# Atomic Models

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

按照愛因斯坦狹義相對論

光子的動量,也就是說光有質量。很難理解。

# The discovery of electron

Cathode ray tubes are glass cylinders that have been vacuum-sealed and have had the majority of the air evacuated. A high voltages is applied to two electrodes at one end of the tube, forcing particles to flow from the cathode (negatively charged electrode) towards the anode (positively charged electrode).[2]

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

由於原子是電中性的,因此有負也有正。西瓜模型。瓜子是電子,瓜肉是正電粒子。猜測的。他的學生Rutherford推翻了他的這種模型假設。 學生做了一個Alpha粒子(+2價的He粒子)的散射實驗,打翻老師的理論。這也是最漂亮的十個實驗之一。

Rutherford gold-foil experiment: In 1909 Rutherford disproved Sir J.J. Thomson's model of the atom as a uniformly distributed substance. Because only very few of the alpha particles in his beam were scattered by large angles after striking the gold foil while most passed completely through, Rutherford knew that the gold atom's mass must be concentrated in a tiny dense nucleus.[5] Image:Encyclopædia Britannica, Inc.
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。

原子核的質量大約是電子的質量7000倍,因此在這個實驗中, 電子是可以忽略的。那麼可能造成影響的就是金的原子核。 金原子核的質量大約是He原子核的50倍,如果原子核像是瓜肉一樣均勻地分佈,那麼應該很容易地穿過去,而不會形成反彈。 以前人認為的原子半徑是101010^{-10} m, 他算出來是原子核在一個很小的體積裡面101410^{-14} m, 半徑縮小了人們認知的1/10000, 這個結果是革命性的。絕代部分的空間是空的。撞到原子核的概率是1/10000, 他的結果是1/8000。大致吻合。如此看來數據也是很有用的啊。

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。

金原子的質量

還沒有知道原子的模型的時候,金原子的質量比He原子的質量重50多倍是如何得到的?

黑體輻射的光是如何產生的

很多的原子電子的集體振動形成的電磁波。

# The spectrum of hydrogen

黑體輻射是原子集體行為的結果。如果我們想要看單個原子的頻譜,我概怎麼做呢? 我覺得這個動機很重要,可以啟發你在做科研的時候應當如何去思考。有人就做了這樣的實驗。

類似Crookes Tube的管子, 抽真空,但是加入少量的氫氣。陰極放出電子,像陽極加速,和氫原子撞擊就會激發氫原子,然後讓氫原子放出電磁波。之前X射線的產生是在電子撞擊金屬靶材得到的光,而這裡是在電子加速的路徑中和氫原子相撞。 所以得到的光譜就是氫原子的光譜。由於是氣體,分子之間的間隔很遠,他們自己不會撞在一起,更不會由於彼此的間隔而像晶體一樣振蕩。因此發出來的光就是電子撞擊氫原子得到的電磁波。這個管子就叫做氣體放電管。

放出的電磁波經過頻譜分析,例如過三稜鏡,發現頻率是離散,在可見光的範圍裡面只會放出四種頻率的光,410.13 nm, 434.01 nm, 486.07 nm, 656.21nm. 這個序列就被叫做Balmer Series。Balmer是一個中學老師,他看到這個序列,發現他們滿足的一個規律。

1λ=R(1/nk21/nj2)\frac{1}{\lambda} = R (1/n_k^2 - 1/n_j^2)

where nk=2,nj=3,4,5n_k = 2, n_j = 3, 4, 5.

最後Rydbergy-Ritz 把整個拓展到可見光的範圍之外, 然後總結出了這個Formula。R=1.096×107m1R=1.096 \times 10^7 \text{m}^{-1}, nk,njn_k, n_j are integers, nj=nk+1,nk+2,...n_j = n_k +1, n_k + 2, ...

Emission Spectrum of Hydrogen [1]

For hydrogen:

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

一旦對H做了類似的實驗,可以想像人們一定會用其他的原子做類似的實驗,得到對應的譜,然後去總結規律。

什麼叫做激發?

電子撞擊H原子放出電磁波為什麼被說成是激發H原子放出電磁波? Because it sounds better. No! The electron hit the atom, and make the electron of this atom jump from the lower energy orbit to a higher energy orbit. This process is called exitation.

波好有趣啊

吉他弦的振動,產生了聲波,有沒有可能吉他弦這種模型可以描述微觀世界中的行為。我不曉得所謂的弦論是說的什麼, 或許和吉他弦有關係也不知道。

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]

量子力學裡面的F=ma對應的公式什麼?

牛頓第二定律統一了經典力學,統一量子力學的公式是哪一個?

Coulomb's law

F=14πε0q1q2r2=keq1q2r2F = \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 ke=14πε0=9×109Nm2C2k_\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 ε0\varepsilon_0 is 8.85418782×10128.85418782 \times 10^{-12} Fm2\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=1.6×1019e = -1.6 \times 10^{-19} C. [6]

Electric potential energy

The electrostatic potential energy, UEU_E of one point charge qq at position rr in the presence of another point charge QQ, taking an infinite separation between the charges as the reference position, is:

UE(r)=keqQrU_E(r) = k_e \frac{qQ}{r}

where kek_e is Coulomb's constant. [7]

接下來的問題是,為什麼會出現離散的分佈, 為什麼不同的原子的光譜線不一樣。 你需要給個解釋。這個重擔就交到了波爾的肩上。 丹麥人,生於小美人魚的故鄉阿姆斯特丹的名門望族。劍橋大學念的大學,阿姆斯特丹念的博士,然後有想要回到劍橋念博士後,照到J. J. Thompson, 後找到其學生Rutherford。 20世紀的科學家愛因斯坦排第一的話,波爾要排第二。

# The Bohr atom

Bohr's Postulates

  1. Electron can take only certain orbits with angular momentum

L=mvr=n,n=1,2,3,...;=h/2πL = 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 EiE_i and goes into another orbit of lower energy EfE_f, electromagnetic radiation will be emitted with a precise frequency given by

ν=EiEfh\nu = \frac{E_i - E_f} {h}

這裡面的第一個假設的傳承了Plank的觀念,第二個原理說轉動的電子居然會不放出電磁波,在古典物理裡面是聞所未聞。而第三個假設是用到了愛因斯坦光子的觀念。但是就是能夠抽出答案。緊接著愛因斯坦得了諾貝爾獎,他的兒子後來也得了諾貝爾獎。明明是原子模型,波爾這裡的三個假設卻都是說的原子裡面的電子。你知道他是怎麼去湊的嗎?

# Hydrogen spectrum explanation

Centripetal acceleration

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

ac=v2ra_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=eq=-e rotates in a circular orbit of radius rr under the electrostatic attraction of the nucleus (q=+eq = +e). For simplicity we will assume that the nucleus remains stationary.

From the circular motion of the electron. Fradial=maradialF_\text{radial} = ma_\text{radial}

kee2r2=mv2rk_e \frac{e^2}{r^2} = m \frac{v^2}{r}

Fron the first postulate, we have

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

Then we have

kee2r2=mrn22m2r2k_e \frac{e^2}{r^2} = \frac{m}{r} n^2 \frac{\hbar^2}{m^2r^2}

We obtain

r=n22e2mker = n^2 \frac{\hbar^2}{e^2mk_e}

Let r0=2m2r2r_0 = \frac{\hbar^2}{m^2r^2}, we have

rn=n2r0r_n = n^2 r_0

We see that only orbits with radius r0,4r0,9r0,...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.

ETOTAL=Ekinetic+Epotential=12mv2kee2rE_\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(1/2)r,

12kee2r=12mv2\frac{1}{2} k_e \frac{e^2}{r} = \frac{1}{2} m v^2

Then, we get

ETOTAL=12kee2r=1214πε0e2rE_\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 rr with rn=n2r0r_n=n^2 r_0

En=1n2e4m8ε02h21n2E0E_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. EnE_n is negative says the energy is 束縛能,which is used to 束縛 electron. The energy is smaller when nn increases. When n=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=1n=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.

When an electron jumps from an allowd orbit to another, photon is emitted.[9] The differences between energyies of the excited states of the hydrogen atom determine the possible wavelengths, or alternately the frequencies, of photons emitted when excited electrons drop to lower energy states. The set of possible photon wavelengths is called the hydrogen atom spectrum.[10]

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?

這裡相當於把電子的軌道量子化了, 只能在特定半徑的軌道上面運動。原子的大小的定義就是這個半徑的範圍。 我絕對不相信這就是世界本來的樣子。這樣之後,雙粒子系統的能量也就被量子化了。 我這裡發現這裡面的推導也是用到了經典物理。

What is angular momentum?

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

角動量也可以量子化

怎麼什麼都可以量子話, 聲音量子化之後是不是就變成聲子了。 做這樣的假設也是在湊答案,量子力學的發展就是在湊答案。

用量子力學和信耶穌是一樣的啊

你用量子力學的模擬去做工作, 例如跑量子力學模擬,你就相信了他的這些假設是對的。這種操作和信耶穌是一樣的東西啊。

我頓時感覺很放鬆

看到全是假設我竟然感覺特別放鬆,原理物理不是那麼神聖。

我怎麼有種感覺吉他線的振動得到的頻率譜

和這裡的頻率譜是一樣的呢? 吉他線的振動也會產生離散的頻譜啊。因為有駐波。

# 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.

In 1914, James Franck and Gustav Hertz performed an experiment which demonstrated the existence of excited states in mercury atoms, helping to confirm the quantum theory which predicted that electrons occupied only discrete, quantized energy states. [15]

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 Ek=eV0E_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 VrV_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 (Ei\text{E}_i - Ef\text{E}_f) cause 'resonance' and induce electron drop from Ei\text{E}_i to Ef\text{E}_f.

(a) Spontaneous emission. (b) Stimulated emission [11]

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. (10310^{-3} sec vs 10810^{-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

The EM Spectrum [13]

[1] Development of Current Atomic Theory (opens new window)
[2] The Discovery of Electron (opens new window)
[3] Wikipedia: Alpha particle (opens new window)
[4] Wikipedia: Alpha decay (opens new window)
[5] Rutherford's nuclear model (opens new window)
[6] Wikipedia: Coulomb's law (opens new window)
[7] Wikipedia: Electric potential energy (opens new window)
[8] Wikipedia: Acceleration (opens new window)
[9] What causes the emission of radiant energy that produces characteristics spectral lines? (opens new window)
[10] Hydrogen atom spectrum (opens new window)
[11] Spontaneous and Stimulated Emission of Radiation (opens new window)
[12] SAMSUNG: What is the defference between Blu-ray Disc, DVD&CD ? (opens new window)
[13] IOWA STATE UNIVERSITY: Electromagenetic Waves (opens new window)
[14] Wikipedia: String Theory (opens new window)
[15] The Frack-Hertz Experiment (opens new window)
[16] Wikipedia: Mercury(element) (opens new window)