Coursework ========== Symmetry -------- The note is from MATSCI214-Structure and Symmetry course at Stanford. Atoms and Bonding I ^^^^^^^^^^^^^^^^^^^ How is the state defined? By its wavefunction. Can you fill the order from n=1 through 5? n=1 has 1s; n=2 has 2s, 2p; n=3 has 3s, 3p, 3d; n=4 has 4s, 4p, 4d, 4f; n=5 and beyond follow the same pattern as n=4. To determine the electron filling order, draw a table with rows for each principal quantum number n (from 1 at the top to 5 at the bottom). Start from 1s, go down, and then up diagonally to the right. Following these diagonals gives the correct order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, etc. Be able to have Fill Order from n=1 to 5 - Some rules: A half-filled or completely filled shell is more stable. Example: [Ar]4s^1 3d^5 What is a molecular orbital? It is a linear combination of atomic orbitals. What are the shapes and notations of molecular orbitals? In the case of :math:`H_2`, :math:`\psi_{\sigma} = \psi_{1s}(r_A) + \psi_{1s}(r_B)`. The antibonding orbital is :math:`\psi_{\sigma^*} = \psi_{1s}(r_A) - \psi_{1s}(r_B)`. In terms of shape, the bonding orbital has two peaks with the middle region shared, so it's greater than 0 in the y-axis. For the antibonding orbital, the two are opposite in direction, with a node (zero) in the middle. What is the concept of bond order? :math:`BO = 0.5 \times (\text{number of electrons in bonding orbitals} - \text {number in antibonding orbitals})` It predicts the "number" of bonds. How would you defined electronegativity? The tendency of a given element to attract electrons when fomring a bond. What are the four types of bonding in solids? Metallic, covalent, ionic, and van der Waals. What is valence? Valence is the ability of an electron to combine wiht other elements. Its ability is determine by the outermost electrons. What is hybridization? Mixing of atomic orbitals with close energies. What is coordination? The number of nearest neighbors. Bonds with less directionality (ionic/metallic) tend to form with maximized coordination. What is the concept of "shielding effect"? Protons in the nucleus attract electrons inward. However, inner electrons repel outer electrons due to like charges. This repulsion partially cancels the nuclear attraction, so the effective pulling force from the nucleus (protons) on the outer electrons becomes weaker. What is effective nuclear charge :math:`(Z_{eff})`? What is the trend of :math:`Z_{eff}` across a period, a group, and as atomic number increases? The effective nuclear charge is the net positive charge experienced by an electron in a multi-electron atom. Across a period (left to right), :math:`Z_{eff}` increases because more protons are added while shielding does not increase as much. Down a group (top to bottom), :math:`Z_{eff}` increases slightly, but the effect is less pronounced due to increased shielding from additional shells. As atomic number increases, :math:`Z_{eff}` generally increases. From the end of Period 2 to the start of Period 3, :math:`Z_{eff}` drops suddenly. Why? A new outer shell is added that is farther from the nucleus and more shielded by inner electrons, so the effective nuclear charge experienced by the outermost electrons decreases. What is "atomic radius"? What is its trend? Atomic radius is the distance from the nucleus to the outermost stable electron orbital. Across a period (left to right), atomic radius decreases because outer electrons experience a stronger pull toward the nucleus without the addition of a new shell. Down a group (top to bottom), atomic radius increases as a new electron shell is added. Define ionization energy The energy required to remove an electron from a gaseous atom or ion. :math:`IE = E(A_g) - E(A^+_g)`. Define electron affinity The energy change when a gas-phase atom gains an electron. :math:`EA = E(A^-_g) - E(A_g)`. Describe the trend of ionization energy in the periodic table Down a group, decrease. :math:`\uparrow` shielding, :math:`\uparrow` atomic radius → electrons easier to remove Acorss a period, increase, :math:`\uparrow` :math:`Z_{eff}`, :math:`\downarrow` atomic radius :math:`\to` electrons more tightly bound How do you compute electronegativitiy? The average between IE and EA. Lecture 2 Be able to sketch two plots of bond force vs. bond length and energy vs. bond length. Check Slide 22 of Lecture 22. Imagine you have CO. Where would bonding electrons (shared electrons) will be relatively? Oxygen has greater effective nuclear charge with greater attraction and (lower-energy/tightly bound). Hence, electrons will be closer to oxygen as it attract them more strongly. Bonding in Crystal Structure Notation Bravais Nets/Lattices, Multi-atom Basis Surface Tension Reciprocal space What is considered "low" or "high" frequency in images? "Low" means smoonth where brightness is smooth, while "high" means brightness flips rapidly. Thermodynamics -------------- Why is the first law? The change of internal energy is the sum of changes in heat and work, commonly expressed as :math:`d U = \delta Q + \delta W`. What is the difference between work and heat? Work and heat can both be measured in the same unit of energy, but they are fundamentally different in nature. On a microscopic level, heat is associated with the random motion of molecules, while work involves organized motion in a specific direction. For example, the random molecular motion in hot water represents heat transfer from the surroundings, whereas when the beaker slides across the desk due to friction, mechanical work is being done. What is the sign convention for work in materials science and engineering? When the system does work on the surroundings (for example, expansion work), energy leaves the system, so the work is considered negative. Can you observe (with your eyes) a revsersible process? No, a reversible process cannot be directly observed with the naked eye. For example, consider ice and water in equilibrium at 0 °C. Although nothing appears to change macroscopically, at the microscopic level there is a continuous, infinitesimal exchange of molecules between the solid and liquid phases—some water molecules freeze while others melt. What's required to have a reversible process to be possible? There must be an external operator which can remove or add heat infinitesimally. What is an isolated system? An isolcated system has no exchagne of energy either heat or work and matter. Hence, its energy is constant. Why is the universe considered "isolated"? We assume that it contains everything so there is no exchange of heat, work, matter. Thermo lecture notes (VERY ROUGH DRAFT) -------------------------------------- Week 3 - Tue (Oct 7, 2025) For an adabatic or isolate system, entropy is either 0 or greater than 0.What is a generic system? For a generic system, entropy can increase or decrease... why is it that? Generic system is an open or non-isolated system. It can exchange :math:`Q` and it can interact with the environment. :math:`(U, V) \equiv const.` .. math:: ds = \frac{dU}{T} + \frac{P}{T}dV \\ .. math:: \begin{align} U(P, V) &\to U(T(V,P),V) \\ F &= U - TS \\ dF &= dU - T\,dS - S\,dT \\ &= T\,dS - P\,dV - T\,dS - S\,dT \\ &= -P\,dV - S\,dT \end{align} Hence :math:`(\partial F/\partial T)_V = -S` and :math:`(\partial F/\partial V)_T = -P`. How does G change across P and T? "At equlibrium, the entropy of the university is not changing." .. math:: \begin{align} ∆G &= ∆U + P∆V - T∆S \\ &= ∆H - T ∆S < 0 \end{align} What do we compute G? It computes the equlibrium state given P and T. It is basically the phase diagram. Crystallography --------------- What do you mean by "equivalent directions" in crystallography? Two directions are *equivalent* if the crystal looks identical when viewed or translated along either one. Why use 4 coordinate system for hexagonal? In cubic systems, planes are described by three indices :math:`(hkl)`, and the family :math:`{100}` includes three mutually perpendicular but equivalent planes: :math:`(100)`, :math:`(010)`, and :math:`(001)`. In the hexagonal system, there are three equivalent axes in the basal plane: :math:`a_1`, :math:`a_2`, and :math:`a_3`. These are 120° apart and equal in length, related by a threefold rotational, so :math:`{100}` includes three distinct but equivalent planes: :math:`(100)`, :math:`(010)`, and :math:`(1\bar{1}0)`. Because the usual :math:`(hkl)` notation cannot specify which in-plane axis (out of 3) is used, a four-index system :math:`(hkil)` is adopted to remove this ambiguity. Common dev phrase