Neutron Scattering from Magnetic Materials part 10 pot

Following a similar procedure as indicated in the case of the spin waves in Heisenberg ferromagnets, one arrives at the spin wave dispersion equations The above equations mean that the H

Fig 9 Double logarithm plot of the linewidth (FWHM) of the spin wave excitations in EuS versus wave vector

at T = 14.1 K The solid line is a guide to the eye The straight parts of the curve indicate a slope proportional to

q3or q4 The change in slope occurs near q ≈ 0.7 Å−1, where¯hω ≈ kBT (from Bohn et al [19]).

Fig 10 Linewidth (FWHM) at T = TCof EuS as a function of q in a double logarithmic scale The straight line represents the best fit of the data giving a dynamical scaling exponent of z = 2.09 ± 0.06 (from Bohn et al [19]).

Fig 11 Double logarithm plot of the linewidth versus q of EuO at TC The prediction of the dynamical scaling

theory (straight line) is obeyed in d decades in energy (from Böni et al [23]).

3.4 Spin waves in Heisenberg antiferromagnets

We already noted that there are only a few insulating Heisenberg ferromagnets But thereexist several insulating ionic antiferromagnets that are well described by a HeisenbergHamiltonian with the addition of some single-ion anisotropy terms It is convenient to startwith a more general Hamiltonian that can describe ferrimagnets and as well as antiferro-

sublat-spins on the m sublattice have magnitude S1 and those on n magnitude S2 J1 and J2areexchange parameters within each sublattice An uniaxial anisotropy is incorporated by the

effective magnetic fields, H A,1 and H A,2 and g1and g2are the gyromagnetic ratios of thetwo types of ions Following a similar procedure as indicated in the case of the spin waves

in Heisenberg ferromagnets, one arrives at the spin wave dispersion equations

The above equations mean that the Hamiltonian (48) has two linear spin wave modes given

by (49) and (50) which are in general not degenerate So far the equations are quite general

in that we have assumed that the magnetic ions in the two sublattices are not the same

For an antiferromagnet, we have g1= g2= g, H A,1 = H A,2 = HA, S1= S2= S and J1=

For cubic crystals and for small values of q

Although the above two equations (62) and (63) hold strictly only for cubic crystals at

small q, it is often a good approximation for other cases So using (62) and (63) in (58) we

Fig 12 Schematic illustration of the crystal and magnetic structures of MnF2.

we will discuss this in some details MnF2crystallizes with tetragonal rutile type structure

(space group D144h , P 42/mnm) with lattice parameters a = 4.873, c = 3.130 Å There are

2 formula units in the unit cell and the atomic coordinates are give by: Mn: (2a) 0, 0, 0;

2 The only reflection

condition is given by h0l: h + l = 2n So nuclear reflections are absent for h0l: h + l =

neighbors along111 and third nearest neighbors along 100 and 010 MnF2orders at

((2a) 0, 0, 0) in the unit cell is parallel to001 and that of Mn atom at the body centered

position ((2a) 12,12,12) is oppositely oriented The propagation vector of this magnetic

structure is k= 0 The reflection condition of the magnetic reflections is h+k +l = 2n+1.

So in general magnetic reflections in neutron diffraction are superimposed on the structural

reflections However, h0l reflections with h + l = 2n + 1 have no structural contributions

and are purely magnetic Figure 12 shows schematically the crystal and magnetic structures

of MnF2

To measure spin wave dispersions along the two principal symmetry directions[100]

scatter-ing plane contains the (h0l) zone of the reciprocal space shown in Figure 13 This zone contains pure magnetic reflections (h + l = 2n + 1) and also some pure nuclear reflec-

tions (h + l = 2n) The large open circles show the positions of the nuclear reflections

whereas the large filled circles show the positions of the magnetic reflections They arealso indicated by the suffixes N and M The dashed lines show the antiferromagnetic zone

boundaries The small circles and squares indicate positions for well-focused constant-Q

scans of spin waves propagating along[100] and [001] directions Figure 14 shows such

constant-Q scans from MnF2 and Figure 15 shows dispersions of spin waves in MnF2along[100] and [001] directions Let us now discuss the exchange couplings in MnF2 Thestrongest exchange interaction in MnF2is between the next nearest-neighbor Mn atoms sit-

Fig 13 (h0l) zone of the reciprocal lattice of MnF2 The large solid circles represent nuclear reflections, whereas the large open circles represent magnetic reflections Small circles and squares represent positions for

well-focused constant-Q scans of spin waves propagating along the[100] and [001] directions, respectively (from

Shirane et al [2]).

Fig 14 Constant-Q scans of spin waves of MnF2at 10 K (from Shirane et al [2]).

uated at (000) and (12,12,12) positions They belong to two different magnetic sublattices This strong interaction, denoted by J2, is the antiferromagnetic superexchange interaction

via the fluorine ligands The nearest-neighbor Mn atoms along[001] are coupled by the

ferromagnetic exchange interaction J1, which is by about a factor five less than the next nearest-neighbor superexchange interaction J2shown from the inelastic neutron scatteringinvestigation of the spin wave dispersion of MnF2by Okazaki et al [24] The same study

Fig 15 (Top) Spin wave dispersion of MnF2along the [100] and [001] directions obtained from scans like those

shown in Figure 14 The solid lines are fits to (67) (Bottom) Corresponding integrated intensities corrected for

the magnetic form factor and also the Q-dependent factor involved in the scan process (from Shirane et al [2]).

also showed that the interaction between the third nearest-neighbor Mn atoms along[100]

has insignificant contribution to the spin wave dispersion The zone center spin wave gap

of the dispersion clearly shows the existence of substantial single-ion anisotropy energy D,

which can be explained by dipole–dipole interactions The spin wave dispersion [24] lation for MnF2corresponding to the Heisenberg Hamiltonian including a dipolar term isgiven by

Here a and c are the tetragonal lattice parameters of MnF2, z1= 2 is the number of nearest

neighbors and z2= 8 is the number of second nearest-neighbors If the dipolar term and the

nearest-neighbor ferromagnetic exchange interaction are neglected, then by setting D= 0

and J1 = 0 we get ζq= 0 The dispersion equation then reduces to

ignoring the tetragonal distortion of the lattice, i.e., assuming a = c So for small q a

linear dispersion is obtained This is typical for the spin wave dispersion of magnets as opposed to the quadratic dispersion in ferromagnets in the same limit The

antiferro-anisotropy energy D introduces an energy gap in the spin wave spectrum, while a finite value of the nearest-neighbor interaction J1causes the dispersion to be different along the

[100] and [001] directions

3.5 Two-magnon interaction in Heisenberg antiferromagnets

The effects of temperature on the spin wave dispersion of Heisenberg antiferromagnetscan be evaluated by extending the method used in Heisenberg ferromagnets For simplicityone considers the case of the nearest-neighbor exchange coupling We recall that for non-interacting spin waves in a nearest-neighbor Heisenberg model the dispersion equation can



The two-magnon interaction also renormalizes the spin wave dispersion of Heisenbergantiferromagnets like in the case of Heisenberg ferromagnets But in the case of Heisenbergantiferromagnets there is the additional feature of renormalizing the anisotropy field Tocalculate the temperature dependence of the dispersion given by (74) one has to solve a

complicated integral equation (75) by numerical methods However one can simplify the

situation by assuming hA= 0 in (75) C(T ) is then given by

wave theory outlined above The two sets of curves for T = 49.5 and 62.0 K correspond

to calculations with the renormalization factor in ¯hωq multiplied by that indicated in thefigure

3.6 Spin waves in Heisenberg ferrimagnets

The ferrimagnetic materials consist of two magnetic sublattices occupied by two differenttypes of ions having unequal moments oriented in the opposite directions They have likeferromagnets net magnetization for a particular domain Equations (48)–(50) are quite gen-eral and are valid for two sublattices which are not in general equal So they are valid forferrimagnets These equations mean that the Hamiltonian given by (48) leads to two linearspin wave modes with energy¯hω q,0and¯hω q,1 These two modes are not equal in the case

of ferrimagnets with two unequal sublattices A ferrimagnet can possess under certain ditions, thermodynamic properties similar to those of a ferromagnet To illustrate this we

con-Fig 16 The measured dispersions of MnF2[24] at T = 4.2, 49.5 and 62.0 K along with the calculations [26]

based on the renormalized spin wave theory outlined in above The two sets of curves for T = 49.5 and 62.0 K

correspond to calculations with the renormalization factor in¯hωqmultiplied by that indicated in the figure (from

Low [26]).

set HA,1 = H A,2 = 0, H = 0 and also J1= J2= 0 The dispersion equation for the two

spin wave modes is then reduced to

S1S2J (2)

S − S



Thus the spin wave dispersion of the acoustic mode for small q has a q2dependence just

as in the case of a ferromagnet discussed before Also under the above conditions thethermodynamic properties of a ferrimagnet can resemble those of a ferromagnet, e.g., the

magnetization decreases with temperature T 3/2 For nearest-neighbor coupling only, D of

a ferrimagnet can be written as

dis-lizes with cubic inverse spinel structure (space group O h7, F d3m) with the lattice parameter

occupied by Fe3+ions only whereas the sixteen octahedral B sites are occupied by equal

numbers of both Fe2+and Fe3 +ions in random distribution at room temperature Fe3O4

undergoes a charge ordering Verwey transition below TV≈ 119 K The resistivity increases

by several orders of magnitude below this transition The crystal structure changes fromcubic to monoclinic symmetry The charge ordering of formally Fe2+and the remaining

Fe3+ ions in octahedral sites become ordered below T

V Recent diffraction experimentssuggest that the charge ordering is only partial and the ordering pattern is still debated

Fe3O4is ferrimagnetic already at room temperature The ferrimagnetic transition ature of Fe3O4is TC≈ 848 K Néel [27] accounted for the observed saturation magnetiza-

temper-tion by postulating that the Fe ions in A and B sites are oriented antiferromagnetically The

Fe3+ and Fe2 + have S = 5/2 and 2, respectively and have therefore 5 and 4µB, tively So the resultant magnetic moment per formula unit at low temperature is expected

respec-to be (5 + 4) − (5) = 4µB The experimental value is about 4.1µB This magnetic structuremodel was verified by a powder neutron diffraction investigation [28]

The spin wave dispersion of Fe3O4single crystal was investigated by Brockhouse andWatanabe [29,30] by a triple-axis and also by a rotating crystal time-of-flight neutron spec-trometer at the Chalk River reactor The triple-axis spectrometer was used to study thehigher energy part of the acoustic branch and the optic branch The rotating crystal spec-trometer with incident cold neutrons was used to study the low energy part of the acousticbranch by using its high resolution The complete acoustic branch and a part of the opticbranch were determined along the[001] direction by constant-Q and constant-E scans.

The resultant dispersion is shown on top of Figure 17 To test to what degree the acoustic

dispersion curve is isotropic, constant-E scans for an energy transfer of E = 49.3 meV

were done along [001], [110] and [111] directions The values for the wave vectors of

the acoustic magnons of the above energy propagating in these directions were equal, to

well within the errors of the measurements, the mean reduced wave vector ( aq2π) being

0.585 ± 0.015 showing the isotropic nature of the spin wave dispersion The dispersion of

the acoustic branch for small q is proportional to q2as in the case of a ferromagnet Wealready discussed that ferrimagnets can also have similar behavior under certain conditions

Fig 17 (Top) Spin wave dispersion of Fe3O4at room temperature determined by inelastic neutron scattering (Bottom) Spin wave dispersion curves of Fe3O4calculated by Kaplan [33] (from Brockhouse [30]).

The theory of spin waves in Fe3O4is complex in algebra due to the complexity of theinverse spinel structure It has been worked out by several authors [31–34] The orderedinverse spinel structure may be considered to consist of six interpenetrating face centeredcubic lattices, two consisting of tetrahedral A sites and four of octahedral B sites There-fore the spin wave spectrum is expected to consist of six branches corresponding to the 6

Fe atoms in the primitive unit cell One of these branches is acoustic whose energy is

pro-portional to q2at small q The other five branches are called optic branches and have finite energies at q = 0 Three nearest-neighbor exchange interactions, JAB, JBBand JAA have

been considered Of these JAB, the exchange interaction between the nearest-neighbor Feions belonging to A and B sublattices, is by far the strongest and is antiferromagnetic

The other two nearest-neighbor exchange interactions JBB and JAA are very small On

bottom of Figure 17 the theoretical spin wave dispersion curves of Fe3O4calculated by

Kaplan [33] are shown The spin wave branches have the following properties at q= 0: The

acoustic branch (designated as 1) and one of the optic branches (designated 2) cause thestill fully-aligned A and B sublattices to precess about each other (with different phases for

1 and 2) Another optic branch (designated 4) disorganizes the A-sites, the B-sites

remain-ing fully aligned durremain-ing the motion A triply degenerate optic branch (designated 3, 5, 6)

disorganizes the B-sites, the A-sites remaining fully aligned The three-fold degeneracy is

removed for q = 0, but under the usual assumption of forces, branches 5 and 6 remain

de-generate Along[100] degeneracies occur at the zone boundary due to symmetry, branch 3

is continuous with branch 1 and so is branch 4 with branch 2 The degenerate branch (5, 6)

is flat for nearest-neighbor interaction only on the B-sites, this would not be so for morecomplicated B–B interactions The solid lines of Figure 17 show the dispersion curves cal-

culated by considering JAA only The resemblance of the lower branches to the measuredcurves is already very good and shows the general correctness of the theory The dashed

lines show the effect of including a small ferromagnetic JBBinteraction on the branches

1 and 3 The branches 2 and 4 are not much affected by JBBand similarly branches 1 and 3

are not affected by JAA For small q we get the following relation for the acoustic mode

The optic mode (2) intercept, i.e., the optic mode energy at q = 0, gives JAB directly

by (90) By fitting the experimental data with the calculations [34] an exchange

interac-tion JABof−2.4 meV was obtained The inclusion of a ferromagnetic exchange interaction

It was also noted that values of JAA smaller than 0.1JAB do not produce any significantchanges in the dispersion curves

To summarize, inelastic neutron scattering investigations have established that theHeisenberg localized model, despite the algebraic complexity due to the complex crystaland magnetic structures, is appropriate for the insulating ferrimagnet Fe3O4

4 Spin waves in itinerant magnetic systems

We have so far discussed spin waves in localized spin systems which can be well described

by the Heisenberg Hamiltonian But a localized Heisenberg model is not expected to workfor the iron-group transition metals such as Fe, Co and Ni and also their alloys, that aregood conductors The experimental saturation magnetization of Fe, Co and Ni is 2.216,

1.715 and 0.616µBper atom, respectively For a first attempt to explain the experimentalsaturation magnetic moment let us look at free Fe, Ni and Co ions for which we deter-mine the magnetic moments from Hund’s rules well known from the quantum mechanics

of atoms and ions Hund’s rules give the sequence of occupation of electronic states The

first rule requires that for a given configuration the term with maximum spin S possesses

the lowest energy The second rule requires for a given configuration and multiplicity the

term with the largest value of angular momentum L possesses the lowest energy In order

to possess the lowest energy, the third rules requires for a given configuration, multiplicity

and angular momentum, the value of the total angular momentum, J is minimum if the

configuration is less than half-filled shell, but is a maximum if the shell is more than halffilled The effective magnetic moment of the ion (from a Curie–Weiss fit of the suscepti-bility data) is then to a good approximation

where g is the Landé g-factor The angular momentum is completely quenched in the

transition metals, but the spin-orbit coupling gives a very small contribution To a good

ap-proximation the magnitude of the magnetic moment is given by the spin S of the electrons.

So depending on the assumed configuration the saturation magnetic moments of Fe, Niand Co are expected to be 5, 4 and 3 (or 4, 3 and 2), respectively The experimental resultsare very different from these values Thus we see that a simple-minded localized approach

to estimate the saturation magnetic moments fails completely for the conducting transitionelements, Fe, Ni and Co The localized picture also fails to explain the large specific heat

at low temperature that has been measured in these materials So the relevant approach is

to use the band formalism The 3d electrons become itinerant and participate in the lic cohesion The 3d electrons which are responsible for magnetism are mobile and like 4s electrons form bands with certain widths but narrower than the 4s bands Calculations

metal-using band formalism can explain such phenomena and the modern band calculations havebeen very successful [35] in giving a wide variety of ground state properties of transitionelements

4.1 Generalized susceptibility and neutron scattering cross-section

Neutron scattering cross-sections given in the previous sections are applicable to localizedelectron systems and have to be modified to deal with itinerant electrons For this purpose

we must first derive a Hamiltonian that incorporates the salient features of electrons in anarrow band We consider only a single band of electrons for simplicity This may appear

odd because the transition metals have five d bands However, the simple arguments

pre-sented below for a single band differ little from those that would be necessary in the case

of degenerate bands, only the latter are algebraically more complex Experiments suggest

that the d electrons of transition metals behave in such a way that in some cases they

appear to be well localized whereas in some other instances they appear to be itinerant.For example the magnetization distribution of transition metals determined by polarized

neutron diffraction shows more or less localized character whereas the large d electron

contribution to the low temperature specific heat and the reduced magnetic moment of thetransition elements are the properties which can only be explained by the band theory Thekey to understanding these conflicting localized and itinerant electron states is the correla-

tion between the electrons In spite of their band character, the d electrons on an atom are

strongly correlated with each other whereas the electrons belonging to different atoms areonly weakly correlated This strong intra-atomic electron correlation leads to a behavior

that appears more or less localized The Hamiltonian for the electrons in a narrow d band

must contain a repulsive interaction that operates between electrons of opposite spin on thesame atom This on-site strong repulsion leads to the ferromagnetic ground state as well asthe existence of spin wave modes The Hamiltonian for the electrons in a solid is given by

where pi and ri are the momentum operator and the position vector, respectively of the

ith electron The first term is the kinetic energy of the ith electron, the second term is

the periodic potential V (r) due to the ionic cores The eigenfunctions of this part of this

(band) Hamiltonian are well-known Bloch functions φ kσ (r) The last term is the Coulomb

interaction of the electrons This is the most difficult term to handle If one only considers

a single narrow band and assumes that the wave function centered about different site issmall (weak inter-atomic correlations) the Hamiltonian can then be reduced to

ergy and the second term, whose strength is given by I , represents the Coulomb interaction

of electrons of opposite spins at the same atom site This repulsive interaction between theelectrons of opposite spins on the same atom leads to ferromagnetism The Hamiltoniangiven by the (94) is called the Hubbard Hamiltonian because its properties were first dis-cussed by Hubbard [36,37] The generalized susceptibility of the magnetic electrons can

of... class="page_container" data-page="15">

4.1 Generalized susceptibility and neutron scattering cross-section

Neutron scattering cross-sections given in the previous sections are applicable... summarize, inelastic neutron scattering investigations have established that theHeisenberg localized model, despite the algebraic complexity due to the complex crystaland magnetic structures,