X- RAY SCATTERING STUDIES OF LANTHANIDE MAGNETISM 81 lanthanide series seems well accounted for. The same arguments apply to the variation of
6. Summary
Starting with the first experiments performed by de Bergevin and Brunel on NiO in 1972, remarkable progress has been made in the study of magnetism using X-rays. Progress was at first steady, but has accelerated rapidly within the last decade as more and better sources of synchrotron radiation have come on line, and a greater understanding of how to exploit them has been won. X-ray studies of the lanthanides in particular have produced a great wealth of information. This includes, on the one hand, a deeper insight into the specific magnetic properties of these elements, while on the other, it has allowed general principles of the X-ray scattering cross-section to be both explored and developed. On the theoretical front the salient features of the non-resonant and the resonant cross-sections are now well understood. Probably the main challenge here is to establish a framework for the resonant cross-section that is capable of explaining the discrepancies from the one-electron view of the resonant process, such as the asymmetry in the branching ratios (L-edges) for the light and heavy lanthanides, and to push into the inelastic regime. Presently, the pace of innovation in experimental techniques shows no sign of slowing, with recent examples including the observation of surface magnetic scattering (Ferret et al. 1996, G.M. Watson et al. 1996), and the application of high-energy X-rays to the study of magnetic phase transitions in transition metal compounds (Brfickel et al. 1993). It is certain that more extensive use will be made of the polarization dependence of the X-ray cross-section, most immediately by exploiting ¼-wave plates on undulator sources (Sutter et al. 1997).
In this regard it is worth making the point that polarization analysis is in many ways easier to perform for X-rays than neutrons, and this gives even greater incentive to develop fully these techniques.
Acknowledgements
We would like to express our deep gratitude to all of our colleagues who have contributed to the work described in this review. In particular we would like to thank to John Hill for his careful reading of the manuscript. The work at Brookhaven is supported under a grant by the US DOE under contract No. DE-ACH02-276CH00016.
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Handbook on the Physics and Chemistry of Rare Earths VoL 26
edited by K.A. Gschneidner, Jr and L. Eyring
© 1999 Elsevier Science B. V All rights reser•ed
Chapter 170
S T A T I C A N D D Y N A M I C S T R E S S E S A.M. TISHIN and Yu.I. SPICHKIN
Faculty o f Physics, M. E Lomonosov Moscow Stare University, Moscow, 119899, Russia
J. B O H R
Department o f Physics, Technical University o f Denmark, Building 307, DK-280Õ Lyngby, Denmark
C o ~ e ~ s
List of symbols Abbreviations 1. Introduction
2. Effect of static pressure on the phase transition to a magnetically ordered state 2.1. Systems with localized magnetic
moments
2.1.1. General considerations
2.1.2. Lanthanide metals and their alloys 2.1.3. Lanthanide nonmagnetic element
compounds
2.2. Lanthanide 3d laansition metal systems 2.2.1. General considerations for the R -
Fe, R - C o and R - N i compounds 2.2.2. First-order transitions in RCo 2 2.2.3. R2Fel7 compounds
3. Influence o f static pressure on the magnetic phase diagrams and magnetic orde~order phase transitions
4. The effect of static pressure on the spin structures o f the lanthanide metals 5. Influence o f static pressure on the
magnetization
6. Sound attenuation and internal friction 6.1. Introduction
6.2. Paramagnetic phase
88 6.2.1. Ultrasound attenuation 122
89 6.2.2. Internal friction 129
89 6.3. Magnetically ordered state 132 6.3.1. Ultrasound attenuation 132
90 6.3.2. Internal friction 138
7. Elastic properties 140
90 7.1. Introduction 140
90 7.2. Anomalies near magnetic transitions 140 94 7.2.1. Thermodynamic consideration 140 7.2.2. Microscopic models 145 101 7.3. Magnetically ordered stare of the heavy 103 lanthanide metals and their alloys 146
7.3.1. Helical phase 146
103 7.3.2. Ferromagnetie phase 152
110 7.4. Gadolinium 156
112 7.5. The effect of commensurate magnetic structures on the elastic properties 158 7.6. Young's moduli of the metals and their
113 alloys 162
7.7. Elastic properties of R-Fe and R - C o
115 intermetallic compounds 163
7.8. Higher-order elastic constants of the
119 metals 167
122 8. Conclusion 170
122 Acknowledgments 170
122 References 170
87
List o f
C/
ai A b
a c~
B12, B22, B r, B ~ C
c«
cd
¢ij A c o e
««
E A E
EF Eù,o E=
F
G
gJ H 7-[
Her Hma~
h
I0, I, Ioù.
I(0)
I(Q) J K
K,,r~S kB
k~
N *
symbols
a-axis lattice constant lattice constaut amplitude of sormd wave b-axis lattice constant
armihilation and creation phonon operators single-ion magnetoelastic coupling constants
c-axis lattice constant
c-axis basis vector of reciprocal unit cell Curie constant of the lanthanide ions d-electron Curie constant
elastic constants change in elastic constant electronic charge
polarization vector o f a sound wave Young's modulus
change in Young's modulus Fermi energy
magnetoelastic energy anisotropic energy free energy
de Germes factor = (~j 1)2j(j + 1) Landé factor
magnetic field Hamiltonian critical magnetic field
maximum value of critical magnetic field Planck's constant divided by 2g exchange interaction integrals effective exchange interaction integral between itinerant electrons
paramagnetic indirect exchange integral Fourier transformation of exchange integral
total angular magnetic moment quantum number
kinetic coefficient
magnetocrystalline anisotropy constants Boltzmann's constant
wave vector
conduction electron wave vector at the Fermi smface
effective mass of the conduction electron
M~j,z« first-order magnetoelastic interaction t e n s o r
nRR, nRM molecular field constants
N number of lanthanide ions per unit volume N ( £ ) density of states per unit volume P, Pi tmiaxial mechanical stress
P pressure
magnetic spin smacture wave vector Q< internal friction
Qs spin-slip magnetic structure ware vector reciprocal space vector
R distance between ions /~~ atom position vector
interatomic distance vector F conduction electron position vector Ro,kloo second-order magnetoelastic interaction
tensor
s 0 elastic compliance constants S spin momentum quantum number
spin angular moment T absolute temperature T c Curie temperature T N Néel temperamre
Tp paramagnetic Curie tempera~lre ATNc interval where helical strucmre exists T~~ spin-reorientation transition temperamre T d Curie temperature of the d-electron system Tcy transition temperature to cycloidal
structure
T D Debye's temperature
Bj components of the mechanical stress tensor
ü lattice displacement vector
V volume
Va atomic volume
U D U t longitudinal and shear sound velocities x concentration
W bandwidth
z coordination number Z ionic charge
a,/3, y, e thermodynamic coefficients in free energy expansion
a T linear coefficient of thermal expansion
STATIC AND DYNAMIC STRESSES 89 a M molecular field constant for collective a 0
d-electrons a~
al, at, ŒÜ sound attenuation coefficients as F(~]), F 0 s-f exchange interaction integrals r A~i degree of relaxation of Young's modulus ¢ 6(r) Dirac's function
e~ mechanical strain tensor components cp
t¢ volume compressibility Xd
A linear magnetostriction
#B Bohr magneton Zs
#s saturation magnetic moment
#«f effective magnetic moment Zd0
v critical exponent of the attenuation
coefficient Xhf
p density X(q)
Pm magnetic contribution to resistivity
a magnetization m
saturation magnetization at T = 0 K component of magnetization vector saturation magnetization
relaxation time
angle between the magnetic moment and c-axis
helical turn angle
magnetic snsceptibility of collective d-electrons
magnetic susceptibility of lanthanide spin system
magnetic susceptibility of noninteracfing eollective d-electrons
high field magnetic susceptibility generalized conduction eleetron magnetic susceptibility
frequency
Abbreviations
FM ferromagnetic phase R
HAFM helicoidal antiferromagnetic structure TM
LSW longitudinal spin wave RKKY
lanthanide (rare earth) metals transition metal
Ruderman-Kittel-Kasuya-Yosida