Total angular momentum and atomic magnetic moments

Journal of Magnetism and Magnetic Materials 65 (1987) 99-122 North-Holland, Amsterdam

TOTAL ANGULAR MOMENTUM

99

AND ATOMIC MAGNETIC MOMENTS

Xavier O U D E T Centre national de la Recherche Scientifique, Laboratoire de Magn(tisme, 1, Place Aristide Briand, [:-92190 Meudon Principal Cedex, France

and Georges LOCHAK Centre National de la Recherche Scientifique, Fondation Louis de Broglie, 1, rue Montgolfier, F-75003 Paris, France

Received 28 July 1986 A new calculation of the magnetic moment of an atom is suggested on the basis of the hypothesis that the total angular momentum of one electron is the same in a complex atom situated in a solid as in a hydrogen-like atom described in Dirac's theory. The latter is first revisited and the quantum states thus defined are compared with those of SchrSdinger's theory. Then, the experimental basis of the notion of spin is recalled and compared to the subshell division of the p, d and f shells in Dirac's theory. Using this division in subsheUs a calculation of the magnetic moment is applied to the heavy rare earth metals, iron, cobalt, nickel and the chromium compounds and compared with experimental data. This leads us to a discussion of the Pauli exclusion principle and to the choice of a convenient electronic configuration of each magnetic element. Finally these configurations are compared to the theoretical magnetic moments.

1. Introduction T h e c a l c u l a t i o n of the m a g n e t i c m o m e n t of an a t o m is closely l i n k e d to o u r n o t i o n of spin a n d to the n u m b e r of q u a n t u m states. But the s t u d y of the e x p e r i m e n t a l results which led to the n o t i o n of spin a n d f u r t h e r to the D i r a c m o d e l d i s p l a y s for these two aspects s o m e i m p o r t a n t d i s c r e p a n c i e s when they are c o m p a r e d with the Schr/Sdinger m o d e l such as it is used n o w a d a y s in the t h e o r y of m a g n e t i s m . F o r this r e a s o n the p u r p o s e of this w o r k is to e x a m i n e h o w D i r a c ' s t h e o r y of the e l e c t r o n c o u l d enlighten the c a l c u l a t i o n o f the m a g n e t i c m o m e n t s at 0 K w i t h an a p p l i c a t i o n to the cases o f h e a v y rare e a r t h metals, c h r o m i u m c o m p o u n d s , iron, c o b a l t a n d nickel. I n our o p i n i o n it w o u l d b e w r o n g to restrict the role o f the D i r a c e q u a t i o n to the relativistic field of high energies a n d to r e d u c e its usefulness in the a t o m i c field to small c o r r e c t i o n s in o p t i c a l spectra. D i r a c ' s t h e o r y is a b l e i n d e e d to p r o v i d e such corrections, b u t m o r e o v e r , it is also the o n l y o n e w h i c h describes c o r r e c t l y the a n g u l a r m o m e n t u m o f an a t o m a n d which gives an exact n o m e n c l a t u r e of the m a g n e t i c states. M o r e precisely it is the only one which establishes the essential p h y s i c a l p o i n t that n e i t h e r o r b i t a l m o m e n t u m n o r spin are c o n s t a n t s o f m o t i o n a n d that o n l y the t o t a l a n g u l a r m o m e n t u m is a constant. This is n o t s i m p l y a relativistic c o r r e c t i o n which c o u l d b e a c c o u n t e d for b y the i n t r o d u c t i o n o f a s p i n - o r b i t c o u p l i n g in the Schr/Sdinger e q u a t i o n : it is a n o t h e r s t a t e m e n t o f the law of c o n s e r v a t i o n of the a n g u l a r m o m e n t u m which leads, for the shells n p , n d a n d n f to a structure in subshells which is different f r o m those o b t a i n e d b y associating s p i n - u p a n d s p i n - d o w n states with each state d e s c r i b e d b y the Schr/Sdinger equation. This structure is also different f r o m the one suggested b y the Pauli e q u a t i o n , in which the o r b i t a l m o m e n t u m a n d spin are s e p a r a t e l y c o n s t a n t s of m o t i o n , s u p p o r t i n g the s p i n - u p a n d - d o w n structure. 0 3 0 4 - 8 8 5 3 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

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Strictly speaking, the properties of the electron angular m o m e n t u m invoked above are proved only for an isolated atom and even in the hydrogen case and it seems obvious at a first glance that such an individual property of the electron cannot be conserved in a m a n y electron atom and afortiori in a crystal field. But precisely, our basic assumption will be that something remains of this individual property of the electron even in the m a n y electron cases, despite the strength of the perturbation. The fact that this is admissible will be suggested by the periodic classification as will be shown and, for a crystal, it will be supported by some curious and even striking results in the domain of magnetism, which will be given below. Let us remind that with the appearance of wave mechanics, the theory of magnetism was built with different models depending on the case considered. In the case of rare earth one admits on one hand a strong coupling between the different orbital m o m e n t a and on the other hand a coupling between the spins: this is the L - S coupling. The discrepancies between theory and experiment are attributed to an auxiliary contribution of the conduction electrons in pure metals [1,2]. For the compounds of the 3d metals it is mainly the spin only contribution which is generally considered [3]. The case of chromium compounds will show us further the limits of this approach. For iron, cobalt and nickel the band model is used to explain their respective moments [4,5]. So we asked ourselves about the necessity of different ways of interpretation for the 3d compounds, for the 4f metals and yet another one for iron, cobalt and nickel. Our attempt consists in asking if such a situation may be improved by applying the Dirac model to the study of magnetism. If, as we think, the conservation of the total angular m o m e n t u m is a clue property of the quantum state, it must be possible to find a calculation of the magnetic moment using this property for the 4f as for the 3d as well as their compounds and their metallic states. It must also be noticed that with the development of techniques, we have, since recently, at our disposal very good measurements of magnetic moments such as those of the heavy rare earths metals or chromium compounds. This situation affords probably for the first time the possibility to verify a same simple model in numerous cases. We have already alluded to the difficulty arising from the fact that contrary to the Schr~Sdinger one, the Dirac equation cannot describe a system of particles: this difficulty is inherent to every relativistic theory. Therefore the Dirac equation just describes correctly the hydrogen atom, and it seems that we shall meet with a number of problems for a complex atom. But we can argue that actually the periodic table hardly takes into account the interactions between the electrons in an atom. In particular, the structure in K, L, M . . . . levels and in ns, np, nd, nf shells is always established making the assumption that each electron of the atom behaves as an electron of a hydrogen like atom, that is, as if it was alone with the nucleus. The presence of other electrons and their behaviour as a whole is just involved in one point, fundamental indeed: the Pauli exclusion principle which is the corner stone of the structure in levels and shells. Thus as far as the Schr~Sdinger equation of a system of particles is not effectively used for the periodic classification *, we may without loosing generality replace the Schr~Sdinger equation of one electron by the Dirac equation. On the other side let us note that when, in agreement with other authors, we admit that it is possible to explain the periodic table using the quantum numbers (n, l) of hydrogen-like atoms, we neglect the ensemble coherency of the electrons into the atom. So when we just invoke here the conservation of the total angular m o m e n t u m of one electron and when we admit the structure in levels and shells, avoiding the structure in spin-up and -down, we work in a similar way. Inf act, we assume the hypothesis that: the coherence of the angular states of each electron is stronger than the ensemble angular coherency of the electrons of the atom. In other words the total angular m o m e n t u m of an electron is considered here as an inalienable property; so is, as a consequence, the projection on the direction of the magnetic field, a property that we shall make use of in the calculation of the magnetic moment. We have

* Another problem would be the calculation of the atomic spectral frequencies which obviously involves the interaction between electrons, but this is not our problem in the present paper.

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presently no general theoretical justification for such a hypothesis, except the experimental support that will be given in the following sections. In section 2 the problem of the quantum states structure in Dirac's equation, will be revisited in order to exhibit some features which do not appear in the conventional treatment of the problem. (Two appendixes are devoted to some technical aspects.) We shall compare in section 3 the atomic states given by the Dirac model to those which are given by the Schrtidinger one. In section 4 we shall briefly remind the experimental basis of the notion of spin and we shall see how the original model of two spin orientations is reflected in an angular momentum splitting in Dirac's model. In section 5 using the division of the shall 3d and 4f in subshell 3d 3/2, 3d 5/2, 4f5/2 and 4f7/2 we will give a first calculation of the magnetic moment, then we will discuss the Pauli exclusion principle and propose the electronic configuration of 3d and 4f elements. In section 6 we will compare the theoretical value of the magnetic moment with the electronic configuration of the corresponding element. Finally in section 7 we give some concluding remarks. Preliminary results of this work have been published before [6-9].

2. The structure of the atomic states in the Dirac equation

For the physicist or the chemist of the solid state the structure of atomic states is essentially known through the Schr6dinger model. This is why after giving the solution of the Dirac equation we shall compare them with those of the schr~dinger equation. Let us examine the problem of the atomic states of the hydrogen. The current integration of the Dirac equation, is actually based on the conception of spin-up and -down model. The solutions are chosen in order to exhibit and to couple up the values j = l - 1 / 2 and j + l + 1 / 2 of the total angular momentum. These couples of solutions are known as corresponding to the alkaline doublets (the Sommerferld doublets) which play an important role in magnetism and we shall go back to them. But this way of working presents the inconvenience of giving a central role to the orbital number l which is not a constant of motion, as was mentioned above. Furthermore, in this way, one brings together by an a priori reason couples of states (the components of alkaline doublets) which are split by a relativistic effect and which do not belong to the same energy level in Dirac's theory. Despite the vicinity of the components of relativistic doublets, this mixing of different energy levels in the same couple of solutions finally obscures the nomenclature of the energy states, which is essential for the periodic table of the elements. So we shall proceed in searching the energy levels and the corresponding degenerate states without any a priori. We~ shall see that each level corresponding to a given value of the total angular momentum j has two kinds of degeneratices: 1) on the one hand a 2 j + 1 order magnetic degenerency corresponding to the different possible values u of the projection of the total angular momentum on one direction of the space; 2) on the other hand appears a structure of degenerated doublets. For each value of j and u there are two states (with the same energy) labeled with a quantum number x, respectively equal to - ( j + 1/2) and + ( j + 1/2) for each of these two states. It is the screening degeneracy which already appears in the Schr~dinger equation, but which is much clear in the Dirac one. This degeneracy is slightly removed even in the hydrogen atom by a radiative effect (the Lambshift), but far more radically a splitting occurs in the many electron atoms, giving the so called screening-doublets. For this reason we suggest to call screening-number the quantum number x. So we hope to give a clearer nomenclature of the hydrogen states that will be introduced in the classification of the energy levels (section 3.3) and in the description of the electronic configuration of the elements (section 5). Before entering into the subject of this section it is of interest to remind in a few words how the Dirac equation was introduced [10,11]~ In 1928 the small energy difference of the alkaline doublets had been explained for a long time by Sommerfeld in 1916 as a relativistic correction in the framework of the old quantum theory. Some years later in 1925, appeared Uhlenbeck and Goudsmit's hypothesis of the electron

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spin. This is why several attempts were made, with only a partial success, in order to introduce relativity and spin separately in quantum mechanics (Klein, Gordon, Pauli, Darwin, etc.). The almost magic feature of Dirac's theory partly lies in the fact that he introduced only relativity, with a kind of principle of simplicity and he got the spin, so to say, as a bonus. He considered the relativistic relation between the rest mass m 0, the energy E and the linear momentum ~r of a particle that is: E 2 / c 2 = ~r2 + rn~c 2 where c is the velocity of light. His original idea was to search for such an expression that the quantities E and ~r do appear in a linear form instead of the quadratic ones E z and qr2. This requirement involves the three main aspects of the Dirac equation: 1) it is of the first order and not of the second one like the Schrtidinger and K l e i n - G o r d o n equations; 2) it brings matrices which are roots of unity; 3) the wave function will have not only one, but four components. Some more technical details are given in appendix A, owing to which it is easily found that, for a hydrogen-like atom, the Dirac equation takes the form of the following system: (ih O~-t- m°c2 + Z @ e 2+) i h c s ' g r X = O ' r

(ih -

Ze ) -~ + mo c2+ r x + ihcs " We~ = O.

(2.1)

s is written for the Panli matrices s 1, s 2, s3; r~ and X are two-component wave fonctions (see appendix A):

Thus the whole wave function is: =

(/)~

The angular momentum operator which is a constant of the motion according to the system of equations (2.1) (that is which commute with the Hamiltonian, see appendix B) is the sum of the angular momentum and of the spin:

where J is a square matrix of rank 2:

J = L + S,

L=-ir×v,

S = ½S,

(2.4)

L and S are not separately constants of the motion. Now we have to find the solutions of the system (2.1), but we shall do it in a slightly different way than the classical one [12,13] for the reason explained above. Let us write the stationary states: • (t, r) = e x p ( - i E T / h ) ~ ( r ) ,

X(t, r) = e x p ( - E T / h ) x ( r ) ,

(2.5)

with E > 0 for the whole relativistic energy. Eq. (2.2) becomes:

E - mo C2 -t-

Ze2

¢~(r) q- ihcs" V x ( r ) = O, (2.6)

( E+mOcz+ Ze2)

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According to the s y m m e t r y of the p r o b l e m we try, as in the case of Schr6dinger's equation, a solution where each function • and X is the p r o d u c t of a radial function and an angular function. In the Schrrdinger equation the angular functions are the spherical h a r m o n i c Ylm(~, ¢p) which are the eigenfunctions of the square angular m o m e n t u m L 2 and of the c o m p o n e n t L z. With x = r sin t~ sin rp, y = r sin ~ sin ~, z = r cos ~ the Y/" m a y be written as: y / , ( ~ , ~p)= ( - 1 ) " ( 2 1 + 2'1!

+m)!]l/2ei"~ dl-m

1)1/2[ (1 (1 -- m ) !

sint 0 dye,_"" sin2tO,

(2.7)

where l = 0, 1, 2 . . . . ; m = - l , - l + 1 , . . . , l - 1, 1. T h e phase choice is useful for what follows. In the case of the Dirac equation, the corresponding angular functions cannot be Yt", but t w o - c o m p o n e n t functions just like the wave functions ~b and X themselves. It is the so-called spherical functions with spin, denoted $2. They are defined as eigenfunctions of j 2 and Jz. But from (2.4) we have [j2, L 2 ] = 0 and it follows that they are also eigenfunctions of L 2. The c o m p o n e n t s of sa are thus p r o p o r t i o n a l to the YIm. One finds two possible forms (appendix B): 1 + m ] 1/2

sa?(+) =

y/m- 1

21+lJ

;

[ l - r1e/ 2+2I l]+ 1

,a?(-)

=

y m

(2.8)

[l+m11" - L 2~7-i-J

Y/~

T h e y respectively correspond to: J212?(+) =j(j+

1)sa~'(+)

with with

4saT'(+) = usa?(+)

j = l + 1/2, u=m-1/2,

(2.9)

with j = l - 1 / 2 , with u = m - 1 / 2 .

(2.10)

and

J2127(-) = j ( j + 1)Ia?(-) ) = usa?(- )

F o r both sets (2.9) and (2.10), the possible values of j and u are: j = 1 / 2 , 3 / 2 , 5 / 2 , . . . , u = - j , - j + 1..... j-l, j. Therefore I 2 ? ( + ) and s a ? ( - ) correspond respectively to the values j = l + l / 2 and j = 1 - 1 / 2 of the total angular m o m e n t u m j 2 . In other words they c o r r e s p o n d to the two possible ways of s u m m a t i o n of the orbital and spin angular m o m e n t a . But for given values of m and l sa?( + ) and 12~'(- ) correspond to the same value u = m - 1 / 2 of the projection of J on Oz. N o w it is i m p o r t a n t to count the n u m b e r of possible values of u in f o r m u l a e (2.8), (2.9), (2.10). In s a ? ( + ) , m takes the values - l, - 1 + 1 . . . . . 0 . . . . , l - 1, l, but it can also take the value l + 1, despite the fact that the function y / + l does not exist, because the corresponding coefficient ( l - m + 1) 1/2 in the first formula (2.8) cancels when m = 1+ 1. Therefore, saC+l(+) exists. Thus we find for u: 2 j + 1 = 2 1 + 2 different values. In I 2 ~ ' ( - ) m can only take the values - l + 1 , - 1 + 2 . . . . . 0 . . . . . l - 1 , 1. The value m = - I is forbidden because the fonction y S -1 does not exist and its factor does not cancel because (l m + 1) 1/2 = 1 =~ 0. The same remark stands for the value m = l + 1. Thus we find for u: 2 j + 1 = 21 different values. N o w we have to solve the system of equations (2.6). Let us search for solutions with the same total angular momentum that is with the same oalue j = l + 1 / 2 . There are two possible types of solutions: -

[~l=Gl(r)sa?(+) g" = t x , = i F l ( r ) I 2 ~ + , ( - )

. '

g,2 = { ~ = = G 2 ( r ) [ 2 ~ + l ( - ) X2 i F 2 ( r ) s a ~ ' ( + )

G ( r ) and F ( r ) are ordinarily scalar functions; the factor i is just in order that F ( r ) will be real.

(2.11)

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Let us remind that the set of values l and rn is the same one in `/1 and '/2. In comparison with the SchriSdinger model, we will see that "/1 belongs to the shell l and "/2 to the shell l + 1. This choice of solutions (2.11) brings together the quantum states giving the screening doublets, instead of the alkaline doublets. Let us introduce ~z and X / i n (2.6):

{

2

[E-moC2+ Zer ]Gl(r)f2t(W)-hcs.17[Fl(r)f2~+l(-)] =0 , (2.12)

[E + moc2 + rZez] F l ( r ) f ~ l ( - ) + hcs'v[Gl(r)~2?( + )] = 0 . A classical way is now to multiply the eq. (2.12) by the operator: 1 s'n = -s-r= r

[cos0 t sin 0 e 1~

s i n ~ e -i~°] . - cos

(2.13)

It is worth to note that:

[J, s.n]=O,

(s'n)2=I.

(2.14)

Thus s- n transforms each $2 with given eigenvalues j and u in another ~2 with the same eigenvalues j and u: s..OT(

+ ) =

(2.15)

+ ) +

where the coefficients A and B do not depend on m. They are calculated in appendix B and one finds: S*tl~'~n(q-) = ~'~+1(--);

S °n~'2~+1(-- ) = ~-~n(+).

(2.16)

Besides, a direct calculation gives: d

(s "n)(s" gr)

dr

1

(2.17)

r s "L.

In consequence, applying s "n to eq. (2.12) and taking into account that s ° L does not act on r, we obtain:

E - m o c2+ Ze2] r ] a l ( "r ) 1 2 ~ + l ( - ) - h c ~ t 2 ~ l

[E+m°c2+ Ze2] r Fl(r)

hC~-2~(-}-)

f27(+) +

d/dr does not act on fa and that

( - ) + h~CrFt(r)s'Lt2~l(- ) = O,

h¢ Gl(r)s.Lf~'~(+)=O" ---7

(2.18)

Furthermore from (2.4) and (B.5) in appendix B we have: = mr"(+),

s.Lo?+I(-)

= -(t+

(2.19)

f2~+1(- ) and ~2~(+) are thus factorized in (2.18) and we get the radial equations: dF -~-r +

l+2 r

FI-(

E - moc2 Za he + - G l )= 0 ' r (2.20)

dG~

dr

1GI+

r

hc

+ -

r

F1 = 0 ,

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105

with a = e2/hc. Introducing ~2 and X2 in (2.6), in the same way, we find:

IF2 r

dF

dr

E

-- m o c2

(

hc

Zot

+

) G2 =

(E+moc2Za) + r he

dG2 l+2 --~--r + - - Gr2 +

(2.21) F2=0.

The usefulness of our choice of solutions will now appear. Let us first introduce a usual notation but which provides here a particular symmetry. It is actually a new quantum number defined as:

x = T-(j+ 1 / 2 ) = ~ ( l + 1).

(2.22)

Where the minus sign corresponds, to the solution '/'1, and the plus sign to the solution '/'2. Therefore we have: 1-x=l+2 1-x=-l,

1+~=-I 1+r=l+2

for for

g'l; x/"2.

(2.23)

Then both systems (2.20) and (2.21) take the same form: dF 1- x -~-r + - - F - J r dG l+r -d--~-r+ r

E

mo c2 Zot hc + - - IG = 0 ' r [E+_moC2 _ ~ ] G+ [

--

+

hc

(2.24)

F=0.

With the two possible values (2.22) for x. Here let us note that K with its algebraic sign includes both types of spin. A classical calculation [12,13] then gives the solutions which correspond to the two kinds of solutions ~1X1 and ~2X2, according to the two sets of values of ~ (2.22): G=N[1

+ (2.25) _

m oc2

Xh

Where p = 0, 1, 2 . . . . is the radial quantum number,

A = _ ~cl (m2c4_ E2)1/2 ' y = ( x 2 _ a 2 Z 2 ) 1/2.

(2.26)

N is a normalization constant and L 2Y are the generalized Laguerre polynomials. In this work we essentially need to count up accurately the quantum states in order to be able to choose the ones which contribute to the magnetic properties of the solids. This is given by the expression of the energy and we thus do not need the explicit form of N and L2L At the nearest a 3 the energy Wnj is given by the following expression:

W,s = RhZ----~2 1 + n2

W n j = E - m o c2,

(2.27) ~

I~-I

Ix l = j + l / 2 ,

R is the Rydberg constant.

4

'

n=[x[+p=j+l/2+p,

(2.28)

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106

N o w let us note that both solutions defined by (2.11) have the same total angular m o m e n t u m and that they only differ by the sign of x = -T-(j + 1/2). The degeneracy is thus easy to find. Discussing the angular states given by (2.8) we have already found a 2 j + 1 degeneracy for each given value of j. N o w for each angular state there are two radial states corresponding to the two signs of K, except for the highest value j + 1 / 2 = n. Indeed in this case, from (2.28), we have p = 0 and the polynomial term L 2v- z does not exist. Therefore, the corresponding term must disappear in (2.25), which implies:

aZmoc/Xh + ~ = 0

for

[~1 = n,

(2.29)

and we see that, for p = 0 we must have x < 0. Hence only the value x = - ( j + 1 / 2 ) = - n is allowed, while the value x = n is forbidden. Finally, on each level E , j there are 2(2j + 1) = 4 ( j + 1 / 2 ) --- 4 1~ I states corresponding each value I x l = J + 1 / 2 = 1, 2, 3 . . . . . n - 1 and only 2 j + 1 = 2 ( j + 1 / 2 ) = 2 1x l for I K I = J + 1 / 2 = n. Then for a given quantum number n there are: n-1

4 ~

]xJ+2n--4

n ( n - 1) 2 +2n=2n

2 states.

(2.30)

[~l=Z Thus we find the classical result but more directly, only making use of the total angular momentum, i.e. of a conservative quantity and without counting more or less artificially introduced up and down spin states in order to explain the fact that the number of quantum states is twice the number given by the Schr~Sdinger equation.

3. Comparison with the Sehr6dinger equation To have a clear idea of the different quantum states of the Dirac equation, the best way is to compare them with the ones of the SchriSdinger equation. For this we make use of the nonrelativistic approximation of the states of the hydrogen-like atom. Let us introduce in (2.24) the notation E = moc2 + W noticing that I W I << moc2. Then E + moc 2 = 2m0 c2 and I F [ << [ G I. N o w eliminating the function F between the two differential equations of (2.24) we find: d2---~G+2dGdr 2 r~

+[ ~-2°h W+ 2Ze2m°lh2 r

~ ( ~G+21=) ] 0 " r

(3.1)

Let us write G = (1/r)g: we find again the familiar form of the Schr~Sdinger radial equation, but with the difference that x replaces l.

1 d2--~g+ [ 2 m 0 W + -2Ze2mo dr 2 [--7 h2 r

~ ( ~ + 1 ) ] g = 0. r2

(3.2)

Now we can compare Dirac's solutions to Schr~Sdinger's ones,

3.1. The screening doublets" Let us consider the two types of solutions defined by (2.11). They have the same angular m o m e n t u m j. As we can see the first degeneracy appearing with (2.27) correspond to the two possible signs of K for the same absolute value, that is the two kinds of solutions of (2.11). Let us express x as a function of l, in (3.2): For

~=-(l+1)

wehave

x(x+l)=l(l+l),

For

~=l+1

we have

x(x + 1) = ( l + 1 ) ( l + 2).

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107

Thus if the two kinds of solutions differ by their angular structures, they also differ by their radial functions even at the nonrelativictic approximation. Indeed the quantum number 1 corresponding to K = - ( l + 1) is replaced by l + 1 for the solutions corresponding to x = l + 1. In the nonrelativictic approximation the "fine components" X disappears and (2.11) becomes:

~1 ---~a l ( r ) ~'2~n( "~-) ;

(3.3)

~2 = a 2 ( r ) ~ ' 2 ~ + l ( - ) "

Thus for ~1 the subscript l in a2~'(+) corresponds to the classical subscript in spherical harmonic Yt" and to the shell l of the Schrrdinger model. In the case ~2 we find the subscript l + 1 and it corresponds to the shell l + 1 in the Schr/Sdinger model. N o w let us consider an atom with several electrons. For the preceding two different radial distributions these electrons screen the attraction of the nucleus in a different way. The degeneracy disappears, we get the "screening doublets" as the spectroscopists say [14]. For this reason we call the quantum number K the

screening quantum number. The splitting of this degeneracy appears even in a hydrogen-like atom as soon as the radiation retroaction and the vacuum polarization are taken into account: it is the Lambshift [15] which is observed as a removing of the degeneracy of the screening doublet 2Sl/2, 2Pl/2 with K = - ( j + 1 / 2 ) = - 1 for 2sl/2 and x = j + 1 / 2 = 1 for 2pl/2. But this splitting is far much smaller than the one which is due to the screening effect in the m a n y electron atoms.

3.2. The alkaline doublets Now, let us consider the states belonging to the same shell s, p, d, f, which will be conventionally labeled by the "orbital number" l. The components of an alkaline (i.e. relativistic) doublet correspond to two successive values of the total angular momentum: j = I + 1 / 2 and j = 1 + 3/2. In order to get the classical "spin-up and spin-down" nomenclature, we just need to take 1 - 1 in place of l and we find the doublet:

J =l-13, with j=l+½,

K= + ( j + ½ ) = l , with

x(x+l)=/(l+l),

x=-(j+l)=-(l+l),

x(x+l)=l(l+l).

The two components are effectively separated in the relativistic case and the lowest level is the one with K = l. But the splitting is of the order of a 2 (see (2.27)), so that at the nonrelativistic Schr~Sdinger approximation, the alkaline doublet is degenerated, as is easily seen in eq. (3.2), due the fact that we have just shown that K(~ + 1 ) = l(l + 1) for both components. At this approximation the "fine components" X disappear and only the "gross components" do remain:

~1 = Gl(r)a27'(+),

~2 = G 2 ( r ) / a T ' ( - ) -

(3.4)

Let us remind that according Dirac's equation the number u of different quantum states on the same subshell is: v=21

for

K=I,

u = 2l+ 2

for

r = -l-

1,

i.e. for

~2,

i.e. for

~z,

so that on the whole shell, we have r = 4l + 2 quantum states.

108

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

Table 1 The energy levels of a hydrogen-like atom according to the quantum numbers (n, l, j )

2sl/2(2, 0, 1/2) 2pl/2(2, 1, 1/2) 2p3/2 (2, 1, 3/2)

lsl/2(1, O, 1/2)

3sl/2(3, 0, 1/2) 3pl/2(3, 1, 1/2) 3p3/2(3, 1, 3/2) 3d3/2(3, 3, 3/2) 3d5/2(3, 2, 5/2)

4sl/2(4, 0, 1/2) 4pl/2(4, 1, 1/2) 4p3/2(4 , 1, 3/2) 4d3/2(4 , 2, 3/2) 4d5/2(4, 2, 5/2) 4f5/2(4, 3, 5/2) 4f7/2(4 , 3, 7/2)

Table 2 The energy levels of a hydrogen-like atom according to the quantum numbers (n, ~, j). We have indicated the alkaline doublets (j, j + 1) with braces and the screening doublet :g ( j + 1/2) with brackets

lsl/2(1, - 1, 1/2)

2sl/2(2, - 1, 1/2)] 2pa/2(2, 1, 1/2) _] 2p3/2(2, - 2, 3/2)

3sl/2(3, - 1, 1 / 2 ) ] f 3pl/2(3,1,1/2) J 3p3/2(3, - 2, 3/2) ] [ 3d3/2(3, 2, 3/2) J I, 3d5/2(3 , - 3, 5/2)

4Sl/2(4, -- 1, 1/2) ] 4pl/2(4, 1, 1/2) 4p3/2(4, -- 2, 3/2) ] 4d3/2(4, 2, 3/2) 4d5/2(4, - 3, 5/2)] 4f5/2(4, 3, 5/2) J 4f7/2 (4, - 4, 7/2)

3.3. The nomenclature of the levels

The comparison between levels respectively given by Dirac's and SchriSdinger's models gives the opportunity of a remark about the nomenclature of the energy levels. We have seen the important role plaid by the screening quantum number K which defines the two types of solutions of Dirac's equation. We may use it in the classification of levels in place of l. Table 1 gives the classical nomenclature with (n, l, j ) and table 2 with (n, x, j).

4. The notion of spin The hypothesis of spin was proposed by Uhlenbeck and Goudsmit in 1925 [16,17] in order to explain the existence of alkaline, or relativistic, doublets. Consider for example in alkaline metals the sharp and principal series. Let v~ and v~ be the frequencies of the respective transitions of these two series, we have: < = R / ( 1 + p ) 2 _ R / ( n + S ) 2, vp = R / ( 1

+

S) 2-

R/(n

+

p)2,

(4.1)

R is the Rydberg constant, n the principal quantum number. The term including n is the running term the other one the fixed term. S and P are two constants. The transition takes place between the shells s and p but in place of obtaining just one line, two lines are observed, thus there are two values of v~ very close and also two very close values v~p. The variation of the frequency splitting of these two lines for sharp and principal series of caesium is plotted in fig. 1.

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

109

[ Princi~ <

v

:

I !

1

!

I

i

I

,I

'

] Sharp

__.,I .... L

Fig. 1. Frequencyplot of the doublet fine structure in the principal and sharp series of cesium (White 1934) [14].

This variation shows an important point: the level p is double while the level s is single. Indeed when the electron goes back from np to s in a principal series the running term R/(n + p)2 diminishes when n increases, so if p is double the frequency splitting is smaller and smaller, furthermore as s is single we just observe two lines. On the reverse, when the electron is going back from ns to p the running term R/(n + S) 2 is single, but the fixed term R/(1 + p)2 is double so the splitting is constant. The study of the diffuse and fundamental series show equally the double character of the nd and nf levels (see White [14], de Broglie [18]). Now we have to identify the running and fixed terms with the eigenstates of the hydrogen-like atom. We have seen that inside the same shell l the solutions of the Dirac equation corresponding to the two lines of an alkaline doublet are characterized by the value x = - l - 1 and x = l. The corresponding levels (relation (2.27)) are splitted with a term of the order of a 2. Thus there are precisely double terms np, nd and n f in the Dirac model. They correspond respectively to the values l = 1, l - - 2 and 1--3 in the Schr/Sdinger model. Now, let us consider the case l = 0. We have seen with the screening doublets that the solution =-1-1 belongs to the shell l of the solutions of Schr/Sdinger's equation, since the solution with x = l + 1 belongs to the shell l + 1. The levels ns are thus single, as one can see in table 2: indeed the second level associated to x = - 1 with an s state would correspond to K = 0. Such a value is forbidden because [K[ = j + 1 / 2 and j > 0 since j is a total angular momentum. Thus, according to the relativistic corrections, the Dirac model correctly accounts for the doublet levels. Besides let us note that both kinds of spin characterized by the two lines of a doublet are observed without application of a magnetic field. The Zeeman effect just removes the u degeneracy of each level and allows to verify that the number of states u of each level is effectively the one given by the Dirac theory, but the doublet structure pre-exists to the application of the field. Since the spin is the hypothesis which explains the doublet series we must consider that both kinds of solutions inside the same shell np, nd and nf correspond to both types of spin. There are the two kinds of solutions that give what we call in the following the subshells: Pl/2, P3/2, d3/2, d5/2 and f5/2, f7/2" Now if we consider the states ns they correspond to just one kind of solution that is: l = 0, x = - 1 . We must admit that they correspond to only one type of spin. It must be pointed out that in such an interpretation there are different numbers of the two types of spin inside the same shell. This number is equal to 2l on the first subshell and to 21 + 2 on the second one. Thus it seems difficult to keep the idea, attractive indeed, but too simple, according to which the two subshells would correspond respectively to symmetric states "spin-up" and "spin-down". In reality the writing j = l - 1 / 2 and j -- 1 + 1 / 2 which suggests this structure is somewhat misleading and should be regarded cautiously from the point of view of Dirac's theory. As we have seen, L and S are quantities which are not conserved and which canr/ot be arbitrarily added or substracted. Only one thing may be asserted: the two subshells correspond to two values j and j + 1 of the total angular momentum.

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

110

5. The magnetic moment of an atom

The magnetization M of a substance arises from the magnetic moments/~ of its atoms. To avoid any ambiguity we shall consider substances with just one kind of magnetic atoms. The magnetization of a substance is anisotropic, so we only consider measurements on a single crystal along the direction of easy magnetization, when such measurements are available. In a high magnetic field, for a temperature close to 0 K, we can write: M=M0+XH,

(5.1)

where X is a susceptibility. At 0 K, for one kind of atom and for a gram atom M 0 = N/,, where N is Avogadro's number and /~ the atomic magnetic moment. Thus the measurement of the magnetic moment of an atom is extrapolated to absolute zero temperature and zero internal magnetic field.

5.1. The experimental results The elements which most generally exhibit magnetic properties are those which correspond to the filling of the shells 3d or 4f, called 3d transition metals or rare earths metals. We shall tackle just these two cases. Besides as it is easier to prepare pure metal than a compound or an alloy we shall first consider measurement for pure metals for which the best experimental values are available. The alloys or the compounds have often been less investigated and furthermore it is often difficult to avoid departure from the stoichiometry, so we shall limit this work to the cases of chromium compounds which is specially interesting in the framework of Dirac's theory. The experimental values #ex of the magnetic moment of the heavy rare earth metals are given in fig. 2 and in table 3 with appropriate references. These works have also been cited in the review papers of McEwen 1978 [1] and Legvold 1980 [2]. All these measurements have been made on single crystals. They have been reported at different times from different laboratories. The coherence between these different measurements is very good. The discrepancy is often less than 1%. In fig. 2 the experimental values are compared with values from L - S model and with the theoretical /~th that we shall determine (see table 6). The magnetic moments of iron, cobalt and nickel are known since long: as early as 1926 and 1928 on single crystals by H o n d a and K a y a [19,20] on polycrystals in 1929 for iron and nickel by Weiss and Forrer [21]. These measurements have been repeated ever since with increasing accuracy. Pauthenet in 1983 has

7 I

Gd I

!

8

IlJl

Er

I

HB

!

1

Dy

10

I

1.71

Tb

Ho

9

I

"i

III ;.oo I ,,I

i 0.43 i"

,~,,£~ i

;,[ii III

Tin41 Fig. 2. The magnetic moment of the heavy rare earths in pure metal. The experimental values are marked with a dash J. The theoretical values /,~th are marked by a dashed vertical line. The vertical arrows $ indicate the product gJ. The horizontal arrows show the contributions which appear between two successive elements. They correspond to contributions of precise quantum states. The references are given in table 3.

111

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

Table 3 The magnetic moment of heavy rare earth metals. The experimental values /~x and the values of the product gJ given in Bohr magneton per atom Gd

Tb

Dy

Ho

Er

Tm

#

ref.

/~

ref.

#

ref.

/~

ref.

/t

ref.

#

ref.

7.55 7.39 7.63

8 4,5 10

9.34 9.28 9.33

6 5 11

10.20 10.4 10.20

3 5 1

10.34 10.22 10.20

12 5 2

8.00 8.05 8.00

2 5 7

7.13

9

7.00

9.00

10.00

10.00

9.00

7.00

[1] D.R. Behrendt, S. Legvold and F.H. Spedding, Phys. Rev. 109 (1958) 1544. [2] R.M. Bozorth, A.E. Clark and J.H. Van Vleck, Intern. J. Mag. 2 (1972) 15. This paper published the curves, the values are given in McEwen (1978) ref. [1] of the text. [3] G.J. Cock, thesis, University of Amsterdam (1976). [4] J.L. Feron and R. Pauthenet, C.R. Acad. Sc. B269 (1969) 549. [5] J.L. Fero, G. Hug and R. Pauthenet, Les 616ments des terres rares, Tome II (Centre National de la Recherche Scientifique, Paris, 1970) p. 19-30, in french. See also: Z. Angew. Phys. 30 (1970) 61 (in german). [6] R.W. Green, S. Legvold and F.H. Spedding, Phys. Rev. 122 (1961) 827. [7] D.E. Hegland, S. Legvold and F.H. Spedding, Phys. Rev. 131 (1963) 158. [8] H.E. Nigh, S. Legvold and F.H. Spedding, Phys. Rev. 132 (1963) 1092. [9] D.B. Richards and S. Legvold, Phys. Rev. 186 (1969) 508. [10] L.W. Roeland et al., J. Phys. F 5 (1975) L233. [11] L.W. Roeland, G.J. Cock and P.-A. Linghrd, J. Phys. C 8 (1975) 3427. [12] D.L. Strandburg, S. Legvold and F.H. Spedding, Phys. Rev. 127 (1962) 2046.

published a very detailed study on single crystals between the temperature of liquid helium and room temperature [22]. The measurements of this author given with the thousandth of Bohr magneton are those to which we compare our theoretical results fig. 3. This paper also gives a large bibliography on the topic. At the beginning of quantum mechanics, it was surprising to discover that the magnetic moment of these elements does not correspond to an integer number of Bohr magnetons. In the Schr~dinger model the number of single electrons is 4 for iron, 3 for cobalt and 2 for nickel and people were expecting to find the same number of Bohr magnetons (Stoner [4]). As it is pointed out by Wohlfarth [5] the interpretation of these results and their complexity gave rise to a great controversy mostly upon the localized or itinerant character of the electron carrying the magnetic moment.

0

1

2

I

~a

J

.t

2.20

F:e

J 1 . 80

Co INli

~

0.60 !

=

=

I

I

Fig. 3. The magnetic moment of iron, cobalt and nickel metals. The experimental values are manked with a dash J ref. [22]. The theoretical values #th are marked by a dashed line, see table 6.

2.4

2.8

3,2

I

I

I

Rb2CrCl4 CsCrCl3

L

CfN

IL J-

ZnCr2 Se4 CdCr2Se4 HgCr25e 4 CrTe RbCrl3

• Pa

I

I JJ

I

II I L

JL

Fig. 4. Chromium magnetic moments in some compounds. Experimental values from magnetization p and from neutron diffraction studies L. The numbers on the scale are the theoretical values.

112

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

Table 4 Chromium magnetic moments in some compounds. Numerical values from magnetization and neutron diffraction studies used in the fig. 4 and references

Rb2CrC14

CsCrC13

/~

Ref.

3.30 3.30 3.10

1 2 3

3.16

4

~

Ref.

ZnCr2Sen

2.79 2.84

5 6

CrTe

CdCr2Se4

2.80 2.81 2.82

7 8 9

RbCrT3 CrN

HgCr2Se4

/~

ReL

2.39 2.45 2.40 2.40 2.38 2.40

9 10 11 12 13 14

[1] K. Anthony et al., J. Chem. Soc., Dalton Trans. (1975) 1306. [2] M.J. Fair et al., Physica 86-88B (1977) 657. [3] G. Mi~nninghoffet al., J. de Phys. 43-C7 (1982) 243. [4] H.W. Zandbergen and D.J.W. Idjo, J. Solid State Chem. 34 (1980) 65. [5] R. Plumier, Thrse Pads (17-12-1968) p. 134. ]6] M. Nogues, our laboratory private communication, /~= 2.84~B on single crystal. [7] K. Menyuk et al., J. Appl. Phys. 37 (1966) 1387. [8] P.K. Baltzer et al. Phys. Rev. 151 (1966) 367. [9] C. Guillaud and S. Berbezat, C.R. Acad. Sci., Paris 222 (1946) 386. [10] F.K. Lotgering and E.W. Gorter, J. Phys. Chem. Solids 3 (1957) 238. [11] G.I. Makovetskii and G.M. Shakhlevich, Kristall und Tecknik 14 (1979) 97. [12] H.W. Zandbergen and D.J.W. Idjo, J, Solid State Commun. 24 (1977) 487. [13] M. Nasr-Eddine and E.F. Bertaut, Solid State Commun. 24 (1977) 487. [14] M. Nasr-Eddine and M. Roubin, Solid State Commun. 32 (1979) 953.

Fig. 4 a n d table 4 give the experimental values of the m a g n e t i c m o m e n t /~ of several c h r o m i u m c o m p o u n d s . The references are given in table 4. W e have i n c l u d e d some a n t i f e r r o m a g n e t i c c o m p o u n d s the m o m e n t s of which have b e e n d e t e r m i n e d b y n e u t r o n diffraction. F o r the halogenides of c h r o m i u m a n d alkaline Rb2CrC14, CsCrC13 a n d R b C r I 3 the m e a s u r e m e n t s have b e e n p e r f o r m e d o n single crystals. F o r the selenide spinels MCr2Se 4 with M = Z n , Cd or H g the m e a s u r e m e n t s have b e e n p e r f o r m e d o n polycrystals except for Z n C r 2 S e 4. F o r the c o m p o u n d s C r T e a n d C r N the m e a s u r e m e n t s have b e e n p e r f o r m e d o n polycrystals. F o r all these c o m p o u n d s the o x y d a t i o n n u m b e r is II or III. A c c o r d i n g to the classical spin only i n t e r p r e t a t i o n the m o m e n t /x m u s t be, respectively, 4/~ B or 3/~B, the theoretical values that we shall propose are: 3.2/~ B, 2.8~ta or 2.4/x B which, as we show, are in good agreement with e x p e r i m e n t a l data. Let us n o w e x a m i n e how the calculation of the m a g n e t i c m o m e n t / ~ is possible o n the basis of Dirac's theory. 5.2. The calculation of the magnetic moment I n the i n t r o d u c t i o n we have plotted out the fact that the q u a n t u m i n t e r p r e t a t i o n of the periodic table is b a s e d u p o n the q u a n t u m n u m b e r s of the hydrogen-like atom, as to say neglecting the electronic interaction. The significance of this i n t e r p r e t a t i o n suggests that p e r h a p s at least o n e q u a n t u m p r o p e r t y of the i n d i v i d u a l electron subsists in a n a t o m with several electrons. As we have indicated in the i n t r o d u c t i o n we make the hypothesis that this p r o p e r t y is the total a n g u l a r m o m e n t u m . If it is the case we can suppose that the m a g n e t i c m o m e n t of each electron is equally retained. Let us t h e n consider the expression giving the energy level in Z e e m a n effect [11,14,18]: x eh , W,j~, = W°.k + u ~ + 1 / ~ 2moC H .

(5.2)

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

113

Table 5 The different quantum states for the shells nd and nf. The screening qua nt um number x. The numbe r m which defines the wave function ff'l or '/'2 through ~2~' and (2~+1. The total observable angular mome nt um u in h units and the corresponding magnetic m oment # = gu. d 3 / 2 : g = 2, g = 4 / 5 m

2

1

u

3/2

1/2

~=gu

1.2

0.4

'd5/2: x = - 3 , g = 6/5 0

-1

3

2

1

-1

-2

-i/2

-3/2

5/2

3/2

1/2

-1/2

-3/2

-5/2

-0.4

-1.2

3

1.8

0.6

-0.6

-1.8

-3

f5/2: ~ = 3 , g = 6 / 7 m

3

2

1

u

5/2

3/2

i/2

2.14

1.29

0.43

= gu

0

f7/2: K = - 4 , g = 8 / 7 0

-1

-2

4

3

2

1

-1

-2

-3

-1/2

-3/2

-5/2

7/2

5/2

3/2

1/2

-1/2

0

-3/2

-5/2

-7/2

-0.43

-1.29

-2.14

4

2.86

1.71

0.57

-0.57

-1.71

-2.86

-4

For a hydrogen-like atom W ° , is given by (2.27). If it is a complex atom in a solid we do not know W ° , but this term does not contribute to the magnetic moment of the electron. We need only the second term with u and x for the calculation. According to our hypothesis we shall keep for u and x the expressions (2.9), (2.10) and (2.22), given for the atom of hydrogen: u=-j,

-j+l

..... j-,

j

and

x=-T-(j+I/2)=

T(I+I).

With the projection uh of the total angular m o m e n t u m is associated the magnetic m o m e n t /~e:

#c= ugl% with

g= K/(x+ 1/2)

and

/~B = h / 2 m 0 c ,

(5.3)

where g is the Land6 factor of the hydrogen atom and #B the Bohr magneton. Let us recall that according to (2.9) and (2.10) u = m - 1/2. Now, the Land6 factor is defined as a function of x alone. As a consequence in the same shell there are two different g factors that is: gl = 4 / 5 and g2 = 6 / 5 for the shells nd, gl -- 6 / 7 and g2 = 8 / 7 for the shells nf. All the data needed for the calculation of the magnetic m o m e n t of an atom are given in table 5. These data are the quantum number m which determines the projection u of the total angular momentum; u in h units, the g factors and the magnetic moment #c = gu of each quantum state relation (5.3). By our hypothesis the magnetic moment of an atom will be the sum of the magnetic moment of each electron which contributes to the amount:

/~th = • ~e"

(5.4)

The problem will be to determine what are these electrons. Inside the same shell the coherency of our interpretation will be tested by the coherency with which we shall be able to determine the electronic configuration itself. Nevertheless as a first step without taking care of the electronic configuration it is, interesting to remark that the relation (5.4), taking into account the relation (5.3), can always be written as: /Ath = 1F/lg I q- ln2g 2.

(5.5)

In this expression the numbers n~ and n 2 are necessary intergers. The subscript 1 or 2 corresponds to the two subshells and the factor 1 / 2 arises from the fact that each value of u is always half an interger. The values of # calculated in this way are compared with the experimental values in table 6. The accuracy is often better than 1%. It also appears in figs. 2, 3 and 4. Now we have to determine the electronic configuration corresponding to n, and n 2, but we thought it was important at first to give prominence to the success of this formula.

X. Oudet, G. Lochak // Total angular momentum and atomic magnetic moments

114

Table 6 1 Comparison between the calculated values of g according to the formula (5.5) g = ~nlg 1 + i1n 2 g 2 with the experimental values of iron, cobalt, nickel, chromium compounds and heavy rare earths metals Fe gl = 4/5 g2 = 6//5

nI n2 g th P'exp

gl = 6/7 g2 = 8/7

nl n2

1 3 2.200 2.226 Gd 11 5

Co 0 3 1.800 1.729 Tb 11 8

Ni 0 1 0.600 0.619 Dy 12 9

CrTe 6 0 2.40 2.41 Ho 12 9

ZnCr2Se4 7 0 2.80 2.79 Er 12 5

Rb2CrC14 8 0 3.20 3.30 Tm 10 5

½nlgl ½n292

4.71 2.86

4.71 4.57

5.14 5.14

5.14 5.14

5.14 2.86

4.29 2.86

P"th

7.57

9.29

10.29

10.29

8.00

7.14

gex

7.55 7.39 7.63

9.34 9.28 9.33

10.20 10.4 10.20

10.34 10.22 10.20

8.00 8.05 8.00

7.13

5.3. T h e m a g n e t i c c o n t r i b u t i o n a n d the P a u l i e x c l u s i o n p r i n c i p l e

N o w we w a n t to p r o p o s e a m o r e c o m p l e t e i n t e r p r e t a t i o n of the m a g n e t i c m o m e n t built u p o n the electronic c o n f i g u r a t i o n s of the atoms. I n this view w e are l e d to a d m i t t h a t a c o n f i g u r a t i o n of filled up subshell is n o t in c o n t r a d i c t i o n with the existence of a m a g n e t i c m o m e n t . I n a m o r e restrictive w a y let us c o n s i d e r two electrons which, in a h y d r o g e n - l i k e a t o m w o u l d have equal b u t o p p o s i t e m a g n e t i c m o m e n t s . W e are led to consider, t h a t two such electrons in a c o m p l e x a t o m b e l o n g i n g to a solid can c o n t r i b u t e to the o b s e r v a b l e m a g n e t i c m o m e n t . This h y p o t h e s i s j u s t c o n c e r n s the 3d a n d 4f shells d u r i n g their filling. M o s t c e r t a i n l y the P a u l i exclusion p r i n c i p l e in b o t h cases a p r i o r i implies that there is n o m a g n e t i c c o n t r i b u t i o n . If we c h o o s e to a d o p t a different p o s i t i o n it is b e c a u s e the a g r e e m e n t b e t w e e n our c a l c u l a t i o n a n d the e x p e r i m e n t l e n d s it s u p p o r t . O n e c a n try to p r o p o s e qualitative s u p p o r t s to this hypothesis. W e k n o w that the m a g n e t i c m o m e n t a s s o c i a t e d to an e l e c t r o n is o b s e r v a b l e in the d i r e c t i o n of the m a g n e t i c field. I n a h y d r o g e n atom, the q u a n t u m states d e f i n i n g a subshell, c a n always b e g r o u p e d b y c o u p l e s in such a w a y t h a t their respective m a g n e t i c m o m e n t s are equal b u t opposite, (see t a b l e 5). Let us call A, B such a couple. Let us n o w c o n s i d e r in a c o m p l e x a t o m two o c c u p i e d states c o r r e s p o n d i n g to a c o u p l e A, B. T h e m a g n e t i c n e u t r a l i t y of such a c o u p l e of electrons is o b t a i n e d o n l y if they are b o t h s u b j e c t e d to the s a m e m a g n e t i c field. This is generally a d m i t t e d s u p p o s i n g at the s a m e t i m e the h o m o g e n e i t y of the m a g n e t i c field a n d a whole d e l o c a l i z a t i o n of the electrons in the atom. But it will n o t b e always the case b e c a u s e s t r o n g m a g n e t i c g r a d i e n t s in the crystal field c a n involve a m o r e or less i m p o r t a n t localization. Such a p o s s i b i l i t y is precisely t a k e n here into account. Let us then c o n s i d e r two q u a n t u m states A, B a n d let b e ]ge I the a b s o l u t e value of their m a g n e t i c m o m e n t . W h e n these are o c c u p i e d we shall s u p p o s e t h a t they c a n c o n t r i b u t e either 0 l g e I or 2 1 g ~ [. S u p p o s e n o w t h a t for o n e of these two electrons we h a v e an a n t i f e r r o m a g n e t i c structure. T h e n o n l y the o t h e r electron b r i n g s its c o n t r i b u t i o n . A s a result three cases are possible; these two electrons b r i n g a c o n t r i b u t i o n of 0 I / ~ [, l l/x ~ [ o r 2 Ig~ 1- O w i n g to these three p o s s i b l e values we shall o b t a i n a c o h e r e n c y b e t w e e n the theoretical values gth a n d the electronic configurations.

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

115

5. 4. T h e electronic configurations

Let us c o n s i d e r s o m e properties of rare e a r t h e l e m e n t s w h i ch will guide us in the choice of the electronic configurations. T h e rare earths are k n o w n to exhibit very similar c h e m i c a l properties. T h e y h a v e all the o x y d a t i o n n u m b e r I I I a n d o f te n only that. But in spite o f this h o m o g e n e i t y it is possible to distinguish two groups: the ceric or light rare earths, the yttric or h e a v y rare earths. This p r o p e r t y is easy to u n d e r s t a n d if o n e admits that the two subshells 4f are successively filled up the first b ei n g the 4f5/2 subshell with six q u a n t u m states. If as we think the shell 4f is filled in this way, s a m a r i u m with six 4f electrons has the first subshell filled up an d the second empty. T h e n o n e is al l o w ed to think that such a c o n f i g u r a t i o n is in a c e rta in way similar to a filled up shell. In such a c o n d i t i o n the o x y d a t i o n n u m b e r II o f s a m a r i u m is to be c o m p a r e d with the o x y d a t i o n n u m b e r II of zinc, c a d m i u m or y t t e r b i u m . The s e r e m a r k s on the rare earths p r o p e r t i e s suggest the h y p o t h e s i s that effectively the 4f5/2 an d 4f7/2 subshells are filled up successively. In a n a l o g y we shall su p p o se that it will be so for the 3d3/2 an d 3d5/2 subshells. Besides into the same subshell we s u p p o s e that the first o c c u p i e d q u a n t u m state is the o n e with the highest value of u, then the second o c c u p i e d q u a n t u m state is the highest a m o n g the r e m a i n i n g q u a n t u m states a n d so on. A m o n g the rare earth e l e m e n t s the case of l a n t h a n u m has to be specified. F o r this e l e m e n t the a d d i t i o n a l e l e c t r o n as c o m p a r e d to b a r i u m is g e n e r al l y s u p p o s e d to be a 5d electron. Actually, o n e has to s u p p o s e that this electron o c c u p i e d a 4f state as has b e e n r e c e n t l y p r o p o s e d by o n e o f us [6,8,9]. A m o n g the different a r g u m e n t s w h i c h s u p p o r t this hypothesis let us recall that it avoids to keep two e l e m e n t s La a n d Lu as h o m o l o g o u s 5d i of the e l e m e n t s Sc a n d Y. Such a situation is in c o m p l e t e c o n t r a d i c t i o n i n d e e d with the driving c o n c e p t which led M e d e l e i e v to the d i s c o v e r y of the p e r i o d i c table.

Table 7 The different 4f quantum states according to the two subshells: 4fs/2, 4f7/2. The number m which defines the wave function 'P. The total observable angular momentum u in h units and the corresponding magnetic moment /Le = gu, in Bohr magnetons, The Ln elements with their corresponding number u of 4f electrons. The place of the Ln elements is such that the additional electron is supposed occupied the quantum state of the column 4f5/2: x = 3, g = 6/7 m u Ln

3 5/2 2.14 La 1

2 3/2 1.29 Ce 2

1 1/2 0.43 Pr 3

4f7/2:

0 -1/2 -0.43 Nd 4

-1 -3/2 -1.29 Pm 5

-2 -5/2 -2.14 Sm 6

4 7/2 4.00 Eu 7

x =

- 4 , g = 8/7

3 5/2 2.86 Gd 8

2 3/2 1.71 Tb 9

1 1/2 0.57 Dy 10

0 -1/2 -0.57 Ho 11

-1 -3/2 -1.71 Er 12

-2 -5/2 -2.86 Tm 13

-3 -7/2 -4.00 Yb 14

Table 8 The different 3d quantum states according to the two subshells 3d3/2, 3d5/2. The number m which defines the wave function '/'. The total observable angular momentum u in h units and the corresponding magnetic moment #e = gu, in Bohr magneton units. The 3d elements M with their corresponding number u of 3d electrons. The place of the 3d elements M is such that the additional electron is supposed occupied the quantum state of the column d3/2: x = 2, g = 4/5 m u /~ M v

2 3/2 1.2 Sc 1

1 1/2 0.4 Ti 2

d5/2: g = - 3 , g = 6/5 0 - 1/2 - 0.4 V 3

-1 - 3/2 - 1.2 Cr 4

3 5/2 3 Mn 5

2 3/2 1.8 Fe 6

1 1/2 0.6 Co 7

0 - 1/2 - 0.6 Ni 8

-1 - 3/2 - 1.8 Cu 9

-2 - 5/2 - 3 Zn 10

116

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

4f

mognetic

stotes m

Lo

3

Ce

2

[]

Pr

E

Nd Pm

0

Sc

-l

Ti P

-2

Sm 4

2

0

3 d

1

2

Pa

magnetic

states

m

[]

,

[]

o

Cr

Eu

4

Gd

3

Tb

2

I

3 Mn

Dy

1

Fe

Ho

0

Co

Er

-1

Tm

-2

Yb

-3

Fig. 5. The 4f quantum states defined through the number m which defines the wave function q'. The lengths of the different horizontal bars are proportional to the magnetic moment of the quantum states. The 4f element for which the quantum state is occupied for the first time. The elements going from La to Sm are those corresponding to the first subshell, those going from Eu to Yb correspond to the second subshell,

Ni

I 2

I 1

I 0

t I

loooooooo .....

ooooooo

I 2

Ib 3

ooooooo

]

o ......

E ....... ~

3

2

o

Co Zn Fig. 6. The 3d quantum states defined through the number m which defines the wave function '/'. The length of the different horizontal bars are proportional to the magnetic moment of the quantum states. The 3d element for which the quantum state is occupied for the first time. The elements going from Sc to Cr are those corresponding to the first subshell, those going from Mn to Zn correspond to the second subshell.

T h e set of electronic configurations to which we are led are recapitulated i n tables 7 a n d 8. Figs. 5 a n d 6 describe the m a g n e t i c characteristics of these configurations.

6. Magnetic moments and electronic configurations F r o m the subshell structure a n d the respective c o n t r i b u t i o n s of the q u a n t u m states we have b e e n able to p r o p o s e a calculation of the magnetic m o m e n t ~th (relationship (5.5), fig. 2 a n d table 6) close to the e x p e r i m e n t a l values. N o w we have to show how these values /xtn can be explained b y the electronic configurations. Let us start with the case of the heavy rare earths. Gadolinium. This element, as we have p o i n t e d out i n the previous section table 7, w o u l d have eight 4f electrons, thus: six in 4f5/z a n d two in 4f7/2 subshells. Therefore if we consider only the two 4f7/2 electron c o n t r i b u t i o n , it would give a n a m o u n t of 4.00 + 2.86 = 6.86~B, which is inferior to the experimentally observed value 7.5/~ B. Therefore one should envisage a m a g n e t i c c o n t r i b u t i o n from electrons i n 4fs/2 subshell also. W e explain the G d m o m e n t as follows. T h e c o n t r i b u t i o n from 4f7/2 is only 2.86/~ B a n d the 4.71~t B arises from the subshell 4f5/2, the way this c a n b e o b t a i n e d is shown o n table 9. I n this case there are two ways it m a y be done. It would be interesting to d e t e r m i n e which one is responsible for the

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

117

Table 9 The different possible magnetic contributions from the subshell 4f5/2 with g = 6/7. The corresponding values for the subshell 4f7/z with g = 8 / 7 4f5/2

0.43 0.86 1.29 1.71 2.14 2.57 3.00 3.43 3.86 4.29 4.71 5.14 5.57 6.~ 6.43 6.86 7.29 7.71

4f7/2

0.5=0.5 1,0=0.5+0.5 1,5=1.5 2.0=1.5+0.5 2.5=2.5orl.5+0.5+0.5 3.0=2.5+0.5orl.5+1.5 3.5=2.5+0.5+0.5orl.5+1.5+0.5 4.0=2.5+1.5orl.5+1.5+0.5+0.5 4.5=2.5+1.5+0.5 5.0=2.5+1.5+0.5+0.5or2.5+2.5 5.5=2.5+2.5+0.5or2.5+1.5+1.5 6.0=2.5+2.5+0.5+0.5or2.5+1.5+1.5+0.5 6.5=2.5+1.5+1.5+0.5+0.5or2.5+2.5+1.5 7.0=2.5+2.5+1.5+0.5 7.5=2.5+2.5+1.5+0.5+0.5 8.0=2.5+2.5+1.5+1.5 8.5=2.5+2.5+1.5+1.5+0.5 9.0=2.5+2.5+1.5+1.5+0.5+0.5

0.57 1.14 1.71 2.29 2.86 3.43 4.00 4.57 5.14 5.71 6.29 6.86 7.43 8.~ 8.57 9.14 9.71 10.29

observed value. From the only interpretation of # this is impossible. Such a situation is actually also observed for almost all the 4f5/2 contribution that we shall consider. An important feature which must be kept in mind is that all the variations of the 4f5/2 contribution when going from an element to the next one comes by steps of 3/~ B = 0.43/x B. It is the essential fact which is expressed by formula (5.5).

Terbium. Here again, like for Gd the contribution of 4.00AtB f r o m 4f7/2 with u = 7 / 2 is absent. This contribution never appears in the following interpretation. This leaves us with 2.86 + 1.71 = 4.57/t B from 4f7/2 subshell and adding to this 4.71/~B from 4f5/2 as before we obtain a moment of 9.29t~B (considering three decimal places) which agrees very well with experimental values of 9.34, 9.28 and 9.35/~ B (table 3). It is important to underline that this step of 1.71/x B between/~(Gd) and/~(Tb) corresponds to the additional electron of Tb with respect to Gd. Dysprosium and holmium. By comparing #(Tb) with /~(Dy), we observe a step of 1/~ B. The additional electron which appears on 4f7/2 has a m o m e n t /~ = 4/~ B = 0.57/~ B. Adding this to the contribution of 0.43/~Bi from 4f5/2, (see table 9) we obtain 1.00/~B, the step between #(Tb) and/L(Dy). The agreement with experiment is better than 1%. The moment of holmium is interpreted in the same way. The additional electron entering into 4f7/2 subshell does not bring any contribution. Erbium. Let us compare Atex(Er) = 8.00/~ B with the m o m e n t of D y and H o #th = 10.29/t~. We observe a decrease of 2.29/~n which can correspond to the disappearance of the contribution 1.71/~ B and 0.57/~ B that is a sum of ~/~B." In erbium, in 4f7/2 there are two electrons with /x = 1.71/~ B and two others with At = 0.57/~ B. We may consider that their contributions are mutually neutralized, then # d~(Er) = 8.00/~ B with (5 + 1 / 7 ) # B o n 4f7/2. But we can equally consider the contributions 3.43/~ B on 4f5/z and 4.57 o n f7/2" In the present state of this research it is impossible to decide a priori which solution is realized.

118

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

Thulium. The m o m e n t of this element i s / ~ x ( T m ) = 7.13/~ B. With respect to the erbium there is a further decrease of 0.87/~w This can just be obtained with 4f5/2, starting from the case of Er we get this result given in table 2 with a theoretical value Of 7.14~t B. Thus for the heavy rare earth pure metals the agreement between theory and experiment is often better than 1%. With the L - S coupling the situation is not so, the most important discrepancies arising with the erbium, where the experimental value of 8.00/~ B is to be compared with 9.00/~B, given by the L - S coupling. Let us now consider the 3d metals. Iron. The experimental value [22] is i~(Fe) = 2.226/~ a. According to the choice of electronic configuration table 8, iron has 3d4/2, 3d~/2, and one of the 3d5/2 states with /~e = 1.80/~ B is occupied. With the contribution of one of the two states I/~ [0-40/~B on 3d3/2 we get/zth = 2.20/~ B. Cobalt. The state with /x~--1.80/~ gives a good interpretation of the experimental value [22] /z(Co)= 1.729/L B. Nickel The experimental value [22] is /~(Ni)= 0.619/~ B. This element has two occupied states with I/~e I = 0.60/~B. One of these states is already occupied with the cobalt but does not bring any contribution. We interpret #(Ni) as the contribution of the other electron with ~e = 0"60/~B" Chromium in its compounds. According to the electronic configuration that we have supposed in table 8, chromium has a particular and interesting position: in this case only the first subshell is filled up and the second one is empty. Hence from this approach a chromium ion can just exhibit moments given by the simplified relation (4.5) with n 2 = 0. That i s : /~ = n (0.40/~B) with 0 ~< n ~ 8, where n is an integer. With n = 8, ~ has the m a x i m u m permissible value of 3.20/~ B for the 3d3/2 subshell. We have collected some experimental data on ferromagnetic compounds where the chromium has the oxydation number II or III (fig. 4) in order to find n. We also report some neutron diffraction results. For numerous ferromagnetic chalcospinels the experimental results are very close to 2.80/x B, the value for n is 7. In these compounds the chromium has oxydation number III. A m o n g the four 3d3/2 electrons just one with u = 0.5 and ~t = 0.4/~ B does not contribute to/x as may be expected for Cr 3+ ions corresponding to the oxydation number III. N o w in a previous paper one of us has shown that even with the o.n. I I I we 2+ , with an additional covalent bond obtained with a 3d electron must consider the ionicity 2 + and Cr m [23]. In this case this covalent bond quenches the contribution of the 3d electron. It is worth noting that the "spin-only" value for Cr 3+ is 3/xB. To explain the experimental value very close to 2.80/~ B one has to propose and additional hypothesis to explain the discrepancy of 0.20/~ B. Further difficulty will appear with /~ex(CrN) --- 2.40#B. In this work it corresponds to n = 6 which gives/xth = 2.40/~ B. It could be surprising that for CrTe where Cr has an o.n. II, we observe the same value. But with CriZi+ or Crx2i~- there is no difficulty. The value 2.40/~ B is obtained by supposing that the two electrons 1/~ [ = 1.20/~ B are adding their moment while the other two with I/~ [ = 0.40/~B are in opposition. Now we want to discuss the case of ferromagnetic Rb2CrC14. F r o m this work the highest possible value for the chromium ion is 3.2#B for n---8. In 1975 anthony et al. published #~x = 3.5/xB at H = 35 kOe. Their extrapolated value at H = 0 is 3.30/~ w N o w for neutron diffraction studies, large single crystals are required. With a new method Garton and Walker [24] got very good results. They underline that going from powder to single crystals the purity is improved. This point is important because in neutron diffraction study, Fair et al found in 1977/~ = 3.9/~a on powder but 3.3/~B on single crystal. This value has recently been confirmed by polarized neutron diffraction: Miamminghoff et al. found in 1982/~ = 3.10btB. Similar results have also been found on the antiferromagnetic compound CsCrC13 in 1980 by Zandbergen and Ijdo with bt = 3.16/~ B by neutron diffraction study. Here again the value is close to the maximum

X. Oudet, G, Lochak / Total angular momentum and atomic magnetic moments

119

theoretical value 3.20#B. As from the classical spin-only value one would expect to obtain 4.00/~ B for Cr n we think that the value of 3.2/~ B as the m a x i m u m value found for chromium is a very good test to check our model.

7. Conclusion The study of Dirac's theory of the electron enables us to propose a calculation of the magnetic moment. Applied to some fifteen metals or chromium compounds the discrepancy between calculation and experiment is in most cases less than one percent. The discussion about the electronic configuration leads us to show that the crystalline field can lead to a filled subshell to exhibit a particular value of magnetic moment depending on the nature of the compound in which the considered element is present, in other words on the crystal field. Nevertheless, we have seen that it m a y be expected that such a situation is not in contradiction with Pauli's exclusion principle. Let us focus our attention on this possibility. It opens the way for the interpretation of others cases. We know for example that the rare earth alloys exhibit for the same element according to the alloy, different values of the magnetic moments [25] and it seems to us that the crystal field is responsible for that. But in place of a continuous variation of the quantum state occupation and as a consequence of the magnetic moment, as proposed in the current models, we think that an interger number of quantum states bring their contributions making a discrete series of atomic magnetic moments. However to verify this, we need to have precise measurements data as for those for the pure metals. The ever increasing accuracy from the technical possibilities allow us to hope for them in the near future.

Appendix A. A few details about Dirac's equation Let us first consider a classical relativistic particle with a proper mass m 0 and an electric charge - e , in presence of electromagnetic potentials V, A. Its linear m o m e n t u m ¢r and mechanical energy ~ are: or=my,

c = m c 2,

m=mo/[1-v2/c2]

x/2,

(A.1)

from which follows: (A.2)

c 2 / c 2 = ¢r 2 + mEc 2.

Defining the total energy E and the total m o m e n t u m p as: E =, - eV,

(A.3)

p = ~r- (e/c)A,

we get from (A.2): (1/c2)( E + eV) 2 - (p + (e/c)A

) 2 - m 2 c 2 = 0.

(A.4)

Dirac's idea may be summarized as a linearization of the latter quadratic equation in the following form: ( 1 / c ) ( E + e V ) - ct . ( p + ( e / c ) A ) - m o c a 4 = 0,

(A.5)

where a = {a 1, a 2, a 3 ) and a 4 are four anticommutative matrices such that: ol~,a~ + a oq, = 0 (p, 4: v),

a~,-2 _ 1 (bt = 1, 2, 3, 4).

(A.6)

X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

120

The left hand part of (A.5) is a kind of square root which gives (A.4) again when multiplied by:

E + eV+ a "('p + ( e / c ) a ) + moCCt4. Now, it is known that if we introduce in (A.4) the operators:

e=ihO/~t,

p=-ih

17".

(A.7)

We get the Klein-Gordon equation, if we do the same with (A.5) we get the Dirac equation:

[l(ih~--~+eV)-a'(-ih

V+eA)-rnoca,)g'=O.

(A.8)

The current representation of the a. is: ak=

sk

0

(k=1,2,3),

t~4=

0

I '

where the s k (k = 1, 2, 3) are the Pauli matrices and I the two-dimensional unit matrix. The wave function '/" is then a four-lines column matrix written as: q ~ = ( X~ )

with:

~=(~,',),

X=(xX;).

(A.X0)

If we introduce these notations in (A.8) supposing that the potential is the one of an atomic nucleus:

V= Ze/r,

A =0.

(A.11)

We find system (2.1).

Appendix B. The angular momentum Owing to (A.8) and (A.9) eq. (2.1) may be written as

-lh-~" ~ ~X = H

, H =c

-r + iha° V - m°c°t4 '

which defines the Hamiltonian H. An elementary calculation then shows that the total angular momentum defined by (2.3) is a constant of motion, while L and S are not because: [H, M ] = 0 ,

[H, L]=~0,

[H, S]4=0.

(B.2)

The calculation of the $2 spin harmonics is quite classical [12,13,26]. Thus we shall give only an outline, adding only a simplified proof to the way the operator s ° n acts on them. From (2.4) one can see that [ j z , L 2] = 0 and [j2, S 2] = 0. Hence, we have simultaneously:

J212=j(j+ l)I2,

L212=l(l+1)(2,

S212=s(s+ l)~2=]~2.

(B.3)

Therefore, owing to (2.4), we have:

JZfa = (L + S)2~2 = ( L 2 + S 2 + 2 L - S ) I 2 = (L 2 + S 2 + L , S ) f a , which means that

j ( j + 1)I2= [l(l+ 1) + 3 / 4 + L - S ] / 2 .

(B.4)

X.. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

Making use of this result and trying the

121

a priori form: (B.5)

~"~= ( C1Y/m) C 2 Yml 1 ,

it is easy to compute the constant C 1 and C 2 owing to elementary properties of the ym and one finds the expressions (2.8) (where normalization constants are included). Now, let us give a simple proof of (2.16) we know already from (2.15) that we have only to prove that:

s.nI2~(+)

= Af2~'(+) + Bl2~+x(-),

(B.6)

where A and B are constants which could depend on l but certainly not on m. We may thus choose particular values of m and of the angles. We shall take: m=l+l,

~=~/2,

cg=0.

(B.7)

Remind that we already know that the value m = l + 1 is allowed. The expressions (2.8) are thus reduced to:

1 ~2~+1(+)=(Y/)0 '

"'l-l~ot+l/--)=

}

[21 + 3] 1/2 _[2l+2] [2l+3

¥'l+1

(B.8)

"z+x

But, from (2.8) and (B.8) we have: Y/+l(~r/2'0)=0'

[[2-T7-3] 21+ 211/2El+l"nr 2 ' ( / ,0)=-Y/(v/2,0)

(B.9)

and on the other side, (2.13) gives:

s'n=(~

~) (if~=~r/2,

q)=0).

(B.lo)

Hence, we must have in (B.7): A =0,

B=I

(B.11)

which gives the first formula (2.16). The second follows from ( s . n) = L

References The references for the experimental data of the magnetic moment at 0 K, of the heavy rare earth metals and the chromium compounds are given in tables 3 and 4. [1] K.A. McEwen, in: Handbook on the Physics and Chemistry of Rare Earths, vol. 1, eds. K.A. Gschneidner, Jr. and L. Eyring (North-Holland, Amsterdam, 1978) pp. 411-488. [2] S. Legvold, in: Ferromagnetic Materials, Handbook on the Properties of Magnetically Ordered Substances, vol. 1, ed. E.P. Wohlfarth (North-Holland, Amsterdam, 1980) pp. 183-295. [3] M.M. Schieber, Experimental Magnetochemistry, in: Selected Topics in Solid State Physics (North-Holland, Amsterdam, 1967). [4] E.C. Stoner, Magnetism and Matter (Methyen & Co, London, 1934) p. 431. [5] E.P. Wohlfarth, Ferromagnetic Materials (North-Holland, Amsterdam, 1980) chap. 1. [6] X. Oudet, 7~me Journ6es de la R.C.P. 520 sur les compos6s de Terres Rares et Actinides h Valence Anormales, Grenoble (7 and 8 November 1984) p. 132-157.

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X. Oudet, G. Lochak / Total angular momentum and atomic magnetic moments

[7] X. Oudet, Proc. of the Intern. Conf. on Solid Compounds of Transition Elements, Vienna Austria (9-13 April 1985), communication 0.12. [8] X. Oudet, New Frontiers in Rare Earth Science and Applications, Proc. of the Intern. Conf. on Rare Earth Development and Applications, vol. 1, eds. Xu Guangxian and Xiao Jimei, Beijing, 10-14 September 1985 (Science Press, Beijing, 1985) pp. 309-313. [9] X. Oudet, ibid. pp. 899-903. [10] P.A.M. Dirac, Proc. Roy. Soc. Al17 (1928) 610. [11] P.A.M. Dirac, Proc. Roy. Soc. Al18 (1928) 351. [12] H.A. Kramers, Quantum Mechanics (Dover, New York, 1964). [13] A.D. Sokolov and I.M. Ternov. The Relativistic Electron (Nauka, Moscow, 1974). [14] H.E. White, Introduction to Atomic Spectra (McGraw-Hill, New York, London, 1934). [15] W. Heitler, The Quantum Theory of Radiation (Clarendon, Oxford, 1954). [16] G.E. Uhlenbeck and S. Goudsmit, Naturwissenschaften 13, (1925) 953. [17] G.E. Uhlenbeck and S. Goudsmit, Nature 117 (1926) 264. [18] L. De Broglie, L'Electron Magnttique (thtorie de Dirac) (Hermann, Paris, 1934). [19] K. Honda and S. Kaya, Sci. Rep. Tohoku Univ. 15 (1926) 721. [20] S. Kaya, Sci. Rep. Tohoku Univ. 17 (1928) 639 and 1157. [21] P. Weiss and R. Forrer, J. de Phys. 12 (1929) 279. [22] R. Pauthenet, High Field Magnetism (North-Holland, Amsterdam, 1983) pp. 77-86. [23] X. Oudet, Ann. de Chim. 8 (1983) 483-507, in French, English version available from the author. [24] G. Garton and P.J. Walker, J. Cryst. Growth 33 (1976) 61. [25] H.R. Kirchmayr and C.R. Poldy, Handbook on Physics and Chemistry of Rare Earths, vol. 2, eds. K.A. Gschneidner, Jr. and L. Eyring (North-Holland, Amsterdam, 1979) chap. 14, pp. 55-230. [26] D. Bohm, Quantum Theory (Prentice Hall, London, 1960).