rmn: local magnetic measurements

RMN: LOCAL MAGNETIC MEASUREMENTS

• allow to probe the electronic environment of the nucleus• Knight shift in metals• Some examples: cuprates and C60 compounds

• Notion of relaxation• rf pulse techniques and Fourier transforms

INTERACTIONS BETWEEN NUCLEAR AND ELECTRONIC SPINS: HYPERFINE COUPLINGS

RELAXATION TIME

• rf pulse techniques and Fourier transforms• Local field fluctuations and dynamic electronic

susceptibility. • Korringa relaxation in metals.• Spins echos and transverse relaxation T2 (spin spin)

NMR APPLICATIONS

Solid state and soft matterChemistry, BiologyMedical imaging (IRM) Industrial chemistry, food control

H. Alloul, EPFL,30/04/09

NMR , ESR RESONANCE IN THE PARAMAGNETIC REGIME

000 S.B. .BSBH zZ γγγγγγγγµµµµ hh −−−−====−−−−====−−−−====

Exciting ac field Act as a perturbation for HZ

tH Lrf ωωωωγγγγ cosS.B1h−−−−====

/z/B0

1/2 1/2- stransition →zB1⊥⊥⊥⊥

01/2-1/2i

1/2 1/2- stransition

≠

→

rfHf

Electric current

at frequency ωωωω

SampleAbsorption spectroscopy

H. Alloul, EPFL,30/04/09

HYPERFINE INTERACTIONSNMR Frequency shifts

Interactions between nuclear moments Iand electronic moments s et l

Dipolar

−−−−−−−−====ΗΗΗΗ 23

2).)(.(3.

rrsrIsI

ren

dd

vrvrrrh γγγγγγγγ

Orbital lIr

enorb

rrh.3

2 γγγγγγγγ−−−−====ΗΗΗΗ

Contact )(.38 2 rsIenc

vrrh δδδδγγγγγγγγππππ====ΗΗΗΗ

0;0 ≡≡≡≡ΗΗΗΗ≡≡≡≡ΗΗΗΗ ddorb

ΙΙΙΙ r

s

• Filled atomic shells :

• Paramagnetic or diamagnetic compounds:

0BBL χχχχ∝∝∝∝>>>><<<<r

• Paramagnetic or diamagnetic compounds:

)(. 0 LncorbddZT BBIrrr

h ++++−−−−====ΗΗΗΗ++++ΗΗΗΗ++++ΗΗΗΗ++++ΗΗΗΗ====ΗΗΗΗ γγγγ

[[[[ ]]]]>>>><<<<−−−−++++>>>><<<<==== LLLL BBBBrrrr

Frequency shift

Mean field Relaxation time

Linear response

Local measurement of the electronic susceptibility

Insulators HorbChemical shift

(orbital currents)

metals Hc χχχχPauliKnight shift

(unpaired electrons)H. Alloul, EPFL,30/04/09

KNIGHT SHIFT IN METALS

∑∑∑∑====ΗΗΗΗi

iienc rsI )(.38 2 vrr

h δδδδγγγγγγγγππππ

)0(.38)(.3

8 2 MIrsI ni

iiencrr

hvrr

h γγγγππππδδδδγγγγγγγγππππ ========ΗΗΗΗ ∑∑∑∑

M(0) magnetization density operator at the nuclear site

Bloch electronic states srki

k erusk rvr

r vrrψψψψ)(, ====

0)(2

)0( BM Pk

ruF

k χχχχvrr

==== )()(21 2

FeP Enγγγγχχχχ h====

Contact hamiltonian for the spin I(for all the electrons of the metallic band)

Pauli susceptibility of the electronic band

Fk

kruenA

2)(2

38 vrh γγγγγγγγππππ====with

one might write

Pne

L ABB

K χχχχγγγγγγγγ20 h========

∑∑∑∑====ΗΗΗΗi

iic rsIA )(.vrr

δδδδ

Pauli susceptibility of the electronic band

• The electrons are then considered as free electrons• The hyperfine coupling A contains informations on

the electronic band structure

Knight shift

H. Alloul, EPFL,30/04/09

CuO2

High Tcsuperconducting cuprates

⊥⊥⊥⊥0B

CuO2 conducting planes

Y

Ba

Cu O

//0B

Tc

KNIGHT SHIFT IN METALS

17O NMR of the CuO2 planes

Vanishing ofχχχχP in the superconducting stateCooper pairs are in a singlet state

H. Alloul, EPFL,30/04/09

ALKALI C 60 COMPOUNDS

Cubic A3C60 Compounds Superconductors with high Tc

(up to 38K)

AC60 Compounds Polymers

K3C60 3 K+ + C603 -

Polymers

1D metal

AF for T< 30K

CsC60 Cs+ + C60-

H. Alloul, EPFL,30/04/09

POLYMERIZED AC 60 PHASES

KC60

RbC60

1D metal

CsC60 Cs+ + C60-

The 13C site differenciation evidences the polymerization

Difference between KC60 and CsC60

-200 0 200 400 600

Shift (ppm)

CsC60

Different 3D ordering of the polymer chains

sp3 carbonsH. Alloul, EPFL,30/04/09

Response to an a.c. applied field

(((( ))))δδδδωωωωδδδδωωωωδδδδωωωωωωωω

sinsincoscos)cos()(

cos)(

00

1

ttMtMtM

tHtH a

++++====−−−−========

( ) ( )[ ] tieHitM ωωωωωωωωχχχχωωωωχχχχ 1"'Re)( −=

Defines a complex magnetic susceptibility

LINEAR REPONSE AND DISSIPATION

Origin of the linewidth in a homogeneous applied field?Dissipation and relaxation

Fil infini (d’axe z)

Absorbed power

χχχχ’’ (ω) (ω) (ω) (ω) absorption

Ha = 0

1NTr avec ====

(((( )))) (((( )))) (((( )))) (((( ))))(((( ))))δδδδδδδδωωωωχχχχωωωωχχχχωωωωχχχχωωωωχχχχ sincos"' ii −−−−====−−−−====

(((( )))) 210"

2HP µµµµωωωωχχχχωωωω====

Magnetic energy absorbed during a period T=2π/ω2π/ω2π/ω2π/ω

(((( )))) (((( ))))[[[[ ]]]](((( )))) 2

10

022

10

0 0

"

sin"cossin'

H

dttttH

dtdtdH

MW

T

T

µµµµωωωωπχπχπχπχ

ωωωωωωωωχχχχωωωωωωωωωωωωχχχχωωωωµµµµ

µµµµ

====

++++====

−−−−====

∫∫∫∫

∫∫∫∫

χχχχ’ (ω) (ω) (ω) (ω) dispersionH. Alloul, EPFL,30/04/09

∫∫∫∫t

∫∫∫∫ ∞∞∞∞−−−−−−−−====

tdttHttmtM ')'()'()(

Response to an excitation H(t’)

Time dependent response to an excitation• Causality(the effect follows the cause)• Stationnary

impulsion in t’ response m(t-t’ )

• Linear

For a sinusoidal excitation )exp()'( 1 tiHtH ωωωω====

PULSE RESPONSE

(((( ))))[[[[ ]]]]

(((( ))))∫∫∫∫

∫∫∫∫

∫∫∫∫

∞∞∞∞∞∞∞∞−−−−

∞∞∞∞−−−−

−−−−====

−−−−−−−−====

−−−−====

01

1

1

exp)()exp(

''exp)'()exp(

')'exp()'()(

dttitmtiH

dtttittmtiH

dttiHttmtM

t

t

ωωωωωωωω

ωωωωωωωω

ωωωω

(((( )))) (((( ))))∫∫∫∫∞∞∞∞

−−−−====0

exp)( dttitm ωωωωωωωωχχχχ

(((( )))) (((( )))) (((( ))))∫∫∫∫∞∞∞∞∞∞∞∞−−−−

−−−−==== ωωωωωωωωωωωωχχχχππππ dtitm exp2)( 1

Frequency response

χχχχ (ω)(ω)(ω)(ω) m(t)Pulse response

Fourier transform

H. Alloul, EPFL,30/04/09

−−−−====ττττττττ

χχχχ ttm exp)( 0

Example : exponential response : relaxation time ττττ

(((( ))))

(((( ))))[[[[ ]]]](((( ))))[[[[ ]]]]ττττχχχχ

χχχχ

χχχχ

ττττττττχχχχ

ττττ

ττττ

/exp1

exp

exp

''

exp)(

0

/00

/

00

00

th

uh

duuh

dtt

htM

t

t

t

−−−−−−−−====

−−−−−−−−====

−−−−====

−−−−====

∫∫∫∫

∫∫∫∫

PULSE RESPONSE

(((( )))) ∫∫∫∫∞∞∞∞

++++−−−−====0

0 1exp dttiωωωω

ττττττττχχχχωωωωχχχχ

220

10

1

11

ττττωωωωωτωτωτωτχχχχωωωω

ττττττττχχχχ

++++−−−−====

++++====−−−− i

i

(((( ))))

(((( ))))22

0

220

1"

1'

ττττωωωωττττωωωωχχχχωωωωχχχχ

ττττωωωωχχχχωωωωχχχχ

++++====

++++====

If ω<<1/τω<<1/τω<<1/τω<<1/τStatic susceptibility χχχχ0

maximal absorption for ω=1/τω=1/τω=1/τω=1/τ

(((( ))))[[[[ ]]]]ττττχχχχ /exp10 th −−−−−−−−====

H. Alloul, EPFL,30/04/09

Angular moment M/γγγγ γγγγ=-gµµµµB

BMM ∧∧∧∧==== γγγγdt

d

const.0

MM0M

M s

========

========

zz M

dt

dMdt

d

0BL γγγγωωωω −−−−====Larmor precession

MAGNETIC RESONANCE

RELAXATION TIME T 1

1

M(B)-M-BM

MTdt

d ∧∧∧∧==== γγγγ

1

--

T

MM

dt

dM szz −−−−====

(((( ))))

−−−−−−−−−−−−====−−−−

1expcos

Tt

MMMM sszs θθθθ

T1 energy exchange with a thermostat

Spin lattice relaxation (ex: Phonons)

H. Alloul, EPFL,30/04/09

Two rotating fields at + ωωωω and -ωωωω

xcos')(B 1 tBt ωωωω====

B-(t) is negligible at the Larmor frequency

B’1=2B1

)ysinx(cos)(B

)ysinx(cos)(B

ttBt

ttBt

ωωωωωωωωωωωωωωωω

−=+=

−

+

1

1

NMR DETECTION

zrr

ωωωω====ΩΩΩΩ

aaA rrrr

∧∧∧∧ΩΩΩΩ++++====dtd

dtd

mz)x'z(mm

10 ∧∧∧∧−−−−++++∧∧∧∧==== ωωωωγγγγ BBdtd

at the Larmor frequency

Change of reference frame x’Oy’ rotating at + ωωωω

mM

aA

Oy'x'xOy

rr

rr

⇒⇒⇒⇒

⇒⇒⇒⇒

⇒⇒⇒⇒

H. Alloul, EPFL,30/04/09

MOTION EQUATIONS IN THE ROTATING REFERENCE FRAME

With relaxation

L−−−−ωωωωωωωωωωωω

1

z-mbm

mT

M

dtd s

eff −−−−∧∧∧∧==== γγγγ

For ω = ωω = ωω = ωω = ωL = -γγγγB0beff = B1 x’

beff is the effective field in the rotating reference frame

x'zx'z)(b 110 BBB Leff ++++−−−−====++++++++====

γγγγωωωωωωωω

γγγγωωωω

zszz m'-MmM ==defineusLet

z)('y'xm zsyx m'-Mmm ++++++++====

(((( ))))

(((( ))))

11

11

1

''

)(

T

mmB

dt

dm

T

mmm'-MB

dt

dm

Tm

mdt

dm

zy

z

yxLzs

y

xyL

x

++++−−−−====−−−−

−−−−−−−−−−−−====

−−−−−−−−====

γγγγ

ωωωωωωωωγγγγ

ωωωωωωωω

H. Alloul, EPFL,30/04/09

LINEAR RESPONSE REGIME

2

Stationary solution BLOCH EQUATIONS

Small B1

01

01

22

/)("

/)('

0)(

µµµµωωωωχχχχµµµµωωωωχχχχ

Bm

Bm

mmOm

y

x

yxz

++++

++++========

≈≈≈≈++++====

(((( ))))

(((( ))))1

1

1

T

mmMB

dt

dm

Tm

mdt

dm

yxLs

y

xyL

x

−−−−−−−−−−−−====

−−−−−−−−====

ωωωωωωωωγγγγ

ωωωωωωωω

21

21

0

21

2

21

0

)(1)("

)(1

)()('

T

TM

T

TM

Ls

L

Ls

ωωωωωωωωγγγγµµµµωωωωχχχχ

ωωωωωωωωωωωωωωωωγγγγµµµµωωωωχχχχ

−−−−++++====

−−−−++++

−−−−====

++++

++++

DISPERSION ABSORPTION

H. Alloul, EPFL,30/04/09

x'z)(b 10 BBeff ++++++++====γγγγωωωω

0BL γγγγωωωωωωωω −−−−====≈≈≈≈

γγγγωωωωωωωω

γγγγωωωω −−−−====++++>>>>>>>> LBB 01Si

1Bbeff ≈≈≈≈

1Bmbmm ∧∧∧∧≈≈≈≈∧∧∧∧==== γγγγγγγγ effdt

d

x

RADIOFREQUENCY PULSES

dt

rf pulse width tw

ttw

B1

In the rotatingreference frame (x ’y ’z)

m rotates around x’ at the Larmor frequency ωωωω1=γ γ γ γ B1

ww tBt 1γγγγθθθθ =⇔ angle an of m of rotationwidth of pulse

magnetization free precessionat the Larmor frequency ωωωω0=γ γ γ γ B0

pulse θ=πθ=πθ=πθ=π/2

Signal sampling

Complex Fourier transform

FREE PRECESSION SIGNAL

Absorption

Dispersion

χχχχ ’’( ω)ω)ω)ω)

χχχχ’(ω)ω)ω)ω)

Pulse response = Fourier transform of the spectrum

H. Alloul, EPFL,30/04/09

DETECTION

(((( ))))tAe

ttSe

g

LL

ωωωωωωωω

cos

cos

2

1

⇒⇒⇒⇒

⇒⇒⇒⇒gL ωωωωωωωω ≈≈≈≈

( )

( ) ( )[ ] ( )[ ] tttAS

tttASe

gLgLL

gLLs

ωωωωωωωωωωωωωωωω

ωωωωωωωω

−++=

=

coscos

coscos

2

1

Low frequency

rf source

gωωωω2≈≈≈≈

COMPLEX FOURIER TRANSFORM

(((( )))) (((( ))))ωωωωχχχχωωωωχχχχ "et'

Low pass filter

tAe gωωωωcos2 ⇒⇒⇒⇒ ( ) ( )[ ]ttAS gL ωωωωωωωω −⇒ cos02

1

tAe gωωωωsin'2 ⇒⇒⇒⇒ ( ) ( )[ ]ttAS gL ωωωωωωωωππππ −⇒ sin/ 22

1

H. Alloul, EPFL,30/04/09

SPIN LATTICE RELAXATION TIME MEASUREMENT

tD

ππππ

t’D

t ’’ D

π/π/π/π/

π/π/π/π/2222

π/π/π/π/2222

π/π/π/π/2222

Ms

(π π π π - tD - ππππ/2) sequence

tD

π/π/π/π/2222

−−−−−−−−−−−−====1

exp2)( TtMMtM ssD

Ms

- MsH. Alloul, EPFL,30/04/09

PHYSICAL ORIGIN OF THE SPIN LATTICE RELAXATION

0.BSH zZ γγγγh−−−−====

rf excitting fieldperturbation for HZ

tH Lrf ωωωωγγγγ cosS.B1h−−−−====

zB1⊥⊥⊥⊥0≠

→

1/2-1/2if

1/2 1/2- stransition

rfH

z//B0

[ ]><−+><= LLLL BBBBrrrr

Relaxation: transverse components of the fluctuating field at the Larmor frequency

[ ]LLLL

(((( ))))∫∫∫∫∞∞∞∞∞∞∞∞−−−−

−−−−++++ −−−−>>>><<<<==== dttiBtBT nLLn ωωωωγγγγ exp)0()(1 21

Transition probability

Correlation function of the local field

T1 results from the coupling with the equilibrium fluctuations of the electron spins degrees of of freedom

H. Alloul, EPFL,30/04/09

T1 IN A METAL: KORRINGA LAW

(((( ))))∫∫∫∫∞∞∞∞∞∞∞∞−−−−

−−−−++++ −−−−>>>><<<<==== dttiBtBT nLLn ωωωωγγγγ exp)0()(1 21

)0()( 2 MArsABnei

iin

Lr

h

vr

h

r

γγγγγγγγδδδδγγγγ −−−−====−−−−==== ∑∑∑∑

∑∑∑∑++++−−−−====++++====i

iincZ rsIAB.IHHH )(.0vrrr

h δδδδγγγγ

(((( ))))∫∫∫∫∞∞∞∞∞∞∞∞−−−−

−−−−++++ −−−−>>>><<<<==== dttiMtMA

T ne

ωωωωγγγγexp)0()(1

24

2

1

rr

h

Fluctuation-dissipation theorem(Transverse dynamic susceptibility of the electron gas)

(((( )))) (((( ))))∫∫∫∫∞∞∞∞∞∞∞∞−−−−

−−−−++++ −−−−>>>><<<<−−−−==== dttiMtMTk nB

nnT ωωωωωωωωωωωωχχχχ exp)0()()exp1(2

1"rrh

h

For a fermion gas

TkBn <<ωωωωhwith

( ) )()( FFeT EniEn 222

2

1 ωωωωππππγγγγωωωωχχχχ hh +=

So that Korringa law for a metal

(((( ))))n

nTB

eTk

A

T ωωωωωωωωχχχχ

γγγγ"2

122

2

1 h====

TkEnAT BF )(1 221 h

ππππ====

22

1 4

====ne

BkTKT γγγγγγγγ

ππππh

)(22 Fne

Pne

EnAAK γγγγγγγγχχχχγγγγγγγγ

========h

logT127Al NMR in Al metal

T1 IN A METAL: KORRINGA LAW

log(1/T)

ΤΤΤΤ1T= 1.85 sec.°KKorringa law

Thermometry

H. Alloul, EPFL,30/04/09

T1 IN A SUPERCONDUCTOR

V3Sn

51V NMR Korringa

(T1T)-1 vanishes in the superconducting state: low T behaviour gives the SC gap

T1 minimum correspond to an increase of relaxation rate below Tc

Hebel –Slichter peak

H. Alloul, EPFL,30/04/09

SPIN ECHOES

t=2τ2τ2τ2τττττ+

B1t====0+

ττττ-

ττττ ττττ t=2τ2τ2τ2τ

B1

δωδωδωδω

δωδωδωδωDistribution of Larmor

frequencies

ττττ- ττττ+ t=2τ2τ2τ2τ

Echo intensity varies with ττττ

2τ2τ2τ2τ12τ2τ2τ2τ2 2τ2τ2τ2τ3

T2*< T2 < T1

T2 transverse relaxation time

∆ω ∆ω ∆ω ∆ω T2*

H. Alloul, EPFL,30/04/09

BLOCH EQUATIONS

Two relaxation times T2 << T1

11

''Tm

mBdtdm z

yz ++++−−−−====−−−− γγγγ

(((( ))))2

1 )( Tm

mm'-MBdtdm y

xLzsy −−−−−−−−−−−−==== ωωωωωωωωγγγγ

(((( ))))2T

mmdtdm x

yLx −−−−−−−−==== ωωωωωωωω

Solutions

2

212

122

22

20

)(1)("

TTBT

TML

s γγγγωωωωωωωωγγγγµµµµωωωωχχχχ

++++−−−−++++====++++

212

122

22

22

0)(1

)()('TTBT

TML

Ls γγγγωωωωωωωω

ωωωωωωωωγγγγµµµµωωωωχχχχ++++−−−−++++

−−−−====++++

Width due to T2

Saturation coefficient

2/1211 )( −−−−<<<<<<<< TTBγγγγ

Linear response

H. Alloul, EPFL,30/04/09

T2

Nuclear spin-spin interactions

Liquid state

−−−−−−−−====ΗΗΗΗ 221

21321

2).)(.(3.

rrIrIII

rdd

vrvrrrh γγγγγγγγ

T2 transverse relaxation time

The transverse relaxation conserves the Zeeman energy ( Mz = constant )

∑∑∑∑−−−−====i

izZ B.IH 0γγγγh

Liquid state

High resolution NMR

I1 +I2

- terms

Do not modify the total spin for γγγγ2 = γγγγ1homonuclear interactions are dominant in T2

Hdd governs the NMR linewidth

Hdd is averaged out by the molecular motion

Solid state

H. Alloul, EPFL,30/04/09

RMN: LOCAL MAGNETIC MEASUREMENTS

• allow to probe the electronic environment of the nucleus• Knight shift in metals• Some examples: cuprates and C60 compounds

• Notion of relaxation• rf pulse techniques and Fourier transforms

INTERACTIONS BETWEEN NUCLEAR AND ELECTRONIC SPINS: HYPERFINE COUPLINGS

RELAXATION TIME

• rf pulse techniques and Fourier transforms• Local field fluctuations and dynamic electronic

susceptibility. • Korringa relaxation in metals.• Spins echos and transverse relaxation T2 (spin spin)

NMR APPLICATIONS

Solid state and soft matterChemistry, BiologyMedical imaging (IRM) Industrial chemistry, food control

H. Alloul, EPFL,30/04/09