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The Nuclear Atom
Wei-Li Chen
11/21/2016
Dispersion
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Snell ' s law n1 sin 1  n2 sin  2
n( ) is a function of wavelengt h
Empirical formula fit
n2
n  364.6 2
nm
n 4
Rydberg - Ritz formula
1
mn
 R(
1
1

) for n  m
m2 n 2
RH  1.096776  107 m1
J. J. Thomson’s Model
Electrons are embedded in positively charged liquid.
If atoms
Rutherford’s Nuclear Model
• Thomson’s atomic model can not explain the
Rydberg-Ritz formula.
• In Geiger and Marsden’s experiment, most α
particles are deflected by very small angles less
than 1 degree.
• Some particles are deflected by large angles near
180 degree, which can not be explained by
Thomson’s atomic model.
• In Rutherford’s atomic model, all the positive
charges and most of the mass are confined in a
very small “nucleus” of the atom.
Rutherford’s Nuclear Model
kqα Q
θ
b
cot
2
mα v
2
scattering angle
Impact parameter
kqα Q
θ
b
cot
2
mα v
2
The smaller b is, the larger  is.
Geiger and Marsden’s Experiment
I 0 is the incident  particles per second per unit area
The number per second scattered by one nucleus through
angles larger tha n  is b 2 I 0
cross section   b 2 is the area for scattering angles larger tha n 
 atoms 
 g 
n
 3 
3 
 cm    cm   n  N A
M
 atoms 
 g 
N A
M



 mol 
 mol 
The fraction f scattered through angles larger tha n  is
f 
b 2 I 0 nAt
I0 A
 b 2 nt
where A is the beam area, t is the film thickness
This relation can be verified by counting the scattered  particles.
Assuming nuclei do not overlap on the beam path.
It is justified by the extreme small size of the nucleus.
b 2
atom
Rutherford deduced another formula :
2
1
 I 0 ASC nt  kZe 


N  

2
4

 r
 2 Ek  sin  2 
N is the number of particles per unit area per second
ASC is the detector area, r is the foil - detector distance
2
Z is different
The Size of the Nucleus
 kq Q

1

2
 
 0 
 0  m v 
2

 large r  rd
 rd
kq Q
1
2
 m v 
2
rd
rd 
2kq Q
m v 2
Putting 7.7 MeV  particle
 rd  3 10 14 m
Deviation from prediction
Electron Orbits
• Based on Rutherford’s experimental results, the
mass of the atom concentrating in a tiny volume at
the center of the atom.
• Electrons must circulate the nucleus on an circular
or elliptical orbits to maintain a stable system.
• However, a negatively charged electron moving
on an circular orbit must radiate electromagnetic
waves according to classical physics. The radius
of the orbit will decrease and the electron will
finally reach the nucleus. This prediction is
contrary to the experimental observation.
Fc  Fe
mv 2
1 e2


r
40 r 2
v
e
40 mr
1 2
e2
KE  mv and PE  
2
40 r
1 2
e2
 E  KE  PE  mv 
2
40 r

e2
80 r

1
r
The frequency of the movement is
v
e
1
f 

 32
2r 2r 40 mr r
According to classical physics, a charged particle
moving on an circular orbit will emit radiation.
The total energy of the system will drop and the
radius will be reduced as the time goes.
Therefore, the electron w ill eventually land on
the nucleus.
Bohr’s Theory
• When an electron is in one of the quantized orbits, it does not
emit any electromagnetic radiation; thus, the electron is said to
be in a stationary state.
• The electron can make a discontinuous emission, or quantum
jump, from one stationary state to another. During this
transition it does emit radiation. When an electron makes a
transition from one stationary state to another, the energy
difference ∆E is released as a single photon of frequency
ν= ∆E /h ( or h ν= Ei-Ef ).
• In the limit of large orbits and large energies, quantum
calculations must agree with classical calculations.
(correspondence principle)
• The permitted orbits are characterized as quantized values of
the orbital angular momentum. This angular momentum is
always an integer multiple of h/2π.
 mv 2
1 Ze 2

the equationof motion

40 r 2
 r
 L  mvr  n quantized angular momentum

n  N is called the quantum number.
40 mvr
40 n 2  2
2 a0
r


n
Ze 2
m
Zme2
Z
2
0
40 
where a0 

0
.
5291771
A
is called the Bohr radius.
me2
The total energy
2
1 2
1 Ze 2 1 1 Ze 2
1 Ze 2
1 1 Ze 2
En  mv 



2
40 r
2 40 r
40 r
2 40 r
1 Ze 2 Zme2
Z 2 me4
Z2
 En  

  E0 2
2 40 40 n 2  2
2(40 ) 2 n 2  2
n
me4
1
1
1
For H atom, En  
  E0 2  13.6eV  2
2 2
2
2(40 )  n
n
n
me4
where E0 
 13.6eV
2(40 ) 2  2
Rydberg - Ritz equation :

Eo Z 2  1
1 
1 Eo 2  1
1 
1 
2 1
v

and

Z


R
Z

H
 n2 n2 
h  n 2f ni2 
 hc  n 2f ni2 
i 
 f
Eo
me4
me4
where RH 


 1096.78m
hc 2(40 ) 2  2 hc 4 (40 ) 2  3c
nf  3
nf  2
Rydberg - Ritz equation :
 1
1
 RH Z 2  2  2 
n


 f ni 
E
me4
me4
where RH  o 

 1096.78m
hc 2(40 ) 2 2 0 h3c
1
nf 1
Spectrum of Atomic Hydrogen
Reduced Mass Correction
In Bohr' s model, we assume that the mass of the nucleus is infinite.
However, the mass of the nucleus is finite. It requires a correction to fix it.
p2
p2 M  n 2 p2
Ek 


p 
2 M 2m
Mm
2
mM
1
where  
m
mM
1 m M


1

R  R 
1 m M 
where R  1.0973731107 m
Isotopes :
Atoms that have the same Z but different masses.
The transitio n lines from isotopes are slightly different
due to the difference in reduced mass.
Correspondence Principle
Correspond ence principle : In the limit of large orbits and large energies, quantum
calculatio ns must agree with classical calculatio ns.
The classical frequency of revolution of the electron is
n  Zme2 
n
n mr
v
1





f rev  
2 2 
2
2m  40 n  
2mr
2r
T 2r
Z 2 me4
 f rev 
2
2 40  n 3  3
2
According to quantum theory, when n is large
 1
Z 2 me4
2n  1
me4
1 
2

Z 2
 2  
f   cRH Z 
2 3 3
2
2
2
h

)

4
(
2
n








4

2
1

n
n
1

n
0


0 n 
Eo
me4

where RH 
hc 2(40 ) 2 hc
c
2
Wilson-Sommerfeld Quantization Rule
For any physical system in which t he coordinate s are
periodic functions of time, there exists a quantum condition
for each coordinate . These quantum conditions are
 p dq  n h
q
q
where q is one of the coordinate s,
pq is the momentum associated with that coordinate ,
nq is a quantum number whi ch takes on integral values,

means that the integratio n is taken over one period
of the coordinate q.
1D Simple Harmonic Oscillator
p x2 Kx 2
E  k .E.  P.E. 

2m
2
p x2
x2


1
2mE 2 E K
 p dx  n h
x
x
which is the area of the ellipse
on the phase diagram.
 2mE
2E
m
 nh  2E
 nh
K
K
K
 K  mw 
 2v
m
 E  nhv which is Planck' s quantizati on law.
2
Px
2mE
2E
K
x
Circular Orbit (Bohr’s H Atom)
1D Free Moving Particle
 p dx  p  2a  nh
x
h
2a  n  n
p
 Ld  nh
L  2  nh
h
Ln
 n
2
h
de Broglie wavelengt h λ 
p
h
mvr  pr  n
2
h
 2r  n  n
p
Standing wave can be used to explain
The stationary state concept in Bohr’s model.
Fine Structures of H atom
Sommerfeld applied relativist ical correction in ellipsoida l orbits
v2
to find the correction should be of the order 2 .
c


e2
v


n3
2
40 
mr1
40 
m
me2
energy splitting
v
e2
1
 


c 40 c 137
n2
n

 is called the fine structure constant.
3
2
1
n

2
1
selection rule
n  1
X-ray Spectra
K  X  ray
Mosley Plot
From Bohr' s H atom model,
Eo Z 2  1
1 
f 

h  n 2f ni2 

me4
1 
2 1

Z

2(40 ) 2  2 h  n 2f ni2 

me4
1 
2 1

Z

4 (40 ) 2  3  n 2f ni2 
Mosley' s formula for multi - electron atoms
me4
1 

2


K series f 
1

Z

1


4 (40 ) 2  3  n 2 
me4
 1 1 
2
L series f 
 2 Z  7.4 
2 3  2
4 (40 )   2 n 
f 1 2  An Z  b 
b is due to shielding effect of the inner electron(s ).
In Mendeleevs periodic table of the element,
atoms are arranged by its atomic weight.
18 th Potassium (K, 39.102amu)
19 th Argon (Ar, 39.948amu)
However, their chemical properties are contrary t o
those elements in the same line.
In Mosley' s plot, it showed the correct atomic numbers :
18 th Argon (Ar, 39.948amu)
19 th Potassium (K, 39.102amu)
Auger Electrons
K.E.  E  E3
n=3
n=2
E  E2  E1
n=1
Different atoms have
different spectrum
KLM Auger process
1. The electron at n  1 is kicked out
by a high energy electron.
2. The electron at n  2 fills this vacancy.
3. The energy difference is transfere d to
a third electron at n  3 state.
4. The electron at n  3 is ejected by
KE  E  E3  E2 -E1  E3
Auger Electron Spectroscopy
Differentiated data
Franck-Hertz Experiment
I
∆V≠0, with gas, the current
drops periodically
V0
∆V=0, no gas, the maximum current
Is limited by the filament
Accelerating voltage
Hg gas
∆E=E2-E1
If the electron energy is
absorbed by an atom to
induce electron transition, the
amount of the energy
transferred will be quantized.
From grid to plate, this
process could happen
multiple times, therefore a
periodical reduction in
current is observed.
Electron Energy Loss Spectroscopy
(EELS)
The energy structures
of various atoms are
different. The energy
loss spectrum of the
incident electrons is
also different. EELS
can be used to identify
atoms and explore the
quantized energy
structures.
Critique of Bohr Theory
• In Bohr’s model, the H atom spectra and Xrays were successfully explained. However,
the quantitative analysis was absent. The
transition rate that a particular transition (ie.
the intensity of the lines) can not be
predicted by Bohr’s model. This lead us to
the development of the quantum mechanics
or wave mechanics.