GENERAL PHYSICS III GENERAL PHYSICS III Optics & Quantum Physics Chapter XXII Chapter XXII Atom Atom ic ic Structure Structure §1. Hydrogen atom §2. Angular momentum of electron §3. Electron spin §4. Many-electron atoms At present it is well known for you that The Atom - electrons confined in Coulomb field of a nucleus  Au  v Rutherford (also Geiger-Marsden) Experiment (1911): Measured angular dependence of particles (He ions) scattered from gold foil. The results: • Mostly scattering at small angles. But… • Occasional scatterings at large angles (even > 90 o )  Something massive in there ! Conclusion: Most of atomic mass is concentrated in a small region of the atom a nucleus! Recall some history: Early sugession for the quantum nature of atoms: (nm) Atomic hydrogen Discrete emission and absorption spectra: • When excited in an electrical discharge, atoms emitted radiation only at discrete wavelengths • Different emission spectra for different atoms. This could not be explained by classical physics. +Ze -e F There is also a BIG PROBLEM with the classical picture: As the electron moves in its circular orbit, it is ACCELERATING  radiates electromagnetic energy  would continuously lose energy and spiral into the nucleus (in about 10 -9 sec). We must apply QUANTUM MECHANICS to ATOMS §1. Hydrogen atom: 1.1 Potential energy of the electron Potential energy of the electron in the hydrogen atom in the hydrogen atom: r e )r(U 2   r U(r) 0 229 0 C/Nm109 4 1    2 2 )( r e r rU f      • Make a test: This potential function implies actually the Coulomb force between the electron & the nucleus x y z r   • The origin of coordinate system is at the center of mass of the system (nucleus & electron). • U depends on r, but not on directions in the space. Such a potential is called spherically symmetrical. • The problem is 3-D. Due to the spherical symmetry  it is more convenient to use the spherical coordinates, than Cartersian. )()()( )( 2 2 22 xExxU dx xd m       EzyxU z x y x x x m                   ),,( )()()( 2 2 2 2 2 2 22          ErU rrr r rrm                                   )( sin 1 sin sin 11 2 2 2 222 2 2 2  1.2 The time independent SEQ: • Recall 1-D time independent SEQ: • The 3-D time independent SEQ is as follows: • For the hyrdrogen atom, the potential function U(x,y,z) = U(r), and the SEQ in the spherical coordinate system is r e )r(U 2   Solving and finding the eigenvalues for E and the eigenfunctions ψ requires some mathematical preparation. In our course of general physics, we will concentrate on the physical implications of mathematical solutions. 1.3 Energy levels of the electron in the hydrogen atom: As you have seen, in the solving the SEQ, the physical conditions for the wave function (finiteness, single valuedness, continuosness) lead to the consequence that the energy of the system can not have continuum values, but must accept a discrete set of values: “the energy levels”. The energy levels of the electron in the hydrogen atom is found to be 22 o 2 n n eV6.13 n 1 a2 e E      , .3,2,1n 2 2 0 em a    = “Bohr radius” = 0.053 nm -15 -10 -5 0 0 5 10 15 20 r/a 0 E (eV) U(r) -15 0 0 20 E 2 E 1 E 3 * Notes: Before Schrödinger, N. Bohr proposed the same scheme of energy levels for the hydrogen atom in his theory (1913). But in the Bohr’s theory the concept of discrete energy levels was introduced as a postulate. The values of the energy levels are determined by fitting in the experimental discrete spectra of the hydrogen atom. 2 13.6 eV n E n   2 2 1 1 13.6 eV i f n n i f E n n             we have 3 2 1 1 13.6 eV 1.9eV 9 4 photon E E            1240eV nm 656nm 1.9eV photon hc E      (nm) Atomic hydrogen Recall that the discrete spectra of the hydrogen atom is related to the optical transitions of between the electron’s energy levels. Example: An electron, initially excited to the n = 3 energy level of the hydrogen atom, falls to the n = 2 level, emitting a photon in the process. What are the energy and wavelength of the photon emitted? -15 -10 -5 0 0 5 10 15 20 r/a 0 E (eV) U(r) -15 0 0 20 E 2 E 1 E 3  From eVEEEE n 6.130 11   Note: The ionization energy of atomic hydrogen at its ground state is 1.4 The eigenfunctions and physical interpretation: Due to the spherical symmetry, the solution to the SEQ in spherical coordinates is found by the method of separation of variables, and has the form: x y z r   ),()(),,(  lmnlnlm YrRr  with quantum numbers: n l and m principal orbital magnetic (angular momentum) • R nl (r) is the ‘radial part’ of wave function. • Y lm (  ,  ) are the angle-dependent functions called “spherical harmonics.” • The principal quantum number n = 1,2, 3,… This number is that in the formula of the energy levels. • For every n, the values of l are 0, 1, 2,…, (n-1). • For every l, the values of m are -l, -(l-1), …, -2, -1, 0, 1, 2,…, (l-1), l. The spherically symmetric states: We write here the wave functions for wavefunctions with no angular dependence. These are called “s-states”. For them, l = 0 and m = 0, the function Y 00 = constant, and the part R n0 (r) obeys the ‘radial SEQ’ which takes the form: r R 30 r 15a 0 0 0 r R 20 10a 0 0 0 0.5 R 10 0 0 4a 0 0 3/ 2 00 0,3 3 2 2 3)( ar e a r a r rR                   o ar e a r rR 2/ 0 0,2 2 1)(           )r(RE)r(R r e r r r 1 m2 0nn0n 2 2 22               )r(R),,r( 0n   s-state wavefunction: (l=0) o ar erR / 0,1 )(   [...]... principle: To have a model for the structure of many-electron atom we need an additional principle - the exclusion principle From spectra of complex atoms, Wolfgang Pauli (1925) deduced the following rule: “Pauli Exclusion Principle” “No two electrons can be in the same quantum state, i.e in a given atom they cannot have the same set of quantum numbers n, l, ml , ms” I.e., every atomic orbital with n,l,ml... 4, 2, -1/2) m= -l, -(l -1), … (l-1), l n>l 2 Which of the following atomic electron configurations violates the Pauli exclusion principle? (a) (b) (c) (d) (e) 1s2, 2s2, 2p6, 3d10 1s2, 2s2, 2p6, 3d4 1s2, 2s2, 2p8, 3d8 1s1, 2s2, 2p6, 3d5 1s2, 2s2, 2p3, 3d11 2(2l +1) = 6 allowed electrons 2(2l +1) = 10 allowed electrons 4.3 Filling the atomic orbitals according to the Pauli Principle:  6 eV 2 13 s p d... to electron-electron interactions Nevertheless, this hydrogenic diagram helps us keep track of where the added electrons go (2l +1) orbitals Z = atomic number = number of protons label s 1 1 1 l 0 1s22s22p63s1 p 3 2 d 5 3 f 7 Handy pocket guide for filling Atomic Orbitals Due to electron-electron interactions, the hydrogen levels fail to give us the correct filling order as we go higher in the periodic...   S 2 g  g (  ) Z  S m S  g S U   ) Z B ( S [Spectral slitting into (2.½+1)=2 lines in a magnetic field] §4 Many-electron atoms: • An atom is a system of nucleus and Z electrons Z is the atomic number • Atom is electrically neutral, the electric charge of the nucleus is +Z and this charge is from the Z protons • Applying the SEQ to this system and finding solutions of such equations are... all the elements (and all the molecules made up from them) is all due to the way the electrons organize themselves, according to quantum mechanics Example: An alkali metal atom What is the electronic structure of lithium (3 electrons)? That is, what quantum numbers do the electrons have? Solution: The guiding principle is to find the lowest energy This involves (for atoms without too many electrons)... electron state is specified by the quantum number (n, l, ml , m S) • The electrons in an atom fill the orbitals from the lowest energy level in according to the Pauli exclusion principle It states that every atomic orbital with (n,l,ml ) can hold 2 electrons: ( )  • For few-electron atom the orbital order is similar to that of the hydrogen As we go higher in the periodic table, the hydrogen levels fail to . GENERAL PHYSICS III Optics & Quantum Physics Chapter XXII Chapter XXII Atom Atom ic ic Structure Structure §1. Hydrogen atom §2. Angular momentum of. of atomic mass is concentrated in a small region of the atom a nucleus! Recall some history: Early sugession for the quantum nature of atoms: (nm) Atomic