PHYS 3313 – Section 001
Lecture #14
Monday, Mar. 3, 2014
Dr. Jaehoon Yu
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•
•
•
•
Bohr’s Hydrogen Model and Its Limitations
Characteristic X-ray Spectra
Hydrogen Spectrum Series
X-ray Scattering
Bragg’s Law
Monday, Mar. 3, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
1
Announcements
• Mid-term exam
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–
–
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In class on this Wednesday, Mar. 5
Covers CH1.1 – what we finish today (CH4.6)+ appendices
Mid-term exam constitutes 20% of the total
Please do NOT miss the exam! You will get an F if you miss it.
BYOF: You may bring a one 8.5x11.5 sheet (front and back) of handwritten
formulae and values of constants for the exam
– No derivations or solutions of any problems allowed!
– No additional formulae or values of constants will be provided!
• Reminder Homework #3
– End of chapter problems on CH4: 5, 14, 17, 21, 23 and 45
– Due: Monday, March 17
• Colloquium this Wednesday at 4pm in SH101
– Dr. Xun Jia of UTSW Medical Center
Monday, Mar. 3, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
2
Monday, Mar. 3, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
3
The Correspondence Principle
Classical electrodynamics
+
Bohr’s atomic model
Determine the properties
of radiation
Need a principle to relate the new modern results with classical ones.
Bohr’s correspondence
principle
Monday, Mar. 3, 2014
In the limits where classical and quantum
theories should agree, the quantum theory
must produce the classical results.
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
4
The Correspondence Principle
• The frequency of the radiation emitted fclassical is equal to the orbital
frequency forb of the electron around the nucleus.
fclassical = fobs
e
æ
ö
e
w
1 v
= 1
=
=
=
3
2p 2p r 2p r 4pe 0 mer 2p çè 4pe 0 me r ÷ø
1
2
12
me e 4 1
=
4 e 02 2 n 3
• The frequency of photon in the transition from n + 1 to n is
fBohr
E0 æ 1
1 ö E0 n 2 + 2n +1- n 2
E0 é 2n + 1 ù
= ç 2=
ê 2
2÷ =
2 ú
2
2
h è ( n ) ( n +1) ø h
h
n
n
+
1
) úû
êë (
n ( n +1)
• For large n the classical limit,
Substitute E0:
fBohr
2nE0 2E0
fBohr »
= 3
4
hn
hn
2 æ e 2 ö me e 4 1
2E0
= 2 2 3 = fClassical
= 3 =
3ç
÷
hn è 8pe 0 a0 ø 4 e 0 n
hn
So the frequency of the radiated E between classical theory and Bohr
model agrees in large n case!!
Monday, Mar. 3, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
5
Importance of Bohr’s Model
• Demonstrated the need for Plank’s constant in
understanding the atomic structure
• Assumption of quantized angular momentum which
led to quantization of other quantities, r, v and E as
follows
4pe 0 2 2
2
• Orbital Radius: rn =
a
n
n = 0
2
mee
• Orbital Speed:
• Energy levels:
Monday, Mar. 3, 2014
n
1
v=
=
mrn ma0 n
e2
E0
En =
= 2
2
8pe 0 a0 n
n
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
6
Successes and Failures of the Bohr Model
• The electron and hydrogen nucleus actually revolve about their
mutual center of mass  reduced mass correction!!
• All we need is to replace me with atom’s reduced mass.
me M
me
me =
=
me + M 1+ me M
• The Rydberg constant for infinite nuclear mass, R∞ is replaced
by R.
me
me e 4
1
R=
R¥ =
R¥ =
2
3
me
1+ me M
4p c ( 4pe 0 )
For H: RH = 1.096776 ´ 10 7 m-1
Monday, Mar. 3, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
7
Limitations of the Bohr Model
The Bohr model was a great step of the new quantum
theory, but it had its limitations.
1) Works only to single-electron atoms
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–
Works even for ions  What would change?
1
1ö
2 æ 1
= Z Rç 2 - 2 ÷
The charge of the nucleus
l
èn n ø
l
u
2) Could not account for the intensities or the fine
structure of the spectral lines
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Fine structure is caused by the electron spin
Under a magnetic field, the spectrum splits by the spin
3) Could not explain the binding of atoms into molecules
Monday, Mar. 3, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
8
Characteristic X-Ray Spectra and Atomic Number
• Shells have letter names:
K shell for n = 1
L shell for n = 2
• The atom is most stable in its ground state.
An electron from higher shells will fill the inner-shell
vacancy at lower energy.
• When a transition occurs in a heavy atom, the radiation
emitted is an x ray.
• It has the energy E (x ray) = Eu − Eℓ.
Monday, Mar. 3, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
9
Atomic Number
L shell to K shell
M shell to K shell
Kα x ray
Kβ x ray
• Atomic number Z = number of protons in the nucleus
• Moseley found a relationship between the frequencies of the characteristic
x ray and Z.
This holds for the Kα x ray
2
f Ka
Monday, Mar. 3, 2014
3cR
=
( Z -1)
4
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
10
Moseley’s Empirical Results
• The x ray is produced from n = 2 to n = 1 transition.
• In general, the K series of x ray wavelengths are
1
1ö
1ö
2æ 1
2æ
= R ( Z -1) ç 2 - 2 ÷ = R ( Z -1) ç 1- 2 ÷
è
ø
è
ø
lK
1
n
n
• Moseley’s research clarified the importance of the
electron shells for all the elements, not just for
hydrogen
• Concluded correctly that atomic number Z, rather than the
atomic weight, is the determining factor in ordering of the
periodic table
Monday, Mar. 3, 2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
11