ECEN 5005 Crystals, Nanocrystals and Device Applications Class 18 Group Theory For Crystals

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1 ECEN 5005 Crystals, Nanocrystals and Device Applications Class 18 Group Theory For Crystals Multi-Electron Crystal Field Theory Weak Field Scheme Stron Field Scheme Tanabe-Suano Diaram 1

2 Notation Convention for Spectroscopic Terms Russell-Saunders couplin scheme - A state is specified by a set of quantum numbers, (L, M L, S, M S ). - Excludin spin-orbit interaction, the states havin the same L and S are usually deenerate. Thus, a term is conventionally represented by L and S only. - L is denoted by a capital letter, i.e. L = 0 S, L = 1 P, L = 2 D, L = 3 F, etc. - S is represented by addin (2S+1) as a superscript in front of L. - Example: The round term for a free V 3+ ion has L = 3 and S = 1. 3 F J-J couplin scheme - A state is specified by a set of quantum numbers, (L, S, J, M J ). - Spin-orbit interaction is not inored and thus the states with different J can have different eneries even thouh they have the same L and S. Thus, a term needs to be represented by L, S and J. - L and S are denoted by the same convention as above. - J is represented as a subscript after L. - Example: The round term for a free Pr 3+ ion has L = 5, S = 1 and J = 4. 3 H 4 2

3 Application of Russell-Saunders Couplin Scheme Consider a V 3+ ion which has 2 electrons in 3d shell. - Each electron has l=2 and s=1/2. - They can have any values of m l = -2, -1, 0, 1, 2 and m s = ±1/2. - However, they cannot have the same values of m l and m s, as prohibited by the Pauli exclusion principle. In order to find multi-electron states, we need to obtain L by addin l 1 and l 2 and S by addin s 1 and s 2. Recall the anular momentum addition rule: - L is the sum of l 1 and l 2, therefore allowed values of L are L = l 1 - l 2, l 1 - l 2 + 1,, l 1 + l 2-1, l 1 + l 2 - Once L is determined, then the allowed values of M L are M L = -L, -L + 1,, L - 1, L - Same principle applies to S and J. For V 3+ ion, l 1 = 2 and l 2 = 2. - The allowed values of L are 0, 1, 2, 3, 4. - Similarly, S = 0, 1 as s 1 = 1/2 and s 2 = 1/2. Pauli exclusion principle prohibits (L=0, S=1), (L=2, S=1), (L=4, S=1), (L=1, S=0) and (L=3, S=0). Thus the allowed terms are (L=0, S=0), (L=1, S=1), (L=2, S=0), (L=3, S=1) and (L=4, S=0), or 1 S, 3 P, 1 D, 3 F and 1 G. 3

4 Application of Russell-Saunders Couplin Scheme If the hydroen-like atom model is valid, then all of the terms obtained before must have the same enery, because we are dealin with a fixed n (principal quantum number) for all electrons. However, in multi-electron systems, there is an important term that was not included in the hydroen-like atom problem. That is the Coulomb repulsion between electrons. The Coulomb interaction between electrons split the enery levels accordin to L and S, which is why we denote a term with L and S in the Russell-Saunders couplin scheme. An empirical rule is - The term with the larest spin quantum number has the lowest enery. - Amon the terms with the same spin quantum number, larest anular momentum quantum number ives the lowest enery. - This rule is very effective in findin the round level. - This is the celebrated Hund s rule. Applyin the Hund s rule to the case of V 3+ ion, we find the round term is 3 F. 4

5 Multi-Electron Crystal Field Theory When an ion with many electrons is placed in a crystal field, the crystal field will shift and split the enery levels of the multi-electron system. In addition to the simple hydroen-like atom Hamiltonian, there are three main interactions that need be included, the Coulomb interaction between electrons, spin-orbit interaction, and the crystal field. In eneral, it is impossible to obtain exact solutions. Thus, we solve for the problem includin only the larest interactions and then add the smaller terms later as small perturbations. Weak field scheme: The strenth of the crystal field is small compared to electron-electron interaction and spin-orbit interaction. - First, obtain the enery levels of free ion without the crystal field. - Then, include the crystal field effect and investiate the splittin of free ion enery levels due to the crystal field. Stron field scheme: The strenth of the crystal field is much larer than the electron-electron interaction and spin-orbit interaction. - First, obtain the enery levels and wavefunctions of one electron system under the crystal field. - Then, include the electron-electron interaction and spin-orbit interaction and investiate the splittin of the one-electron crystal field levels due to the additional interactions. 5

6 Application of weak field scheme for d 2 system As shown before, the allowed terms for free V 3+ ion are 1 S, 3 P, 1 D, 3 F and 1 G. For each term, there are (2L+1) allowed values of M L. That is, each term has a deeneracy of (2L+1). These deeneracy is partly lifted by crystal field. The reduction scheme may be obtained by usin the reduction formula just as we did for the sinle-electron case. In an octahedral crystal field, free ion term (deeneracy) 1 S (1) 3 P (3) 1 D (5) 3 F (7) 1 G (9) terms in an octahedral field (deeneracy) 1 A 1 (1) 3 T 1 (3) 1 E (2) + 1 T 2 (3) 3 A 2 (1) + 3 T 1 (3) + 3 T 2 (3) 1 A 1 (1) + 1 E (2) + 1 T 1 (3) + 1 T 2 (3) 1 S 1 A 1 1 E 1 G 3 P 1 D 1 T 1 1 T 2 1 A 1 3 T 1 1 E 1 T 2 3 A 2 3 F 3 T 2 3 T 1 free ion weak crystal field 6

7 Stron Field Scheme Start with the one-electron enery levels and wavefunctions determined by the sinle-electron crystal field theory. As shown previously, the sinle d- electron states split into two levels in an octahedral crystal field. Let us denote the triply deenerate lower level (E = 4Dq) as t 2 and the 5-fold deenerate d-shell φ u, φ v e 6Dq E = 0 4Dq φ ξ,φ ζ, φ η free ion octahedral field doubly deenerate upper level (E = 6Dq) as e. There are three possible confiurations, (1) (t 2 ) 2 - both electrons in t 2, (2) t 2 e - one in t 2 and one in e, (3) e 2 - both electrons in e. t 2 e 6Dq E = 0 4Dq t 2 Inorin the electron-electron interaction, the eneries of the three confiurations are E = -8Dq for (t 2 ) 2, 2Dq for t 2 e and 12Dq for e 2. 7

8 Reduction of Direct-Product Representation The two-electron states are represented by the direct-product representations. - (t 2 ) 2 = T 2 T 2, t 2 e = T 2 E, and e 2 = E E - These direct-product representations are reducible. Recall the discussion on direct-product representation in Class 9. The character of the direct-product matrix is the product of characters of individual matrices. χ ( A B) = χ( A) χ( B) The above equation allows us to determine the characters of the direct-product representation. Then, we can apply the usual reduction formula, as done formally in Class 9. For example, the characters for T 2 T 2 are E 8C 3 3C 2 6C' 2 6C 4 i 8iC 3 3iC 2 6iC' 2 6iC 4 T T 2 T Now use the reduction formula to find T1 T1 = A1 + E + T1 + T2 E T1 = T1 + T2 E + E E = A1 + A2 8

9 Stron Field Scheme For each irreducible representation, we may have spin sinlet (S = 0) and triplet (S = 1) states. However, some of these states are forbidden by the Pauli exclusion principle. stron field confiuration (deeneracy, enery) (t 2 ) 2 (9, E = -8Dq) t 2 e (12, E = 2Dq) e 2 (4, E = 12Dq) terms in an octahedral field (deeneracy) 1 A 1 (1) + 1 E (2) + 1 T 2 (3) + 3 T 1 (3) 1 T 1 (3) + 1 T 2 (3) + 3 T 1 (3) + 3 T 2 (3) 1 A 1 (1) + 1 E (2) + 3 A 2 (2) 1 A 1 e 2 1 E 3 A 2 1 T 1 1 T 2 t 2 e 3 T 1 3 T 2 1 A 1 1 E (t 2 ) 2 stron field confiuration 1 T 2 3 T 1 stron crystal field 9

10 Correlation Diaram Both weak field scheme and stron field scheme yield the same set of final terms. However, their order in enery is very different. So in order to et the correct enery levels, one must use the appropriate scheme that is riht for the system of interest. free ion level weak field enery level stron field enery level stron field confiuration 10

11 Tanabe-Suano Diaram The methodoloy developed for two-electron system may be extended for multi-electron systems. Enery levels of a multi-electron system in a crystal field, calculated by usin the stron field scheme, are expressed as a function of crystal field strenth. - First devised by Tanabe and Suano, Excellent description for transition metal ions in solids. Tanabe-Suano diaram for Cr 3+ ion in octahedral field 11

12 Tanabe-Suano Diaram 12

13 Tanabe-Suano Diaram for Mn 2+ (5 d-electrons) When crystal field is moderate, the round term is 6 A 1 which has a 3 2 stron field confiuration of t 2 e - accordin to Hund s rule. At extremely stron crystal field, 2 T 2 term which has a stron field 5 confiuration, t, becomes the round 2 term. crystal field enery becomes reater than spin pairin enery. e 10Dq t 2 10Dq 6 A 2 1 T 2 13

* 1s. --- if the sign does change it is called ungerade or u

* 1s. --- if the sign does change it is called ungerade or u Chapter Qualitative Theory of Chemical Bondin Backround: We have briefly mentioned bondin but it now time to talk about it for real. In this chapter we will delocalied orbitals and introduce Hückel MOT.