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Nuclear Magnetic Resonance NMR (PowerPoint download)

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					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 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 eln     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

     dI                        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  13 cos   1
                        ℏω-1/2→-3/2                                                  3e 2 qQ
-1/2                                                                                 2

                                                                                   8I2I  1
                                               1


1/2                     ℏω1/2→-1/2

                        ℏω3/2→1/2
                                                                     1/21/2    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     4k B   N    4k 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     4k B   N    4k 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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