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Conjugation properties of double tensor operators and selection rules for their matrix elements
Physica 131A (1985) 289-299 North-Holland, Amsterdam
CONJUGATION
PROPERTIES OF DOUBLE TENSOR OPERATORS
AND SELECTION
RULES FOR THEIR MATRIX ELEMENTS J.A. TUSZVrjSKI
Department
of Physics,
Memorial
University of Newfoundland, Canada A 1B 3X7
St. John’s,
Newfoundland,
Received 28 May 1984 Revised 26 November 1984
The application of parity, time, charge, hermitian and quasi-spin conjugations to the set of irreducible tensor operators separates two classes of tensor operators: polar and axial. Various product tensor operators, in particular one- and two-body double tensor operators, are subsequently examined as to their conjugation properties. As a result, new selection rules are found for their matrix elements which emphasize the need for a more precise labelling of tensorial ranks in order to account for the polar content and enable configuration mixing effects. The derived results are of particular importance to the form of phenomenological and equivalent Hamiltonians of atomic and molecular physics which are often presented as series of doubie tensor operators.
1. Introduction Racah’) defined irreducible tensor operators (ITO’s) Tk through the commutation rules of their (2k + 1) components with J, and J+ = J, 2 iJ,, :
1-LT:l = d-i, [J,, T:] = [(k 2
(1) q + l)(k
r q)]1’2T:,,
.
(2)
In a recent paper by Chatterjee et al.‘) based on the division of classical vectors into two classes according to their parity-reversal properties, ITO’s have been divided into axial tensor operators (ATO’s) and polar tensor operators (PTO’s). ATO’s describe magnetic properties and satisfy’)
PT;(B)P-’ = T:(B),
(3)
TT@)T-’
(4)
= (-l)k-qTkq(B),
037%4371/85/!$03.30 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
J.A. TUSZYhKI
290
where
P is the parity- and T is the time-reversal
for any axial vector, and satisfy2)
B is the prototype
operation.
i.e. B, H, M, J, L or S. PTO’s describe
electric
properties
PT;(E)P-'=(-l)kTi(E).
(5)
TTt(E)T-'=(-I)-"T!,(E),
(6)
where
E is the prototype
for any polar
vector,
i.e. E, 0,
P or r. Familiar
examples of an AT0 are the tensor operators built on the components of angular momentum: J,, J+ and J_, denoted T:(J)whose explicit form has been provided by Buckmaster et a1.3). A well-known normalized spherical harmonics4):
example
of a PTO is the set of
since Y,, are built using x, y and z. Of course, their polar nature is only manifested when k is odd. The two classes of ITO’s constitute inequivalent representations of the Racah algebra, eqs. (1) and (2) having the same rotation properties but different space- and time-reflection properties. The combined PT-reversal and the charge conjugation (C) effects are identical cyQ?-‘=
(-l)"+qT"q
(ro
and so is the PCT-product which is an identity transformation. The hermitian adjoint of a tensor operator’s component is also defined’), irrespective of the class, as
(9
HTtH-'= (-l)YTkq. the quasi-spin The many-body theory employs another conjugation, jugation (Q) defined as an antiunitary operato?) which, when applied ITO, yields QTtQ-’
conto an
= (++q+‘Tf,
provided the particles involved are fermions. ITO’s act on the eigenstates of angular momentum decomposed into their spin- and orbital part$)
(10)
/s&r)
which
can
be
CONJUGATION PROPERTIES OF DOUBLE TENSOR OPERATORS
where (. . .I . . .) is a Clebsch-Gordan coefficient. Since I determines of a state, it is conventional to require that Pls~m) = (-l)‘lsbm) .
291
the parity
(12)
The time conjugation of a state produces a phase which is arbitrary since Tlsljm) transforms contragradiently to lsljm) ‘). For our purposes we set Tlsm,) = (-l)S-*sl~ - m,),
TJlm,) = (-1)‘-“‘11-
m,)
(13)
and use eq. (11) together with the property’) (sm, lm, I sljm) = (- l)‘-“-‘(s - m, 1 - m, I slj - m)
(14)
to obtain Tldjm)
= (-l)‘-”
lslj - m) .
Repeating the action minus identity if j degeneracy. Finally, quasi-spin valent electrons with
(15)
of T on ldjm) is an identity operation if j is integer or a is half-integer. The latter case exhibits Kramer’s9) conjugation of a many-particle the orbital number 1 yields”)
QllN SL M, ML Q M,) = (-1)o-“+4’+2-N
state involving N equil-
SLM, ML Q - M,) ,
(16)
where Q and MO are quasi-spin labels. Thus, only the states of half-filled shells are eigenfunctions of the operator Q.
2. Conjugations
of product tensor operators
Racah’) defined a product of two tensor operators as T(k,k2)3a,
p)
E
(T”(a)
X Tk2(P)): (17)
292
J.A. TUSZYIkSKI
where
in our
products
case
a and /3 are either
are distinguished
eqs. (3)-(6), are derived
(8)-(10)
according
together
for the three
E or B. Therefore,
three
classes
of
to ((Y,p): (E. E), (B, B) and (E, I?). Using
with eq. (14) P-, T-, C-, Q- and H-conjugations
classes of products l)P’“J)T(klkd);(a,
pT(k,k2)&
P)P-.’
= (_
TT(ktkzJ);(a,
/j&t-’
= (-l)‘(a.B)T(klkz)liq(a,
CT(kIkz);(a,
p)c-’
=
(-
~)““~“‘T’k~kz)~q(a,
QT(“lkzJ;((y,
P)Q-'
=
(-
l)qs(n.pJT(klkz)liq(Ly,
0).
p),
p),
p)
(W (21)
,
(22)
where p, t, c, qs and h are the appropriate exponents which depend on cy and fi and whose values are summarized in table I. It is concluded that the products with (Y = /3, i.e. (E, E) and (B, B) behave like the corresponding single tensor operators and the mixed product (E, B) combines the properties of axial and polar operators, i.e. it behaves like polar under P- and like axial under T-conjugation. Apart from the usual triangular rule imposed on the ranks of product tensor operators: (k, - k,( 4 k s k, + k,, all hermitian products must comply with an additional requirement that (k, + k2 + k) be an even integer.
3. Selection rules for matrix elements The Wigner-Eckart tors can be calculated
theorem’) as
states
that matrix
TABLE
elements
of tensor
I
The conjugation exponents for product tensor operators.
P
0
k-q
kl kz- q
k,+kz
?
k+q k,+k2+l+q
k+q k,+kztl+q
k+q k,+kZ+l+q
k,ikz+k+q
k,+kZ+k+q
k,+k2+k+q
c w h
k,ikzik-q
opera-
CONJUGATION PROPERTIES OF DOUBLE TENSOR OPERATORS
(sljmIT$sl’j’m’)= (-l)‘_” (_‘,
“,
;,)(sQIIT’llsr’i’)
)
293
(23)
where (. . . II. . . II.. .) is a reduced matrix element and the 3j-symbol satisfies the following property: j
k
m
-q
j’
(24)
-m’ >
Using eqs. (15) (4) (23) and (24) yields (sljm IT-‘(TT~(B)T-‘)Tlsr’j’m’)
1T~(B)lsl’j’m = (- 1)W+i’+k)+m-m’-9(slj,,,
f> ,
(25) which is an identity since 2(j + j’ + k) is always an integer due to the superselection rule and (m - m’- q) must be zero due to the conservation of angular momentum. Similarly, using eqs. (11) (13) (6), (23) and (24) yields (slimIT_‘(TT~(E)T-‘)Tlsl’j’m)
I Ti(E)lsl’j’m = (-1) 2(l+~+k)+m-m’-q(Sljm
‘) , (26)
which, again, is an identity. Hence, time-conjugation produces no constraints on the matrix elements of ITO’s. To check the role of parity-conjugation we use eqs. (12) and (3) and obtain (slirnIP_‘(PT:(B)P-‘)PIsl’j’m’)
= (-l)‘+“(s~m~T~(B)Jsl’j’m’),
(27)
requiring that (1+ I’) must be even for a nonzero matrix element of an ATO. Similarly, using eqs. (12) and (5) we find that (sljm IP-‘(PT~(E)P-‘)Plsrl’m’)
= (-l)‘+“‘k(sljm IT~(E)lsl’j’m’)
,
(28)
which implies that (I + I’+ k) must be an even integer. However, we must realize that these selection rules have been derived without consideration given to the reduced matrix elements. The two classes of ITO’s differ in this respect due to their distinct commutation relationships with the square of the total angular momentum. F’TO’s preserve the total angular momentum”)
[.P, T;(B)] = 0 whereas ATO’s do not”):
294
J.A. TUSZYhSKI
[J’, T:(E)] Therefore, matrix
= h*k(k + 1)7-;(E).
it is not surprising
elements.
Following
to find different Rotenberg
(.sljlJTk(E)J(sl’j’) = (- l)j+’
selection
rules for their reduced
et al.6), it is found
[(2j + 1)(2j’+ 2
1)]“2 j s
k o
that j’ _,j,,
+
(-u’+“+kl,
which implies that (I + l’+ k) must be even, in agreement with our previous result. However, Buckmaster et al.3) found that for PTO’s only fully diagonal matrix elements are nonzero (I = I’ and j = j’) and they are expressed by
(32) Hence, the previous constraint: (I + I’) is even should be changed to I= I’. Consequently PTO’s must act within a given configuration unlike ATO’s which can participate in configuration mixing. This leads to the conclusion that the two classes of operators can only be interchanged (as for example in equivalent operator transformations) within a manifold of constant angular momentum. Off-diagonal operator equivalents of different class are not allowed. It should be interesting to analyze the relationship between the two classes of ITO’s and the classes of diagonal and by Witschel’*) and Atkins that an important analogy Quasi-spin conjugation
off-diagonal second quantized operators discovered and Seymou?). Based on our findings it is evident between the two classifications exists. can only be useful within the states of half-filled
shells where Q(+) = t]+) and the “?” sign distinguishes between two possible types of states. With the aid of eq. (lo), it can be deduced that diagonal matrix elements of half-filled shells do not vanish only when (k + 1) is even. This requirement
can never
be satisfied
for PTO’s
since
P-invariance
reads
now:
(21+ k) even. This is a well-known fact 4.‘4) which complicates the analyses of EPR spectra of S-state ions (e.g. Gd3+). A generalization of the selection rules derived in this section to embrace tensor operator products is automatic when eqs. (18)-(21) are applied. The results are combined in table II. On the basis of our derivations it should be reemphasized that ATO’s and ITO’s are quite distinct having different P- and T-conjugation properties and, as a result, different selection rules. Therefore, it is only natural to require that they must not be interchanged except within a manifold of constant angular momentum. This restriction has been previously invoked by other
CONJUGATION PROPERTIES OF DOUBLE TENSOR OPERATORS
295
TABLEII The selection rules for matrix elements of one-body tensor operators. P
T’(B)% T’(E)
T “qB,
B)
T
Q*
(k + 1)
I= 1’
= T”lk~k(E, E)
Tckl’dk(E, B)
even
even
-
(k + 1) even
(I+ I’+ kl) even I= I’ if kl= 0
-
(k+ 1) even
(l+I’+k)
* Relevant for diagonal matrix elements of half-filledshells only.
authors4*‘5,16 ); however, no comprehensive equivalent operator replacement”)
x+Jx,
y-+J,,
r2+ J(J+
X-+JZ,
rationale has been given. The usual
1)
is an example of such procedures. Since it is increasingly apparent”) that schemes involving configuration mixing effects are necessary, our recommendation to keep the two classes separate may acquire practical qualities. In particular, new spin Hamiltonians with more terms should be devised to better describe crystalline and magnetic field effects.
4. Double tensor operators A particular example of product tensor operators tors”) (DTO’s) usually defined as
are double tensor opera-
WCklk$ = b(Tkl(S) x Tk’(L))t , where b = b(k,k,k)
(33)
is a numerical factor needed for normalization:
(sill WCklk2)(1i, IJlsl’> = S1,,,6,*,,.[(2k, + 1)(2k, + l)]“*
(34)
and I,, 1, are used to specify the configurations on which DTO acts; however, in light of the distinct properties of ATO’s and PTO’s the present rank labelling scheme is inadequate. It is because both S and L are axial and, as demonstrated before, are incapable of connecting different configurations. In order to render eq. (34) justified, a new definition for DTO’s is needed: ~[~1(~21~22)‘21: z B[T~I(S)
x
(Tkqjq
x
+.z(jg)“qk
,
(35)
296
J.A. TUSZYk3KI
where B = B(k, k,, k,,k,k). This generalized definition embraces all the allowed classes of tensor operator products: axial, polar and mixed and thus allows for configuration operators,
mixing
effects.
eqs. (18)-(22),
p=k
22,
and Selection
Based on the derivations for the following exponents are obtained
t=kik,,-q,
c=k+q,
product tensor for DTO’s:
qs=k,+k,+k+l+q
h=k,+k,,+k,,+k2+q. rules for the matrix
elements
of DTO’s.
, I, = (lNSLJM1 W rk,(kz,k2:)k21~)1N~‘1’S’L,J,MI) are subsequently
found
applying
the procedures
(36) outlined
in section
3. The
present findings are summarized in the upper part of table III. The triangular rules for ranks and state labels are too numerous to be listed in detail but their form is quite obvious’). It is concluded that the value of k,, determines the existence of configuration mixing effects as it describes the polar component of a DTO. Hence, it is the individual ranks kZ, and k,, and not the combined rank k, that are of foremost importance to conjugation properties and selection rules, Our conclusions about the new rank notation are reinforced. Two-body DTO’s are products of two one-body DTO’s which act on different particles. Therefore, using eq. (35) as a pattern we define a generalized two-body DTO in the j-j scheme of coupling as
dklk2)); -; ; I,
A{[Tfll(S)
(7+(L)
x
x T;24(E))k21]kl
x ( 7-;12(S) x (Tf2’(L) x T;26(E))k2qk2}; . where
A depends
on all the ranks.
of coupling is straightforward (34.3) of Yutsis”). Adopting following
conjugation
p= kzt&,
An extension
(37)
of eq. (37) to the U-scheme
and the two schemes can be related through eq. the same procedures as before results in the
exponents
for the operator
t=k,,tk,,tk-q,
of eq. (37):
c-ktq,
qs=k+q,
11=k,,tk23tk24tk,2fk25tk2h+ktq. Consequently, operator,
selection
rules
for an arbitrary
I2 = (1NSLJM(w~~~k2’~~lN-21’l”S’L’J’M’),
matrix
element
involving
this
(W
CONJUGATION
PROPERTIES
OF DOUBLE TENSOR OPERATORS
297
TABLE III Selection rules
for matrix elements of double tensor operators. Selection rules
Symmetry One-body DTO’s 1)
2)
conservation of angular momentum hermiticity
3) parity invariance 4)
(a) (b)
M=Q+M’
triangular rules for ranks and state labels
(kl + kZI+ kz + k2)
(a)
(I+ I’+ k&
(b)
I = I’
even
even if ka = 0
time invariance
5) Quasi-spin invariance (N = 2i+ 1, I = /‘)
(kt + k2)
even
Two-body DTO’s 1)
M=Q+M’
conservation of angular momentum
triangular rules for ranks and state labels
2) hermiticity (jj)
(kl + kz + k) even (kll+k~+k~+klZ+k~+k~+k)
3)
parity invariance
(I + I’ + I” + kx + k26) even
4)
time invariance
5)
quasi-spin invariance (N = 21+ 1, 1 = /‘)
I’= k
I if ka= 0;
I” = I’
even if kz = 0
even
can be deduced with ease using the standard methods. They are presented in the lower part of table III. The importance of the results of this section rests with the fact that DTO’s are the most general entities used to construct symmetry-adapted effective Hamiltonians of atomic and nuclear physics. In view of our results these Hamiltonians should be written in general as
+
c c c cw kIlWt4 %klku Wx!n ‘[z’q
([kI,(k~kza)k211k,IkIz(kukzdk,Jk2}k 9’
(39)
where parameters D, and D, depend on all the respective summation indices. Not all of these indices, however, are physically meaningful since the Hamiltonian is still subject to invariance requirements, i.e.
2Y8
J.A. TUSZY&-SKI
g,sIpg;‘=X where
(lSi=Gn+7),
in the most general
g={T,H,R,
,,...,
We have employed for a rotation
about
(40)
case
R,,,u~,uV,u,~,u~,I}={gr;i=l
the following a p-fold
notation
. . . . . n-t-7).
for symmetry
axis, uk denotes
a reflection
elements
(41)
g, : R, stands
by a plane
k and I is
an inversion. Of course, the particular choice of symmetry elements varies for each of the 32 point groups. Prathe?) published an exhaustive study on the forms of invariant polar operators for all non-cubic point groups. He summarized his results in table 5 of his paper. A similar table can be constructed for axial operators. Since they are insensitive to space inversion (see eq. (3)) any two point groups with the same rotation and reflection properties but differing in their inversion signature become equivalent for ATO’s. For example C,, and C, or C, and S, become equivalent for ATO’s. With this in mind the Prather table can be used for ATO’s as well. An additional restriction is related to the time-invariance of X and it requires that the sum of all axial component ranks in each term must always be an even integer. A similar requirement for the ranks of polar components to the given point group.
will be in effect only if an inversion
centre
belongs
The end result of our analysis is that since the selection rules on ATO’s are more relaxed, more terms than usually allowed will be included in the effective Hamiltonian, eq. (39). Since parameters D, and D2 are often obtained from best fits to experimental hyperfine spectra, this calls for a modification in the parameterization schemes used to generate these coefficients. The sensitivity of these parameterization schemes to the number of adjustable parameters has been pointed out*‘.“) and is now being supported.
Acknowledgement This research is supported Engineering Research Council
by a grant of Canada.
from
the
Natural
Sciences
References 1) G. Racah, Phys. Rev. 62 (1942) 438. 2) R. Chatterjee, J.A. Tuszyliski and H.A. Buckmaster, Can. J. Phys. 61 (1983) 1613. 3) H.A. Buckmaster, R. Chatterjee and Y.H. Shing, Phys. Stat. Sol. (a) 13 (1972) 9.
and
CONJUGATION PROPERTIES OF DOUBLE TENSOR OPERATORS
299
4) H. Watanabe, Operator Methods in Ligand Field Theory (Prentice-Hall, Englewood Cliffs, NY, 1966). 5) J.S. Bell, Nucl. Phys. 12 (1959) 117. 6) M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr., The 3m-j Symbols and 6j-Symbols (The Technology Press, MIT, Cambridge, MA, 1959). 7) D.M. Brink and G.R. Satchler, Angular Momentum (Oxford Univ. Press, London, 1968). 8) A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, NJ, 1957). 9) H.A. Kramers, Proc. Acad. Sci. (Amsterdam) 33 (1930) 953. 10) B.R. Judd, Second Quantization and Atomic Spectroscopy (The Johns Hopkins Univ. Press, Baltimore, 1%7). 11) D.C. Mattis, The Theory of Magnetism (Harper and Row, New York, 1965). 12) W. Witschel, Mol. Phys. 20 (1973) 419. 13) P.W. Atkins and P.A. Seymour, Mol. Phys. 25 (1973) 113. 14) B.G. Wyboume, Phys. Rev. 148 (1966) 317. 15) A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon, Oxford, 1970). 16) W. Low, Paramagnetic Resonance in Solids (Academic Press, New York, 1960). 17) K.W.H. Stevens, Spin Hamiltonians in Magnetism, vol. 1, G.T. Rado and H. Suhl, eds. (Academic Press, New York, 1964). 18) B.R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw-Hill, New York, 1967). 19) A.P. Yutsis, LB. Levinson and V.V. Vanagas, Mathematical Apparatus of Angular Momentum (NSF Translation, Washington, DC, 1960). 20) J.L. Prather, Atomic Energy Levels in Crystals (U.S. National Bureau of Standards, Washington, DC, 1961). 21) S.S. Bishton and D.J. Newman, J. Phys. C3 (1970) 1753. 22) B.R. Judd, J.E. Hansen and A.J.J. Raassen, J. Phys. B15 (1982) 1457.
CONJUGATION
PROPERTIES OF DOUBLE TENSOR OPERATORS
AND SELECTION
RULES FOR THEIR MATRIX ELEMENTS J.A. TUSZVrjSKI
Department
of Physics,
Memorial
University of Newfoundland, Canada A 1B 3X7
St. John’s,
Newfoundland,
Received 28 May 1984 Revised 26 November 1984
The application of parity, time, charge, hermitian and quasi-spin conjugations to the set of irreducible tensor operators separates two classes of tensor operators: polar and axial. Various product tensor operators, in particular one- and two-body double tensor operators, are subsequently examined as to their conjugation properties. As a result, new selection rules are found for their matrix elements which emphasize the need for a more precise labelling of tensorial ranks in order to account for the polar content and enable configuration mixing effects. The derived results are of particular importance to the form of phenomenological and equivalent Hamiltonians of atomic and molecular physics which are often presented as series of doubie tensor operators.
1. Introduction Racah’) defined irreducible tensor operators (ITO’s) Tk through the commutation rules of their (2k + 1) components with J, and J+ = J, 2 iJ,, :
1-LT:l = d-i, [J,, T:] = [(k 2
(1) q + l)(k
r q)]1’2T:,,
.
(2)
In a recent paper by Chatterjee et al.‘) based on the division of classical vectors into two classes according to their parity-reversal properties, ITO’s have been divided into axial tensor operators (ATO’s) and polar tensor operators (PTO’s). ATO’s describe magnetic properties and satisfy’)
PT;(B)P-’ = T:(B),
(3)
TT@)T-’
(4)
= (-l)k-qTkq(B),
037%4371/85/!$03.30 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
J.A. TUSZYhKI
290
where
P is the parity- and T is the time-reversal
for any axial vector, and satisfy2)
B is the prototype
operation.
i.e. B, H, M, J, L or S. PTO’s describe
electric
properties
PT;(E)P-'=(-l)kTi(E).
(5)
TTt(E)T-'=(-I)-"T!,(E),
(6)
where
E is the prototype
for any polar
vector,
i.e. E, 0,
P or r. Familiar
examples of an AT0 are the tensor operators built on the components of angular momentum: J,, J+ and J_, denoted T:(J)whose explicit form has been provided by Buckmaster et a1.3). A well-known normalized spherical harmonics4):
example
of a PTO is the set of
since Y,, are built using x, y and z. Of course, their polar nature is only manifested when k is odd. The two classes of ITO’s constitute inequivalent representations of the Racah algebra, eqs. (1) and (2) having the same rotation properties but different space- and time-reflection properties. The combined PT-reversal and the charge conjugation (C) effects are identical cyQ?-‘=
(-l)"+qT"q
(ro
and so is the PCT-product which is an identity transformation. The hermitian adjoint of a tensor operator’s component is also defined’), irrespective of the class, as
(9
HTtH-'= (-l)YTkq. the quasi-spin The many-body theory employs another conjugation, jugation (Q) defined as an antiunitary operato?) which, when applied ITO, yields QTtQ-’
conto an
= (++q+‘Tf,
provided the particles involved are fermions. ITO’s act on the eigenstates of angular momentum decomposed into their spin- and orbital part$)
(10)
/s&r)
which
can
be
CONJUGATION PROPERTIES OF DOUBLE TENSOR OPERATORS
where (. . .I . . .) is a Clebsch-Gordan coefficient. Since I determines of a state, it is conventional to require that Pls~m) = (-l)‘lsbm) .
291
the parity
(12)
The time conjugation of a state produces a phase which is arbitrary since Tlsljm) transforms contragradiently to lsljm) ‘). For our purposes we set Tlsm,) = (-l)S-*sl~ - m,),
TJlm,) = (-1)‘-“‘11-
m,)
(13)
and use eq. (11) together with the property’) (sm, lm, I sljm) = (- l)‘-“-‘(s - m, 1 - m, I slj - m)
(14)
to obtain Tldjm)
= (-l)‘-”
lslj - m) .
Repeating the action minus identity if j degeneracy. Finally, quasi-spin valent electrons with
(15)
of T on ldjm) is an identity operation if j is integer or a is half-integer. The latter case exhibits Kramer’s9) conjugation of a many-particle the orbital number 1 yields”)
QllN SL M, ML Q M,) = (-1)o-“+4’+2-N
state involving N equil-
SLM, ML Q - M,) ,
(16)
where Q and MO are quasi-spin labels. Thus, only the states of half-filled shells are eigenfunctions of the operator Q.
2. Conjugations
of product tensor operators
Racah’) defined a product of two tensor operators as T(k,k2)3a,
p)
E
(T”(a)
X Tk2(P)): (17)
292
J.A. TUSZYIkSKI
where
in our
products
case
a and /3 are either
are distinguished
eqs. (3)-(6), are derived
(8)-(10)
according
together
for the three
E or B. Therefore,
three
classes
of
to ((Y,p): (E. E), (B, B) and (E, I?). Using
with eq. (14) P-, T-, C-, Q- and H-conjugations
classes of products l)P’“J)T(klkd);(a,
pT(k,k2)&
P)P-.’
= (_
TT(ktkzJ);(a,
/j&t-’
= (-l)‘(a.B)T(klkz)liq(a,
CT(kIkz);(a,
p)c-’
=
(-
~)““~“‘T’k~kz)~q(a,
QT(“lkzJ;((y,
P)Q-'
=
(-
l)qs(n.pJT(klkz)liq(Ly,
0).
p),
p),
p)
(W (21)
,
(22)
where p, t, c, qs and h are the appropriate exponents which depend on cy and fi and whose values are summarized in table I. It is concluded that the products with (Y = /3, i.e. (E, E) and (B, B) behave like the corresponding single tensor operators and the mixed product (E, B) combines the properties of axial and polar operators, i.e. it behaves like polar under P- and like axial under T-conjugation. Apart from the usual triangular rule imposed on the ranks of product tensor operators: (k, - k,( 4 k s k, + k,, all hermitian products must comply with an additional requirement that (k, + k2 + k) be an even integer.
3. Selection rules for matrix elements The Wigner-Eckart tors can be calculated
theorem’) as
states
that matrix
TABLE
elements
of tensor
I
The conjugation exponents for product tensor operators.
P
0
k-q
kl kz- q
k,+kz
?
k+q k,+k2+l+q
k+q k,+kztl+q
k+q k,+kZ+l+q
k,ikz+k+q
k,+kZ+k+q
k,+k2+k+q
c w h
k,ikzik-q
opera-
CONJUGATION PROPERTIES OF DOUBLE TENSOR OPERATORS
(sljmIT$sl’j’m’)= (-l)‘_” (_‘,
“,
;,)(sQIIT’llsr’i’)
)
293
(23)
where (. . . II. . . II.. .) is a reduced matrix element and the 3j-symbol satisfies the following property: j
k
m
-q
j’
(24)
-m’ >
Using eqs. (15) (4) (23) and (24) yields (sljm IT-‘(TT~(B)T-‘)Tlsr’j’m’)
1T~(B)lsl’j’m = (- 1)W+i’+k)+m-m’-9(slj,,,
f> ,
(25) which is an identity since 2(j + j’ + k) is always an integer due to the superselection rule and (m - m’- q) must be zero due to the conservation of angular momentum. Similarly, using eqs. (11) (13) (6), (23) and (24) yields (slimIT_‘(TT~(E)T-‘)Tlsl’j’m)
I Ti(E)lsl’j’m = (-1) 2(l+~+k)+m-m’-q(Sljm
‘) , (26)
which, again, is an identity. Hence, time-conjugation produces no constraints on the matrix elements of ITO’s. To check the role of parity-conjugation we use eqs. (12) and (3) and obtain (slirnIP_‘(PT:(B)P-‘)PIsl’j’m’)
= (-l)‘+“(s~m~T~(B)Jsl’j’m’),
(27)
requiring that (1+ I’) must be even for a nonzero matrix element of an ATO. Similarly, using eqs. (12) and (5) we find that (sljm IP-‘(PT~(E)P-‘)Plsrl’m’)
= (-l)‘+“‘k(sljm IT~(E)lsl’j’m’)
,
(28)
which implies that (I + I’+ k) must be an even integer. However, we must realize that these selection rules have been derived without consideration given to the reduced matrix elements. The two classes of ITO’s differ in this respect due to their distinct commutation relationships with the square of the total angular momentum. F’TO’s preserve the total angular momentum”)
[.P, T;(B)] = 0 whereas ATO’s do not”):
294
J.A. TUSZYhSKI
[J’, T:(E)] Therefore, matrix
= h*k(k + 1)7-;(E).
it is not surprising
elements.
Following
to find different Rotenberg
(.sljlJTk(E)J(sl’j’) = (- l)j+’
selection
rules for their reduced
et al.6), it is found
[(2j + 1)(2j’+ 2
1)]“2 j s
k o
that j’ _,j,,
+
(-u’+“+kl,
which implies that (I + l’+ k) must be even, in agreement with our previous result. However, Buckmaster et al.3) found that for PTO’s only fully diagonal matrix elements are nonzero (I = I’ and j = j’) and they are expressed by
(32) Hence, the previous constraint: (I + I’) is even should be changed to I= I’. Consequently PTO’s must act within a given configuration unlike ATO’s which can participate in configuration mixing. This leads to the conclusion that the two classes of operators can only be interchanged (as for example in equivalent operator transformations) within a manifold of constant angular momentum. Off-diagonal operator equivalents of different class are not allowed. It should be interesting to analyze the relationship between the two classes of ITO’s and the classes of diagonal and by Witschel’*) and Atkins that an important analogy Quasi-spin conjugation
off-diagonal second quantized operators discovered and Seymou?). Based on our findings it is evident between the two classifications exists. can only be useful within the states of half-filled
shells where Q(+) = t]+) and the “?” sign distinguishes between two possible types of states. With the aid of eq. (lo), it can be deduced that diagonal matrix elements of half-filled shells do not vanish only when (k + 1) is even. This requirement
can never
be satisfied
for PTO’s
since
P-invariance
reads
now:
(21+ k) even. This is a well-known fact 4.‘4) which complicates the analyses of EPR spectra of S-state ions (e.g. Gd3+). A generalization of the selection rules derived in this section to embrace tensor operator products is automatic when eqs. (18)-(21) are applied. The results are combined in table II. On the basis of our derivations it should be reemphasized that ATO’s and ITO’s are quite distinct having different P- and T-conjugation properties and, as a result, different selection rules. Therefore, it is only natural to require that they must not be interchanged except within a manifold of constant angular momentum. This restriction has been previously invoked by other
CONJUGATION PROPERTIES OF DOUBLE TENSOR OPERATORS
295
TABLEII The selection rules for matrix elements of one-body tensor operators. P
T’(B)% T’(E)
T “qB,
B)
T
Q*
(k + 1)
I= 1’
= T”lk~k(E, E)
Tckl’dk(E, B)
even
even
-
(k + 1) even
(I+ I’+ kl) even I= I’ if kl= 0
-
(k+ 1) even
(l+I’+k)
* Relevant for diagonal matrix elements of half-filledshells only.
authors4*‘5,16 ); however, no comprehensive equivalent operator replacement”)
x+Jx,
y-+J,,
r2+ J(J+
X-+JZ,
rationale has been given. The usual
1)
is an example of such procedures. Since it is increasingly apparent”) that schemes involving configuration mixing effects are necessary, our recommendation to keep the two classes separate may acquire practical qualities. In particular, new spin Hamiltonians with more terms should be devised to better describe crystalline and magnetic field effects.
4. Double tensor operators A particular example of product tensor operators tors”) (DTO’s) usually defined as
are double tensor opera-
WCklk$ = b(Tkl(S) x Tk’(L))t , where b = b(k,k,k)
(33)
is a numerical factor needed for normalization:
(sill WCklk2)(1i, IJlsl’> = S1,,,6,*,,.[(2k, + 1)(2k, + l)]“*
(34)
and I,, 1, are used to specify the configurations on which DTO acts; however, in light of the distinct properties of ATO’s and PTO’s the present rank labelling scheme is inadequate. It is because both S and L are axial and, as demonstrated before, are incapable of connecting different configurations. In order to render eq. (34) justified, a new definition for DTO’s is needed: ~[~1(~21~22)‘21: z B[T~I(S)
x
(Tkqjq
x
+.z(jg)“qk
,
(35)
296
J.A. TUSZYk3KI
where B = B(k, k,, k,,k,k). This generalized definition embraces all the allowed classes of tensor operator products: axial, polar and mixed and thus allows for configuration operators,
mixing
effects.
eqs. (18)-(22),
p=k
22,
and Selection
Based on the derivations for the following exponents are obtained
t=kik,,-q,
c=k+q,
product tensor for DTO’s:
qs=k,+k,+k+l+q
h=k,+k,,+k,,+k2+q. rules for the matrix
elements
of DTO’s.
, I, = (lNSLJM1 W rk,(kz,k2:)k21~)1N~‘1’S’L,J,MI) are subsequently
found
applying
the procedures
(36) outlined
in section
3. The
present findings are summarized in the upper part of table III. The triangular rules for ranks and state labels are too numerous to be listed in detail but their form is quite obvious’). It is concluded that the value of k,, determines the existence of configuration mixing effects as it describes the polar component of a DTO. Hence, it is the individual ranks kZ, and k,, and not the combined rank k, that are of foremost importance to conjugation properties and selection rules, Our conclusions about the new rank notation are reinforced. Two-body DTO’s are products of two one-body DTO’s which act on different particles. Therefore, using eq. (35) as a pattern we define a generalized two-body DTO in the j-j scheme of coupling as
dklk2)); -; ; I,
A{[Tfll(S)
(7+(L)
x
x T;24(E))k21]kl
x ( 7-;12(S) x (Tf2’(L) x T;26(E))k2qk2}; . where
A depends
on all the ranks.
of coupling is straightforward (34.3) of Yutsis”). Adopting following
conjugation
p= kzt&,
An extension
(37)
of eq. (37) to the U-scheme
and the two schemes can be related through eq. the same procedures as before results in the
exponents
for the operator
t=k,,tk,,tk-q,
of eq. (37):
c-ktq,
qs=k+q,
11=k,,tk23tk24tk,2fk25tk2h+ktq. Consequently, operator,
selection
rules
for an arbitrary
I2 = (1NSLJM(w~~~k2’~~lN-21’l”S’L’J’M’),
matrix
element
involving
this
(W
CONJUGATION
PROPERTIES
OF DOUBLE TENSOR OPERATORS
297
TABLE III Selection rules
for matrix elements of double tensor operators. Selection rules
Symmetry One-body DTO’s 1)
2)
conservation of angular momentum hermiticity
3) parity invariance 4)
(a) (b)
M=Q+M’
triangular rules for ranks and state labels
(kl + kZI+ kz + k2)
(a)
(I+ I’+ k&
(b)
I = I’
even
even if ka = 0
time invariance
5) Quasi-spin invariance (N = 2i+ 1, I = /‘)
(kt + k2)
even
Two-body DTO’s 1)
M=Q+M’
conservation of angular momentum
triangular rules for ranks and state labels
2) hermiticity (jj)
(kl + kz + k) even (kll+k~+k~+klZ+k~+k~+k)
3)
parity invariance
(I + I’ + I” + kx + k26) even
4)
time invariance
5)
quasi-spin invariance (N = 21+ 1, 1 = /‘)
I’= k
I if ka= 0;
I” = I’
even if kz = 0
even
can be deduced with ease using the standard methods. They are presented in the lower part of table III. The importance of the results of this section rests with the fact that DTO’s are the most general entities used to construct symmetry-adapted effective Hamiltonians of atomic and nuclear physics. In view of our results these Hamiltonians should be written in general as
+
c c c cw kIlWt4 %klku Wx!n ‘[z’q
([kI,(k~kza)k211k,IkIz(kukzdk,Jk2}k 9’
(39)
where parameters D, and D, depend on all the respective summation indices. Not all of these indices, however, are physically meaningful since the Hamiltonian is still subject to invariance requirements, i.e.
2Y8
J.A. TUSZY&-SKI
g,sIpg;‘=X where
(lSi=Gn+7),
in the most general
g={T,H,R,
,,...,
We have employed for a rotation
about
(40)
case
R,,,u~,uV,u,~,u~,I}={gr;i=l
the following a p-fold
notation
. . . . . n-t-7).
for symmetry
axis, uk denotes
a reflection
elements
(41)
g, : R, stands
by a plane
k and I is
an inversion. Of course, the particular choice of symmetry elements varies for each of the 32 point groups. Prathe?) published an exhaustive study on the forms of invariant polar operators for all non-cubic point groups. He summarized his results in table 5 of his paper. A similar table can be constructed for axial operators. Since they are insensitive to space inversion (see eq. (3)) any two point groups with the same rotation and reflection properties but differing in their inversion signature become equivalent for ATO’s. For example C,, and C, or C, and S, become equivalent for ATO’s. With this in mind the Prather table can be used for ATO’s as well. An additional restriction is related to the time-invariance of X and it requires that the sum of all axial component ranks in each term must always be an even integer. A similar requirement for the ranks of polar components to the given point group.
will be in effect only if an inversion
centre
belongs
The end result of our analysis is that since the selection rules on ATO’s are more relaxed, more terms than usually allowed will be included in the effective Hamiltonian, eq. (39). Since parameters D, and D2 are often obtained from best fits to experimental hyperfine spectra, this calls for a modification in the parameterization schemes used to generate these coefficients. The sensitivity of these parameterization schemes to the number of adjustable parameters has been pointed out*‘.“) and is now being supported.
Acknowledgement This research is supported Engineering Research Council
by a grant of Canada.
from
the
Natural
Sciences
References 1) G. Racah, Phys. Rev. 62 (1942) 438. 2) R. Chatterjee, J.A. Tuszyliski and H.A. Buckmaster, Can. J. Phys. 61 (1983) 1613. 3) H.A. Buckmaster, R. Chatterjee and Y.H. Shing, Phys. Stat. Sol. (a) 13 (1972) 9.
and
CONJUGATION PROPERTIES OF DOUBLE TENSOR OPERATORS
299
4) H. Watanabe, Operator Methods in Ligand Field Theory (Prentice-Hall, Englewood Cliffs, NY, 1966). 5) J.S. Bell, Nucl. Phys. 12 (1959) 117. 6) M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr., The 3m-j Symbols and 6j-Symbols (The Technology Press, MIT, Cambridge, MA, 1959). 7) D.M. Brink and G.R. Satchler, Angular Momentum (Oxford Univ. Press, London, 1968). 8) A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, NJ, 1957). 9) H.A. Kramers, Proc. Acad. Sci. (Amsterdam) 33 (1930) 953. 10) B.R. Judd, Second Quantization and Atomic Spectroscopy (The Johns Hopkins Univ. Press, Baltimore, 1%7). 11) D.C. Mattis, The Theory of Magnetism (Harper and Row, New York, 1965). 12) W. Witschel, Mol. Phys. 20 (1973) 419. 13) P.W. Atkins and P.A. Seymour, Mol. Phys. 25 (1973) 113. 14) B.G. Wyboume, Phys. Rev. 148 (1966) 317. 15) A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon, Oxford, 1970). 16) W. Low, Paramagnetic Resonance in Solids (Academic Press, New York, 1960). 17) K.W.H. Stevens, Spin Hamiltonians in Magnetism, vol. 1, G.T. Rado and H. Suhl, eds. (Academic Press, New York, 1964). 18) B.R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw-Hill, New York, 1967). 19) A.P. Yutsis, LB. Levinson and V.V. Vanagas, Mathematical Apparatus of Angular Momentum (NSF Translation, Washington, DC, 1960). 20) J.L. Prather, Atomic Energy Levels in Crystals (U.S. National Bureau of Standards, Washington, DC, 1961). 21) S.S. Bishton and D.J. Newman, J. Phys. C3 (1970) 1753. 22) B.R. Judd, J.E. Hansen and A.J.J. Raassen, J. Phys. B15 (1982) 1457.