PHYS 3313 – Section 001
Lecture #13
Wednesday, Feb. 26, 2014
Dr. Jaehoon Yu
•
•
•
•
•
•
Bohr Radius
Bohr’s Hydrogen Model and Its Limitations
Characteristic X-ray Spectra
Hydrogen Spectrum Series
X-ray Scattering
Bragg’s Law
Wednesday, Feb. 26,
2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
1
Announcements
• Mid-term exam
–
–
–
–
–
In class on next Wednesday, Mar. 5
Covers CH1.1 – what we finish on coming Monday, Mar. 3 + 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!
• Homework #3
– End of chapter problems on CH4: 5, 14, 17, 21, 23 and 45
– Due: Monday, March 10
• Colloquium today at 4pm in SH101
– Dr. Billy Quarles of NASA Ames Research Center
Wednesday, Feb. 26,
2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
2
Uncertainties
• Statistical Uncertainty: A naturally occurring
uncertainty due to the number of measurements
– Usually estimated by taking the square root of the
number of measurements or samples, √N
• Systematic Uncertainty: Uncertainty occurring
due to unintended biases or unknown sources
– Biases made by personal measurement habits
– Some sources that could impact the measurements
• In any measurement, the uncertainties provide
the significance to the measurement
Wednesday, Feb. 26,
2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
4
Bohr’s Quantized Radius of Hydrogen
• The angular momentum is
L = r ´ p = mvr = n
n
• So the speed of an orbiting e can be written ve =
me r
• From the Newton’s law for a circular motion
2
e
1 e2 meve
Þ ve =
Fe =
=
2
r
4pe 0 r
4pe 0 mer
• So from above two equations, we can get
n
e
4pe 0 n 2
ve =
=
Þ r=
2
mer
4pe 0 mer
mee
Wednesday, Feb. 26,
2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
2
5
Bohr Radius
• The radius of the hydrogen atom for stationary states is
4pe 0 n 2
rn =
2
mee
2
= a0 n 2
Where the Bohr radius for a given stationary state is:
4pe 0
=
a0 =
2
me e
2
( 8.99 ´10 N × m C ) × (1.055 ´10 J × s )
( 9.11 ´10 kg ) × (1.6 ´10 C )
9
2
-34
2
-31
-19
2
2
= 0.53 ´10-10 m
• The smallest diameter of the hydrogen atom is
d = 2r1 = 2a0 »10-10 m » 1A
– OMG!! The fundamental length!!
• n = 1 gives its lowest energy state (called the “ground” state)
Wednesday, Feb. 26,
2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
6
Ex. 4.6 Justification for nonrelativistic treatment of orbital e
• Are we justified for non-relativistic treatment of
the orbital electrons?
– When do we apply relativistic treatment?
• When v/c>0.1
• Orbital speed:
• Thus
4pe 0 mer
(1.6 ´ 10 ) × ( 9 ´ 10 )
» 2.2 ´10 ( m s )< 0.01c
( 9.1 ´ 10 ) × ( 0.5 ´ 10 )
-16
ve =
ve =
e
Wednesday, Feb. 26,
2014
9
6
-31
-10
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
7
The Hydrogen Atom
• The energies of the stationary states
e2
e2
E0
En = = =
- 2
8pe 0 rn
8pe 0 a0 n 2
n
E0 = -
e2
8pe 0 a0
=
- (1.6 ´10 -19 C )
2
8p ( 8.99 ´10 9 N × m 2 C 2 ) × ( 0.53 ´10 -10 m )
= -13.6eV
where E0 is the ground state energy
 Emission of light occurs when the atom is
in an excited state and decays to a lower
energy state (nu → nℓ).
hf = Eu - El
where f is the frequency of a photon.
æ 1 1ö
f Eu - El E0 æ 1 1 ö
= =
=
- 2 ÷ = R¥ ç 2 - 2 ÷
2
ç
l c
hc
hc è nl nu ø
è nl nu ø
1
R∞ is the Rydberg constant. R¥ = E0 hc
Wednesday, Feb. 26,
2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
8
Transitions in the Hydrogen Atom
• Lyman series: The atom will
remain in the excited state for a
short time before emitting a
photon and returning to a lower
stationary state. All hydrogen
atoms exist in n = 1 (invisible).
• Balmer series: When sunlight
passes through the atmosphere,
hydrogen atoms in water vapor
absorb the wavelengths (visible).
Wednesday, Feb. 26,
2014
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
9
Fine Structure Constant
• The electron’s speed on an orbit in the Bohr model:
n
ve =
=
mern
n
4pe 0 n 2
me
2
me e
2
=
1 e
n 4pe 0
2
• On the ground state, v1 = 2.2 × 106 m/s ~ less than
1% of the speed of light
• The ratio of v1 to c is the fine structure constant,  .
2
e
v1
=
=
=
aº
ma0 c 4pe 0 c
c
( 8.99 ´ 10
Wednesday, Feb. 26,
2014
(1.6 ´ 10 -19 C )
9
2
N × m 2 C 2 ) × (1.055 ´ 10 -34 J × s ) × ( 3 ´ 10 8 m s )
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
»
1
137
10
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
Wednesday, Feb. 26,
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
11