1
Relaxation Times
Relaxation = return to equilibrium (Boltzmann) after a pulse, redistribution of
energy
Relaxation can be described for isolated spins by the Bloch Equations,
the total relaxation is determined by two characteristic time constants:
T1
Longitudinal, Spin-Lattice Relaxation
Build up of longitudinal magnetisation via energy
exchange between spins and their environment („lattice“)
enthalpy
T2
Transversal, Spin-Spin Relaxation
Dephasing of transversal magnetisation without energy
exchange between spins and their environment, entropy
Two important relations between T1 and T2 :
 T2 cannot be longer than T1 : T2 ≤ T1
 In the „extreme narrowing limit“: T2 = T1
2
T1 and T2 in Data Acquisition
exp(-t/T1)
exp(-t/T2)
relaxation delay
repetition time
T1 governs the repetition frequency for subsequent transients (scans)
Relaxation delay = 5 T1
T2 governs the decay time constant of individual FID’s
Optimum sensitivity of the NMR experiment is obtained if T1 = T2
T1 and T2 in Data Acquisition
3
4
Magnetization
More nuclei point in parallel to the
static magnetic field.
The macroscopic magnetic
moment, M0
M0 = Σ μi
In-Field
5
Longitudinal Magnetization
6
Spin-Lattice Relaxation Time
R1 = 1/T1 [Hz] longitudinal relaxation rate constant
T1 [s] longitudinal relaxation time
spin-lattice relaxation time
enthalpy
7
Transverse Magnetisation
8
Spin-Spin Relaxation Time
R2 = 1/T2 [Hz] transverse relaxation rate constant
T2 [s] transverse relaxation time constant
spin-spin relaxation time
entropy
9
Relaxation = Return to Equilibrium
10
Relaxation
Relaxation in other types of spectroscopy:
• spontaneous emission (not in NMR) fluorescence, phosphorescence
• collisional deactivation
(not in NMR, molecular tumbling does not change orientation of I, always
along B0)
• stimulated emission
lasers
magnetic interactions of nuclear spin with external fluctuating mg. field
(dipolar) containing many different frequencies, when it contains L,
resonance causes relaxation = emission of excess energy, transition from
exited to ground state
Spectral Density Function
11
Nuclear spin relaxation
• not a spontaneous process
• requires stimulation by suitable fluctuating fields to induce the spin
transitions
Longitudinal relaxation requires a time-dependent magnetic field fluctuating
at the Larmor frequency
Time-dependence = motions of the molecule (vibration, rotation, diffusion...)
Molecules “tumble” in solution - characterized by
a rotational correlation time - C
The probability function of finding motions at a given angular frequency ω can
be described by the spectral density function - J(ω)
Spectral Density Function
12
Frequency distribution of the fluctuating magnetic fields
13
Correlation Time C
Correlation Time C describes molecular tumbling
1. Look at one molecule
C = average time during which a molecule stays in one
orientation, until a collision changes its orientation
small molecules, low viscosity short 1012 s
polymers, high viscosity long 108 s
14
Correlation Time C
2. Look at a group of molecules ( 1 mole)
All molecules oriented in the same way,
then C is time in which the orientation is dispersed to 1 rad
(~60º)
t < C molecules are close to the original orientation
t >> C random distribution
1/C = tumbling rate
15
Correlation Time C
Time
16
Correlation Time C
Correlation function describes molecular tumbling
How the actual orienation correlates with the original one
Correlation function
0
0,2
0,4
0,6
0,8
1
1,2
0 200 400 600 800 1000 1200
time, ps
k(t)
)exp()0()(
C
t
ktk


C
1/e
17
Correlation Time C
C <> 1/0 poor energy transfer, T1 long, narrow lines
C = 1/0 effective energy transfer, T1 short, fast
relaxation, wide lines
= viscosity, high  = slow tumbling, long C, wide lines
a = molecular diameter, large particles = long C, wide lines
T = temperature, high T = fast tumbling = short C, narrow lines
Tk
a
Tk
V
D BB
C
3
4
6
1 3

 
18
Correlation Time C
Approximate rule
C [ps] ~ Mr in H2O at room temp.
Supercritical CO2 is a good NMR solvent
@65 ºC and 65 bar has low viscosity, narrow lines
19
Correlation Time C
1/0
short C = fast
tumbling,
small
molecules,
low viscosity
long C = slow
tumbling
rigid molecules,
high viscosity
extreme
narrowing
T1 = T2 = long
sharp lines
Relaxation Times vs Correlation Time
20
long C =
slow tumbling
rigid molecules
high viscosity
extreme narrowing
T1 = T2 = long
sharp lines
short C =
fast tumbling
small molecules
low viscosity
21
The Influence of Correlation Times
on Relaxation
 Correlation times are not molecular constants, but depend
on a number of factors, e.g. temperature, effective
molecular size, solvent viscosity…
 Variation of these factors may induce changes in c of
several ordes of magnitude.
 These changes may lead to violation of the „extreme
narrowing“ conditions, and introduce the necessity for a
more concise treatment of the correlation time dependence
of relaxation times.
22
Linewidth
T1 = lifetime of a nucleus in a certain energy state
Heissenberg uncertainity principle
E t ≥ h/2 h = 6.626 1034 J s
h ½  ≥ h/2
½ ≥ 1/T1 ½ ≥ 
High relaxation rate = short relaxation times
= wide lines in spectra
Werner Heisenberg
(1901-1976)
NP in physics 1932
23
Linewidth
Relaxation rate 
i iT
R
1 Linewidth
21
2/1
11
TT

long C = slow tumbling = long T1 , short T2
Linewidth is given by T2
2
2/1
1
T
 







21
2/1
11
TT
short C = fast tumbling = long T1 ≥ T2
Relaxation
24
Solution NMR
usually T2 = T1 (small/medium molecules, fast tumbling rate, non-viscous liquids)
Some process like scalar coupling with quadrupolar nuclei, chemical exchange,
interaction with a paramagnetic center, can accelerate the T2 relaxation such that
T2 becomes shorter than T1.
Solid state NMR
T1 is usually much larger than T2. The very fast spin-spin relaxation time provide
very broad signals
Measuring the experimental line width to determine the T2 relaxation time,
the experimental line width depends also on the inhomogeneity from the magnetic
field:
1/T2
* = 1/T2 + 1/T2(inhomogeneity)
Inhomogeneity is more critical for nuclei with higher frequency  =  B0
25
Relaxation Mechanisms
 Direct dipolar interaction of a nuclear spin with other nuclear spins
 Molecular motion in the presence of large chemical shielding
anisotropies
 Interaction of a nuclear spin with a nuclear quadrupole
 Scalar coupling of nuclear spins
 Paramagnetic relaxation by unpaired electrons
 Spin rotation
1 1, 1, 1,
1 1 1 1
.....
DD CSA QT T T T
   
Fluctuating magnetic fields (of the right amplitude and frequency) make
spins exchange energy with their environment
Important mechanisms to generate these fluctuating magnetic fields are:
The individual contributions combine to make the total relaxation
Relaxation Mechanisms
26
Interaction
Range of
interaction
(Hz)
Relevant parameters
Dipolar coupling 104 - 105
abundance of magnetically active nuclei
size of their magnetogyric ratio
Quadrupolar coupling 106 - 109
size of quadrupolar coupling constant
electric field gradient at the nucleus
Paramagnetic 107 -108
concentration of paramagnetic
impurities
Scalar coupling 10 - 103 size of the scalar coupling constants J
Chemical Shift
Anisotropy (CSA)
10 - 104
size of the chemical shift anisotropy CSA
symmetry at the nuclear site
Spin rotation
27
Dipolar Relaxation T1,DD
INTRAMOLECULAR
The magnetic moment of a nuclear spin B influences
the local field at the position of a neighbouring nucleus A:
B loc(A) = Bloc,0(A) + D
D denotes the dipolar coupling constant which is
defined as
Brownian motion of the sample containing nuclei A and B induces a
fluctuation of  which leads in turn to a time dependent modulation of the
local magnetic field Bloc(A).

A
B
rAB
B0
The Direct Interaction of a Nuclear Spin with other Spins
)cos31(
8
2
62
0





AB
BA
r
D

28
Dipolar Relaxation T1,DD
The contribution of this modulation to the T1 relaxation of nucleus A can be
expressed in terms of a characteristic time constant T1,DD:
c = molecular correlation time
0 = vacuum permeability
S = spin of nucles B
B = magnetogyric ratio, large value = faster relaxation of A, shorter T1,DD
nuclei with large (e.g. H) relax neighbouring nuclei
1/r6
AB = only directly bound nuclei contribute = intramolecular
 
   C
AB
BA
DD
SS
rAT



1
12
1
62
2222
0
,1


Substitution H/D: If H is replaced by D, in the X-D bond, the X-nuclei
relax much slower than in the corresponding X-H due to the lack of dipoledipole
relaxation, H is 6.5 time larger than D.
(in the extreme narrowing limit)
29
13C - 1H Dipolar Relaxation T1,DD
(in the extreme narrowing limit)
T1 relaxation of 13C by directly attached protons :
c = molecular correlation time
nH = number of attached protons
H = magnetogyric ratio of H
1/r6
CH = 109 pm
  C
CH
HCH
DD r
n
CT



62
2222
0
13
,1 16
1 

Protonated carbons relax faster in 13C NMR than quarternary carbons
30
Dipolar Relaxation T1,DD
INTERMOLECULAR
Tk
N
Da
hN
T BerDD
0
24242
0
)(int,1
3
2
1  

N0 = number of molecules in m3
D = difussion coefficient
T = temp, high T narrows lines
Protons relax both inter and intramolecularly
C6H6 neat T1(H) = 19 s
C6H6 diluted in CS2 T1(H) = 90 s
a
Tk
D B
6

31
I: The Influence of the observed nucleus A in a A-H fragment:
A 31P 13C 29Si 15N 103Rh
(X) 10.84 6.73 -5.32 -2.71 -0.85
rAH [Å] 1.4 1.1 1.4 1.0 1.6
T1,DD (c=10-11) 8 s 5 s 33 s 17 s 48 min
T1,DD (c=10-9) 80 ms 50 ms 330 ms 170 ms 29 s
II: The Influence of the neighboring nucleus X in an A-X fragment (A=15N):
X 1H 31P 13C 11B 51V
(X) 26.75 10.84 6.73 8.59 7.05
rAX [Å] 1.0 1.7 1.4 1.3 1.8
S(S+1) 0.75 0.75 0.75 3.75 3.75
T1,DD (c=10-11) 17 s 42 min 34 min 160 s 400 s
T1,DD (c=10-9) 170 ms 25 s 20 s 1.6 s 4 s
III: The Influence of the internuclear distance in a N…H fragment:
rAX [Å] 1.0 2.1 2.7
T1,DD (c=10-11)
(N-H)
8 s
(N-C-H)
24 min
(N-C-C-H)
110 min
c = 10-11s: medium sized (in)organic molecule
c = 10-9 s: small polymer
32
Quadrupole Induced Relaxation T1,Q
Nuclei with I > ½ possess an electric quadrupole
moment eQ which is quantized according to its
oriention in the electric field gradient (efg) of the
electrons if the local symmetry is less than spherical.
nucleus
e-clond
The Interaction of a nuclear spin I with a quadrupole moment
Nuclei I > ½
Due to strong coupling between eQ and I,
the nuclear magnetic spin levels depend
on both B0 and the efg.
Electric quadrupole moment eQ = nonspherical distribution of the
positive nuclear charge
Q > 0Q < 0 Q = 0
33
Quadrupole Induced Relaxation T1,Q
mi=1
mi=0
mi=-1
B0
I = 1

BROWNIAN MOTION of sample molecules
modulates the different mI energies which leads
to a stochastic modulation of the local magnetic
field Bloc(A).
Tumbling = spread of energy levels
in solution the average transition energy does not
change but the spread contributes to relaxation
34
c = correlation time
I = nuclear spin
Q = nuclear quadrupole moment (Q ≠ 0 for I > ½ )
qZZ = electric field gradient
qZZ = 0 for high symmetry (spherical, Cl-, cubic Td, Oh, ClO4
-, SO4
2-, AsF6
η
= asymmetry parameter (η = 0 for axial symmetry)
The contribution to T1(A) can be expressed in terms of a characteristic
time constant T1,Q (extreme narrowing limit)
Quadrupole Induced Relaxation T1,Q
C
zz
QQ h
Qqe
II
I
TT

 2
22
2
2
,2,1
))(
3
1(
)12(10
)32(311




zz
xxyy
q
qq 

Quadrupole Induced Relaxation T1,Q
35
Q - the electric quadrupole moment of the nucleus
A large moment results in efficient relaxation of the nucleus by molecular
motion, very broad lines.
125I- in water (I = 5/2, Q = 0.79) has ν½ = 1800 Hz
All other iodo compounds are much less symmetric and have lines so broad
their NMR signals cannot be detected.
Deuterium (I = 1, Q = 0.00273) and 6Li (I = 1, Q = 0.0008) have among the
smallest electric quadrupole moments of all isotopes and are easy to
observe and usually give lines sharp enough to resolve J coupling.
55Mn (I = 5/2, Q = 0.55)
Quadrupole Induced Relaxation T1,Q
36
qZZ - the electric field gradient (EFG)
The Quadrupole coupling vanishes in a symmetrical environment
symmetrical [NH4]+ : qzz = 0 and therefore has very long T1 = 50 s
CH3CN : qzz = 4 MHz and T1 = 22 ms
(I = 1, Q = 0.016)
37
Quadrupole Induced Relaxation T1,Q
2
/zze q Q  Nuclear Quadrupole Coupling Constant, NQCC
)12(
)32(
2
2



II
IQ
l
Linewidth factor
I 1 3/2 5/2 3 7/2 4
l [Q2] 5 1.33 0.32 0.20 0.16 0.10
The larger the I nuclear spin, the faster relaxation, the shorter T1,Q, the broader lines
38
I. The Influence of the electric field gradient qzz:
14N relaxation times:
Bu4N+ (Td) NaNO3 (D3h) NNN-(Ch) MeSCN(Cv) DABCO(Cv)
c[MHz] 0.04 0.745 1.03 3.75 4.93
T1,Q 1.8 s 85 ms 29 ms 2 ms 0.6 ms
55Mn relaxation times:
Mn2CO10 BrMn(CO)5 HMn(CO)5 CpMn(CO)3
c[MHz] 3.05 17.46 45.7 64.3
T1,Q 3.8 ms 0.46 ms 74 s 32 s
II. The Influence of Q and I: T1,Q in [M(CO)6]
M = 95Mo 97Mo 187Re(+) 185Re(+) 181Ta(-)
Q[10-28 m2] 0.12 1.1 2.6 2.8 3
I 5/2 5/2 5/2 5/2 7/2
Q(2I+3)/(2I-1) 0.30 2.75 6.50 7.00 10.5
T1,Q >450 ms 53 ms 141 s 122 s 48 s
W1/2 [Hz] <0.7 6 2250 2600 6700
39
CSA Induced Relaxation, T1,CSA
Magnetic shielding is anisotropic and may vary for different
orientations of the magnetic field B0 with respect to the
molecular frame.
H2O
Pt
H2O
OH2
OH2
2+
  -500
||  +14500
BROWNIAN MOTION of sample molecules induces time dependent
modulation of  and thus a stochastic fluctuation of the effective local
magnetic field B0,loc(A).
Tumbling of molecules with large chemical shielding anisotropies
Important for nuclei with wide range of chemical shifts: 31P, 195Pt, 113Cd




//
2
1
)( yyxxzz
Chemical Shielding Anisotropy CSA
40
CSA Induced Relaxation, T1,CSA
The contribution to T1(A) can be expressed in terms of a characteristic
time constant:
c = molecular correlation time
 = shielding anisotropy
B0 = magnetic field strength = wide lines in strong magnets !!!!
CA
CSA
B
AT
 222
0
,1
)(
)(
1

(in the extreme narrowing limit)
41
I: The Influence of the observed nucleus A in ( = 100 ppm; B0 = 7 T):
A 31P 13C 15N
(X) 10.84 6.73 -2.71
T1,CSA (tc=10-11) 130 s 340 s 35 min
T1,CSA (tc=10-9) 1.3 s 3.4 s 21 s
II: The Influence of the magnetic field B0 (nucleus 195Pt;  = 1000 ppm):
B0 [T] 4.7 7.1 11.7 17.6
(1H) [MHz] 200 300 500 750
T1,CSA (tc=10-11) 10 s 4 s 1.6 s 0.7 s
T1,CSA (tc=10-9) 100 ms 40 ms 16 ms 7 ms
III: The Influence of the shielding anisotropy (nucleus 195Pt; B0 = 7 T):
 [ppm] 15 150 1500 15000
T1,CSA (tc=10-11) 5.5 h 3.3 min 2 s 20 ms
T1,CSA (tc=10-9) 3.3 min 2 s 20 ms 0.2 ms
c = 10-11s: medium sized (in)organic molecule; c = 10-9 s: small polymer;
42
Spin Rotation Induced Relaxation, T1,SR
Tumbling molecule = bonding electrons move and
induce magnetic field around the molecule. Important
for small fast rotating molecules with high symmetry:
SF6, PCl3, PtL4
j
B
SR
TCVk
T
2
2
,1 3
21


V = moment of inertia
C = SR constant
j = time in which a molecule changes its angular momentum, e.g.
time between collisions
Hubbard (if j << C, valid for small molecules below b. p.)
Tk
V
B
Cj
6

43
Contributions of CSA versus SR
[Pt(PtBu3)2] [Pt(PEt3)3] [Pt{(P(OEt)3}4]
symm linear trigonal tetrahedral
T1 [s]
@ 9.4 T
0.03 2.4 5.6
CSA % 100 50 10
SR % 0 50 90
SR important at high T, high symmetry
CSA important at high B0, low symmetry
44
Scalar Coupling Induced Relaxation,
T1,SC
Two nuclei coupled through JAB and one of them relaxes fast =
the fast spin orientation change of B is transferred to A
•exchange of B nucleus (e.g. H exchange)
 = lifetime of the exchange process
•quadrupolar nucleus B
 = T2q quadrupolar relaxation time
22
22
,1 )(1
)1(
3
81


SISC
SS
J
T 

S = spin of B
Spin A
45
Paramagnetic Relaxation, T1,e
Dipolar relaxation by electron magnetic moment
Transfer of unpaired electron density onto a nucleus
O2 in the solvent
TM ions
Tk
N
T B
effp
e
222
,1
41 

Np = concentration of paramagnetic species in m3
eff = magnetic moment of e, thousand times larger than
magnetic moment of nuclei, even small conc. of paramagnetic
species shortens considerably relaxation time, wide lines
= viscosity
Relaxation agent Cr(acac)3 can be added to the solution of a
slow relaxing compound (13C, 29Si,..) to shorten the acq.
delay
46
Paramagnetic Relaxation, T1,e
e
NS
C
SI
e
SS
a
SS
rT






)1(
24
)(
)1(
12
)(1
2
2
0
62
2
0
,1


c = molecular correlation time
e = electron correlation time
aN = electron-nucleus spin coupling constant
dipole-dipole term contact term
Relaxation Mechanisms
47
Approaches to distinguish the various relaxation mechanisms:
1. by the strength of the interaction :
Paramg > Q > DD > CSA > J
2. by the use of isotopic substitution to identify the DD
3. by the field dependence:
CSA is proportional to B0 (applied field).
Quadrupole interaction is inversely proportional to B0
4. by their temperature dependence
48
Away from Extreme Narrowing
Conditions
Theoretical analysis of relaxation processes under conditions which fail
to fulfil the requirements of extreme narrowing revealed that dependence
of T1 on c follows frequently a relation
2 2
1
1
1
c
cT

 

  = Zeeman/2
This relation allows a more detailed analysis of temperature effects on
relaxation.
49
The Temerature Dependence
of T1 Relaxation
T1 is independent of B0 in the extreme narrowing regime (22 1)
T1 goes through a minimum (optimum relaxation conditions, 22 1, very
efficient relaxation)
T1 depends on B0 if 22 1
fast motion
short C
long C
large molecules
50
Measurement of T1
The Inversion Recovery Experiment
180o
Inversion
Pulse
variable
Delay 
(incremrnted)
90o
Read
Pulse
recovery of z-magnetisation
Non-equilibrium z-magnetisation recorvers during delay 
Mz() is converted into observable magnetisation by the read pulse
Performing a series of experiments and incrementing  allows to sample
Mz() at different times
T1 is obtained from a fit of observed signal intensities as a function of 
1/0
: ( ) (1 2 )T
z zM M e 
  
 
51

0.00 1.26ms
Shift: 717.502 ppm
T1 : 0.5755 ms
Measurement of T1(51V) for a Vanadium
Complex
52
Relaxation Time T2
T2 relaxation occurs without energy transfer „entropic process“.
The characteristic time constant T2 is connected with the linewidth:
1/ 2*
2 2, 2
1 1 1
true
w
T T T
   
*
2T
describes the effect of magnetic field inhomogeneities,
i.e. mostly bad shimming
53
Relaxation Time T2
For most I = n/2-nuclei,
1/T2,true >> 1/T2
*
T2 may be determined directly from measured linewidth:
w1/2= 1/T2 1/T2,true
For I = 1/2-nuclei,
1/T2,true ≤ 1/T2
*
T2 must be measured by dedicated experiments (spin echo or CPMG)
54
Relaxation Time T2
Mg. field inhomogeneity refocused at the end of the 2nd delay.
Echo after the 2t delays - the size of this echo will only be affected by
the spin-spin relaxation processes
55
23Na Relaxation Time
Methyl methacrylate (MMA) with NaClO4 in propylene carbonate (PC)
Polymerization initialized with UV radiation and the BBE initiator
23Na nucleus (I = 3/2) has a large electric quadrupole moment, which
causes its extreme sensitivity to the nearest neighbor coordination
T1 - the inversion recovery
T2 - the spin-echo technique
R. Korinek, J. Vondrak, K. Bartusek, M. Sedlarikova J Solid State Electrochem (2013) 17:2109