NMR, ESR, Mssbauer (SR) (for solid- state physics magnetism) P. - PowerPoint PPT Presentation

NMR basics (6): quadrupole interaction + - - + I=1/2        2 2 2 eQ V V V        2 2 2 2   H ( I I ) ( 3 I I )       Q x y z 2 2 2   4 I ( 2 I 1 )  x y z    2 e qQ 1        2 2 2  H 3 I I ( I 1 ) ( I I )  Q z x y   4 I ( 2 I 1 ) 2   2 2    V V     2    2 2   V   2 2 2    x y      V V V eq , ,     2 2     2 2 2    z V z y x 2  z

Quadrupole interaction: back to the spectrum  =0, axial sym. - 1 st order 1- H 0  0 δ𝐹 (1) ~ n Q (3 𝑑𝑝𝑡 2 𝜄 − 1) 3𝑛 2 − 𝐽 𝐽 + 1 n NMR  n Q Degeneracy of the transitions lifted by quadrupolar effects EFG H 0 Ppal axis n NMR = g /2 p H 0 ~ n Q f( q ) q I = 3/2 n NMR = g /2 p H 0 n NMR = g /2 p H 0 2I+1 levels Quadrupolar nuclei: lifting the multiplicity of transitions on single crystals

Cobaltates Na 0.66 CoO 2 : charge segregation Spectra taken in two field directions on oriented powders • Different charge environments • Na + is driving the charge state and physical properties •

Quadrupole interaction only: back to the spectrum  =0, axial sym. - 1 st order 1- H 0  0 δ𝐹 (1) ~ n Q (3 𝑑𝑝𝑡 2 𝜄 − 1) 3𝑛 2 − 𝐽 𝐽 + 1 n NMR  n Q Degeneracy of the transitions lifted by quadrupolar effects n NMR = g /2 p H 0 I = 3/2 n NMR = g /2 p H 0 2I+1 levels Quadrupolar nuclei: distribution of angles  powder average

Averaging on angles: EFG, hyperfine tensor The EFG and hyperfine tensors may not have the same principal axis! One can manage, playing with isotopes, field … Single crystals are best. Fitting routines for powders …

Quadrupole interaction: NQR (6)  =0, axial sym. - 1 st order 2- H = 0 δ𝐹 (1) ~ n Q /6 3𝑛 2 − 𝐽 𝐽 + 1 Degeneracy of the transitions lifted by quadrupolar effects m=  3/2 n Q  0 m=  1/2 I = 3/2 n NMR = n Q 2 levels Quadrupolar resonance: powders = single crystals

NQR of Cu in cuprates: 2 sites, 2 isotopes, I = 3/2 1.0 x=1,0 Cu plane 0.8 Cu chain 0.6 0.4 0.2 0.0 18 20 22 24 26 28 30 32 34 ×àñòî òà (Ì Ãö) n (MHz)

One example of a difficult spectrum If I>1/2, nuclear spin I is sensitive to any Electric Field Gradient from the lattice NMR Intensity (normalized) 1.0 300 K 0.5 150 K 200 K 250 K 300 K 0.0 6.4 6.6 6.8 6.60 6.65 6.70 H (Tesla)

M.I.T., 2005 Cu 20  ac 3 /mol Cu) 2 17 O lineshift (%) Zn/Cu -4 cm 10 OH SQUID (10 1  17 O NMR Herbertsmithite 0 0 0 100 200 300 Cl ZnCu 3 (OH) 6 Cl 2 T (K) Cu 2+ , S=1/2 A. Olariu et al., Phys. Rev. Lett (2008)

Magnetic ordering • for a ferromagnet : « enhancement factor »: collective response from electronic spins; associated with the existence of a magnetization • very strong local fields : in the paramagnetic phase, need of a field H 0 so that <S> ≠ 0; in an ordered phase H ~ A hf <S>, <S> ≠ 0 • ZERO FIELD NMR : if hyperfine field is strong enough, no need of an applied field : S n 0  A hf S

Magnetic ordering S S Zero Field H 0 =0 NMR under an applied fieldH 0 T>T C T>T C Null ! n 0 n 0 (1+K) T<T C T<T C n 0  A hf S n 0 +A hf S n 0 -A hf S n 0 +A hf S

Magnetic ordering: local field vs T , structure pnictides BaFe 2 As 2 Fe • H is parallel to c Kitagawa, Takigawa, JPSJ 08 • Dipolar coupling • Discussion of the magnetic structure

Magnetic ordering: various types • Field distribution gives information on the type of ordering Antiferromagnetic ordering Spin density wave eg: Na0.5CoO2 eg: Cr: distributed field Splitting of the lines n (kHz) 78500 78750 79000 79250 Intensity (arbitrary units) T=80 K 0.4 0.0 200 T N  (G) T MIT 100 0 0 20 40 60 80 100 T (K)

Ferromagnets Cu/Co ferromagnetic multilayers Panissod et al., PRB 1992 Co surrounded by Co Co surrounded by Co and Cu Co entouré de Co Cr inside the domain Cr in the domain wall NArath, Phys Rev 1965

NMR in ferromagnetic multilayers Co-Cu multilayers: Co resonance depends on local environment • 12 Co • 11 Co, 1 Cu, intensity ~ c Cu • 10 Co, 2 Cu, intensity ~ c 2 Cu Work from Panissod Co/Cu mutilayers (1992) Marginal as compared to the world of thin films

NMR in ferromagnetic multilayers Work from Panissod Co/Cu mutilayers (1992) Marginal as compared to the world of thin films

Quantum dimers: model Hamiltonians T=25mK H=27 Tesla Grenoble High Magnetic Field Kodama, Science (2002)

Summary: observables Static Orbital susceptibility • • Spatially resolved static susceptibility Inhomogeneities, distribution of local fields • Charge effects • Ordered phases (charge or magnetic order) • Techniques In applied field: NMR: easy for I=1/2 on powders • For I>1/2, quadrupolar effects, much better with single crystals Zero applied field: NQR (no probe of  ), ZFNMR • ~ single crystals Dynamics <h + loc (t) h - loc (0)> Magnetic correlations  (T) • Excitations (gapped or not gapped) D • Critical regime • Compare timescales of the probes vs coupling constant

What about an experiment? H 0

H 0 Radiofrequency pulse ~ few m sec H 0 M

H 0 U t

Why pulsed NMR? Nb of nuclei frequency or local field p D n  n sin( t ( )) n  0 f ( ) f p D n  n 0 t ( ) 0 D t A pulse has a spectral width in Fourier space. Fourier transform yields the response of the sample in the frequency domain of the pulse.

Experimental set-ups Field range: 1T – 45 T T-range: 10 mK – 1000 K Sensitivity: 1 mMole … depends on sensitivity Misc: pressure (few GPa), in-situ rotation

z T 1 H 0 T 2 ( ) transverse relaxation : T 2  dM M   g  X , Y X , Y M H Enegy is conserved X , Y dt T 2 ( ) Longitudinal relaxation : T 1  M M dM   g  equilibriu m Z Z M H Energy exchange Z dt T with the lattice 1

Relaxation time T 1 Local magnetic fluctuations at  n (Fermi golden rule)  1 ( )        ~ B ( t ) B ( 0 ) exp i t dt L L n   T 1   B ( t ) A ( r ) I . S ( r , t ) hf i i coupled nuclei r i 1  ( )   2       Fourier transform ~ A ( q ) s ( q , t ) s ( q , 0 ) exp i t dt hf n   T q 1 Fluctuation    1  ( ) ( )             " - n ( 1 exp ) S ( q , t ) S ( q , 0 ) exp i t dt q , n t n    2 k T Dissipation B   " 1 1 k T ( q , )    2  n  B t n  A ( q ) k B T ( )  m 2 2 T g q 1 n B  n is small ≈ 0 as compared to neutrons, integrate over q

Relaxation time T 1 : electronic spins      " 1 1 k T  ( q , ) A ( q ) A ( r ) e ( i q . r )   2 B t n ( ) A q i i ( )  m 2 2 T g r i q 1 n B A(q) form factor and favours some q.  ( )   1 2 g   A ( q ) ~ 2 1 cos( q a ) cos( q b )  g x y   2 Cu favours q=0, ferromagnetic fluctuations O between Cu   ( ) 2     2 A ( q ) ~ 2 cos( q a ) cos( q b ) x y favours q= p,p, antiferromagnetic  Cu  fluctuations  O   Underdoped cuprate Takigawa et al., PRB (1991)

Magnetic transition: divergence of T 1 Slowing down of fluctuations In a weak metallic antiferromagnet • Above T Neel : T 1 TK=cst • At T Neel : divergence of 1/T 1  n Kyogaku et al., JPSJ (1993)

T 1 : Gapped magnetic insulator – Haldane chain (S = 1) AgVP 2 S 6 Shimizu et al., PRB (1995) D Haldane gap  1 ~ T e T 1

T 1 : High-T regime for an insulator  exchange (J)

T 1 : High-T regime for an insulator  exchange (J)

T 1 : High-T regime for an insulator  exchange (J) 0.6 -1 ) 1/T 1 (ms 0.4 0.2 0.0 1 10 100 T(K) Frustrated 2DHAF S=1/2

Single molecule magnets M.-H. Julien, Phys. Rev. Lett. 83, 227 (1999)

Mössbauer spectroscopy (Or nuclear g – ray spectrocopy) 1958: Discovery and interpretation by Rudolf Mössbauer 1961: Nobel Prize Born 1929 • Transition between nuclear levels • Emitter (source)  Absorber → transmission geometry for bulk samples → Surface studies: Conversion Electron Spectroscopy Thanks to P . Bonville, CEA Saclay

Is nuclear g ray spectrocopy possible ? Conservation of energy and momentum  = ħ/  |e  h n E 0 p g = h n / c |g  nucleus/atom h n = E 0 + P 2 /2M n h n /c = P 2 E 1 Recoil energy E R = P 2 /2M n  0 2 2 M c n M n c 2 ~ 100 GeV ; E 0 ~ 100 keV; E R ~ 1 meV Care about recoil energy

Is nuclear g ray spectrocopy possible ? Energy conservation: • emission: h n =E 0 -E R • absorption: h n =E 0 +E R if  E R , no overlap Optics At temperature T: moving emitter/abs  Doppler broadening D  2  k B T E R g rays X rays for g rays:  << E R Impossible in free atoms / nuclei ≠ Atomic spectroscopy

Is nuclear g ray spectrocopy possible ? For g rays,  << E R and D ~ E R  weak overlap On cooling, not on heating (D  as T  ), R.M. observes an increase of the resonant absorption in 191 Ir … 2 1 E Interpretation: nucleus bound in a solid  E R  0 2 2 M c solid free /N A <<  = E R photon emitted without recoil of the nucleus (for rigid atomic bonds)

In practice Lamb-Mössbauer factor: f(T) = |J 0 (kx 0 )| 2  1  ½ k 2 x 0 2 f(T)  1  k 2 <x 2 > T  exp( )  finite probability f(T) of nuclear resonant absorption of a photon with no phonons absorbed or emitted allows Mössbauer spectroscopy of hyperfine (electro-nuclear) interactions (~10 -6 eV) if  <<  hf Recoilless is for source and absorber!

In practice 1 mm/s ~30 MHz Vary the speed of the source ~ sweep the frequency (Doppler)

In practice 46 elements, 89 isotopes, 104 Mössbauer transitions ~10 used in condensed matter!

In practice: 57 Fe, no EFG, no field -3/2 E e = 14.4 keV -1/2 E e I e =3/2 1/2 µ e = 0.153 µ n 3/2 -1/2 E g 1/2 E g = 0 D m I = 0, ±1 I g =1/2 µ g = 0.0903 µ n

In practice: 57 Fe, EFG, no field E e = 14.4 keV ±3/2 E e I e =3/2 ±1/2 µ e = 0.153 µ n -1/2 E g 1/2 E g = 0 D m I = 0, ±1 I g =1/2 µ g = 0.0903 µ n

In practice: 57 Fe  field (no EFG) -3/2 E e = 14.4 keV -1/2 E e I e =3/2 1/2 µ e = 0.153 µ n 3/2 -1/2 E g 1/2 E g = 0 D m I = 0, ±1 I g =1/2 µ g = 0.0903 µ n

In practice: 57 Fe  field (no EFG) -3/2 E e = 14.4 keV -1/2 E e I e =3/2 1/2 µ e = 0.153 µ n 3/2 -1/2 E g 1/2 E g = 0 D m I = 0, ±1 I g =1/2 µ g = 0.0903 µ n

In practice: 57 Fe  field (no EFG)

Static: Orbitals, surrounding charges, fields Magnetic hyperfine field Quadrupole splitting Isomer shift δ Δ B hf Symmetric charge Asymmetric charge Symmetric or asymmetric charge No magnetic field No magnetic field Magnetic field (internal or external) V (mm/s)

Isomer shift  local environment Relative Transmission D Fe 3+ Fe 2+  -4 -4 -2 -2 0 0 2 2 4 4 Velocity (mm/s) Velocity (mm/s)

Isomer shift  local environment Use of Mössbauer spectroscopy as a “fingerprinting” technique Isomer shifts and quadrupole splittings of 4 Fe-bearing phases vary [6] Fe 2+ systematically as a [8] Fe 2+ 3 function of Fe oxidation, [5] Fe 3+ [4] Fe 2+ Fe spin states, and Fe coordination. [4] Fe 3+ 2 [6] Fe 3+ [6] Fe(III) Knowledge of the Mössbauer parameters [5] Fe 2+ can therefore be used 1 to “fingerprint” an unknown phase. [6] Fe(II) [sq] Fe 2+ 0 -0.5 0.5 1.0 1.5 0.0 Isomer shift (mm/s)

Isomer shift  local environment (Murad et al.)

Magnetic properties of Fe-pnictides Isomer shift typical of Fe(II) low or • intermediate spin state • Small internal field Fe2+ hyperfine coupling well known • Extraction of a small moment 0.25(5) µ B : • first indication in favour of a commensurate Spin Density Wave • Note: disorder fitted witn a double sextet (Klauss, Luetkens et al.)

Magnetic properties of Fe-pnictides (Klauss, Luetkens et al.)

Dynamics: Linewidth DE D t ~ ħ • Lifetime , linewidth  : DE   ~ ħ/  •  ~ 10 5  10 11 s • Slow relaxation  Note: effects are not the same on all lines: outer are more « protected »

Dynamics: Linewidth DE D t ~ ħ • Lifetime , linewidth  : DE   ~ ħ/  •  ~ 10 5  10 11 s • Intermediate  relaxation

Example: Fe 4 molecular magnet S=5 AF interactions 3 ineq sites, EFG  0

Example: persistent fluctuations in frustrated magnets 155 Gd in Gd 3 Ga 5 O 12 No magnetic order of Gd 3+ moments due to geometrical frustration P.Bonville et al PRL 92 (2004) 167202

Example: phase transition in Yb 2 Ti 2 O 7 2 5 lines  with equal 170 Yb intensities 0

NMR/Mössbauer: a comparative summary Mössbauer NMR Which sample? Needs a source Many… needs time Fluctuation rate Few 10 GHz… MHz 100 MHz – fraction of Hz Location/coupling At. Site, hyperfine At. Site, hyperfine 0.1 T – 10 T/ µ B 0.1 T – 10 T/ µ B Observables Magnetic transitions Magnetic susceptibilities Temperature range 10 mK – … K 10 mK – 1000 K Field range 0 – (few T) 1 – 45 T Intrinsic drawback Need a source r.f. field needed, field needed Tuning of the probe Fast fluctuations 1/T 1 ~ A 2  c

ESR: principle    ˆ  ˆ ˆ ˆ • Angular momentum   m   g Γ   S S e e  e m       m • Magnetic moment S ( S 1 ) S ( S 1 ) g e B m e e  • Bohr magneton 24 Am m    2 9 . 274 10 B 2 m e • Landé factor g=2(1+  /2 p +...) (1/2)g  e H m s =+1/2 • D m s = + 1 D E = g  e H D m s = 1 h n = g  e H E (-1/2)g  e H m s =-1/2 H

ESR and NMR comparison! electron proton ratio Rest mass m e =9.1094*10 -28 g m p =1.6726*10 -24 g 5.446*10 -4 m S =-g e m e S m S =-g N m N S Magnetic dipole moment g e = 2.002322 g N = 5.5856 m e =eh/4 p m e c = m N =eh/4 p m N c = 1836.12 9.274*10 -21 erg/G 5.0504*10 -24 erg/G Frequency: Factor 1000 larger in EPR ! (GHz instead of MHz) Dipolar coupling: Factor 1 000 000 larger in EPR ! (MHz instead of Hz) Relaxation Times: Factor 1000 000 smaller in EPR ! (ns instead of ms) = much higher techniqual requirements, but unique sensitivity to molecular motion Sensitivity : Factor 1 000 000 better than in NMR !! (1nM instead of 1mM ) An ideal case, though

ESR: in practice ~ 33 GHz / Tesla • Traditional frequencies, n , used are microwave bands originally developed for radar: – X band; ~9-9.5 GHz, in most widespread use ( l ~3 cm). – K band; ~ 24 GHz ( l ~1 cm) – Q band; ~ 35 GHz ( l ~0.8 cm) – W band; ~ 95 GHz • Traditional electromagnets with fields up to 3 Tesla. – At g=2, about which most spectra are centered, X-band setups have resonances at 3,000-3,500 Gauss. • Cutting edge EPR is going to ever higher and ever lower n .

ESR apparatus

ESR apparatus Use of a cavity (except at high frequencies) → sweep the field

ESR detection • Modulation of magnetic field • Phase sensitive detection • Spectrum = derivative • Intensity by double integration ~ static susceptibility

Spin(s) Hamiltonian for EPR      H H H H H H eZ ZFS ee en nZ H eZ = Electron Zeeman interaction: g tensor H ZFS = Zero-field splitting interaction: anisotropy, dipolar H ee = Interactions between electron moments: exchange H en = Electron - nucleus interaction H NZ = Nuclear zeeman interaction Hilbert space of coupled electrons and nuclear spins has   a dimension    ( 2 1 ) ( 2 1 ) n S I H m n m n

e-Zeeman interaction    m   H B g S eZ B • « Effective » g • Crystal Field + Spin Orbit • Different hierarchy for 3d and 4f • 3d: < L > = 0 Crystal field dominates, Spin orbit = small corr.            m    l   m    H H g B ( L S ) L S g B g S S . D . S eZ LS e B e B 4f: use g J (free atom) instead of g (takes into account the spin orbit term)

e-Zeeman interaction            m    l   m    H H g B ( L S ) L S g B g S S . D . S eZ LS e B e B     l  g e = 2.0023  m   g g ( 1 2 ) H B g S e eZ B  l  2 D   g 0 0       xx L L     g 0 i n n j 0   g 0 0    yy ij E E    g 0 n 0 n 0 0   zz Cubic symmetry: g xx = g yy = g zz • h n = g xx m B B, B//x, h n = g yy m B B, Axial symmetry (trigonal, tetragonal, B//y • h n = g zz m B B, B//z. hexagonal): g xx = g yy = g  and g zz = g || Orthorhombic symmetry: g xx  g yy  g zz •

g-tensor: axial case B // z (g zz = g // ) E    m   H B g S eZ B B  z (g xx = g  ) B 99

g-tensor: axial case, powder lineshape S=1/2, I=0, g x =g y <>g z Axially symmetric g-factor n n h h    q  q 2 2 2 2 1 / 2 B [ g cos g sin ]  m m r II g eff B B q is the angle between a z-principal axis and the magnetic field direction The given solid angle W is defined to be the ratio of the surface area S to the total surface area on the sphere: W S / 4 p r 2 : d W/W 2 p r 2 sin q d q/ 4 p r 2  sin q d q/2 1   q d q f ( B ) f ( B ) dB sin q / cos dB d m q  q 2 2 2 2 3 / 2 ( g cos g sin )   Absorption B II f ( B ) n  q 2 2 h ( g g ) cos  II m 1  g  B f ( B ) g // n  q 3 2 2 h B ( g g ) cos  r II B