Atomic Mgnetic Moment

8/8/2019 Atomic Mgnetic Moment

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3A T O M I C M A G N E T I C M O M E N T S

3.1 STRUCTURE OF ATOMSIn the classical Bohr model of the atom, Z electrons are circulating about the atomicnucleus wh ich carries an electric charge Ze (C), where Z is the atomic number, and e(C) is the elementary electric charge. One of the origins of the atomic magneticmoment is this orbital motion of electrons. Suppose that an electron moves in acircular orbit of radius r (m) at an angular velocity t o (s"1) (Fig. 3.1). Since theelectron makes /27r turns per second, its motion constitutes a current of ea)/2Tr(A), w here e is the electric charge of a single electron. The magnetic moment of aclosed circuit of electric current i whose included area is S (m 2) is known fromelectromagn etic theory to be n0iS (Wbm). Therefore th e magnetic moment producedby the circular motion of the electron in its orbit is given by

Since the angular mom entum of this m oving electron is given bywhere m (kg) is the mass of a single electron, (3.1) may be written as

Thus we find that the magnetic moment is proportional to the angular momentum,although their sense is opposite.

Fig. 3.1. Bo hr's atom ic mo del.


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5 4 ATOMIC MAGNETIC M O M E N T SNow it is well known that th e orbital motion of the electron is quantized, so thatonly discrete orbits can exist. In other words, th e angular momentum is quantized,and is given by

where h is Planck's constant h divided by 2 7 7 , or

and / is an integer called the orbital angular momentum quantum number. Substituting(3.4) for P in (3.3), we have

Thus the magnetic moment of an atom is given by an integer multiple of a unit

which is called the Bohr magneton.W hy the angular momen tum of an orbital electron is quantized by hIn quantum mechanics, th e s component of the angular momentum ps is expressed by theoperator*

where s is the circular ordinate taken along the circular orb it, and is related to the azim uthalangle of the electron byThe mo me ntum along the orbit is given by

Therefore the angular momentum along the z-axis, which is normal to the orbital plane, isgiven by

The eigenvalue of the angular momentum, mh, is obtained by solving the wave equation

The general solution of this equation is given by

Although the angular momentum is a physical quantity related to time, it can be expressed in terms oionly the ordinate s as long as we treat only stationary states.


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S T R U C T U R E OF ATOMS 55

Fig. 3.2. Ex plan ation of a stationary state.The necessary condition fo r this solution to be a stationary state is that

Otherwise the wave function cannot be a unique function as shown in Fig. 3.2. Applying (3.14)to (3.13), we haveor

In order to satisfy (3.16), we have

or

Thus the eigenvalue of L2 is an integer multiple of h.Besides th e orbital angular momentum, th e electron has a spin angular momentum.This concept was first introduced by Uhlenbeck and Goudsm it1 for the purpose ofinterpreting th e hyperfine structure of the atomic spectrum. In 1928, Dirac2 provideda theoretical foundation fo r this concept by making a relativistic correction to thewave equation.The magnitude of the angular momentum associated with spin is ft/2, so that itsangular momentum is given by

where s is the spin angular mom entum quantum n umber and takes th e values . Themagnetic moment associated with spin angular momentum P is given by

Comparing this equation with that of the orbital magnetic moment or (3.3), w e findthat a factor 2 is missing in the denominator. However, substituting P in (3.19) into(3.20), we find that the magnetic moment is again given by the Bohr magneton. Theseconditions were proved by Dirac 2 using th e relativistic quantum theory.


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5 6 ATOMIC MAGNETIC MOMENTSGenerally the relationship between M and P is given by

where the g-factor is 2 for spin and 1 for orbital motion. The coefficient of P in (3.21)is calculated to be

which is called the gyromagnet ic constant. Using (3.22), we can express (3.21) as

Thus w e find the magnetic moment of an atom is closely related to the angularmo mentum of the electron motion. A more exact definition of the g-factor is given by(3.39).Now let us examine the relationship between the electronic structure of atoms andtheir angular momentum. As mentioned above, in a neutral atom Z electrons arecirculating about a nucleus having an electric charge Ze (C). The size of the orbit ofthe electron is defined by the principal quantum number n, which takes the numericalvalues 1,2,3,4... . The groups of orbits corresponding to n = 1,2,3,4,... are calledthe K, L, M,N,... shells, respectively. The shape of the orbit is determined by theangular momentum, or in classical mechanics by the areal velocity. The angularmom entum is defined by the orbital ang ular momentum quantum number in (3.5). Theorbits which belong to the principal quantum number n can take n angular momentacorresponding to / = 0,1,2,..., n 1. The electrons with / = 0,1,2,3,4,... are calledthe s,p,d,f,g,... electrons. The spin angular momentum is defined by (3.19) inwhich s can take the values |.According to the Pauli exclusion principle,3 only two electrons, with s = +\and \,can occupy the orbit defined by n and /. The total angular momentum of oneelectron is defined by the sum of the orbital an d spin angular momenta, so thatwhere ;' is the to tal angular momentum quantum number.When a magnetic field is applied to an atom, the angular momentum parallel to themagnetic field is also quantized and can take 21 + 1 discrete states. This is calledspatial quantization. Intuitively this corresponds to discrete tilts of the orbital planesrelative to the axis of the magnetic field. The component of / parallel to the field, m ,or the magnet i c quantum number can take the values

For instance, in the case of the d electron (/ = 2), the orbital moment can take fivepossible orientations corresponding to m 2,1,0, 1 , 2 as shown in Fig. 3.3. In thiscase it is meaningless to discuss the azimuthal orientation of the orbit about themagnetic field, because the orbit precesses about the magnetic field. Figures 3.3(a)and 3.4 show such precessions for m = 2, 1, and 0, according to the classical model. It

tas


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S T R U C T U R E O F A T O M S 57

Fig. 3.3. Spatial quantization fo r orbital (a) and spin (b ) angular momenta.is interesting to note that these figures have some similarity with the atomic wavefunctions shown in Fig. 3.18. T he spin can take up an orientation either parallel(s = +\) or antiparallel (s = -\) to the magnetic field (Fig. 3.3(b)).Wh e n an atom contains many electrons, each electron can occupy one state definedby n, I, and s . Figure 3.5 shows possible states belonging to the M shell. Since theprincipal quantu m num ber n of the M shell is 3, possible orbital angular m om enta are/ = 0, 1, and 2. In other words, there are s, p, and d orbital states. According tospatial q uan tizat ion , each o rbital states consists of 21 + 1 orbits with different mag-netic quantum numbers m. That is, the number of orbits is 1, 3, and 5 for s, p, and d,respectively. Therefore th e total number of orbits belonging to one atom is given by

Fig. 3.4. Shapes of orbits fo r various magnetic quantum numbers.


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58 ATOMIC MAGNETIC M O M E N T S

Fig. 3.5. Various electronic state s belonging to the M electron shell.Since twoelectrons with + and - spins can enter into one orbit, the total number ofelectrons belonging to one neutral atom is equal to In2. In the case of the M shellwith n = 3, the total number is 2n2 = 2 X 32 = 18.In an actual atom with atomic number Z, Z electrons occupy the possible orbitalstates starting from the lowest energy state. Table 3.1 lists the electron configurationsof the atoms which are most important in connection with magnetism. As seen in thistable, the electrons occupy the states in the normal order, from the lower n states toth e higher ones, up to argon (Z = 18). It must be noted that th e energy defined by nis that of one isolated non-interacting electron. When a number of electrons arecirculating around the same nucleus, we must take into consideration the interactionbetween these electrons. However, as long as the inner electrons are distributed withspherical sym metry about the n ucleus, the ordering of the energy levels rem ainsunchanged, because the effect of the inner electrons is simply to shield the electricfield from the nucleus. Therefore, if the inner Z - 1 electrons form a spherical chargedistribution about the nucleus, the outermost electron feels the difference in electricfield between the nuclear charge +Ze and the charge of the inner electron cloud (Z l)e. This is the same as the electric field produced by a proton of charge +e.This is the reason w hy the size of the atom remains almost unchanged from that ofhydrogen, irrespective of the number of electrons, up to heavy atoms with manyelectrons.If, however, the charge distribution of the orbit deviates from spherical symmetry,the situation is changed. Figure 3.6 shows the shapes of various Bohr orbits. We seethat the 3d orbit is circular, whereas the 3s orbit is elliptical, so that part of the orbitis close to the nucleus. In other w ords, the atomic wave function of s electrons is verylarge in the vicinity of the nucleus, as shown in Figure 3.21(c). Thus the energy of the4s electron is lowered, because of a large Coulomb interaction with the unshieldednuclear charge. For this reason, the 4s orbits are occupied before the 3d orbits are


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VECTOR M O D E L 59

Fig. 3.6. Various Bohr orbits.occupied for atoms heavier than potassium (Z = 19). As seen in Table 3.1, the 5sorbits are occupied before th e 4d orbits fo r atoms heavier than rubidium (Z = 37).Similarly, th e 5s, 5p, 5d, and 6s orbits are occupied before th e 4/ orbits fo r atomsheavier than lanthanum (Z = 57). The same thing happens fo r atoms heavier thanhafnium (Z = 72), in which th e 6s orbits are filled before th e 5d orbits are occupied.The elements which have incomplete electron shells exhibit abnormal chemical andmag netic properties, and are called transition elements. In the order mentioned above,they are called th e 3d, 4d, rare-earth and 5d transition elements, respectively. Themagnetic elements are found among these transition elements.

3 .2 VECTOR MODELIn this section w e discuss how the orbital and spin magnetic moments of electrons inan incomplete electron shell form an atomic magnetic moment. Let the spin andorbital angular momentum vectors of the j'th an d yth electrons be sf, /, , s;- and /,.,respectively. These vectors interact in a local scale. The most important of theseinteractions are those between spins (s, and ) , and those between orbitals (/ , andlj). As a result, the spins of all the electrons are aligned by the strong spin-spininteractions, thus forming a resultant atomic spin angular momentum

Similarly, the orbitals of each electron are aligned by the strong orbit-orbit interac-tions, thus forming a resultant atomic orbital angular momentum


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Table 3.1. Electronic configuration of elements.Levels an d number of states

K Z M N O P Q ~2 8 18 32 50 72 I s 2s 2p 3s 3p 3 d i l 4p 4d 4/ 5s 5p 5d 5 f 5 g IT s 6p 6d 6 / 6 5 6 A 7i G r o u n dElem ents 2 2 6 2 6 10 2 6 10 14 2 6 10 14 18 2 6 10 14 18 22 2 term s

123

101118

,19 200 211 223 23B / 24.2 \ 25g 262 27- 28

< * > 2 9V 3 036

,37I 38a 395 400 41 { 42143i44- 45 I 46v 47

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HHeLiNeNaArKCaScTiVCrM nFeCoNiCuZnKrRbSrYZrMbMoTcRuRhPdAgXe

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12355678101010101010101010101010101010

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122211211012 6


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Table 3.1. (contd.)Levels and number of states_ _ _ _ _ _ __

2 8 18 32 50 72 Is 2s 2p 3s 3p 3d 4s 4 p 4d 4f 5s 5p 5d 5f 5g 6s 6p 6d 6f 6g 6ft 7s G r o u n dE l e m e n t s 2 2 6 2 6 10 2 6 10 14 2 6 10 14 18 2 6 10 14 18 22 2 t e r m s

55 Cs56 Ba, 57 La58 Ce59 Pr, 60 Ndg 61 Pm 62 Sm3 63 Eu- < 64 Gd 65 Tbu 66 Dy" 67 Ho

c 68 Er69 Tm70 YbV 71 Lu12 Hf573 Ta|74 Wo 75 Reg { 76 Os5j 77 Ir5 78 Ptg 79 Au^ ^ 80 Hg>n .

86 Rn87 Fr

102 No

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6

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6

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1010101010101010101010101010101010101010101010101010101010

1(3)4(5)6778(9): i o ):n)131414141414141414141414141414

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22

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6

11(0)0(0)0011( 1)( 1 )( 1 )00123456791010101010

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6 2 ATOMIC MAGNETIC MOMENTS

Fig. 3.7. Russ ell-Saunders coupling.Now the senses of these resultant vectors S and L are governed by the so-calledspin-orbit interaction energy

thus forming th e total resultant angular momentum

This interaction is called the Russell-Saunders interaction4 (see Fig. 3.7).When a number of electrons exist in an atom, the arrangement of these vectors aregoverned by Hund's rule5, which is described as follows:(1) The spins s, are arranged so as to form a resultant spin 5 as large as possiblewithin the restriction of the Pauli exclusion principle. The reason is that the electronstend to take different orbits owing to the C oulomb repu lsion, and also the intra-atomicspin-spin interaction tends to align these spins parallel to each other. For instance, inthe case of the 4/ shell, which has the capacity of accepting 14 electrons, theelectrons occupy states in the order of the numbers in Fig. 3.8. For example, when the4/ shell has 5 electrons, they occupy the states with a positive spin, thu s form ing 5 = f.When the number of electrons is 9, 7 electrons have positive spin, w hile the remaining2 have negative spin, so that the resultant spin becomes 5 =\\=f.(2) The orbital vectors /, of each electron are arranged so as to produce themaximum resultant orbital angular momentum L within th e restriction of the Pauliexclusion principle and also of condition (1). The reason is that the electrons tend tocirculate about the nucleus in the same direction so as to avoid approaching oneanother, which would increase the Coulomb energy. In the case of the 4/ shell, themagnetic quantum number m can take the values 3,2,1,0, 1 , 2 , -3. In the case of5 electrons, the maximum resultant orbit is L = 3 + 2 + 1 + 0 - 1 = 5. In the case of 9electrons, the first 7 electrons occupy the half shell w ith positive spins, produ cing noorbital moment, while the remaining 2 electrons occupy the states with m = 3 and 2,thus resulting i n L = 3 + 2 = 5 (Fig. 3.8).(3) The third rule is concerned with the coupling between L and S. When thenumber of electrons in the 4/ shell, n, is less than half the maximum number, or


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VECTOR MODELSpin angular momentum

63

Fig. 3.8. Spin and orbital states of electrons in the 4/ electron shell.

n < 1, J = L - S. When the shell is more than half filled, or n > 1, J = L + S. Sowhenthe number is 5, / = 5 f = f, while when the number is 9, / = 5 + f = . Such aninteraction is based on the s I interaction of the same electron. When an electron iscirculating about th e nucleus (Fig. 3.9), this electron sees th e nucleus circulatingabout itself on the orbit shown by the broken circle in the figure. As a result, th eelectron senses a magnetic field H pointing upwards produced by the circulatingnucleus with positive electric charge. Then its spin points downwards, because th espin angular momentum is opposite to the spin magnetic moment (see (3.20)). Sinceth e orbital angular momentum of this electron points upwards (see Fig. 3.9), it followsthat / and s of a single electron are always opposite. Therefore when the number ofelectrons is less than half th e maximum number , / and s are opposite for all theelectrons, so that it follows that L and S are also opposite. However, when the

Fig. 3.9. Explanation of the l-s coupling.


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64 A T O M I C M A G N E T I C M O M E N T Snumber of electrons is more than half the maximum number, the orbital mom entumfor the 7 electrons with positive spin is zero, so that the only orbital momentum Lcomes from electrons with negative spin which point opposite to the resultant spin S,thus resulting in parallelism between L and S. In terms of w in (3.29), the sign of wis positive when the number of electrons n < 7, while it is negative wh en n > 7.Now let us calculate the values of 5, L and J fo r rare earth ions, which haveincomplete 4/ shells. The rare earth elements have an electronic structure expressedbyin which the incomplete 4/ shell is well protected from outside disturbance by theouter (5s)2(5p)6 shell, so that its orbital magnetic moment is well preserved or'unquenched' by the crystalline field. T he outermost electrons (5d) l(6s)2 are easilyremoved from the neutral atom, thus producing trivalent ions in ionic crystals, andconduction electrons in metals or alloys. T herefore the atomic magnetic moments ofrare earth elements are more or less the same in both compounds and in metals.A s mentioned above, as the number of 4/ electrons, n, is increased, S increaseslinearly with n from La to Gd, which has 4/7, and then decreases linearly to Lu,which has 4/14. The value of L increases from 0 at La to values of 3, 5, 6; and thendecreases towards 0 at Gd with 4/7, where a half-shell is just filled. This state andalso a completely filled 4/14 state are called 'spherical'. Further increase in n resultsin a repetition of the same variation. T he total angular momentum, /, changes asshown in Fig. 3.10, because /= L - S for n = 0 to 7,while /= L + S for n = 1 to 14.So / is relatively small for n < 7, while it is relatively large for n > 1.

Fig. 3.10. Spin S, orbital L, and total angular momentum 7 as functions of the number of 4/electrons of trivalent rare earth ions.

2


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VECTOR MODEL 65

Fig. 3.11. Composition of atomic magnetic mom ent M .N ow let us discuss the magnetic moments associated with these angular momenta.Referring to (3.6) and (3.7), w e have th e orbital magnetic moment

Referring to (3.19) and (3.20), we have the spin magnetic moment

Therefore, the total magnetic moment MR is given by

When vectors L and S take different orientations, th e vector L + 25 takes adirection different from /(Fig. 3.11). Since, however, L and S precess about /, thevector L + 2S also precesses about /. Therefore the average magnetic momentbecomes parallel to /, and its magnitude is given by the projection on / or

The magnetic moment given by (3.34) is called the saturation magnetic moment.Comparing (3.34) and (3.33), and also referring to the geometrical relationship shownin Fig. 3.11, we haveFrom th e relationship between three sides and the angle ^ABO in the triangleA ABO, we haveEliminating cos ZABO, we have

from which w e obtain an expression for g:


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6 6 ATOMIC MAGNETIC M O M E N T SIn quantum mechanics, we must replace S2, L2 and J2 by 5(5 + 1), L(L + 1) and/(/+ 1), respectively.* Then we have

This relationship was first introduced by Lande empirically to explain the hyperfinestructure of atomic spectra.6 If 5 = 0, then J = L, so that it follows from (3.39) thatg=l,while if L = 0, then /= 5, so that g 2. This is exactly w hat we find in (3.1).When a magnetic atom is placed in a magnetic field, Jz can take the followingdiscrete values as a result of spatial quantization of the vector /:

This fact affects the calculation of the statistical average of magnetization, as will bediscussed in Part III. As a result, the m agnitude of the atomic mom ent deduced fromthe thermal average of magnetization is given by

which is called the e f f e c t i v e magnetic moment . The calculation will be shown inChapter 5.Figure 3.12 shows the effective magnetic moment calculated using (3.39) and (3.40)as a function of the number of 4/ electrons. The solid curve represents the calcula-tion based on Hund's rule. The shape of the curve is similar to that of L (see Fi3.10), except that the magnetic moment is much more enhanced by the g-factor forheavy rare earths than for light ones. Experimental values observed for trivalent ions* Let us deduce this relationship for orbital angular momentum L. Let the x-, y- and z-components of Lbe Lx, Ly and Lz, respectively. Then

On the other hand, these components can be expressed in terms of the components of mo mentu m p, or px,py and pz, and the position coordinates x, y, and z, as

In qua ntum mechanics, p must be replaced by ih(d/dq) (q is the positional variable), so that we have

In polar coordinates, we can write

Executing this operator to the atomic wave function given by (3.65), we have

(Examine this relationship fo r desired va lues of / and m in (3.65).) Thus it is concluded that the eigenvalueof L2 is L(L + 1).


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Fig. 3.12. Effective magnetic mom ent as a function of the n u mb er of 4/ electrons measured fortrivalent rare earth ions in compounds and rare earth metals, and comparison with the Hu n dand Van Vleck-Frank theories.in th e compounds are also shown as open circles in the figure. The agreementbetween theory and experiment is excellent, except for Sm and Eu. This discrepancyw as explained by Van Vleck and Frank7 in terms of multiple! terms of these elements.In these elements, S and L almost compensate one another in the ground state, whilein the excited states, S and L make some small angle, thus producing a non-zero J.The thermal excitation and the mixture of these excited states, therefore, result in anincrease in the magnetic moment. The broken curve in the figure represents thiscorrection, an d reprodu ces the e xperim ent w ell.The experimental points for metals, shown as crosses in the figure, are also inexcellent agreement with the theory, except for Eu and Yb. The reason is that theseions are divalent in metals, because there is a tendency for atoms or ions to becomespherical. Therefore, Eu3+, whose electronic structure is 4/6, tends to become Eu2+or 4/7 by accepting an electron from the conduction band, whereas Yb3+, which is4/13, tends to become Yb2+ or 4/14 in the same way. These divalent ions have thesame electronic structures as Gd3+ and Lu3+, respectively, thus showing the sameeffective magnetic moments, as seen in the f igure.The electronic structures discussed in this section are often expressed in spectro-scopic notation such as 2S1/2, 5DQ , or *F g/2, in which S,P,D,F,G,... signify that

vector modeldeell.


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6 8 ATOMIC M A G N E T I C M O M E N T SL = 0,1,2,3,4, respectively. A prefix to the capital letter represents 2S + 1, and asuffix represents /. Electronic configuration and spectroscopic ground terms are givenin Table 3.1.

3.3 GYROMAGNETIC EFFECT AND FERROMAGNETICRESONANCEAs noted in the preceding section, there are two possible origins for magnetism inmaterials; spin and orbital magnetic moments. Many attempts* have been made tomeasure the g-factors of various magnetic materials in order to determine th econtribution of these possible origins. The first such experiment was done by Maxw ell.8This measurement was based on the simple idea: if a bar magnet supported horizon-tally at the center on pivots is rotated about the vertical axis, it is expected to tilt fromthe horizontal plane if it has an ang u lar mo men tu m associated w ith its magnetization.This experiment was, however, unsuccessful, because the effect is extremely small.In 1915, Barnett9 first succeeded in determining the g-factor by comparing themagnetization of two identical magnetic rods, one rotating abou t its long axis and theother magnetized by an external magnetic field applied along its long axis. Thisexperiment was based on the idea that if the magnetic atoms in the bar have anangular momentum, th e rotation of the bar should drive this momentum towards th eaxis of rotation. This relationship between rotation an d magnetization is called thegyromagnetic e f f e c t .The most successful measurement technique is known as the Einstein-de Haase f f e c t , which was developed more precisely by Scott.11 The principle is illustrated inFig. 3.13. The specimen is suspended by a thin elastic fiber and is magnetized by avertical field which reverses direction at a frequency corresponding to the naturalfrequency of mechanical oscillation of the system. Consider the moment when thefield (and therefore th e magnetization) change direction from upward to downward.Then the associated angular momentum must also change. Since this system isisolated mechanically, th e total angular momentum must be conserved. Therefore th ecrystal lattice must rotate so as to compensate for the change in angular momentumof the magnetic atoms. The mechanism by which the magnetic atoms transfer theirangular momentum to the crystal lattice will be discussed later in this section. In theactual experiment done by Scott, the n atu ral period of oscillation was 26 seconds. Therate of decay of the amplitude of the oscillation was measured with and without analternating magnetic field applied in synchronism with the mechanical oscillation. Thedifference gives the change in angula r momentum. The magnetization was alsomeasu red, so that the g yromag netic constant in (3.23), and accordingly the g-factor in(3.22), could be calculated. Since the values of the g-factor determined in thisexperiment are different from th e values determined from a magnetic resonance* The attempts are based on the gyroscopic effect: when a force is applied to change the axis of rotationof a spinning top, the axis always rotates perpend icu lar to the direction in which the force is applied (seeFig. 3.14).


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GYROMAGNETIC EFFECT 69

Fig. 3.13. Apparatus for observing the Einstein-de Haas effect.experiment, as will be discussed below, th e former is referred to as the g ' - factor, whileth e latter is called simply th e g-factor. Values of these tw o quantities are listed fo rvarious magnetic materials in Table 3.2.The g-factor can also be determined by a magnet ic resonance experiment. Supposethat the atomic magnetic moment, M , deviates from the direction of the appliedmagnetic field, H , by the angle 6. Then a torque

will act on M . Since an angular momentum, P, is associated with the magneticmoment, the direction of this vector must change by L per unit second or

Table 3.2. Comparison between g- and g'-values. g ( % )Material g g/(g ~ 1) g' from g from g'Fe 2.10 1.91 1.92 5 4Co 2.21 1.83 1.85 10.5 7.5Ni 2.21 1.83 1.84 10.5 8FeNi 2.12 1.90 1.91 6 4.5CoN i 2.18 1.85 1.84 9 8Su perm allo y 2.10 1.91 1.91 5 4.5Cu2MnAl 2.01 1.99 1.99 0.5 0.5MnSb 2.10 1.91 1.98 5 1NiFe2O4 2.19 1.84 1.85 9.5 7.5


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Fig. 3.14. Processional motion of a magneticmoment. Fig. 3.15. Ferromagnetic resonance experi-ment.(This relationship is a modification of Newton's second law of motion saying that timederivative of momentum is a force. Applying this law to a rotational system, we havethat the time derivative of the angular momentum must be given by a torque.) Asshown in Fig. 3.14, the vector L is perpendicular to P, and also to H, so that P mustrotate about H without changing its angle of tilt 6 and its magnitude. The trace ofthe point of the vector P is a circle of radius P sin 6, so that the angular velocity ofthe precession is given by

It is interesting to note that the angular velocity given by (3.44) is independent of theangle 6. Accordingly, when a magnetic field is applied to a magnetic material, themagnetic moments of all the magnetic atoms in the material precess with the sameangular frequency, a > , no matter how the magnetic moment tilts from the direction ofthe field. Therefore, if an alternating magnetic field of this frequency is appliedperpendicular to the static magnetic field, precessional motion will be induced for allthe magnetic atoms. In the actual experiment a specimen is attached to the wall of amicrowave cavity, and the intensity of the magnetic field applied perpendicular to theH vector of the m icrowave is increased gradually (Fig. 3.15). When the intensity H ofthe magnetic field reaches the value which satisfies the condition (3.44), a precessionis induced, so that the radio frequency (r.f.) permeability shows a sharp maximum.


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G Y R O M A G N E T I C EFFECT 71

Fig. 3.16. Ferromagnetic resonance curve (H r: resonance field).Figure 3.16 shows experimental results for Permalloy. This phenomenon is known asferromagnetic resonance (FMR) . From the field at which resonance occurs, we candetermine v from (3.44), and accordingly th e value of g from (3.22).The precessional motion described above is somewhat idealized. The actual preces-sion is associated with various relaxation processes by which the system loses energy.A more detailed discussion will be given in Chapter 20. In the preceding discussion ofth e Einstein-de Haas effect, we assumed that th e magnetization reverses it s senseupon the application of the magnetic field. This is, however, true only if somerelaxation process absorbs energy from th e precessional motion and allows th e

Fig. 3.17. Ferrom agnetic resonance of a cylindrical ferromagnetic specimen.


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7 2 ATOMIC MAGNETIC MOMENTSmagnetization to relax towards the direction of the field. The reaction of thisrelaxation mechanism causes the rotation of the crystal lattice of the specimen.The ferromag netic resonance experiment was first performed by Griffith12 in 1946.The correct g-values were determined by Kittel13 by taking into consideration thedemagnetizing field caused by the precessional motion. For simplicity, let us considera cylindrical specimen (Fig. 3.17). When the magnetization tilts from the cylinder axisby the angle 6, a demagnetizing field 7s(sin 0)/2/i0 is produced perpendicular tothe axis. The torqu e acting on the ma gnetization is, therefo re, given by the sum of thetorque from the external field and that from th e demagnetizing field, or

Accordingly the resonance frequency (3.44) is modified to

If the specimen is a flat plate and the external field is applied parallel to its surface, asimilar calculation shows that

Equation (3.47) is obtained as follows: Set the x-axis normal to the sample surface, and thez-axis parallel to the external field. Then we have for the equations of motion

Eliminating P y from the above equations by using (3.23), we have

Solving this equation, we have the angular frequency

Using equations (3.46) and (3.47), we can calculate the gyromagnetic constant andaccordingly the g-factor.


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G Y R O M A G N E T I C EFFECT 73In addition to the demagnetizing field, the magnetic anisotropy also influences the

resonance frequency, because it produces a restoring torque tending to hold th emagnetization parallel to the easy axis (see Chapter 12).The restoring force acting on the magnetization can be equated to a hypotheticalmagnetic field acting parallel to the easy axis. Let this field be Hl when themagnetization deviates towards the *-axis, and H 2 when the magnetization deviatestowards the y-axis. Then the resonance frequency is given by

A high-freque ncy m agnetic m aterial called Ferroxplana takes advantag e of this effectto increase the resonance frequency (see Section 20.3).The g-values obtained from ferrom agnetic resonance m easurem ents are listed inTable 3.2 together with th e g '-values determined from gyromagnetic experiments.Generally speaking, the values observed for 3d transition elements are close to 2,which tells us that the origin of the atomic m agnetic mo ment is not orbital m otion butmostly spin. We also note that the sign of the deviation from a value of 2 is differentfor g- and g'-values.The deviation of the g'-factor from 2 is apparently caused by a small contributionfrom th e orbital magnetic mom ent. Let the part of the saturation mangetization dueto spin motion be (/s)spin, and that from orbital motion by (/s)orb. The correspondingangular momenta will be (P s) spin and (Ps)orb, respectively. Then we have

Considering that g' = 1 for orbital m otion and g' = 2 for spin, (3.52) is modified to

On the other hand, th e g-values obtained from magnetic resonance are greaterthan 2, as seen in Table 3.2. If the magnetic atoms do not interact with the crystallattice, th e g-values determined from magnetic resonance should be equal to theg' -values determined from the gyrom agnetic experim ent for the same m aterial. Aspointed out by Kittel, Van Vleck and others,13"16 when a magnetic atom is under thestrong influ ence of the crystal lattice, it orbital angu lar m om entu m is not conserved,so that only spin angular momentum contributes to the expression for the g-factor:

Adding both sides of (3.53) and (3.54), we have

which can be writ ten as


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7 4 A T O M I C M A G N E T I C M O M E N T SThe g' -values calculated using (3.56) are listed in Table 3.2,together with the valuesdirectly determined from the gyromagnetic experiment. The values are in excellentagreement.If we denote the ratio of the orbital contribution to the spin contribution by

we can write from (3.52)

and from (3.53)

assuming s 1. The values calculated from (3.58) and (3.59) are also listed in Table3.2. These values are all less than about 10%.The quantity g is often referred toas the spectroscopic splitting factor, while the g'-factor is known as the magneto-mechanical factor.In addition to the ferromagnetic resonance discussed in this section, we can observeparam agnetic spin resonance, antiferro m agn etic resonance, and ferrom agnetic reso-nance (see Section 20.5). These resonances are all concerned with electron spins, sothat they are called electron spin resonance (ESR).

3 .4 C R Y S T A L L I N E FIELD AND Q U E N C H I N G OF ORBITALA N G U L A R M O M E N T U MW e have discussed the magnetism of atoms so far mainly in terms of Bohr's classicalquantum theory. W e have learned that the orbital magnetic moment is mostlyquenched in materials composed of 3d mag netic atoms. The mechanism of quenchingof the orbital moment must be known more precisely in order to understand theorigin of magn etocrystalline anisotropy and magnetostriction, wh ich will be discussedin Chapters 12-14.

Now let us consider the wave function of a hydrogen atom, in which a singleelectron is circulating abou t a proton. Since the electron is under the influence of theCoulomb field produced by a proton w ith electric charge +e (C), the potential energyis given by

where r (m ) is the distance between the electron and the proton, and s0 is thepermittivity of vacuum


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Q U E N C H I N G OF ORBITAL ANGULAR M O M E N T U M 75The state of the orbital electron is described by the wave function fy(.r), which mustsatisfy th e Schrodinger equation

where % ? is the Hamiltonian given by

The first term is the operator corresponding to the kinetic energy of the electron. Inclassical dynamics, the kinetic energy is given by \mv2 = (l/2m)p2, where p is themomentum. In quantum mechanics, th e momentum p is replaced by the operator \fi(d/ds\ so that the kinetic energy is given by

Since the operator A is expressed by

we recognize that (3.64) is the same as the first term in (3.63).The solution of the Schrodinger equation (3.62) is given in all textbooks onqua ntum mechanics . Here we simply refer to the final solution, w hich is

where Rn!(r) is a function of the radial distance from th e nucleus r, lm(d) is afunction of the polar angle 9 from the axis of quantization, and < & m ( < p ) is a function ofthe azimuthal angle < p about the axis of quantization. The suffixes n, I, and m arerespectively the principal, orbital, and magnetic quantum numbers, as already ex-plained in Section 3.1. The reason why these functions are characterized by thesequantum numbers is that only functions characterized by these n um bers express thestationary states, as shown for example by Fig. 3.2.The specific forms of these functions involving the angles 6, ( p are given by

and


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76 A T O M I C M A G N E T I C M O M E N T SThe term Pjm |(cos 6) is the /th transposed Legendre polynomial and given for/ = 0,1,2,3 or s, p, d, f electrons

The functions (3.69) express the eigenstates involving the azimuthal angle < p asdiscussed in Section 3.1. When the orbital magnetic moment remains unquenched,such wave functions are circulating about th e axis of quantization, thus producing acircular current, either clockwise or counterclockwise. Accordingly, the azimuthalvariations are smeared out, so that the directional distribution of the wave functions isexpressed simply by rotating the functions (3.69) about the axis of quantization asshown in Fig. 3.18.In this figure, we recognize that the wave functions for m = 0 in d or / electronsstretch along th e z-axis or the axis of quantization, while those fo r m = 1 or themaximum values spread along the z-plane or the plane perpend icular to the axis ofquantization. T he reader m ay recognize a similarity between these pictures and thoseof the Bohr orbits as shown in Fig. 3.4. It is, however, meaningless to consider furtherdetails of this correspondence. No te that if the square of wave functions with differen tm are added, the sum becomes isotropic. This is also the case if the square of wave


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Q U E N C H I N G OF ORBITAL ANGULAR M O M E N T U M 77

Fig. 3.18. Angular distribution of atomic wave functions with various orbital and magneticquantum numbers.functions with a finite positive m and a half with m = 0 are added. W e call such anelectron shell spherical, as already mentioned.


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Fig. 3.19. The 3d atomic wave functions with m + 2 stabilized by neighboring anions ornegatively charged ions.When such magnetic atoms are assembled into crystals, their magnetic properties

are influenced by these shapes. The magnetocrystalline anisotropy of most rare-earthions can be interpreted in terms of the atom shapes (Chapter 12). In the case of 3delectron shells, the orbital magnetic moments are strongly influenced by the crys-talline field, because the magnetic shells are exposed to the influence of neighboringatoms in the crystal environment. Sometimes their orbital angular momentum istotally quenched. We will elucidate the mechanism of quenching below.

Suppose that tw o wave fun ctions w ith m = + 2 and m = 2 are superposed, thuscancelling their angular momenta. Referring to (3.68), the resultant wave function isgiven by

Figure 3.19 illustrates the angular distribution of this wave function, drawn by using 2 + 2 m(3.69) and (3.70). This figure represents a standing wave constructed by asuperposition of the oppositely circulating wave functions.* Since this wave functionspreads to avoid the anion s (negatively charged ions) in the e n vironm en t, as shown inthe figure, the Coulomb energy will be lowered. Thus the q ue nch ed state of orbitalmoment s is stabilized by the crystalline field.*The fact that (3.70) is real means that th e wave function is independent of time. Therefore neitherangular mom entum nor electric current is produced, an d accordingly no orbital magnetic moment arises.


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Q U E N C H I N G O F O R B I T A L A N G U L A R M O M E N T U M 79In general, when d electrons are placed in a crystalline field with cubic symmetry,th e d wave function can be written in terms of the following orthogonal realfunctions:

Figure 3.20 shows the angular distribution of these wave functions. As seen in thefigure, the wave functions < / ^ , i/^, i / ^ extend along the (110> directions, avoidingthe principal axes with fourfold symm etry, while the functions i [ / u , i f / u extend along theprincipal axes.* The former are called ds, and the latter dy. Since th e 3d magneticatoms or ions have different neighbors, or the same neighbors at different distances,along the (110) and directions, the energy of the nearest-neighb or interactionis different for ds an d dy. The origin of this interaction is not only the Coulombinteraction, but also includes exchange interaction, covalent bonding, etc. A bettername for this interaction is the l igand field rather than th e crystalline field. If thestate expressed by one of these wave functions is stabilized by a ligand field, theorbital magnetic moment will be quenched. We shall also use the ligand theory inSection 12.3 to discuss the mechanism of magnetocrystalline anisotropy.Finally, we shall elucidate the radial part Rn,(r) in (3.66), which is given by thegeneral form

where x = Zr/a0, a 0 = 4Tre0h2/me2, and L2n'^(t) is the Laguerre polynomial givenby

* This ( / > is the same as that shown in Fig. 3.19, because


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80 ATOMIC MAGNETIC M O M E N T SFor a number or orbital electrons the func tional form of (3.72) is given by

Figure 3.21 shows a computer-generated pattern representing the spatial distribu-tion of the atomic wave function (3.66) calculated from (3.67), (3.68) and (3.72). Inthese patterns, the vertical axis is the z-axis or the axis of quantization. Themagnitude of the wave function is expressed by the density of drawing, and its sign isdistinguished by the direction of the lines: radial lines signify positive values andcircumferential lines show negative values. The graphs shown below each patternrepresent the radial variation of R(r). It is seen that the s function is large at thenucleus, as already discussed with the Bohr models shown in Fig. 3.6.


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PROBLEMS 81

Fig. 3.20. The d(e) an d d ( y ) wave functions of 3d electrons in a cubic ligand field.PROBLEMS3.1 Calculate the spectroscopic splitting factor (g-factor) for the neodymium (Nd) atom.3.2 Assuming that a thin Permalloy rod can be magnetized to its saturation magnetization byapplying a magnetic field of 100 Am"1 (= 1.2Oe), calculate the angular velocity of the rotationof this ro d about it s long axis necessary to magnetize it to its saturation magnetization withoutapplying an y magnetic field. Assume that g = 2.3.3 How strong a magnetic field must be applied parallel to the surface of an iron plate to


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Fig. 3.21. Computer-drawn patterns fo r spacial d istribution of atomic wave functions in a planeincluding the z-axis or the axis of quantization. (The magnitude of i /> is expressed by density oflines, while the sign of i /> is distinguished by the direction of drawing. Rad ial lines signify i f / > 0,while the circum ferential line signifies i / < 0.)


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REFERENCES 83observe ferromagnetic resonance, using microwaveswith a wavelength of 3.0 cm? Assume thatg = 2.10 and 7S = 2.12T, and ignore the effect of magnetocrystalline anisotropy.3.4 Show that th e 3d5 electron shell is spherical, by calculating the sum of the square of thewave functions given by (3.69) for m = 2, 1, 0, -1, and 2.

REFERENCES

1. G. E. Uhlenbeck and S. Goudsmit, Die Naturwissenschaften, 13 (1925), 953.2. P. A. M. Dirac, Proc. Roy. Soc., All? (1928), 610; A118 (1928), 351.3. W. Pauli, Z. Physik, 31 (1925), 765.4. H. N. Russell and F. A. Saunders, Astrophy. J. 61 (1925), 38.5. F. Hund, Linien Spektren und periodisches System der Elemente (Julius Springer, Berlin,

1927).6. A. Lande, Z. Physik, 15 (1923), 189.7. J. H. Van Vleck, Theory of electric and magnetic susceptibilities (Clarendon Press, Oxford,

1932), p. 245.8. J. C. Maxwell, Electricity and magnetism(Dover, New York, 1954), p. 575.9. S. J. Barnett, Phys. Rev., 6, 111 (1915), 239.

10. A. Einstein and W. J. de Haas, Verhandl. Deut. Physik. Ges., 17 (1915), 152; A. Einstein,Verhandl. Deut. Physik. Ges., 18 (1916), 173; W. J. de Haas, Verhandl. Deut. Physik. Ges., 18(1916), 423.

11. G. G. Scott, Phys. Rev., 82 (1951), 542; Proc. Int. Con/. Mag. and Cryst., Kyoto (7. Phys.Soc. Japan, 17, Suppl. B-l) (1962), 372.

12. J. H. E. Griffith, Nature, 158 (1946), 670.13. C. Kittel, Phys. Rev., 71 (1947), 270; 73 (1948), 155; 76 (1949), 743.14. D. Polder, Phys. Rev., 73 (1948), 1116.15. J. H. Van Vleck, Phys. Rev., 78 (1950), 266.16. C. Kittel and A. H. Mitchell, Phys. Rev., 101 (1956), 1611.

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Atomic Mgnetic Moment