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author | Prefetch | 2021-10-12 14:29:59 +0200 |
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committer | Prefetch | 2021-10-12 14:29:59 +0200 |
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parent | e28d2a982d0c65fcad9a2d2a4c20d06a9848fa8f (diff) |
Expand knowledge base
-rw-r--r-- | content/know/concept/kubo-formula/index.pdc | 2 | ||||
-rw-r--r-- | content/know/concept/lawson-criterion/index.pdc | 133 | ||||
-rw-r--r-- | content/know/concept/lindhard-function/index.pdc | 420 |
3 files changed, 554 insertions, 1 deletions
diff --git a/content/know/concept/kubo-formula/index.pdc b/content/know/concept/kubo-formula/index.pdc index 9e52835..f0208da 100644 --- a/content/know/concept/kubo-formula/index.pdc +++ b/content/know/concept/kubo-formula/index.pdc @@ -223,5 +223,5 @@ $$\begin{aligned} *Many-body quantum theory in condensed matter physics*, 2016, Oxford. 2. K.S. Thygesen, - *Linear response theory*, + *Advanced solid state physics: linear response theory*, 2013, unpublished. diff --git a/content/know/concept/lawson-criterion/index.pdc b/content/know/concept/lawson-criterion/index.pdc new file mode 100644 index 0000000..59801ee --- /dev/null +++ b/content/know/concept/lawson-criterion/index.pdc @@ -0,0 +1,133 @@ +--- +title: "Lawson criterion" +firstLetter: "L" +publishDate: 2021-10-06 +categories: +- Physics +- Plasma physics + +date: 2021-10-04T14:49:24+02:00 +draft: false +markup: pandoc +--- + +# Lawson criterion + +For sustained nuclear fusion to be possible, +the **Lawson criterion** must be met, +from which some required properties +of the plasma and the reactor chamber can be deduced. + +Suppose that a reactor generates a given power $P_\mathrm{fus}$ by nuclear fusion, +but that it leaks energy at a rate $P_\mathrm{loss}$ in an unusable way. +If an auxiliary input power $P_\mathrm{aux}$ sustains the fusion reaction, +then the following inequality must be satisfied +in order to have harvestable energy: + +$$\begin{aligned} + P_\mathrm{loss} + \le P_\mathrm{fus} + P_\mathrm{aux} +\end{aligned}$$ + +We can rewrite $P_\mathrm{aux}$ using the definition +of the **energy gain factor** $Q$, +which is the ratio of the output and input powers of the fusion reaction: + +$$\begin{aligned} + Q + \equiv \frac{P_\mathrm{fus}}{P_\mathrm{aux}} + \quad \implies \quad + P_\mathrm{aux} + = \frac{P_\mathrm{fus}}{Q} +\end{aligned}$$ + +Returning to the inequality, we can thus rearrange its right-hand side as follows: + +$$\begin{aligned} + P_\mathrm{loss} + \le P_\mathrm{fus} + \frac{P_\mathrm{fus}}{Q} + = P_\mathrm{fus} \Big( 1 + \frac{1}{Q} \Big) + = P_\mathrm{fus} \Big( \frac{Q + 1}{Q} \Big) +\end{aligned}$$ + +We assume that the plasma has equal species densities $n_i = n_e$, +so its total density $n = 2 n_i$. +Then $P_\mathrm{fus}$ is as follows, +where $f_{ii}$ is the frequency +with which a given ion collides with other ions, +and $E_\mathrm{fus}$ is the energy released by a single fusion reaction: + +$$\begin{aligned} + P_\mathrm{fus} + = f_{ii} n_i E_\mathrm{fus} + = \big( n_i \expval{\sigma v} \big) n_i E_\mathrm{fus} + = \frac{n^2}{4} \expval{\sigma v} E_\mathrm{fus} +\end{aligned}$$ + +Where $\expval{\sigma v}$ is the mean product +of the velocity $v$ and the collision cross-section $\sigma$. + +Furthermore, assuming that both species have the same temperature $T_i = T_e = T$, +the total energy density $W$ of the plasma is given by: + +$$\begin{aligned} + W + = \frac{3}{2} k_B T_i n_i + \frac{3}{2} k_B T_e n_e + = 3 k_B T n +\end{aligned}$$ + +Where $k_B$ is Boltzmann's constant. +From this, we can define the **confinement time** $\tau_E$ +as the characteristic lifetime of energy in the reactor, before leakage. +Therefore: + +$$\begin{aligned} + \tau_E + \equiv \frac{W}{P_\mathrm{loss}} + \quad \implies \quad + P_\mathrm{loss} + = \frac{3 n k_B T}{\tau_E} +\end{aligned}$$ + +Inserting these new expressions for $P_\mathrm{fus}$ and $P_\mathrm{loss}$ +into the inequality, we arrive at: + +$$\begin{aligned} + \frac{3 n k_B T}{\tau_E} + \le \frac{n^2}{4} \expval{\sigma v} E_\mathrm{fus} \Big( \frac{Q + 1}{Q} \Big) +\end{aligned}$$ + +This can be rearranged to the form below, +which is the original Lawson criterion: + +$$\begin{aligned} + n \tau_E + \ge \frac{Q}{Q + 1} \frac{12 k_B T}{\expval{\sigma v} E_\mathrm{fus}} +\end{aligned}$$ + +However, it turns out that the highest fusion power density +is reached when $T$ is at the minimum of $T^2 / \expval{\sigma v}$. +Therefore, we multiply by $T$ to get the Lawson triple product: + +$$\begin{aligned} + \boxed{ + n T \tau_E + \ge \frac{Q}{Q + 1} \frac{12 k_B T^2}{\expval{\sigma v} E_\mathrm{fus}} + } +\end{aligned}$$ + +For some reason, +it is often assumed that the fusion is infinitely profitable $Q \to \infty$, +in which case the criterion reduces to: + +$$\begin{aligned} + n T \tau_E + \ge \frac{12 k_B T^2}{\expval{\sigma v} E_\mathrm{fus}} +\end{aligned}$$ + + + +## References +1. M. Salewski, A.H. Nielsen, + *Plasma physics: lecture notes*, + 2021, unpublished. diff --git a/content/know/concept/lindhard-function/index.pdc b/content/know/concept/lindhard-function/index.pdc new file mode 100644 index 0000000..d38dc2e --- /dev/null +++ b/content/know/concept/lindhard-function/index.pdc @@ -0,0 +1,420 @@ +--- +title: "Lindhard function" +firstLetter: "L" +publishDate: 2021-10-12 +categories: +- Physics +- Quantum mechanics + +date: 2021-09-23T16:21:57+02:00 +draft: false +markup: pandoc +--- + +# Lindhard function + +We start from the [Kubo formula](/know/concept/kubo-formula/) +for the electron density operator $\hat{n}$, +which describes the change in $\expval{\hat{n}}$ +due to a time-dependent perturbation $\hat{H}_1$: + +$$\begin{aligned} + \delta\expval{{\hat{n}}}(\vb{r}, t) + = -\frac{i}{\hbar} \int_{-\infty}^\infty \Theta(t - t') \expval{\comm{\hat{n}_I(\vb{r}, t)}{\hat{H}_{1,I}(t')}}_0 \dd{t'} +\end{aligned}$$ + +Where $\Theta$ is the [Heaviside step function](/know/concept/heaviside-step-function/), +and the subscript $I$ refers to the [interaction picture](/know/concept/interaction-picture/). +Notice from the limits that the perturbation is switched on at $t = -\infty$. +Now, let us consider the following harmonic $\hat{H}_1$ in the Schrödinger picture: + +$$\begin{aligned} + \hat{H}_{1,S}(t) + = g(t) \: \hat{V}_S + \qquad + g(t) + \equiv \exp\!(- i \omega t) \exp\!(\eta t) + \qquad + \hat{V}_S + \equiv \int_{-\infty}^\infty V(\vb{r}) \: \hat{n}(\vb{r}) \dd{\vb{r}} +\end{aligned}$$ + +Where $\eta$ is a tiny positive number, +which represents a gradual switching-on of $\hat{H}_1$, +eliminating transient effects +and helping the convergence of an integral later. + +We assume that $V(\vb{r})$ varies slowly compared to the electrons' wavefunctions, +so we argue that $\hat{V}_S$ is practically time-independent, +because the total number of electrons is conserved, +and $\hat{n}$ is only weakly perturbed by $\hat{H}_1$. + +Because $\hat{H}_1$ starts at $t = -\infty$, +we can always shift the time axis such that the point of interest is at $t = 0$. +We thus have, without loss of generality: + +$$\begin{aligned} + \delta\expval{{\hat{n}}}(\vb{r}) + = \delta\expval{{\hat{n}}}(\vb{r}, 0) + &= -\frac{i}{\hbar} \int_{-\infty}^\infty \Theta(- t') + \Big( \expval{\hat{n}_I \hat{V}_I}_0 - \expval{\hat{V}_I \hat{n}_I}_0 \Big) g(t') \dd{t'} +\end{aligned}$$ + +The expectation values $\expval{}_0$ are calculated for $\ket{0}$, +which was the state at $t = -\infty$. +Note that if $\ket{0}$ is an eigenstate of $\hat{H}_{0,S}$, +there is no difference which picture (Schrödinger or interaction) $\ket{0}$ is in, +because any operator $\hat{A}$ then satisfies: + +$$\begin{aligned} + \matrixel{0_I}{\hat{A}_I}{0_I} + &= \matrixel**{0_S}{\exp\!(-i \hat{H}_{0,S} t / \hbar) \:\: \hat{A}_I \: \exp\!(i \hat{H}_{0,S} t / \hbar)}{0_S} + \\ + &= \matrixel{0_S}{\hat{A}_I}{0_S} \:\exp\!\big(i (E_0\!-\!E_0) t / \hbar\big) + = \matrixel{0_S}{\hat{A}_I}{0_S} +\end{aligned}$$ + +Therefore, we will assume that $\ket{0}$ is an eigenstate of $\hat{H}_{0,S}$. +Next, we insert the identity operator $\hat{I} = \sum_{j} \ket{j} \bra{j}$, +where $\ket{j}$ are all the eigenstates of $\hat{H}_{0,S}$: + +$$\begin{aligned} + \delta\expval{\hat{n}}(\vb{r}) + &= -\frac{i}{\hbar} \sum_{j} \int \Theta(-t') + \Big( \matrixel{0}{\hat{n}_I}{j} \matrixel{j}{\hat{V}_I}{0} - \matrixel{0}{\hat{V}_I}{j} \matrixel{j}{\hat{n}_I}{0} \Big) g(t') \dd{t'} +\end{aligned}$$ + +Using the fact that $\ket{0}$ and $\ket{j}$ +are eigenstates of $\hat{H}_{0,S}$, +and that we chose $t = 0$, we find: + +$$\begin{aligned} + \matrixel{j}{\hat{V}_I(t')}{0} + &= \matrixel{j}{\hat{V}_S}{0} \:\exp\!\big(i (E_j \!-\! E_0) t' / \hbar\big) + \\ + \matrixel{j}{\hat{n}_I(0)}{0} + &= \matrixel{j}{\hat{n}_S(0)}{0} \:\exp\!(0 - 0) + = \matrixel{j}{\hat{n}_S(0)}{0} +\end{aligned}$$ + +We define $\omega_{j0} \equiv (E_j \!-\! E_0)/\hbar$ +and insert the above expressions into $\delta\expval{\hat{n}}$, +yielding: + +$$\begin{aligned} + \delta\expval{\hat{n}}(\vb{r}) + &= -\frac{i}{\hbar} \sum_{j} \bigg( + \matrixel{0}{\hat{n}_S}{j} \matrixel{j}{\hat{V}_S}{0} \int \Theta(-t') \exp\!(i \omega_{j0} t') \: g(t') \dd{t'} + \\ + &\qquad\qquad\:\, + - \matrixel{0}{\hat{V}_S}{j} \matrixel{j}{\hat{n}_S}{0} \int \Theta(-t') \exp\!(-i \omega_{j0} t') \: g(t') \dd{t'} \bigg) +\end{aligned}$$ + +These integrals are [Fourier transforms](/know/concept/fourier-transform/), +and are straightforward to evaluate. The first is: + +$$\begin{aligned} + \int_{-\infty}^\infty \Theta(-t') \exp\!(i \omega_{j0} t') \: g(t') \dd{t'} + &= \int_{-\infty}^0 \exp\!\big(\!-\! i (\omega - \omega_{j0}) t' + \eta t' \big) \dd{t'} + \\ + &= \bigg[ \frac{\exp\!\big(\!-\! i (\omega - \omega_{j0}) t' + \eta t' \big)}{- i (\omega - \omega_{j0}) + \eta} \bigg]_{-\infty}^0 + \\ + &= \frac{1}{- i (\omega - \omega_{j0}) + \eta} + = \frac{i}{\omega - \omega_{j0} + i \eta} +\end{aligned}$$ + +The other integral simply has the opposite sign in front of $\omega_{j0}$. +We thus arrive at: + +$$\begin{aligned} + \delta\expval{\hat{n}}(\vb{r}, \omega) + &= \frac{1}{\hbar} \sum_{j} \bigg( + \frac{\matrixel{0}{\hat{n}_S}{j} \matrixel{j}{\hat{V}_S}{0}}{\omega - \omega_{j0} + i \eta} + - \frac{\matrixel{0}{\hat{V}_S}{j} \matrixel{j}{\hat{n}_S}{0}}{\omega + \omega_{j0} + i \eta} \bigg) +\end{aligned}$$ + +Inserting the definition $\hat{V}_S = \int V(\vb{r}') \:\hat{n}(\vb{r}') \dd{\vb{r}'}$ +leads us to the following formula for $\delta\expval{\hat{n}}$, +which has the typical form of a linear response, +with response function $\chi$: + +$$\begin{aligned} + \boxed{ + \begin{gathered} + \delta{n}(\vb{r}, \omega) + = \int_{-\infty}^\infty \chi(\vb{r}, \vb{r'}, \omega) \: V(\vb{r}') \dd{\vb{r}'} + \qquad\quad \mathrm{where} + \\ + \chi(\vb{r}, \vb{r'}, \omega) + = \sum_{j} \bigg( \frac{\matrixel{0}{\hat{n}_S(\vb{r})}{j} \matrixel{j}{\hat{n}_S(\vb{r}')}{0}}{(\omega + i \eta) \hbar - E_j + E_0} + - \frac{\matrixel{0}{\hat{n}_S(\vb{r}')}{j} \matrixel{j}{\hat{n}_S(\vb{r})}{0}}{(\omega + i \eta) \hbar + E_j - E_0} \bigg) + \end{gathered} + } +\end{aligned}$$ + +By definition, $\ket{j}$ are eigenstates +of the many-electron Hamiltonian $\hat{H}_{0,S}$, +which is only solvable if we crudely neglect +any and all electron-electron interactions. +Therefore, to continue, we neglect those interactions. +According to tradition, we then rename $\chi$ to $\chi_0$. + +The well-known ground state of a non-interacting electron gas +is the Fermi sea $\ket{\mathrm{FS}}$, given below, +together with $\hat{n}_S$ in the language of the +[second quantization](/know/concept/second-quantization/): + +$$\begin{aligned} + \ket{\mathrm{FS}} + = \prod_\alpha \hat{c}_\alpha^\dagger \ket{0} + \qquad \quad + \hat{n}_S(\vb{r}) + = \hat{\Psi}{}^\dagger(\vb{r}) \hat{\Psi}(\vb{r}) + = \sum_{\alpha \beta} \psi_\alpha^*(\vb{r}) \psi_\beta(\vb{r})\: \hat{c}_\alpha^\dagger \hat{c}_\beta +\end{aligned}$$ + +For now, we ignore thermal excitations, +i.e. we set the temperature $T = 0$. +In $\chi_0 = \chi$, we thus insert the above $\hat{n}_S$, +and replace $\ket{0}$ with $\ket{\mathrm{FS}}$, yielding: + +$$\begin{aligned} + \matrixel{0}{\hat{n}_S}{j} + \quad\longrightarrow\quad \matrixel**{\mathrm{FS}}{\hat{\Psi}{}^\dagger \hat{\Psi}}{j} + = \sum_{\alpha \beta} \psi_\alpha^* \psi_\beta \matrixel**{\mathrm{FS}}{\hat{c}_\alpha^\dagger \hat{c}_\beta}{j} +\end{aligned}$$ + +This inner product is only nonzero if +$\ket{j} = \hat{c}_a \hat{c}_b^\dagger \ket{\mathrm{FS}}$ +with $a = \alpha$ and $b = \beta$, +or in other words, +only if $\ket{j}$ is a single-electron excitation of $\ket{\mathrm{FS}}$. +Furthermore, in $\ket{\mathrm{FS}}$, +$\alpha$ must be filled, and $\beta$ must be empty. +Let $f_\alpha \in \{0,1\}$ be the occupation number of orbital $\alpha$, then: + +$$\begin{aligned} + \matrixel{0}{\hat{n}_S}{j} + \longrightarrow \sum_{\alpha \beta} \psi_\alpha^* \psi_\beta \: f_\alpha (1 - f_\beta) \: \delta_{a \alpha} \delta_{b \beta} +\end{aligned}$$ + +In $\chi_0$, the sum over $j$ becomes a sum over $a$ and $b$ +(this implicitly eliminates all $\ket{j}$ that are not single-electron excitations), +and $E_j\!-\!E_0$ becomes the cost of the excitation $\epsilon_b \!-\! \epsilon_a$, +where $\epsilon_a$ is the energy of orbital $a$. +Therefore, we find: + +$$\begin{aligned} + \chi_0 + &= \sum_{a b} \bigg( \sum_{\alpha \beta \kappa \mu} f_\alpha (1 \!-\! f_\beta) f_\kappa (1 \!-\! f_\mu) + \frac{\psi_\alpha^*(\vb{r}) \psi_\beta(\vb{r}) \psi_\kappa(\vb{r}') \psi_\mu^*(\vb{r'})}{(\omega + i \eta) \hbar - \epsilon_b + \epsilon_a} + \delta_{a \alpha} \delta_{b \beta} \delta_{a \kappa} \delta_{b \mu} + \\ + &\qquad\:\:\: - \sum_{\alpha \beta \kappa \mu} f_\alpha (1 \!-\! f_\beta) f_\kappa (1 \!-\! f_\mu) + \frac{\psi_\alpha^*(\vb{r'}) \psi_\beta(\vb{r'}) \psi_\kappa(\vb{r}) \psi_\mu^*(\vb{r})}{(\omega + i \eta) \hbar + \epsilon_b - \epsilon_a} + \delta_{a \alpha} \delta_{b \beta} \delta_{a \kappa} \delta_{b \mu} \bigg) + \\ + &= \sum_{a b} \bigg( f_a^2 (1 \!-\! f_b)^2 + \frac{\psi_a^*(\vb{r}) \psi_b(\vb{r}) \psi_a(\vb{r}') \psi_b^*(\vb{r'})}{(\omega + i \eta) \hbar - \epsilon_b + \epsilon_a} + \\ + &\qquad\:\: - f_a^2 (1 \!-\! f_b)^2 + \frac{\psi_a^*(\vb{r'}) \psi_b(\vb{r'}) \psi_a(\vb{r}) \psi_b^*(\vb{r})}{(\omega + i \eta) \hbar + \epsilon_b - \epsilon_a} + \bigg) +\end{aligned}$$ + +Because $f_a, f_b \in \{0, 1\}$ when $T = 0$, we know that $f_a^2 (1 \!-\! f_b)^2 = f_a (1 \!-\! f_b)$. +We then swap the indices $a$ and $b$ in the second term, leading to: + +$$\begin{aligned} + \chi_0 + &= \sum_{a b} \Big( f_a (1 \!-\! f_b) - f_b (1 \!-\! f_a) \Big) + \frac{\psi_a^*(\vb{r}) \psi_b(\vb{r}) \psi_a(\vb{r}') \psi_b^*(\vb{r'})}{(\omega + i \eta) \hbar - \epsilon_b + \epsilon_a} + \\ + &= \sum_{a b} \Big( f_a - f_b \Big) + \frac{\psi_a^*(\vb{r}) \psi_b(\vb{r}) \psi_a(\vb{r}') \psi_b^*(\vb{r'})}{(\omega + i \eta) \hbar - \epsilon_b + \epsilon_a} +\end{aligned}$$ + +To proceed, we make the radical assumption that $\vb{H}_{0,S}$ +has continuous translational symmetry, +or in other words, that the external potential is uniform in space. +Clearly, this is not realistic, +so our conclusions will be more qualitative than quantitative. + +In that case, the wavefunction of a non-interacting particle is simply a plane wave, +so we insert $\psi_a(\vb{r}) = \exp\!(i \vb{k}_a \cdot \vb{r})$ +and $\psi_b(\vb{r}) = \exp\!(i \vb{k}_b \cdot \vb{r})$, yielding: + +$$\begin{aligned} + \chi_0 + &= \sum_{a b} \Big( f_a - f_b \Big) + \frac{\exp\!\big( \!-\! i \vb{k}_a \cdot \vb{r} + i \vb{k}_b \cdot \vb{r} + i \vb{k}_a \cdot \vb{r}' - i \vb{k}_b \cdot \vb{r}' \big)} + {(\omega + i \eta) \hbar - \epsilon_b + \epsilon_a} + \\ + &= \sum_{a b} \Big( f_a - f_b \Big) + \frac{\exp\!\big( \!-\! i (\vb{k}_a \!-\! \vb{k}_b) \cdot (\vb{r} \!-\! \vb{r}')\big)} + {(\omega + i \eta) \hbar - \epsilon_b + \epsilon_a} +\end{aligned}$$ + +Here, we see that $\chi_0$ only depends on the differences +$\vb{r}\!-\!\vb{r}'$ and $\vb{k}_a\!-\!\vb{k}_b$. +Therefore, we define $\vb{q}' \equiv \vb{k}_b\!-\!\vb{k}_a$ +and rename $\vb{k}_a \to \vb{k}$. +We thus have: + +$$\begin{aligned} + \chi_0(\vb{r}\!-\!\vb{r}') + &= \sum_{\vb{k} \vb{q}'} \Big( f_k - f_{k+q'} \Big) + \frac{\exp\!\big( i \vb{q'} \cdot (\vb{r} \!-\! \vb{r}')\big)} + {(\omega + i \eta) \hbar - \epsilon_{k+q'} + \epsilon_k} +\end{aligned}$$ + +The summation goes over all $\vb{k}$ and $\vb{q}'$ +where $\vb{k}$ is inside the Fermi sphere, and $\vb{k}\!+\!\vb{q}'$ is outside. +Let $k_F$ be the Fermi radius, +then we convert this sum into an integral, +which means introducing a factor of $1/(2 \pi)^{3}$ +as usual in solid state physics: + +$$\begin{aligned} + \chi_0(\vb{r}) + &= \sum_{\substack{|\vb{k}| < k_F \\ |\vb{k}+\vb{q}'| > k_F}} + \frac{(f_k - f_{k+q'}) \exp\!\big( i \vb{q}' \cdot \vb{r} \big)} + {(\omega + i \eta) \hbar - \epsilon_{k+q'} + \epsilon_k} + \\ + &= \frac{1}{(2 \pi)^3} \iint_{\substack{|\vb{k}| < k_F \\ |\vb{k}+\vb{q}'| > k_F}} + \frac{(f_k - f_{k+q'}) \exp\!\big( i \vb{q}' \cdot \vb{r} \big)} + {(\omega + i \eta) \hbar - \epsilon_{k+q'} + \epsilon_k} \dd{\vb{k}} \dd{\vb{q}'} +\end{aligned}$$ + +Fourier transforming the position $\vb{r}$ into the wavevector $\vb{q}$, +we recognize an integral that can be evaluated +to a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta(\vb{q})$: + +$$\begin{aligned} + \chi_0(\vb{q}) + &= \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \iint_{\substack{|\vb{k}| < k_F \\ |\vb{k}+\vb{q}'| > k_F}} + \frac{(f_k - f_{k+q'}) \exp\!\big( i (\vb{q}' \!-\! \vb{q}) \cdot \vb{r} \big)}{(\omega + i \eta) \hbar - \epsilon_{k+q'} + \epsilon_k} + \dd{\vb{k}} \dd{\vb{q}'} \dd{\vb{r}} + \\ + &= \iint_{\substack{|\vb{k}| < k_F \\ |\vb{k}+\vb{q}'| > k_F}} + \frac{(f_k - f_{k+q'}) \:\delta(\vb{q}'\!-\!\vb{q})}{(\omega + i \eta) \hbar - \epsilon_{k+q'} + \epsilon_k} \dd{\vb{k}} \dd{\vb{q}'} +\end{aligned}$$ + +This delta functions eliminates the integral over $\vb{q}'$, +giving the following linear response $\chi_0$ +of a non-interacting electron gas in a uniform potential: + +$$\begin{aligned} + \boxed{ + \chi_0(\vb{q}, \omega) + = \int_{\substack{|\vb{k}| < k_F \\ |\vb{k}+\vb{q}| > k_F}} + \frac{f_k - f_{k+q}}{(\omega + i \eta) \hbar - \epsilon_{k+q} + \epsilon_k} \dd{\vb{k}} + } +\end{aligned}$$ + +The resulting electron density change $\delta{\expval{\hat{n}}}$ is as follows, +where we use the [convolution theorem](/know/concept/convolution-theorem/) +to convert the convolution in $\vb{r}$-space into a product in $\vb{q}$-space: + +$$\begin{gathered} + \delta\expval{\hat{n}}(\vb{r}, \omega) + = \int_{-\infty}^\infty \chi_0(\vb{r}\!-\!\vb{r}', \omega) \: V(\vb{r}') \dd{r'} + \\ + \implies \qquad + \boxed{ + \delta\expval{\hat{n}}(\vb{q}, \omega) + = \chi_0(\vb{q}, \omega) \: V(\vb{q}) + } +\end{gathered}$$ + +So far, we have neglected electron-electron interactions, +but now we approximately correct this. +We split the effective potential $\vb{V}_\mathrm{eff}$ felt by the electrons +into the external potential $V_\mathrm{ext}$ +and the internal interactions $V_\mathrm{int}$, +such that: + +$$\begin{aligned} + V_\mathrm{eff}(\vb{r}) + = V_\mathrm{ext} + V_\mathrm{int} +\end{aligned}$$ + +We approximate $V_\mathrm{int}$ as follows, +where $V_{ee}$ represents electron-electron interactions: + +$$\begin{aligned} + V_\mathrm{int}(\vb{r}) + \approx \int_{-\infty}^\infty V_{ee}(\vb{r} \!-\! \vb{r}') \: \delta{n}(\vb{r}') \dd{\vb{r}'} + \qquad\quad + V_{ee}(\vb{r} \!-\! \vb{r}') + = \frac{e^2}{4 \pi \varepsilon_0} \frac{1}{|\vb{r} - \vb{r}'|} +\end{aligned}$$ + +Consequently, $V_\mathrm{int}$ satisfies Poisson's equation, +which has a well-known Fourier transform: + +$$\begin{aligned} + \nabla^2 V_\mathrm{int}(\vb{r}) + = - \frac{\delta{n}(\vb{r})}{\varepsilon_0} + \quad \implies \quad + V_\mathrm{int}(\vb{q}) + = \frac{e^2}{\varepsilon_0 |\vb{q}|^2} \delta{n}(\vb{q}) +\end{aligned}$$ + +Meanwhile, from all of the above calculations, +we can write $\delta{n}$ as follows, +where $\chi$ and $\chi_0$ are the +(unknown) interacting and (known) non-interacting response functions: + +$$\begin{aligned} + \delta{n}(\vb{q}) + = \chi V_\mathrm{ext} + \approx \chi_0 V_\mathrm{eff} +\end{aligned}$$ + +Keep in mind that we are treating interactions as a perturbation to $V_\mathrm{ext}$, +therefore $V_\mathrm{ext} \approx V_\mathrm{eff}$. +With this, $V_\mathrm{eff}$ becomes as follows in $\vb{q}$-space, +where we have used the convolution theorem +to get the product $\delta{n} (\vb{q}) V_{ee}(\vb{q})$: + +$$\begin{aligned} + V_\mathrm{eff}(\vb{q}) + = V_\mathrm{ext} + \chi_0 V_\mathrm{eff} V_{ee} + \qquad \quad + V_{ee}(\vb{q}) + = \frac{e^2}{\varepsilon_0 |\vb{q}|^2} +\end{aligned}$$ + +Isolating this equation for $V_\mathrm{ext}$ +yields the definition of the relative permittivity $\varepsilon_r$: + +$$\begin{aligned} + V_\mathrm{ext} + = (1 - \chi_0 V_{ee}) V_\mathrm{eff} + \equiv \varepsilon_r V_\mathrm{eff} +\end{aligned}$$ + +Therefore, by inserting all the above expressions, +we arrive at the following dielectric function $\varepsilon_r$ +for a non-interacting electron gas in a uniform potential: + +$$\begin{aligned} + \boxed{ + \varepsilon_r(\vb{q}, \omega) + = 1 - \frac{e^2}{\varepsilon_0 |\vb{q}|^2} \sum_{k} \frac{f_{k-q} - f_k}{\hbar (\omega + i \eta) + E_{k-q} - E_k} + } +\end{aligned}$$ + + + +## References +1. K.S. Thygesen, + *Advanced solid state physics: linear response theory*, + 2013, unpublished. +2. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. +3. G. Grosso, G.P. Parravicini, + *Solid state physics*, + 2nd edition, Elsevier. |