Magnetic anisotropy or better Magnetocrystalline anisotropy The - - PowerPoint PPT Presentation

• The magneocrystalline anisotropy energy is

expressed by the formula :

• where represents the Legendre polynoms,
• anisotropy constants,

 and  are the polar and azimutal angles (sphéric coordinates)

• f magnetization direction.

Ea ,

 

Kn mPn m cos

 

cos m m0 n



n0

 

Pn m

Kn m

M C axis





Magnetic anisotropy or better Magnetocrystalline anisotropy

• K.H.J. Buschow and F.R. de Boer (2003) “Physics of Magnetism

and materials” Kluver Academic/Plenum publisher

Magnetic anisotropy or better Magnetocrystalline anisotropy

uniaxial symmetry

 

         

4 2 2 1

sin sin , K K Ea

 

... ) 6 cos( sin sin sin sin ,

4 4 6 3 4 2 2 1

            K K K K Ea

Hexagonal symmetry E

a

is an important parameter for soft and hard magnetic materials

C axis M  M C axis





Crystal electric field

• Coupling with the 4f electronic shell
• Favors the orientation of the shell in the

electric field gradient

V(r)  Al m ml

l 

l



rlYl m(r)

Al m

Yl m

Spherical harmonics Crystal electric field coefficients

• Hutchings, M. T. (1964) Solid state phys.,

16, 227.

• Stevens coefficients

Anisotropy constant may vary with T !

May even change of sign

How to identify the Easy Magnetization Direction

200 400 600 800 1000 1200 30 40 50 60 70 80

Intensity (a.u.) 2  (°) field-oriented free powder Si Si Si ( 1 0 1 ) ( 0 0 1 ) ( 1 1 0 ) ( 1 1 1 ) ( 2 0 0 ) ( 0 0 2 ) ( 1 1 2 ) ( 2 1 1 ) ( 2 0 1 )

40 80

H0 104 A/m M 106 A/m

1.2 0.8 0.4 X-ray diffraction Neutron diffraction experiments Magnetization curves

Anisotropy field

Magnetization (B / u.f.) Applied field (kOe) Nd2Fe13GaBH3.7 (300 K)

 M//c  Mc

Ha = 2 K1 / µ0 Ms +… Ha = 2 K1 / µ0 Ms

H

c

< H

a C axis M  Higher order : See talk O. Fruchart

Rare earth contribution to magnetocrystalline anisotropy

)) 1 ( 3 (   J J J z

J second order Stevens coefficient (J) for R 3+ J > 0 J < 0

Purely Atomic parameter

K1 T  0

   3

2 J r2 4f (3Jz  J(J 1))A2

r2 4f

A2

Crystal electric field gradient

J r2 4f

(3Jz  J(J 1))

Quadrupolar moment 4f shell

Atomic environment electric charges

Remark :

• References
• K.N.R. Taylor et M.I. Darby, Physics of Rare Earth Solids, edited by

Chapmam and Hall Ltd, London, 1972, p. 22. ISBN 0 412 101602.

• K.W.H. Stevens, Proc. Phys. Soc., 65, (1951), p. 209.
• K.H.J. Buschow and F.R. de Boer (2003) “Physics of Magnetism

and materials” Kluver Academic/Plenum publisher

• K.H.J. Buschow, Magnetism and Processing of Permanent Magnet

Materials, Handbook of Magnetic Materials Volume 10, édited by K.H.J. Buschow, Elsevier Science B.V., Amsterdam (1997), Ch. 4,

• p. 463. ISBN 0 444 853138.

Ion Ce 3+ Pr 3+ Nd 3+ Sm 3+ Tb 3+ Dy 3+ Ho 3+ Er 3+ Tm 3+ Yb 3+ J 102

• 5,71
• 2,10
• 0,64

+4,13

• 1,01
• 0,63
• 0,22

+0,25 +1,01 +3,17

2

nd

• rder Stevens coefficient

J > 0 J < 0 Similar equation for K2; K3 and so on

200 400 600 800 1000

YCo5 YCo4Ga PrCo5 PrCo4Ga

T (K)

c-axis canted (a, b) plane

20 40 60 80 100 50 100 150 200 250 300

 (EMD- c axis) (°) T(K) TSR = 160 K PrCo4Ga

200 400 600 800 1000 1200 30 40 50 60 70 80

Intensity (a.u.) 2  (°) field-oriented free powder Si Si Si ( 1 0 1 ) ( 0 0 1 ) ( 1 1 0 ) ( 1 1 1 ) ( 2 0 0 ) ( 0 0 2 ) ( 1 1 2 ) ( 2 1 1 ) ( 2 0 1 )

2 4 6

4 K 168 K 250 K Mz (µB/f.u.)

• 4
• 2

2 4 6 90 180 270 360

4 K 168 K 250 K Mx (µB/f.u.)  (°)

More complex behaviour :

Complex behaviour

100 200 300 400 500 600 700 CeFe

11Ti

PrFe

11Ti

NdFe

11Ti

SmFe

11Ti

GdFe

11Ti

TbFe

11Ti

DyFe

11Ti

HoFe

11Ti

ErFe

11Ti

TmFe

11Ti

LuFe

11Ti

YFe

11Ti

Axial Conical Planar Temperature ( K )

Magnetic phase diagram