Rare Earth Magnetism

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0	PREFACE			i

1	ELEMENTS OF RARE EARTH MAGNETISM			1

	1.1	A brief history		2
	1.2	Rare earth atoms		8
	1.3	The metallic state		16
	1.4	Magnetic interactions		39
	1.5	Rare earth magnetism		50

2	MAGNETIC STRUCTURES			68

	2.1	Mean-field theory of magnetic ordering		68
		2.1.1	The high-temperature susceptibility	71
		2.1.2	The mean-field approximation	74
		2.1.3	Transversely ordered phases	79
		2.1.4	Longitudinally ordered phases	83
		2.1.5	Competing interactions and structures	85
		2.1.6	Multiply periodic structures	89
	2.2	The magnetic anisotropy		94
		2.2.1	Temperature dependence of the Stevens operators	94
		2.2.2	Anisotropic contributions to the free energy	101
	2.3	Magnetic structures of the elements		111
		2.3.1	Bulk magnetic structures	112
		2.3.2	The magnetization of Holmium	125
		2.3.3	Films and superlattices	130

3	LINEAR RESPONSE THEORY			134

	3.1	The generalized susceptibility		134
	3.2	Response functions		137
	3.3	Energy absorption and the Green function		142
	3.4	Linear response of the Heisenberg ferromagnet		149
	3.5	The random-phase approximation		154
		3.5.1	The generalized susceptibility in the RPA	154
		3.5.2	MF-RPA theory of the Heisenberg ferromagnet	160

4	MAGNETIC SCATTERING OF NEUTRONS			163

	4.1	The differential cross-section in the dipole approximation		164
	4.2	Elastic and inelastic neutron scattering		173

5	SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS			181

	5.1	The ferromagnetic hcp-crystal		182
	5.2	Spin waves in the anisotropic ferromagnet		186
	5.3	The uniform mode and spin-wave theory		198
		5.3.1	The magnetic susceptibility and the energy gap	198
		5.3.2	The validity of the spin-wave theory	206
	5.4	Magnetoelastic effects		211
		5.4.1	Magnetoelastic effects on the energy gap	211
		5.4.2	The magnon-phonon interaction	219
	5.5	Two-ion anisotropy		231
		5.5.1	The dipole-dipole interaction	232
		5.5.2	General two-ion interactions	240
	5.6	Binary rare earth alloys		247
	5.7	Conduction-electron interactions		256
		5.7.1	The indirect-exchange interaction	256
		5.7.2	The mass-enhancement of the conduction electrons	270
		5.7.3	Magnetic contributions to the electrical resistivity	275

6	SPIN WAVES IN PERIODIC STRUCTURES			285

	6.1	Incommensurable periodic structures		286
		6.1.1	The helix and the cone	287
		6.1.2	The longitudinally polarized structure	300
	6.2	Commensurable periodic structures		305

7	CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS			312

	7.1	MF-RPA theory of simple model systems		313
	7.2	Beyond the MF-RPA Theory		322
	7.3	Perturbations of the crystal-field system		334
		7.3.1	Magnetoelastic effects and two-ion anisotropy	334
		7.3.2	Conduction-electron interactions	342
		7.3.3	Coupling to the nuclear spins	349
	7.4	Magnetic properties of Praseodymium		352
		7.4.1	Induced magnetic ordering	352
		7.4.2	The magnetic excitations	361

8	PERSPECTIVES FOR RARE EARTH RESEARCH			370

	REFERENCES			381

	INDEX			398

	APPENDIX (REVIEW)			404

Martin Rotter,
McPhase is a program package for the calculation of magnetic properties,
McPhase: A Mean Field Monte Carlo Phase Diagram Program.

J. Jensen,
Linear Response Theory and the Random Phase Approximation (International workshop on McPhase, Heinz Maier - Leibnitz Zentrum, Munich, 5.-7. November 2014)
.

R. A. Cowley and J. Jensen,
Magnetic structures and interactions in erbium,
J. Phys. Condens. Mat. 4, 9673-96 (1992).

J. Jensen and R. A. Cowley,
Magnetic non-planar structures and trigonal couplings in erbium,
Europhys. Lett. 21, 705-10 (1993).

K. N. Clausen, K. A. McEwen, J. Jensen, and A. R. Mackintosh,
New mode of magnetic excitation in praseodymium,
Phys. Rev. Lett. 72, 3104-7 (1994).

J. Jensen,
1/z-renormalization of the mean-field behavior of the dipole-coupled singlet-singlet system HoF₃,
Phys. Rev. B 49, 11833-41 (1994).

K. N. Clausen, S. Aagaard Sørensen, K. A. McEwen, J. Jensen, and A. R. Mackintosh,
Crystal fields and the conduction electrons in praseodymium,
J. Magn. Magn. Mater. 140-144, 735-8 (1995).

R. M. Nicklow and N. Wakabayashi,
Spin excitations in the longitudinally modulated magnetic phase of erbium,
Phys. Rev. B 51, 12425-32 (1995).

J. Jensen,
Theory of commensurable magnetic structures in holmium,
Phys. Rev. B 54, 4021-32 (1996).

J. Jensen,
Threefold symmetric magnetic two-ion coupling in hcp rare-earth metals,
Acta Phys. Pol. A 91, 89-103 (1997).

Magnetism in metals - A symposium in memory of Allan Mackintosh, Copenhagen, 26-29 August, 1996, eds.: D. F. McMorrow, J. Jensen, and H. M. Rønnow,
Mat. Fys. Medd. Dan. Vid. Selsk. 45 (1997), 443 p.

D. A. Tindall, M. J. Small, M. O. Steinitz, and Z. Tun,
Experimental verification of the significance of trigonal coupling for the 1/4 lock-in in holmium,
Physica B 241-243, 699-701 (1998).

M. Ellerby, K. A. McEwen, and J. Jensen,
Magnetoresistance and magnetization study of thulium,
Phys. Rev. B 57, 8416-23 (1998).

J. Jensen,
Magnetic structures and excitations in rare-earth metals.
In: Magnetic and Superconducting Materials, eds. M. Akhavan, J. Jensen, and K. Kitazawa (World Scientific, Singapore, 2000) Vol. B, 711-26.

M. Ellerby, K. A. McEwen, E. Bauer, R. Hauser, and J. Jensen,
Pressure dependent resistivity and magnetoresistivity of erbium,
Phys. Rev. B 61, 6790-97 (2000).

R. S. Sarthour, R. A. Cowley, R. C. C. Ward, M. R. Wells, D. F. McMorrow, and J. Jensen,
The magnetic phase diagram of Ho_xTm_1-x alloys,
Phys. Rev. B 62, 14943-9 (2000).

H. Ohsumi
Magnetoelastic effect in holmium studied by x-ray diffraction,
J. Phys. Soc. Jpn 71, 1732 (2002).

I. D. Hughes, M. Dãne, A. Ernst, W. Hergert, M. Lõders, J. Poulter, J. B. Staunton, A. Svane, Z. Szotek, and W. M. Temmerman,
Lanthanide contraction and magnetism in the heavy rare earth elements,
Nature 446, 650 (2007).

K. M. Dõbrich, A. Bostwick, J. L. McChesney, K. Rossnagel, E. Rotenberg, and G. Kaindl,
Fermi-surface topology and helical antiferromagnetism in heavy lanthanide metals,
Phys. Rev. Lett. 104, 246401 (2010).