PHYS 141: Principles of Mechanics

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PHYS 141 Principles of Mechanics
PART THREE QUANTUM MECHANICS
PART THREE QUANTUM MECHANICS
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I. Blackbody Radiation A. Properties

• 1. Definition
• a) A blackbody is a perfect absorber of light at
  all wavelengths.
• b) Wiens Law The peak wavelength of emission
  for a bb decreases as temperature increases
• lp 2.898 x 10-3 (m-K)/T, (I.A.1)
• c) The Stefan-Boltzmann Law The power (E/s)
  P/m2 of a blackbody is given by
• Pbb sAT4, (I.A.2)
• where s 5.67 x 10-8 W/(m2K4).

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I. Blackbody Radiation A. Properties

• c) The Stefan-Boltzmann Law
• Further define an energy Flux, F P/A
• F sT4 (I.A.3)
• Hotter blackbodies emit more total power per
  area.
• Properties of blackbody described principally by
  T.
• d) Radiance radiated power as a function of
  wavelength and temperature (Max Planck).
• (I.A.4)
• (I.A.5-6)

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I. Blackbody Radiation A. Properties
Light bulb off No light at any l.
Radiance
Visible
IR
Radio
UV
5
I. Blackbody Radiation A. Properties
The combined brightness at each l
(color) determines what color a BB appears.
Light bulb on T 3000 K.

Dim in Violet And Blue
Radiance
Fairly bright in Red Orange
Visible
IR
Radio
UV
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I. Blackbody Radiation A. Properties
Light bulb on T 6000 K.

Bright in YG
Radiance
Fairly bright in Red Violet
Visible
IR
Radio
UV
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I. Blackbody Radiation A. Properties
Light bulb on T 30,000 K.

Very Bright in V,I,B
Radiance
Bright in G,Y,O,R
Visible
IR
Radio
UV
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I. Blackbody Radiation A. Properties

• e) The Ultraviolet Catastrophe!
• Classical thermal physics predicted infinite
  flux and infinite intensity at small
  wavelengths
• The solution Plancks hypothesis
• Electric resonators responsible for bb
  radiation have discrete (quantized) energies
• En nhf. (I.A.7)
• where h is Plancks constant,

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I. Blackbody Radiation A. Properties

• B. The Photoelectric Effect
• Einstein proposed the particle model of light
  photons.
• Ephoton hf nhc/l. (I.B.1)
• 2. Extension of Plancks work on molecular
  oscillators

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I. Blackbody Radiation A. Properties

• B. The Photoelectric Effect
• Einstein proposed the particle model of light
  photons.
• Ephoton hf nhc/l. (I.B.1)
• 2. Extension of Plancks work on molecular
  oscillators

-e
-e
-e
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I. Blackbody Radiation A. Properties

• B. The Photoelectric Effect
• Einstein proposed the particle model of light
  photons.
• Ephoton hf nhc/l. (I.B.1)
• 2. Extension of Plancks work on molecular
  oscillators

-e
-e
-e
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I. Blackbody Radiation A. Properties

• B. The Photoelectric Effect
• Einstein proposed the particle model of light
  photons.
• Ephoton hf nhc/l. (I.B.1)
• 2. Extension of Plancks work on molecular
  oscillators

-e
-e
-e
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I. Blackbody Radiation A. Properties

• B. The Photoelectric Effect
• Einstein proposed the particle model of light
  photons.
• Ephoton hf nhc/l. (I.B.1)
• 2. Extension of Plancks work on molecular
  oscillators

-e
-e
-e
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I. Blackbody Radiation A. Properties

• B. The Photoelectric Effect
• Einstein proposed the particle model of light
  photons.
• Ephoton hf nhc/l. (I.B.1)
• 2. Extension of Plancks work on molecular
  oscillators

-e
-e
-e
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I. Blackbody Radiation A. Properties

• B. The Photoelectric Effect
• Einstein proposed the particle model of light
  photons.
• Ephoton hf nhc/l. (I.B.1)
• 2. Extension of Plancks work on molecular
  oscillators

-e
-e
-e
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I. Blackbody Radiation A. Properties

• B. The Photoelectric Effect
• Einstein proposed the particle model of light
  photons.
• Ephoton hf nhc/l. (I.B.1)
• 2. Extension of Plancks work on molecular
  oscillators

-e
-e
-e
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I. Blackbody Radiation A. Properties

• 3. Particle-Theory of Light predictions RIGHT!
• a) As the frequency of light increases, the
  maximum K of electrons also increases
• Kmax hf - f, (B.2)
• Where f is the Work Function that holds the
  electrons to the plate.
• b) If f lt f0 cutoff frequency f/h, no
  electrons ejected.
• c) Maximum electron energy (as measured by the
  stopping potential , VS) is independent of
  intensity.

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I. Blackbody Radiation A. Properties
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I. Blackbody Radiation A. Properties

• Example Light shines on a substance with f 1
  eV.
• If ? 500 nm, are electrons ejected from the
  surface?
• Step 1 Find the energy of the photons. For ?
  500 nm, we have
• E hc/? (6.6 x 10-34 J-s)(3.0 x 108 m/s)/(5
  x 10-7 m)
• 4.0 x 10-19 J
• E (4.0 x 10-19 J)(1 eV/1.6 x 10-19 J) 2.5
  eV.
• Step 2 Find Kmax 2.5eV - 1.0 eV 1.5 eV gt
  0.
• So, yes, electrons are ejected.

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II. Light, More Light

• A. Compton Effect (1923)
• Scattered light has lower frequency than incident
  light.
• Recall that relativistic energy is E pc for a
  massless particle. Thus, the relativistic
  momentum for a photon is
• p E/c hf/c h/?. (A.1)

?
? ?
e-
?
e-
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II. Light, More Light
?

• Conserve momentum energy for electron initially
  at rest

? ? (h/mc)(1 - cosq). (A.2) Dl (h/mc)(1
- cosq). (A.3)
and (h/mc) ?C Compton Wavelength. (A.4)
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II. Light, More Light

• 3. Example How much energy is lost by a 1 MeV
  photon that scatters off an electron with an
  angle q 60o?
• Ei hf hc/? 1.6 x 10-13 J ?? 0.0012 nm.
• ? ? (h/mec)(1 - cosq).
• 1.2 x 10-12 m (2.4 x 10-12 m)(1 - .5)
• 2.4 x 10-12 m 0.0024 nm.
• Since the wavelength doubled, the Ef hc/?
  1/2Ei 0.5 MeV.
• 1 MeV 1.602 x10-13 J.

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II. Light, More Light B. Light as a Wave

• 1. And yet light also behaves as a wave
• a) Young Double Slit experiment interference
• b) Maxwells Equations
• b) Connection between frequency and wavelength
• lf c. (B.1)
• c) Electromagnetic Spectrum

Low Energy
High Energy
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II. Light, More Light C. Waves of Matter?

• de Broglie (1923) If light is observed to have
  wavelike properties some times (interference) and
  particle-like properties other times
  (photoelectric effect), then what about matter?
• ? h/mv de Broglie Wavelength (I.C.1)
• Example what is ? for an electron moving at
  .01c?
• ? (6.6 x 10-34 J-s)/(9.11 x 10-31 kg3.0 x 106
  m/s)
• ? 2.4 x 10-10 m 0.24 nm.

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II. Light, More Light D. Atomic Spectra

• 1. Spectral Analysis
• a) SA The identification of a chemical
  substance by its unique spectral lines.
• b) Joseph Fraunhofer (1815) Hundreds of dark
  lines in the Solar Spectrum
• c) The Value of Spectral Analysis
• Composition, abundance
• Temperature
• Motion

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II. Light, More Light

• 2. The visible Hydrogen series (Balmer Series)
• 1/l R(1/22 -1/n2), n 3, 4, 5 (I.D.1)
• Where R 1.097 x 107 m-1 is the Rydberg
  constant.

Balmer lines 656 nm (B?, n 3). are
transitions 486 nm (B?, n 4). to/from n
2 434 nm (B?, n 5).
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II. Light, More Light E. The Bohr Model

• 1. General aspects of the model
• a) Hydrogen Atom as simple example
• b) Electron orbits are quantized not all orbits
  stable.
• c) To move from one orbit to the next, an
  electron needs to absorb or emit an exact DE.
• d) Larger orbital differences mean more energy
  required to move
• e) Packet of energy a photon!
• f) Model later modified from strict orbits
  about the nucleus to energy levels

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II. Light, More Light

• 2. The mathematics of quantization
• a) Frequency of radiation absorption/emission
• Eu - El hf. (I.E.1)
• b) Quantum condition only specific values of
  electron orbital angular momentum allowed
• L mevrn n(h/2p), n 1, 2, 3, (I.E.2)

Eu
hf
e-
El
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II. Light, More Light

• c) Electrostatic Force
• F kq1q2/r2 (I.E.2a)
• PE -kq1q2/r. (I.E.2b)
• with
• k 8.99 x 109 Nm2/C2 is the Coulomb constant
• and the unit of charge is the Coulomb
• e 1.602 x 10-19 C.

q1
r
q2
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II. Light, More Light

• d) Total energy of hydrogen atom
• PE -ke2/rn. (Electric Potential energy)
  (I.E.3)
• KE (1/2)mev2. (I.E.4)
• Thus, E PE KE -ke2/r (1/2)mev2.
• Now assume that the electron orbit is circular
• F ke2/rn2 mev2/rn 2rn(KE), so (I.E.5)
• KE ke2/(2rn), and (I.E.6)
• E -ke2/(2rn). (I.E.7)

-e
rn
e
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II. Light, More Light

• e) Orbital sizes use the quantum condition
  (E.1)
• rn n(h/2p)/mev, and
• rn2 n2(h/2p)2/me2v2.
• From (E.5), ke2/rn2 mev2/rn, so v2
  (ke2/me)/rn. Thus,
• rn n2(h/2p)2/(ke2me). (I.E.8)

Define the Bohr radius r1 r(n1) 0.0529
nm, so rn n2r1. (I.E.9)
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II. Light, More Light

• Energy levels of orbits
• En -mek2e4/(2(h/2p)2(1/n2), or (I.E.9)
• En (-13.6 eV)/n2 E1/n2. (I.E.10)

Energy required for ionization DEi E(n
infinity) (13.6 eV)/n2 (13.6
eV)/n2. (I.E.11)
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II. Light, More Light

• F. Quantization and atomic spectra (H)
• Ground state n 1, E1 -13.6 eV.
• First excited state n 2, E2 -3.4 eV.
• Energy required to move from E1 to E2
• DE -3.4 eV - -13.6 eV 10.2 eV.
• Hydrogenic atoms Z number of protons in
  nucleus.
• En (Z/n)2E1. (F.1)
• rn (n2/Z)r1. (F.2)

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II. Light, More Light

• Photon frequency/wavelength
• f DE/h meke2e4/4p(h/2p)3(1/nf2 -
  1/ni2),
• or
• f cR(1/nf2 - 1/ni2), and (I.F.3)
• 1/? R(1/nf2 - 1/ni2). (I.F.4)

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II. Light, More Light

• Quantization and atomic spectra
• Spectral types explained!
• Unique electron orbits gt unique energy
  differences gt unique patterns
• Absorption/Emission spectra understood
  as upward/ downward transitions of
  electrons.

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II. Light, More Light

• G. The Wave Nature of Matter (revisited)
• 1. de Broglies wave description of electrons can
  now be understood in terms of standing waves
  surrounding a nucleus only waves that close
  back on themselves can constructively interfere
  and survive

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III. Quantum Mechanics

• The Failure of the Bohr Model
• 1. Couldnt be applied to more complicated atoms
• 2. Couldnt explain fine structure
• 3. Couldnt explain solids, liquids, molecules
• 4. Quasi-classical mechanics

The Bohr Model, while substantial, was incomplete.
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III. Quantum Mechanics B. Wave Mechanics

• Define a Wave Function, ?, that describes a
  particle.
• a) Wave function contains information about a
  particles state position, speed, momentum
  energy, spin, etc.
• b) Matter Wave or Matter Field
• c) Depends on position and time ??????(x,y,z,t).
• d) Probabilities
• ? Probability density
• ?2 Probability of finding an electron at
  position (x,y,z) and time t.

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III. Quantum Mechanics B. Wave Mechanics

• Calculations Solution to the Schroedinger Wave
  Equation
• a) One dimensional case, Cartesian coordinates.
  Probability of finding a particle between x and x
  dx
• b) Normalization The particle must be
  SOMEWHERE. Therefore

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III. Quantum Mechanics B. Wave Mechanics

• Calculations Solution to the Schroedinger Wave
  Equation
• c) Radial Probability distribution. Probability
  of finding a particle between r and r dr
• d) Normalization The particle must be
  SOMEWHERE. Therefore

(B.1)
(B.2)
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III. Quantum Mechanics B. Wave Mechanics

• 3. Example Ground State of Hydrogen
• a) Wave function for ground state
• b) Step 1 Find the constant by normalization

(B.3)
(B.4a,b)
(B.5)
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III. Quantum Mechanics B. Wave Mechanics

• 3. Example Ground State of Hydrogen
• c) Now ask What is the probability of finding
  an electron inside the Bohr Radius, a0? (see pg.
  649 and Appendix A the for integral)

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III. Quantum Mechanics C. HUP

• Heisenberg Uncertainty (Indeterminancy) Principle
• There is an inherent uncertainty in pairs of
  correlated variables
• Momentum-Position Uncertainty Principle
• ?x?p (h/2?). (III.C.1)
• The original argument comes from measurement of
  an electron. The uncertainty in the electrons
  position is roughly the wavelength of observation
  (?), and the photon will give some unknown
  portion of its momentum to the electron (h/?)
  when it strikes the electron. Trying to narrow
  the position means using smaller ?, but that
  imparts more (unknown) momentum to the electron.

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III. Quantum Mechanics C. HUP

• 3. Energy-Time Uncertainty Principle
• ?E?t (h/2?). (III.C.2)
• In this UP, the energy of a particle is
  uncertain (?E), over a time scale ?t (h/2?)/?E.
• Warning Part of the problem is that we tend to
  view electrons, etc. exclusively as particles.
  Indeterminancy is a direct result of electrons,
  protons, etc. having both wave AND particle
  properties.

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III. Quantum Mechanics C. HUP

• Example What is the maximum precision for
  measuring the position of an electron moving with
  a speed equal to
• v (3.00 0.01) x 106 m/s?
• ?x?p (h/2?) gt
• ?xmin (6.6 x 10-34 J-s/2?)/(9.1 x 10-31
  kg)(104 m/s),
• 1.2 x 10-8 m,
• 12 nm, or about 100x the size of an atom.

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III. Quantum Mechanics C. HUP

• 5. Example What is the lifetime of a particle
  with a spread in energy equal to 1kev?
• ?E?t (h/2?) gt
• ?t (6.6 x 10-34 J-s/2?)/(1000eV)(1.6 x 10-19
  J/eV),
• 6.6 x10-19 s.

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III. Quantum Mechanics D. Quantum Numbers

• Principal quantum number n (e.g., En E1/n2).
• a) Related to energy of electron
• b) n 1, 2, 3, ?
• Orbital Quantum number l
• a) related to orbital angular momentum vector of
  electron
• L l(l 1)1/2 , where h/(2?), and
  (D.1)
• l 0, 1, 2, , n-1. (D.2)

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III. Quantum Mechanics D. Quantum Numbers

• 3. Magnetic quantum number ml.
• a) Related to direction of electrons angular
  momentum
• - l, - l 1, - l 2, , 0, l - 2, l - 1,
  l. (D.3)
• b) Angular momentum along the z-axis
• Lz ml (D.4)

Lz
ml 1.
Total number of states for a given value of l
depends on projection Lz.
0
ml 0.
-
ml -1.
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III. Quantum Mechanics D. Quantum Numbers

• 3. Magnetic quantum number ml.
• c) Energy levels are split in the presence of
  magnetic field.

ml 1.
n 2 l 1
ml 0.
ml -1.
n 1 l 0
ml 0.
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III. Quantum Mechanics D. Quantum Numbers

• 4. Spin quantum number ms.
• a) originally thought of as intrinsic spin of
  electron
• b) ms 1/2, but s ½ always (for an electron)
• c) fine structure

Electron spin vector is given by
(D.5).
n 2 l 0
ml 0.
n 1 l 0
ml 0.
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III. Quantum Mechanics D. Quantum Numbers

• 5. Total Angular Momentum Vector sum of J L
  S

For l ? 0 j l ½ (parallel vs. antiparallel)
(D.6) For l 0 j ½ (D.7)
(D.8) Total number of angular momentum states
depends on projection Jz -j, -j1, , j-1, j
gt 2j 1 states. Nomenclature
n(l)j, (D.9) Where l 0, 1, 2, 3, 4 gt S, P,
D, F, G
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III. Quantum Mechanics D. Quantum Numbers

• Selection Rules and Probability Distributions
• a) Allowed transitions ?l 1.

l
n
l and ml change the shape of the probablitiy
distribution function.
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III. Quantum Mechanics E. Other Atoms

• 1. The Pauli Exclusion Principle
• a) Electrons in the same atom cannot have the
  same quantum state--each electron has a unique
  set of quantum numbers.
• b) Usually electrons are found in the lowest
  possible state.
• c) Similar exclusion principle for any particle
  that has half integer values of spin (e.g.,
  protons neutrons) Fermions.
• d) Particles with integer values of spin do not
  obey EP
• (e.g., photons) Bosons.

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III. Quantum Mechanics E. Other Atoms

• e) Examples
• He (Z 2). 2e (n, l, ml, ms ) (1, 0, 0,
  1/2).
• (n, l, ml, ms ) (1, 0, 0, -1/2).
• Li (Z 3). 3e (n, l, ml, ms ) (1, 0, 0,
  1/2).
• (n, l, ml, ms ) (1, 0, 0, -1/2).
• (n, l, ml, ms ) (2, 0, 0, 1/2).
• Be (Z 4). 4e (n, l, ml, ms ) (1, 0, 0,
  1/2).
• (n, l, ml, ms ) (1, 0, 0, -1/2).
• (n, l, ml, ms ) (2, 0, 0, 1/2).
• (n, l, ml, ms ) (2, 0, 0, -1/2).

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IV. Nuclear Physics A. Radioactive Decay

• 1. Radioactive Decay Equation
• N/N0 e-?t, (A.1)
• where ? is the decay rate.
• 2. Half-life The time for half the sample to
  decay from parent isotope to daughter isotope
  t1/2.
• (A.2)
• (A.3)

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IV. Nuclear Physics A. Radioactive Decay

• Rewriting the RDE, we have
• N/N0 exp(-0.693t/t1/2). (A.4)
• 3. Define activity as the number of
  decays/second,
• ?N/?t -?N (0.693/t1/2)N (?N/?t)0e-?t.
• 4. Decay channels
• a-particle emission He nucleus (2p, 2n) gt Z
  - 2, A - 4
• b-particle emission e- gt Z1, A
• g-particle emission high energy photon gt Z, A

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IV. Nuclear Physics A. Radioactive Decay

• Example 238U as a half-life of 4.5 billion
  years. If an initial sample contains 1018 atoms,
  what is the initial activity?
• ? 0.693/t1/2 5 x 10-18 s-1.
• The initial activity for the sample is
• (?N/?t)0 ?N (2 x 10-18 s-1)(1018 nuclei) 5
  decays/second.
• What is the activity after 13.5 billion years?
• (?N/?t) (?N/?t)0 e-?t 0.6 decays/second
  1/8(?N/?t)0

58
IV. Nuclear Physics A. Radioactive Decay

• Example 238U as a half-life of 4.5 billion
  years. If an initial sample contains 1018 atoms,
  what is the initial activity?
• After 13.5 billion years, how much 238U is left?
• N/N0 exp(-0.693t/t1/2) exp(-0.693(3))
  1/8.
• 3 half-lives gt (1/2)(1/2)(1/2).

59
IV. Nuclear Physics B. Nuclear Structure

• 1. Nomenclature
• Z proton (element) number
• N neutron number
• A atomic mass number, e.g., 147N or 136C
• 2. Spin
• Each nucleon (p,n) is a fermion
• Nuclear spin is quanitized (I, Iz) resonances
  NMR
• Each nucleus may be a fermion or a boson
  depending on the number of nucleons. For even
  and equal numbers of p n, nucleons are pairedgt
  spin 0
• 42He gt spin 0 (boson)
• 32He gt spin ½ (fermion)

60
IV. Nuclear Physics B. Nuclear Structure

• 3. Nuclear stability
• a) Nuclei have shell structure analogous to
  atomic shells
• b) Z,N 2, 8, 20, 28, 50, 82, 126 stable
• c) PEP applies to n,p gtforces nucleons into
  energy levels
• Ground State configurations
• 42He p n s 0
• 32He p n s ½
• 126C
• p n

Even Z, N minimize total energy and
increase stability of nucleus.
E3
E2
E1
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IV. Nuclear Physics C. Energy

• 1. Reactions
• a X gt Y b. (IV.C.1)
• X(a,b)Y. (IV.C.2)
• 2. Reaction energy
• Q (Ma MX - Mb -MY)c2. (IV.C.3)
• ?KE KEb KEY - KEa - KEX. (IV.C.4)
• Q gt 0 Exothermic (energy released).
• Q lt 0 Endothermic (energy required).

62
IV. Nuclear Physics C. Energy

• 1. Reactions
• a X gt Y b. (IV.C.1)
• X(a,b)Y. (IV.C.2)
• 2. Reaction energy
• Q (Ma MX - Mb -MY)c2. (IV.C.3)
• ?KE KEb KEY - KEa - KEX. (IV.C.4)
• Q gt 0 Exothermic (energy released).
• Q lt 0 Endothermic (energy required).

63
IV. Nuclear Physics C. Energy

• 3. Binding Energy
• a) Unified mass
• 1u 931.5 MeV/c2.
• b) Definition The energy required to hold a
  nucleus together.
• M(proton) 1.00728 u M(42He) 4.00153 u
• M(neutron) 1.00866 u
• BE (2Mp 2Mn)c2 MHec2
• (4.03188 uc2 - 4.00153 uc2 )(931.5 MeV/uc2)
• 28.3 MeV.

64
IV. Nuclear Physics C. Energy

• 4. Reaction Example n 147N ? 21H X. What is
  X?
• In this example, a neutron strikes a 147N
  nucleus producing deuterium (21H) and another
  element. To find X, conserve nucleon number and
  charge
• 14 (nitrogen) 1 (neutron) A 2 (deuterium).
• 7e (in nitrogen nucleus) qX 1e (deuterium).
• Thus, A 13 and qX 6, and X 136C.

65
IV. Nuclear Physics C. Energy

• 5. Example What is the minimum proton kinetic
  energy required for the reaction (11H 136C ?
  137N n) to take place?
• Step one compute rest energy on both sides
• M(136C) 13.003355 u M(137N) 13.005738 u
• M(11H) 1.007825 u M(n) 1.008665 u
• (Mafter - Mbefore)c2 (0.003223 uc2)(931.5
  MeV/uc2) 3.00 MeV.
• Thus, the proton would have to have
• kinetic energy of at least 3.00 MeV.

66
IV. Nuclear Physics D. Fission

• 1. Definition the splitting of one nucleus into
  multiple nuclei after particle bombardment.
• 2. Example 23592U
• a) n 23592U ? 23692U ? 14156Ba 9236Kr
  3n.
• b) n 23592U ? 8838Sr 13654Xe 12n.
• 3. Energy released in single fission event
• Efission ?Mc2 (D.1)

67
IV. Nuclear Physics D. Fission

• Example What is the energy released by the
  reaction n 23592U ? 8838Sr 13654Xe 12n?
• Before Mi MU Mn 235.0439 u 1.008665 u
  236.053 u.
• After Mf MSr MXe M12n 235.917 u.
• Efission ?Mc2 (0.135 uc2)(931.5 MeV/uc2)
  126 MeV.

68
IV. Nuclear Physics D. Fission

• Chain Reactions
• Sustained nuclear reaction requires chain
  reaction
• Moderator to slow neutrons
• Enrichment
• Critical mass kilograms

69
IV. Nuclear Physics E. Fusion

• Definition the fusing of two or more nuclei
  into new nucleus.
• Example Proton-Proton Chain in the Sun.

70
First Step in Proton-Proton Chain 11H 11H ?
21H e ?e 0.42 MeV. Likelihood of
interaction cross-section of interaction between
two protons (t 109 years.)
71
Second Step in Proton-Proton Chain 21H 11H ?
32He ? 5.49 MeV. t 1 second.
?
72
Third Step in Proton-Proton Chain 32He 32He
? 42He 11H 12.86 MeV t 106 years. Total
products 42He 2?e 2? 2e 26.7 MeV
73
Bottlenecks Deuterium bottleneck H to He
Fusion High Mass Bottleneck no stable nuclei
with 5 or 8 nucleons (protons or neutrons) e.g,
42He 42He ? 84Be is endothermic the product
has a half-life of 10-16 s. Fusion reactions
tend to favor steps built on 4He, rather than
proton capture, (but process is SLOW.)
?
?
74
IV. Nuclear Physics E. Fusion

• 2. Example Proton-Proton Chain in the Sun.
• a) Deuterium Bottleneck
• b) Energy produced in one chain 27 MeV.
• c) 2 gamma rays positrons 2 neutrinos
• d) Requires high temperature (fast moving
  particles) Tcore 15 million K

Use of plasma with similar parameters as solar
interior plasma for energy production on Earth is
completely impractical - as even modest 1 GW
fusion power plant would require about 170
billion tons of plasma occupying almost one cubic
mile.
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IV. Nuclear Physics E. Fusion

• 3. Solar life-time estimate How long can the
  Sun shine?
• Lsun 4 x 1026 W 4 x 1026 J/s 2.5 x 1039
  MeV/s.
• The mass of the Suns core is 5 x 1029 kg, or
  2 x1056 protons.
• In the p-p chain, 4p gt He 27 MeV. The number
  of reactions per
• second is then 2.5 x 1039 MeV/s/(27 MeV)
  1038.
• Every second, 4 x 1038 H become He (billion
  metric tons).
• How long before there is no more hydrogen?
• 2 x 1056 p/(4 x 1038 p/s) 5 x 1017 s 10
  billion years.

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IV. Nuclear Physics E. Fusion

• 4. Other Fusion reactions
• C-N-O cycle
• H-fusion in higher mass
• stars and later evolution
• of the Sun.
• 12C is a nuclear catalyst.
• Yield is 27 MeV.

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IV. Nuclear Physics E. Fusion

• Example How much energy is
• released in the reaction
• 136C(11H,?)147N?
• ?M M(13C) M(1H) - M(14N),
• 0.008106 u
• E (0.008106uc2)(931.5 MeV/uc2),
• 7.55 MeV 1.2 x 10-12 J.
• ? gt hc/E (6.6 x 10-34 Js)(3 x 108 m/s)/(1.2 x
  10-12 J) 0.000165 nm.

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IV. Nuclear Physics E. Fusion

• 4. Other Fusion reactions
• b) Triple-alpha He-Fusion
• Ignition T 108 K
• Occurs in the Sun when
• H core is depleted.

4He 4He gt 8Be 4He gt 12C ? 7.367 MeV.
Side process 12C 4He ? 16O ?
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5. Massive Stars Nucleosynthesis 25 MSUN Star
Saturn Orbit 3 billion km
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5. Massive Stars Nucleosynthesis 25 MSUN Star
Earth Diameter 12, 000 km
H fusion
He fusion

• C fusion
• O fusion
• Ne fusion
• Si fusion

Fe
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5. Massive Stars Nucleosynthesis 25 MSUN Star
Fe
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5. Massive Stars Nucleosynthesis 25 MSUN Star
10 km
n
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Stellar fusion
Built from alpha capture
Neutron capture, decay products
BBN
spallation
Built from identical nucleus fusion n, p release
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V. Cosmology A. The Pillars of Modern Cosmology

• 1. General Relativity
• a) Spacetime is curved by the mass-energy in it.
  Matter tells space how to curve, Space tells
  Matter how to move.
• 2. The Cosmological Principle
• b) At the largest scales, the
• Universe is both isotropic
• and homogeneous.

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V. Cosmology A. The Pillars of Modern Cosmology

• The Shape of Space Given GR and the CP,
  mass-energy density of the Universe allows only
  three possible geometries

1.Spherical (positive curvature) 2.Hyperboli
c (negative curvature) 3.FlatEuclidean
(zero curvature)
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V. Cosmology A. The Pillars of Modern Cosmology

• 1. General Relativity
• a) Spacetime is curved by the mass-energy in it.
  Matter tells space how to curve, Space tells
  Matter how to move.
• 2. The Cosmological Principle
• b) At the largest scales, the
• Universe is both isotropic
• and homogeneous.

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V. Cosmology B. Observations

• 1. Observational Data and Cosmological Theory
• a) Galaxy Recession
• b) Elemental Abundances
• c) Cosmic Microwave Background

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V. Cosmology B. Observations

• 2. Galaxy Recession Why are Galaxies Moving
  Away from Us?
• a) Vesto Slipher
• (1910s)
• b) Hubble Humason
• (1920s)
• c) Doppler Shift

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(B.1)
(B.2)
Similar shift in frequency to smaller frequencies
for recession and larger frequencies for approach.
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V. Cosmology B. Observations

• 3. The Hubble Law
• vCosmo H0d, (B.4)
• where v is in km/s and
• d is in Mega-light years
• or Mega-parsecs.
• Current WMAP measurements give

(km/s)/Mly.
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V. Cosmology B. Observations

• 4. Age of the Universe
• We can estimate the age
• of the Universe using the
• Hubble constant
• H0 s-1, so
• t(Universe) 1/H0 13.7 0.1 Gyr.

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V. Cosmology C. The Big Bang

• 1. Standard Model (prior to 1980)
• a) Combines CP, GR
• b) Hubble Constant gt Approximate Age
• c) Temperature decreases with time
• d) Expansion gt mass-energy Dark Energy

t lt 10-43 s after Big Bang General
Relativity breaks down. t 10-43 s
strong-electro-weak force gravity, T 1032
K. t 10-35 s electro-weak force strong
force gravity, T 1027 K. t 10-12 s four
forces separate, T 1015 K.
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V. Cosmology C. The Big Bang

• 1. Standard Model (prior to 1980)

t 10-6 s Quark confinement production of p
n, T 1013 K. t 1 s Universe becomes
transparent to neutrinos, T 1010 K. t 3 min
Cosmic nucleosynthesis everywhere, T 109 K.
DE h/Dt.
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V. Cosmology C. The Big Bang

• Side note on particles

Leptons (F) Hadrons Baryons (F) Mesons
(B) Field Particles (B) Higgs (B)
2/3
2/3
-1/3
Also particles and anti-particles
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V. Cosmology C. The Big Bang

• 1. Standard Model (prior to 1980)

t 10-6 s Quark confinement production of p
n, T 1013 K. t 1 s Universe becomes
transparent to neutrinos, T 1010 K. t 3 min
Cosmic nucleosynthesis everywhere, T 109
K. t 4 x 105 yrs Universe becomes transparent
to photons, T 3000 K t 4 x 108 yrs First
galaxies, T 30 K.
DE h/Dt.
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V. Cosmology D. Universal Composition
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VI. The END

• A. The Ultimate Fate of the Universe
• 1. Before 1999
• The Big Good Bye (?)
• The Big Crunch (100 billion years)
• The Big Freeze (1014 years)
• 2. Post 1999
• The Big Rip?
• (20 Billion years)

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VI. The END

• A. The Ultimate Fate of the Universe
• 1. Before 1999
• The Big Good Bye (?)
• The Big Crunch (100 billion years)
• The Big Freeze (1014 years)
• 2. Post 1999
• The Big Rip?
• (20 Billion years)