Tag Archives: 诺丁汉大学代写

半导体物理学 Semiconductor Physics PHYS4014

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半导体物理学 Semiconductor Physics PHYS4014

Here we see that a complex $\chi(\omega)$ results from a real $\chi\left(t-t^{\prime}\right)$. Inserting
$$
\frac{1}{\varepsilon_{0}} \boldsymbol{P}(\omega)=\int_{-\infty}^{+\infty} \mathrm{e}^{\mathrm{i} \omega t}\left[\int_{-\infty}^{t} \chi\left(t-t^{\prime}\right) \boldsymbol{E}\left(t^{\prime}\right) \mathrm{d} t^{\prime}\right] \mathrm{d} t .
$$
Introducing
$$
1=\mathrm{e}^{-\mathrm{i} \omega t^{\prime}} \mathrm{e}^{\mathrm{i} \omega t^{\prime}}
$$
in the inner integral and rearranging the terms gives
$$
\frac{1}{\varepsilon_{0}} \boldsymbol{P}(\omega)=\int \boldsymbol{E}\left(t^{\prime}\right) \mathrm{e}^{\mathrm{i} \omega t^{\prime}}\left[\int \chi\left(t-t^{\prime}\right) \mathrm{e}^{\mathrm{i} \omega\left(t-t^{\prime}\right)} \mathrm{d} t\right] \mathrm{d} t^{\prime}=\chi(\omega) \boldsymbol{E}(\omega) .
$$
This is identical
With the knowledge of (6.11) we can apply Cauchy’s theorem, which connects the real and imaginary parts of the Fourier transforms of analytic functions. This theorem leads us to
$$
\varepsilon_{1}(\omega)-1=\frac{1}{\pi} P \int_{-\infty}^{+\infty} \frac{\varepsilon_{2}\left(\omega^{\prime}\right)}{\omega^{\prime}-\omega} \mathrm{d} \omega^{\prime}
$$
and
$$
\varepsilon_{2}(\omega)=-\frac{1}{\pi} \int_{-\infty}^{+\infty} \frac{\varepsilon_{1}\left(\omega^{\prime}\right)-1}{\omega^{\prime}-\omega} d \omega^{\prime}
$$

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PHYS4014 COURSE NOTES ：

and cyclic permutations of the indices $V_{u c}$ is the volume of the unit cell given by
$$
V_{\mathrm{uc}}=\boldsymbol{a}{1}\left(\boldsymbol{a}{2} \times \boldsymbol{a}{3}\right) . $$ A general translation vector of the reciprocal lattice is usually called $\boldsymbol{G}$ $$ \boldsymbol{G}=l{1} \boldsymbol{b}{1}+l{2} \boldsymbol{b}{2}+l{3} \boldsymbol{b}{3} \quad l{i}=0, \pm 1, \pm 2, \ldots \quad i=1,2,3 .
$$
Without trying to be complete, we give some properties of the reciprocal lattice and its connections with the real one.

Every periodic function in real space which is sufficiently smooth and has a periodicity given by $f(\boldsymbol{r}+\boldsymbol{R})=f(\boldsymbol{r})$ and $\boldsymbol{R}$ defined by (7.6) can be written as a Fourier series summing over all vectors of the reciprocal lattice
$$
f(r)=\sum_{G} f_{G} \mathrm{e}^{\mathrm{i} G r}
$$
with
$$
f_{G}=V_{u c}^{-1} \int_{u c} f(\boldsymbol{r}) \mathrm{e}^{-\mathrm{i} \boldsymbol{G r}} \mathrm{d} \tau .
$$
The scalar product of $R$ and $G$ always fulfills
$$
\boldsymbol{R} \cdot \boldsymbol{G}=2 \pi m ; \quad m=0, \pm 1, \pm 2, \ldots
$$









基本粒子物理学 Elementary Particle Physics PHYS4013

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基本粒子物理学 Elementary Particle Physics PHYS4013

$$
N_{\nu}^{\bar{\mu}}=\frac{2}{\nu}-\gamma+\ln 4 \pi+\ln \frac{\bar{\mu}^{2}}{\kappa^{2}} .
$$
Apparently $Z_{3}(\nu)$ possesses the pole divergence on the dimension $\nu=4-d$. It corresponds to the logarithmic divergence on the momentum $L$
$$
N_{\nu}^{\bar{\mu}} \rightarrow \ln \left(\frac{L^{2}}{p^{2}}\right)
$$
resulting in the momentum cut-off in the integral over $q$. The renormalization constant choice in the form $\kappa^{2}=Q^{2}$ ensures the natural result
$$
\left.\Gamma_{\mu \nu}^{(2,0) R}(p)\right|{p^{2}=\kappa^{2}}=i\left[\frac{-g{\mu \nu}+p_{\mu} p_{\nu} / p^{2}}{p^{2}}\right]
$$
It is clear that the divergent part associated with the gluon self-energy may be removed from the theory by means of introducing the following counter-term into the initial Lagrangian:
$$
\delta \mathcal{L}{G{a}}=\left(Z_{3}-1\right)\left[\frac{1}{2}\left(\partial^{\mu} G_{a}^{\nu}-\partial^{\nu} G_{a}^{\mu}\right)^{2}\right]
$$
where in the case of an arbitrary gange $Z_{3}$ is given as
$$
Z_{3}=1+\frac{g_{s}^{2}}{16 \pi^{2}}\left[\left(\frac{13}{6}-\frac{\xi}{2}\right) c_{V}-\frac{2 N_{f}}{3}\right] N_{\nu}^{\bar{\mu}}
$$
and the argument $\nu$ is omitted. There is no need to introduce the counter-terms of the kind $\left(G_{a}^{\mu}\right)^{2}$ and $\left(\partial_{\mu} G_{a}^{\mu}\right)^{2}$ into the Lagrangian because of transversality of the gluon field polarization tensor.

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PHYS4013 COURSE NOTES ：

The Feynman diagram describing the ghost self-energy.
$$
G^{(g h)}(p)=G_{0}^{(g h)}(p)\left[1+\frac{g_{s}^{2}}{16 \pi^{2}} c_{V}\left(\frac{3}{4}-\frac{\xi}{4}\right)\left(\frac{2}{\nu}-\gamma-\ln \frac{p^{2}}{4 \pi \bar{\mu}^{2}}\right)\right]
$$
where $G_{0}^{(g h)}(p)$ is the free ghost propagator. Singling out the divergent factor in Eq. (3.125), we obtain
$$
G^{(g h)}(p)=\tilde{Z}{3} G^{(g h) R}(p), $$ where $$ \tilde{Z}{3}=1+\frac{g_{s}^{2}}{16 \pi^{2}} c_{V}\left(\frac{3}{4}-\frac{\xi}{4}\right) N_{\nu}^{\bar{\mu}}
$$
and the expression for the renormalized propagator is written in the form
$$
G^{(g h) R}(p)=G_{0}^{(g h)}(p)\left[1-\frac{g_{s}^{2}}{16 \pi^{2}} c_{V}\left(\frac{3}{4}-\frac{\xi}{4}\right) \ln \frac{p^{2}}{\kappa^{2}}\right]
$$
The Feynman rules are such that the momentum of a coming ghost line may be always factorized. As a consequence, the divergences connected with the ghost self-energy do not demand introducing any mass counter-term and we simply have
$$
\delta \mathcal{L}{\omega^{\dagger} \omega}=\left(\tilde{Z}{3}-1\right)\left[-\omega_{a}^{\dagger}(x) \partial^{2} \omega_{a}(x)\right] .
$$









极端天体物理学 Extreme Astrophysics PHYS4009

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极端天体物理学 Extreme Astrophysics PHYS4009

In wavelength units, the flux emitted by a blackbody of temperature $T$ is given by the Planck function:
$$
B_{\lambda}(T)=2 h c^{2} \lambda^{-5}\left(e^{\frac{\text { he }}{\Lambda x}}-1\right)^{-1}
$$
If the blackbody is a stellar surface then the Eddington flux or spectral flux emitted per unit area per unit time per unit wavelength per unit solid angle is
$$
H(\lambda){\mathrm{BB}}=\frac{1}{4} B{\lambda}(T)
$$
In units of $\lambda_{100}=100 \AA$ and $T_{100}=100000 \mathrm{~K}$ the above relation becomes
$$
H(\lambda){\mathrm{BB}}=2.98 \times 10^{26} \lambda{100}^{-5}\left(e^{\frac{14.4}{2_{1000} 100}}-1\right)^{-1}
$$
where the Eddington flux is given in units of ergs $\mathrm{cm}^{2} \mathrm{~s}^{-1} \AA^{-1} \mathrm{sr}^{-1}$. The Eddington flux is introduced because it is a frequently used quantity which is the product of model stellar atmosphere calculations. It is related to the observed flux at the Earth by
$$
f(\lambda)=4 \pi\left(\frac{R^{2}}{D^{2}}\right) H(\lambda)
$$
where $R$ is the stellar radius and $D$ the distance to the star. It is also related to the effective temperature $\left(T_{\text {eff }}\right)$ through Stefan’s law,
$$
4 \pi \int_{0}^{\infty} H(\lambda) d \lambda=\sigma T_{\mathrm{eff}}^{4}
$$
where $\sigma$ is the Stefan-Boltzmann constant.

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PHYS4009 COURSE NOTES ：

$$
Q(T) \approx\left(n_{\mathrm{e}}^{2} / T\right)(d s / d \ln T)=A\left(T / T_{\max }\right)^{\alpha}
$$
where $T_{\max }$ is the upper limit of the range over which the distribution is non-zero and $A$ is a normalisation constant derived from the data. The power law index $(\alpha)$ is related to the radiative cooling function and has a value $\approx 1$ has considered the response of the EUVE scanner and deep survey telescope to radiative loss models for coronal plasma, providing tables of the unit count rate flux calibration as a function of temperature and interstellar column density.









理论工具箱 Theory Toolbox PHYS3015

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理论工具箱 Theory Toolbox PHYS3015

Let $f: D \subseteq \mathbb{R} \rightarrow \mathbb{R}$ be a continuously differentiable function, and let $[x] \in I R R$ be an interval with $[x] \subseteq D$. Then
$$
N([x]):=m([x])-\frac{f(m([x]))}{f^{\prime}([x])}
$$
has the following properties:

  1. Every zero $x^{} \in[x]$ of $f$ satisfies $x^{} \in N([x])$.
  2. If $N([x]) \cap[x]=0$, then there exists no zero of $f$ in $[x]$.
  3. If $N([x]) \stackrel{\circ}{\subset}[x]$, then there exists a unique zero of $f$ in $[x]$ and hence in $N([x])$.Remark: The conditions of Theorem $6.1$ can be checked on a computer. For example, if $N_{\odot}([x])$ denotes the evaluation of $N([x])$ in floating-interval arithmetic and if condition 3 is satisfied for $N_{\diamond}$, then we have
    $$
    N([x]) \subseteq N_{\circ}([x]) \stackrel{\circ}{\subset}[x]
    $$
    Thus, the condition is fulfilled for $N([x])$, too. On the other hand, if we cannot fulfill conditions 2 and 3 of the Theorem, then $[x]$ may contain one or more zeros.
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PHYS3015 COURSE NOTES ：

In our global optimization method, we apply one step of the extended interval Newton’s method to the problem
$$
f^{\prime}(y)=0, \quad y \in[y]
$$
Let
$$
N^{\prime}([y]):=m([y])-\frac{f^{\prime}(m([y]))}{f^{\prime \prime}([y])}
$$
and compute
$$
\left[y_{N}\right]:=[y] \cap N^{\prime}([y])
$$









物理学中的对称性与作用 Symmetry & Action in Physics PHYS3013

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物理学中的对称性与作用 Symmetry & Action in Physics PHYS3013

The change of variable
$$
y_{j}^{\prime}=x_{i}^{\prime} B_{j}^{i}
$$
(shorthand $y^{\prime}=x^{\prime} B$ ) defined by transposition
$$
\left\langle y^{\prime}, y\right\rangle=\left\langle x^{\prime}, x\right\rangle, \quad \text { i.e. } y_{j}^{\prime} y^{j}=x_{i}^{\prime} x^{i},
$$
implies
$$
B_{j}^{i} A_{k}^{j}=\delta_{k}^{i}
$$
now reads
$$
\int_{\mathbb{R}^{D}} D_{a, Q} x \exp \left(-\frac{\pi}{a} Q(x)-2 \pi \mathrm{i}\left\langle x^{\prime}, x\right\rangle\right):=\exp \left(-a \pi W\left(x^{\prime}\right)\right)
$$
where
$$
\begin{aligned}
W\left(x^{\prime}\right) &=\delta^{i j} x_{k}^{\prime} x_{\ell}^{\prime} B_{i}^{k} B_{j}^{\ell} \
&=: x_{k}^{\prime} x_{\ell}^{\prime} W^{k \ell}
\end{aligned}
$$
The quadratic form $W\left(x^{\prime}\right)=x_{k}^{\prime} x_{l}^{\prime} W^{k l}$ on $\mathbb{R}{D}$ can be said to be the inverse of $Q(x)=Q{k l} x^{k} x^{l}$ on $\mathbb{R}^{D}$ since the matrices $\left(W^{k l}\right)$ and $\left(Q_{k l}\right)$ are inverse 泊रi吕 to each other.

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PHYS3013 COURSE NOTES ：

  • They are the inverses of each other:
    $$
    D G=1 \quad\left(\text { on } \mathbb{X}^{\prime}\right), \quad G D=\mathbf{1}(\text { on } \mathrm{X}) .
    $$
  • They are symmetric:
    $$
    \begin{array}{ll}
    \langle D x, y\rangle=\langle D y, x\rangle & \text { for } x, y \text { in } \mathrm{X} \
    \left\langle x^{\prime}, G y^{\prime}\right\rangle=\left\langle y^{\prime}, G x^{\prime}\right\rangle & \text { for } x^{\prime}, y^{\prime} \text { in } \mathrm{X}^{\prime}
    \end{array}
    $$
  • The quadratic forms are given by
    $$
    Q(x)=\langle D x, x\rangle, \quad W\left(x^{\prime}\right)=\left\langle x^{\prime}, G x^{\prime}\right\rangle .
    $$
    We set also $W\left(x^{\prime}, y^{\prime}\right):=\left\langle x^{\prime}, G y^{\prime}\right\rangle$ for $x^{\prime}, y^{\prime}$ in $X^{\prime}$.









银河系的架构 The Structure of Galaxies PHYS3011

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银河系的架构 The Structure of Galaxies PHYS3011

Noting that $R_{0} \cos l=R \sin \alpha+d$, we have
$$
V_{\mathrm{t}}=R_{0} \cos l\left(\frac{V}{R}-\frac{V_{0}}{R_{0}}\right)-V \frac{d}{R} .
$$
Close to the Sun, we can substitute $R_{0}-R \approx d \cos l$, to show that $V_{\mathrm{t}}$ varies almost linearly with the distance $d$ :
$$
V_{\mathrm{t}} \approx d \cos (2 l)\left[-\frac{R}{2}\left(\frac{V}{R}\right)^{\prime}\right]{R{0}}-\frac{d}{2}\left[\frac{1}{R}(R V)^{\prime}\right]{R{0}} \equiv d[A \cos (2 l)+B]
$$
where the constant $B=-12.4 \pm 0.6 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{kpc}^{-1}$. In Section 3.3, we will see another method to estimate $B$. The Oort constants $A$ and $B$ measure respectively the local shear, or deviation from rigid rotation, and the local vorticity, or angularmomentum gradient in the disk.

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PHYS3011 COURSE NOTES ：

$$
\Phi(\mathbf{x})=-\int \frac{G \rho\left(\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} \mathrm{d}^{3} \mathbf{x}^{\prime}
$$
and the force $\mathbf{F}$ per unit mass is
$$
\mathbf{F}(\mathbf{x})=-\nabla \Phi(\mathbf{x})=-\int \frac{G \rho\left(\mathbf{x}^{\prime}\right)\left(\mathbf{x}-\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{3}} \mathrm{~d}^{3} \mathbf{x}^{\prime}
$$
The integral relation can be turned into a differential equation. Applying $\nabla^{2}$ to both sides, we have
$$
\nabla^{2} \Phi(\mathbf{x})=-\int G \rho\left(\mathbf{x}^{\prime}\right) \nabla^{2}\left(\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right) \mathrm{d}^{3} \mathbf{x}^{\prime}
$$
In three dimensions, differentiating with respect to the variable $\mathbf{x}$ gives, for $\mathbf{x} \neq \mathbf{x}^{\prime}$ (check by trying it in Cartesian coordinates),
$$
\nabla\left(\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)=-\frac{\mathbf{x}-\mathbf{x}^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{3}}, \quad \text { and } \nabla^{2}\left(\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)=0
$$









固体物理学入门 Intro to Solid State Physics PHYS3002

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固体物理学入门 Intro to Solid State Physics PHYS3002

$$
i \hbar \frac{\partial}{\partial t} \int d^{3} r u_{k}^{}(\mathbf{r}) \Psi(\mathbf{r}, t)=\int d^{3} r u_{k}^{}(\mathbf{r})\left[H^{\prime} \Psi^{\prime}(\mathbf{r}, t)\right]
$$
on the right,
$$
\begin{aligned}
i \hbar \frac{\partial}{\partial t} \psi_{k^{k}}(t) &=\int d^{3} r u_{\mathbf{k}^{}}(\mathbf{r}) H^{\prime} \sum_{\mathbf{k}^{\prime}} \psi_{\mathbf{k}^{\prime}}(t) u_{k^{\prime}} \ &=\sum_{k^{k}} \psi_{k^{\prime}}(t)\left[\int d^{3} r u_{k}^{}(\mathbf{r}) H^{\prime} u_{\mathbf{k}^{\prime}}(\mathbf{r})\right]
\end{aligned}
$$ can be rewritten as
$$
i \hbar \frac{d}{d t} \psi_{k}(t)=\sum_{k^{\prime}} H_{k_{k}}^{\prime}(t) \psi_{k}(t)
$$
where the matrix elements $H_{\mathrm{k}, \mathrm{k}}^{\prime}$ are defined by
$$
H_{k, k^{\prime}}^{\prime}(t)=\int d^{3} r u_{k}^{*}(\mathbf{r}) H^{\prime} u_{k^{\prime}}(\mathbf{r})
$$

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PHYS3002 COURSE NOTES ：

$$
\mathbf{V} \cdot \mathbf{V}=\sum_{1} V_{i}^{*} V_{i}
$$
There is an analogous result for state vectors as well.
$$
\begin{aligned}
\int d^{3} r \Psi * \Psi &=\int d^{3} r\left[\sum_{r} \psi_{i} u_{i}(\mathbf{r})\right]^{} \sum_{i} \psi_{j} u_{j}(\mathbf{r}) \ &=\sum_{i} \psi_{i}^{} \psi_{j} \int d^{3} r u_{i}^{}(\mathbf{r}){j}(\mathbf{r}) \end{aligned} $$ Using eq. $(2,16)$, $$ \int d^{3} r \psi r \psi \psi=\sum{1} \psi_{i}^{} \psi_{i}
$$









原子、光子和基金粒子 Atoms, Photons & Fund Particle PHYS3001

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原子、光子和基金粒子 Atoms, Photons & Fund Particle PHYS3001

$$
K_{0}\left(t, \boldsymbol{r}-\boldsymbol{r}^{\prime}\right)=\left(\frac{m}{2 \pi \mathrm{i} t}\right)^{3 / 2} \mathrm{e}^{\mathrm{im} r^{2} /(2 t)}\left[\mathrm{e}^{-\mathrm{i} m \boldsymbol{r} \cdot \boldsymbol{r}^{\prime} / t}+I\left(t, \boldsymbol{r}, \boldsymbol{r}^{\prime}\right)\right]
$$
where
$$
I\left(t, \boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=\mathrm{e}^{-\mathrm{i} m \boldsymbol{r} \cdot \boldsymbol{r}^{\prime} / t}\left[\mathrm{e}^{\mathrm{i} m r^{2} /(2 t)}-1\right] .
$$
Substituting $(9.22 \mathrm{a})$ into $(9.21 \mathrm{a})$, we find that
$$
\chi(t, \boldsymbol{r})=\left(\frac{m}{\mathrm{i} t}\right)^{3 / 2} \mathrm{e}^{\mathrm{i} m r^{2} /(2 t)}[\tilde{\chi}(m \boldsymbol{r} / t)+R(t, \boldsymbol{r})]
$$
where $\tilde{\chi}(m \boldsymbol{r} / t)$ is the Fourier transform of the wave packet $\chi\left(\boldsymbol{r}^{r}\right)$ from (9.21b), i.e.
$$
\tilde{\chi}(m \boldsymbol{r} / t)=(2 \pi)^{-3 / 2} \int \mathrm{d} \boldsymbol{r}^{\prime} \mathrm{e}^{-\mathrm{i} m \boldsymbol{r} \cdot \boldsymbol{r}^{\prime} / t} \chi\left(\boldsymbol{r}^{\prime}\right)
$$
Here, the so-called residual function $R(t, \boldsymbol{r})$ has the form
$$
R(t, \boldsymbol{r})=(2 \pi)^{-3 / 2} \int \mathrm{d} r^{\prime} \mathrm{e}^{-\mathrm{i} m \boldsymbol{r} \cdot \boldsymbol{r}^{\prime} / t}\left[\mathrm{e}^{\mathrm{i} m r^{2} /(2 t)}-1\right] \times\left(\boldsymbol{r}^{\prime}\right)
$$

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PHYS3001 COURSE NOTES ：

$$
\left|\left\langle\boldsymbol{r} \mid U_{0}(t) \psi_{0}\right\rangle\right| \leq \frac{C}{t^{3 / 2}} \quad(t>0)
$$
where $C$ is a positive constant. This enables us to write
$$
\left|V U_{0}(t) \psi_{0}\right| \leq \frac{C}{t^{3 / 2}}|V|
$$
which implies that
$$
\begin{gathered}
I\left(t_{0}, \psi_{0}\right)=\int_{t_{0}}^{\infty} \mathrm{d} t|\xi(t)| \leq C|V| \int_{t_{0}}^{\infty} \mathrm{d} t t^{-3 / 2}=2 \frac{C|V|}{t_{0}^{1 / 2}}=\frac{C_{0}|V|}{t_{0}^{1 / 2}} \
I\left(t_{0}, \psi_{0}\right)<\infty \quad\left(t_{0}>0\right)
\end{gathered}
$$
Hence, in the case of short-range potentials $(|V|<\infty)$, the integral $I\left(t_{0}, \psi_{0}\right)$ exists for a certain $t_{0}>0$. Then, the relations $(9.14 \mathrm{~b}, \mathrm{c})$ and $(9.27)$ give
$$
\left|\Omega\left(t_{0}\right) \psi_{0}-\Omega(\infty) \psi_{0}\right| \leq I\left(t_{0}, \psi_{0}\right) \leq \frac{C_{0}|V|}{t_{0}^{1 / 2}}<\infty
$$









傅里叶方法 Fourier Methods PHYS2008

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光学和电磁学 Optics and Electromagnetism PHYS2007

Suppose that $f: \mathbb{T} \rightarrow \mathbb{C}$ is continuous.
(i) $\sum_{n=-\infty}^{\infty}|\hat{f}(n)|^{2}$ converges and
$$
\sum_{n=-\infty}^{\infty}|\hat{f}(n)|^{2}=\frac{1}{2 \pi} \int_{\mathbb{T}}|f(t)|^{2} d t
$$
(ii) The expression
$$
\frac{1}{2 \pi} \int_{\mathbb{T}}\left|f(t)-\sum_{n=-N}^{N} a_{n} e^{i n t}\right|^{2} d t
$$
has a unique minimum (over choices of $a_{n}$ ) when $a_{n}=\hat{f}(n)$.
(iii) We have
$$
\frac{1}{2 \pi} \int_{\mathbb{T}}\left|f(t)-\sum_{n=-N}^{N} \hat{f}_{n} e^{i n t}\right|^{2} d t \rightarrow 0
$$
as $N \rightarrow \infty$.

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PHYS2008 COURSE NOTES ：

The following statements are equivalent.
(i) If $\alpha$ is an irrational number and $0 \leq a \leq b \leq 1$, then
$$
\frac{\operatorname{card}{1 \leq n \leq N \mid\langle n \alpha\rangle \in[a, b]}}{N} \rightarrow b-a
$$
as $N \rightarrow \infty$.
(ii) If $\alpha$ is an irrational number and $0 \leq a \leq b \leq 2 \pi$, then
$$
\frac{\operatorname{card}{1 \leq n \leq N \mid 2 \pi n \alpha \in[a, b]}}{N} \rightarrow \frac{b-a}{2 \pi}
$$
as $N \rightarrow \infty$.
(iii) If $\alpha$ is an irrational number and $f: \mathbb{T} \rightarrow \mathbb{C}$ is continuous then
$$
\sum_{n=0}^{N} f(2 \pi n \alpha) \rightarrow \frac{1}{2 \pi} \int_{\mathbb{T}} f(x) d x
$$
as $N \rightarrow \infty$.









光学和电磁学 Optics and Electromagnetism PHYS2007

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光学和电磁学 Optics and Electromagnetism PHYS2007

previously presented for phase noise case except that the interference is downconverted by spurious signal, which is considered as a single tone. With the SNR of the baseband signal being,
$$
S N R=\left(P_{S i g}+P_{L O}\right)-\left(P_{I n t}+P_{S_{p}}\right)>S N R_{\min }
$$
After rearrangement,
$$
P_{S p}-P_{L O}<P_{S i g}-P_{I n t}-S N R_{\min }
$$
where $P_{S_{p}}-P_{L O}$ denotes the power of spurious signal in $\mathrm{dBc}$, relative to the carrier power. For example, Bluetooth standard specifies an interferer of $+30 \mathrm{~dB}$ at $2 \mathrm{MHz}$ away from the desired signal. The reference spur can be also at $2 \mathrm{MHz}$ away from the carrier if the frequency of the reference signal is $2 \mathrm{MHz}$. The minimum SNR requirement is $18 \mathrm{~dB}$, same as the previous example. Substituting the numbers, the spurious signal requirement results in $-48 \mathrm{dBc}$ at $2 \mathrm{MHz}$ from carrier.

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PHYS2007 COURSE NOTES ：

The stability factor for an amplifier is given as:
$$
K=\frac{1+|\Delta|^{2}-\left|S_{11}\right|^{2}-\left|S_{22}\right|^{2}}{2\left|S_{21}\right|\left|S_{12}\right|}
$$
where
$$
|\Delta|=\left|S_{11} S_{22}-S_{12} S_{21}\right|
$$
For unconditional stability, $K>1$ and $\Delta<1$. Thus, higher reverse isolation $\left(S_{l 2}\right)$ improves the stability of an amplifier. In a common-gate LNA, since the gate of the MOSFET is conventionally connected to an AC ground, there is no Miller effect associated with the feed-forward capacitor $C_{g d}$. This improves the reverse isolation and hence, the stability.