information from magnetisation curvesmagnetism.eu/esm/2007-cluj/slides/pop-slides.pdf · 1 p.r....
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![Page 1: Information from magnetisation curvesmagnetism.eu/esm/2007-cluj/slides/pop-slides.pdf · 1 P.R. Rhodes, E.P. Wolfarth, Proc. R. Soc. 273 (1963) 347. 2 O. Isnard, V. Pop, K.H.J. Buschow,](https://reader034.vdocuments.us/reader034/viewer/2022051909/5ffcfe1746f16d3fff4c6e4c/html5/thumbnails/1.jpg)
Information from magnetisation curves
Viorel PopFaculty of Physics, Babes-Bolyai University, 4000480 Cluj-Napoca, Romania
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Fundamental properties of matter and Applications
Physics is DIFFICULT and HARD to understand
keep away of physic
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Magnetic moments of the electrons
Magnetism in mater
Magnetic moment of the atoms
Magnetic study of the mater
Fundamental properties and Applications
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Magnetic moment of the atoms
nucleusLr
lµr
vr -e
Sr
Kinetic moment of a charge
Magnetic moment: Jgm B rr µ=
Lr
Sr
Orbital kinetic moment
Spin kinetic moment
)1(2 )1()1()1(1+
+−++++= JJ LLSSJJgLandé factor
( )
( )CGScme SIme eB eB 22 hh=
=
µ
µ
μμμμB ≡≡≡≡ spinspinspinspin magnetic momentul of free electronμB = 9,2742⋅10-24 A⋅m2
2g-g sss == ;Smem e r
r
2
1g-g == ll ;Lmem el r
r
2
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V∑=
mM
r
r
magnetisation M magnetic susceptibility χχχχmagnetic permeability μμμμ
( )MHB 0 rrr
+= µ
μμμμ0 = 4π⋅π⋅π⋅π⋅10-7 H/m
HM=χ
B=µ0(H+χχχχH)= µ0(1+χχχχ)H= µH HB=µ
( )χ1µ µ 0 +=
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C, Cu, Pb, H2O, NaCl, SiO 2
M Tχ
χ H(a) (b)Diamagnetic
Paramagnetic
Magnetic ordered
0m =r
χχχχ < 0
0m ≠r
J ij = 0χχχχ > 0
0m ≠r
J ij ≠ 0χχχχ >> 0
χ ≠ f(H)
CBeff ⋅⋅⋅⋅==== 8)(µµµµµµµµ CBeff ⋅⋅⋅⋅==== 4664,)(µµµµµµµµ)( 1+= JJg Beff µµ
Na, Al, CuCl 2
χ
M
H
1/C
1/χT
lawCurie;1 CT=−χ
JCN kBeff 03
µµ
⋅=
if χ(emu/mole) if χ(µB/T∙f.u)
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a) ferromagneticjijJH SSi rr⋅−= 2 J ij > 0
Fe, Co, Ni, Gd…
Ms ≠ 0
1/χχχχM
TTc θ
θχ
−=
T
C
T1<T2<Tc<T3
T1
T2
T3
M
H
Ms(0) = gJ µB J
Molecular field approximation
Curie – Weiss law
θθθθ = Tc
MNH iim ====
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b) antiferromagnetic
Ms=0
χχχχ
T
χχχχχχχχ⊥⊥⊥⊥
TN
1/χ
TTNθθθθ
θθθθ < 0
θχ
+=+= TCHMM BA
Η=0Η Η ΗΗ
J ij < 0BA MM rr=
MnO, Mn, Cr…
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T>Tc
1/χ
T0 Tcθ
'θθθθσσσσ
χχχχχχχχ −−−−−−−−++++==== TCT011
M=MA-MB
M=MA-MB
M=MA-MB
M=MA-MB0
T0
T0
TMB MB
MB
MA MAMA
M0
M0M0
Tc
Tc
Tc
Tcomp
M MMNAA≈NBB NAA<NBBNAA>NBB
T<Tc
(a) (b) (c)
c) ferrimagnetism
Ms≠≠≠≠0
J ij < 0
Fe3O4, ferrites, GdCo5,…
MA ≠ MB
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M
H
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HM=χχχχ
M
Hχiχm
Hsat
MstM0 χ0
Msat
Ms
0== Hi dHdMχ
χ
H
T<Tcχi
Mst = saturation magnetisationMs = spontaneous magnetisationχm
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1/χχχχMs
TTc θ
θχ
−=
T
C
Ms(0)
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1/χχχχMs
TTc θ
θχ
−=
T
C
Ms(0)
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M
H
Ms(T1)
Ms(T2)
Ms(T3)
χχχχp
T1
T2
T3
T1 < T2 < T3
HMM ps χχχχ++++====
01234560 100 200 300 400 500
Al5Mn3Ni2Ms (µB/f.u.) T (K)Ms = 5.03 µB/f.u.Tc = 401 K 0123450 2 4 6 8 10
Al5Mn3Ni2T = 10 K M(µB/f.u.) µ0H (T)3.63.844.24.44.64.855.25 6 7 8 9 10T = 10 K
T = 100 KT = 200 KT = 300 K
y = 5.0293 + 0.00043122x R= 0.10622
y = 4.7774 + 0.009374x R= 0.96475
y = 4.218 + 0.023918x R= 0.99702
y = 3.535 + 0.022673x R= 0.99868
M(µB/f.u.) µ0H (T)
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HHaMM ps χχχχ++++ −−−−==== 1
0
0,5
1
1,5
2
2,5
3
0 2 4 6 8 10
GdCo4SiT = 4 KM(µB/f.u.)
µ0H (T)2,22,32,42,52,62,72,82,932 3 4 5 6 7 8 9 10M(µB/f.u.) µ0*H (T)
Ms(T)T→0KFor the rare earth (Gd for example)M0 = NgµB J
Ms(0) ≡ M0For 3d transition metals (Fe, Co, Ni…), the orbital moment is blocked by crystalline field:M0 = NgµB S; g ≈ 2
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1/χχχχM
TTc
θχ
−=
T
C
?BoBiic k JJNgNT 3 122 )( ++++
====µµµµµµµµ
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Curie temperature evaluation
T →→→→ Tc; T < Tc −−−−++++++++
++++⋅⋅⋅⋅==== cTTJJ JM TM 1113100 22 22 )( )()( )(
T2sM
Tc
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Curie temperature evaluation
T →→→→ Tc; T < Tc −−−−++++++++
++++⋅⋅⋅⋅==== cTTJJ JM TM 1113100 22 22 )( )()( )(
05101520
012345
300 320 340 360 380 400LuFe4B
M2 (a.
u.) M (a.u.)
T (oC) Tc010203040506070
0246810
0 200 400 600 800M2 (a.
u.) M (a.u.)T(oC) T
2sMTc
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Curie temperature evaluation
T →→→→ Tc; T < Tc −−−−++++++++
++++⋅⋅⋅⋅==== cTTJJ JM TM 1113100 22 22 )( )()( )(
0246810
200 400 600 800 1000 1200
ThFe11C1.5M(a.u.)
T(K)01020304050300 350 400 450 500 550 600 650 700M2 (
a.u.
)
T(K)O. Isnard, V. Pop, K.H.J. Buschow, J. Magn. Magn. Mat. 256 (2003) 133
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234567890 200 400 600 800 1000M (a.u.) T (oC)
0246810
200 400 600 800 1000 1200SmCo5+20Fe_8hMMM(a.u.)T(K)
Phase analysis and thermal evolution
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MHMbMaMFm 042 42 µµµµ−−−−⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅++++++++====)(In the low magnetisation region - for example T →→→→ Tc; T < Tc
0====dMdFm HbMaM 03 µµµµ====++++ or babHMM −−−−==== 02 µµµµ
(((( ))))(((( ))))
HMCJM TJJMT TTN c cii 03220200 110 1223 µµµµµµµµµµµµ ====++++
++++++++++++−−−− )(molecular field approximations:
Nii = Tc/C
Hm = Nii M
(((( ))))c ciiT TTNa −−−−==== 0µµµµ
(((( )))) CJM TJJb 22020 110 1223++++
++++++++====)(µµµµ
T <<<< TcT = Tc T >>>> Tc.
a < 0a = 0a >>>> 0
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020406080100120
0 0.2 0.4 0.6 0.8 1ThFe11C1.5
M2 (µB/f.u.)2µ0H/M (T*f.u./µB)
400 K 440 K420 KO. Isnard, V. Pop, K.H.J. Buschow, J. Magn. Magn. Mat. 256 (2003) 133
M2 T2 T3Tc T4T5T1H/M1/χ
Ms2 H/M3 H/M4H/Mc02sMT1 T2 T3
TcT4
T5
T1 <<<< T2 <<<< T3 <<<< Tc <<<< T4 <<<< T5.
M2
−−−−
−−−−++++ −−−−==== 34 4334 MHMH MHMHTMHMHTT ccc
Arrott plotA. Arrott, Phys. Rev. 108, 1394 (1957)
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1/χχχχM
TTc θ
θχ
−=
T
C
Ms(0)
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M
H
HM====χχχχ
Paramagnetic sampleor
Ferromagnetic sample at T>Tc
If there are some ferromagnetic impurityscMHM ++++==== χχχχ HMcHM s++++==== χχχχ
HMH1
χχχχ
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CBeff ⋅⋅⋅⋅==== 8)(µµµµµµµµ CBeff ⋅⋅⋅⋅==== 4664,)(µµµµµµµµ)( 1+= JJg Beff µµ JCNkBeff 03
µµ
⋅=
if χχχχ(emu/mole) if χχχχ(µµµµB/T∙∙∙∙f.u) TTN Tc Tc
θθθθχχχχ
++++==== TC
θθθθχχχχ
−−−−==== TC
'θθθθσσσσ
χχχχχχχχ −−−−−−−−++++==== TCT011
TC====χχχχ
θθθθ θθθθθθθθ
1/χ
Ferro
para
Antiferro
Ferri
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Ms(0) = gJ µB J0)( 1++++==== ppBeff JJgµµµµµµµµ
T→0K T > Tc
Ms(0) ≈ 2 µB S0 )1( +≈ ppBeff SSgµµT→0K T > Tc10 >>>>==== SSr pFor 3d transition metals (Fe, Co, Ni…), the
orbital moment is blocked by crystalline field:
For the rare earth (Gd for example): J0=Jp
1/χχχχM
TTc θ
θχ
−=
T
C
Ms(0)
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1 P.R. Rhodes, E.P. Wolfarth, Proc. R. Soc. 273 (1963 ) 347.2 O. Isnard, V. Pop, K.H.J. Buschow, J. Magn. Magn. M at. 256 (2003) 1333 D. Bonnenberg, K.A. Hempel, H.P.J. Wijn, Landolt-B. orsntein new series, Vol. III, 19a,
Springer, Berlin, 1986, p. 142.4 N. Coroian, PhD thesis5 R. Ballou, E . Burzo, and V. Pop, J. Magn. Magn. Ma t. 140-144 (1995) 945.
r = 1 local moment limitr →∞∞∞∞ total delocalisation limit →∞∞∞∞2.031.691.51.321.011.00r
YCo3B25HoCo 4Si4Fe3C3ThFe11C1.5
2Co1Fe1Gd1
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HH || easy magnetisation direction
HaM
∆MH ⊥⊥⊥⊥ easy magnetisation directionH
H || easy magnetisation directionHa
M∆MH ⊥⊥⊥⊥ easy magnetisation direction
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low anisotropy energyspin – flop transition
H(z)(z) H(z)H MbMa
MaMaMb Mb
M
H0 (a) (b)
T < TN, antiferromagnetic materials, χχχχ⊥⊥⊥⊥ >>>> χχχχ
E. Du Trémolet de Lacheisserie (editor), Magnetisme, Presses Universitaires de Grenoble, 199 9
exac HHH ====
Density of energy in magnetic field H,
E = -χµχµχµχµ0H2/2
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M
H0(a)(b)(z) H MbMa
H(z)Ma MbH(z)Ma Mb
High anisotropy energyspin – flip transition
spin–flip
andspin–flop
also in ferrimagnetic materials
metamagnetic transition
exc HH ≈≈≈≈
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( ) ( )( )( )
( )χµµµχµµ
µχµχµµ
+=⋅=+⋅=
⋅=+⋅=+=+=
1
1
1
00
0
00
r
r
HH
HHMHB
µµµµi µµµµµµµµm
B
H
H
µiµmµ
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M
H-Hc
HcH
Mr-Mr
B
B’A
CD
E F
O
reversible
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µ >> 0magnetic coresmagnetic circuitssoft
Hc – small~0,001÷10 A/m
permanentmagnetshard
Hc – BIG ~102÷106 A/m
MH-Hc Hc
HM r-M r
BB ’ A CDE FO
( )MHBrrr
+= 0µ
Curie temperature, Tc
Fundamental research M
Application research B
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Hard magnetic materials
-HB
BH(BH)max0P(BH)=const
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MH
MH
Rotation mechanism Pinning mechanism
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Exchange bias field
E. Girgis et al, J. Appl. Phys. 97 (2005) 103911
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M HM H
Spring magnets
-100-50050100
-4 -3 -2 -1 0 1 2 3 4SmCo5 + 20Fe/8h MM
as milled
450oC 0.5h
500oC 1.5h
550oC 1.5h
600oC 0.5h
650oC 0.5hSmCo
5/2h MM
M (emu/g) H (T)SmCo 5
Fe
Hr0H ====r
SmCo 5
Fe
θ
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-50050
-2 -1.5 -1 -0.5 06h+450C/0.5h2h+450C0.5hM (emu/g)
H (T)T = 4 K
050100150200250-8 -6 -4 -2 0
SmCo5_20 wt%FeT = 300K 2h/450C30'4h/450C30'6h/450C30'8h/450C30'dM/dH
H (T)
The reversibility curvesand the dM/dH variationvs H are very fine instruments in qualitative evaluation of the inter-phase hard/soft exchange coupling.
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Ha
+ ++
++++
- -- - - - -
Ms Hd
+ ++ + + ++
- -- - - - -
Ms HdMH rr dd N−−−−====
Ha
sd MNH II==== sd MNH ⊥⊥⊥⊥====
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dai HHHH rrrr++++======== Ha = applied field
(((( )))) (((( ))))ad dadaN NH1M MHHHHMχχχχ
χχχχχχχχχχχχχχχχ
++++====
−−−−====++++========
M
H
M
Ha
1/Nd
MHrr dd N====
The influence of the demagnetising field on the mag netisation curves
Ndx ==== Ndy ==== Ndz ====1/3.
(((( )))) MHd rr⋅⋅⋅⋅−−−−−−−−==== θθθθcos1l
θO
Mrdsphere d << l Nd = 0d >> l Nd = -1
H II easy magnetisation direction
Nd = 0
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M
H
Ms(T1)
Ms(T2)
Ms(T3)
T1
T2
T3
T1 < T2 < T3
HMM ps χχχχ++++====
Hd ≠ 0
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M
HHsat = Ms
H
H
NO MAGNETOCRYSTALLINE ANISOTROPY
Magnetic measurements give magnetisation (A/m)
magnetic measurements on plate shape samples
Hi = H -NMs
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M
HHsat = Ha
H
M
HHsat = MsH
PERPENDICULAR ANISOTROPY
Magnetic measurements give magnetisation (A/m)
Magnetic measurements give magnetocrystalline anisotropy
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-50050
-2 -1.5 -1 -0.5 0
6h+450C/0.5h2h+450C0.5h
M (emu/g)H (T)
T = 4 K
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χχχχ
M H T1T2T3χχχχ
M H M HT1<T2<Tc<T3(a) (b) (c)MHχ i χm Hst
MstM0 χ0
HM=χ 0====
==== HdHdMMs
MHχ i χm Hst
MstM0 χ0
HM=χ 0====
==== HdHdMMs
M H-Hc HcMr-Mr
BB’’ A CDE FO
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HH || easy magnetisation directionHaM
∆MH ⊥⊥⊥⊥ easy magnetisationdirection M HH
MHa
M 1/NdHM
HaM
χχχχ/(1+Ndχχχχ)χχχχ
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MHHsat= Ms
H || thin layerH ⊥ thin layer MHHsat= Ha
H || thin layer MHHsat= Ms
H ⊥ thin layerHM H1χχχχ
scMHM += χ HMcHM s+= χ2sM T1 T2 T3Tc T4T5T1 <<<< T2 <<<< T3 <<<< Tc <<<< T4 <<<< T5.M2 T2 T3Tc T4T5T1
H/M1/χ
Ms2 H/M3 H/M4H/Mc0
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-50050
-2 -1.5 -1 -0.5 06h+450C/0.5h2h+450C0.5hM (emu/g)
H (T)T = 4 K
050100150200250-8 -6 -4 -2 0
SmCo5_20 wt%FeT = 300K 2h/450C30'4h/450C30'6h/450C30'8h/450C30'dM/dH
H (T)
The reversibility curvesand the dM/dH variationvs H are very fine instruments in qualitative evaluation of the inter-phase hard/soft exchange coupling.