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