Transcript Atomic Spectra - Ohio Wesleyan University

Atomic Spectra
• Early experimental evidence for discrete energy
states in atoms: observation of the line spectra of
hydrogen (Balmer, Rydberg, Ritz, 1885 – 1890)
– Experimental setup:
(From Modern Physics for scientists and
engineers, Thornton and Rex, 2nd ed.)
• Light emitted at discrete wavelengths, separated by grating
• Principle behind neon signs, sodium street lamps, etc.
Line Spectra of Hydrogen
• Balmer, Rydberg, and Ritz deduced an empirical
formula that predicted the observed wavelengths of
lines in the hydrogen emission spectrum:
1 1
 RH  2  2 

n k 
1
–
–
–
–
RH = Rydberg constant = 1.0973732  107 m–1
n, k are integers
k > n (always)
Understanding this equation theoretically was a hot topic in
the early 20th Century
Bohr Model of Hydrogen
• To explain these results, Niels Bohr suggested
(1913) that the electron in hydrogen moved around
nucleus (proton) in circular orbits
– Similar to a miniature planetary system
– Coulomb attractive force keeps the electron in orbit
(analogous to gravity)
Bohr Model of Hydrogen
• To maintain stable orbits, Bohr also suggested that
the electron’s orbital angular momentum was
quantized:
nh
me vr 
2
 n n  1,2,3...
• In combination with Newton’s
2nd law, this leads to discrete
radii for the electron orbits
n 2 2
2
rn 

a
n
0
2
me ke e
n  1,2,3...
– a0 = Bohr radius = 0.0529 nm
Bohr Model for Hydrogen
• From conservation of energy, the model predicts the
discrete total energies of the electron in hydrogen:
ke e 2
me ke2e 4 1
13.6
E

  2 eV n  1,2,3...
2
2
2r
2 n
n
• Thus the energy levels form a
series of states (like steps in a
ladder), with transitions that can
take place between the levels
– Wavelengths of transitions are in
exact agreement with empirical
result!
 1

1
1
 RH  2  2 
n


n
f
i


Interactive Example Problem:
The Bohr Model
Animation and solution details given in class.
(ActivPhysics online problem 18.1, copyright Addison Wesley, 2006)
Bohr Model for Hydrogen
• Although this model predicts the energies correctly,
it has several deficiencies:
– Only useful for one-electron atoms (only considers e– –
nucleus force, not e– – e– forces)
– Unable to account for doublets of spectral lines in
emission spectra (2 or more very closely spaced lines)
– Cannot calculate transition intensities
– Violates Uncertainty Principle (for circular orbit, we would
know r and pr exactly, which is not possible)
• The correct picture of the atom includes the wave
attributes of the electron
– Electron’s position is “smeared out,” and we are left with
electron “clouds” surrounding the nucleus
Examples of Electron Cloud Distributions
How Lasers Work
• Laser = Light Amplification by Stimulated Emission
of Radiation
• Stimulated
emission:
– Note difference from spontaneous emission
• To produce more stimulated-emission transitions
than absorption transitions, need more atoms in
excited metastable state (“population inversion”) 
then probability of absorbtion  probability of
stimulated emission
How Lasers Work
• For helium – neon laser:
1.
2.
E2
3.
4.
E4 = 20.6 eV
1.96 eV
E3
E2
5.
20.6 eV
E1
E1
Helium (He)
Neon (Ne)
1. Absorption from electric current
2. He atoms collide with Ne atoms
3. E4 (metastable state) in Ne is populated (pop. inversion)
4. Spontaneous emission of 1.96 eV photons ( = 632.8 nm, red
light) produces stimulated emission of other 1.96 eV photons
5. Spontaneous emission transitions “recycle” process
• Schematic of He – Ne laser:
Laser tube
Rear
Flat mirror, 100% reflective
Front
Parallel laser beam
Concave mirror, reflects 99%, transmits 1%
The Atomic Nucleus
• Nuclei consist of both protons and neutrons
(collectively called nucleons)
Designate nucleus by: AZXN (example: 126 C6 )
Z = atomic number = # protons (names the element)
N = neutron number = # neutrons
A = mass number = Z + N
proton (p)
neutron (n)
mass = 1.0072765 u mass = 1.0086649 u
charge = +e
charge = 0
spin = ½
spin = ½
(1 u = 1.66054  10–27 kg = 931.49 MeV / c2)
• For a given element, there are typically several
nuclear isotopes having different A but same Z
Image courtesy of ANL
Nuclear Sizes
• Nuclear sizes can be inferred from scattering
experiments, where the “probe” must have a deBroglie
wavelength on the order of the size being investigated
(~ 1 fm)
Scattered particles
Beam of particles
(e– , p, a, n, etc.)
Nucleus
• From these experiments, we learn:
– Nuclear density is roughly a constant (2.3  1017 kg/m3)
– Nuclear force radius  Mass radius  Charge radius
– Can approximate nuclear radius assuming a spherical
charge distribution as R = r0 A1/3 where r0 = 1.2 fm
– 1 fm = 1  10–15 m (“femtometer” or “fermi”)
Nuclear Binding Energy
The Nuclear Force
• The nuclear force must be strong in magnitude and
attractive to prevent Coulomb repulsion from blowing
the nucleus apart (known as strong force because it’s
the strongest of the known forces)
• Nucleons themselves provide the forces
• The most straightforward way to probe this force is
through scattering experiments
• There are n–p, p–p, and n–n forces (and potentials)
V(r)
n
n
p
n–p
V(r)
p
r
p
p
p
p
p–p
r
The Nuclear Force and Models
• Protons and neutrons experience potential wells in
nucleus, and fill discrete energy states
neutrons
r
~ – 43 MeV
V(r)
protons
r
~ – 37 MeV
• However, we do not fully understand the nuclear
force or how nucleons interact inside the nucleus 
no unified descriptive theory exists for the nucleus!
• Variety of models are used to describe nuclear
behavior
– Nucleons move nearly independently in a common
nuclear potential (Independent Particle Models)
– Nucleons are coupled strongly together and move
collectively (Strong Interaction Models)
Nuclear Models
• Independent Particle Models
– Nucleons move nearly independently in a common
nuclear potential
– Nucleons can be thought of as exerting forces on one
another through the exchange of pions (an elementary
particle with rest energy of 140 MeV)
– Spherical Shell Model
– Deformed Shell Model
• Strong Interaction Models
– Nucleons are strongly coupled together and move
collectively
– Liquid Drop Model
– Rotational Model
– Vibrations
(image above from Krane, Modern Physics, 2nd ed.)
Basic Research in Nuclear Physics
• Nuclear structure studies
– Individual particle motion: How occupation of single-particle
orbits affect shape of nucleus
– Collective motion: How rotations and vibrations affect shape
of nucleus
– Interplay between individual-particle and collective motion
– Deformation studies from charge-distribution/mean lifetime
measurements
– Pairing between protons and neutrons
• Nuclear reaction studies
– Cross section (probability) measurements
– Use polarized projectiles to learn how spin affects reactions
• Nuclear Astrophysics
– Origin and synthesis of the elements
Applications of Nuclear Physics
•
•
•
•
•
•
•
Medicine (gamma radiation used for imaging)
Time dating (lead and carbon dating)
Art (determines trace elements of paints)
Agriculture (gamma radiation kills bacteria)
Small power systems (used in Apollo crafts)
Fission/fusion reactors
Knowledge of the origin of the elements
(production via nuclear reactions)
Image courtesy of ANL
and NASA
How Nuclei are Made in the Lab
Radioactive Decay
• A nucleus can release three forms of radiation
– Gamma (g) decay
• Nucleus is in an excited energy state
• A series of gamma rays (high-energy photons) is released until the
nucleus reaches its lowest-energy (ground) state
• Nuclear isotope remains the same (no change in Z or A)
– Alpha (a) decay
• Heavy nucleus has too large a ratio of neutrons to protons relative
to stable nuclei
• Nucleus emits a particle (4He nucleus)
• Nucleus loses two protons and two neutrons
– Beta (b) decay
• Nucleus has too many protons or neutrons relative to stable nuclei
• If too many protons, p  n and a positron (e+) is emitted to
conserve charge (b+ decay)
• If too many neutrons, n  p and an electron (e–) is emitted to
conserve charge (b– decay)
Radioactive Decay
• Example of b+ decay
– 148O6  147N7 + b+ + n (14O unstable, 14N stable)
– b+ particle is a positron
• Example of b– decay
– 146C8  147N7 + b– + n (14C unstable, 14N stable)
– b– particle is an electron
• Example of a decay
– 23092U138  22690Th136 + a (230U unstable, 226Th
unstable and will also undergo a decay)
– a particle is a 4He nucleus
Radioactive Decay
• All three forms of radioactive decay
follow the exponential decay law
–
–
–
–
N  N0et
 = decay constant
N = number of radioactive nuclei present at time t
N0 = number of radioactive nuclei present at time t = 0
e = 2.718 … = Euler’s constant
• The decay rate (or activity R) follows the same
exponential decay law
R  R et
– R = activity at time t
– R0 = activity at time t = 0
• R and N are related by:
R  N
0
Radioactive Decay
• Plot of N (or R) vs. time:
• The time it takes for half of a
given number of radioactive
nuclei to decay = half life
= T1/2
T1/ 2 
ln 2


0.693

Radioactive Decay Experiment
Example Problem #29.36
A living specimen in equilibrium with the atmosphere
contains one atom of 14C (half-life = 5730 yr) for every
7.70  1011 stable carbon atoms. An archeological
sample of wood (cellulose, C12H22O11) contains
21.0
mg of carbon. When the sample is placed inside a
shielded beta counter with 88.0% counting efficiency,
837 counts are accumulated in one week. Assuming
that the cosmic-ray flux and the Earth’s atmosphere
have not changed appreciably since the sample was
formed, find the age of the sample.
Solution (details given in class):
9.96  103 yr
Example Problem #29.51
A patient swallows a radiopharmaceutical tagged
with phosphorus–32 (32P), a b– emitter with a half-life
of 14.3 days. The average kinetic energy of the
emitted electrons is 700 keV. If the initial activity of
the sample is 1.31 MBq, determine (a) the number of
electrons emitted in a 10–day period, (b) the total
energy deposited in the body during the 10 days,
and (c) the absorbed dose if the electrons are
completely absorbed in 100 g of tissue.
Solution (details given in class):
(a) 8.97  1011
(b) 0.100 J
(c) 100 rad