HAL Id: jpa-00247677
https://hal.archives-ouvertes.fr/jpa-00247677
Submitted on 1 Jan 1992
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Matrix-element proportionalities for the atomic f shell from Labarthe’s groups
B. Judd, G. Lister
To cite this version:
B. Judd, G. Lister. Matrix-element proportionalities for the atomic f shell from Labarthe’s groups.
Journal de Physique II, EDP Sciences, 1992, 2 (4), pp.853-863. �10.1051/jp2:1992171�. �jpa-00247677�
Classification Physics Abstracts
31.10 31.90
Matrix-element proportionalities for the atomic f shell from Labarthe's groups
B. R. Judd and G. M. S. Lister
Henry A. Rowland Department of Physics and Astronomy, The Jouns Hopkins University, Baltimore, Maryland 21218, U-S-A-
(Received 23 September 199I, accepted 27 November 1991)
R4sumd.- On d£montre que le groupe SO(7) utilis£ par Racah pour d£finir les dtats et les
op£rateurs dans la couche f peut dtre augmentd par deux groupes similaires, SO(?I' et SO (7)", ddcouverts par Labarthe it y a plus de dix ans. Leurs propri£tds sont dtudi£es par deux
m£thodes (I) en utilisant [es op£rateurs multi-dlectroniques, et (2) en regardant les 16 384 £tats de la couche atomique f comme ceux de la configuration is + f)~, compl£tde par deux Etiquettes
de pant£, oh [es symboles s et f repr£sentent des analogues h des quarks. En mettant [es repr£sentations irrdductibles de SO(?) et de SO(?I' entre celles de SO(8) et G~, on peut donner
une explication pour quelques-unes des relations inattendues entre des dldments de matrice. Un
exemple d£taill£ est dorm£ pour la configuration £lectronique f.
Abstract. It is shown that the SO(7) group used by Racah to define states and operators in the f shell can be usefully augmented by the two similar groups, SO(?I' and SO(7)~', discovered over
ten years ago by Labarthe. Their properties are studied by two approaches : ii) using multi- electron operators, and (2) regarding the 16 384 states of the atomic f shell to be generated by the
quark-like configuration (s + f)~, augmented by two parity labels. By sandwiching the irreducible representations of SO(?) and SO(71' between representations of SO (8) and G~, an explanation
can be found for some of the unexpected relations between matrix elements. The method is illustrated by a detailed example in the electron configuration f~.
1. Introduction.
In 1979, Jean-Jacques Labarthe took a leave of absence from the Laboratoire Aimd Cotton and spent some months with the atomic theory group here at Johns Hopkins. His principal
role was to examine whether useful Lie groups could be constructed by taking multiple
products of the creation and annihilation operators for electrons. Until that time, the generators of Lie groups were almost always formed by taking at most pairs of such operators.
This choice guarantees that the commutator of two generators yields operators of the same
type, thereby facilitating the construction of the various group generators. Racah's classic analysis Iii of the atomic f shell can be cast in this form, for example. However, it had been clear for some time that the scope for (ntroducing Lie groups into atomic shell theory would be enormously widened if this constraint on the generators could be relaxed. In his analysis,
Labarthe took advantage of a four-fold factorization of the atomic f shell that had been established some years earlier [2, 3] in order to limit the possible multiple products to a
manageable set. This involved taking four linear combinations A~, ~c~, v~, and f~ of the basic
annihilation and creation operators rather than the operators themselves. The index
q runs over the 2 f + I values associated with the orbital states of
an I electron, and the collection of operators 6~(0 m A, ~c, v, or f) forms a spherical tensor 0 of rank I in the sense of Racah [4]. These four quasiparticle tensors anticommute, one with another, and provide
the mathematical basis for the four-fold factorization of the shell.
In the process of studying the commutation properties of coupled tensors of the type
(00 0), in which an even number of o's appear, Labarthe noticed that a remarkable
mathematical structure appears for f electrons [5]. In addition to the operators (00)l~l
(k =1, 3, 5), which close under commutation and form the generators for the Lie group
SO~(7), the tensor (00)l~l can be combined with the sextuple product (000000)l~l in two different ways to form, with (00)ill and (00
)~~l> the generators for two new SO(7) groups, which we shall call here SO~(71' and SO~(7)". By summing over the four possibilities for 6 (with appropriate phases (- 1)~ drawn from earlier work [3]), we obtain the generators for SO (7 ), SO (71' and SO (7) ". The first of these three groups is identical to that used by Racah Ill ; but the other two lie outside the normal scope of atomic spectroscopy. At the time when Labarthe made his discovery, he did not recognize its great potential value ; and neither did
we. He wrote, referring to the construction of a variety of groups from multi-electron tensors,
« though these groups are small enough, we have not found useful applications for them » [5].
It is the function of the present paper to alert the reader to progress that has been made in
putting Labarthe's groups to work, and to give some new examples of their usefulness.
2. Automorphisms of SO(8).
From the perspective of the present day, the three SO(7) groups have their origin in the
automorphisms of SO(8). The three basic irreducible representations of this group, namely
(1000), and ), all possess a dimension of 8 and can be
2 2 2 2 2 2 2 2
converted, one to another, under rotations in weight space. This is described in detail by Georgi in the section « Fun with SO(8) » in his book on Lie algebras for particle physics [6]. In
descending from SO(8) to SO(7), we can either imagine a given representation of SO(8) being projected into three different directions, corresponding to three different SO(7) groups, or we
can rotate the representation into its three forms and take a common projection to yield three
representations of a single SO(7) group. The various embeddings of SO(7) in SO(8) have been used in the theories of supergravity and superstrings [7] in particular, they have very
recently been introduced to study octonionic superstring solitons [8].
3. Factorizing the f shell.
To generate the states of the atomic f shell from our four-fold factorization, we need two
parity labels (to indicate whether the number of electrons with spins up or spins down, N~ and NB, are even or odd) and four statistically independent quark-like objects belonging
to the 8-dimensional spinor of SO(7) [2]. For electrons with spins up,
2 2 2
~ yields the representations (000), (loo), (I lo) and (I II) of total dimension 64
2 2 2
that Racah introduced to describe the terms of maximum multiplicity of f°, f~, f~ and
f~ (for N
~ even) or f~, fl, f~ and f~ (for N~ odd). The 128 states produced in this way are then
coupled with similar states with their spins down to provide all 16 384 possible states of the f
shell. If we use the fact that the SO(3) structure of is s + f [2], we can easily
2 2 2
confirm that the quadruple Kronecker product (s + f)~ gives all L values, with their correct
degeneracies 2 I + I, for the spectroscopic terms of the f shell coming from a particular parity
choice gg, gu, ug or uu for (N~, NB).
The 28 generators of our SO(8) are the components of the tensors
£ '(00 )ill, £ '(00)l~~, z '(00 )l~l, £ '(000000 )l~l (I)
g g g e
Table I. Branching rules for U (8 )
- SO (8 ) - SO (7 and SO (71'. The branching rules for
SO (8 - SO (7 " are identical to those for SO 8)
- SO (7 (with the addition ofdouble primes
to the representations of SO (7)") except that II I-I) yields (000) " (100) " (200)" and ( II II) yields II I)". The rules for the branching of the repi-esentations of all three SO (7 groups to
G~ can be read off from table E-2 of Wybourne [9].
U(8) SO(8) SO(7) SO(71'
~~~ ~~~~~~ (0001'(1001'
[11] (1loo) (100) (1lo) (lool' (I lol'
12j jo000) (000) (o001'
(2000) (lll) (0001' (lool' (2001'
[1 1] (11 lo) (I lol' I I II'
I I I
[21] (1000) j j j (0001'(1001'
3 3 3
(2100) j j j j § § (lool'(l101'(2001'(2101'
[3] (1000) j (0001'
(1001'
(3000) (0001' (1001'(2001'(3001'
[1111] (1 11-1) (1 11) (1111'
(11 11) (000) loo) (200) (1111'
[211] (1loo) (100) (1lo) (1001' (1lol'
(21lo) (I lo) (I I1) (210) (211) (1lol'(I IIl' (2101'(2111'
[22] (0000) (000) (0001'
(2000) (1II (0001' (lool' (2001'
(2200) (200) (210) (220) (2001' (2101' (2201'
[31] (1 100) (100) (110) (1001' (1 101'
(2000) (111) (0001' (1001' (2001'
(3100) (211) (221) (1001' (1101' (2001' (2101' (3001' (3101'
[4] (0000) (000) (0001'
(2000) (1II (0001' (lool' (2001'
(4000) (222) (0001' ( lool' (2001' (3001' (4001'
The primes on the summation signs indicate that appropriate phases (for example, the factors (- 1)~ in the original quasiparticle formulation [3]) should be included. As Labarthe showed, the operators (I) change the electron number by 0 or ± 4 [5]. The first three tensors (I)
commute with the total spin S, but this is not true of the last one. Thus the irreducible
representations of SO(8) involve states with mixtures of Ms values and electron numbers
differing by multiples of 4. It is convenient to embed of SO(7) in (1000) of
2 2 2
SO(8) and lo.. 0] of U(8), for then the leading weight « I » suggests a single quark, q. If we delete the final string of zeros in the representations of U(8), we can write those occurring in the four-quark configuration q~ as [1111], [211 ], [22], [31], and [4]. The decompositions of these representations, via those of SO(8), provide the irreducible representations W of SO(7) used by Racah [I] to label the states of the f shell. The relevant branching rules are set out in
table1.
4. The common subgroup G~.
One of the unexpected features of Racah's analysis Ill is the occurrence, as a subgroup of SO(7), of the exceptional Lie group G~ that itself contains the SO(3) group corresponding to rotations in ordinary 3-dimensional space. The generators of G~ are the 14 components of the
tensors of ranks I and 5 appearing in (I). Of course, this result is independent of whether or
not we take linear combinations of the two tensors of rank 3. It follows that G~ is also a
subgroup of both SO (71' and SO (7)". Thus, in descending from SO(8) to G~, and thence to SO(3), we have three distinct routes. The options for a single quark run
(1000) - j
- (00) + lo)
- s + f,
(1000)
- (0001' + (1001' - (00) + (10) - s + f, (2)
(1000)
-
"
- (00) + (10)
- s + f.
These fairly modest differences at the I-quark level become much more striking in
q~. For example, with the aid of table I and the branching rules describes by Wyboume [9] we find
(3100)
- (211) - (lo) + (II) + (20) + (21) + (30)
+ (221)
- (lo) + (II) + (20) + (21) + (30) + (31)
for SO(7) as an intermediary and
(3100)
- (lool'
- (lo)
+ (l101'- (lo) + (II)
+ (2001'- (20)
+ (2101'- (II) + (20) + (21)
+ (3001'- (30)
+ (3101'- (21) + (30) + (31)
for SO (71'. The same representations of G~ occur in both decompositions of (3 loo) as, of
course, they must; but the representations of SO(7) and SO (71' are different in the two
cases. The sandwiching of SO(71' between SO(8) and G~ is an important tool for our
subsequent analysis.
5. Quarks as basic entities.
From a mathematical point of view we need not continue to use the operators based on the
creation and annihilation of electrons. We could equally well employ the creation and
annihilation operators q+ and q of our quarks. Their tensorial character is specified in the reductions (2). Since the generators of SO(71' belong to the representation (I lol', their
construction from the s and f quarks necessarily takes the form £~f( f~)l~l, where
g
k
= 1, 3, and 5. In this coupled product the operators f( and f~ stand for quarks and not electrons. Of course, the similarity between these generators and those for SO(7), where electron operators are coupled to orbital ranks of1, 3, and 5, is very striking : but such a result is all part of the symmetry associated with the automorphisms of SO(8). Labarthe noticed the
possibility of using quark-like generators for the various SO(7) groups, without, however, naming the basic entities [5] beyond using 3 for f and 0 for s. His R'(7) is our
SO~(7)" ; his R"(7) is our SO~(71'. We have interchanged the single and double primes
because the most substantial change in the analysis from SO(7) occurs for SO (71', and the continual use of a double prime would clutter up the mathematics more than is necessary.
One of the advantages of using quarks instead of quasiparticles is that our generators are
products of pairs of annihilation and creation operators, rather than the multiple products of the 0 that they are equivalent to. It is also not very difficult to specify the states of
q~ in terms of the irreducible representations of U(8), SO(8), SO(7) (or SO(71'),
G~ and SO(3). The necessary tables of fractional parentage have been completed and will be published elsewhere. However, we are faced with a difficulty when we tum to operators of physical interest. For example, Racah showed that the Coulomb interaction between the f electrons possesses a component e~ that belongs to the irreducible representation (400) of
SO(7). Since a single creation or annihilation operator for a quark belongs to ),
2 2 2
we would need a product of 8 such operators to produce the leading weight of 4. In other words, part of the two-body Coulomb interaction must be equivalent to a four-quark
interaction of the type (q( q( q) q/ q~q~ q~ q~). On the other hand, some interactions
undergo a simplification. Thus the three-electron operator t~, which describes some of the effects of configuration interaction within the f shell [10], belongs to (222) of SO(7) and can be
reproduced by some suitable collection of two-quark interactions. It is thus apparent that the approach via quarks possesses both advantages and drawbacks that must be balanced against
each other to maximize the gain.
6. Representations of SO(8).
The obvious way to find the states of the f shell that form the bases for representations of SO(8) is to diagonalize the generators of the group. Whether we use electron operators or
quark operators, we are faced with calculations of some intricacy. However, they can be circumvented by taking advantage of the idea of complementary groups Ill ]. Just as Racah's
SO(7) states can be found by diagonalizing the operators S (the spin) and Q (the quasispin)
that commute with the generators of SO(7) and form the generators of the complementary
group SOS(3) x SO~(3), so our SO(8) states can be found by diagonalizing the generators of the complementary group X to SO(8). The group X must be a subgroup of the SO(4) group
formed by SOS(3) x SO~(3) if, as seems likely, it is a finite group, its operations must leave invariant the generators of SO(8). One candidate stands out : the group whose operations interchange the tensors X, ~, v, (, with all possible changes of sign (or multiplication by I)
consistent with preserving the basic anticommutation relations. However, we might not need
as large a group as this if no degeneracies occur in the diagonalizing process. The interchanges
of the tensors by themselves form the permutation group S~, which is isomorphic to the
octahedral group. This last group is very familiar to physicists working in crystal-field theory,
and it is straightforward to construct operators from S and Q that are octahedral invariants. If
we take the octahedral scalar
~( 2 XlX2+YlY2+ZlZ22 2 2 2 2) ( 2 2 YlZ2+ZlY2+ZlX2+XlZ2+XlY2+YlX22 2 2 2 2 2 2 2 2 2)
and replace ri by a vector acting in the spin space and r~ by a vector acting in the quasispin
space, we obtain the operator E given by
~
~
~ sj2jQj2j ~ j~j2) ~ ~12jjjQj2j ~ Qj2jj
~
(~) where Sl~l and Ql~l
are spherical tensors of rank 2 acting in the spin and quasispin spaces respectively.
The operator E has proved to be successful in generating most of the SO(8) states we
require. It can be put to work by letting it first act on a representation of SO(7) for which the
SO(8) assignment is unique. Consider, for example, the representation (221) appearing in f~ with Ms
= I. The branching rules for U (8) - SO (8 )
- SO (7) given in table I indicate that (221) can only derive from (3 loo) of SO(8). Thus the label (3 loo) can be attached to (221) without further ado. On the other hand, (211) occurs in both (3100) and (21lo). If we seek (211) states that can be connected by 3£ to the (221) state of f~ with Ms
= I, we find there are
just two : (211) of f~ with Ms
= I and (211) of f~ with Ms
=
I. When we ask for the linear combination of these two states for which the connection vanishes, the combination belonging
to (21lo) must necessarily be produced. To work out the matrix elements we need the spins
and quasispins of the states. They are given by S
= I, Q =1/2 for (221) and S
= I,
Q
=
3/2 for (2II) ; the M~ values for f~ and f~ are 1/2 and 3/2 [12]. An application of the
Wigner-Eckart theorem yields the result
(21lo) j (211))
= (1/2) f~(211), Ms
= 1) (3/4)~/~[f~(211), Ms = 1). (4) The classificatory symbol j is included because two other states belonging to (21lo) (211) and the same parity combination uu can be constructed (that for which Ms = 0, and that for which the Ms values are reversed). We could generate three more sets with similar coefficients
corresponding to ug, gu, and gg. We also note that each state in (4) labelled by (2II) stands
for the collection of the 189 components of this representation with their various
L and M~ values.
These methods can be extended to generate states with other SO(8) labels. A complete
listing for the d shell, for which U(4) and its isomorphic form SO(6) are important, is being given elsewhere [13].
7. Transformations between SO(?) and SO (71'.
In order to take full advantage of the existence of the three SO(7) groups, we should be able to transform our states from one description to another. If we choose the pairing of SO(7) with SO (71' for special study, the obvious way to proceed is to diagonalize Casimir's operator for SO(7) in the basis provided by the states of SO (71'. Of course, we could carry out the analysis with the roles of SO(7) and SO (7l'reversed : but the first way has the advantage that
