nuclear magnetic resonance (nmr) - isu sites · 2020. 4. 29. · n：nuclear gyromagnetic...

Basics of nuclear magnetic

resonance and its application to

condensed matter physics

Zaffarano A121

Yuji Furukawa

Fuji (Japanese restaurant), Ames

NMR Lab.

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 Its application to condensed matter physics superconducting and magnetic materials

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 a 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. Hahn, 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 NNNuclear 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

241005.52

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 e (electron spin)

Magnetism of materials 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 the 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、Magnetic materials, Superconductors、and so on ・Chemistry Analysis and/or identification of materials ・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 the magnetic and electronic properties of materials from a microscopic point of view. (nucleus as a probe)

Hyperfine interaction between nuclear 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 electrons

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 about the symmetry of SC gap

full gap ⇒ 1/T1~exp(-Δ/kBT)

anisotropic gap ⇒ 1/T1~Tα

Investigation of spin dynamics

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 CANNOT 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 ; electromagnet 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 ～4.2K) ( ~$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 = 0.05-650 K, P = 2.0 GPa

NMR laboratories (condensed matter physics) in the world

There are many NMR labs in the world !

USA & Canada: ~7 NMR groups

Europe: ~10 groups but in Dresden 6 groups

Japan: ~20 NMR groups

(…NMR city)

(someone called … NMR country)

NMR laboratory in the world

NMR spectrometer with DR refrigerator

very low temperature

one of the extreme conditions


High pressure NMR

Ames: up to 2 GPa

Tokyo, Kyoto, Chiba Univ. : 6 GPa

(diamond anvil and/or bridgemann)

(one of the extreme conditions)


As far as I know, only four NMR labs in the world.

NMR under high pressure with dilution refrigerator

Ames: high pressure NMR down to 0.1K

other NMR labs. (Tokyo, Osaka) , China

& (multiple extreme conditions)

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 can 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 perturbations and non-zero matrix elements

)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 the 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.

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

given by the vector

sum of and L

1: dt

pd

dt

vdmamF

dt

Ld

:law sNewton' of analog

2: dt

LdL

g Bl

BB

precession


Classical treatment

HNdt

Id

H

dt

dN

μ

H



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),


Larmor precession expression in rotating coordinate system


)( Ht

effH


No change in time ! (since we are looking at spin moment on

rotating frame with the same frequency of γH0）

,

z y

z y

t t

d

dt

i j k

ij k Ω i

, , ,

,, ,, , ,

, , ,

, ,

(in rotating frame)

x y z

yx zx y z

x y z

d d d

dt dt dt

dd dd d d d

dt dt dt dt dt dt dt

t

t

t

i j kΩ i Ω j Ω k

μ i j k

i j kμ i j k

μ Ω i j k

μ Ω μ

μ μ H Ω μ μ H Ω

x y z Ω i j k

i

y

z

x

y

z

i

z t j

y t k


Classical treatment

HNdt

Id

H

dt

dN

μ

H




Ω

)( Ht

effH

(With a simple assumption H=H0k)


No change in time ! (since we are looking at the spin moment on

the rotating frame with the same frequency of γH0）

If H=(0,0,H0),


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 the z-axis

When ω＝-γH0, you have resonance and have only H1 magnetic field along to the x-axis

This means spin rotates about the x-axis with a frequency of 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 the 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 90°pulse (just after the pulse, all

nuclear spins are along

the x-axis)

(finite magnetization in the

xy plane)

=> FID

t

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

Spin echo 90°pulse

FID

180°pulse

spin echo

t 2

spectrum

NMR spectrum

H0 = 0

H

m = -1/2

m = +1/2

0H

H0

Signal intensity

(Spin echo intensity)

HI・NZeemanH

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


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 the xy-plane

Ｉ(0)＝0

When ｔ~0

t= ∞

x

y

z

2/π

π I(t)=I0

Signal intensity is proportional to the xy-component of nuclear magnetization

How to measure nuclear spin lattice relaxation T1

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)

+ + + +

Nucleus 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

22

signal intensity ~ transition probability (5:8:9:8:5)

1| | ( )( 1)m I m I m I m

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

1283

13122

2

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

Spin

echo inte

nsity

H ( kOe )

93Nb-NMR

in NbO

93Nb-NMR in NbO (field swept spectrum)

Textbook like typical powder pattern spectrum

I=9/2

(1) NMR shift (Knight shift)

Hyperfine field

(sensitive to magnetic

phase transition)

From NMR spectrum

(1) spacing the lines

Quadrupole interaction

(sensitive to structural

phase transition,

charge ordering)

H0

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

seFH

53

*3

rrH edip

rrss

3

* 1

rH eorb l

Core-poratization

interaction

i

ii

e

cpH22

)0()0(3

8

s

⊿ω=γＨhf

In materials, nuclei experience additional fields due to hyperfine interactions

3d system

~-100 kOe/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

-6

-4

-2

0

2

4

6

ll c

ll b

ll a

K (

%)

T (K)

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 =A Since is expressed by (thermal average value of electron magnetization), =A～A (=AH0) Knight shift is given by K = Hhf/H0 = AH0/H0 ～A K is proportional to ！！

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)

total(T)=spin(T)+orb+・・・+impurity Ktotal(T)=Kspin(T)+Korb

What is ground state ?

Spin singlet ? or magnetic?

From the NMR measurements, the increase of 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-plot K ＝ A/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 the value at the P site per one Bohr magneton of V4+ spins (Vanadium spin produces the hyperfine field at P-site)

The origin of this hyperfine field is “transferred hyperfine field”

NMR in simple metals

1) NMR shift (Knight shift) K=(A/μB)pauli since pauli is expressed by (1/2)g

2μB2NEf

2) Nuclear spin lattice relaxation time T1 Relaxation mechanism

scattering of free electrons from ┃k,↑> to ┃k’,↓> nuclear spin can flop from ↓ to ↑ states

Pauli paramagnetism pauli 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

22

B 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 materials !

Korringa Relation

However deviation from the Korringa relation

is observed in many materials.

Model is so simple

importance of interaction between electrons

(electron correlation)

22

B 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

NMR example

Spin fluctuations at q=Q

V3Se4

VSe1.1

Magnetic phase transition can

be detected by 1/T1.

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

Two electron system

Consider 2 e’s, and ignore their Coulomb repulsion – what will their total wavefunction be ?

A = [(1)b(2) - b(1)(2)]/2

But [total wavefunction] = [space wavefunction][spin wavefunction]

So, either the spatial term is antisymmetric and the spin term is symmetric, or vice versa

Since the total wavefunction must be antisymmetric,

the spatial term must be antisymmetric

if the system is in the spin-antisymmetric singlet state,

if the system is in the spin-symmetric triplet state,

the spatial term must be symmetric

One antisymmetric spin wavefunction [(+½, -½) (-½, +½)]/2 (singlet)

(Exchange => [(-½, +½) (+½, -½)]/2 = - [(+½, -½) - (-½, +½)]/2; antisymmetric)

(+½, +½)

[(+½, -½) (-½, +½)]/2

(-½, -½)

Three symmetric spin wavefunctions (triplet)

NMR study of superconductors

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


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)


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

NMR in magnetic material

In some cases, the answer is No!

In magnetically ordered state, you have spontaneous magnetization (M) without applying external magnetic field. =A～A≠0

hfIHH nTherefore, Hamiltonian for nuclear is not zero without external field

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

┃Hint ┃ = 5.5 Tesla

59Co-NMR under H=0

If you know Ahf,

You can estimate ordered

magnetic moment

=Hint/Ahf

Internal field

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

f=γNHint

Do we always need to apply magnetic field to observe NMR signal?

Thank you for your attention.

I hope that you get some ideas about what NMR is.

If you are interested in NMR, please contact me.

nuclear magnetic resonance (nmr) - isu sites · 2020. 4. 29. · n：nuclear gyromagnetic...

Documents