Magnetic Resonance in magnetic materials

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1 Ferdinando Borsa, Dipartimento di Fisica, Universita di Pavia Magnetic Resonance in magnetic materials Information on static and dynamic magnetic properties from Nuclear Magnetic Resonance and Relaxation

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3 Both NMR-NQR and EPR are radiofrequency spectroscopies NMR-NQR in Mhz range EPR in GHz range In NMR the magnetic moment interacting with the local magnetic field is fixed and not affected by the environment µ = g µ N I = γ h bar I ( g factor fixed Nucleus is passive probe of internal fields in matter!!!!!) In EPR the moment depends on the environment µ = g β I = γ h bar I ( g factor depends on orbital contributions and thus depends on material )

4 Study of local magnetic properties by NMR spectra H loc =H dip + H contact H dip = I A S H contact = B I S f res = γ (H+ H+H loc ) H EN r r r r r r S I ( Si rij )( Ij rij ) 2 γ N γ eh i j = 3 + I 3 5 j A% ij S i 2 ij rij rij r r

5 Ordered magnetic state or superparamagnetic state: local internal field even in zero external f =γh + g <s> A magnetic field The local field H int = g <s> A is seen as static by the resonant nucleus if the fluctuation time is longer than the inverse interaction energy ( in frequency) τ > A -1

6 NMR study of the critical exponent for M(T) order parameter in 3D Heisemberg magnets The time-averaged sublattice magnetization M in zero external field obeys the relation M ~ (T N T) 1/3 with remarkable accuracy just below T N. This is strikingly similar to the behavior of the difference in densities between the coexisting liquid and vapor phases observed experimentally near fluid critical point ( UNIVERSALITY OF CRITICAL PHENOMENA ) First determinaton of the critical exponent for 3D Heisemberg paramagnet: P.Heller and G. Benedeck, PRL 8, 428 (1962) from NMR of 19 F in MnF 2

7 H 0 H local ( α ) = H I (T ) + H 0 + p (T ) H 0 H local ( β ) = - H I (T ) + H 0 + p (T ) H 0 υ ( T ) = γ 19 H I (T)

8 υ(t) / υ(0) = M(T) / M(0) M(T) / M(0)= A ( T N T ) 1/3

9 Easy magnetization axis Fe8 Ground state spin configuration Fe (3+) s=5/2 S=10 (giant spin)

10 57 Fe-NMR spectrum in 57 Fe8 from S.H.Baek et al. 5 7 F e F e - N M R Spin echo intensity (a.u) H = 0 T T = 1. 5 K f r e q u e n c y ( M H z ) Observation of 57 Fe-NMR signal under zero magnetic field resonance frequency is proportional to internal field (H int ) ω res =γ N H int The direction of H int is opposite to the that of spin moments H int µ S Core- polarization

11 Internal spin structure of Fe8 (parallel field) Y.Furukawa et al. PRB 68, (R) (2003) frequency (M Hz) T=1.5K Fe1, Fe Fe3,Fe4 Fe5~Fe8 H (T) ω res =γ N H eff H eff = H int +H ext For two of eight Fe 3+ ions H int is parallel to H ext (spin direction is antiparallel) For other six Fe 3+ ions H int is antiparallel to H ext (spin direction is parallel) H ext

12 Spin moments on Fe 3+ ions in Fe8 spin moments (µ B ) Comparison with neutron diffraction results Y. Pontillon et al., J. Am. Chem. Soc, 121 (1999)5342 NMR results F e 1 F e 2 F e 3 F e 4 H int =A<S> A=-126kOe/µ B F e 5 F e 6 F e 7 F e 8 NMR N.D. Total spin moments NMR = 16.9 µ B N.D. = µ B (Magnetization measurements ~20 µ B ) Spin density is slightly distributed over the cluster including organic framework

13 1 H in External Field H o m I = -1/2 m I = +1/2 E = hω = hγ H 12 N N 0 H 0 N 2 N 1 = exp hωn k T B Disproportion between UP than DOWN spins leads to nuclear magnetization M 0 = N ( γ h) N We define the process of growth towards the equilibrium magnetization as SPIN LATTICE RELAXATION 2 4k T B H 0

14 Nuclear spin lattice relaxation in the weak collision limit + - hν L H (t)= H Z + H P (t) W( - + ) =W -+ W( + - ) = W +- W -+ = W +- exp( - hν L /kt) W +- =W 1/T 1 = 2W H P (t) H P (0) exp( -i ν L t) dt 1 / T

15 Relationship between nuclear spin-lattice relaxation rate and spectral density of spin fluctuation (a) weak collision approximation (i) Fluctuation time τ <<T 1 (ii)time dependent perturbation theory (a) (b) strong collision approximation (i) Direct Exchange of Energy Moriya s Formula i( ωn ± ωe) t 1 = 2( γ Nγ eh) ( + 1) { α ij i ( ) i (0) +... ij T s s s t s e dt + z z iωn t...+ β s ( t) s (0) e dt} ij i i Longitudinal Term Transversal term

16 In presence of SRO and strong correlation among interacting electron spins, magnetic systems are better described in terms of collective variables in q-space obtained by Fourier transforming the local spin variables in real space 1/T 1 = (hγ n γ e ) 2 /(4π) dt cos(ω n t) dq (1/4 Α ± (q) < S ± q (t) S± -q (t)> + Α z (q) ) < S z q (t) Sz -q (t)>) Or by using the fluctuation dissipation theorem: < S q (t) S -q (t) > cos(ωt) dt = 2 kt/ω Im χ ( q, ω ) 1/T 1 =(hγ n γ e ) 2 /(4πg 2 µ B2 ) k B T [1/4Σ q Α ± (q) χ ± (q) f q± (ω e ) + Σ q Α z (q)χ z (q)f qz (ω n )]

17 Critical spin dynamics in FeF2 ( T N = K ) δυ(t) = A [( T T N ) / T N ] n ( n = ± 0.01 ) A.M. Gottlieb and P.Heller, PRB,3, 3615 ( 1971)

18 SUPER-PARAMAGNETIC PARAMAGNETIC BEHAVIOR Magnetic energy for single domain particles with uniaxial anisotropy is: E = K V Sin 2 ( θ ) K is anisotropy V is volume 2 mimima θ = 0 θ =180 When the energy barrier is comparable or smaller than the thermal energy the particles magnetization fluctuates between two minima with a characteristic time given by τ. τ = τ 0 e kv kbt

19 CI CONCENTREREMO SU: Fe30 Fe10 Cr8 Fe6:Li

20 Magnetic properties of molecular rings and clusters ALCUNI MAGNETI MOLECOLARI E LORO PRINCIPALI CARATTERISTICHE

21 6 5 4 H=0.47 T H=0.73 T H=1.23 T H=2.74 T H=4.7 T A field dependent peak is observed A field dpendent peak is observed 1/T 1 (ms -1 ) T (K) The peak has universal features Cr8 1/( T 1 χt ) (emu/mol K ms) Cr8 H = 0.47 T Cr8 H = 0.73 T Cr8 H = 1.23 T 0.8 Fe6(Na) H = 0.5 T Fe6(Na) H = 1 T Fe6(Li) H = 1.5 T Fe10 H = 1.28 T 0.6 Fe10 H = 2.5 T Fe30 H = 0.47 T T/T 0 (H)

22 The quantum description can be used also for overdamped states by replacing the delta functions with Lorenzian defined by the lifetime width of the quantum states <S i (t)s j (0)> = Tr exp(-βh)exp(iht/h)s j exp(-iht/h)s i /Tr exp(-βt) 1/T 1 =Σ i,j Σ n,m exp(-βe n )[A<n S j m> <m S i n>δ(e n -E m -ω L )]/Σ n exp(-βe n ) Tχ A τ / (1+ω L2 τ 2 )

23 1 ω ( T) ω ( T) = A = T T T 1 c AχT c 2 ( ) ( ) + 2 c L c L ω ω ω ω, The proton spin lattice relaxation rate 1/T1 depends in first approximation from: 1) A i.e. Magnitude of proton-electron interaction 2) χt = C i.e. Magnitude of the local magnetic moment on the ion 3) ω c (T) i.e. Magnitude and temperature dependence of the characteristic frequency for the fluctuations ( relaxation) of the ring magnetization

24 The characteristic frequency ω c appears to have a power law T dependence and to scale with the gap between ground state and first excited state: S = 1 S = 0 ω c (MHz) 10 4 ~ (T/ ) T/ Cr T Cr T Fe6(Li) 1.5 T Fe6(Na) 1 T Fe T

25 Characteristic frequency derived from NMR scaled by the ground state gap 10 4 Cr 8 ~ (T/ ) Γ (MHz) Cr8 H=0.47 T Cr8 H=0.73 T Cr8 H=1.23 T Cr8 H=2.73 T Cr8 H=4.7 T T/ Γ (MHz) ~ (T/ ) T/ Cr T Cr T Fe6(Li) 1.5 T Fe6(Na) 1 T Fe T - ω c is the lifetime Γ of the magnetic levels? - A BASSA T, Γ DIPENDE ANCHE DA H!!!