Download Bohr Atom, Atomic spectra

PHYS274 Spring 2017
Homework #6 due this Friday, March 3
– Starting with Photons
– Chap 39 in its entirety
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Q19.1
In the Rutherford scattering experiment, Ernest
Rutherford’s students shot what kind of particles at
the gold foil target?
A.
B.
C.
D.
Electrons
Protons
Neutrons
Alpha particles
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Q19.1
In the Rutherford scattering experiment, Ernest
Rutherford’s students shot what kind of particles at
the gold foil target?
A.
B.
C.
D.
Electrons
Protons
Neutrons
Alpha particles
Helium nuclei from a radioactive source
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Q19.2
If you increase the kinetic energy of the electrons
in an electron microscope, you will
A.
B.
C.
D.
Improve the resolution of the microscope
Degrade the resolution of the microscope
The resolution will remain unchanged.
The electrons cannot be captured by the
magnetic focus.
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Q19.2
If you increase the kinetic energy of the electrons
in an electron microscope, you will
A.
B.
C.
D.
Improve the resolution of the microscope
Degrade the resolution of the microscope
The resolution will remain unchanged.
The electrons cannot be captured by the
magnetic focus.
Why?
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Q19.3
If you want to reduce the wavelength of the electron
by a factor of two in an electron microscope, you
A. Decrease the accelerating voltage by half
B. Increase the accelerating voltage by 2 times
C. Increase the accelerating voltage by 4 times
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Q19.3
If you want to reduce the wavelength of the electron
by a factor of two in an electron microscope, you
A. Decrease the accelerating voltage by half
B. Increase the accelerating voltage by 2 times
C. Increase the accelerating voltage by 4 times
V_ba = h^2/(2m e lambda^2)
lambda^2 = h^2/(2 m e V_ba)
Lambda ~ 1/(sqrt[ V_ba])
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“Ultra-violet catastrophy!”
Why doesn’t this happen ?
Cliffhanger…
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Review: Breakdown of classical physics (Crisis)
• Rutherford’s experiment
suggested that electrons orbit
around the nucleus like a
miniature solar system.
• However, classical physics
predicts that an orbiting
electron (accelerating charge)
would emit electromagnetic
radiation and fall into the
nucleus. So classical physics
could not explain why atoms
are stable.
Question: What is the
solution to this crisis ?
There is a ground
state energy level
OK, what does THAT mean?
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Quantization of atomic energy levels (Experimental)
Three classes of
spectral features:
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Quantization of atomic energy levels (visual evidence)
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Quantization of atomic energy levels
• Niels Bohr explained atomic line spectra and the stability of
atoms by postulating that atoms can only be in certain discrete
energy levels. When an atom makes a transition from one
energy level to a lower level, it emits a photon whose energy
equals that lost by the atom.
• An atom can also absorb a photon, provided the photon energy
equals the difference between two energy levels.
Insert Figure 39.16
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Quantization of atomic energy levels
• An atom can also absorb a photon, provided the photon energy
equals the difference between two energy levels.
The master equation for the photon
energy in these transitions is
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Next exercise
Are you ready for Bohr Model Bootcamp ?
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The Bohr model of hydrogen (original argument)
• Bohr explained the line
spectrum of hydrogen with a
model in which the single
hydrogen electron can only be
in certain definite orbits.
Ln=rp=m vn rn
• In the nth allowed orbit, the
electron has orbital angular
momentum nh/2π (see Figure
on the right).
• Bohr proposed that angular
momentum is quantized (this
will turn out to be correct in
general in quantum mechanics
but is not right for the hydrogen
atom).
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The Bohr model of hydrogen
Let’s use a different argument based on
deBroglie waves to obtain the same conclusions.
Think of a standing wave with wavelength λ
that extends around the circle.
Q: How is the momentum of the atomic electron
related to its wavelength ? (remember the Prince)
Same as the Bohr
quantization
condition
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The Bohr model of hydrogen
Now let’s use a Newtonian
argument for a planetary
model of the atom but use
the Bohr quantization
condition. (A little hokey).
(The mass m is that
of the electron.)
Balance electrostatic
and centripetal forces
Here we used the Bohr quantization condition
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The Bohr model of hydrogen (Bohr radius)
Here n is the “principal quantum number” and a0 is the “Bohr
radius”, which is the minimum radius of an electron orbital.
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The Bohr model of hydrogen (iClicker Interlude)
A muon is a “heavy electron” (~100 MeV/c^2 vs. 0.5 MeV/c^2)
and we are continually bombarded by them
They rain down continuously (will
cover later), and some of them lose
energy and slow down near an atom.
Will its orbit?
A. Further away
B. Nearer to nucleus
C. Accelerate inward, destroy the
nucleus and all life as we know
it !
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The Bohr model of hydrogen (Energy levels, derivation)
Note that E and U are
negative (1/8-1/4=-1/8)
This expression for the allowed energies can be rewritten and used
to predict atomic spectral lines !
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The Bohr model of hydrogen (iClicker check)
The velocity of the electron in the ground
state, compared with the first excited state
(n=2) is
A.
B.
C.
D.
E.
¼ as fast
½ as fast
The same
Twice as fast
4 times as fast
1
vn = v1
n
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The Bohr model of hydrogen (Energy levels)
Here R is the “Rydberg
constant”, R=1.097 x 107 m-1
Also hcR = 13.60 eV is a useful result.
Question: How can we find the energies of
photon transitions between atomic levels ?
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The Bohr model of hydrogen (It works)
Here R is the Rydberg
constant, R=1.097 x 107 m-1
If nupper=3, nlower=2, let’s calculate the
wavelength.
Balmer Hα line, agrees with experiment within 0.1%
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Hydrogen spectrum (also has other spectral lines)
•
The line spectrum at the bottom of the previous slide is not the entire
spectrum of hydrogen; it is just the visible-light portion.
•
Hydrogen also has series of spectral lines in the infrared and the
ultraviolet.
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Hydrogen-like atoms
•
The Bohr model can be applied to any atom with a single electron. This
includes hydrogen (H) and singly-ionized helium (He+). See the Figure
below.
Question: How should this
formula be modified for
singly-ionized helium ?
Ans: He has 2p. If an
atom is singly ionized,
then rnrn/ZEnZ2E
But the Bohr model does not
work for other atoms; need QM
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For next time
• Photons as Quantum Mechanics gateway
 Read material in advance
 Concepts require wrestling with material
• Homework #6 available
– Due date this Friday, Mar. 3rd
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