436 |
> |
function of internal coordinates of the molecule. The energy and |
| 437 |
> |
quadrupole moment can be transformed into the body frame using a |
| 438 |
> |
rotation matrix $\mathsf{\eta}^{-1}$, |
| 439 |
> |
\begin{align} |
| 440 |
> |
q_{\alpha\beta} &= \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
| 441 |
> |
H &= H_o - q:{\nabla}\mathbf{E} \\ |
| 442 |
> |
&= H_o - q_{\mu\nu}\;\partial_\nu E_\mu \\ |
| 443 |
> |
&= H_o |
| 444 |
> |
-\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu |
| 445 |
> |
E_\mu. \label{energyQuad} |
| 446 |
> |
\end{align} |
| 447 |
|
Here the starred tensors are the components in the body fixed |
| 448 |
< |
frame. Substituting equation (\ref{energyQuad}) in the equation (\ref{boltzQuad}) |
| 449 |
< |
and taking linear terms in the expansion we get, |
| 450 |
< |
\begin{equation} |
| 451 |
< |
\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)}, |
| 448 |
> |
frame. Substituting equation (\ref{energyQuad}) in the equation |
| 449 |
> |
(\ref{boltzQuad}) and taking linear terms in the expansion we obtain, |
| 450 |
> |
\begin{equation} |
| 451 |
> |
\braket{q_{\alpha\beta}} = \frac{\displaystyle \int q_{\alpha\beta} \left(1 + |
| 452 |
> |
\frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu |
| 453 |
> |
E_\mu }{k_B T}\right)\; d\Omega}{\displaystyle \int \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)\; d\Omega}. |
| 454 |
|
\end{equation} |
| 455 |
< |
where $\eta_{\alpha\alpha'}$ is the inverse of the rotation matrix that transforms |
| 456 |
< |
the body fixed co-ordinates to the space co-ordinates. |
| 455 |
> |
Recall that $\eta_{\alpha\alpha'}$ is the inverse of the rotation |
| 456 |
> |
matrix that transforms the body fixed co-ordinates to the space |
| 457 |
> |
co-ordinates. |
| 458 |
|
% \[\eta_{\alpha\alpha'} |
| 459 |
|
% = \left(\begin{array}{ccc} |
| 460 |
|
% cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
| 462 |
|
% sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
| 463 |
|
% \end{array} \right).\] |
| 464 |
|
|
| 465 |
< |
Integration of 1st and 2nd terms in the denominator gives $8 \pi^2$ |
| 466 |
< |
and $8 \pi^2 /3\;{\nabla}.\mathbf{E}\; Tr(q^*) $ respectively. The |
| 467 |
< |
second term vanishes for charge free space, ${\nabla}.\mathbf{E} \; = \; 0$. Similarly integration of the |
| 468 |
< |
1st term in the numerator produces |
| 469 |
< |
$8 \pi^2 /3\; Tr(q^*)\delta_{\alpha\beta}$ and the 2nd term produces |
| 470 |
< |
$8 \pi^2 /15k_B T (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
| 471 |
< |
{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta$, |
| 472 |
< |
if ${\nabla}.\mathbf{E} \; = \; 0$, |
| 473 |
< |
$ \partial_\alpha E_\beta = \partial_\beta E_\alpha$ and |
| 474 |
< |
${q}^*_{\alpha'\beta'}= {q}^*_{\beta'\alpha'}$. Therefore the |
| 475 |
< |
Boltzmann average of a quadrupole moment can be written as, |
| 476 |
< |
|
| 465 |
> |
Integration of the first and second terms in the denominator gives |
| 466 |
> |
$8 \pi^2$ and |
| 467 |
> |
$8 \pi^2 ({\nabla} \cdot \mathbf{E}) \mathrm{Tr}(q^*) / 3 $ |
| 468 |
> |
respectively. The second term vanishes for charge free space (where |
| 469 |
> |
${\nabla} \cdot \mathbf{E}=0$). Similarly, integration of the first |
| 470 |
> |
term in the numerator produces |
| 471 |
> |
$8 \pi^2 \delta_{\alpha\beta} \mathrm{Tr}(q^*) / 3$ while the second |
| 472 |
> |
produces |
| 473 |
> |
$8 \pi^2 (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
| 474 |
> |
{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta / |
| 475 |
> |
15 k_B T $. |
| 476 |
> |
Therefore the Boltzmann average of a quadrupole moment can be written |
| 477 |
> |
as, |
| 478 |
|
\begin{equation} |
| 479 |
< |
\braket{q_{\alpha\beta}}\; = \; \frac{1}{3} Tr(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
| 479 |
> |
\langle q_{\alpha\beta} \rangle = \frac{1}{3} \mathrm{Tr}(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
| 480 |
|
\end{equation} |
| 481 |
< |
where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupole polarizablity and ${\bar{q_o}}^2= |
| 482 |
< |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. |
| 481 |
> |
where $\alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular |
| 482 |
> |
quadrupole polarizablity and |
| 483 |
> |
${\bar{q_o}}^2= |
| 484 |
> |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ |
| 485 |
> |
is the square of the net quadrupole moment of a molecule. |
| 486 |
|
|
| 487 |
|
\section{Gradient of the field due to quadrupolar polarization} |
| 488 |
|
\label{singularQuad} |
| 489 |
< |
In this section, we will discuss the gradient of the field produced by |
| 490 |
< |
quadrupolar polarization. For this purpose, we consider a distribution |
| 491 |
< |
of charge ${\rho}(\mathbf r)$ which gives rise to an electric field |
| 492 |
< |
$\mathbf{E}(\mathbf r)$ and gradient of the field ${\nabla} \mathbf{E}(\mathbf r)$ |
| 493 |
< |
throughout space. The total gradient of the electric field over volume |
| 494 |
< |
due to the all charges within the sphere of radius $R$ is given by |
| 495 |
< |
(cf. Jackson equation 4.14): |
| 489 |
> |
In section IV.C of the main text, we stated that for quadrupolar |
| 490 |
> |
fluids, the self-contribution to the field gradient vanishes at the |
| 491 |
> |
singularity. In this section, we prove this statement. For this |
| 492 |
> |
purpose, we consider a distribution of charge $\rho(\mathbf{r})$ which |
| 493 |
> |
gives rise to an electric field $\mathbf{E}(\mathbf{r})$ and gradient |
| 494 |
> |
of the field $\nabla\mathbf{E}(\mathbf{r})$ throughout space. The |
| 495 |
> |
gradient of the electric field over volume due to the charges within |
| 496 |
> |
the sphere of radius $R$ is given by (cf. Ref. \onlinecite{Jackson98}, |
| 497 |
> |
equation 4.14): |
| 498 |
|
\begin{equation} |
| 499 |
< |
\int_{r<R} {\nabla}\mathbf{E}\;d^3r = -\int_{r=R} R^2 \mathbf{E}\;\hat{n}\; d\Omega |
| 499 |
> |
\int_{r<R} \nabla\mathbf{E} d\mathbf{r} = -\int_{r=R} R^2 \mathbf{E}\;\hat{n}\; d\Omega |
| 500 |
|
\label{eq:8} |
| 501 |
|
\end{equation} |
| 502 |
|
where $d\Omega$ is the solid angle and $\hat{n}$ is the normal vector |
| 503 |
|
of the surface of the sphere, |
| 486 |
– |
$\hat{n} = sin[\theta]cos[\phi]\hat{x} + sin[\theta]sin[\phi]\hat{y} + |
| 487 |
– |
cos[\theta]\hat{z}$ |
| 488 |
– |
in spherical coordinates. For the charge density ${\rho}(\mathbf r')$, the |
| 489 |
– |
total gradient of the electric field can be written as, ~\cite{Jackson98} |
| 504 |
|
\begin{equation} |
| 505 |
< |
\int_{r<R} {\nabla}\mathbf {E}\; d^3r=-\int_{r=R} R^2\; {\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; {\nabla}\;\left(\int \frac{\rho(\mathbf r')}{|\mathbf{r}-\mathbf{r}'|}\;d^3r'\right) \hat{n}\; d\Omega |
| 505 |
> |
\hat{n} = \sin\theta\cos\phi\; \hat{x} + \sin\theta\sin\phi\; \hat{y} + |
| 506 |
> |
\cos\theta\; \hat{z} |
| 507 |
> |
\end{equation} |
| 508 |
> |
in spherical coordinates. For the charge density $\rho(\mathbf{r}')$, the |
| 509 |
> |
total gradient of the electric field can be written as,\cite{Jackson98} |
| 510 |
> |
\begin{equation} |
| 511 |
> |
\int_{r<R} {\nabla}\mathbf {E}\; d\mathbf{r}=-\int_{r=R} R^2\; {\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; {\nabla}\;\left(\int \frac{\rho(\mathbf r')}{|\mathbf{r}-\mathbf{r}'|}\;d\mathbf{r}'\right) \hat{n}\; d\Omega |
| 512 |
|
\label{eq:9} |
| 513 |
|
\end{equation} |
| 514 |
|
The radial function in the equation (\ref{eq:9}) can be expressed in |
| 520 |
|
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, |
| 521 |
|
\begin{equation} |
| 522 |
|
\begin{split} |
| 523 |
< |
\int_{r<R} {\nabla}\mathbf{E}\;d^3r &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; {\nabla}\;\left(\int \rho(\mathbf 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 \\ |
| 524 |
< |
&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(\mathbf 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 |
| 523 |
> |
\int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; {\nabla}\;\left(\int \rho(\mathbf 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\mathbf{r}'\right) \hat{n}\; d\Omega \\ |
| 524 |
> |
&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(\mathbf 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\mathbf{r} |
| 525 |
|
' |
| 526 |
|
\end{split} |
| 527 |
|
\label{eq:11} |
| 535 |
|
\end{split} |
| 536 |
|
\label{eq:12} |
| 537 |
|
\end{equation} |
| 538 |
< |
Using equation (\ref{eq:12}) we get, |
| 538 |
> |
where $Y_{l,l+1,m}(\theta, \phi)$ is a vector spherical |
| 539 |
> |
harmonic.\cite{Arfkan} Using equation (\ref{eq:12}) we get, |
| 540 |
|
\begin{equation} |
| 541 |
|
{\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}}, |
| 542 |
|
\label{eq:13} |
| 543 |
|
\end{equation} |
| 544 |
< |
where $ Y_{l,l+1,m}(\theta, \phi)$ is a vector spherical harmonics \cite{Arfkan}. Using Clebsch-Gorden coefficients $C(l+1, 1, l|m_1,m_2,m) $, equation \ref{eq:14} can be written in spherical harmonics, |
| 544 |
> |
Using Clebsch-Gordan coefficients $C(l+1,1,l|m_1,m_2,m)$, the vector |
| 545 |
> |
spherical harmonics can be written in terms of spherical harmonics, |
| 546 |
|
\begin{equation} |
| 547 |
< |
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}. |
| 547 |
> |
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}. |
| 548 |
|
\label{eq:14} |
| 549 |
|
\end{equation} |
| 550 |
|
Here $\hat{e}_{m_2}$ is a spherical tensor of rank 1 which can be expressed |
| 555 |
|
\end{equation} |
| 556 |
|
The normal vector $\hat{n} $ is then expressed in terms of spherical tensor of rank 1 as shown in below, |
| 557 |
|
\begin{equation} |
| 558 |
< |
\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). |
| 558 |
> |
\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). |
| 559 |
|
\label{eq:16} |
| 560 |
|
\end{equation} |
| 561 |
|
The surface integral of the product of $\hat{n}$ and |
| 562 |
< |
${Y_{l+1}}^{m_1}(\theta, \phi)$ gives, |
| 562 |
> |
$Y_{l+1}^{m_1}(\theta, \phi)$ gives, |
| 563 |
|
\begin{equation} |
| 564 |
|
\begin{split} |
| 565 |
< |
\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 \\ |
| 566 |
< |
&= \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 \\ |
| 565 |
> |
\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 \\ |
| 566 |
> |
&= \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 \\ |
| 567 |
|
&= \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), |
| 568 |
|
\end{split} |
| 569 |
|
\label{eq:17} |
| 570 |
|
\end{equation} |
| 571 |
< |
where ${Y_{l}}^{-m} = (-1)^m\;{{Y_{l}}^{m}}^* $ and |
| 572 |
< |
$ \int {{Y_{l}}^{m}}^*\;{Y_{l'}}^{m'}\;d\Omega = |
| 571 |
> |
where $Y_{l}^{-m} = (-1)^m\;{Y_{l}^{m}}^* $ and |
| 572 |
> |
$ \int {Y_{l}^{m}}^* Y_{l'}^{m'}\;d\Omega = |
| 573 |
|
\delta_{ll'}\delta_{mm'} $. |
| 574 |
|
Non-vanishing values of equation \ref{eq:17} require $l = 0$, |
| 575 |
|
therefore the value of $ m = 0 $. Since the values of $ m_1$ are -1, |
| 578 |
|
modified, |
| 579 |
|
\begin{equation} |
| 580 |
|
\begin{split} |
| 581 |
< |
\int_{r<R} {\nabla}\mathbf{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( |
| 582 |
< |
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d^3r' \\ |
| 583 |
< |
&= -\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)\\ |
| 584 |
< |
&= - \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). |
| 581 |
> |
\int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} = &- \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( |
| 582 |
> |
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d\mathbf{r}' \\ |
| 583 |
> |
&= -\sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;d\mathbf{r}'\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right)\\ |
| 584 |
> |
&= - \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\;C_\mathrm{total}\;\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right). |
| 585 |
|
\end{split} |
| 586 |
|
\label{eq:19} |
| 587 |
|
\end{equation} |
| 588 |
< |
In the last step, the charge density was integrated over the sphere, yielding a total charge $\mathrm{C_total}$.Equation (\ref{eq:19}) gives the total gradient of the field over a sphere due to the distribution of the charges. |
| 589 |
< |
For quadrupolar fluids the total charge within a sphere is zero, therefore |
| 590 |
< |
$ \int_{r<R} {\nabla}\mathbf{E}\;d^3r = 0 $. Hence the quadrupolar |
| 588 |
> |
In the last step, the charge density was integrated over the sphere, |
| 589 |
> |
yielding a total charge $C_\mathrm{total}$.Equation (\ref{eq:19}) |
| 590 |
> |
gives the total gradient of the field over a sphere due to the |
| 591 |
> |
distribution of the charges. For quadrupolar fluids the total charge |
| 592 |
> |
within a sphere is zero, therefore |
| 593 |
> |
$ \int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} = 0 $. Hence the quadrupolar |
| 594 |
|
polarization produces zero net gradient of the field inside the |
| 595 |
|
sphere. |
| 596 |
|
|
| 572 |
– |
\section{Geometric Factors for Two Embedded Point Charges} |
| 573 |
– |
|
| 597 |
|
\bibliography{dielectric_new} |
| 598 |
|
\end{document} |
| 599 |
|
% |
