原子模型

Atomic Model有了正確的原子模型，才會發明了雷射

( 1856 – 1940 )

原子結構中的電子是如何被發現的 ?

可以參考美國物理學會 ( American Institute of Physics ) 網站For in-depth information, check out the American Institute of Physics' History Center exhibit on the discovery of the electron at http://www.aip.org/history/electron/

西瓜模型 ( Plum Pudding Model )

• by J. J. Thompson

帶負電的西瓜子( 電子 )

帶正電的西瓜肉

西瓜模型 : negatively charged plum ( electron )

atomic size~ Å

positively charged pudding

Plum Puddingmodel

Rutherford 模型 ( Rutherford Model )

scattering of α particles ( experiment by Rutherford ) :

• most particles scattered at very small angles

• about 1 in 8000 particles scattered at angles > 90o

( 1871 – 1937 )

Thomson and Rutherford in 1930s

• mα ∼ 7000 me electrons negligible• mAu,positive part ∼ 50 mα , but if the mass was spread uniformly

over the whole atom ( r ∼ 10-10 m , Å )Au positive part could not cause large deflection ( 大散射角度的機會 ∼ 1 / 103500 )The positive part of the atom has to be concentrated in an extremely small volume ( r’ ∼ 10-14 m )

原子核( nuclei )

原子半徑

~ 10-10 m原子核半徑

~ 10-14 m

α 粒子撞到原子核的

機會 ~ 10-14/ 10-10

1/104

原子半徑中絕大部分

的空間是空無一物的 !

Robert P. Crease, a member of the philosophy department at the State University of New York at Stony Brook and the historian at Brookhaven National Laboratory, asked physicists to nominate the most beautiful experiment of all time. The New York Times listed the 10 winners of this polling and Rutherford’s experiment is one on them.

Planetary Model ( 行星模型 )

行星模型的問題 ( 從古典物理的觀點來看 ) :if e- moves in circle at constant frequency

equivalent to 2 mutually perpendicular SHM of chargecharge undergoing SHM would radiate EM waveEM wave carries away energye- loses energy and spirals into the nucleus ( within 10-8 s )

( 1 ) atom would be unstable( 2 ) radiate EM wave of continuous frequency ( because e-

circulating frequency ↑ as radius ↓ )

氫原子光譜 ( Spectrum of Hydrogen Atom )

• each element has a characteristic set of lines• visible spectrum of H consists of 4 lines :

410.12 nm , 434.01 nm, 486.07 nm, 656.21 nm ( Balmer series )• an empirical relation :

1/λ = R ( 1/nk2 – 1/nj2 ) – Rydberg-Ritz formula

R = 1.096 x 107 m-1 , nk , nj : integers , nj = nk+1, nK+2, …..

Low pressure H gas

例 : for 氫原子nk =2 , nj = 3, 4, 5 frequency lines in the visible range

Balmer seriesnk = 1 , nj = 2, 3, 4 frequency lines in the UV range

Lyman seriesnk = 3 , nj = 4, 5, 6 frequency lines in the infrared

Paschen series

為什麼原子會發射不連續的光譜線 ?

為什麼每一種元素的原子的發射譜線都不一樣 ?

⇒ 古典物理 ( 如簡單的原子行星模型 ) 無法解釋 !

波耳原子模型 ( The Bohr Atom )

(1) 假設 1 : electron can take only certain orbits with angular momentumL = mvr = nh , n = 1, 2, 3, …..h = h / 2π

(2) 假設 2 : electron in the allowed orbits does not radiate EM radiation

(3) 假設 3 : radiation occurs only when e- goes from one allowed orbit of energy Ei to another lower energy Ef. The radiated frequency is υ = ( Ei – Ef )/h

波耳假設 ( Bohr’s Postulates )

波耳氫原子模型的光譜線

L = mvr = nh v = n ——

Fradial = maradial ( 等速率圓周運動 )

——— = mv2 / r

——— = — n2 ———

rn = n2ro , ro = ————

only radius ro, 4ro, 9ro ….. are allowed

hmr

e2

4πεor2

e2

4πεor2mr

h2

m2r2

4π h2 εoe2m

例 : 電子環繞原子的最小半徑

ro = ——— ≈ 0.53 Å4π εo h2

e2m

轉動角動量的量子化 ( quantization of angular momentum L ) 同時意味了

電子軌道半徑 的量子化 ( quantization of radius r ) 能量 的量子化 ( quantization of energy ) ( 或 “不連續性的變化” )

Etotal = K.E. + P.E.= ½ mv2 – ———— ( 氫原子只有一個電子及一個質子 )

since ———— = mv2 / r ( Fradial = m aradial )

½ mv2 = ½ ———— Etotal = - ½ ————

E = – ———— ( r = n2ro )

En = – ————— —— = – Eo / n2 , Eo = ———— , n = 1, 2, 3, …

e24π εor

e24π εor2

e24π εor

e24π εor

e2

8π εon2ro

e4 m8 εo2 h2

1n2

e4 m8 εo2 h2

例 : 氫原子中電子的能階En=1 = …… = - 13.56 eV ground state ( 基態 )En>1 : excited state ( 激發態 )

所有的物理系統都趨向於最低的能量狀態

( All physical systems tend to seek the lowest energy state available to them. )

- 13.56 eV

- 3.39 eV

- 1.51 eV

- 0.85 eV

Excited State

Ground State

原子光譜線 ( Spectral Lines )

En = – Eo / n2

If electron jump from Ei Ef and emits a photon

υ = ( Ei – Ef ) / h

= ——————————

= —— ( ——— - ——— )

1/λ = —— ( 1/nf2 – 1/ni2 )

= RBohr ( 1/nf2 – 1/ni2 )

RBohr = Eo / hc ≈ 1.097 x 107 m-1

與 Rydberg-Ritz formula 的結果大致相同

( - Eo/ni2 ) – ( - Eo/nf2 )h

Eoh

1 1nf2 ni2Eo

hc

Typical x-ray Spectrum

I

υmax

, V ( 幾千伏特 )

–

–

––

X-ray

原子的發射光譜

原子的吸收光譜

The Franck-Hertz 實驗 ( Franck-Hertz Experiment )

e- reaching G Ek = eVo

In vacuum Vr > Vo e- turned back

Vr < Vo e- reaches P ⇒ i

GridPlate

Hg vapor

Vo Vr(~1V)

i

with vapor of element

e- lose K.E. through collisions with atoms

( energy absorbed by the vapor )

For the case of Hg vapor :

e- e-

0 eV

- 3.7 eV

- 5.5 eV

- 10.4 eV

4.9 eV 6.7 eV

GridPlate

Hg vapor

Vo Vr(~1V)

i

雷射 ( Laser ) ( Light Amplification by Stimulated Emission or Radiation )

波耳原子模型 原子中的電子能階

氫原子

• e- absorb energy ( through photon or e- collision )excited statedecay to lower energy state through

spontaneous emissionstimulated emission

• spontaneous emission : in all directions

• an incoming photon of the correct energy ( Ei – Ef )

cause “resonance” and induce e- drop from Ei to Efstimulated emission

3 features of stimulated emission : (1) 1 photon in, 2 photons out light amplification(2) emitted photon in the same direction as the incoming photon(3) emitted photon in step ( in phase 同相, coherent ) with the

incoming photon

除了 stimulated emission 外 , 要產生雷射還有兩個重要的條件 :

(1) population inversion

(2) metastable state

a higher energy state in which e- can stay for a much longer time than in an ordinary excited state ( 10-3 sec vs. 10-8 sec for example )

more time to enhance population inversion

stimulated emission occur before spontaneous emission

雷射的操作與特性( Operation and Characteristics of a Laser )

Characteristics of a laser : (1) unidirectional (2) high intensity

例 : He-Ne laser : ~ 100 W/m2

( ~ 4000 times of sunlight )(3) nearly monochromatic

( 單一頻率 )(4) coherent

T.H. Maimanholding the first laser ( 1960 )

紅寶石雷射 ( Ruby Laser ) – 一種固態雷射

E3, E4 : short-lived excited state ( ~ 10-8 sec )E2 : long-lived ( ~ 3x10-3 sec ) metastable stateE1 : ground state

e- decay readily from E3, E4 E2 , but not from E2 E1

(1) optical pumping : E1 E3, E4(2) E3, E4 decays to E2 readily(3) a spontaneously emitted photon

( E2 E1 ) triggers series of stimulated emission laser at 694.3 nm

(4) pulse laser ( several pulses per flash )

E1

E2E3

E4

氦氖雷射 ( He-Ne Laser ) – 一種氣體雷射

(1) collision between e- and He ions ⇒ ( E1,He E2,He )(2) collision between He and Ne ⇒ ( E2,He E3,Ne )

collision between e- and Ne ⇒ ( E1,Ne E3,Ne )(3) ( E3,Ne E2,Ne ) ⇒ red laser ( 632.8 nm wavelength )

E2,Ne decays quickly to E1,Ne

1st continuous-wave laser ( 1960 ) ( 15% He, 85% Ne )

~ 1 % transparent

100 % reflective

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