Lanthanide Calculations
This document provides interactive calculations of quanities useful for lanthanide electronic spectroscopy. (The lanthanides are also known as the rare earth elements.) There are 13 lanthanides that contain a partially full 4f subshell in their trivalent ions - lanthanum (4f0) and lutetium (4f14) can have no 4f-4f transitions and so are of less interest.
Electronic configurations
The electronic configuration of lanthanide atoms are shown below. All lanthanides are most stable as trivalent ions. 6s and 5d electrons are ionised before the 4f and so the trivalent lanthanide ions have n f electrons where n is the lanthanide number in the left hand column of the table.
| No. | Symbol | Name | Electronic configuration |
|---|---|---|---|
| 1 | Ce | Cerium | [Xe] 6s25d14f1 |
| 2 | Pr | Praseodymium | [Xe] 6s24f3 |
| 3 | Nd | Neodymium | [Xe] 6s24f4 |
| 4 | Pm | Promethium | [Xe] 6s24f5 |
| 5 | Sm | Samarium | [Xe] 6s24f6 |
| 6 | Eu | Europium | [Xe] 6s24f7 |
| 7 | Gd | Gadolinium | [Xe] 6s25d14f7 | 8 | Tb | Terbium | [Xe] 6s24f9 | 9 | Dy | Dysprosium | [Xe] 6s24f10 | 10 | Ho | Holmium | [Xe] 6s24f11 | 11 | Er | Erbium | [Xe] 6s24f12 | 12 | Tm | Thulium | [Xe] 6s24f13 | 13 | Yb | Ytterbium | [Xe] 6s24f14 |
Configuration weight
The configuration weight is the total number of states in an electron configuration. This would be the total number of energy levels if there were no degeneracy. The total number of states is the number of ways the electrons in the atom can be arranged in its orbitals - bearing in mind that electrons are indistinguishable. In the ground state configuration of the trivalent lanthanides we can neglect the xenon core as there is only one way the electrons can be arranged in full shells, so we need only consider the 4f subshell, in which there are 14 spin-orbital combinations for a single electron.
For a subshell of orbital angular momentum quantum number l (0=s, 1=p, 2=d, 3=f, 4=g) there are 2l+1 orbitals, and so 2(2l+1) spin-orbitals. It follows that the total number of ways of arranging n electrons in a subshell of orbital angular momentum l is:
The form of the denominator above shows that the configuration weights take value that are symmetrical about the half-full shell. These weights are as follows:
| fn | Configuration weight |
|---|---|
| f0, f14 | 1 |
| f1, f13 | 7 |
| f2, f12 | 91 |
| f3, f11 | 364 |
| f4, f10 | 1001 |
| f5, f9 | 2002 |
| f6, f8 | 3003 |
| f7 | 3432 |
However, when considering excited configurations, when a 4f electron is excited to another subshell, we also need to consider the spin-orbital combinations of the other subshell. The total number of states is then equal to the product of the number of states in each of the different partially-full subshells.
The ground state level of fn configurations
The ground state level of fn configurations is determined by Hund's rules. First S is determined by maximising the electron spin projection, then L is determined by maximising the orbital magnetic quantum numbers. Finally J, which is the vector sum S+L, takes its minimum value for less than half-full shells and its maximum value for more than half-full shells. (When a shell is half-full the ground state is an S state with no orbital angular momentum, so J can only take one value.) The ground state levels for the fn configurations are as follows:
| fn | Ground state level |
|---|---|
| f0, f14 | 1S |
| f1 | 2F5/2 |
| f2 | 3H4 |
| f3 | 4I9/2 |
| f4 | 5I4 |
| f5 | 6H5/2 |
| f6 | 7F0 |
| f7 | 8S7/2 |
| f8 | 7F6 |
| f9 | 6H15/2 |
| f10 | 5I8 |
| f11 | 4I15/2 |
| f12 | 3H6 |
| f13 | 2F7/2 |
Remember how these levels are described as 2S+1LJ where the L value is denoted by a letter: 0=S, 1=P, 2=D, 3=F, 4=G, 5=H, 6=I, 7=K, 8=L, 9=M 10=N, 11=O, 12=Q.
Note how the S and L quantum numbers are symmetrical around the half-full shell. This is because the electron-electron repulsion interaction is two-body in nature and so the sign of the interaction is not changed when positive holes are considered instead of negative electrons. However, the spin-orbit interaction, coupling S and L to give J is one-body in nature and so does change sign when considering positive holes in the second half of the lanthanide series.
The coupling of angular momenta
We started by considering the electrons individually when looking at electronic configurations and their weights. However, when considering observable states of an atom we use quantum numbers denoted by capital letters that represent the ensemble of electrons. This is because the angular momenta that electrons possess - which is what interacts with electromagnetic radiation - couple together to form a resultant. It is most convenient to use the coupling scheme that best reflects the relative importance of the different interactions experienced by the electrons. In the lanthanides the main interactions, in decreasing order of importance, are:
Electrostatic repulsion > spin-orbit coupling > crystal field
Since spin-orbit coupling is a relativistic phenomenon its magnitude increases with (the fourth power of) atomic number, and so it dominates in actinide electronic spectroscopy but plays a very minor role in the first-row d block elements. The crystal field interaction is only minor in the lanthanides as the 4f orbitals are core-like, but plays a more significant role in other transition series, especially the heavier d block metals.
A total angular momentum state can often be built up from two individual angular momenta in more than one way. For example an S term ['term' is used to denote an SL level], ie one with L=0, in the p2 configuration can result either from the two individual electrons both having ml=0, ie spin-paired in that orbital, or the electrons taking ml=+1 and ml=–1. (Let's not worry about the different spin combinations of this state for the time being.) It is often necessary to express a two-electron state as a linear combination of single-electron states, but to do so we need to calculate the coefficients in front of each of the single-electron states in the expansion. These coefficients are the Clebsch-Gordan coefficients, shown in the expansion below:
J, used in the expression above, is the letter used for a general angular momentum (ie not specifying if it is spin or orbital angular momentum). |(j1j2)JM> represents the coupled two-electron state with angular momentum J, the resultant of angular momenta j1 and j2 from electrons 1 and 2 (ie the uncoupled states), which have projections m1 and m2 respectively. The Clebsch-Gordan coefficient is the term at the far right-hand side, which is multiplied by the two single-electron states as part of the sum of single-electron states that make up the two-electron state on the left-hand side. In the earlier example, J=M=0 as it is a singlet state, and j1=j2=1 as the electrons are p electrons.
Let's consider a couple of special cases of Clebsch-Gordan coefficients. When J=j1+j2 and M=J then there is only one case when the Clebsch-Gordan coefficient takes a non-zero value, and that value (necessarily) is 1. This is because the only way of coupling j1 and j2 to the resultant is when m1=j1 and m2=j2. Therefore the expansion of JM contains only only one term and so its coefficient must equal one. Another interesting special case is when the resultant has zero angular momentum, as in our earlier example. The general result for this type of Clebsch-Gordan coefficient is:
The Greek deltas are Kronecker deltas, which take the value of zero when the two variables following them are different, and take the value of 1 when they are the same. The two Kronecker deltas follow from the triangle rule and the conservation of angular momentum respectively (these are explained in the following section on 3-j symbols). In the case of our two p electrons all non-zero values of the Clebsch-Gordan coefficient will be equal to 3–1/2, so all possible uncoupled functions must occur in the linear combination with equal weight. There are in fact three possible allowed uncoupled functions: m1=+1 m2=–1, m1=–1 m2=+1, and m1=m2=0. So why don't the coefficients add to 1? It's because these are the coefficients for the wave function for the coupled state, and it is the square of the wave function that represents the probabilities. In the square of the wave function the sum of the squares of the coefficients comes to 1, while all the cross terms can be neglected because the angular momentum functions are defined to be orthonormnal, ie different functions multiply to give zero and like functions multiply to give 1.
3-j symbols
It can be quite time-consuming calculating Clebsch-Gordan coefficients. It is easier to calculate them from the value of the 3-j symbols, which are closely related. The way 3-j symbols are defined, they also have cyclic permutation symmetry, which is a useful property. The following expression relates how Clebsch-Gordan coefficients can be found from the relevant 3-j symbol.
It is clear from the above relation that j3 in the 3-j symbol represents the total angular momentum resultant J in the Clebsch-Gordan coefficient, and that m3 is minus the projection of J. If the columns of the 3-j symbol are permuted cyclically then its value remains unchanged. A non-cyclic permutation introduces a phase factor of (–1)j1+j2+j3. A similar phase factor results from changing the signs of the three m values in the 3-j symbol. The 3-j symbol has several properties which reflect the addition of two vectors to form a resultant. The values of the three projection quantum numbers must sum to zero, or else the 3-j symbol vanishes. This is because m3 takes minus the value of M, and so this reflects the conservation of angular momentum. Similarly the modulus of an m value may not exceed its associated j value, as the projection of a vector may not be greater than the length of the vector. Similarly the value of the resultant, j3, must fall in the range between the sum and the positive difference of the vectors j1 and j2. This last rule is known as the triangle rule. It is the conservation of angular momentum as a vector sum. Due to the cyclic permutation symmetry of the 3-j symbol, j1 and j2 are under the same kind of constraint as j3. This is described succintly in the final selection rule in the 3-j Calculator below.
The general expression below for calculating 3-j symbols is found by applying angular momentum raising and lowering operators to general coupled and uncoupled states and equating the common vector-coupling coefficients. Then the expression is symmetrised so that all three angular momenta are on an equal footing, in other words by considering three angular momenta coupling to zero rather than two angular momenta coupling to a third. The derivation of the expression below is very long, and given by Edmonds (1996).
The sum is over all the integers k such that all the factorials in f(k) have non-negative arguments (Weisstein).
3-j calculator
Input values of the different j and m values and press Calculate to see the value of the 3-j symbol. For calculations involving the orbital angular momentum of f2 states j1=j2=3. For 3-j symbols involving electron spin, when half-odd integers will be required, express halves using a decimal point. Note that the calculation will return zero if j and m values do not conform to the selection rules:
- j and m values must be integers or half-odd integers.
- j1+j2+j3 must be an integer.
- m1+m2+m3 must be zero.
- j+m must be an integer.
- j values may not be negative.
- mi must take one of the values ji, ji–1, ..., –ji+1, –ji.
- j1,j2 and j3 must obey the triangle conditions by satisfying: j1+j2–j3>=0 ; j1–j2+j3>=0 ; –j1+j2+j3>=0.
6-j symbols
3-j symbols are employed when three angular momenta couple to zero. If three angular momenta couple to a non-zero resultant then a 6-j symbol is required to describe the coupling. One might suspect a '4-j' to be what is needed but the extra two angular momenta reflect the fact that - unlike the 3-j case - there is more than one way of coupling three angular momenta to a non-zero resultant. For example, j1 may first couple with j2 to give j12 which then couples with j3, or j2 may couple with j3 to give j23 before then coupling with j1.
6-j symbols are required to find the spin-orbit matrix elements. The operator acts on the wave function in the |JM> basis. The operator itself is the scalar product S.L. It acts on S and L while leaving J unchanged. The vector operators S and L are tensors of rank 1 and so they can connect spin and orbital angular momentum states differing by up to one unit. The resulting 6-j symbol is written as:
where the primes denote the quantum numbers for the ket wave function, following the nomenclature of Judd (1998). Note that 6-j symbols are written in curly brackets (not parentheses like 3-j symbols) and are independent of m quantum numbers and so these are omitted. Generalised 6-j symbols are typically written:
6-j symbols are highly symmetrical: they are invariant under the exchange of any two columns and any two numbers in one row with the two corresponding numbers in the other row. There is a formula that can be used to compute 6-j symmbols. It is similar in form to the 3-j formula, except longer:
The sum is over all the integers k such that all the factorials in f(k) have non-negative arguments (Weisstein).
6-j calculator
Input values of the different j and l values and press Calculate to see the value of the 6-j symbol. When half-odd integers are required, express halves using a decimal point. Note that the calculation will return zero if j and l values do not conform to the selection rules:
- Four triangle rules have to be satisfied: (j1 j2 j3), (l1 l2 j3), (l1 j2 l3), (j1 l2 l3).
- j and l values must be integers or half-odd integers.
- j and l values must be non-negative.
- In each of the triads of the triangle rules above, the sum of the elements of each triad must be an integer (ie all three elements should be integers or two of them should be half-odd integers). For example, the first triangle rule means that the following must be satisfied: j1+j2–j3>=0 ; j1–j2+j3>=0 ; –j1+j2+j3>=0.
Bibliography
- Edmonds, A. R., Angular Momentum in Quantum Mechanics, Princeton University Press, 1996
- Judd, Brian R., Operator Techniques in Atomic Spectroscopy, Princeton University Press, 1998
- Rotenberg, Manuel, Bivins, R., Metropolis, N., Wooten, John K. Jr, The 3-j and 6-j Symbols, The Technology Press, Massachusetts Institute of Technology, 1959
- Weisstein, Eric W., "Wigner 3j-Symbol" and "Wigner 6j-Symbol" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Wigner3j-Symbol.html and /Wigner6j-Symbol.html
Last updated 27/08/08
These pages are still under development.
Future work to include making an online database of 3-j and 6-j values, looking up the 3-j values in online calculations of electrostatic matrix elements, and looking up the 6-j values in online calculations of spin-orbit matrix elements and coefficients of fractional parentage. Also to write a 9-j calculator in order to express the coefficients in the expansion of f2 |JM> in |s1ms,1l1ml,1> |s2ms,2l2ml,2>, where of course s1=s2=1/2 and l1=l2=3.