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Chapter 4 Bohr’s model of the atom 4.1 Thomson’s model (plum pudding) (1) An atom in which the negatively charged electrons were located within a continuous distribution of positive charge. (2) At its lowest energy state, the electrons would be fixed at their equilibrium positions. (3) In excited atoms, the electrons would vibrate about their equilibrium positions. Plum pudding (4) A vibrating electrons emit electromagnetic radiation. A Thomson hydrogen atom has only one characteristic emission frequency conflict with the very large number of different frequencies observed in the spectrum of hydrogen. Chapter 4 Bohr’s model of the atom Rutherford : The positive charge (nucleus) is concentrated in a very small region. (1908 Nobel prize for the chemistry of the radioactive substances) α particle scattering N: the number of atoms that deflect an α particle θ: the angle of deflection in passing through one atom Θ: the net deflection in passing through all the atoms ( 2 )1 / 2 N ( 2 )1 / 2 2 I 2 / 2 e d is the number of particle scattered 2 within the angular range to d N ( )d r 10 10 m 10-4 rad by Thomson' s model prediction Chapter 4 Bohr’s model of the atom 6 Ex: In a typical experiment, α particles were scattered by a gold foil 10 m thick. The average scattering angle was found to be ( 2 )1 / 2 1o 2 102 rad Calculate ( 2 )1 / 2 (a) the number of atoms N 10 6 m/ 10 10 m 10 4 ( 2 )1 / 2 2 10 2 ( ) 2 10 4 rad 2 10 N agree with the Thomson' s atom estimation 2 1/ 2 (b) the fraction of particle scattering : experiment al prediction : N( 90 o ) / I 10 4 theoretica l prediction for Thomson' s atom : N( 90 ) / I o 180o 90o N( )d / I e ( 90) 10 3500 2 An atom has a very small nucleus with positive charge. Thomson’s atom model must be corrected. Chapter 4 Bohr’s model of the atom 4.2 Rutherford’s model All the positive charge of the atom, and consequently essentially all its mass, are assumed to be concentrated in a small region in the center called the nucleus (1) Nucleus radius: Thomson : r 10 10 m Rutherford : r 10 14 m (2) Maximum deflection angle: Thomson : 10 4 rad Rutherford : 1 rad Chapter 4 Bohr’s model of the atom α particle scattering trajectory b: impact parameter M: α particle mass v : incident particle velocity total angular momentum conservati on : Mvb Mv 'b' L 2 total kinetic energy conservati on : Mv 2 / 2 Mv ' / 2 v v ' b b' The scattering trajectory of a light positive +ze particle by a heavy nucleus +Ze can be solved by Newton’s law. zZe 2 d 2r d 2 F ma M [ r ( ) ] 4 0 r 2 dt 2 dt Chapter 4 Bohr’s model of the atom d 2 r / dt 2 : the radial acceleration r (d / dt )2 2 r : the centripetal acceleration r 1/ u dr dr d dr du d 1 du L du 2 dt d dt du d dt u d M d d 2r d dr d L d 2 u Lu 2 L2 u 2 d 2 u ( ) put into (1) dt 2 d dt dt M d 2 M M 2 d 2 L2 u 2 d 2 u 1 Lu 2 2 zZe 2 u 2 ( ) 2 2 M d u M 4 0 M d 2u zZe 2 M zZe 2 M 2 2 u set D ( zZe / 4 ) /( Mv / 2) 0 2 2 2 2 2 d 4 0 L 4 0 M v b d 2u D u the solution is u A cos B sin D / 2b 2 2 2 d 2b for coefficients A and B consider the initial condition : (1) 0 as r (2) dr / dt v as r u 1 / r 0 A cos 0 B sin 0 D / 2b 2 A D / 2b 2 Chapter 4 Bohr’s model of the atom dr L du L v ( A sin 0 B cos 0) dt M d M Mv Mv 1 B L Mvb b D 1 D the solution is u cos sin 2b 2 b 2b 2 1 1 D the scattering trajectory is sin 2 (cos 1) r b 2b 2b as r and set cot - - - - - - - (2) 2 D - 1 1 D the closest distance R ( for ) sin( ) 2 [cos( ) 1] 2 R b 2 2b 2 D 1 from (2) R [1 ] 2 sin( / 2) head - on collision : b 0 no deflection : 0 b and R Chapter 4 Bohr’s model of the atom the number N ( )d scattered into to d the number incident from b to b db area of incident ring : 2bdb the number of ring : t scattering probabilit y : P (b )db t 2bdb D d / 2 2 sin 2 ( / 2) d P (b )db tD 2 sin 8 sin 4 ( / 2) P (b )db is the scattering probabilit y from to d N ( )d sin d P ( b )db tD 2 I 8 sin 4 ( / 2) from (2) db R180o zZe 2 D the nucleus radius 4 0 ( Mv 2 / 2) Chapter 4 Bohr’s model of the atom rnuleus 10 14 m d d Ind : differenti al cross section d d d 2 sin d dN zZe 2 2 1 N ( )d dN ( ) ( ) Ind 2 4 4 0 2 Mv sin ( / 2) 1 2 Rutherford scattering differenti al cross section : d 1 2 zZe 2 2 1 ( ) ( ) d 4 0 2 Mv 2 sin 4 ( / 2) Chapter 4 Bohr’s model of the atom 4.3 The stability of the nuclear atom The serious difficulties in the previous atomic model: (1) The charged electrons constantly accelerate in their motion around the nucleus, radiate energy in the form of electromagnetic radiation. The atom would rapidly collapse to nuclear dimension. (2) The continuous spectrum of radiation is not in agreement with the discrete spectrum observed in experiments. 4.4 Atomic spectra An apparatus for measuring atomic spectra Chapter 4 Bohr’s model of the atom n2 Balmer (1885) : 3646 2 ex : n 3( H ), n 4( H ) n 4 1 1 1 Ryberg (1890) : RH ( 2 2 ) n 3,4.. 2 n RH 1.097 10 7 m -1 : Ryberg constant for H For alkali elements (Li, Na, K,...) : κ 1 1 1 R[ ] λ ( m a )2 ( n b)2 Chapter 4 Bohr’s model of the atom 4.5 Bohr’s postulate Bohr’s postulate (1913): (1) An electron in an atom moves in a circular orbit about the nucleus under the influence of the Coulomb attraction between the electron and the nucleus, obeying the laws of classical mechanics. (2) An electron move in an orbit for which its orbital angular momentum is L n nh / 2 , n 1,23.., h Planck’s constant (3) An electron with constant acceleration moving in an allowed orbit does not radiate electromagnetic energy. Thus, its total energy E remains constant. (4) Electromagnetic radiation is emitted if an electron, initially moving in an orbit of total energy Ei, discontinuously changes its motion so that it moves in an orbit of total energy Ef. The frequency of the emitted radiation is ( E i E f ) / h . Chapter 4 Bohr’s model of the atom 4.6 Bohr’s model Ze 2 v2 m for L mvr n , n 1,2,3... 4 0 r 2 r 1 n 2 n 2 2 Ze 4 0 mv r 4 0 mr ( ) 4 0 mr mr n 2 2 r 4 0 mZe 2 n 1 Ze 2 v mr 4 0 n 2 2 Potential energy : V r Ze 2 Ze 2 dr 4 0 r 2 4 0 r ground state 1 Ze 2 2 Kinetic energy : K mv 2 4 0 2r Ze 2 mZ 2 e 4 1 Total energy : E K V K E (4 0 ) 2 2r ( 4 0 ) 2 2 2 n 2 Chapter 4 Bohr’s model of the atom Ei E f ( h 1 Pachen 2 4 0 1 )2 4 mZ e 1 1 ( ) 4 3 n 2f ni2 Lyman c me 4 Z 2 1 1 ( ) ( ) 3 2 2 4 0 4 c n f ni 1 R Z 2 ( Balmer 2 1 1 ) n 2f ni2 me 4 for R ( ) RH 4 0 4 3 c 1 2 Chapter 4 Bohr’s model of the atom 4.7 Correction for finite nuclear mass the reduced mass of the system : mM mM L vr n RM Z 2 ( 1 1 ) n 2f ni2 M M R R , RM R , as mM m m M For hydrogen atom : 1836 m 1 RM R 2000 RM Chapter 4 Bohr’s model of the atom Ex: The positronium atom, consisting of a positron and an electron revolving about their common center of mass, which lies halfway between them. (a) In such system were a normal atom, how would its emission spectrum compare to that of the hydrogen atom? (b) What would be the electron-positron separator, D, in the ground state orbit of the positronium. mM m2 m m R RM R m M 2m 2 mm 2 RM hcZ 2 R hcZ 2 E positronium n2 2n 2 1 R 1 1 Z 2( 2 2 ) c 2 n f ni the electron - positron separator D in ground state is : Dpositronium 4 0 n 2 2 4 0 n 2 2 2 2rhydrogen Ze 2 mZe 2 Chapter 4 Bohr’s model of the atom Ex: A muonic atom contains a nucleus of charge +Ze and a negative muon μmove about it, The μ- is an elementary particle with charge –e and a mass that is 207 times as large as an electron mass. (a) Calculate the muon-nucleus separation, D, of the first Bohr orbit of a muonic atom with Z=1. (b) Calculate the binding energy of a muonic atom with Z=1. (c) What is the wavelength of the first line in the Lyman series for such an atom? (a) m 207me , M 1836me Dn 1 207me 1836me 186me 207me 1836me o 4 0 2 1 3 11 5.3 10 m 2.8 10 A 2 186 186me e me e 4 186 13.6 eV 2530 eV (b) E 186 (4 0 ) 2 2 2 is the ground state energy. The binding energy is 2530 eV. o 1 1 1 1 RM ( 2 2 ) 186 R (1 ) 139.5 R 6.5 A (c) 4 n f ni Chapter 4 Bohr’s model of the atom Ex: Ordinary hydrogen contains about one part in 6000 of deuterium, or heavy hydrogen. This is a hydrogen atom whose nucleus contains a proton and a neutron. How does the doubled nuclear mass affect the atomic spectrum? R 109737 cm 1 RH R 109678 cm 1 m (1 m / M ) (1 1 / 1836) R 109737 cm -1 RD R 109707 cm -1 m (1 m / M ) (1 1/2 1836) RD RH D H D H The spectral lines of the deuterium atom are shifted to slightly shorter wavelength s compared to hydrogen. Chapter 4 Bohr’s model of the atom 4.8 Atomic energy states Franck -Hertz experiment (1914): the quantized atomic energy 9.8 eV Hg V : accelerating potential Vr : retarding potential 4.9 eV Energy level of Hg vapor Chapter 4 Bohr’s model of the atom 4.8 Interpretation of the quantization rules Some Mysteries: Bohr’s quantization of the angular momentum? Planck’s quantization of the energy? Wilson-Sommerfeld quantization rules: For every physical system in which the coordinates are periodic functions of time, there exists a quantum condition for each coordinate. The quantum conditions are pq dq nq h q: one of the coordinate pq : the momentum associated with the coordinate q nq : the integer quantum number : the integration over one period of the coordinate q Chapter 4 Bohr’s model of the atom For one-dimensional simple harmonic oscillation: x ( t ) A cos t a( t ) dx ( t ) A sin t v ( t ) dt dv ( t ) 2 A cos t dt F a ( t )m kx ( t ) 2 m k k 2 m p x2 kx 2 p x2 x2 E K V 1 2m 2 2mE 2 E / k p x2 x 2 1 for b 2mE , a 2 E / k b2 a 2 p dx ab x 2mE 2 E / k 2E / E / n x h nh E nh E E ( n 1) E ( n) ( n 1)h nh h h 0 E 0 continuous energy Chapter 4 Bohr’s model of the atom The angular momentum quantization for Bohr’s atom: p dq n h Ld L q q 2 0 d 2L nh n 2 nh L mvr pr n 2 2L nh L for de Broglie wavelength h r h h p p nh 2r n , n 1,2,3... 2 de Broglie standing wave Chapter 4 Bohr’s model of the atom 4.10 Sommerfeld’s model Fine structure: a splitting of spectral lines due to spin-orbit interaction Sommerfeld’s explanation for an elliptical orbit: Ld n h L2 n h L n / , n 1,2,3.. p dr n h L(a / b 1) n h, n 0,1,2,... r r r r 4 0 n 2 2 n 1 2 Z 2 e 4 a ,b a E ( ) Ze 2 n 4 0 2n 2 2 : reduced mass nr : radial quantum number n : azimuthal quantum number n nr n principal quantum number (1) n n circular orbit (2) n nr elliptical orbit For the same n, but different nr and n energy is degenerate. Chapter 4 Bohr’s model of the atom Sommerfeld removed the degeneracy by treating the problem relativistically. for hydrogen atom v / c 102 E (v / c )2 10 4 (eV) energy splitting Z 2 e 4 2Z 2 1 3 E [ 1 ( )] 2 2 2 (4 0 ) 2n n n 4n e2 1 fine structure constant 4 0 c 137 1 Selection rule: ni nf 1 Chapter 4 Bohr’s model of the atom 4.11 The correspondence principle Bohr (1923): (1) In a limit of very large quantum number, the prediction of quantum theory corresponds to that of classical theory. (2) Any selection rule hold true in the quantum theory, which also apply in the classical limit (very large quantum number). Ex: blackbody radiation: Planck' s theory : nh n h Classical theory : 0 kT constant n h kT as 0 and h 0 n Chapter 4 Bohr’s model of the atom Ex: Apply the correspondence principle to hydrogen atom radiation in the classical limit. The classical radiation frequency of 0 in Bohr orbit n is v 1 2 me 4 2 0 ( ) 2r 4 0 4 3 n 2 Bohr' s radiation theory for ni n f 1 me 4 1 1 1 2 me 4 2n 1 ( ) [ ] ( ) [ ] 3 2 2 3 2 2 4 0 4 ( n 1) n 4 0 4 ( n 1) n 1 2 me 4 2 n ( ) 0 4 0 4 3 n 2 1 2 Transitions are observed to occur between states of low n, in which the old quantum theory cannot always be made to agree with experiment. Chapter 4 Bohr’s model of the atom 4.12 A critique of the old quantum theory (1) The Wilson-Sommerfeld quantization is just used to treat the periodic system (2) It can be used to calculate the energy of the allowed states, but cannot be used to calculate the transition rate. (3) It is successful only for one-electron system, fails badly for two (many) electron. (4) The entire theory seems to lack coherence.