# Presentation: 544019

## Nuclear Magnetic Resonance (NMR) 

- Yuji  Furukawa
- A121 Zaffarano
- furukawa@ameslab.gov

## 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 complicated

- But, NMR measurements give us very important information
- which can not be obtained by other experimental techniques

## H i s t o r y

-    Prof. Gorter, first attempt to detect nuclear magnetic spin
-                (But he did not succeed,
-              1H in K[Al(SO4)2]12H2O and 19F in LiF))
-    Prof. Rabi, First detection of nuclear magnetic spin
-                (1944 Nobel prize)
-    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)
-    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)
-      I=1/2
-       γN/2π=42.577 MHz/T

- gN:g-factor (dimension less)
- γN:nuclear gyromagnetic ratio (rad/sec/gauss)
-                                         (erg/gauss)

- c.f. electron spin moment
-     **μ**e=-gμB**S**

- (erg/gauss)

- |μB/μN|~1800

## Nuclear magnetism

- Nuclear magnetic moment

- Much less than χe (electron spin)

- Magnetism of material is mainly dominated by χe!!

**Nuclear magnetism**

- Curie law

## NMR (Nuclear Magnetic Resonance)

- NMR (Nuclear Magnetic **R**esonance)

- Nucleus has magnetic moment (nuclear spin)
- nucleus is very small magnet

**Zeeman interaction**

- 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.

- ・ 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

- For example;

## NMR in condensed matter physics

- Fermi contact

- dipole interaction

- orbital
- interaction

- NMR measurements
-    investigation of static and dynamical properties of hyperfine field (electron spins)

- 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

## NMR spectrum 

- NMR spectrum measurements
- (static properties of hyperfine field)

- 1 magnetic system
-   spin structure, spin moments and so on
- 2 metal
-   local density of state at Fermi level

- H

- H0
- =ω/γ

- ⊿H

- NMR shift: K=ΔH/H

- ΔH:contribution from electron

- H

- H=H0+ΔH

## Nuclear spin-lattice relaxation time(T1)

- Nuclear spin-lattice relaxation time

- Dynamical properties of hyperfine field

*I*z=1/2

- -1/2

- 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

- 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...)
- 2)  Low energy excitation
-          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
-             (in principle, you can NOT modify the equipment)

- For example
-  f=100MHz
-    ⇒5mK

## 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 $10/liter)
-                       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 !

- Tallahassee (USA), Grenoble (France), Tohoku,(Japan), Tsukuba (Japan )...

- NMR spectroscopy with Hybrid magnet (~35T)

- NMR spectrometer with DR refrigerator

## NMR laboratory in the world  

- Pulse NMR spectroscopy with pulse magnet
-            Japan         project of “100T spin science”
-            Germany     Dresden

- Exciting new challenge!

##          H0

- In the case of I=1/2 and **H**=(0, 0, H0),
- Eigen energies for two quantum levels are
- given

- To make a resonance, one needs time dependent perturbation
- and non-zero matrix element

- Magnetic transition

-          *H*0

- alternating current
-  ⇒ alternating field

- 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)
- Need to think about motion of nuclear magnetic moment

## Motion of magnetic moment

- Classical treatment

**μ**

- H

- Larmor precession
-    ω=γNH

- (Time variation of angular momentum is equal to  torque)

- Rotating coordinate system (Ω)

- Ω

- (With a simple assumption H=H0**k)**

- 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)

- If H=(0,0,H0),
-     then  μx=Asin(ωt+a),  μy=Acos(ωt+a),  μz=const.

## Effects of alternating field

**H**x=Hx0 cosωt **i**

**H**x=**H**R+**H**L

**H**R=H1(**i** cosωt + **j** sinωt )
**H**L=H1(**i** cosωt - **j** sinωt )

- H1=H0/2

- Laboratory frame

- Coordinate system rotating about z-axis

- 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

- x

- y

- z

- spin

- H0

- without H1

- with H1 (rotating frame)

- H1

- You can control the direction
- of spins!

- Manipulation of spin

## 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

- Spin rotes in xy-plane in laboratory frame
-  (spin rotates in the coil)
-      ⇒ this induces voltage

- You can detect the voltage
-   -> observation of  signal from nuclear spin!
- Typically the induced voltage is ~10-6 V
- We need to amplify the voltage to observe easily
- (with amplifier)

- H1

- t=0

- t=π/2γH1 (π/2 pulse)

- t=π/γH1 (π pulse)

- If you stop to give H1 just after t (π/2 pulse)

## FID signal

## 

- Spin echo method

- a

- b

- c

- e

- d

- π/2 pulse

- π
- pulse

- Spin echo signal

- Two pulse sequence

- ω+⊿ω

- ω-⊿ω

- t

## 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

## Nuclear spin lattice relaxation T1

- Nuclear spin lattice relaxation T1

- Boltzmann
- distribution

- thermal
- equilibrium
- state

- Resonance
- (absorption)
- nonequilibrium
- state

- Relaxation
- (energy
- emission
-  to lattice
- (electron system)
- -> 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

- t-dependence of signal intensity
-          I(t)=I0(1-exp(-t/T1))

- T1 can be estimated

- No mag. in xy-plane
-  I(0)=0

- When t~0

- t= ∞

- 2/π

- π

- 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

## NMR spectrum 

- Zeeman interaction
-      (interaction between magnetic moment and magnetic field)

- Electric quadrupole interaction  (I>1/2)
-     ( interaction between electric field gradient and nuclear quadrupole moment)

- Nuclear is NOT spherical but ellipsoidal body (I>1/2)

- For η=0

- η: assymmetry parameter

## NMR spectrum

- 1. Hquadrupole≠0, H=0

- 2.     Hzeeman >> Hquadrupole

- Hq=0

- I=5/2

- NQR (nuclear quadrupole resonance)

- ω

- 5/2

- 3/2

- 1/2

- -1/2

- -3/2

- -5/2

## NMR spectrum in powder sample

- θ=0

- θ=90

- Hz>>HQ

- (I=3/2)

- Center line is affected
- in 2nd order perturbation

## NMR spectrum in powder sample

- 93Nb-NMR in NbO (field sweep spectrum)

- Textbook like typical powder pattern spectrum

- I=9/2

- Central transition line
-         Opposite?!

## NMR spectrum

- (1) ω-sweep ( H=constant;H0)

- NMRspectrum
-      Magnetic field sweep and frequency sweep

- H0

- (2) H-sweep (ω=constant; ω0)

- ω0

- HA

- HB

- Opposite!!
-       Need to pay attentions !!

## Hyperfine field at nuclear site

- These give additional field (Hhf) at nuclear site
-     -> shift in spectrum (NMR shift)

- ω

- ω0

- ω0+⊿ω

- Fermi contact

- Dipole interaction

- orbital interaction

- S-electron

- Core-poratization
- interaction

- ⊿ω=γHhf

- In the material, nuclear experiences additional field due to hyperfine interaction

- 3d system
-  ~-100kOe/μB

## Relation between NMR shift and magnetic susceptibility

- H=Hz+Hhf

- Hamiltonian

- Hz=Hzeeman           (H=H0)

- Hhf=Hdipole+HFermi+Hcore-polarization+.....
-     =A**I****・****S**

- A: hyperfine coupling constant

- NMR shift originates from thermal average value of Hhf
-               <Hhf>=A

- Since 
 is expressed by <M> (thermal average value of electron magnetization),
-           <Hhf>=A
~A<M>  (=AχH0)
- Knight shift is given by  K = Hhf/H = AχH/H ~Aχ
-                           K is proportional to χ !!

- <M> increases with
- increasing H
- -> high accuracy

- (hyperfine field)

## Example 

- Spin dimer system VO(HPO4)0.5H2O

- V4+ (3d1: s=1/2)

- AF interaction

- Magnetic susceptibility

- NMR shift  (31P-NMR)

- χtotal(T)=χspin(T)+χorb+・・・+χimpurity

- Ktotal(T)=Kspin(T)+Korb

- What is ground state ?
-        Spin singlet ? or magnetic?

- 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!!)

- Y. Furukawa et al., J. Phys. Soc. Japan 65 (1996) 2393

## Example of K-χ plot

- K-χplot      K = Aχ/NμB,

- Good linear relation
-    K is proportional to χ
- Hyperfine coupling constant can
- be estimated from the slope

- Ahf =3.3 kOe/μB

- 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

- NMR shift (Knight shift)
-    K=(A/μB)χpauli
-    since χpauli is expressed by (1/2)g2μB2NEf2
- 2)Nuclear spin lattice relaxation time T1
-     Relaxation mechanism
-         scattering of free electron from ┃k,↑>  to ┃k’,↓>
-       nuclear spin can flop from ↓ ⇒ ↑ state

- Pauli paramagnetism χpauli
- No electron correlation

- Simple metal  (like Cu, and so on)

- 1/T1 is proportional to T

- T1T= constant

- K is independent of T

## Korringa relation

- This does not depend on material !

- Korringa Relation

- 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

- Modified Korringa Relation

- Kα>>1:AF spin correlation
- Kα<<1:F spin correlation

- Stoner enhancement

- χ= χ0/(1- α0 )  I=U/N0

- α0=Iχ0/2

- RPA

## 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
~A<M>≠0

- Therefore, Hamiltonian for nuclear is not zero without external field

- For example, AF insulator spinel Co3O4 :TN=33K)

- ┃Hint ┃ = 5.5Tesla

- 59Co-NMR under H=0

- If you know Ahf,
- You can estimate ordered
- magnetic moment
- <S>=Hint/Ahf

**Internal field**

- T. Fukai, Y.F., et al., JPSJ 65 (1996) 4067.