Nuclear Magnetic Resonance (NMR) Yuji Furukawa A121 Zaffarano firstname.lastname@example.org Jan 26 Introduction of NMR Jan 28 Basics of NMR I Jan 30 Basics of NMR II Feb. 2 Example I (low-D spin system) Feb. 4 Example II (superconductors) Feb. 6 Introduction of ESR Principle of NMR ・・・・・ a little bit complicated NMR experiments ・・・・・ a little bit complicated Data analysis of NMR results ・・・・・・ a little bit complicat But, NMR measurements give us very important information which can not be obtained by other experimental techniques H i s t o r y 1936 Prof. Gorter, first attempt to detect nuclear magnetic spin (But he did not succeed, 1H in K[Al(SO ) ]12H O and 4 2 2 19F in LiF)) 1938 Prof. Rabi, First detection of nuclear magnetic spin (1944 Nobel prize) 1942 Prof. Gorter, First use of terminology of “NMR” (Gorter, 1967, Fritz London Prize) 1946 Prof. Purcell, Torrey, Pound, detected signals in Paraffin. Prof Bloch, Hansen, Packard, detected signals in water (Purcell, Bloch, 1952 Nobel Prize) 1950 Prof. Haln, Discovery of spin echo. -> Spin echo NMR spectroscopy Remarkable development of electronics, technology and so on -> Striking progress of NMR technique!! Nuclear property Nuclear magnetic moment c.f. Proton (three quarks) μ n gN N I n I I=1/2 γN/2π=42.577 MHz/T gN:g-factor (dimension less) c.f. electron spin moment γN:nuclear gyromagnetic ratio (rad/sec/gauss) μe=-gμBS e e B 0.92 10 20 N 5.05 10 24 (erg/gauss) 2me c (erg/gauss) 2m p c |μＢ/μＮ|~1800 Nuclear magnetism Nuclear magnetic moment μ n gN N I n I U H g N I z H z Nuclear magnetism I U g N I z exp M I z I k BT NgI B x U I N I k T exp I Z B M Ng 2 N I I 1 2 N Curie law H 3k BT Much less than χe (electron spin) Magnetism of material is mainly dominated by χe!! NMR （Nuclear Magnetic Resonance) Nucleus has magnetic moment (nuclear spin) nucleus is very small magnet Zeeman interaction H Zeeman N I・ H N H (ｈ：Planck’s constant、ν：frequency、γN：nuclear gyromagnetic ratio、H：magnetic field) Magnetic resonance can be induced by application of radio wave whose energy is equal to the energy between nuclear levels Application of NMR NMR is utilized widely not only Physics and/or chemistry but also medical diagnostics (MRI) and so on. For example; ・ Physics Condensed matter physics、Magnet, Superconductor、and so on ・Chemical Analysis and/or identification of material ・Biophysics Analysis of Protein structure ・Medical MRI (Magnetic Resonance Image) Brain tomograph NMR in condensed matter physics One of the important experimental method for the study on magnetic and electronic properties of the materials from the microscopic point of view. (nucleus as a probe) Hyperfine interaction between nuclear spin and electron spins 8 I･ S 3( I･ r )(S･ r ) I･ H eln N g B [( (r )I･ S ( 3 5 ) 3 ] 3 r r r Fermi contact dipole interaction orbital interaction NMR measurements investigation of static and dynamical properties of hyperfine field (electron spins) NMR spectrum NMR relaxation time (T1, T2) ⇒ static properties of spins ⇒dynamical properties NMR spectrum NMR spectrum measurements ⊿H （static properties of hyperfine field） ΔＨ：contribution from electron NMR shift： Ｋ＝ΔＨ／Ｈ H0 Ｈ H Ｈ＝Ｈ０＋ΔＨ =ω/γ ΔＨ ① magnetic system spin structure, spin moments and so on H0 Ｈ ② metal local density of state at Fermi level Nuclear spin-lattice relaxation time（Ｔ１） Nuclear spin-lattice relaxation time Dynamical properties of hyperfine field H - N I H hf t -1/2 - N I H hf (t ) I H hf (t ) 2 I I x ± y , H hf H hf ± hf ± ± x iI iH y Iz=1/2 1 N H 2 hf , H hf t exp i N t dt T1 2 A2 N S H 2 i , S i t exp i N t dt hf AS i 2 Ex. Metal ⇒ T1T＝const. （Korringa relation） Superconductor ⇒ T-dependence of T1 provides information of symmetry of SC gap full gap ⇒ 1/T1~exp(-Δ/kbT) anisotropic gap ⇒ 1/T1~Tα Characteristics of NMR 1) Local properties information at each nuclear site (e.g., local density of states, spin state for each site…) microscopic measurements (NMR, μＳＲ，ESR, Mossbauer ND, ) macroscopic measurements (Magnetization, specific heat, resistively…) For example ｆ＝100ＭＨｚ 2) Low energy excitation ⇒5ｍＫ information of low energy (electron) spin excitation (energy scale in different experiments NMR, μSR : MHz, Mossbauer：γ-ray, ND: ～meV） 3) Laboratory size NMR spectrometer can be set up in lab space. (you can modify the spectrometer as you like!) μＳＲ measurements －＞ need to go facility NMR spectroscopy in condensed matter physics NMR spectroscopy Continuous wave (CW) NMR Pulse NMR (FT (Fourier transform) –NMR) ←mainstream ・Spectrometer frequency range 1～500MHz ・Magnetic field up to 2Ｔ ; electron magnet up to 9T ; superconducting magnet (Nb3Ti) up to 23T ; superconducting magnet (Nb3Sn) up to 35T ; Hybrid magnet more than 40 T ; pulse magnet Temperature down to 77K ; liquid N2 (less than $1/liter) down to 1.5K ; liquid He (boiling T ～4.3K) (more than down to 0.3K ; 3He cryostat ($100K) down to 0.01K ; 3He-4He dilution refrigerator ($300K) NMR lab at ISU (at present, just a couple of months after I moved in) f=1-500MHz, H=9T, T=1.5K Plan to purchase DR refrigerator One 3He cryostat: not available now NMR laboratory in the world There are many NMR labs in the world ! NMR spectrometer with DR refrigerator NMR spectroscopy with Hybrid magnet (~35T) Tallahassee (USA), Grenoble (France), Tohoku,(Japan), Tsukuba (Japan )… NMR laboratory in the world Pulse NMR spectroscopy with pulse magnet Exciting new challenge! Japan project of “100T spin science” Germany Dresden Magnetic resonance H Zeeman N I・ H In the case of I=1/2 and H=(0, 0, H0), Eigen energies for two quantum levels are m = -1/2 given 1 1 E1 / 2 N H 0 E 1 / 2 N H 0 2 2 H E n H 0 m = +1/2 H0 = 0 H0 ≠ 0 To make a resonance, one needs time dependent perturbation and non-zero matrix element I I H ' (t ) N H 1 I x cos( N t ) Ix alternating current 2 ⇒ alternating field m 1 H ' (t ) m 0 Magnetic transition Using a coil perpendicular to H0, you can apply an alternating field which induces magnetic transition. But how can you detect the signal (magnetic transition) H0 Need to think about motion of nuclear magnetic moment Motion of magnetic moment Classical treatment H dI d N H N H dt dt μ (Time variation of angular momentum is equal to torque) If H=(0,0,H0), Larmor precession then μｘ=Asin(ωt+a), μy=Acos(ωt+a), μz=const. ω＝γNH Rotating coordinate system (Ω）(With a simple assumption H=H0k) (H ) t H eff Ω If Ω＝ｰγH0 then Heff=0 -＞δμ/δｔ ＝ 0 No change in time ! (since we are looking at spin moment on rotating frame with same frequency of γH0） Effects of alternating field y Hx=HR+HL Hx=Hx0 cosωt i HR=H1(i cosωt ＋ j sinωｔ ） x HL=H1(i cosωt - j sinωｔ ） Hx H1=H0/2 Laboratory frame Coordinate system rotating about z-axis d ( H 0 H1 ) ( H 0 )k H1i dt t When ω＝-γH0, you have resonance and have only H1 magnetic field along to x-axis This means spin rotates about x-axis with frequency γH1 z z H0 You can control the direction of spins! spin y H1 y Manipulation of spin x x without H1 with H1 (rotating frame) Motion of magnetic moment Motion of magnetic moment Larmor precession Motion of magnetic moment Motion of magnetic moment Motion of magnetic moment Motion of magnetic moment Motion of magnetic moment Effects of alternating field t=0 t=π/2γH1 (π/2 pulse) t=π/γH1 (π pulse) z z z H1 y H1 y H1 y x x x If you stop to give H1 just after t (π/2 pulse) z Spin rotes in xy-plane in laboratory frame (spin rotates in the coil) ⇒ this induces voltage You can detect the voltage y -> observation of signal from nuclear spi Typically the induced voltage is ~10-6 V x We need to amplify the voltage to observe easily (with amplifier) FID signal Spin echo method Two pulse sequence t t a b c d e ｔ π/2 pulse π pulse Spin echo signal ω-⊿ω ω+⊿ω Quantum treatment of Spin echo Quantum treatment of Spin echo Absorption energy and spin lattice relaxation T1 Nuclear spin lattice relaxation T1 Nuclear spin lattice relaxation T1 Iz= -1/2 H Iz = 1/2 H0 = 0 H0 ≠ 0 thermal Resonance Relaxation equilibrium (absorption) (energy state emission nonequilibrium to lattice Boltzmann state (electron system) distribution -> thermal equilibrium state T1 is a time constant (from nonequilibrium to equilibrium states) Nuclear spin lattice relaxation T1 Relaxation is induced by fluctuations of hyperfine field with NMR frequency How to measure nuclear spin lattice relaxation T1 1.0 0.8 Spin echo intensaity 0.6 0.4 0.2 0.0 time t-dependence of signal intensity I(t)=I0(1-exp(-t/T1)) When ｔ~0 z T1 can be estimated t= ∞ z Saturation z π 2/π H1 y H1 y y x 2/π x x No mag. in xy-plane π Ｉ(0)＝0 I(t)=I0 Signal intensity is proportional to xy-component of nuclear magnetization block diagram (NMR spectrometer) Receiver Amp PSD LPF PSD Multiplication of Input frequencies -> out put frequency difference and sum sin(1t ) sin( 2 t ) cos(1 2 )t 1 2 cos(1 2 )t 1 2 NMR spectrum Zeeman interaction (interaction between magnetic moment and magnetic field) H Zeeman - H n H 0 I Z Electric quadrupole interaction (I>1/2) ( interaction between electric field gradient and nuclear quadrupole moment e 2 qQ 2 2 1 2 2 HQ (3I z I ) ( I I ) η: assymmetry parameter 4 I (2 I 1) 2 2V 2V x 2 2V y 2 For η＝0 q 2 z V z 2 2 Em A 3m 2 I ( I 1) e 2 qQ + A + + + 4 I (2 I 1) Nuclear is NOT spherical but ellipsoidal body (I>1/2) NMR spectrum 1. Hquadrupole≠0， Ｈ＝0 2. Hzeeman >> Hquadrupole 2 Em A 3m I( I 1) A e 2qQ 4I( 2I 1) I=5/2 Hq＝0 m=±5/2 -5/2 0 12 A 12A -3/2 0 6 A m=±3/2 -1/2 0 0 1/2 eq=0 6A m=±1/2 0 6 A 3/2 eq≠0 0 12 A 5/2 ω 6Ａ 12Ａ ω NQR (nuclear quadrupole resonance) NMR spectrum in powder sample Hz>>HQ (I=3/2) -3/2 n 2m 13 cos 1 ℏω-1/2→-3/2 3e 2 qQ -1/2 2 8I2I 1 1 1/2 ℏω1/2→-1/2 ℏω3/2→1/2 1/21/2 0 A2 1 - 9cos2 1 - cos2 3/2 A2 9 2 I 3 e 2 qQ 2 64 4 I 2 2 I 1 0 θ=90 A1=1/4e2qQ/ℏ Center line is affected in 2nd order perturbation θ=0 ωn-2A1 ωn-A1 ωn ωn+A1 ωn+2A1 ωn－16A2/9ℏ ωn ωn+A2/ℏ 2nd oeder splitting of central powder pattern (I=3/2) transition for powder pattern spectruim NMR spectrum in powder sample 93Nb-NMR in NbO (field sweep spectrum) I=9/2 93 Nb-NMR in NbO Spin echo intensity ωn－16A2/9ℏ ωn ωn+A2/ℏ 60 65 70 75 80 H (T) Central transition line Textbook like typical powder pattern spectrum Opposite?! NMR spectrum ＮＭＲspectrum Magnetic field sweep and frequency sweep ω (1) ω－sweep ( H=constant;H0) signal (A) ωＡ signal (B) ωＢ ω ωＢ ωＡ H H0 ω (２) Ｈ－ｓｗｅｅｐ (ω＝constant; ω0） signal (A) ω0 signal (B) Ｈ ＨＡ ＨＢ H Opposite!! ＨＡ ＨＢ Need to pay attentions !! Hyperfine field at nuclear site In the material, nuclear experiences additional field due to hyperfine intera 8 e HF s (0) 2 Fermi contact S-electron 3 s 3 s r r Dipole interaction H dip e * r3 r5 orbital 1 H orb e l * interaction r 3 μS Core-poratization 8 e interaction H cp s i (0) i (0) 2 2 3 i Hint These give additional field (Hhf) at nuclear site 3d system -> shift in spectrum (NMR shift） ~-100kOe/μB ⊿ω=γＨhf ω ω0 ω0+⊿ω Ｒｅｌａｔｉｏｎ between NMR shift and magnetic susceptibilit Hamiltonian H=Hz+Hhf Hz=Hzeeman (H=H0) Hhf=Hdipole+HFermi+Hcore-polarization+….. =AI・S A: hyperfine coupling constant H n I ( H 0 H hf ) H hf AS (hyperfine field) NMR shift originates from thermal average value of Hhf <Hhf>=A<s> Since <s> is expressed by <M> (thermal average value of electron magnetization), <Hhf>=A<s>～A<M> (=AχH0) Knight shift is given by K = Hhf/H = AχH/H ～Aχ <M> increases with K is proportional to χ increasing H ！！ -> high accuracy Example Spin dimer system VO(HPO4)0.5H2O What is ground state ? V4+ (3d1: s=1/2) Spin singlet ? or magnetic? AF interaction Magnetic susceptibility NMR shift (31P-NMR) -5 1.8x10 0.6 -5 1.6x10 magnetic susceptibility (emu/g) -5 0.5 1.4x10 -5 1.2x10 0.4 -5 1.0x10 K (%) 0.3 -6 8.0x10 -6 6.0x10 0.2 -6 4.0x10 0.1 -6 2.0x10 0.0 0.0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 T (K) T (K) Y. Furukawa et al., J. Phys. Soc. Japan 65 (1996) 2393 χtotal(T)=χspin(T)+χorb+・・・+χimpurity Ktotal(T)=Kspin(T)+Korb From the NMR measurements, increase of χ at low temperature is concluded to be due to magnetic impurities NMR can see only intrinsic behavior (exclude the impurity effects!! Example of K-χ plot K-χplot K ＝ Aχ/NμB, Good linear relation K is proportional to χ 0.6 Hyperfine coupling constant can 0.5 be estimated from the slope 0.4 dK A K (%) d N B 0.3 0.2 0.1 Ahf =3.3 ｋＯｅ/μB 0.0 -6 -5 -5 0.0 5.0x10 1.0x10 1.5x10 (emu/g) This is a value at P site per one Bohr magneton of V4+ spin (Vanadium spin produces the hyperfine field at P-site) The origin of this hyperfine field is “transferred hyperfine field” NMR in simple metal Simple metal (like Cu, and so on) Pauli paramagnetism χpauli No electron correlation 1) NMR shift (Knight shift) K=(A/μB)χpauli since χpauli is expressed by (1/2)g2μB2NEf2 K is independent of T K g B N F A 2 2 2)Nuclear spin lattice relaxation time T1 Relaxation mechanism scattering of free electron from ┃k,↑> to ┃k’,↓> nuclear spin can flop from ↓ ⇒ ↑ state 1 N A I f k f k Ek E k 2 2 2 s 1 T1 k ,k f f k f k k BT 1 k BT Ek E F 1/T1 is proportional to T 1 ( N ) 2 A 2 g 2 N F k BT 2 T1 T1T= constant Korringa relation g B N F A 2 K 2 1 ( N ) 2 A 2 g 2 N F k BT 2 T1 2 2 1 4k B N 4k B N g S T1TK 2 B B Korringa Relation This does not depend on material ! However deviation from the Korringa relation is observed in many material. Model was simple importance of Interaction between electrons (electron correlation) Modified Korringa relation Korringa Relation 2 2 Stoner enhancement 1 4k B N 4k B N g S T1TK 2 B B χ＝ χ0/（1- α0 ） Ｉ＝Ｕ/Ｎ0 α0＝Ｉχ0/2 Modified Korringa Relation ＲＰＡ S 0 ( q, ) K ( q, ) T1TK 2 1 0 [ 0 (q, ) / 0 (Q0 ,0)] 1 K~~ Kα>>1：AF spin correlation (1 0 ) Kα<<1：F spin correlation (1 0 ) 2 K ( q ) (1 q ) 2 FS NMR in magnetic material Do we always need to apply magnetic field to observe NMR signal? In some case, the answer is No! In magnetically ordered state, you have spontaneous magnetization (M) without applying external magnetic field. <Hhf>=A<s>～A<M>≠0 Therefore, Hamiltonian for nuclear is not zero without external field H n IH hf (1) For example, AF insulator spinel Co3O4 :TN=33K) 59Co-NMR under H=0 Internal field ┃Hint ┃ = 5.5Tesla If you know Ahf, You can estimate ordered magnetic moment <S>=Hint/Ahf T. Fukai, Y.F., et al., JPSJ 65 (1996) 4067.
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