# # Atomic Models

With atomic models, we have lasers. --By Professer.

## # The discovery of electron

In the early 1900s, J. J. Thompson 第一個發現了電子，他利用cathode ray tube， 在Cathode Ray的路徑上家電場和磁場，射線的路徑會發生偏移，說明帶了負電荷。用電荷計算出的粒子的質量比氫原子小很多，說明是一種比氫原子下很多的粒子。嘗試了不同的陰極材料，發現得到的結果都是相同的。JJ發現了一種新的粒子。他叫它Electron。他得到了諾貝爾獎獎，他的兒子也得到了諾貝爾獎。卡文迪許研究所主持了30多年，產生了7個諾貝爾獎。

## # The discovery of atom

Gold was used because it is a soft metal that can be beaten very thin. 金的延展性特別好，很軟，可以打到很薄一層，做成金薄膜, 原子層數不多。 然後用Alpha粒子去打這個薄膜。然後我們去看穿過薄膜的Alpha粒子的數量，分佈。螢光幕會留下粒子的痕跡，通過環形的多個螢光幕的擋板，就可以測出散射的角度，大部分走直線。而有些1/800的粒子甚至會彈回來。這個用西瓜模型不能夠解釋。

Alpha 射線的得到

Alpha particle consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha deday, but may also be produced in other ways.[3] Alpha decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle (helium nucleus). [4] Alpha射線的得到是通過反射性元素例如U235,的衰變得到Alpha粒子。就是He的原子核。

## # The spectrum of hydrogen

$\frac{1}{\lambda} = R (1/n_k^2 - 1/n_j^2)$

where $n_k = 2, n_j = 3, 4, 5$.

For hydrogen:

• If $n_k = 2$, then $n_j = 3, 4, 5$, frequency lines in the visible range. Balmer series;
• If $n_k = 1$, then $n_j = 2, 3, 4$, frequency lines in the UV (Ultraviolet) range. Lyman series;
• If $n_k = 3$, then $n_j = 4, 5, 6$, frequency lines in the infrared. Paschen series.

String Theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity.[14]

Coulomb's law

$F = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} = k_\text{e} \frac{q_1 q_2}{r^2}$

The constant $k_\text{e} = \frac{1}{4\pi\varepsilon_0} = 9\times10^9 \text{N}\cdot\text{m}^2\cdot\text{C}^{-2}$ is called Coulomb's constant. Where $\varepsilon_0$ is $8.85418782 \times 10^{-12}$ $\text{F}\cdot\text{m}^{-2}$ is the electric constant. The symbol C stands fo Coulomb and is the unit of charge, and F stands for Farad is the unit for electrical capacitance. It is important to note at this time that 1 C is not the charge of an electron. The charge of the electron in coulomb is $e = -1.6 \times 10^{-19}$ C. [6]

Electric potential energy

The electrostatic potential energy, $U_E$ of one point charge $q$ at position $r$ in the presence of another point charge $Q$, taking an infinite separation between the charges as the reference position, is:

$U_E(r) = k_e \frac{qQ}{r}$

where $k_e$ is Coulomb's constant. [7]

## # The Bohr atom

Bohr's Postulates

1. Electron can take only certain orbits with angular momentum

$L = mvr = n \hbar, n=1, 2, 3, ...; \hbar = h/2\pi$

1. Electron in the allowed orbits does not radiate electromagenetic radiation. No reason.
2. If an electron is initially in an allowed orbit of energy $E_i$ and goes into another orbit of lower energy $E_f$, electromagnetic radiation will be emitted with a precise frequency given by

$\nu = \frac{E_i - E_f} {h}$

## # Hydrogen spectrum explanation

Centripetal acceleration

For a given speed $v$, the magnitude of the centripetal acceleration is inversely proportional to the radius $r$ of the circle and increases as the square of this speed[8]:

$a_c = \frac{v^2}{r}$

Using Bohr Atom we can try to explain the spectrum of Hydrengen. Let's us consider the simplest of atoms, the hydrogen atom. An electron of charge $q=-e$ rotates in a circular orbit of radius $r$ under the electrostatic attraction of the nucleus ($q = +e$). For simplicity we will assume that the nucleus remains stationary.

From the circular motion of the electron. $F_\text{radial} = ma_\text{radial}$

$k_e \frac{e^2}{r^2} = m \frac{v^2}{r}$

Fron the first postulate, we have

$v = n \frac{\hbar}{mr}$

Then we have

$k_e \frac{e^2}{r^2} = \frac{m}{r} n^2 \frac{\hbar^2}{m^2r^2}$

We obtain

$r = n^2 \frac{\hbar^2}{e^2mk_e}$

Let $r_0 = \frac{\hbar^2}{m^2r^2}$, we have

$r_n = n^2 r_0$

We see that only orbits with radius $r_0, 4r_0, 9r_0, ...$, are allowed, indicating the quantization of orbit.

Next, we need to explain the energy are discretized. So that we can explain the spectrum of hydrogen. The total energy of the atom.

$E_\text{TOTAL} = E_\text{kinetic} + E_\text{potential} = \frac{1}{2} m v^2 - k_e \frac{e^2}{r}$

Multiplying both sides of previous Newton's formula by $(1/2)r$,

$\frac{1}{2} k_e \frac{e^2}{r} = \frac{1}{2} m v^2$

Then, we get

$E_\text{TOTAL} = -\frac{1}{2} k_e \frac{e^2}{r} = -\frac{1}{2} \frac{1}{4\pi\varepsilon_0} \frac{e^2}{r}$

Replace $r$ with $r_n=n^2 r_0$

$E_n = -\frac{1}{n^2} \frac{e^4m}{8\varepsilon_0^2h^2} \triangleq -\frac{1}{n^2}E_0$

This equantion indicates the quantization of energy. $E_n$ is negative says the energy is 束縛能，which is used to 束縛 electron. The energy is smaller when $n$ increases. When $n=\infty$, this energy equals to zero, the electron leaves the atom. Now I know why the values of atomic energy levels are all negative.

We call the electron in Ground State when $n=1$, and sometimes the electron in ground state can be transfered to some other orbit radius. In this case we say the elctron is exited, and it is in the Excited State, and this process is calle Excitation. All physical systems tend to seek the lowest energy sate available to them. Therefore the electron in the excited state would eventually fall to the lower energy states. And this process is called De-excitation, in which lights are emitted.

How beautiful!

It seems that I can feel the excitment when Bohr did this job.

Use the third postulate, we will know the reason behind the spectrum of hydrogen. Beautiful.

Can you give your postulates to explain this?

Can you?

What is angular momentum?

Angular momentum is compared to momentum. It is a equivalent property can be associated with a rotating body.

## # X-ray explaination

Bohr Atom model can also be used to explain why there are some spikes in the X-ray spectrum. The electrons of atoms on the target is exited by the the electron stream. These electrons will emit light with they fall back to the lower energy levels. The spikes are caused by de-exitation.

## # The Franck-Hertz Experiment

By postulating the existence of a dicrete spectrum of energy levels through the angular momentum condition, Bohr was able to predict the correct electromagenetic spectrum for hydrogen. The existence of discrete energy levels in atoms was demonstrated directly by James Franck and Gustav Hertz in 1914.

The essential part of the equipment consists of a tube containing vapor of Mercury(Hg). The tube contains three electrodes: a filament that provides electrons when heated, a grid, and a plate. A grid is a charge screen, and most of is open space. The electrons will reach the grid (in the absence of collisions) with a kinetic energy $E_k = eV_0$. After reaching the grid the majority of these electrons will go through the holes in the grid, be collected by the plate, and contribute to the plate current. A small, constant retarding voltage $V_r$ (1V) is applied between the plate and grid.

Mercury(element)

Mercury is a chemical elemen with the symbol Hg and atomic number 80. Mercury is the only metallic element that is known to be liquid at standard conditions for temperature and pressure. [16]

Why the drops happend at 4.9, 2x4.9, 3x4.9?

Can this be explained by standing wave?

## # Laser

Laser is Light Amplification by Stimulated Emission of Radiation.

Electron absorb energy (through photon or electron collision)

• exited state
• decay to lower energy state through Spontaneous emission or Stimulated emission.

Sponteneous emission: in all directions

Stimulated emission: an incoming photon of the correct energy ($\text{E}_i$ - $\text{E}_f$) cause 'resonance' and induce electron drop from $\text{E}_i$ to $\text{E}_f$.

There are three features of stimulated emission:

1. One photon in, two photons out, causing light amplification
2. Emitted photon in the same direction as the incoming photon
3. Emitted photon in step (in phase, coherent) with the incomimg photon.

The two conditions to create laser.

1. Population inversion. This is process to move the electons at the lower level into high level. Normally the majority of electrons are in the resting state. Polulation inversion revert the electrons distribution.

2. Metestable state. A higher energy state in which electrons can stay for a much longer time than in an ordinary excited state. ($10^{-3}$ sec vs $10^{-8 }$ sec for example) Think about the Erxianqiao Unicle.

The development of laser: Solid laser, gas laser，simiconductor laser. The laser pen is a type of simiconductor laser.

DVD, CD is the application of laser.

What is the difference between Blu-ray Disc, DVD & CD?

Just as DVD meant a 5 to 10 times increase in storage capacity compared to CD, Blu-ray Disc will increase DVD capacity by 5 to 10 times. This is due to the usage of a blue(405nm) instead of a red(650nm) laser and improved lens specifications , allowing for a much smaller focus laser beam which enables the recording of much smaller and higher density pits on the disc.[12]

Resonance

The stimulated emission is like the resonance. Therefore the keyword of laser is resonance.

TIP