Atoms, Molecules and Solids (selected topics)

Atoms, Molecules and Solids (selected topics)

Part I: Electronic configurations and transitions

Transitions between atomic states (Hydrogen atom) Transition probabilities are different depending on the combination of the quantum numbers n, l, m l, m % of the initial and final states Selection rules: n = anything l = ±1 m l = 0, ±1 The photon carries one unit of angular momentum (ħ)

Allowed / forbidden transitions n = anything l = ±1 m l = 0, ±1 Allowed transitions: Electrons absorbing or emitting photons to change states when l= ±1. Forbidden transitions: Other transitions still possible (not truly forbidden) but occur with much smaller probabilities when l ±1

PHGN310: Total angular momentum Solving the Schödinger equation introduces the orbital angular momentum as quantum number. A relativistic treatment would also yield the spin angular momentum as quantum number Orbital angular momentum L Spin angular momentum S Total angular momentum J J = L + S

If j and m j are quantum numbers for the single electron (hydrogen atom). Quantization of the magnitudes. J = j j + 1 J 3 = m 4 ħ L = l l + 1 S = s s + 1 J = j j + 1 The total angular momentum quantum number for the single electron can only have the values j = l ± s L, L z, S, S z, J, J z are all quantized. PHGN310: Total angular momentum

PHGN310: New quantum numbers New quantum numbers: j = l ± s Total angular momentum quantum number. Note: j=+1/2 (if l=0) j < m 4 < +j with j, m 4 : half-integers Notation: nlm l changed to nlm 4 Example: n = 2, l = 1, m 4 = 3 2 gives 2P C/E

PHGN310: Precession Only J 3 can be known, because the uncertainty principle forbids J F or J G from being known at the same time as J 3. No external magnetic field Random Orientation External Magnetic field J cannot align with B IFJ (also true for L and S ) Complex precession motion

PHGN310: Spin-orbit coupling An effect of the spins of the electron and the orbital angular momentum interaction is called spin-orbit coupling. Similarly than in the case of the magnetic moment: Spin magnetic moment µ S B KLJIMLNO L V l% = μ % S B KLJIMLNO Level splitting due to spin-orbit coupling: j = l 1/2 lower in energy than j = l + 1/2 2P 2P 3/2 (4) -3/2 m j 3/2; m j = ±1/2, ±3/2 2P 1/2 (2) -1/2 m j 1/2; m j = ±1/2

PHGN310: Expanded selection rules Now the selection rules for a single-electron atom become n = anything l = ±1 m 4 = 0, ±1, j = 0, ±1

PHGN310: Many-electron atoms Two-electron atom: J = L T + L E + S T + S E It gets messy very quickly! Couplings: LS, jj Hund s rules (1925): The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle. Insofar as rule 1 is not violated, L should also be maximized. For atoms having subshells less than half full, J should be minimized.

PHGN310: Simplification? For example: Alkalis: core + one electron Similar to H

Many electron atoms Dealing with many-electron atoms becomes quickly very complicated: Electrons interact with nucleus (+Ze) Electrons interact with each other, but they also induce screening effects à Complex potential interactions It is impossible to solve analytically the Schrödinger equation for many-electron atoms. Need help of (powerful) computers But: it is possible to understand some chemical properties of the elements by looking in quantum mechanics.

The Pauli exclusion principle 1925 Wolfgang Pauli (1900-1958) formulates the exclusion principle: No two electrons in an atom may have the same set of quantum numbers nlm l m s The Pauli exclusion principle applies to all half-integer spin particles (known as Fermions ). This is very important, since protons and neutrons are fermions too (Nuclear Physics) Note: the electrons in an atom tend to occupy the lowest energy levels available to them.

How many electrons can we put in an energy level defined by n? Atomic shells n = 1 l = 0 m l = 0 m % = ±1/2 (2) n = 2 l = 0 m l = 0 m % = ±1/2 (2) l = 1 m l = 1 m % = ±1/2 l = 1 m l = 0 m % = ±1/2 (6) l = 1 m l = +1 m % = ±1/2 n = 3 l = 0 m l = 0 m % = ±1/2 (2) l = 1 m l = 1 m % = ±1/2 l = 1 m l = 0 m % = ±1/2 (6) l = 1 m l = +1 m % = ±1/2 l = 2 m l = 2 m % = ±1/2 l = 2 m l = 1 m % = ±1/2 l = 2 m l = 0 m % = ±1/2 (10) l = 2 m l = +1 m % = ±1/2 l = 2 m l = +2 m % = ±1/2 ( )

Designation n label of the electron shell l electron sub-shell Fine structure is changing slightly the ordering of the shells that are filled. Large l -value shells appear more shielded (do not feel the full intensity of the attractive force) than small l -value shells à higher energy levels.

The periodic table of elements Dimitri Mendeleev 1834 1907 Creates the periodic table of elements, by ranging the 60 known elements (at the time) according to their chemical properties Not well accepted, until elements are discovered with the chemical properties he predicted. Gallium (1875) Scandium (1879) Germanium (1886)

Electron configurations in the periodic table

Groups / periods Groups: Vertical columns. Same number of electrons in an l orbit. Can form similar chemical bonds. Periods: Horizontal rows. Correspond to filling of the subshells. Some properties of elements are compared by the ionization energies of elements and atomic radii.

Elements in the periodic table: noble gases Chemical reactivity of certain elements reveal the basic factors controlling binding atoms to form molecules Noble gases / Inert gases are chemically inert: He (Z=2), Ne (Z=10), Ar (Z=18), etc He: (1s) 2 Ne: (1s) 2 (2s) 2 (2p) 6 Ar: (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 6 Etc Rn Xe Kr Ar à Their electronic configurations correspond to completely filled major shells. Ne He

Elements in the periodic table: alkalis and halogens Alkalis: Single s electron outside an inner core Easily form positive ions with a charge +1e Lowest ionization energies, largest atomic radii Electrical conductivity is relatively good H: (1s) 1 Li: (1s) 2 (2s) 1 Na: (1s) 2 (2s) 2 (2p) 6 (3s) 1 Etc Halogens: Need one more electron to fill outermost subshell Form strong ionic bonds with the alkalis More stable configurations occur as the p subshell is filled F: (1s) 2 (2s) 2 (2p) 5 Cl: (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 5 Etc

Elements in the periodic table: alkaline earths, lanthanides and actinides Alkaline Earths: Two s electrons in outer subshell Large atomic radii (similar to Alkaline metals) High electrical conductivity Lanthanides (rare earths): Have the outside 6s 2 subshell completed As occurs in the 3d subshell, the electrons in the 4f subshell have unpaired electrons that align themselves The large orbital angular momentum contributes to the large ferromagnetic effects Actinides: Inner subshells are being filled while the 7s 2 subshell is complete Difficult to obtain chemical data because they are all radioactive Have longer half-lives

Elements in the periodic table: transition metals Transition Metals: Three rows of elements in which the 3d, 4d, and 5d are being filled Properties primarily determined by the s electrons, rather than by the d subshell being filled Have d-shell electrons with unpaired spins As the d subshell is filled, the magnetic moments, and the tendency for neighboring atoms to align spins are reduced

Part II: Chemical bonds and molecules

The ionic bond How do atoms bond together? Coulomb force Electromagnetic / Long Range Repulsive component (nuclei) Force related to potential energy: F = dv dr Negative slope: dv dr < 0 repulsive force Positive slope: dv dr > 0 attractive force A,B>0 and n>m to have a potential well Attractive component (electrons)

The ionic (or electrovalent) bond Cl (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 5 Na (1s) 2 (2s) 2 (2p) 6 (3s) 1 Na gives up its (3s) electron, while Cl accepts the electron to fill up its (3p) shell Na à Na + ; Cl à Cl - ; Attractive potential (NaCl molecule)

The covalent bond Cl (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 5 Cl 2 Molecule Atoms not as easily ionized Share their outer electrons (ex: diatomic molecules)

Wave Functions Overlap (covalent bond) in the hydrogen ion H 2 + Two hydrogen ions far apart Concentration of negative charge Two hydrogen ions closer Overlap of their wave functions (y 1 +y 2 or y 1 -y 2 ) Electron probability density depends on the relative sign of the two wave functions.

Coulomb potential energy of the protons Bonding The Hydrogen ion (cont d) Antibonding Total energy of the ion = energy protons + energy electron (U p + E + ) or (U p + E - ) The Hydrogen Molecule Minimum: E = -16.3eV r = 0.106 nm E = (-13.6eV)Z 2 /n 2 with Z=2 and n=1 No minimum à No bound state Energy required to break the H 2 + molecule: B = E(H+H + ) E(H 2 + ) = -13.6eV (-16.3eV) = 2.7 ev

Hydrogen bond +d +d H H -d O H +d -d O H +d Binding between molecules due to weak electric and magnetic forces High boiling points in liquid molecules don t easily separate

Other bonds Van der Waals bond: Found in liquids and solids at low temperatures Ex: graphite Atoms in a sheet held together by strong covalent bonds Adjacent sheets held together by Van der Waals bonds Metallic bond: (Quasi-)free valence electrons shared by a number of atoms

Bonding pp covalent bond HC CH sp-hybrid bond H 2 O sp bond Oxygen: 1s 2 2s 2 2p 4 Opportunity for 2 covalent bonds with H Note: the larger the overlap, the tighter the bound

3-D Ammonia NH 3 Methane CH 4 Nitrogen: 1s 2 2s 2 2p 3 Opportunity for 3 covalent bonds with H Benzene

Molecules At the atomic level, electromagnetic radiations may induce transitions between electronic levels At the molecule level, they can induce (through emission, absorption, scattering ) transitions between molecular states, e.g. collective modes in the molecule. Rotational states Vibrational states

Rotational states in a molecule Rotational States in a simple case Diatomic molecule: two atoms connected with a massless and rigid rod E M^J = LE 2I Angular Momentum (Quantum): Moment of inertia L = l l + 1 ħ E M^J = l l + 1 ħe 2I

Exercise Calculate the moment of inertia I of the N 2 molecule (use m=2.33x10-26 kg and R=10-10 m) Estimate the value of E rot (in ev) for the lowest rotational energy state of N 2

Vibrational states in a molecule Vibrational states Two atoms oscillating around their equilibrium position Two mass connected with a massless spring [model: Harmonic Oscillator] E dkem = n + 1 with 2 ħω ω = κ μ k: spring constant µ: reduced mass à μ = `a`b `ac`b Assuming a pure ionic bond, we can estimate k: κ = df dr d dr e E 4πε m r E = ee 2πε m r C Application: r~10-10 m à k~460 N/m

Vibrational modes ω e E 2πε m μr C κ = df dr d dr e E 4πε m r E = ee 2πε m r C

Exercise 1. Given the spacing between the vibrational energy levels of the HCl molecule is 0.36 ev, calculate the effective force constant. 2. Find the classical temperature associated with the difference between vibrational energy levels in HCl

Vibration and rotation Total Energy: E = E M^J + E dkem = l l + 1 ħe 2I + n + 1 2 ħω Transitions between states: DE = E ph à Energy of the photon emitted/absorbed in the process Example: from l + 1 to l (with Dn = 0) E no = ħe 2I l + 1 l + 2 l l + 1 = ħe I l + 1 BAND SPECTRUM: DE n

Band spectrum Vibrational energies typically greater than rotational energies Allowed transitions l = ±1: Photon carries away its intrinsic momentum of one quantum unit (ħ)

Absorption spectrum In the absorption spectrum of HCl, the spacing between the peaks can be used to compute the rotational inertia I. The missing peak in the center corresponds to the forbidden l = 0 transition. The central frequency: f = 1 2π k μ

Example Part of the emission spectrum of N 2

Part III: Properties of solids (abbreviated)

Condensed matter physics Condensed matter physics: The study of the electronic properties of solids. Crystal structure: The atoms are arranged in extremely regular, periodic patterns. Max von Laue proved the existence of crystal structures in solids in 1912, using x-ray diffraction. The set of points in space occupied by atomic centers is called a lattice.

Structural properties of solids Perfect crystals are rare Most solids are in a polycrystalline form Made up of smaller crystal structures [from a few atoms to a few thousands atoms on a side] Solids with no significant lattice structure are called amorphous ( without form ) Common glass is amorphous Amorphous solids can also be referred as glasses Why solids crystallize? When the material changes from the liquid to the solid state, the atoms can each find a place that creates the minimum energy configuration.

Model Pauli exclusion principle The net potential energy felt by each ion on the crystal is the result of an attractive potential [Coulomb] and a repulsive potential [Pauli principle + overlap of the electron shells]. V = V NJJ + V MIn = αee 4πε m r + λetm u

Madelung constant NaCl crystal Attractive potential: V NJJ = αee 4πε m r Similar to the Coulomb force, a: Madelung constant The Madelung constant is calculated by estimating the mean-field charge surrounding a given atom. Example: NaCl crystal (looking at central Na + ): - attractive force with 6 Cl - neighbors (dist=1) - repulsive force with 12 Na + neighbors (dist= 2) - attractive force with 8 Cl- neighbors (dist= 3) - Madelung Constant: α = 6 12 2 + 8 3 1.7476 (The Madelung constant is crystal-dependent)

Net potential Repulsive potential: V MIn = λe tm u Screening effect à The force diminishes rapidly for r>r (r ~ range of the repulsive force) From the potential: V = V NJJ + V MIn = αee 4πε m r + λetm u Net force at the equilibrium: F = dv dr = 0 0 = αee E 4πε m r λ m ρ etm u e tm u = ραee E 4πε m λr m V r = r m = αee 4πε m r m 1 ρ r m with r << r 0 Example: NaCl r/r 0 = 0.11

Thermal expansion modelling (in brief) r 0 : equilibrium position at T=0 r T : equilibrium position at T>0 Tendency of a solid to expand when T increases Model: Oscillations around the equilibrium position x = r r m Potential: V = ax E bx C Harmonic Oscillator Anharmonic component

Other properties of solids Thermal Conductivity Magnetic properties Ferromagnetism Materials with a net magnetic moment even in zero applied magnetic field Paramagnetism Net magnetic moment only in the presence of an applied field in the direction of the field Diamagnetism Net (but weak) magnetic moment opposite to an applied magnetic field Antiferromagnetism / Ferrimagnetism Superconductivity ( )