Transcript Nuclear Magnetic Resonance (NMR)

Nuclear Magnetic Resonance
(NMR)
Yuji Furukawa
A121 Zaffarano
[email protected]
Jan 26
Jan 28
Jan 30
Feb. 2
Feb. 4
Feb. 6
Introduction of NMR
Basics of NMR I
Basics of NMR II
Example I (low-D spin system)
Example II (superconductors)
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
History
1936 Prof. Gorter, first attempt to detect nuclear magnetic spin
(But he did not succeed,
1H in K[Al(SO ) ]12H O and F in LiF))
4 2
2
19
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
μn  gNNI   nI
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)
e
N 
 5.05 1024 (erg/gauss)
2m p c
c.f. electron spin moment
μe=-gμBS
e
 0.921020
2me c (erg/gauss)
|μＢ/μＮ|~1800
B 
Nuclear magnetism
Nuclear magnetic moment
μn  gNNI   nI
U    H   gN I z H z
Nuclear magnetism
 U
 k BT
I
M
  g N I z exp
I z  I
 U 


exp

k
T
I Z  I
 B 
I


  NgI B x 
N
M Ng 2  N2 I I  1
N  
H
3k BT
Curie law
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
H el  n
8
I･ S 3( I･ r )( S･ r )
I･ 
  N g B [(  (r )I･ S  ( 3 
) 3 ]
5
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
⇒ static properties of spins
NMR relaxation time (T1, T2)
⇒dynamical properties
NMR spectrum
NMR spectrum measurements
（static properties of hyperfine field）
⊿H
ΔＨ：contribution from electron
NMR shift： Ｋ＝ΔＨ／Ｈ
Ｈ＝Ｈ０＋ΔＨ
Ｈ
H0
=ω/γ
H
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
 
H   - N I  H hf t 
Dynamical properties of hyperfine field
-1/2
Iz=1/2
-  N   


I H hf (t )  I H hf (t ) 
2
±
±
I  I x ±iI y , H hf  H hfx ±iH hfy
1  N2

T1
2

 H


A2 N2
2

hf
 S



, H hf t  expi N t dt

i

, S i t  expi N t dt
H
hf

 AS i
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…)
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)
・Spectrometer
・Magnetic field
Temperature
←mainstream
frequency range 1～500MHz
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
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
One 3He cryostat: not available now
Plan to purchase DR refrigerator
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
m = -1/2
In the case of I=1/2 and H=(0, 0, H0),
Eigen energies for two quantum levels are
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
alternating current
⇒ alternating field
H ' (t )  N H1 I x cos( N t )
I  I
Ix 
2
 m  1 H ' (t ) m  0
Magnetic transition
H0
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
H
Classical treatment
d
   N H
dt
dI
 N   H
dt
μ
(Time variation of angular momentum is equal to torque)
If H=(0,0,H0),
then μｘ=Asin(ωt+a), μy=Acos(ωt+a), μz=const.
Rotating coordinate system (Ω）
Larmor precession
ω＝γNH
(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=Hx0 cosωt i
x
Hx
Hx=HR+HL
HR=H1(i cosωt ＋ j sinωｔ ）
HL=H1(i cosωt - j sinωｔ ）
H1=H0/2
Coordinate system rotating about z-axis
Laboratory frame
d
    ( H 0  H1 )
dt




    ( H 0  )k  H1i 
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
without H1
x
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=π/2γH1 (π/2 pulse)
t=0
z
H1
x
t=π/γH1 (π pulse)
z
H1
y
x
z
y
H1
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
y
x
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)
y
FID signal
Spin echo method
Two pulse sequence
t
a
t
b
c
d
e
ｔ
π
pulse
π/2 pulse
ω-⊿ω
ω+⊿ω
Spin echo signal
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
equilibrium
state
Boltzmann
distribution
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
1.0
Spin echo intensaity
0.8
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
π
H1
x
t= ∞
Saturation
z
2/π
No mag. in xy-plane
Ｉ(0)＝0
x
z
2/π
y
H1
T1 can be estimated
y
y
x
I(t)=I0
π
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   )
1
cos(1   2 )t     
2
1
 cos(1   2 )t     
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  )
4 I (2 I  1) 
2

η: assymmetry parameter
  2V
 2V  x 2   2V y 2 

 q  2  
2
2
 V z
 z

+
+
+
+
For η＝0


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

e 2qQ
Em  A 3m  I( I  1) A 
4I( 2I  1)
2
I=5/2
Hq＝0
m=±5/2
0  12A 
-5/2
-3/2
12A
m=±3/2
eq=0
6A
-1/2
0  6 A 
0
0
1/2
m=±1/2
0  6 A 
0 12A 
3/2
eq≠0
5/2
ω
6Ａ
12Ａ
NQR (nuclear quadrupole resonance)
ω
NMR spectrum in powder sample
Hz>>HQ
(I=3/2)
-3/2
ℏω-1/2→-3/2
-1/2
1/2

 1
3e 2 qQ
 n  2m  13 cos   1
8I2I  1
ℏω1/2→-1/2
1/21/2
ℏω3/2→1/2
3/2
2


A2
 0 
1 - 9cos2 1 - cos2 




9 2 I  3 e 2 qQ
A2 
64 4 I 2 2 I  1 0
2
θ=90
A1=1/4e2qQ/ℏ
Center line is affected
in 2nd order perturbation
θ=0
ωn-2A1
ωn-A1
ωn
ωn+A1
powder pattern (I=3/2)
ωn+2A1
ωn－16A2/9ℏ
ωn
ωn+A2/ℏ
2nd oeder splitting of
transition for powder
spectruim
central
pattern
NMR spectrum in powder sample
93Nb-NMR
in NbO (field sweep spectrum)
I=9/2
93
Spin echo intensity
Nb-NMR
in NbO
ωn－16A2/9ℏ
60
65
70
75
H (T)
Textbook like typical powder pattern spectrum
ωn
ωn+A2/ℏ
80
Central transition line
Opposite?!
NMR spectrum
ＮＭＲspectrum
Magnetic field sweep and frequency sweep
ω
(1) ω－sweep ( H=constant;H0)
signal (A)
ωＡ
ωＢ
ωＡ
ω
ωＢ
H
Ｈ
H0
ω
(２) Ｈ－ｓｗｅｅｐ (ω＝constant; ω0）
ＨＡ
signal (B)
signal (A)
signal (B)
ω0
ＨＢ
Opposite!!
Need to pay attentions !!
ＨＡ
ＨＢ
H
Hyperfine field at nuclear site
In the material, nuclear experiences additional field due to hyperfine interaction
Fermi contact
Dipole interaction
orbital
interaction
HF  
8 e
2
 s  (0)
3
H dip  e   *
s

r3
3 s  r r
r5

1

3
r
H orb  e  l  *
Core-poratization
8 e
H cp  
 s
interaction
3
S-electron
 
(0)    i (0) 
2
i
2
μS

i
Hint
These give additional field (Hhf) at nuclear site
-> shift in spectrum (NMR shift）
⊿ω=γＨhf
ω0
3d system
~-100kOe/μB
ω
ω0+⊿ω
Ｒｅｌａｔｉｏｎ between NMR shift and magnetic susceptibility
Hamiltonian
H=Hz+Hhf
Hz=Hzeeman
(H=H0)
Hhf=Hdipole+HFermi+Hcore-polarization+…..
=AI・S
H   nI ( H0  H hf )
A: hyperfine coupling constant
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
K is proportional to χ ！！
<M> increases with
increasing H
-> high accuracy
Example
Spin dimer system VO(HPO4)0.5H2O
V4+
(3d1:
What is ground state ?
Spin singlet ? or magnetic?
AF interaction
NMR shift (31P-NMR)
s=1/2)
Magnetic susceptibility
-5
1.8x10
0.6
-5
0.5
-5
1.4x10
-5
1.2x10
0.4
-5
1.0x10
K (%)
magnetic susceptibility (emu/g)
1.6x10
-6
8.0x10
-6
6.0x10
0.3
0.2
-6
4.0x10
0.1
-6
2.0x10
0.0
0.0
0
50
100
150
200
250
300
T (K)
0
50
100
150
200
250
300
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
be estimated from the slope
0.5
K (%)
0.4
dK
A

d N B
0.3
0.2
0.1
0.0
0.0
-6
5.0x10
-5
1.0x10
-5
1.5x10
Ahf =3.3 ｋＯｅ/μB
 (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
A
K  g B2 N  F 
2
K is independent of T
2)Nuclear spin lattice relaxation time T1
Relaxation mechanism
scattering of free electron from ┃k,↑> to ┃k’,↓>
nuclear spin can flop from ↓ ⇒ ↑ state
1 
2
  N A   I  
T1 
k ,k 
f k 1  f k    k BT
2

s 
2
f k 1  f k   E k   E k  
f
 k B T  E k  E F 

1 
2
 ( N ) 2 A 2 g 2 N  F  k BT
T1 
1/T1 is proportional to T
T1T= constant
Korringa relation
A 2
g B N  F 
2
1 
2
 ( N ) 2 A 2 g 2 N  F  k BT
K
T1

2
2
1
4k B   N 
4k B   N 


   S


2


T1TK
  g 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
1
4k B   N 
4k B   N 


   S


2


T1TK
  g B 
  B 
Stoner enhancement
χ＝ χ0/（1- α0 ）
Ｉ＝Ｕ/Ｎ0
α0＝Ｉχ0/2
Modified Korringa Relation
ＲＰＡ
S
K 
T1TK 2
Kα>>1：AF spin correlation
Kα<<1：F spin correlation
 (q,  ) 
 0 (q,  )
1   0 [  0 (q,  ) /  0 (Q0 ,0)]
1
(1   0 )
(1   0 ) 2
K ( q ) 
(1   q ) 2
K~~
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IHhf
(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.