Chapter 73 Chapter 73
2.2. Theory
In 1935 theory came in two forms: with group theory and without group theory.
Every theorist was conscious of this dichotomy. It is probably impossible to recapture the partisan character of the early articles on atomic theory, though the passions that the methodology aroused are obvious enough. Slater (1929), in his fundamental article on the energies of spectroscopic terms, announced in the first sentence of his abstract that 'Atomic multiplets are treated by wave mechanics, without using group theory'. In the text he stressed the simplicity of the mathematics he proposed to use. C o n d o n and Shortley (1935), in a famous passage in the introduction to their book on atomic spectra, stated, in regard to group theory, that they were 'able to get along without it'. They argued that the effort to understand group theory was an obstacle to the physicist anxious to apply quantum mechanics to atomic spectroscopy, though they conceded the power of group theory in the hands of a skilled practitioner. White (1934) made no mention of group theory at all in his introductory text. On the other hand, the development of atomic theory as presented in the books of Weyl (1931), Wigner (1931) and van der Waerden (1932) depends crucially on the theory of groups. Much is made of the homomorphism between the unitary group in two dimensions, U(2), and the special orthogonal group SO(3) corresponding to rotations in physical three-dimensional space. As long as the group theorists stayed with SO(3), the difference between their approach and that of the more physically inclined was merely a matter of language. The difficulties arose as soon as the group U(2) was used to derive actual formulas, for at that point the abstraction proved too much for the average physicist. One of the reasons for this was the cumbersome apparatus that had to be wheeled into place to derive what seemed to be quite obvious results: for example, that the acceptable quantum numbers J can be obtained from the compositional S and L by simply running in integral steps from S + L down to IS - L]. To prove this, Weyl (1931, p.
190) appealed to a 'Clebsch-Gordan equation' previously derived for U(2) using monomials as basis functions for the two irreducible representations of dimensions 2S + 1 and 2 L + 1. Physicists, however, had a much more potent image in the vector
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 89 model as described, for example, by White (1934). All they had to do was take two vectors with lengths S and L and find the lengths of the possible resultants, bearing in mind that these resultant lengths differed from one another by integers.
In order to appreciate the subsequent developments in the theory, we need to explore the relationships between the different theoretical approaches in more detail.
This entails specifying the purely semantic differences and looking into the origin and working of U(2). We have also to describe the status of the theory with respect to rare-earth spectroscopy in 1935.
2.2.1. Semantics
Consider the basic commutation relations satisfied by the components Jx, Jr and J~ of an angular-momentum vector J:
[Jx, Jr] = iJz, [Jy, J=] = iJ x, [J=,Jx]=iJy. (3)
T o the physicist, the origin of these equations lies in such commutation relations as Ix, Px] = i, where the momentum p of a particle is measured in units of h. The construction of / from r × p leads to [lx, ly] = il~ and its cyclic permutations, from which the general form of the commutation relations for the components of any angular-momentum vector (including spin) is hypothesized. T o a group theorist, eqs.
(3) specify the Lie algebra that underlies the group SO(3). Operations of the group are performed by exp(iOkJk), where k -- x, y or z. The angles of rotation, Ok, are the parameters of the group: the operators
Jk
a r e the 9enerators.In the language of angular momentum theory, the equations
j 2 [j, Mj ) = J(J + 1)l J, Mj ), (4)
J~IJ, M j ) = m j l J , m j ) , (5)
J+IJ, M j ) = [J(J + 1 ) - Mj(Mj ± 1)]I/21J, M J ± 1), (6) where J_+ - Jx -+ iJy, specify the action of the components of an angular-momentum vector on kets defined by the pair of quantum numbers J and Mj (Condon and Shortley 1935). F r o m the standpoint of group theory, eqs. (5) and (6) represent the action of a generator of SO(3) on a basis function for the irreducible representation ~ j that possesses a dimension 2J + 1. The operator j z commutes with all the generators Jk and (apart from a possible constant factor) is Casimir's operator for SO(3). G r o u p theorists visualize the kets and the shift operators J+ in the following way. The basis functions I J,
mj
~ are represented by the 2J + 1 points, - J , - J + 1 . . . J, in a one-dimensional weight space. The action of J+, as given by eq. (6), is to allow trips to be taken in the weight space, one step at a time.A state of a rare-earth atom can be written as 17 J M j ) , where 7 denotes all the additional quantum numbers that are required to make the definition complete. In 1935, a mechanism was available for calculating the matrix elements of any operator H. The s t a t e s
[TJMj)
had to be expressed as linear combinations of Slater determinants involving the quantum numbers (nlmsmt)j for each electron j. The required matrix elements thus became sums over integrals involving a pair of Slater determinants and H. The procedures and tables of Condon and Shortley (1935)lightened this formidable task. If S and L were well-defined (that is, if Russell- Saunders coupling obtained), the state I,/JMj) could first be expressed as a superposition of the states 17SM s LML) by writing
1 7 J M j ) = ~ ( S M s , L M L I J M j ) I T S M s L M L ) , (7) Ms, ML
where the coefficients are the so-called Clebsch~]rordan (CG) coefficients. A general form for them had been given by Wigner (1931), and his result was quoted by Condon and Shortley (1935). They need not have done this: their own techniques were adequate to the task in hand had they but applied a few simple theorems from combinatorics (see, e.g., Griffith 1961, or Judd 1963). In the language of group theory, eq. (7) amounts to expressing a basis function for 9 j of SOj (3) as a linear combination of bases for the Kronecker product 9 s x 9t~ of the direct product SOs(3) x SOL(3). The notation being used here implies that the vector A forms the generators for the group SOA (3).
The next step involves expressing 17 S M s L ML) as a linear combination of the Slater determinants {K1 g2"" "Ks}, where K j - (nlmsm~) J. Mathematically, this entails the construction of antisymmetric tensors from products of N spin-orbitals transforming as
( 9 1 / 2 X 9 / 1 ) X ( 9 1 / 2 X (..~/2) X "'" X (C-~1/2 X (.~lN).
Weyl (1931, pp. 369-377) outlined how this can be done, but he limited his example to the classification of the terms of l 3 rather than giving an explicit calculation of their wavefunctions. F r o m his general approach, however, it seems clear that he would have brought the group U(2) into play to do so. Condon and Shortley (1935, p. 226), on the other hand, proposed the method of Gray and Wills (1931), and they gave as an example the expansion of ]d 3 2D, M s = ½, ML = 2). As written, this ket is ill-defined because there are two 2D terms in d 3. The limited freedom in expanding the ket is removed by picking one state at random. This effectively defines a particular 7, say 7a. The one remaining 2D term can be found from the orthogo- nality constraint, thereby defining a second 7, say 7b. The states ]2D~a ) and 12DTb) are now defined and available for use in calculating matrix elements. In 1935, no group-theoretical significance could be attached to 7a and 7b" Fortunately, terms like 2s + t L occurring in most configurations of immediate interest could be separated by specifying a coupling scheme. For example, there are two 2D terms in pds, but they can be formally separated by writing I(Pd)3D, s, 2D) or [(pd)lD, s, 2D). This is not possible for configurations of equivalent electrons because it is not clear how the Pauli Exclusion Principle can be properly satisfied. It is apparent from table 17 of Condon and Shortley (1935, p. 208) that the frequent occurrence of like terms in the rare-earth configurations fN was well known. The ad hoc procedure of Gray and Wills (1931) in fixing 7a, °A, "; .. . . must have seemed highly unsatisfactory, to say the least.
2.2.2. The unitary group U(2)
The great appeal of U(2) over SO(3) lies in the fact that our basic equations (3)-
ATOMIC THEORY AND OPTICAL SPECTROSCOPY 91 (6) remain valid if the following replacements are made:
J+ --' - t / ~ , J - --" - ~ , J~ - , - k ~ + ½t/~, (8) [ J , m ) -'~(--1)J-M ~J-Mt/J+M[(J -- M ) ! ( J + m ) ! ] - 1 ' 2 , (9) where ~ - ~ / ~ , etc. Thus the three components of J as well as the 2J + 1 states ]J, M j ) c a n be represented by functions of just two quantities (( and q) and the corresponding differential operators ( ~ and ~,). We say that ( and t/ form the components of a spinor. The connection between U(2) and SO(3) can be seen by writing the general unitary transformation in two dimensions for which the de- terminant is + 1 as
~' = ~ eil~+'~"2cos½fl + t/e-il~-'/2sin½fl, (10)
t/' = -¢eil~-~'~/Zsin½fl + t/e il~ 7)~2cos½fl. (11) It is not difficult to show that the spinor ( - 0 , , ©~) transforms in an identical way to (~, t/). We can now use (8) to find how J transforms, and we get
J'x = Jx(COS~COSflCOS7 - sin~sinT)
+ Jy (sin c~ cos fl cos 7 + cos c~ sin 7) - J~ sin fl cos 7, (12) together with similar equations for J'y and J'~. These transformations correspond to a rotation in physical space through the Euler angles (~, fl, 7)- Thus an element of U(2), acting in the space of the spinor (~, t/), induces a coordinate rotation in the laboratory space (x, y, z).
This approach impresses at once with its power. F o r example, we can immediately find how ]J, M j ) behaves under a rotation characterized by (e, fl, 7) by substituting the transformed spinor (¢',t/') into (9). The theory can be further developed by introducing an invariant formed from the two spinors (~, 1/) and (a, b). Under the transformations given by eqs. (10) and (11) we find
a't/' - b'~' = a t / - b~,
and the combination a t / - b~ is called a spinor invariant. The orbital states of an 1 electron can be represented by ( a t / - b ~ ) 2t, since we have only to ask for the coefficient of a t + " b t - " to produce the (unnormalized) state ~t ,,t/t +,,, which, accord- ing to (9), corresponds to ]l, m). Spinor invariants are a key ingredient in what has become known as the symbolic method of Kramers (1930a, 1931). The method of using C G coefficients to couple angular momenta is replaced by the formation of the appropriate products of spinor invariants. Consider, for example, the two-electron configuration 1l'. In addition to the spinor (~, ~/) for the orbital angular momentum l of the first electron, we need (/~, v) for the second. We can form three spinor invariants, namely ( a t / - b~), (av - bl~) and (~v - r//~). The product @c, given by
@L = (at/ -- b ~ ) t + L - r ( a v -- b/~)r+c-'(~v - t/#)t+r-L, (13) is a sum of terms of the type
(ax+YbZL-x-y)(~l+L-l'+z-xt/l+l'-L-z+x)(lAZl' y-z yy+z). (14)
The powers of ~ and/] add to 2/, and the powers of p and v add to 2l': hence q)L is a linear combination of the orbital states of ll'. Since the powers of a and b add to 2L, we know that the products aX+Yb 2L-x-y transform under U(2) like the components of a state with angular-momentum quantum number L. But q)L is an invariant:
hence the transformation properties of the part of q~L depending on ~, r/, p and v must also be characterized by L. That is, ~L is the orbital part of the wavefunction of ll' with total angular moment6m L.
The representation of operators by spinors is a rather more delicate matter. By 1935 only one detailed application of Kramers's symbolic method to atomic spectroscopy appears to have been carried out: that of Wolfe (1932) to the atomic configuration Is. Some of the difficulties of the method are discussed below in section 5.1. F o r the moment we need only say that the abstract nature of the method did not prove appealing to most theoretical spectroscopists. N o one in the 1930's or since, as far as the writer is aware, has used the functions
~6, ~5/], ~4/]2, ~3/]3, ~2/]4, ~/]5 and /]6 (15)
for the seven (unnormalized) orbital states of an f electron. Of course, the spinor whose two components c( and fl represent the spin-up and spin-down states of an electron are familiar to all. But students find these quantities difficult enough to come to grips with. Some wonder where ~ and fl are located, unaware that a special space has been created for them. A mathematician might well be tempted to take the fourteen spin-orbitals of an f electron in the form ~6Gt, ~6fl, ~5/]~ . . . q6fl. T o a physicist, the loss of a mental image of a wavefunction, such as that provided by the spherical harmonics Ya,,(0,(h), is too high a price to pay for mathematical homogeneity.
2.2.3. Application of the theory to lanthanide spectroscopy
The attraction of the second spectrum of lanthanum for atomic spectroscopists has been mentioned in section 2.1. The analyses of the two-electron configurations 4fSd and 4f 2 carried out by Condon and Shortley (1931) achieved great importance by being the only lanthanide spectrum described in detail in their subsequent monograph (Condon and Shortley 1935). The least-squares fits to the experimental data provided by Russell and Meggers (1932) yielded Slater parameters given (in cm 1 ) by
F2(4f, 5d) = 12075, Gl(4f, 5d) = 12500,
F4(4f, 5d) = 11100,
G3(4f, 5d) = 9350, G 5(4f, 5d) = 5750, (16) for La II 4f5d, and by
F/(4f, 4f) = 21000, F4(4f,4f) = 23500, F6(4f,4f) = 1930 (17) for L a l I 4f 2. It is highly interesting to compare these values with those of the modern analysis of Goldschmidt (1978). Taking the effects of the influence of 4f6s,
A T O M I C T H E O R Y A N D O P T I C A L S P E C T R O S C O P Y 93
5d6p and 6s6p on 4f5d into account, she obtained
FZ(4f, 5d) = 15855 _+ 105, F4(4f, 5d) = 12266 + 208, Gl(4f, 5d) =11025 _+ 35, G3(4f, 5d) = 10080 _ 157,
GS(4f, 5d) = 6098 + 305, I18)
for L a l I 4f5d + 4f6s + 5d6p + 6s6p. A similar analysis for LaI1 5d 2 + 5d6s + 6s 2 + 4f6p + 4f 2 + 6p 2 + 6d6s yielded
F2(4f,4f) = 24043, Fg(4f,4f) = 21964, F6(4f,4f) = 14668, (19) with mean errors of around 2};. It is evident that the numbers obtained by Condon and Shortley (1931, 1935) have stood the test of time pretty well, with one exception:
F6(4f,4f). This parameter is almost an order of magnitude too small in eqs. (17), a result which Goldschmidt (1978) attributes to 6p 2 ID2 being mistaken for 4f 2 ID 2.
A clue that something is amiss is the failure of the F k of eqs. (17) to satisfy the requirement F z > F 4 > F 6, which follows from the very definition of the Slater integrals (see, e.g., Condon and Shortley 1935, p. 177). Goldschmidt's analysis confirms that the lowest term of 4f5d has 1G as its principal component rather than 3H. The latter would be expected from a naive application of Hund's rule. Since the lowest configuration of La II is 5d z, whose Hund term, 3F, is indeed the lowest for that configuration, not much attention has been paid to the apparently anomalous situation for 4f5d in La II. The appearance of 4f5d as the ground configuration of CeI and C e I I I is much more striking (see sections 6.2 and 6.2.1).
The hyperfine structures of E u I and L u I I reported by Schiller and Schmidt (1935a,b) were analyzed by Casimir in his 1936 prize essay for the Teyler's Foundation in Haarlem. The later reprint (Casimir 1963) has made this work more accessible. Racah (1931a,b) had worked out a method for using the electronic spin- orbit coupling constants (which could be easily deduced from experiment) together with some relativistic correction factors to find the hyperfine interaction strengths.
The procedure was further elaborated by Breit and Wills (1933). Casimir (1963) was able to use this method to deduce actual numerical values for the magnetic-dipole and electric-quadrupole moments of the europium and lutetium isotopes. His results for the quadrupole moments provided the first quantitative figures for the non- spherical distribution of electric charge within the nucleus.
Nothing as striking as this occurred at that time in the interpretation of the spectra of rare-earth ions in crystals. The deceptively low value of F 6 given in eqs.
(17) was to impede progress for several decades. N o detailed analyses of crystal spectra had been attempted by 1935, though Bethe (1930) had examined in a qualitative way some Zeeman data of Becquerel (1929). Bethe (1929) had also set up a formal theory for the splittings of atomic levels produced by the crystal field, but no applications to rare-earth ions were possible.