2 Today I asked the best qual exam question ever: Where do atoms come from? -MatSci Qual Committee Member
3 Physics Nobel Prize 1967 Alpher, Bethe, Gamow ( αβγ ) Phys. Rev. 73, 803 (1948): The origin of chemical elements Physics Nobel Prize 1983 Burbidge, Burbidge, Fowler & Hoyle (B 2 FH) Rev. Mod. Phys. 29, 547 (1957): Synthesis of the Elements in Stars
4 Some preliminary rules of nuclear synthesis A ZX 1. Nomenclature: Atomic number Z=# protons in nucleus Mass number A = # protons + # neutrons 2. Conservation laws Z1+Z2=Z3+Z4 A1+A2=A3+A4A A A4 3. Energy Liberated Q Q=[(m1+m2)-(m3+m4)]c 2
5 Big Bang Nucleosynthesis T 9 universe ~ 10 K. Within 3 minutes of the Big Bang, the primordial quark-gluon plasma froze out to form fundamental subatomic particles: electrons, protons, neutrons, photons, neutrinos, positrons, α particles ( 4 2He 2+ nucleus), β particles (e - ejected from nucleus), γ photons (emitted dfrom nucleus) Strong force binds protons and neutrons Electromagnetic force binds electrons to nuclei to form atoms Within 17 minutes, first elements (mass abundance): 1 H(75%), 4 He (23%), 2 H (0.003%), 3 He (0.004%), trace amounts of Li and Be (10-10 %)
6 Big Bang Nucleosynthesis At t=20 min after the Big Bang, the temperature of the universe had fallen to T=8K: nucleosynthesis ends Big Bang nucleosynthesis produced no elements heavier than beryllium, due to a bottleneck: the absence of a stable nucleus with 8 or 5 nucleons (A=5 or A=8) A=8 A=5 No way to bridge the gap through capture of nucleons.
7 Nucleosynthesis of light elements After 150 million to 1 billion years had passed The earliest stars resulted from the gravitational force condensing clouds of hydrogen and helium. The compression of these clouds increased the temperature and density of the clouds, allowing nuclear fusion reactions to occur. Eagle nebula (green=h, red=s, blue=o)
8 Young Stellar Nucleosynthesis: H-burning T ~1 7 ~1 3 sun 1.6x10 K, ρ sun 1.4g/cm 1 H 1 H ν 2 H γ 3 He 10 9 years 1 second 4 He ν γ 1 million years Most starts spend their lives burning H in their core 1 H proton neutron positron Because stellar reactions involve mainly charged particles, stellar nucleosynthesis is a slow process.
9 Stellar Nucleosynthesis: He-burning gives 12 C Older star: T star ~6x10 8 K, ρ~2x10 5 g/cm 3 (α particle) 4 He + 4 He 8 Be Unstable decays or reacts within s! 8 Be He C 12 C formation in stars set the stage for the nucleosynthesis of all heavy elements.
10 Oxygen and nitrogen production Oxygen results from reaction of Carbon and an α particle ( 4 He): 12 6 C + α 16 8O + γ (7.2 MeV energy released ) Proton capture by C-12: 12 6 C + 1 1p 13 7N+γ 6 1p 7 γ Positron decay accompanied byneutrino emission: 13 7 N 13 6C + e + + ν Nitrogen results from proton capture by C-13: 13 6 C + 1 1p 14 7N + γ 1 ev=96.48 kj/mol
11 Nucleosynthesis up to Fe (Z=26) A massive star near the end of it s life has an onion-ring structure. The inward force of gravity counteracts the interior thermal pressure. Carbon burning: T star ~6x10 8 K, ρ~2x10 5 g/cm C C 20 10Ne + α +46MV 4.6 MeV 23 11Na + 1 H MeV Neon burning: T star ~1.2x10 9 K, ρ~4x10 6 g/cm Ne + γ 16 8O 8 + α Ne + α 24 12Mg+ γ
12 Nucleosynthesis up to Fe (Z=26) A massive star near the end of it s life has an onion-ring structure. The inward force of gravity counteracts the interior thermal pressure. Oxygen burning: T star ~1.5x10 9 K, ρ~10 7 g/cm O O 28 14Si + α + 10 MeV 31 15P + 1 H MeV Silicon burning: T star ~3x10 9 K, ρ~10 8 g/cm 3 Major Ash: Fe
13 Nucleosynthesis of heavier elements Requires a supernova, which induces a succession of rapid neutron capture events (free neutrons are not present in the earliest stages of stellar evolution) Example: Technetium Neutron capture: 98 42Mo + 1 0n Mo + γ Followed by β decay and neutrino emission: 99 42Mo 99 43Tc + e - + ν
14 Technetium: the first synthetic element Transition i metal with Z=43, between Mo and Ru Does not naturally occur on Earth (usually only in red giant stars), but can be synthesized in nuclear reactions Half-life of 6 hours Emits gamma rays that can be used for medical imaging and functional studies of the brain, heart, kidneys, etc.
15 Abundance of elements in the universe Why does the elemental abundance oscillate?
16 Nuclear binding energies Fe and Ni are very abundant in the universe: most stable nuclei These elements have the highest binding energy (BE) Binding energy : the energy required to disassemble a nucleus into protons and neutrons It is derived from the strong nuclear force In atoms, BE~eV. In nuclei, BE~MeV
17 While the elements are formed on stellar- scales, their properties derive from their electronic structure. Quantum mechanics is needed to understand these properties.
18 by contrast/
19 Why quantum mechanics is important and fun! Atomic transitions Neon Lighting Nanoparticle fluorescence
20 Why quantum mechanics is important and fun! Atomic transitions Neon Lighting Molecular bonding Nanoparticle fluorescence
21 Why quantum mechanics is important and fun! Atomic transitions Periodic table Neon Lighting trends Molecular bonding Nanoparticle fluorescence
22 Electron Wavefunctions Steady-state total wavefunction: ( x, t) ( x)exp iet E=energy of the electron t=time ψ(x) = electron wavefunction that describes only the spatial behavior of the electron Experimentally, we measure the probability of finding an electron in a given position at time t (like an intensity): 2 ( x, y,z,t) ( x, y, z ) 2
23 Time independent Schrodinger equation Schrodinger s equation for one dimensioni 2 d 2m 2 2 dx E V 0 Schrondinger s equation for three dimensions 2 2 x y z 2m 2 ( E V ) 0 A mathematical crank : we input the potential V of the electron (i.e., the force it experiences, F=-dV/dx), and can obtain the electron energies E and their wavefunctions / probability distributions. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw Hill, 2005)
24 Wavefunctions of Hydrogenic Atoms (1 e - ) 2 2m m E V 0 2 The electron in the hydrogenic atom is atom is attracted by a central force that is always directed toward the positive nucleus. Spherical coordinates centered at the nucleus are used to describe the position of the electron. The PE of the electron depends only on r.
25 The potential of 1-electron atoms, V(r) The electron s potential energy V(r) in a hydrogenic atom (i.e., just one electron) is used in the Schrödinger equation. Recall: r 2 =x 2 +y 2 +z 2
26 Solutions to Schrodinger s equation in an atom 2 2m m 2 E V 0 V ( r) 2 Ze 4 o r 2 f : ( r,, ) R( r) Y (, ) R: Radial wavefunction depends on two quantum numbers, n and l d w dp d w q b, d Y: Angular wavefunction depends on another quantum number, m l (A fourth quantum number, also in Y, arises from relativity: m s )
27 Radial wavefunctions, R(r) Radial wavefunctions of the electron in a hydrogenic atom for various n and values. n=1,2,3 =0,1,2,3, (n-1) =0: s =1: p =2: d =3: f
28 Radial Probability r 2 R n, 2 gives the radial probability density., i.e., where are we most likely to find the electron around the nucleus
29 Radial Probability Bohr radius (a 0 ): nm r 2 R n, 2 gives the radial probability density., i.e., where are we most likely to find the electron around the nucleus
30 Angular Wavefunctions, Y(θ,φ) The polar plots of Y(, ) for 1s and 2p states.
31 Angular Probability Distribution The angular dependence of the probability distribution, which is proportional to Y(, ) 2.
32 Radial and Angular Nodes Radial nodes occur where the radial component of the wavefunction passes through zero. An orbital with quantum numbers n and in general has n- -1 radial nodes. Nodal planes occur where the angular wavefunction passes through zero. An electron will not be found anywhere on a nodal plane. Nodal planes separate regions of positive and negative signs of the electron wavefunction. In general an orbital with the quantum number has nodal In general, an orbital with the quantum number has nodal planes.
33 Nodal Planes of p orbitals ( =1)
34 Nodal Planes of d orbitals ( =2)
35 Total Wavefunction: ψ(r, θ, φ)=r(r)y(θ,φ) n=1,2,3 =0,1,2,3, (n-1) =0: s =1: p =2: d =3: f m =-,0,
38 5f0 orbital n=5, l=2, m=0 (i.e., Berkelium or Californium)
39 5g4 orbital n=5, l=4, m=4 (future Stanfordium?)
40 Electron energies Knowing ψ, we can use the Schrodinger equation to find the electron energies. The electron energy in the hydrogenic atom is quantized. E n me 4 Z h 2 n 2 oh (Z is atomic number, n is the quantum number, 1,2,3, ) Ionization energy of hydrogen: energy required to remove the electron from the ground state in the H-atom E I me J 2 2 o h 13.6 ev
41 Radial electron position & ionization energies in a hydrogenic atom (Z 1) ) r max n 2 a Z o Z 2 E effective I, n 2 (13.6 n ev) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw Hill, 2005)
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