13.2: Preliminary Ideas

Derivatives and Integrals of Vectors in Spherical Coordinates

The derivation of the Thomas Fermi equation involves derivatives of vectors in spherical coordinates. For more details on derivatives and vectors see [11, ch. 1]. Consider a scalar function described in spherical coordinates,

\[V=V(\overrightarrow{r})=V(r, \theta, \phi). \nonumber \]

The gradient of \(V (r, \theta, \phi)\) is defined

\[\overrightarrow{\nabla} V=\frac{\partial V}{\partial r} \hat{a}_{r}+\frac{1}{r} \frac{\partial V}{\partial \theta} \hat{a}_{\theta}+\frac{1}{r \sin \theta} \frac{\partial V}{\partial \phi} \hat{a}_{\phi}. \nonumber \]

Gradient was introduced in Section 1.6.1. It returns a vector which points in the direction of largest change in the function. The Laplacian is defined in spherical coordinates as

\[\nabla^{2} V=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial V}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial V}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} V}{\partial \phi^{2}}. \nonumber \]

Qualitatively, the Laplacian of a scalar is the second derivative with respect to spatial position. In the derivations of this chapter, we encounter only functions which are uniform with respect to \(\theta\) and \(\phi\). For functions of the form \(V = V (r)\), the formulas for gradient and Laplacian simplify significantly.

\[\overrightarrow{\nabla} V=\frac{\partial V}{\partial r} \hat{a}_{r} \nonumber \]

\[\nabla^{2} V=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial V}{\partial r}\right) \nonumber \]

We will also need the vector identity of Equation 1.6.8,

\[\nabla^{2} V=\overrightarrow{\nabla} \cdot \overrightarrow{\nabla} V. \nonumber \]

A differential volume element in spherical coordinates is given by

\[d \mathbb{V}=r^{2} \sin \theta \;d r \;d \theta \;d \phi. \nonumber \]

A volume integral of the function \(V (r, \theta, \phi)\) over a sphere of radius 1 centered at the origin is denoted

\[\int_{r=0}^{1} \int_{\theta=0}^{\pi} \int_{\phi=0}^{2 \pi} V(r, \theta, \phi) r^{2} \sin \theta \;d r \;d \theta \;d \phi. \nonumber \]

Assuming \(V\) doesn't depend on \(\theta\) or \(\phi\), the integral is separable.

\[\left(\int_{\theta=0}^{\pi} \int_{\phi=0}^{2 \pi} \sin \theta \;d \theta \;d \phi\right) \int_{r=0}^{1} V(r) r^{2} d r=4 \pi \int_{r=0}^{1} V(r) r^{2} d r \nonumber \]

A sphere of radius \(r\) has volume \(\frac{4}{3}\pi r^3\).

Notation

Writing this text without overloading variables has been a challenge. For example, \(V\) is the logical choice for denoting voltage, volume, and velocity. Up until now, the context offered clues to the meaning of symbols. However in this chapter, we will encounter equations involving both energy and electric field, equations involving both voltage and volume, and equations involving both mass and momentum. To help avoid confusion from the notation, Table \(\PageIndex{1}\) shows an excerpt of the variable list from Appendix A. This table does not list all quantities we will encounter. However, it highlights some of the more confusing ones.

In this chapter, we will encounter many quantities which vary with position. We will not encounter any quantities which vary with time. Therefore, voltage is denoted by a capital letter, not a lowercase letter. Voltage is a function of \(r\), which denotes position in spherical coordinates. Assume that the origin of the coordinate system is at the center of the atom under consideration. Voltage is always specified with respect to some reference level called ground, so assume this zero volt reference level occurs at \(r = \infty\). Also assume there is no \(\theta\) or \(\phi\) dependence of the voltage. Therefore, \(V (\overrightarrow{r}) = V (r)\) represents voltage.

Table \(\PageIndex{1}\): Variable list.

Symbol	Quantity	SI Units	S/V/C	Comments
\(E\)	Energy	J = Nm	S
\(\overrightarrow{E}\)	Electric field intensity	\(\frac{V}{m}\)	V
\(E_f\)	Fermi energy level	J	S	Also called Fermi level
\(\vec{k}\)	Wave vector	\(m^{-1}\)	V
\(k_f\)	Fermi wave vector	\(m^{-1}\)	S
\(m\)	Mass	kg	S
\(\mathbb{M}\)	Generalized momentum	*	S	Many authors use \(p\)
\(\overrightarrow{M}\)	Momentum	\(\frac{kg \cdot m}{s}\)	V	Many authors use \(\overrightarrow{p}\)
\(N\)	(Total) number of electrons per atom	\(\frac{\text{elections}}{\text{atom}}\)	S
\(v\)	Voltage (AC or time varying)	V	S
\(\vec{v}\)	Velocity	\(\frac{m}{s}\)	V
\(V\)	Voltage (DC)	V	S
\(\mathbb{V}\)	Volume	\(m^3\)	S
\(\mu_{chem}\)	Chemical potential	\(\frac{J}{\text{atom}}\)	S
\(\rho_{ch}\)	Charge density	\(\frac{C}{m^3}\)	S

Reciprocal Space Concepts

The idea of reciprocal space was introduced in Section 6.3 in the context of crystalline materials. We can describe the location of atoms in a crystal, for example, as a function of position where position \(\overrightarrow{r}\) is measured in meters. In this chapter, we are interested in individual atoms instead of crystals composed of many atoms. We can plot quantities like energy \(E (\overrightarrow{r})\) or voltage \(V (\overrightarrow{r})\) as a function of position. Figure 6.4.2, for example, plots energy versus position inside a diode. In Section 6.3, the idea of wave vector \(\overrightarrow{k}\) in units of \(m^{-1}\) was introduced. Wave vector represents the spatial frequency. We saw that we could plot energy or other quantities as a function of wave vector, and Fig 6.3.1 is an example of such a plot. We will need the idea of wave vector in this chapter because we describe a situation where we do not know how the energy varies with position, but we do know something about how the energy varies with wave vector.