What holds a crystal together? Objectives Heads up: Notes on the board won’t be available online (lots today) What holds a crystal together? Objectives By the end of this section you should be able to: Apply a van der Waals-London interaction Use the Lennard-Jones potential to understand equilibrium lattice parameters (noble gases) Begin to explain why other types of bonding might exist (will develop more next time) Determine the # of “valence” electrons/atom Identify the sign of electronegativity

Models of Atoms K shell L shell M shell For example, I work a lot on L-edge spectroscopy. We are excited electrons from their 2p orbitals into empty 3d states. (It can be very useful for studying magnetism and/or atomic valence.) Since the electrons come from n=2, this is referred to as the L-edge (even those the electrons go into n=3). I also do some K-edge spectroscopy, which means I use x-rays of higher energy that kick out the core n=1 electrons. K-edge work is also ok at studying valence and is also used in a technique called EXAFS to determine bond lengths and the type of atoms that are nearest neighbors. You might want to know the atom type if you are doping a material with an “impurity” and you want to know where it goes. Of course, there is also M shell spectroscopy. You are more likely to use it if you are studying systems with larger atomic numbers (which I typically don’t). I do a lot of work on Mn, Fe and Ti. K shell L shell M shell

Understanding Elemental Properties by the filling of the orbitals 1s2 2s2 2p6 1s2 2s2 2p6 3s2 3p6 Closed-shell elements: noble gases What would you say about the properties of noble gases? What causes these elements to stick together?

van der Waals Bond graphite Arise from charge fluctuations in atoms due to zero-point motion (due to Heisenberg uncertainty principle); these create dipole moments that are attractive Depends on p2/r6, short ranged Typical strength of 0.2 eV/atom ~1% of other bonds Always present, but significant only when other types of bonding not possible (closed electron shells, saturated molecules) What do weak bonds mean in terms of properties? Work out why each coulomb interaction goes as R^3 on the board Low melting temperature, not hard. Temperatures are so low we refer to them as noble gases, since they aren’t solid at room temp. While the electrons on average might have a spherical distribution, there will be points in time when the electron distribution causes a dipole moment. Typical dipole moment goes as p/r^3, but we have two interacting ones, so it gets squared You might think these dipoles p should cancel out since they could be positive or negative, but the interaction depends on the dipole squared (kind of like how thermal energy is related to velocity squared)

Repulsive force: What works against bringing atoms closer? The exact dependence with R is found experimentally. For inert gases ~B/R12 where the power of 12 and B found by experiment As atoms are brought close, the Pauli exclusion principle will prevent two electrons from having the same quantum numbers. Overlap increases the total energy and causes repulsive term.

Lennard-Jones Potential: joining attractive and repulsive energies Parameter definitions

What about the properties of non-noble gases? Closed-shell elements: noble gases

Consider the case of sodium (atomic # 11) On your own write down the electronic configuration: 1s2…

Closed-shell elements: noble gases Other Properties Closed-shell elements: noble gases Would group 1 elements react to noble gases? No. Their orbits are full. However, in the adjacent column are atoms that would really like one more electron to fill their orbits. Closed-shell –plus one (alkali) elements: reactive due to loosely-bound outer electron in s-shell

Why mobile electrons appear in some solids and not others? According to the very simple Drude model, the valance electrons are responsible for the conduction of electricity, thus termed conduction electrons. Na11 ￫ 1s2 2s2 2p6 3s1 This valance electron, which occupies the third atomic shell, is the electron which is responsible chemical properties of Na. Valance electron (loosely bound) Metallic 11Na, 12Mg and 13Al are assumed to have 1, 2 and 3 mobile electrons per atom respectively. Core electrons

What would a group I element most easily bind with? Closed-shell elements: noble gases Does this group have a valence of 7? Valence of 7? No. Our term valence here refers to loosely bound electrons. The halogens would rather gain an electron than give any away. (We will discuss later the confusion of the term valence as it’s used in chemistry too.) Would group 1 elements react to noble gases? No. Their orbits are full. However, in the adjacent column are atoms that would really like one more electron to fill their orbits. Closed-shell–minus-one elements (halogens): elements with high electron affinity A (energy gained when an additional electron is added to a neutral atom); will easily form negative ions (take additional electron) in remaining p-shell state; very reactive Closed-shell –plus one (alkali) elements: reactive due to loosely-bound outer electron in s-shell

Warning: Confusing Terminology There exist many forms of valence! By valence electrons here we mean outer electrons...the ones that could easily be pulled from the core to form an electron gas (In chemistry) Valence = Maximum number of bonds formed by atom For high group numbers these terms don’t match well, but then those electrons are not very easy to pull away and are not metallic Where would these definitions differ? Sometimes even the experts mix up this terminology even in papers

Organization of Periodic Table Dmitri Mandeleev Columns: groups with similar shells, similar properties Rows: periods with elements with increasingly-full shells Metallic/insulating properties can be understood by how loose (i.e. low ionization energy) outer electrons are. So, on which side of the table are the metals?

How the mobile electrons become mobile When we bring Na atoms together to form a Na metal, the orbitals overlap slightly and the valance electrons become no longer attached to a particular ion, but belong to both. A valance electron really belongs to the whole crystal, since it can move readily from one ion to its neighbor and so on. This is another type of bonding called metallic bonding. (More bonding types next time.) + + + + + + Na metal

Electronegative and Electropositive An atom is electronegative if it gains energy by gaining an extra electron. What group of elements are very electronegative? Why are the Group I elements electronegative? What if you join Na and Cl together?

Group: Find the # of Valence Electrons for Si Group: Find the # of Valence Electrons for Si. Is it metallic or insulating? What is the valence of silicon, atomic Number 14? 1s2 2s2 2p6 3s2 3p2 How many electrons are in the n=3 state? Valence=4 Have them do 1s2, 2s2, 2p6, 3s2, 3p2

Valence often confused/used for oxidation state Let’s determine the type, valence and oxidation states of the atoms and ions in the material BaMnO3. Anion = Negative ion (atom with extra electrons) Cation = Positive ion (atom missing electrons) Which is oxygen? Warning: The "oxidation state" of an atom in a molecule gives the number of valence electrons it has gained or lost through bonding. In contrast to the valence number, the oxidation state can be positive or negative. But sometimes called valence.

Group: Find the Valence of Fe (26) What is the valence of iron, atomic Number 26? 1s2 2s2 2p6 3s2 3p6 4s2 3d6 Number of 3d electrons = 6 Have them do Doesn’t show 4s, but you get the idea. Maybe mention SrMnO3 to discuss how can be related to oxidization state. What about in compounds? (SrFeO3) Except, are these the outer most? No, the 4s is farther! So elemental valence =2!

Why do bonds form? Bonding can be understood from coulomb energy Reminder: k=1/4o The binding energy is released when molecules form. Binding energy = bond strength. Bonding can be understood from coulomb energy Assume that charge (one electron) is completely transferred: - + If this potential energy is greater in magnitude than the energy required to ionize the atoms, a molecule will tend to form

Ionic Solids (Finding Binding Energy) Let’s find the energy required to transfer an electron from Na to Cl and then to form a NaCl molecule To remove an electron from Na (ionize the atom) one needs to “spend” 5.14eV (compare with the ionization energy of a hydrogen atom?) Na + 5.14eV  Na+ + e- When a Cl atom captures an electron, 3.62eV of energy is released (electron affinity of atom) Cl + e-  Cl- + 3.62eV The energy cost to transfer the electron from the alkali to the halogen is DANGER! Electron affinity is the amount of energy released when an electron is added to a neutral atom. In solid state physics, you can also talk about EA for a solid semiconductor surface. Compare to workfunction. In this case, it is the energy obtained by moving an electron from the vacuum just outside the semiconductor to the bottom of the conduction band just inside the semiconductor. This is one reason it is hard to make pure semiconductors. The electron affinity of a Si surface is about 4 times that of a Si atom, thus, they are not the same thing. X + e− → X− + energy

With a partner, find Rc for NaCl. The Ionic Bond Reminder: k=1/4o = 9 x 109 Vm/C, ke2=1.44 eV nm + Na+ - Cl- R Since DE >0, this will not happen if the atoms are far away As the atoms move closer, forming an ionic bond becomes energetically favorable due to coulomb potential energy Total energy of ion as a function of separation R is Ke2=1.44eVnm so Rc is a little less than 1 nm. The lattice parameter for NaCl is actually 0.564 nm (and that’s ~twice R). The ionic bond will form when E(R) becomes negative. With a partner, find Rc for NaCl. This happens when R is below the critical distance at which E(R) = 0.

Will the ions stay at Rc apart? Rc is the minimum distance needed to encourage bonding but they can continue to move closer NaCl critical distance is

The Ionic Bond In reality, ions will not get infinitely close to each other because ? repulsion between nuclei electrons in the same region can not occupy the same quantum states (Pauli exclusion principle), thus increasing the energy of the ion Effective potential 2nd term describes repulsion (two free parameters: B and 6<n<10) This leads to a minimum energy as a function of R which defines distance between ions in molecule: Is that the whole picture? If it were, what would happen to these ions? They would get a lot closer to minimize energy. A reasonable approximation for the binding energy can be obtained from R0:

Reminder: k=1/4o = 9 x 109 Vm/C, ke2=1.44 eV nm Let’s try that for NaCl What do we need to know? ENaCl = 1.5 eV (Na ionization – Cl affinity) Is Ro = Rc? How else might we determine it? How could we find the radius of Na+? Cl-? Nope

Ionic crystals 3D crystals can be produced in this manner. In NaCl, each chlorine atom is surrounded by sodium neighbors, and vice versa. The exact structure is determined by the optimal use of space for the given ionic radii

Impenetrable Spheres Ionic crystals are often modeled as hard spheres - + Impenetrable Spheres Ionic crystals are often modeled as hard spheres If the spheres touch, the sum of the two diameters is approximately the lattice parameter (may not touch if size very different) Due to Pauli exclusion a Due to Coulomb attraction

Attractive Force Rather than Binding Energy Reminder: k=1/4o = 9 x 109 Vm/C Attractive Force Rather than Binding Energy You might also see this done in terms of force. Calculate the coulombic force of attraction between Na+ and Cl-. Z is the effective charge of the ion. What are the affective charges of Na and Cl?

Ionic crystals have strong bonds Most ionic compounds are brittle; a crystal will shatter if we try to distort it. This happens because distortion causes ions of like charges to come close together then sharply repel. Brittleness (Bendable?) Most ionic compounds are hard; the surfaces of their crystals are not easily scratched. This is because the ions are bound strongly to the lattice and aren't easily displaced. Hardness (Scratchable?) Solid ionic compounds do not conduct electricity when a potential is applied because there are no mobile charged particles. No free electrons causes the ions to be firmly bound and cannot carry charge by moving. Electrical conductivity The melting and boiling points of ionic compounds are high because a large amount of thermal energy is required to separate the ions which are bound by strong electrical forces. Melting point and boiling point Explanation Property

Types of Crystals Closed-shell elements: molecular crystals Group IV prefer covalent. I-VII ionic crystals III-V crystals are more covalent and semiconducting. II-VI ionic crystals Closed-shell elements: molecular crystals 3-5 are in zincblende structure Closed-shell–minus-one elements (halogens): elements with high electron affinity A (energy gained when an additional electron is added to a neutral atom); will easily form negative ions (take additional electron) in remaining p-shell state due to large nuclear charge; these elements are very reactive (e.g., F- with e.a.=3.4 eV) Closed-shell –plus one (alkali) elements: reactive due to loosely-bound outer electron in s-shell

Bonding Elemental Na would be metallic, but you can change that if brought in contact with an electronegative element (e.g. Cl) Draw the K, L & M shells of Na and Cl before and after bringing in contact. Anion = Negative ion (atom with extra electrons) Cation = Positive ion (atom missing electrons) Which type of ions are Na and Cl? Figure on page 9 of Holgate

Ions and Cations and Anions, Oh My! NaCl (salt) Ions and Cations and Anions, Oh My! Draw the K, L & M shells of Na and Cl before and after bringing in contact. Anion = Negative ion (atom with extra electrons) Cation = Positive ion (atom missing electrons) Which is the anion? Which is the cation? Which is/are electronegative? Which is/are the ion(s)?

Group Problem: How many atoms? Let’s say you are growing a 10 nm thin film of NaCl which has a density of 2.16 g/cm3. The film is grown on a substrate of 5 mm x 5 mm. How many sodium atoms are in the film? 6.022 x 1023 atoms per mole For effect: Could bring some salt and ask them for the amount of Na.