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A SECOND USE FOR THE GROUP G2
B. Judd
To cite this version:
B. Judd. A SECOND USE FOR THE GROUP G2. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-9-C4-16. �10.1051/jphyscol:1970402�. �jpa-00213857�
JOURNAL DE PHYSIQUE Colloque C4. supplément au u" 11-12. Tome 31. Nov.-Déc. 1970. /w#t' C4-9
A SECOND USE FOR THE GROUP G2
by B. R. JUDD
Physics Department, Johns Hopkins University, Baltimore, Maryland 21218, U. S. A.
Résumé. — Le groupe G2, quand on l'utilise pour les électrons f, montre bien des traits bizarres et surprenants. Pour essayer de les comprendre, on étudie l'emploi du groupe Gi pour la couche p + h. La structure L des représentations (30) et (22) devient évidente. Des analogies sont trouvées du fameux symbole 6-7 nul qui est associé avec l'existence de G2 dans la couche f. Le fait que les éléments de matrice du couplage spin-orbite entre les états .F et G de la représentation (21) soient nuls est expliqué au moyen des règles de sélection sur les moments angulaires P> et /p des quasi- particules qui correspondent aux électrons h et p. On démontre que les opérateurs Q et X2 intro- duits par Racah pour l'étude de l'interaction coulombienne dans la couche f sont complémentaires, dans le sens qu'ils sont les deux opérateurs qui sont seulement exigés pour exprimer les valeurs propres de lh.V> pour (21). Les propriétés de la couche p -f h sont analysées, en particulier la nécessité de l'introduction des phases complexes dans le formalisme des quasi-particules.
Abstract. — The group Gi, when used for f electrons, exhibits many unusual and surprising features. In an effort to understand them, the use of Gj is studied for the p i h shell. The L struc- ture of the irreducible representations (30) and (22) becomes apparent. Analogues are found of the celebrated null 6-y symbol associated with the existence of Gi in the f shell. The vanishing of the matrix elements of the spin-orbit interaction between /"and G states that belong to (21) is under- stood in terms of simple selection rules on the quasi-particle angular momenta /" and /" of the h and p electrons. The operators Q and xi that Racah introduced for studying the Coulomb interac- tion within the f shell are shown to be complementary, in that they are the only two operators required to express the eigenvalues of lh.I" for (21). Properties of the p -j h shell are analyzed, particularly the necessity for introducing complex phases in the quasi-particle description.
1. Introduction. — The classic role of the conti- nuous group G2 in atomic spectroscopy is that of defining states of the f shell [1]. The extension to nuclear f states is straightforward [2]. Mention should be made of the tentative steps taken to test its useful- ness in elementary-particle theory [3, 4], but such applications lie outside our present range of interest.
Because the atomic spectroscopist is familiar with G2
in only one context, it is difficult to put many features of the analysis in any kind of perspective. At the most elementary level, we may wonder whether the possi- bility of using G2 at all is merely a happy accident.
When detailed calculations are carried out, other questions of a more specific kind arise. For example, all matrix elements of the spin-orbit interaction Hio vanish between the F and the G states belonging to the irreducible representation (21) of G2 [5]. This result would not be predicted from the usual group- theoretical arguments, and seems to suggest that some hidden structure remains to be uncovered.
It is the purpose of this article to explore the use- fulness of G, for mixed configurations of the type (p + h)*. The reason for choosing such configurations is the following. The familiar use of G2 in the f shell corresponds to associating the 7-dimensional irre-
ducible representation (10) of G2 with the 7 states of an f electron. The irreducible representation of G2
that possesses the next highest dimension is (11), which, under the same reduction scheme for G2 -> R3, yields the states p and h. Thus we can use G2 again for the configurations (p + h)N with the knowledge that no changes in the branching rules will be neces- sary (though the existing tables [1,6] may have to be extended). Although the parallelism is close, many properties of G2 that have been noticed for f electrons become cast in a different form for the p + h system.
Light is thus thrown on various peculiarities of the f shell analysis. In particular, we shall see that one way of interpreting the null matrix elements of Hso mentioned above is simply to observe that the angular- momentum triad (5/2, 1, 9/2) does not satisfy the triangular condition.
2. Mixed configurations. — The use of groups for configurations of the type (/ + l')N dates from the analysis of Elliott [7]. Detailed work on the confi- gurations (s + d)N has been carried out by Feneuille [8], Wybourne [9], and by these authors in respective collaboration with Pelletier-Allard [10] and Butler [11].
In the general case, Elliott [7] established that the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970402
C4- 1 0 €3. R. JUDD rotation group R(2 1 + 2 1' + 2) C C I L I I ~ b: ~ ~ s e d to defi!~c the states. For our case, I = 5 and I' = 1. so the gro:lp is R(14). The 14 states arising from the 1 I states o f an h electron and the 3 states of a p electron belong to the irreducible representation (1000000) of R(14). which must of course decompose into (1 I) of G,. This precise form for the inclusion of G , in R(14) is listed as [V, in Table 5 of Dynkin [12].
The principal irreducible representations W of R(14j and ti of G , of interest to us are set out in tables I and I1 together with their decompositions.
Brariching rules for R,, -+ G,
Bmi~chirlg rules for G, 4 R,
D ( U ) u L
- - -
1 (00) S 7 (10) F 14 (11) PH 27 (20) DGI 64 (21) DFGHKL 77 (22) SDGHILN 77 (30j P F G H I K M
189 (3 1 ) PDF? GH2 I2 K2 L M N O 286 (32) PDFI G ? H' 12 K? L2 MZ N O Q R 273 (33) PF? G H ? I KZ L M W O Q R U 182 (40) SDFG' HI? KL2 M N Q
148 (41) PD? F'G-1 H-1 1-1 K3 Ll M-' Nz 0 2 Q R T 729 (42) SPD' FI G4 H3 15 K4 L4 M3 N4 0 3 Q j RTZ U V 378 (50) PDF? GH3 12 K 3 L2 M W N ' 0 2 QRU
For convenien:e, th-ir dimensions D ( W j and D ( t i ) a1.c also given. Thc terms of maximum multiplicity
(, f- p.y ti - .y (for all acceptable xi are contained in the reduction of ( 1 " o ' - ~ ) of R(14) when 0 < N < 6.
Both representations ( I I I l I 1 I) and ( I 1 1 1 1 1-1) have to bc taken to give the octets of p" hi-". One interesting I'cature of table I is at once apparent : the triplets of h', lip and p2 bclong to (I I ) and (30) of G,. Now a configuration I' can only provide triplets with odd L (the total orbital angular momentum quantum n ~ ~ n i b e r ) . Since hp can only produce even L terms for L = 4 and L = 6, we would expect (30) to have many more odd values of L in its decomposition than even
~ a l u c s . Conversely. the singlets should have predomi- nantly even L. This is true of the combination
(20) + (22), which arises in the decomposition of the representation (2000000) of R(14), to which all singlets (other than L = 0) must be associated. Unlike the situation in the f shell, where (22) and (30) arise first in f4, these two representations now correspond to two-electron configurations, and their L structure is transparently obvious.
3. Quasi-particles. - It is evident from tables I and I1 that states of the type
are not completely defined even for terms of maximum spin S ; indeed (42) of G, contains no fewer than five I terms. (The symbols Ms and ML above have their usual spectroscopic significance.) I t has recently been shown [13, 141 that multiplicities of this kind can often be resolved by the introduction of quasi- particles. For a given I, four tensors kt, ~ t , vt, and kt are defined through their components as follows :
The annihilation and creation operators (for an electron I j are suffixed with values of In, and m,.
The adjoint operators lead to tensors A, 1, v and 5
for which
Using the generic symbol 8 (- A, 11, \ I , or t), we can
show that the coupled products (07 O)'k' (with odd k) form the generators for the group R0(2 1 + 1) ; and,
since coupled products for different 0 commute with each other [13], the states of the 1 shell can be classi- fied by the quadruple direct product
Only the sin~plest half-integral representation (f 9 ...+) of each group intervenes ; however, to actually define a state, two parity symbols p and y' are also needzd. They apply to the spin-up (ip) and spin- down (I!() spaces respectively, alld indjcate whether the vacuum state:, for the two spaces possess an even ( g ) or an odd (11) number of electrons [13]. For each R0(2 I + I), we select a particular I, that occurs in the decomposition of (4 +...$I under the reduction R,(2 1 f 1) -t R,(3), so that a state of IN is defined by writing
in which the intermediate angular momenta LA and L, are c o ~ ~ p l e d to L.
The extension of these ideas to mixed configurations has been made by Cunningham and Wybourne [I 5, 161
.4 SECOND USE FOR T H E GROUP G2 C4-I I
and by Feneuille [17]. For us, a first step is to add suffixes p and h where required to indicate whether I = 1 o r 5. The 91 generators of R,(14) can now be written as the components of the tensors
Together, they form the components of a single tensor belonging to (1 100000) of R,(14). This much can be readily obtained from the literature. The next step is to form the generators of [G,),.
4. Generators. - Under commutation with the generators of R,(14), the tensors 0th and 0, can only t
yield components of themselves. They can thus be labelled by WU = (1000000) (1 1). The generators of (G,), also belong to (1 1) and must thus be formed from the linear combinations
of the generators of R,(14). The isoscalar factors can be conveniently deduced from the two-particle frac- tional parentage tables of Donlan [IS], and we obtain the following tensors for the generators of (G,), :
As is to be expected, the tensors v(" satisfy the same commutation relations among themselves as the familiar f-electron tensors V(') [5]. However, for this t o occur, several remarkable relations must be satisfied by the 6:j symbols arising in the commuta- tion process. The condition that no tensors of ranks 3, 7, or 9 be produced is perhaps the most striking. It yields the equation
rhis is evidently the analogue for (p + hjN of the equation
for fN, which ensures that no tensors of rank 3 are produced from the commutators [ V 5 ' , V(5)].
Having made the analogy between eq. (1) and (2), we are in a better position t o assess the connection between the existence of the group G2 for f electrons and the vanishing of the 6:j symbol of eq. (2). For,
although we might reasonably say, for f electrons, that the group G2 can be used as a result of the accidental vanishing of a 6-j symbol, it seems quite implausible to suppose that (G,), can be used for the p + h shell merely because eq. (I) happens to be valid.
It is much more natural to suppose that the existence of the group iniposes conditions on the 6-j symbols.
This view is supported by the way in which the group F, enters into the analysis of the configurations (S + d + g + h).' [19].
5. Subgroups. - For a given 0 , the coniponents V'" form the generators of the subgroup R,(3) of (G,), The branching rules of table I1 remain valid because the tensors v("' satisfy the same commutation relations as the V'". However, the sequence
does not represent the only possible descent from R,(14) to R,(3). The tensors (0; Oh)(') (k = 1, 3, 5, 7, 9) and (0; 0,)"' form the generators for R;(I 1) x Ri(3), and this group is obviously a subgroup of R,(14).
Moreover, we can withdraw from the generators of R;(I 1) x Ri(3) the vectors (0: Oh)(l) and (0: 0,)"' ; these form the generators for ~ ; ( 3 ) x Rtj(3). Finally, these two vectors can be combined to give v(". So the sequence
is an alternative t o (3).
As has been pointed out [13], the quasi-particle classification for mixed configurations of the type (I + /'IN stenis from the two basic half-integral repre- sentations (4; ... t $. 4) of R,(2 1 + 2 1' + 2). It can be seen from tables I and I1 that their decompositio~is corresponding to the sequence (3) are
For the sequence (4), however, we get (A11111 7 2 2 2 2 2 + - l) 2 + (f llll 2222) X (t)
-+ (512 + 912 + 1512) x (4)
+ DFGHKL .
So the states D, F, ..., L of (21), which occur many times within the f shell, assume a different character for the p + h shell : they can be regarded as the resultants of the pairs of angular momenta (512, 1/2), (912, 112) and (1 512, 112).
6. States. - As a preliminary t o a discussion of the characterization of states by the two basic half-integral irreducible representations of R,(14), we make the abbreviations
and
w- = (++&++ - T ) '
C4- 1 2 B. R . JUDD for them. For a given 0 , the 91 generators of R0(14) can be augmented with the 14 components of 0; and 0;, thereby forming the 105 generators of R0(15).
From dimensional considerations it is apparent that W+ o r W - can only derive from the representation
(iii;::;) of Ro[l 5 ) ; and slnce this representation reduces to W + + W - of R,(14), it is clear that the two basic half-integral irreducible representations of R,(14) must occur in pairs. In the spin-up space, for example, this gives us the following four possibilities :
where $ stands for
The new parity symbols mi can be associated in a similar way with the various combinations W l W $ i n the spin-down space.
Each ai labels 212 states, and since the total number of states in the spin-up space of (p + h)N is 214, the four mi symbols cover the entire spin-up space.
Similar remarks apply to the spin-down space, so a state of (p + 11)" can be completely defined by writing
Of course, a n y one of the I, can be regarded as the resultant of the ~ ~ n i c l ~ ~ e coupling (1: ) I! I, if it is advan- tageous to d o so.
7. Bras and kets. - It is usual to interpret a ket such as I m , 111 > as meaning that the collection of them for various ~b transforms like the representation IY: W t of Rj,(14) x R,(14). The bras < a , 4, 1, which are properly written as < W: W: I,// 1, trans- form like W ? W" since it is only in the products IY, W - and W - W + that the identity representation (0000000) occurs. Iri fact,
These results are the analogues of the extremely well- known properties of the simplest even-dimensional rotation group R(2). For this, we know that < 1 - m ( transforms like 1 1/17 >, since the same exponential
(, i!f!@
appears in both bra and ket if they are expressed is spherical harmonics.
The gcncrators of R,(14) are coupled products of
t t
thc elementary tensors €4, and 0,, and transform like
(1 100000). This representation is found only i n the products Wi W = , indicating that the generators are diagonal with respect to mi. The elementary tensors themselves belong to (1000000), and, since this is found only in Wi W,, they must be off-diagonal with respect to m i . So IIL;, for example, can only connect m, to m, or a, to m,, and the same is true of 1;. This result has important consequences, as will be seen immediately.
8. Complex parts. - Perhaps the most striking f e a t ~ ~ r e of the analysis of Cunningham and Wybourne 1161 for the configurations (I + l')N is their conclusion that phases cannot be chosen to avoid complex numbers. In general, matrix elements comprise both real and imaginary parts. T o see how this result comes about, we first note that o : ~ Oio is a number, independent of I. So
since elo anticornmutes with OLo. We now take 0 E A and consider the matrix element
where
It is evidently equal to
but the sum over @' comprises the single term for which a' = a,. So
either for (ijj = (13) or for (ijj = (31). These two possibilities are equivalent t o the two possible choices for the sign of i. Taking the first, and noting that I, = 8 is equivalent to
we obtain
, (143/15 504) (a, 1512 )I /I:, 11 a, 1512) -
This shows that the reduced matrix elements of hL and h,f possess a relative phase of i.
It has already been shown [I41 that
(g 1512 11 1; I[ u 1512) = ,/(7 7521143) , (8) and, by the methods that led to that result, we would also find that
The fact that the reduced matrix elements in eq. (8) and (9) are both real does not necessarily conflict
A SECOND USE FOR T H E G R O U P f;? C-1-I3
with eq. (a, since we have yet to specify in detail how the states labelled by the mi are related to those in which the old parity symbols appear. In fact, it is convenient (but not necessary) to replace eq. (9) by
in which case the new states can be related to the old by the following equations :
The subscripts h and p make it clear to which kind of electron the parity symbols u and g refer.
9. Phases. - Before continuing with the analysis, several technical details should be mentioned. States have been written with the quantum numbers des- cribing the h electrons set before those describing the p electrons. (In a similar way, Condon and Shortley set S before L for a state in Russell-Saunders coupling [20].) Within either the spin-up o r the spin-down space, this kind of separation is not possible [13]. A state such as
for example, should be visualized as a sum over states of the type
weighted by the appropriate Clebsch-Gordan coeffi- cients. It is now essential to note that a component of 8th can act directly on I 1 >, since the h-electron part stands t o the left ; in contrast, a component of 0, has t
t o be passed through the h-electron part of I >. In the example above, this part is a superposition of states possessing an odd number of h electrons. (This is the significance of u,, of course.) A n odd number of anticommutations are thus necessary to bring the p-electron operator next to the p-electron part of ( x >, and so a minus sign must be introduced t o allow for this.
Because of the intel-nal structure of the separate p and h shells, eq. (8) and (10) remain valid if i. is replaced by 0. I n fact, the reduced matrix elements of 8th can be read off immediately from the existing table [14], while eq. (10) is sufficient for the entire p shell, once 0 has been substituted for i..
I t is important to realize that the relative phase of i between the reduced matrix elements of 0; a n d 8; is characteristic only of the p + h shell. Similar analyses t o the one presented in sec. 8 would yield relative phases of i between the reduced matrix elements of 0;
and 0: in the p + f shell, and also between the reduced
matrix elements of 0: and 0th in the f + h shell. These results cannot have an absolute significance, since they could be put together to give a relative phase of i Z between the reduced matrix elements of 8: and 0th. in conflict with the result we have just obtained.
10. Weight spaces. - It is of interest to study the geometric projection scheme corresponding to
The seven numbers w i that define the co-ordinates of a weight for a representation of R,(14) can be conve- niently taken as the eigenvalues of the seven commuting operators
The last operator contains i to ensure that its eigen- values are rcal. A natural choice for the two commuting operators of (G,), is V ? ) and VF'. The fact that
J28 v p = L,
for f electrons [5] suggests that the factor J28 should
be introduced. Writing out the operators in full, we obtain
We immediately notice that 7b5' is not a linear colnbination of the commuting operators of R,(14).
This is a rather surprising result, for, in atomic spec- troscopy, we are accustomed to find the commuting operators of a subgroup among the commuting opera- tors of the covering group. The eigenvalues of Vb5' cannot be immediately read off as linear combinations of the w i : instead, a matrix has t o be diagonalized.
Fortunately, this matrix is very small, since the offending combination 0L1 0,l + 0:l O h l can only connect the weight
to the weight
( ~ 1W 2 W 3 W 4 T 5 f 4 ~ 7 ) .
- -
For any such pair the eigenvalues of J28 ~ a ' are
whereas for either of the two weights
C4- 14 B. R . JUDD
only the diagonal parts (+ 1 + - L + - ... + 3) (even number of plus signs) occur. All possibilities can be accounted for by
replacing the + 7 and the f 2 by the expression
- 9 w j + 5 KT6.
Hence a weight
( ~ ' 1 w2 wg w4 w5 wg w7)
of Ro(14) projects into the weight of (G,), whose co- ordinates are given by
- -
< ,128 v;" > = [ 5 w,+4 w2+3 w3+2 w,+w5+w6]
- -
< \ 28 vA5' > = ,,'(1/3)[-3 w , + 6 w,+w,-
When this is done for the 64 weights
(+ + + t f + 3) , (odd number of plus signs) corresponding to the highest weight W , , a pattern of weights is obtained that displays the hexagonal symmetry characteristic of G, (see Fig. I). These weights form the irreducible representation (21) of (G2jo. An identical pattern is obtained from the 64 weights
FIG. I . - T h e weight space of (GI)*. The dots represent the weights of the irreducible representation (21). These are of three possible multiplicities : one, two, and four. They are represented by dots of sizes that increase with the multiplicity.
The dimension of (21) is 64. The projections of the weights on the v\" axis are indicated by the vertical lines in the right-hand part of the diagram. The multiplicities of the projections, running out from the origin. are 6 , 6 , 6, 5, 4, 3, 2, 7, 1. They can be rcsolvcd into the weights in Ro(3) of th: ~~~~~~~~nts (for which
I I I I : , 0) of thc terms corresponding to lo = 2, 3. 4. 5. 7, and 8.
which correspond to W - .
The reduction (G,), + R0(3) is accomplished very simply by projecting the weights of (21) on to t h e F V ' axis. As can be seen from figure I , the pattern of weights resolves itself into a superposition of the weights of the terms D, F, G, H, K and L.
I I. Matrix elements. - The reduced matrix ele- ments of 0: and 0th between states labelled by (21) of (G2jo can be rapidly found by standard tensorial methods [ 5 ] . For example, we find
where x = 1: + I, + 3. The detailed results are set out in table 111.
Equations for the reduced matrix elements of other 0 f , ( ~ E p, 11, 5 ) taken between states characte- rized by w i and a j either vanish (see sec. 7) or give values for the reduced matrix elements that differ from those of a t most by an overall sign.
The tensor 0; belongs to ( I l j of (G,), as does L and the spin-orbit interaction H,, for f electrons. Since
( I 1) x (21)
contains (21) twice in its decomposition [5], a straight- forward application of the Wigner-Eckart theorem might be thought to imply that the entire matrix of one of these operators within the representation (21) could be expressed as some superposition of the matrices for other two. This is not true. Table III cannot be expressed as a combi~lation of the reduced matrices of L and V'"' (the double vector whose scalar component is proportional to H,,) such as occur for the (21) representation in f3. Such a result is very surprising to any spectroscopist who is accustomed t o using the Wigner-Eckart theorem in his work.
The origin of the difficulty lies in the presence of complex quantities in the analysis of the p + h shell.
The phase diffzrence of i between the reduced matrix elements of 0; and 0; means that the reduced matrix elements --of V(j) comprise a real part [coming from - \,'(13128j (0; and an imaginary part [coming from J(15/14) (0:, O,i(j)]. Hence the matrix of V"' must be complex. On the other hand, every matrix of the corresponding operator V t 5 ) is real in the f shell.
Both V(" and V(5' play the role of generators for their resp-ctive groups, and any difference in their reduced matrix elements implies that the basis functions on whichthey act [i. e., the statesof(21)]are1nismatched.
An actual comparison of the reduced matrix ele- ments of V'jl and V(" shows that the states of (21) as they occur in thr: f shr:ll (indicattd by primes)
A SECOND USE FOR T H E G R O U P G z
TABLE I11
Reduced matrix elements of I:
correspond, n o t t o the simple quasi-particle states, but rather t o phase-transformed states. The actual correspondence is as follows :
I D ' > + e i a I w j D > , IF' > -+e-'" I w j F > 3 I G' > -+ e i P ] w j G > ,
I H ' > + e P i " w j ~ > , ] K t > - + e i Y I w j K > , I L ' > + e - ' ? I w j L >
where
j = 1 o r 2 (A and v spaces) , j = 3 o r 4 ( p a n d 5 spaces) .
These conditions account for half of all the possible combinations ; the remainder correspond to replacing i by - i above. The actual angles u , /3, and y are given by
e'" = 4(27/28) + i ,/(1/28),
eio = J(3/28) + i J(25/28) ,
ei" ,,/(16/28) + i J(12/28) .
Quite: remarkably,
p - y = y - u = n / 6 .
We have-no explanation for these highly intriguing equations. Apparently, the hexagonal character of G, has been carried over into the phases in some way.
12. Vanishing matrix elements of the spin-orbit interaction in the f shell. - W e are now ready to take u p a problem mentioned in sec. 1, that matrix elements of H,, within the f shell vanish between F and G states belonging to (21). We have only to note that 8: possesses this property (see table 110 and so must V('), which is diagonal with respect to I,. The matrices of these two operators, both of which
belong t o (1 1) of (G,),, are independent. Since, as already mentioned, (11) x (21) contains (21) twice in its decomposition, any operator transforming like the P state of (1 1) - as does H,, - must also vanish.
The phase transformations of sec. 1 1 are irrelevant here.
It is interesting to examine the crucial point in the proof. This is the fact that 8; has vanishing matrix elements between the F and the G states of (21). It would not be obvious at all unless the F and G states were expressed in their coupled forms (512, 1/21 F, and (912, 112) G. The fact that 1: is different for these two states is enough to guarantee the vanishing of the matrix element.
We could, of course, have used the vector (0; Oh)(l) i n place of 86. The proof would go through as before ; but now we would appeal to the fact that the triad (512, 1, 9/2), does not satisfy the triangular condition.
The selection rule applies to all operators transform- ing like (11) P. The spin-other-orbit interaction has recently been decomposed into its group-theoretical parts [21], and no fewer than six of them (z,, z,, z , , z , , ,
z , , and r,,) all correspond to (I I) P, and must
therefore vanish between the F and C states of (21).
The fact that the selection I-ule becomes con~prehen- sible (and, incidentally, trivial) only when G , is used in the p + 11 shell goes a long way to justifying the study of configurations which, a t first sight, might be thought of as being only of slight interest.
13. S t r ~ ~ c t u r e of certain operators chosen by Racah in his analysis of the coulomb interaction in the f shell. - The special properties of the representa- tion (21) of G, are not confined to the spin-orbit interaction. In his analysis of the Coulomb interac- tions within the f shell, Racah introduced several scalar operators [I], namely e i (i = 0, 1, 2, 3) and the auxiliary operator Q. The (21) representation was found to be unique in that it is the only representation that requires two basic matrices to express the matrix elements o f any one of the four operators e,, e l , e,, and e , + Q taken between states labelled by rcprescnta-
C4- 16 B. R. JUDD tions of G,. The two basic matrices (called z, and %,
by Racah [I]) are required for the diagonal elements of e, within the representation (21). Nowl Racah could have chosen these two basic matrices in an infinity of ways ; all that is required are two linearly independent sets of elements possessing the same group labels as e , . In fact, he chose X , to give the elements of e , in f3, while X , was apparently chosen a t random. But some choices are more natural than others ; and the actual values of X , for the states of (21) are reasonably simple. In fact,
for
L = 2, 3, 4, 5, 7 and 8 .
It can now be pointed out that this choice for X ,
is not so arbitrary as might appear a t first sight. For, with
for the corresponding L values, we may readily verify that
where the six values of L are matched with the same values of I,. The equation [I 31
indicates that 1; transforms like a superposition of the vector parts of (I I) x (I I), i. e., like (I I) P and (30) P.
Hence 1 i . l ; should transform like some combination of the scalar parts of (I 1) x (1 I), (I I) x (30), and (30) x (30) . However, not every scalar component appears. This is because 8; and 0; belong to (1000000) of R0(14), and any quadruple form such as
must belong to the totally antisymmetric part of (1000000)4, i. e., to (1 1 11000). F r o m tables I and 11, we see that the only possible scalars are (22) S and (40) S . The remarkable fact is that the part trans- forming like (22) S is Q (and not some mixture of Q and e,) and that the part transforming like (40) S is X ,
(and not some mixture of X , and x,). Thus X , is, in some strange way, a kind of complementary opera- tor to Q. This was not previously suspected.
14. Conclusion. - An attempt has been made to show how the use of C, for the p + h shell throws light on its use for f electrons. In the course of the analysis, a number of new and surprising features have arisen. Perhaps the most remarkable are connected with the necessity for complex quantities when treating the p + h shell in the quasi-particle formalism ; in particular, the occurrence of phase angles differing by 746 which arise when the (21) states of (G,), are matched with those used in the f shell. Actually, the need for complex quantities suggests that the analysis is in a tighter and more concise form than is usual in shell theory, where simple phase choices of + 1 abound. The richer field of the complex plane is evidently required t o express the fact that states result from the coupling of relatively few angular momenta.
This work was partially supported by the United States Atomic Energy Commission.
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