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Quantum mechanical operator equivalents used in the theory of magnetism
Danielsen, O.; Lindgård, Per-Anker
Publication date:
1972
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Danielsen, O., & Lindgård, P-A. (1972). Quantum mechanical operator equivalents used in the theory of
magnetism. (Denmark. Forskningscenter Risoe. Risoe-R; No. 259).
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Riso Report No. 259
9
£
Danish Atomic Energy Commission
"8
Research Establishment Riso
Quantum Mechanical Operator
Equivalents Used in the Theory of
Magnetism
by O. Danielsen and P.-A. Lindgård
Match 1972
Uu *n*mam Jll. OJlllmy, t7, UhfM* DfeUOT CnftilMPB K, Dnmifc
JhdUbk M cwlMff« fl9mt LUnry, DanUi Atoalc BMfnr Conhrim, KJ«, DK40M KMUM4 Dmwark
U.D.C
530.145.6:531
March 1972
Risø Report No. 259
Quantum Mechanical Operator Equivalents
Used in
the Theory of Magnetism
by
O. Danielsen
and
P . - A . Lindgård
Danish Atomic Energy Commission
Research Establishment Risø
Physics Department
Abstract
Two s e t s of operator equivalents, the Kacah operators and the Stevens
operators are treated. Their definitions a s angular momentum operators,
the transformation properties under rotations, the approximate Bose operator expansions, and the mutual connection of the two sets of operators
are treated in detail. The information i s presented in a number of detailed
tables to enable direct u s e in hand calculations.
ISBN 87 550 0135 I
CONTENTS
Page
1.
Introduction
7
1 . 1 . Expansion in Spherical Harmonics
7
1.2.
7
Angular Momentum Operator Equivalence
1 . 3 . B o s e Operator Equivalence
9
2.
Racah Operator Equivalents, 3 ,
3.
Racah Operator Equivalents Expanded in Bose Operators
3.1.
9
m
....
Crystal Field Calculations
11
3 . 2 . Spin Wave Calculations
4.
12
Transformations under Rotations of Spherical Harmonics
and Racah Operatoi*
14
4 . 1 . Description of the Result of a Rotation
4 . 1 . 1 . Transformation of Cartesian Coordinates
14
15
4.2.
5.
11
4.1.2.
General Transformations
17
4.1.3.
Rotation of Racah Operators
21
Calculations of the Rotation Matrices
d (p)
21
4 . 3 . Rotation of the Angular Momentum Vectors
4 . 4 . »-Transformation of Racah Operators
22
24
Stevens Operator Equivalents, o f 1
24
5.1.
Transformation of the Stevens Operators under
Rotation of the Frame of Coordinates
6.
Crystal Potential Energy in Cubic and Hexagonal Symmetry . .
7.
Definitions and Relations for Spherical Harmonics and
25
26
Racah Operators
28
7 . 1 . Spherical Harmonics
28
7.2.
Racah Operators
7 . 2 . 1 . Non-commuting Racah Operators
29
31
7.2.2.
33
7.3.
3 j - a n d 6j-Symbols
Commuting Racah Operators
34
References
37
Tables
39
LIST OF TABLES
Table No.
Page
1. Racah Operator Equivalents
2. Racah Operator Equivalents Expanded in B o s e Operators
39
....
41
3. Rotation Matrices d (p)
44
4. Rotation Matrices Ol[a, -3, -£) _1
48
5. Coefficients Relating Stevens Operators to Racah Operators . .
52
6. Stevens Operator Equivalents
53
7. Stevens Operator Equivalents Expanded in B o s e Operators . . .
54
8. Transformation of Stevens Operators by D (a, -x, -K )"
55
9. Crystal Potential Energy Expressed in Racah Operators
56
10. Crystal Potential Energy Expressed in Stevens Operators . . . .
57
11. 3 j-Symbols, Integer Values up to 6
58
12. 63-Symbols, Integer Values up to 6
78
1.
1.1.
INTRODUCTION
Expansion in Spherical Harmonics
In physics it i s in many c a s e s convenient to perform an expansion of a
function in spherical harmonics or linear combinations thereof. It i s then
possible to utilize the extensive work done on the transformation and combination properties of +he spherical harmonics (angular momenta). We shall
throughout u s e definitions and conventions as used by Edmonds '.
The function to be expanded may be a potential energy function, a wave
function, or any other function which in general is subject to some symmetry
constraints. The spherical harmonics Y,
form a natural ortho-normal
set of basis functions for rotations and are therefore particularly useful in
an expansion of a function with a number of rotation invariances. If, in addition, the function has inversion and reflection invariances, it i s often
more convenient to use the tesseral harmonics, being a ortho-normal s e t
of functions defined in terms of the spherical harmonics as
c
h»"7j«*I..«+«-,>mTJ.»>
C
lo " Y l„:
S
lo * °
m
> °
m =
<'•"
°
Thus we may expand a function V(r) as
l,m
1.2.
1, m
Angular Momentum Operator Equivalence
Angular momentum operators ' operate in the following way on spherical
harmonics ';
' We use, as did Edmonds, J to denote a generalized angular raompitum;
L i s restricted to denote an orbital momentum.
J
z
Y
lm -
m
Y
lm
0.2)
J+ Y
J
l m * <Jx +
U
~ Ylm • <Jx-
U
J
Y
lm * Vl(l+1)-m(m+1)
Ylm+,
y » Y t o * Vl(l+1)-m(m-1)
Ylm..,
With these properties it is possible to transform a function into a function of angular momentum operators, the so-called operator equivalents.
The operator equivalents transforming a s y ^ * Yj m are called Rrcah
operators and denoted 3 ,
. Oj
i s a function of angular momenta
8,
• O, mt^rfiv"^)- Functions transforming a s J jpr^-Cj m and
jraTSl.maredefinedas
^m = ^ro,- m M-.ro 1>m ,
^
Stevens ' was the first to invent the operator equivalence method in
crystal field calculations. Stevens introduced a different set of operator
equivalents which have later been widely used in the literature. These
"Stevens operators" denoted by 6 ? have the disadvantage of not having the
same systematic transformation properties as the Racah operators, however, they are convenient in "hand calculations" because they are defined
such that a number of square-root factors disappear. They are therefore
included in this report. Thus we can write the exact operator equivalent
of V(r) considered as an operator as
V(r) = v(r) I
\m\m^.lylz)
l,m
B
*« 1, I<
U°U + B U°U>
m
v(r, I B»0j».
l,m
(K4)
- g -
1.3. Bose Operator Equivalence
The commutator relation for angular momenta is
[J-»» J«]
jr
y
* &_
and cyclic permutations.
(1.5)
z
It is inconvenient that the commutator is a new operator. In approximate calculations it is often advantageous to use Bose operators having the
following commutator relation:
[». * + ]
= 1•
0.6)
It is then possible to utilize the extensive theory in the literature for
non-interacting and interacting Bose systems. In spinwavc calculations
this was first utilized by Holstein and Primakofi ' who found the Bose operator expansion of single angular momentum operators. Here we shall go
further and include all the O,
operators. In crystal field calculations a
transformation to approximate Bose operators was first done by Trammel
and Grover41'. For this purpose it is necessary to calculate matrix elements
of the Racah operators. These have been tabulated by B. Birgeneau '.
2. RACAH OPERATOR EQUIVALENTS Oj
m
A set of angular momentum operators O, , which transform under
rotations of the frame of coordinates in the same way as u -mrr times the
spherical harmonics*'.
\ m
transform as f ^ -
Y ^ .
(2.1)
has been tabulated in terms of angular momentum operators J , J J by
6)
x y- z
Buckmaster '. The angular momentum operators can be obtained by the
Stevens operator equivalence method in which the spherical harmonics Y,
expressed in 2,
_ of Jx', Jy*, J.
i" 2r 2rL are translated into functions O,
1, m
z
with due respect to the non-commutativity of the angular momentum components. It is, however, convenient to use this method only for the simplest
case, Y, j - 3 , ,, where no commutation problems occur, and generate
O,
by successive commutations with J". This is essentially the method
described by Racah '.
*' Operators with these transformation properties are per definition tensor
operators.
The commutator between J* * J x - U
according to Racah
[J". <\m]
and a Racah operator O,
• Vl(l+1)-m(m-l) O j ^ , .
is
(2.2)
As a starting point the operator with the maximum m-value, namely
m = 1, is calculated. The operators with lower m-values then are generated
by a straight-forward, but lengthy calculation using (2.2). According to the
definition of the spherical harmonics (7.1) we have
^•»M-n'f^-rfcoseje"'
and from the associated Legendre functions Pf^cose) we have according to
Jahnke and Emde81'
pJ(cose) = iSili- (sine)1
I
2*1!
When we introduce Cartesian coordinates, we find from these two relations
It is clearly convenient to multiply by V ^prr and by the operator
equivalence method, whereby r 7 is replaced by Jx + iJ y = J ; we have
then in accordance with (2.1)
or
a , - kp-fim: (J*)1
2 i l!
The operators O,
(2.3)
II
are obtained by means of the relation
In table I the Racah operator equivalents for all 1 up to 1 » 8 are
tabulated.
As an example we shall explicitly perform the calculations of the Racah
operator equivalents for 1 = 2 . From (2.3) we find
°2.2
|fg (J >
and by Hermitian conjugation we obtain from (2.4)
2
>
•6.2 . - 2 " - iyf g* »< -J »
.
Using the commutator relation (2.2) we find the following operators
[J". 0 2 2 ]
\i
=V2-3 - 2 - 1 0 2 > )
• i l l fJ". <J+>2]
and with (2.4)
\.i
ii^^v')
With one additional commutation we obtain O Q
-s, o
[J", 0 2 ) ] = f ^ 3 0 2 o
1 I i / I r , - r,-
,T+.2I
°2.o=vV^tl"*tj"'(J,]]
5
2o = !<3J!-J(J+lll
3. RACAH OPERATOR EQUIVALENTS EXPANDED IN BOSE OPERATORS
In approximate calculations it is useful to expand the operator equivalents O,
in Bose operators.
3.1.
Crystal Field Calculations
In crystal field calculations it is possible to express approximately an
excitation operatur between the ground state and an excited state as a matrix
element of the operator between the states times a Bose operator. In this
theory it is necessary to know the matrix element of the 6, m operators
between various crystal field states. These have been calculated by Birgeneau ', and we shall refer to that article for the numerical values.
3.2.
Spin Wave Calculations
Suppose a system with total angular momentum J has a sequence of
states beginning with the ground state as follows:
|j), |j-l), . . . . |j-n), ... | - j )
angular momentum states
|0>, |1>
|n>, . . . |2J+1>
deviation states
E Q < E, < . . .
<E n < . . .
energies
<E2J+,
(3.1)
Let us introduce Bose operators a( a + as the spin deviation operators
acting on a state | p )with p deviations: annihilation operator, a |p) =Vp[p-1)
and creationoperator, a |p )= Vp+1 |p+1). This i s to be compared with the
action of the operators O,
on the state | J - p ) .
The idea of expanding an angular momentum operator in a power series
of Bose operators was first used by Holstein and Primakofr' for a single
angular momentum operator. This method was used by Goodings et al. ' i n
finding the Bose operator expansion for a number of ft operators. Each
angular momentum in the expression for O,
(table 1) was replaced by its,
Holstein-Primakoff-expansion, and finally all a* operators were commuted
to the left of the expression (well ordering). This i s a very cumbersome
method, and only a few Oj m operators were expanded. We shall use a
different method which is physically cleaer and easier to perform. We expand the O.
formally in a well-ordered series of Bose operators and
require the matrix elements between corresponding states to be identical.
If we only require correct matrix elements between the ground and the first
excited state, it can be shown that the result of the expansion is identical to
the well-ordered Holstein Primakoff transformation as done by Goodings et
al. 9 >.
The well-ordered expansion of 8 ,
is
^.m ' < < o
+
*m.1«**+ < 2 »
+
"
+
"
+
•••>am
(3-2)
We find the coefficients of O,
using (2.4).
The coefficients are real and determined by matching the matrix elements in the following way:
< J -Pl°l
ml
J
-<PfIn>>=<P|<Am.o+Am,la+a
+A^2a+a+aa>am|P+m>
(3.3)
nence according to (7.9)
H,J
" P Cp
l /p
> ' ^ > J > ^ < o + P <*, + P <* P - ' >
-J+p m
J-p-m'
A m ^ - )
where we have used the standard formula for the matrix elements of the
angular momentum and Bose operators. From (3.3) we find the coefficients.
*
-J m J-m
^ 1 =(-D J - 1 < J ||o l || J >{( J
^m»l
=
Am2
n 2
^ '
l
tN
X
)
- J m J-m''
i4i J < J „o 1 ,i J >j( J
2
l J
x J
)
'^-JmJ-m'
+
« (J
Vm+1
N
l
-j+l
» (J
+
VmTi
V (m+1)(m+2)
m
' jj
J-1-nf
x J
J
(^-J+2
l
J
m J-2-m')!
(3.4)
The reduced matrix element is
)
S.J+lmJ-1-nf
J
and the 3j symbols can be found either in tables or by means of recursive
formulae. In the latter way we find the first coefficients in (3.2)
AL
V«
= (-i)m -L,f> m > ! -pl
l
'
m!
where Sj = 4 ( 2 7 ^ 7
' 2 m (l-m)! V£
=
'CJ-2-)tMMJ-|)-"
^ T
1
>
By means of these formulae the Racah operator equivalents are tabulated for odd 1 as well as even 1 up to 1 - 8 - table 2.
It should be added that the operators O, Q> O 2 0 , • . . O g „ are finite e x pansions, whereas all other operators are infinite expansions in Bose operators. In all operators only terms with up to five Bose operators are
written out because of the limited validity of the spin deviation representation.
4. TRANSFORMATIONS UNDER ROTATIONS OF
SPHERICAL HARMONICS AND RACAH OPERATORS
We shall in this section summarize the transformation properties of
the spherical harmonics and of the Racah operators under rotations. Although this transformation theory has been well developed (Edmonds ',
R o s e , 0 \ Tinkham 11 ', Judd 12 ', and Rothenberg' 3 '), it often causes confusion that the term rotation in the literature sometimes refers to the physical body, sometimes to the axes of the coordinate system. This point will
therefore be discussed in some detail.
4.1.
Description of the Result of a Rotation
An arbitrary rotation of the frame of coordinates i s most conveniently
expressed by the three Euler angles a, B, Y. The definitions of the Euler
angles are shown in fig. 4 . 1 .
- »5 -
Fig. A.I.
The Euler angles
1. A rotation a(0 * a * 2«) about the
z-axis, bringing the frame of axes
from the initial position S into the
position S'. The axis of this rotation is commonly called the vertical axis.
2. A rotation p(0 * p * *) about the yaxis of the frame S', called the
line of nodes. Note that its position is in general different from the
initial position of the y-axis of the
frame S. The resulting position of
the frame of axes is symbolized by
"S".
3. A rotation Y(0 * Y * 2«) about the
z-axis of the frame of axes S",
called the figure axis; the position
of this axis depends on the previous
rotations a and p. The final position of the frame is symbolized by
It should be noted that the polar coordinates • and 6 with respect to
the original frame S of the z-axis in its final position are identical with the
Euler angles o and p respectively.
4.1.1.
Transformation of Cartesian Coordinates
Before dealing with the general transformation theory we shall set up
the much Bimpler formalism of transformation of Cartesian coordinates
which may serve as a guideline for the more complex general transformations.
If the coordinate system is rotated through the Euler angles a, p, Y,
the Cartesian coordinates x, y, z of a point P which is stationary under
the rotation will be changed to x : , y', z'. Let the two sets of Cartesian
coordinates be written as column vectors r and r'
x'
y
-16and let tue matrix PC, P, a) be the 3 x 3 matrix which connects them, then
£
- P
r
(4.1)
and
P(Y.p.a) = P^.OO • Py. (p) • P z (o)
where first P (a) performs a rotation about the original s-axis, then P-jfP)
sZ
-J
a rotation about the y-axis in S'ry', and finally j^tiOO a rotation about the zaxis in S":z". The three rotation matrices are given by
r>>
=
cosa
sina
0
-sin a
coso
0
0
0
1
(4.2)
Py.(P)
cosp
0
0
1
-
sinp
cosY
f>(Y) =
-sinY
0
0
-sinp
a
cosp
sinY
0
COBY
0
0
1
The resulting transformation matrix is therefore
(4.3)
W.P.e) •
(cosa cosp cosY-sina sinY) (sina cosp cosY+cosasinY)-8inpcoa
(-cosa cosp sinY-sina cosY) (-sinacospsinY-)- cosacosY) sinp sin
cosa sinp
sine sinp
cosp
If one wants to rotate the physical body (the point P) instead of rotating
the frame of coordinates, this is simply done by using the inverse rotation
matrix, constructed in the following way
PCY.P.a)-' * (PZ„(Y) Py,(p) Pz(a))-' » P^af' P y ,(pr' P^OO"'
-&<->£.<-« &»<-*>
?(-<•» -P. -V)
P(Y, p, a)" is the transposed of P(Y, p. a).
(4.4)
- 17 -
4.1.2. General Transformations
Conventionally, there are four equivalent ways of describing the result
of a rotation depending on the choice of the object to be rotated and on the
choice of the frame of reference.
Either (1) the function (the body) or (2) the axes of frame are rotated.
The rotations may be performed either (a) with respect to a rotated coordinate system (local system) or (b) with respect to a fixed coordinate
system (global system).
It should be mentioned that the rotated functions are considered to be
represented in terms of basis functions set up with respect to the global
system irrespective of how the rotations are performed (cases 1 a and 1 b).
The various cases are described in the following, and a summary presented
in fig. 4.2.
(a) Rotated Coordinate System
In fig. 4.1 the successive rotations are made about the rotated axes
(local system). Starting with the x, y, z coordinate system (S) we rotate
by a about the z-axis, and denote the new axes x', y*, z' (S'). Then we
rotate by p about the y'-axis, and denote the resulting axes x", y", z" (S").
Finally we rotate by Y about the z"-axis, and label the final axes x'", y'",
z>" (S ,M ). In operator form we express these rotations as
P ^ C ) Py.(P) P z (») •
(b) Fixed Coordinate System
The rotations can also be described with respect to a permanent set of
space-fixed axes (global system). It can be shown, Tinkham ', that the
same final axes x'", y"', z"' might have been obtained by carrying out the
rotations in reverse order about fixed axes. That is, we might have rotated
by Y about z, then by p about y, and finally by a about z again. In operator form we have for these rotations which define the combined rotation
operator D(o, p, Y)
D(o.p.Y) s P z (o) Py(p) PZ(Y) .
(4.5)
(1) Rotation of a Function with Respect to a Fixed (Global) Coordinate System
A finite rotation of a function with respect to a fixed coordinate system
can be looked upon as a succession of infinitesimal rotations.
- IB A rotation through an angle • about a fixed i - a z i s of a one-particle
wave function
?{x, y, z) can be described by
P z (?)*(x.y,z) • Y(xcose + ysine, - xsin» + ycos«, z).
In order to find the operator F z ( •) we first consider an infinitesimal
rotation d?
P z (dT)»(x,y.»i)~J'(x»-yd». y-xde, z)
- 1 (x, y. z) * d,(y " ^ '> -* " f r ? . ')
hence the infinitesimal rotation operator is
When rotating about a direction specified by u, the infinitesimal rotation
operator i s
P„(°» - O - Q » f l " u)
If £ i s the total angular momentum of a many-body system, then i t s
component along any axis u i s related to the operator of infinitesimal rotation about the axis by the relation, Messiah '
P u (d>) = 1 - d e | J . u
(4.6)
Considering a finite rotation P (•) about the axis in the u-direction,
and putting J • u » J
~
P„l*+ dt) = PH(dT) • P„(,) = <L which is equivalent with the equation
aP„(f)
d
4 J„» PJJ(»)
- 19-
This equation has the following solution under the condition P (o) = 1
P„(.) - e
u
l
'r
h
(4.7)
This is the rotation operator for the rotation around the axes u.
From (4.7) we immediately find for the combined rotation operator D(o, p, Y)
J
D(o.p,Y) = P z (o)P y (p)P z (Y)=e
J
J
-ia-f
-ip-r*
-iY-£
h
h
h
e
e
(4.8)
Now we want to find the matrices D (a, p, Y) which describe how the
angular momentum eigenfunctions Y 1 _( % •) transform under the general
rotation operator D(o, p, Y). Because of the choice of the z-axis as quantization axis, J is a diagonal operator and the effect of the rotations by a
and Y is easily written by using the exponential form of the rotation operator.
The rotation P (p) will hare a non-diagonal representation, which is denoted
The transformed spherical harmonic function Y l m (8 l , f') (analogue to
the vector r' in (4.1)) is then expressed by
1
*lnS». *') = »(«• M ) "We.,) • I
Y^.t«, f) DJ»m (a, P, Y)
m'=-l
(4.9)
The matrix elements of the rotation operator D(o, p, Y) is
D m , , m (o,p,Y)= < t a ' | D ( o . p . Y ) | l m ) = e - l m , » d m ) m ( - p ) e - i m Y
< 4 - ,0 >
where
,si4) 21 - 2 -
(4 n)
-
(the summation is over all positive • such that the factorial term? are nonnegative).
For d_'. (P) we have the following symmetry relations
- 30 -
Cm'-W
ni ni
' C nmm
.'W
<*-,2>
«£[»<»
(4.14)
= (-^'-"-»..„W
The matrix elements D*/. (Q, p, Y) are therefore
m ro
D 2 L U . M )
=e" i m , '>d<'>
(pje-^
(4.15)
(2) Permanent Function and Rotated Axes
Consider the case where the function i s held fixed in space, and the
coordinate frame is rotated. We note that if the same rotation was applied
to both the axes and the function, the relative position would be unchanged,
which can be expressed by the relation
P a x e s f c - f ^ a c t i o n *•**
"F
»•">
hence
Junction« 0 '?- 1 " • H L . f c M ) " 1
• Paxes«"*'"!»'-«>
That i s , we can use the same D (a, p, Y) to represent rotation of axes by
-a, -p, -Y in the reverse order to that used if the functions were being
rotated.
The discussion of this section can be summarized in the following figure
4.2.
- 21 -
With respect to
Cos« Rotated
Operator
lo
function
rotated axes
IVtørøftM
lb
function
fixed axes
2a
axes
rotated axes
2b
axes
fixed axes
3(a)5(f»g(Y)
3-ta)l>.tø)l|W
£tY)5t$?M
Fig. 4.2.
The alternative possibilities in transformation theory4.1.3. Rotation of Racah Operators
The Racah operators transform per definition under the unitary transformation comprising the rotation with respect to a fixed frame in the same
way as the spherical harmonics, and thus according to fig. 4. 2: case 1 b we
have
=
1
y
d^ m l (J T .J y .J z )<lm'|D(o.a.V)|lm>
(4.17)
The same relation can in a short hand matrix notation be written as
Oj = Oj • D^o.p.Y)
(4.18)
(The primed operators are angular momentum components in the tripleprimed coordinate system, fig. 4.1).
4.2. Calculations of the Rotation Matrices dX(p)
In table 3 the d (p) matrices for all values of 1 - 1, 2 , . . . 8 are given in
the form
4« m <M • C • <.in§) 2 1 {A 0 + A,cotf + A 2 (co4)V . . + A 2 1 ( c o t | ) 2 1 L , m
(4.19)
. 22 -
where C and A_ are numerical constants.
As an example of the use of table 3 the d (ft) - matrix is constructed
dj,(p) « 1 • (sinf) 2 (cotf) 2
d]0(P> '
V5(sin§) 2 cotf
•= ^(1 + cosp)
s
-^BinP
(4. 20)
d]_,(p) - I • (sinf) 2 - 1
*^1
-cosp)
djjp) .- 1 • (sinf) 2 [-1 + (cot f ) 2 ] = cosp
The rest of the matrix is obtained using the symmetry relations (4.13)
and (4.14)
d'(p):
1
3<1 + cosp)
^sinp
0
Y2
-T^sinp
cosp
-1
^(1 - cos?)
5<1 - cosp)
V5
™
V2 . .
--j-sinp
(4.21)
Blnp
^(1 + cosp)
4. 3. Rotation of the Angular Momentum Vectors
As an example of the transformation of the Racah operators we shall
express the relations between —
J' • ( Jx*, Jj .r JZ ) in the rotated coordinate
system - the local coordinate system - and J = (J,, J , J.) in the stationary
or global coordinate system. This transformation gives a check on the
formalism as it can easily be calculated directly.
From table 1 we have
tflf
[J*',.'« 1 '?
°10- 5 1-l ] ' [ J * . J y J * ]
0
1
0
or in shorthand notation
°i • i • y
(4. 22)
23and the opposite
—
-1
~
+
J = O, • U = O, • U
(4.23)
The corresponding relations of course hold for J' and 8"..
The relation between Oj and O. is for a rotation through the Euler
angles a and p:
Oj • 6,
• D'(O.P.O)
By use of (4.22) we find the relation between J' and J
J' • U = J • U •
DVO,
p. 0)
ji
DV,
p, o) • y"1
=j • y •
or
From (4.15) and (4.21) we find for D'(O, P. 0)
e -ia
1+cosp
2
If
sing
Y2
D^a.p.O)
B io
-io 1-coBp
-ia flinp
'
(4.24)
cosp
1 -cosp
io sinp
ia 1 + cosP
VT
so the final relation between J and J' is
[j
* J y J z ] • [J5.^.JC]
cosa cosp
- sina
cosa sinp
sina cosp
cosa
sina sinp (4. 25)
-sinp
Q
cosp
which is in accordance with the direct calculation using P(-a, -p, 0) in
equation (4.4) and the result (4.16).
-24 -
4.4. K/2 - Transformation of Racah Operators
In practice it is often convenient to transform a set of operators given
in an axially symmetric system with the z-axis along the symmetry axis
into a coordinate system with the z'-axis in the perpendicular plane xy so
that zf makes an angle wim x, and y1 is parallel to z.
The transformation relating the operators in the new (rotated) frame of
reference is
, (4. 26)
Si-a-sH-i-i)-'
where
er'vAi-i
v->
For 1 - 1 the expression gives
COH.O,*3!.-!3-
(old)
[OJj.O^Oj.,]
(new)
1
B
•
In table 4 the matrices D (a. j , ^ )
hu
-rV
V
t
i
0
i
n
-ia
1 -la
i -ia
"7 e
(4.27)
-1
are calculated for 1 - 1,2,3..., 8,
5. STEVEMS OPERATOR EQUIVALENTS, OJ*
The operator equivalents defined by Stevens ' are closely related to the
tensor operators defined by Racah '. The Stevens operators do not have the
systematic transformation properties of the tensor operators; however, they
are convenient in hand calculations as a number of numerical factors are included in the definition.
The Stevens operators are in terms of the Racah operators defined as
follows
- 25 -
r
21+1
1
»F« - ^ m l P ^ i ^ - m - H r ^ J
oo(c) . i.ywL
OJtø
(5..)
^
= 0
where K™ are the normalizing coefficients in the tesseral harmonics. These
coefficients and —— V - y j - are given in table 5 for 1 up to 8. In table 6 the
Stevens operators for all even values of 1 up to 8 are given explicitly in terms
of angular momentum operators. In table 7 the same Stevens operators are
expanded in Bose operators.
We shall give an example of the method of finding the Stevens operators
in terms of the angular-momentum operators using tables 5 and 6:
°2<C> * ^ " f n
°2<C> " ^
°2. o " 2 °2. o - 3 4 - J<J+' >
6^S2,-2+52,2)
°2<"> " ^ - f i l
=T|^<52,-2+52.2,
^<52.-2-S2.2)'
<5- 2>
= 7(<J+)2+«J")2)
"|^\.2-52.2>a-|(<J+)2-(J")2)
5.1. Transformation of the Stevens Operators under Rotation of the Frame
of Coordinates
The transformation properties of the Stevens operators are found by
using their relations to the Racah operators. Table 8 gives the transformations of the Stevens operators unUer the frequently occurring rotation of
frame where the new z-axis is perpendicular to the old one and makes an
angle a with the old x-axis. This transformation is generated by D(a,£, £)"
as described In section 4.
We shall give a few examples using tables 4 and 5
26-
Ojtc) - operator
i o » ( c ) > - ^ ^ O g ( c ) ) - f ^0^(c)
or
(5.3,
0°<c) - - i 0 S « c > - | 0 2 « c '
2
0 2 (c) - operator
<32,-2+32.2' - ( ? ^ . o - f ø . ^ O ^ ) l e o ' t a , -l : «%.-1-\l* , t a W '
^
og(c) -
( ^ i 0°<c) - ^^c))t
»2« - - ^
oJ(c)2sto2o
°2<c> "* ( i ° 2 ( c ) " 7 oi<c))cos2a - 2 oJ(c) sin2a
6. CRYSTAL POTENTIAL ENERGY IN
CUBIC AND HEXAGONAL SYMMETRY
An electron in a crystal is under the influence of the Burrounding electric
charges constituting the so-called crystal field. The crystal field is restricted by the crystal symmetry and i s commonly expressed with respect to
some principal or high-symmetry direction in the crystal. Using the operator
equivalence method we find tor the crystal potential energy from the crystal
field potential either by means of Stevens operators
H
cf- I » r ° r
(6.D
l,m
or Bacah operators
*<*• l*?\m
1, m
(6-2)
- 27-
The potential functions in (6.1) and (6.2) are phenomenological and quite
general. The actual calculation of the parameters is very difficult, and one
is forced to use simplifying models such as the effective point charge model
or ligand field theory. In this section we shall tabulate the crystal potential
energy for the cubic and the hexagonal symmetries in the principal directions in terms of both Stevens and Racah operators, using the tables for
their rotation properties.
Cubic Symmetry
In fig. 6.1 the principal directions in the cubic symmetry are shown,
namely the (001 )-direction which is a 4-fold axis, the (110)-direction which
is a 2-fold axis, and the (111 )-direction which is a 3-fold axis. The crystal
potential energy expressed with respect to the 4-fold axis is given by Hutdungs ' and by means of table 8 the expressions for the energy with respect
to the (110)- and (111 )-directions are calculated. The expressions are given
in tables 9 and 10.
font
FiJ.il.
Th» principal dTucliow of lh> cubic lattice
Hexagonal Symmetry
The crystal potential energy is in the hexagonal symmetry expressed
with respect to the principal directions (0001), (1000), and (1200), see fig.
6.2. The (0001 {-direction is a 6-fold axis, and the (1000)- and (1200)-directions are 2-fold axis. The expressions are given in tables 9 and 10.
-28.
Ftg-S.1.
Hm priftciprt i t m S g i n «• th
7. DEFINITIONS AND RELATIONS FOR
SPHEBICAL HARMONIC FWNCTIOMS AND RACAH OPERATORS
In this section we shall Bummarize the definitions and conventions for
the spherical harmonics and the Racah operators.
7.1. Spherical Harmonics
The spherical harmonics are defined by Edmonds 1)
*«.«« - ^ [ ^ | ^ J / 2 l - r C ^ r " (We(7.1)
•(-uPfHSU^J7'-?!—I.*"'
where Pj(cos6) are the generalized (including negative-valnes of m)aseociated Legendre functions of the first kind. The spherical harmonic«
satisfy the ortho-normality relation
/
d
» / « « ^to< 8 ' •> *i. m .t"-»> sln « 3 -
hi- »,mm1
(7.2)
and furthermore
Y
l.-m <9 ' ,) ' ( - ,)mV i.m< 9 -'>
<7-3>
The product of two spherical harmonics which have the same arguments
is given in terms of the 3j-symbols by
Y
L m ( e ' f ) Y l m <e'f>
2 2
'
=
h
11
Tn
(7.4)
J
L
i, m
m.
i
J l o 0 (>)1We•f,
£
By use of (7.4) and the orthogonality properties of the Sj-symbols we
obtain the reverse formula
(7.5)
^"V-WiP" l C 'o DC, mJ W W '
m. m«
1
^
7.2. Racah 'Operators
Definition of the Operators
The Racah operators are Irreducible tensor operators, which means
that the set of 21+1 operators ^>lm (m = 1, 1-1, l-2 r . . . . . -1) transforms
under rotations of the frame of coordinates as U « m ^im' 8 , *)» namely
D(..P.v)6lmP(o.p.Y)-1 =
V
mni.l
tthn,Dm»m(..P.v)
(7.6)
-SO -
An Equivalent Definition
Since the operators of total angular momentum of the system are multiples of the infinitesimal rotation operators, we may replace the unitary
transformation on the left by a commutator, giving for any component of
angular momentum J
tv3^]*
X
I aL.<»»,i*.ii»>
PV
In that way we find the equivalent definition of the irreducible tensor
operators given by Racah ', namely the commutator relations
(7.8)
fJz-
°h»l
= m 5
lm
The matrix element of a Racah operator i s given by
<jm|5.
| j - m . > - ( - I ) J - m ( J k J ' ) <J I I S J U ' )
* ^.
-m a var
(7.9)
It should be noted that a tensor operator is characterized by its reduced
matrix element, here ( J II < \ || J ' ) for the Racah operators.
When dealing with the Racah operators one has to distinguish between
operators that commute and operators that do not commute. If the operators
are acting on different parts of the system, say spin and orbit, they commute.
This can also be described by saying that the operators act on part i and j
in the system. If the operators are acting on the same dynamical variable,
part 1 in the system, the commutator relation is not in general zero.
In magnetic systems the interactions may be expressed as tensor
products of Racah operators. General expressions for tensor products of
Racah operators and matrix elements of tensor products of both commutingand non-commuting Racah operators are therefore given.
3t
7.2.1. Non-commuting Racah Operators
The product of two non-commuting Racah operators O,
(i) and 6,
(i)
by Judd 'fortensor operators of order zero. This result is here
ralized to tensor operators of order k.
J J ,5V
Av^
%W
<r||aL»IUXJ|l^»l|J>
\iiVU) 2
^
Using the symmetry relationfor3j-symbols, namely
1l <*2 V
<>2 "Jl
V
we find the commutator relation
[6^(1,. o ^ i , ] -
7
ki+k2
TlH^^l^nl^^^H
< j | | & W l l J X J l l ^ W U J>
2 y
L_^
<*iie^«ii'>
S t (i)
w
(7.11)
The reduced matrix element is
wi^no-iV^S 5 -
(7.12)
It should be pointed out that the commutator relation does not depend
on the magnitude of J. This can be seen directly by using the angular
momentum expressions for the O ^ operators (table 1) and the commutator
relations for J ^ J ^ J z .
The tensor product of two non-commuting Racah operators is defined by
(K)
/„(k,)^(k,)N
(O
O
)
Q
(7.13)
r
1
v
2
k +k
- i
Q
2- „
q,»-k, q ^ - k j
k
k
/ i 2
»I "a
K
\~
"*
and for the scalar product of two non-commuting Racah operators we have
( 3 « • o/ K »)= (-1 ) K Y2RM (S<K> 5<K> ) < 0 )
(7.14)
The matrix element of the tensor product of two non-commuting Racah
operators is
(K)
<Jm|(S,k1)5(k2,)
|jm.)
Q
•'•" , " n C^.) w|K,sri C?*j < ' || \» i|| '>< j " 3 '» m «'>
Q
(7.15)
from which we have for the reduced matrix element
N(K)„
<j||^<Wk2>)
„,
( k . k , K,
»j^M)15*^)'
2
}<J||afeW||J><*||?SkW||J>
(7.16)
7.2.2.
Commuting Racah Operators
As the two operators O t
(i) and ( V
(j) commute, we immediately
*!*1
*2l2
have
t6
k,q,W- ^ q ^ = °
<7-,7>
The tensor product of two commuting Racah operators is defined by
k,
k,
I I
V l
q = k
2"2
(7.18)
and for the scalar product of two commuting Racah operators we have
( o f • 3 W ) • (-1 ) K ^
15<K> S<K> I ( 0 >
(7.1 9)
The matrix element of the tensor product of two commuting Racah
operators i s
<JlJ2Jm|J5^>0<k2>j(K)|i.j.J.m>
H) J - m (
)V(2j+i)(2JM-i)<2K+j)k
v2 *tø,||«L nm\)
l
-m Q m '
H)J-m (J
K
J
^\Pk.om>
ki k j K
' ) <i,j2J || {5<ki >3<k2> | ( K , | | n t y >
<7-2")
So the reduced matrix element i s
uvii^'a^iiw')
(7.2!)
,
, c »1 h
= V(3J+1)(2J'+1)(2K+1) J jj j'3
J
,
J'j<j,1|Ot I 0)l|ji><j 2 l|O^U)l|J , 2 >
7. 3. 3j- and 6j-Symbols
The 3 ] - and 6j-symbols occur in many of the preceding expressions.
Their definitions and symmetry properties are therefore given following
13)
Rothenberg
i s equivalent to that of Edmonds '.
henberg et al. ',, whose definition
defil
The 3j-symbol i s defined as
v
m" ,I "*2
m_ *"3
m,'
' ) - (-i/1
' 0l+3^3>!«l-J2+J3>!<Jl+J2+J3>,-0l+m1 ) ! 0|^°l )!02Hn2>:02-m2»!03+m3>!(J8-ln3»!
a, + i,+ i,+ m
* 1 «(Jj+Jj-Jj-k)! 0,-«,-*)! U24^2-k)! (ij-lj+n^+kjl U r i,-m 2 +k)!
k
(7.22)
Under an even permutation of the columns the symmetry properties of
the 3j-symbols are given by
rh
h
S
m
h\
fh
2 H
Km
h
H\fh
2 "»3 »S S
and under an odd permutation by
h ij\
"l "5
(723)
- 35 -
V,
m 2 m^
^m2 m, m£
^m, m 3 m^
Sn 3 m 2 m^'
(7.24)
When changing the signs of the m's we have
A
h h\
=H )
W i 3 (h
h h\
(725)
The 3j-symbol automatically equals zero unless both m. + m 2 + m~ - 0
and j , . j 2 > and j 3 satisfy the triangle conditions
h
+
h-h
' °'
h-h+h'
°> ">i + ^ + i 3
?
«
<7-26>
Besides the sum jj + j , + iq must be an integer.
The 6j-symbol is defined as
Y
^
(-i)ka,+i2+i1+i2+'-k)!
kMj1+j^3^)!(lI%J3^!U,+l2V)!(l1+J2434)!(-j,4,+J3+k)!(.J2-l2+J3+l3+kJ!
where! A/ K-i - rT i(a-t-b-c):
M-MI i-e+ht-^v
~\ ' ' 2
+h
v /-_
(a-b+c)!
(-a+b+c):
(abc =
L 1
*
>
L
(a+b+c*1)\
J
(7.27)
The 6j-symbol stands invariant under interchange of columns. It is
also invariant at interchange of any two numbers in the bottom row with the
corresponding two numbers in the top row. The 6j-symbol is automatically
zero unless each of the four triads (Jj, j 2 , J3). (lj, 12. J3). Oj. *2« V« a n d
(If, J2, 13) satisfy the triangle conditions (7.26). The elements of each
triad also must sum to an integer. The four triangle conditions for the nonvanishing of the 6j-symbol might be given in a diagram form easier to
remember
- 86-
{'/Sl\:
I " j ; £y.b 1X4
«••••
In tables 11 and 12 the 3]- and 6j-symbols are calculated by means of
the Borroughs 6700 computer for all integers up to 6. A more extensive
table including all integers and half-integers up to 8 bas been given by
Rothenberg et aL' 3 '.
- 37 REFERENCES
1)
A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton
University P r e s s , Princeton, 1957). 146 pp.
2)
K. W. H. Stevens, Proc. Phys. Soc. A65 (1 952) 209-21 5.
3)
T. Holstein and H. Primakoff, Phys. Rev. 58(1940)1098-1113.
4)
G. T. Trammell, J. Appl. Phys. 31_ (1 960) 362S-363S.
G. T. TrammeU. Phys. Rev. 131^(1963) 932-948.
5)
R. J. Birgeneau, Can. J. Phys. 45_ (1 967) 3761 -3771.
6)
H. A. Buckmaster, Can. J. Phys. 40 (1 962) 1670-1676.
7)
G. Racah, Phys. Rev. 6£ (1 942) 438-462.
8)
E. Jahnke and F. Emde, Tables of Functions with Formulae and
B. Grover, Phys. Rev. A140 (1 965) 1 944-1 951.
Curves (Dover, New York, 1945).
9)
380 pp.
D. A. Goodings and B. W. Southern, Can. J. Phys. 49 (1 971) 11 37-1161.
10)
M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New
York, 1 957). 248 pp.
11)
M. Tinkham, Group Theory and Quantum Mechanics (Mc-Graw-Hill,
12)
B. R. Judd, Operator Techniques in Atomic Spectroscopy (Mc-Graw-
New York, 1964).
340 pp.
Hill, New York, 1 963). 242 pp.
13)
M. Rothenberg, R. Bivins, N. Metropolis and J. K. Wooten, The 3-j
and 6-j-symbols (Massachusetts Institute of Technology, Cambridge,
M a s s . , 1 959). 498 pp.
14)
A. Messiah, Quantum Mechanics 2 (North Holland Publishing Co.
15)
M. T. Hutchings, Sol. State Phys. 1_6 (1 964) 227-273.
Amsterdam, 1962).
- 39 -
Table 1
Racah operator equivalents
X = J(J + 1)
°4 0
5
4,±i
°4.t2
= {f[35 J^- 130X- Z5l J^+3X2- 6xJ
• ; H ( r e T [ i , J ' - l 3 x * " J I ; j * * ^'• — •\
-
Ull^'l-X-S*'*?
r |J*)J,—ij
5«... •»fftiC'.f*)' • n*>''J
54,t4 -iRIr«1'4
06
0
°» I I
« Tffl 231 Jl ~ l 3 1 5 X - 73»}j 4
" ''/raJll'33
J
+ ( 1 0 5 X 2 - 525X+ 294 ! J 2 - 3 X 3 + 4 0 X 2 - 6 0 x 1
i - ( 3 » X - 1 5 ( J 3 + (5X 2 - 1»X + U H / j *
2
2
+ 2
\ ± 1
'fimCTl
57
= ^ 1 429 J^- 1693 X - 231 Olj* + J 3 I 3 X 2 - 2205 X - 2121 f J^ - .35 X 3 - 385 X 2 + 882 X - I 8 0 } j a j
0
*V+1
S
7 ±2
B
3 3 J * - d » X + I 2 3 ) j t X 4 1 0 X * 103!(J )
J-v!--)]
+
*1WTZB?
" /Aijj
*
M*" I* • - - -} J
p ] 7 1 6 J 8 - ( 1 9 8 0 X - 2310)J4 + (54flX a - I8O0X+ 2I84)J 2 -(20 X 3 - 1 3 0 X 2 + 2 7 0 X + 2 2 5 } /
1 4 3 J
n
3
2
! - < ° X + 8 2 5 ) J + t ' 5 X + I 7 0 X * 2092U 2 I(J*)
2
° 7 ±3
" * •'ITS! 2 " [ t 2 M J * - < 1 3 2 X * 2992)J2 + 6 X 2 * 222 X + 54811 | J + ) 3
%ti
•^in2-[l' 3J '-( 3x+133 ) J ,l( J *) 4 * w+i4 I---']
\±.
-'l&iD'"!-*-141^)5
°a o
+
+
+• J * i
!
(J*) !•••!]
<J*) 3 " . . . > J
• (,*)5 •;•••;]
"TJIIM'5JS"',2"12X"54I>S4'J'+''69,(,X2"64,8I,X+93555'4"
9 260 X 3 - 18270 X 2 + 58388 X - 21390} J 2 + 35 X4 - 709 X 3 1- 3780 X 2 - 5040 X I
5J
B " ( " " " X-2002)J^+(385X 2 -1925X+2695)J 3 -(35X 3 -3l5 X2+854 X+STaU^O*)
°8,±l
" *T3JIB"2"I ' "
°S,±2
' I ^ T O j l ll43j5-(143X+H44W*t(S3X 2 +407X+595)Wf -B 3 +l3X 2 *372Xt4B06))(J*) i l
°8,±S
" : f | f f i i L l 3 8 J ! - < 2 8 X + M 3 ' J ' t (3XStl28X+3648Mz:(J*)3 +
°8,±4
*|/Si|*"J*-(28X''
\±.
- ^ æ i i i ^ f - i x ^ 3 ^ ) « ^ • w*)5i--}j
5
T3 3
' M 2 * X2+ 86X+4284!(J*)4 +
8,t. •|Ki[ii5a*-x-"3> B *)«
5.,±, • *flBHv>»' *(^)''J
+
(j*) 8 |-)J
<J*) 3 (---lJ
(J*)4l"-lJ
+
+
(J*)\"-
(J*) 2 ) • • • ) ]
T»bla 2
Btcah operator »quivalente expanded in Bone opa-rntora
0oo= 1
V ^f1-^"* Sj •*•*«••:•]
vti[~gfr.|=.5|i*-.-.]
Vq-^.'.^.v
,
]
i
Pi
V -i i5«r5—.....
°«= -
V |B5 i—....
V-ies^
V-
43 •
V «i [i - i* • ' • • I * •*•*—...]
V
^pss^—....
V -
v- V-
^
df(f)
Table X The Rotation Matrktt
I«
1
RI
M
p
1
0
»
1*0000
-1-
ft
»
-1
•
0
Vf_
*_
HI
H
r-
1
1*0000
>*O0AO
2**4*5
2.0000
1.0000
1
1
1
0
-
.;
.*_
I-
J
•1
H
ii
N
o
(•0000
1
0
0
o
0
0
0
1
1*0000
1*5011
**i2sr
1*5011
l.OOOfl
0
0
0
0
i.i5«r
o
-
i
0
i
i
-«
9
1
HI 01 HI 11 HI 21 H| 3} H| « ) A| 51 HI é ]
1*0000
i>«m
3.0730
4.4721
3.0710
2*4499
1*0000
1IOODO
•
0
0
o
0
0
1
0
1
0
0
1
0
0
0
1
0
°
0
3
0
i
0
4
3
-
0
ft
0
ft
0
5
0
0
••
0
*
0
-"»
-
1
1
1
0
0
ft
0
3
O
0
0
0
0
1
1
o
0
«
0
«
0
4
0
0
1
o
I
1
1
II
0
-
0
6
o
0
0
0
0
0 - 3
2
0
5
0
*
4
o
0
0
0
0
0
1
0
S
1
0
1
0
It0000
!-?
a
Ht 01 Ht 1 | Ht SI Ht 31 Ht 41
1.0000
1.2247
l.Oftno
r
!
-I
0
*
0
1
l*94»
o
_
I'
1
H]
H
p
0
0
0
0
0
0
0
1
0
0
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Tablet
The rotation-matrices
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Table 5
Coefficients relating Stevens operators to Racah operators
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Table 1C Crystal potential entrgy expressed in Aacah operators
5,
Cubic
S
£
Symmetry Direction Form of crystal potential
(0001)
(1000)
Hexagonal
(1200)
energy
-58-
7bble 11.
3J - symbols
| J, J* J> \
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J, ÅJy
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