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
N ( 2 )1 / 2
2 I   2 /  2
N ( )d 
e
d i s th en u m be rof  parti cl escatte re d
2
wi th i nth ean gu l arran ge to   d
(  2 )1 / 2 
r  1010 m    10-4 rad by Th om son s' m ode lpre di cti on
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  102 rad
Calculate ( 2 )1 / 2
(a) th en u m be rof atom sN  106 m /1010 m  104
(  2 )1 / 2 2  10 2
( ) 

 2  10 4 rad
2
10
N
agre ewi th th e Th om son s' atome sti m ati on
2 1/ 2
(b) th efracti onof  parti cl escatte ri n g:
e xpe ri m e nalt pre di cti on: N(   90o ) / I  10 4
th e ore ti ca
l pre di cti onfor Th om son s' atom:
N(   90 ) / I 
o

180 o
90 o
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
Ruthe rford: r  1014 m
(2) Maximum deflection angle: Th om son:   104 rad
Ru th e rford:   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
totalan gu l arm om e n tu m
con se rvati
on : Mvb  Mv'b'  L
2
totalk i n e ti ce n e rgycon se rvati
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.

zZe2
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 Lu2
L2 u 2 d 2 u

( )


pu t i n to(1)
dt 2
d dt dt
M d 2 M
M 2 d 2
L2 u 2 d 2 u 1 Lu2 2 zZe 2 u 2

 (
) 
2
2
M d
u M
4 0 M
d 2u
zZe 2 M
zZe 2 M

u

se t D  ( zZe 2 / 4 0 ) /( Mv 2 / 2)
2
2
2 2 2
d
4 0 L
4 0 M v b
d 2u
D


u


th esol u ti oni s u  A cos  B si n  D / 2b 2
2
2
d
2b
for coe ffi ci e ts
n A an d B  con si de rth e i n i ti alcon di ti on:
(1)  0 as r   (2) dr / dt   v as r  
 u  1 / r  0  A cos 0  B si n0  D / 2b 2  A  D / 2b 2
Chapter 4 Bohr’s model of the atom
dr
L du
L

 v  
(  A si n0  B cos0)
dt
M d
M
Mv
Mv
1
B


L
Mvb b
D
1
D
th esol u ti oni s u 
cos


si
n


2b 2
b
2b 2
1 1
D
th escatte ri n gtraje ctoryi s
 si n  2 (cos  1)
r b
2b
 2b
as r   an dse t      cot 
- - - - - - - (2)
2
D
 -
1 1
 
D
 
th ecl ose stdi stan ceR ( for  
)
 si n (
)  2 [cos(
)  1]
2
R b
2
2b
2
D
1
from (2)  R  [1 
]
2
si n ( / 2)
h e ad- on col l i si on
:    b  0
n o de fl e cti on:   0  b   an d R  
Chapter 4 Bohr’s model of the atom
th en u m be rN ( )d scatte re di n to to   d
 th en u m be ri n ci de n from
t
b to b  db
are aof i n ci de n ri
t n g: 2bdb
th en u m be rof ri n g: t
scatte ri n gprobabi l iyt : P (b )db  t 2bdb
from (2)  db  
d
8
si n4 ( / 2)
 P (b )db i s th escatte ri n gprobabi l iyt from to   d
 P (b )db  


D d / 2
2 si n2 ( / 2)
tD 2 si n
N ( )d

si nd
  P (b )db  tD 2
I
8
si n4 ( / 2)
R180 o
zZe 2
D
 th en u cl e u sradi u s
2
4 0 ( Mv / 2)
Chapter 4 Bohr’s model of the atom
rnuleus  1014 m
d
d
Ind
: di ffe re n ti
al crossse cti on
d
d
d  2 si nd
dN 
zZe 2 2
1
 N ( )d  dN  (
) (
)
Ind
2
4
4 0
2 Mv
si n ( / 2)
1
2
Ru th e rfordscatte ri n gdi ffe re n ti
al crossse cti on:
d
1 2 zZe 2 2
1
(
) (
)
d
4 0
2 Mv 2 si n4 ( / 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
Bal m e r(1885):   3646 2
e x : n  3( H  ), n  4( H  )
n 4
1
1
1
Rybe rg(1890):    RH ( 2  2 ) n  3,4..

2
n
RH  1.097 107 m -1 : Rybe rgcon stan tfor H
For al k al ie l e m e n ts
(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   ( Ei  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
mZe2
n
1 Ze 2
 v

mr 4 0 n
2
2
Pote n ti ale n e rgy: V   

r
Ze 2
Ze 2
dr  
4 0 r 2
4 0 r
ground state
1
Ze 2
2
Ki n e ti ce n e rgy: K  mv 
2
4 0 2r
Ze 2
mZ 2 e 4
1
Total e n e rgy: 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
3
4 0 4 c
1
2
Chapter 4 Bohr’s model of the atom
4.7 Correction for finite nuclear mass
th ere du ce dm assof th esyste m:  
mM
mM
 L  vr  n
   RM Z 2 (
1
1

)
n 2f ni2
M

M
R 
R , RM  R , as

mM
m
m
M
For h ydroge natom:
 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
mm
2
RM hcZ 2
R hcZ 2
E positroniu m  

n2
2n 2
1 
R
1
1
     Z 2( 2  2 )
 c
2
n f ni
th ee l e ctron- posi tronse paratorD i n grou n dstatei s :
Dpositroniu m
4 0 n 2  2
4 0 n 2  2

2
 2rhydrogen
Ze 2
mZe2
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 e V  2530e V
(b) E  186
(4 0 ) 2 2 2
i s th e grou n dstatee n e rgy.Th e bi n di n ge n e rgyi s 2530e V.
o
1
1
1
1
 RM ( 2  2 )  186R (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
109737cm1
RH  R


 109678cm1
m (1  m / M ) (1  1 / 1836)

R
109737cm-1
RD  R


 109707cm-1
m (1  m / M ) (1  1/2 1836)
RD  RH   D   H   D   H
The spe ctralline sof thede ute rium
atomare shifte d
to slightl yshorte rwave le ngth
s compare dto hydroge n.
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 : acce l e rati
n g pote n tial
Vr : re tardin gpote n tial
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 si n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 kx2
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  2E / 
 E /   n x h  nh  E  nh
E  E ( n  1)  E ( n)  ( n  1)h  nh  h
h  0  E  0 continuous
e ne rgy
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  2L
nh
 n
2
nh
 L  mvr  pr  n 
2
 2L  nh  L 
for de Brogl i ewave l e n gth  

h

r
h
h
 p
p

nh
 2r  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  (
)
Ze2
n
4 0 2n 2  2
 : re du ce dm ass
nr : radi alqu an tu mn u m be r
n : az i m u th al
qu an tu mn u m be r
n  nr  n pri n ci palqu an tu mn u m be r
(1) n  n ci rcu l arorbi t
(2) n  nr e l l i pti cal
orbi t
 For th esam en, bu t di ffe re n tnr an d n e n e rgyi s de ge n e rate
.
Chapter 4 Bohr’s model of the atom
 Sommerfeld removed the degeneracy by treating the problem relativistically.
for hydroge natom v / c  102
 E  (v / c )2  10 4 (e V)e ne rgyspl itti ng
Z 2 e 4
 2Z 2 1
3
E
[
1

(

)]
2
2 2
(4 0 ) 2n 
n
n 4n
e2
1


fi n estru ctu recon stan t
4 0 c 137
1
Selection rule:
ni  nf  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:
Pl an cks' th e ory:   nh    n h
C l assi cal
th e ory:  0    kT con stan t

n h  kT as  0 an d h  0  n  
Chapter 4 Bohr’s model of the atom
Ex: Apply the correspondence principle to hydrogen atom radiation in the
classical limit.
Th e cl assi calradi ati onfre qu e n cyof  0 i n Boh rorbi tn i s
v
1 2 me 4 2
0 
(
)
2r
4 0 4 3 n 2
Boh r's radi ati onth e oryfor ni  n f  1
me 4
1
1
1 2 me 4
2n  1
 (
)
[

]

(
)
[
]
4 0 4 3 ( n  1) 2 n 2
4 0 4 3 ( n  1) 2 n 2
1
2
me 4 2
n     (
)
0
3
2
4 0 4 n
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.