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Principle of NMR ・・・・・ a little bit complicated (quantum mechanics) NMR experiments ・・・・・ a little bit complicated (Low T, RF, magnetic field, Pressure….) Data analysis of NMR results
・・・・・・ a little bit complicated
But, NMR measurements give us very important information which cannot be obtained by other experimental techniques
Plan Basics of NMR (this week) Its application (if I have time ……) low dimensional spin system superconductors and so on
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(SO4)2]12H2O and 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
IIμn ng NN
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
μe=-gμBS
241005.5
2
cm
e
p
N
201092.02
cm
e
e
B
(erg/gauss) |μＢ/μＮ|~1800
Explanation of “magic number” (1949 Mayer and Jensen independently,
by introducing an idea of a strong inverted nuclear spin-orbit interaction)
spuds if pug dish of pig
spdsfpgdshfpig
The energy level structure originates from potential energy of nucleus due to nuclear force
(eat) potatoes if the pork is bad
Nuclear shell model
178O (Z = 8 and N=9) is doubly magic except for an
extra neutrons in the 1d5/2 subshell, so it should
have i = 5/2, as observed.
15N (Z = 7 and N=8) is doubly magic except for a
proton hole in the 1p1/2 subshell, so it should
have i = 1/2, as observed.
example
Nuclear magnetism
IIμn ng NN
Nuclear magnetic moment
zzN HIgHU
xBNgI
Tk
U
Tk
UIg
M NI
II B
I
II B
zN
Z
z
exp
exp
Tk
IINg
H
M
B
NN
3
122
Much less than χe (electron spin)
Magnetism of material is mainly dominated by χe!!
Nuclear magnetism
Curie law
(ｈ：Planck’s constant、ν：frequency、γN：nuclear gyromagnetic ratio、H：magnetic field)
NMR （Nuclear Magnetic Resonance)
Nucleus has magnetic moment (nuclear spin) nucleus is very small magnet
HI・NZeemanH
Zeeman interaction
H N
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 ・Chemistry Analysis and/or identification of material ・Biophysics Analysis of Protein structure, and so on ・Medical MRI (Magnetic Resonance Image)
Brain tomograph
For example;
NMR in condensed matter physics
])))((3
()(3
8[(
353 r
I
r
rSrI
r
SIrgH BNnel
･････ SI
Fermi contact dipole interaction orbital
interaction
NMR measurements
investigation of static and dynamical properties of hyperfine field (electron spins)
One of the important experimental methods for the study on magnetic and electronic properties of materials from the microscopic point of view. (nucleus as a probe)
Hyperfine interaction between nuclear spin and 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）
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
H0
ΔＨ
Ｈ
Ｈ＝Ｈ０＋ΔＨ
Nuclear spin-lattice relaxation time（Ｔ１）
Nuclear spin-lattice relaxation time
Dynamical properties of hyperfine field tHI hfN
-H
y x
y x iH H H iI I I
t H I t H I
hf hf hf
hf hf N
,
) ( ) ( 2
-
± ±
± ±
Iz=1/2
-1/2
iii SAHdttitSSA
dttitHHT
hfN
2
N
2
Nhfhf
2
N
1
exp,2
exp,2
1
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 spin (electron) 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 = 100 MHz ⇒ 5 mK
NMR spectroscopy in condensed matter physics
NMR spectroscopy Continuous wave (CW) NMR Pulse NMR (FT (Fourier transform) –NMR) ←mainstream
・Spectrometer frequency range 5～400MHz ・Magnetic field up to 2Ｔ ; electric magnet up to 9T ; superconducting magnet (NbTi) 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) ( ~$7/liter )
down to 0.3K ; 3He cryostat ($100K) down to 0.01K ; 3He-4He dilution refrigerator ($300K)
My NMR lab at ISU
f = 3.5-500MHz, H = 0-9T, T = 0.05-300 K, P = 2.0 GPa
NMR spectroscopy in condensed matter physics
NMR spectroscopy Continuous wave (CW) NMR Pulse NMR (FT (Fourier transform) –NMR) ←mainstream
・Spectrometer frequency range 5～400MHz ・Magnetic field up to 2Ｔ ; electron magnet up to 9T ; superconducting magnet (NbTi) 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) ( ~$7/liter )
down to 0.3K ; 3He cryostat ($100K) down to 0.01K ; 3He-4He dilution refrigerator ($300K)
My NMR lab at ISU
f = 3.5-500MHz, H = 0-9T, T = 0.05-300 K, P = 2.3 GPa
NMR spectrum
NMR spectrum measurements （static properties of hyperfine field）
H H0
=ω/γ
⊿H
NMR shift： Ｋ＝ΔＨ／Ｈ
ΔＨ：contribution from electron
Ｈ
Ｈ＝Ｈ０＋ΔＨ
How do we measure NMR spectrum ?
Magnetic resonance
H0 = 0 H0 ≠ 0
m = -1/2
m = +1/2
HI・NZeemanH In the case of I=1/2 and H=(0, 0, H0),
Eigen energies for two quantum levels are
given
02/1
2
1HE N 02/1
2
1HE N
0HE nH
To make a resonance, one needs time dependent perturbation and non-zero matrix element
)cos()(' 1 tIHtH NxN 2
II
I x
0)('1 mtHm
Magnetic transition
H0
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
HNdt
Id
H
dt
dN
μ
H
Larmor precession ω＝γNH
(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.
Clarification (classical dipole in a field):
there’s a force to align m & B
Consider a simple dipole (ex. bar magnet) in a field
However!
What do we expect if our magnet is
spinning ?
Due to the angular momentum, it will
not simply line up with the field
Since ,
U l B
– just like the precession of a spinning top
(which is due to the torque created by the
gravitational force)
Bl
Rotation axis is
direction of
Rotation axis is NOW
vector sum of and L
1: dt
pd
dt
vdmamF
dt
Ld
:law sNewton' of analog
2: dt
LdL
g Bl
BB
precession
Motion of magnetic moment
Classical treatment
HNdt
Id
H
dt
dN
μ
H
Larmor precession ω＝γNH
(Time variation of angular momentum is equal to torque)
Rotating coordinate system (Ω）
Ω
)( Ht
effH
(With a simple assumption H=H0k)
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 the same frequency of γH0）
If H=(0,0,H0),
then μｘ=Asin(ωt+a), μy=Acos(ωt+a), μz=const.
Effects of alternating field
Hx=Hx0 cosωt i
x
y
Hx
Hx=HR+HL
HR=H1(i cosωt ＋ j sinωｔ ）
HL=H1(i cosωt - j sinωｔ ）
H1=H0/2
)( 10 HHdt
d
iHkH
t10 )(
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
x
y
z
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
Effects of alternating field
Hx=Hx0 cosωt i
x
y
Hx
Hx=HR+HL
HR=H1(i cosωt ＋ j sinωｔ ）
HL=H1(i cosωt - j sinωｔ ）
H1=H0/2
)( 10 HHdt
d
iHkH
t10 )(
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
x
y
z
with H1 (rotating frame)
H1
You can control the direction
of spins!
Manipulation of spin
Effects of alternating field
x
y
z
H1
x
y
z 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 amplifiers)
x
y
z
H1
x
y
z
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
NMR spectrum
H0 = 0 H0 ≠ 0
Iz= -1/2
Iz = 1/2
Nuclear spin lattice relaxation T1
Boltzmann
distribution
thermal
equilibrium
state
Resonance
(absorption)
nonequilibrium
state
H
Relaxation
(energy
emission
to lattice
(electron system)
-> thermal
equilibrium
state
T1 is a time constant (from nonequilibrium to equilibrium states)
Absorption energy and spin lattice relaxation T1
Nuclear spin lattice relaxation T1
Nuclear spin lattice relaxation T1
Relaxation is induced by fluctuations of hyperfine field with NMR frequency
How to measure nuclear spin lattice relaxation T1
x
y
z
H1
0.0
0.2
0.4
0.6
0.8
1.0
Sp
in e
ch
o in
ten
sa
ity
time
t-dependence of signal intensity
I(t)=I0(1-exp(-t/T1))
T1 can be estimated
x
y
z
H1
Saturation
2/π
π
No mag. in xy-plane
Ｉ(0)＝0
When ｔ~0
t= ∞
x
y
z
2/π
π I(t)=I0
Signal intensity is proportional to xy-component of nuclear magnetization
How to measure nuclear spin lattice relaxation T1
block diagram (NMR spectrometer)
QPSK (heterodyne)
(Quadrature Phase Shift Keying)
block diagram (NMR spectrometer)
Receiver
Amp
Phased shifter
PSD
LPF
t
t
tt
)(cos2
1
)(cos2
1
)sin()sin(
21
21
21
PSD Multiplication of Input frequencies
-> out put
frequency difference and sum
cross diode (back to back)
・parallel => only low voltage signal can be put into preamp.
・series => low voltage noise can be cut.
two out put signals
=> cosine and sine components
block diagram (NMR spectrometer)
Parallel connection
Series connection
FFT and complex detection
FFT and complex detection
NMR spectrum
QH
22
2222
2
2
22222
)(2
1)3(
)12(4
zV
yVxV
z
Vq
IIIIII
qQez
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)
)12(4
)1(3
2
2
II
qQeA
IImAEm
ZnZeeman IHHH 0-
For η＝0
η: assymmetry parameter
NMR spectrum
0
A120
A60
0
A60
A120
m=±5/2
m=±1/2
m=±3/2
12A
6A eq=0
eq≠0
)I(I
qQeA)I(ImAEm
12413
22
1. Hquadrupole≠0， Ｈ＝0
2. Hzeeman >> Hquadrupole
ω 6Ａ 12Ａ
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
-3/2
3/2
-1/2
1/2
ℏω3/2→1/2
ℏω-1/2→-3/2
ℏω1/2→-1/2
128
31312
22
n1
II
qQecosm
powder pattern (I=3/2)
ωn ωn-2A1 ωn-A1 ωn+A1 ωn+2A1
A1=1/4e2qQ/ℏ
ωn－16A2/9ℏ ωn+A2/ℏ ωn
2nd oeder splitting of central transition for powder pattern spectruim
0
22
22
222
01/21/2
124
32
64
9
cos-19cos-1
qQe
II
IA
A
θ=0
θ=90
Hz>>HQ (I=3/2)
Center line is affected
in 2nd order perturbation
NMR spectrum in powder sample
60 65 70 75 80
Sp
in e
ch
o in
ten
sity
H ( T )
93Nb-NMR
in NbO
93Nb-NMR in NbO (field sweep spectrum)
Textbook like typical powder pattern spectrum
I=9/2
ωn－16A2/9ℏ ωn+A2/ℏ ωn
Central transition line
Opposite?!
NMR spectrum
ω
signal (A)
H
signal (B)
(1) ω－sweep ( H=constant;H0)
ＮＭＲspectrum Magnetic field sweep and frequency sweep
H0
ω ωＢ ωＡ
(２) Ｈ－ｓｗｅｅｐ (ω＝constant; ω0） ω
signal (A)
H
signal (B)
ωＡ
ωＢ
ω0
ＨＡ ＨＢ
Ｈ ＨＡ ＨＢ
Opposite!! Be careful !
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 2
)0(3
8
se
FH
53
*3
rrH e
dip
rrss
3
* 1
rH e
orb l
Core-poratization
interaction
i
ii
e
cpH22
)0()0(3
8
s
⊿ω=γＨhf
In materials, nuclear experiences additional field due to hyperfine interaction
3d system
~-100kOe/μB
μS
Hint
Example (T-dependence of hyperfine field)
70.0 70.2 70.4 70.6 70.8
120 K
95 K
75 K
58 K
48 K
34 K
23 K
19 K
14 K
10 K
9 K
8 K
7 K
Inte
nsity (
arb
. un
its)
(MHz)
Hllc Hllb Hlla
Re
f
6 K
220 K
Temperature dependence of spectrum
31P-NMR in Pb2VO(PO4)2
10 100
-4000
-2000
0
2000
4000
6000
P1
ll c
ll b
ll a
Kiso
HTSE fitK
(ppm
)
T (K)
J1 = -5.4 K
J2 = 9.3 K
g = 1.95 (EPR)
16K to 260K
T-dep of NMR shift
100(%)0
0
f
ffK
100(%) 0
res
res
H
HHK
f0
H
H0 Hres
Ｒｅｌａｔｉｏｎ between NMR shift and magnetic susceptibility
H=Hz+Hhf
Hamiltonian
Hz=Hzeeman (H=H0)
Hhf=Hdipole+HFermi+Hcore-polarization+…..
=AI・S A: hyperfine coupling constant
)( hf0 HHIH n ASH hf
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
(hyperfine field)
Example
0 50 100 150 200 250 3000.0
0.1
0.2
0.3
0.4
0.5
0.6
K (%
)
T (K)
Spin dimer system VO(HPO4)0.5H2O
V4+ (3d1: s=1/2)
0 50 100 150 200 250 3000.0
2.0x10-6
4.0x10-6
6.0x10-6
8.0x10-6
1.0x10-5
1.2x10-5
1.4x10-5
1.6x10-5
1.8x10-5
ma
gn
etic s
uscep
tib
ility
(e
mu
/g)
T ( K )
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,
0.0 5.0x10-6
1.0x10-5
1.5x10-5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
K (%
)
(emu/g)
Good linear relation K is proportional to χ
Hyperfine coupling constant can be estimated from the slope
BN
A
d
dK
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
1) NMR shift (Knight shift) K=(A/μB)χpauli
since χpauli is expressed by (1/2)g2μB2NEf
2)Nuclear spin lattice relaxation time T1 Relaxation mechanism
scattering of free electrons from ┃k,↑> to ┃k’,↓>
nuclear spin can flop from ↓ ⇒ ↑ states
Pauli paramagnetism χpauli
No electron correlation
Simple metal (like Cu and Al and so on)
kkkk
N EEkfkfsIAT
11
,
222
1
Fk EETkf
Tkkfkf
BB1
TkNgAT
FN B
2222
1
)(1
1/T1 is proportional to T
T1T= constant
K is independent of T
2
2B F
AK g N
Korringa relation
22
N NB B
2
1 B e
4 41 k kS
TTK g
TkNgAT
FN B
2222
1
)(1
This does not depend on material !
Korringa Relation
However deviation from the Korringa relation
is observed in many material.
Model is so simple
importance of Interaction between electrons
(electron correlation)
2
2B F
AK g N
Modified Korringa relation
Sk
g
k
TKT
2
B
NB
2
B
NB
2
1
441
Korringa Relation
Modified Korringa Relation
Kα>1：AF spin correlation
Kα<1：F spin correlation
Stoner enhancement
χ＝ χ0/（1- α0 ）
enhancement factor α0
FSq
K2
2
0
)1(
)1(
)1(
1~~
0
K
)]0,0(/),([1
),(),(
000
0
q
RPA (random phase approximation)
q
χq
q
χq
0 Q
Ferro. correlations
AF correlations
2
1
)1(~1
qFNTT
α
SKTKT
1
2
1
)(1
T1 and K measurements give us
information of electron correlation!
NMR example (itinerant AF magnet)
Itinerant antiferromagnet V3Se4
Y. Kitaoka et al. JSPJ 48 (1980)1460
NMR example
Spin fluctuations at q=Q
SCR theory
V3Se4
VSe1.1
NMR in superconducting state
Symmetry of cooper pair
s-wave
(l=0, s=0)
p-wave
(l=1, s=1)
d-wave
(l=2, s=0)
Isotropic gap
Anisotropic gap
Anisotropic gap
S-wave
d-wave
NMR study of superconductor
Symmetry of cooper pair
s-wave
(l=0, s=0)
p-wave
(l=1, s=1)
d-wave
(l=2, s=0)
Isotropic gap
Anisotropic gap
Anisotropic gap
)/exp(/1 1 kTT
Knight shift 1/T1
TT 1/1
TT 1/1
Just below Tc
Hebel-Slichter peak
NMR example (Superconductor)
Al metal
Knight shift
Enhancement of transition probability
Divergence behavior of DOS
Hebel-Slichter peak
Above Tc
1/T1~T
Below Tc
1/T1 ~exp(-⊿/ｋＴ）
S-wave SC !
Decrease of spin susceptibility
T-dependence of 1/T1
NMR example (Superconductor)
Ru(Cu)
Sr
O
RuO2面
c
a
bRu4+(4d4)
Crystal structure Sr2RuO4
Sr2RuO4 Tc~1.5K
No change! 1/T1~T3
suggesting P-wave SC!!
K. Ishida et al, Nature 396 (1998)658
Ru4+ (4d4)
NMR example (Superconductor)
Kanoda, Miyagawa, Kawamoto et al., d-wave SC
Pairing symmetry of Cooper pair
can be determined by NMR
measurement
Important information of
origin for the SC appearance
Spin gap (SG) behavior at L-region
Strong AF spin fluctuations in metallic region
SG
Anomalous
Metallic state
~ (0,0)K q=0 comp.
q-sum of the dynamical susceptibility
T dependence of (q,) with
respect to that of q = 0 comp.
no increase of 1/T1T
no obvious AF spin
fluctuations
22 "
02
1
21| ( ) | ( , )N B
M
qA
kA q q
TT N
NMR studies in High Tc Cuprates
NMR is sensitive to not only static properties but also “magnetic fluctuations”
