8–1
8 Relaxation†
Relaxation is the process by which the spins in the sample come to equilibrium
with the surroundings. At a practical level, the rate of relaxation determines
how fast an experiment can be repeated, so it is important to understand how
relaxation rates can be measured and the factors that influence their values.
The rate of relaxation is influenced by the physical properties of the molecule
and the sample, so a study of relaxation phenomena can lead to information on
these properties. Perhaps the most often used and important of these
phenomena in the nuclear Overhauser effect (NOE) which can be used to probe
internuclear distances in a molecule. Another example is the use of data on
relaxation rates to probe the internal motions of macromolecules.
In this chapter the language and concepts used to describe relaxation will be
introduced and illustrated. To begin with it will simply be taken for granted
that there are processes which give rise to relaxation and we will not concern
ourselves with the source of relaxation. Having described the experiments
which can be used to probe relaxation we will then go on to see what the source
of relaxation is and how it depends on molecular parameters and molecular
motion.
8.1 What is relaxation?
Relaxation is the process by which the spins return to equilibrium. Equilibrium
is the state in which (a) the populations of the energy levels are those predicted
by the Boltzmann distribution and (b) there is no transverse magnetization and,
more generally, no coherences present in the system.
In Chapter 3 we saw that when an NMR sample is placed in a static
magnetic field and allowed to come to equilibrium it is found that a net
magnetization of the sample along the direction of the applied field
(traditionally the z-axis) is developed. Magnetization parallel to the applied
field is termed longitudinal.
This equilibrium magnetization arises from the unequal population of the
two energy levels that correspond to the α and β spin states. In fact, the zmagnetization,
Mz, is proportional to the population difference
M n nz ∝ −( )α β
where nα and nβ are the populations of the two corresponding energy levels.
Ultimately, the constant of proportion just determines the absolute size of the
signal we will observe. As we are generally interested in the relative size of
magnetizations and signals we may just as well write
M n nz = −( )α β [2]
†
Chapter 8 "Relaxation" © James Keeler 1999 and 2002
α
β
x y
z
A population difference gives
rise to net magnetization along
the z-axis.
8–2
8.2 Rate equations and rate constants
The populations of energy levels are in many ways analogous to concentrations
in chemical kinetics, and many of the same techniques that are used to describe
the rates of chemical reactions can also be used to describe the dynamics of
populations. This will lead to a description of the dynamics of the z-
magnetization.
Suppose that the populations of the α and β states at time t are nα and nβ,
respectively. If these are not the equilibrium values, then for the system to
reach equilibrium the population of one level must increase and that of the other
must decrease. This implies that there must be transitions between the two
levels i.e. something must happen which causes a spin to move from the α state
to the β state or vice versa. It is this process which results in relaxation.
The simplest assumption that we can make about the rate of transitions from
α to β is that it is proportional to the population of state α, nα, and is a first
order process with rate constant W. With these assumptions the rate of loss of
population from state α is Wnα. In the same way, the rate of loss of population
of state β is Wnβ.
However, the key thing to realize is that whereas a transition from α to β
causes a loss of population of level α, a transition from β to α causes the
population of state α to increase. So we can write
rate of change of population of state α = – Wnα + Wnβ
The first term is negative as it represents a loss of population of state α and in
contrast the second term is negative as it represents a gain in the population of
state α. The rate of change of the population can be written using the language
of calculus as dnα/dt so we have
d
d
n
t
Wn Wnα
α β= − + .
Similarly we can write for the population of state β:
d
d
n
t
Wn Wnβ
β α= − + .
These equations are almost correct, but we need to make one modification.
At equilibrium the populations are not changing so dnα/dt = 0; this immediately
implies that at equilibrium nα = nβ, which simply is not correct. We know that
at equilibrium the population of state α exceeds that of state β. This defect is
easily remedied by replacing the population nα with the deviation of the
population from its equilibrium value n nα α−( )0
, where nα
0
is the population of
state α at equilibrium. Doing the same with state β gives us the final, correct,
equations:
d
d
d
d
n
t
W n n W n n
n
t
W n n W n nα
β β α α
β
α α β β= −( )− −( ) = −( )− −( )0 0 0 0
.
α
β
W
α
β
W
A transition from state α to
state β decreases the
population of state α , but a
transition from state β to state α
increases the population of
state α.
8–3
You will recognise here that this kind of approach is exactly the same as that
used to analyse the kinetics of a reversible chemical reaction.
Using Eqn. [2], we can use these two equations to work out how the zmagnetization
varies with time:
d
d
d
d
d
d
d
d
M
t
n n
t
n
t
n
t
W n n W n n W n n W n n
W n n W n n
W M M
z
z z
=
−( )
= −
= −( )− −( )− −( )+ −( )
= − −( )+ −( )
= − −( )
α β
α β
β β α α α α β β
α β α β
0 0 0 0
0 0
0
2 2
2
where M n nz
0 0 0
= −( )α β , the equilibrium z-magnetization.
8.2.1 Consequences of the rate equation
The discussion in the previous section led to a (differential) equation describing
the motion of the z-magnetization
d
d
M t
t
R M t Mz
z z z
( )
= − ( ) −( )0
[7]
where the rate constant, Rz = 2W and Mz has been written as a function of time,
Mz(t), to remind us that it may change.
What this equations says is that the rate of change of Mz is proportional to
the deviation of Mz from its equilibrium value, Mz
0
. If Mz = Mz
0
, that is the
system is at equilibrium, the right-hand side of Eqn. [7] is zero and hence so is
the rate of change of Mz: nothing happens. On the other hand, if Mz deviates
from Mz
0
there will be a rate of change of Mz, and this rate will be proportional
to the deviation of Mz from Mz
0
. The change will also be such as to return Mz to
its equilibrium value, Mz
0
. In summary, Eqn. [7] predicts that over time Mz will
return to Mz
0
; this is exactly what we expect. The rate at which this happens
will depend on Rz.
This equation can easily be integrated:
d
d
const.
M t
M t M
R t
M t M R t
z
z z
z
z z z
( )
( ) −( )
= −
( ) −( )= − +
∫ ∫0
0
ln
If, at time zero, the magnetization is Mz(0), the constant of integration can be
determined as ln M Mz z0 0
( ) −( ). Hence, with some rearrangement:
8–4
ln
M t M
M M
R tz z
z z
z
( ) −
( ) −





 = −
0
0
0
[7A]
or
M t M M R t Mz z z z z( ) = ( ) −[ ] −( )+0 0 0
exp [7B]
In words, this says that the z-magnetization returns from Mz(0) to Mz
0
following
an exponential law. The time constant of the exponential is 1/Rz, and this is
often called T1, the longitudinal or spin-lattice relaxation time.
8.2.2 The inversion recovery experiment
We described this experiment in section 3.10. First, a 180° pulse is applied,
thereby inverting the magnetization. Then a delay t is left for the magnetization
to relax. Finally, a 90° pulse is applied so that the size of the z-magnetization
can be measured.
We can now analyse this experiment fully. The starting condition for Mz is
M Mz z0 0
( ) = − , i.e. inversion. With this condition the predicted time evolution
can be found from Eqn. [7A] to be:
ln
M t M
M
R tz z
z
z
( ) −
−





 = −
0
0
2
Recall that the amplitude of the signal we record in this experiment is
proportional to the z-magnetization. So, if this signal is S(t) it follows that
ln
S t S
S
R tz
( ) −
−





 = −
0
0
2
[7C]
where S0
is the signal intensity from equilibrium magnetization; we would find
this from a simple 90° – acquire experiment.
Equation [7C] implies that a plot of ln S t S S( ) −( ) −[ ]0 0
2 against t should be
a straight line of slope –Rz. This, then, is the basis of a method of determining
the relaxation rate constant.
8.2.3 A quick estimate for Rz (or T1)
Often we want to obtain a quick estimate for the relaxation rate constant (or,
equivalently, the relaxation time). One way to do this is to do an inversion
recovery experiment but rather than varying t systematically we look for the
value of t which results in no signal i.e. a null. If the time when S(t) is zero is
tnull it follows immediately from Eqn. [7C] that:
ln
ln
ln
1
2
2
2
1



 = − = =R t R
t
T
t
z znull
null
null
or or
Probably the most useful relationship is the last, which is T1 ≈ 1.4 tnull.
This method is rather crude, but it good enough for estimating T1. Armed
with this estimate we can then, for example, decide on the time to leave
between transients (typically three to five times T1).
t
Mz
Mz(0)
Mz
0
t
S(t)–S0
–2S0
ln[ ]
Plot used to extract a value of
Rz from the data from an
inversion recovery experiment.
8–5
8.2.4 Writing relaxation in terms of operators
As we saw in Chapter 6, in quantum mechanics z-magnetization is represented
by the operator Iz. It is therefore common to write Eqn. [7] in terms of
operators rather then magnetizations, to give:
d
d
I t
t
R I t Iz
z z z
( )
= − ( ) −( )0
[8]
where Iz(t) represents the z-magnetization at time t and Iz
0
represents the
equilibrium z-magnetization. As it stands this last equation seems to imply that
the operators change with time, which is not what is meant. What are changing
are the populations of the energy levels and these in turn lead to changes in the
z-magnetization represented by the operator. We will use this notation from
now on.
8.3 Solomon equations
The idea of writing differential equations for the populations, and then
transcribing these into magnetizations, is a particularly convenient way of
describing relaxation, especially in more complex system. This will be
illustrated in this section.
Consider a sample consisting of molecules which contain two spins, I and S;
the spins are not coupled. As was seen in section 2.4, the two spins have
between them four energy levels, which can be labelled according to the spin
states of the two spins.
|αα>
|ββ>
|αβ> |βα>
1
32
4
WI
(1)
WI
(2)
WS
(1)
WS
(2)
W2
W0
|αα>
|ββ>
|αβ> |βα>
1
32
4
(a) (b)
Diagram (a) shows the energy levels of a two spin system; the levels are labelled with the spin of I first and
the spin of S second. The dashed arrows indicate allowed transitions of the I spin, and the solid arrows
indicate allowed transitions of the S spin. Diagram (b) shows the relaxation induced transitions which are
possible amongst the same set of levels.
It turns out that in such a system it is possible to have relaxation induced
transitions between all possible pairs of energy levels, even those transitions
which are forbidden in normal spectroscopy; why this is so will be seen in
detail below. The rate constants for the two allowed I spin transitions will be
denoted WI
1( )
and WI
2( )
, and likewise for the spin S transitions. The rate constant
for the transition between the αα and ββ states is denoted W2, the "2" indicating
that it is a double quantum transition. Finally, the rate constant for the
transition between the αβ and βα states is denoted W0, the "0" indicating that it
is a zero quantum transition.
8–6
Just in the same was as was done in Section 8.2, rate equations can be
written for the flow of population from any of the levels. For example, for level
1
d
d
n
t
W n W n W n W n W n W nS I S I
1 1
1
1
1 2 1
1
2
1
3 2 4= − − − + + +( ) ( ) ( ) ( )
The negative terms are rates which lead to a loss of population of level 1 and
the positive terms are ones that lead to a gain in its population. As was
discussed in section 8.2 the populations ought to be written as deviations from
their equilibrium values, n ni i−( )0
. However, to do this results in unnecessary
complexity; rather, the calculation will be carried forward as written and then at
the last stage the populations will be replaced by their deviations from
equilibrium.
The corresponding equations for the other populations are
d
d
d
d
d
d
n
t
W n W n W n W n W n W n
n
t
W n W n W n W n W n W n
n
t
W n W n
S I S I
I S I S
S I
2 1
2
2
2 0 2
1
1
2
4 0 3
3 1
3
2
3 0 3
1
1
2
4 0 2
4 2
4
2
= − − − + + +
= − − − + + +
= − −
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
44 2 4
2
3
2
2 2 1− + + +( ) ( )
W n W n W n W nS I
All of this can be expressed in a more compact way if we introduce the I and S
spin z-magnetizations. The I spin magnetization is equal to the population
difference across the two I spin transitions, 1–3 and 2–4
I n n n nz = − + −1 3 2 4 [9]
As discussed above, the magnetization has been represented as the
corresponding operator, Iz. Likewise for the S-spin magnetization
S n n n nz = − + −1 2 3 4 [10]
A third combination of populations will be needed, which is represented by the
operator 2IzSz
2 1 3 2 4I S n n n nz z = − − + [11]
Comparing this with Eq. [9] reveals that 2IzSz represents the difference in
population differences across the two I-spin transitions; likewise, comparison
with Eq. [10] shows that the same operator also represents the difference in
population differences across the two S-spin transitions.
Taking the derivative of Eq. [9] and then substituting for the derivatives of
the populations gives
8–7
d
d
d
d
d
d
d
d
d
d
zI
t
n
t
n
t
n
t
n
t
W n W n W n W n W n W n
W n W n W n W n W n W n
W
S I S I
I S I S
S
= − + −
= − − − + + +
+ + + − − −
−
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 3 2 4
1
1
1
1 2 1
1
2
1
3 2 4
1
3
2
3 0 3
1
1
2
4 0 2
11
2
2
2 0 2
1
1
2
4 0 3
2
4
2
4 2 4
2
3
2
2 2 1
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
− − + + +
+ + + − − −
n W n W n W n W n W n
W n W n W n W n W n W n
I S I
S I S I
[12]
This unpromising looking equation can be expressed in terms of Iz, Sz etc. by
first introducing one more operator E, which is essentially the identity or unit
operator
E n n n n= + + +1 2 3 4 [13]
and then realizing that the populations, ni, can be written in terms of E, Iz, Sz,
and 2Iz Sz:
n E I S I S
n E I S I S
n E I S I S
n E I S I S
z z z z
z z z z
z z z z
z z z z
1
1
4
2
1
4
3
1
4
4
1
4
2
2
2
2
= + + +( )
= + − −( )
= − + −( )
= − − +( )
where these relationships can easily be verified by substituting back in the
definitions of the operators in terms of populations, Eqs. [9] – [13].
After some tedious algebra, the following differential equation is found for Iz
d
d
z
z
I
t
W W W W I
W W S W W I S
I I
z I I z z
= − + + +( )
− −( ) − −( )
( ) ( )
( ) ( )
1 2
2 0
2 0
1 2
2
[14]
Similar algebra gives the following differential equations for the other operators
d
d
d2
d
z
z
z
z
S
t
W W I W W W W S W W I S
I S
t
W W I W W S
W W W
z S S S S z z
z
I I z S S
I I S
= − −( ) − + + +( ) − −( )
= − −( ) − −( )
− + +
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
2 0
1 2
2 0
1 2
1 2 1 2
1 2 1
2
++( )( )
W I SS z z
2
2
As expected, the total population, represented by E, does not change with time.
These three differential equations are known as the Solomon equations.
It must be remembered that the populations used to derive these equations
are really the deviation of the populations from their equilibrium values. As a
result, the I and S spin magnetizations should properly be their deviations from
their equilibrium values, Iz
0
and Sz
0
; the equilibrium value of 2IzSz is easily
shown, from its definition, to be zero. For example, Eq. [14] becomes
8–8
d
d
z
z
z
I I
t
W W W W I I
W W S S W W I S
z
I I z
z I I z z
−( ) = − + + +( ) −( )
− −( ) −( )− −( )
( ) ( )
( ) ( )
0
1 2
2 0
0
2 0
0 1 2
2
8.3.1 Interpreting the Solomon equations
What the Solomon equations predict is, for example, that the rate of change of
Iz depends not only on I Izz − 0
, but also on S Szz − 0
and 2IzSz. In other words the
way in which the magnetization on the I spin varies with time depends on what
is happening to the S spin – the two magnetizations are connected. This
phenomena, by which the magnetizations of the two different spins are
connected, is called cross relaxation.
The rate at which S magnetization is transferred to I magnetization is given
by the term
W W S Sz2 0
0
−( ) −( )z
in Eq. [14]; (W2–W0) is called the cross-relaxation rate constant, and is
sometimes given the symbol σIS. It is clear that in the absence of the relaxation
pathways between the αα and ββ states (W2), or between the αβ and βα states
(W0), there will be no cross relaxation. This term is described as giving rise to
transfer from S to I as it says that the rate of change of the I spin magnetization
is proportional to the deviation of the S spin magnetization from its equilibrium
value. Thus, if the S spin is not at equilibrium the I spin magnetization is
perturbed.
In Eq. [14] the term
W W W W I II I z
1 2
2 0
0( ) ( )
+ + +( ) −( )z
describes the relaxation of I spin magnetization on its own; this is sometimes
called the self relaxation. Even if W2 and W0 are absent, self relaxation still
occurs. The self relaxation rate constant, given in the previous equation as a
sum of W values, is sometimes given the symbol RI or ρI.
Finally, the term
W W I SI I z z
1 2
2( ) ( )
−( )
in Eq. [14] describes the transfer of IzSz into I spin magnetization. Recall that
WI
1( )
and WI
2( )
are the relaxation induced rate constants for the two allowed
transitions of the I spin (1–3 and 2–4). Only if these two rate constants are
different will there be transfer from 2IzSz into I spin magnetization. This
situation arises when there is cross-correlation between different relaxation
mechanisms; a further discussion of this is beyond the scope of these lectures.
The rate constants for this transfer will be written
∆ ∆I I I S S SW W W W= −( ) = −( )( ) ( ) ( ) ( )1 2 1 2
According to the final Solomon equation, the operator 2IzSz shows self
relaxation with a rate constant
8–9
R W W W WIS I I S S= + + +( )( ) ( ) ( ) ( )1 2 1 2
Note that the W2 and W0 pathways do not contribute to this. This rate combined
constant will be denoted RIS.
Using these combined rate constants, the Solomon equations can be written
d
d
d
d
d2
d
z
z z
z
z z
z
z z
I I
t
R I I S S I S
S S
t
I I R S S I S
I S
t
I I S S R
z
I z IS z I z z
z
IS z S z S z z
z
I z S z IS
−( ) = − −( )− −( )−
−( ) = − −( )− −( )−
= − −( )− −( )−
0
0 0
0
0 0
0 0
2
2
σ
σ
∆
∆
∆ ∆ 22I Sz z
[15]
The pathways between the different magnetization are visualized in the diagram
opposite. Note that as d dI tz
0
0= (the equilibrium magnetization is a constant),
the derivatives on the left-hand side of these equations can equally well be
written d dI tz and d dS tz .
It is important to realize that in such a system Iz and Sz do not relax with a
simple exponentials. They only do this if the differential equation is of the
form
d
d
I
t
R I Iz
I z z= − −( )0
which is plainly not the case here. For such a two-spin system, therefore, it is
not proper to talk of a "T1" relaxation time constant.
8.4 Nuclear Overhauser effect
The Solomon equations are an excellent way of understanding and analysing
experiments used to measure the nuclear Overhauser effect. Before embarking
on this discussion it is important to realize that although the states represented
by operators such as Iz and Sz cannot be observed directly, they can be made
observable by the application of a radiofrequency pulse, ideally a 90° pulse
aI aIz
I
y
xπ 2( )
 → −
The subsequent recording of the free induction signal due to the evolution of
the operator Iy will give, after Fourier transformation, a spectrum with a peak of
size –a at frequency ΩI. In effect, by computing the value of the coefficient a,
the appearance of the subsequently observed spectrum is predicted.
The basis of the nuclear Overhauser effect can readily be seen from the
Solomon equation (for simplicity, it is assumed in this section that ∆I = ∆S = 0)
d
d
z
z z
I I
t
R I I S S
z
I z IS z
−( ) = − −( )− −( )
0
0 0
σ
What this says is that if the S spin magnetization deviates from equilibrium
there will be a change in the I spin magnetization at a rate proportional to (a)
the cross-relaxation rate, σIS and (b) the extent of the deviation of the S spin
σIS
∆I
∆S
2IzSz
Iz Sz
8–10
from equilibrium. This change in the I spin magnetization will manifest itself
as a change in the intensity in the corresponding spectrum, and it is this change
in intensity of the I spin when the S spin is perturbed which is termed the
nuclear Overhauser effect.
Plainly, there will be no such effect unless σIS is non-zero, which requires the
presence of the W2 and W0 relaxation pathways. It will be seen later on that
such pathways are only present when there is dipolar relaxation between the
two spins and that the resulting cross-relaxation rate constants have a strong
dependence on the distance between the two spins. The observation of a
nuclear Overhauser effect is therefore diagnostic of dipolar relaxation and
hence the proximity of pairs of spins. The effect is of enormous value,
therefore, in structure determination by NMR.
8.4.1 Transient experiments
A simple experiment which reveals the NOE is to invert just the S spin by
applying a selective 180° pulse to its resonance. The S spin is then not at
equilibrium so magnetization is transferred to the I spin by cross-relaxation.
After a suitable period, called the mixing time, τm, a non-selective 90° pulse is
applied and the spectrum recorded.
After the selective pulse the situation is
I I S Sz zz z0 00 0
( ) = ( ) = − [16]
where Iz has been written as Iz(t) to emphasize that it depends on time and
likewise for S. To work out what will happen during the mixing time the
differential equations
d
d
d
d
z
z z
z
z z
I t
t
R I t I S t S
S t
t
I t I R S t S
I z IS z
IS z S z
( )
= − ( ) −( )− ( ) −( )
( )
= − ( ) −( )− ( ) −( )
0 0
0 0
σ
σ
need to be solved (integrated) with this initial condition. One simple way to do
this is to use the initial rate approximation. This involves assuming that the
mixing time is sufficiently short that, on the right-hand side of the equations, it
can be assumed that the initial conditions set out in Eq. [16] apply, so, for the
first equation
d
d
z
init
I t
t
R I I S S
S
I z z IS z z
IS z
( )
= − −( )− − −( )
=
0 0 0 0
0
2
σ
σ
This is now easy to integrate as the right-hand side has no dependence on Iz(t)
mτ
90°180°
S
90°
(a)
(b)
Pulse sequence for recording
transient NOE enhancements.
Sequence (a) involves selective
inversion of the S spin – shown
here using a shaped pulse.
Sequence (b) is used to record
the reference spectrum in
which the intensities are
unperturbed.
8–11
d dz
z m z m
z m m
m m
I t S t
I I S
I S I
IS z
IS z
IS z z
( ) =
( )− ( ) =
( ) = +
∫ ∫0
0
0
0
0 0
2
0 2
2
τ τ
σ
τ σ τ
τ σ τ
This says that for zero mixing time the I magnetization is equal to its
equilibrium value, but that as the mixing time increases the I magnetization has
an additional contribution which is proportional to the mixing time and the
cross-relaxation rate, σIS. This latter term results in a change in the intensity of
the I spin signal, and this change is called an NOE enhancement.
The normal procedure for visualizing these enhancements is to record a
reference spectrum in which the intensities are unperturbed. In terms of zmagnetizations
this means that I Iz z,ref = 0
. The difference spectrum, defined as
(perturbed spectrum – unperturbed spectrum) corresponds to the difference
I I S I I
S
z IS z z z
IS z
z m ref m
m
τ σ τ
σ τ
( )− = + −
=
, 2
2
0 0 0
0
The NOE enhancement factor, η, is defined as
η =
intensity in enhanced spectrum - intensity in reference spectrum
intensity in reference spectrum
so in this case η is
η τ
τ σ τ
m
z m ref
ref
m
( ) =
( )−
=
I I
I
S
I
z
z
IS z
z
,
,
2 0
0
and if I and S are of the same nuclear species (e.g. both proton), their
equilibrium magnetizations are equal so that
η τ σ τm m( ) = 2 IS
Hence a plot of η against mixing time will give a straight line of slope σIS; this
is a method used for measuring the cross-relaxation rate constant. A single
experiment for one value of the mixing time will reveal the presence of NOE
enhancements.
This initial rate approximation is valid provided that
σ τ τIS SRm mand<< <<1 1
the first condition means that there is little transfer of magnetization from S to I,
and the second means that the S spin remains very close to complete inversion.
8.4.1.1 Advanced topic: longer mixing times
At longer mixing times the differential equations are a little more difficult to
solve, but they can be integrated using standard methods (symbolic
mathematical programmes such as Mathematica are particularly useful for this).
Using the initial conditions given in Eq. [16] and, assuming for simplicity that
I Sz z
0 0
= the following solutions are found
enhanced
reference
difference
S I
Visualization of how an NOE
difference spectrum is
recorded. The enhancement is
assumed to be positive.
8–12
I
I R
z
z
IS
τ σ
λ τ λ τm
m m
( ) = −( )− −( )[ ]+0 2 1
2
1exp exp
S
I
R R
R
z
z
I S
τ
λ τ λ τ
λ τ λ τ
m
m m
m m
( ) =
−


 −( )− −( )[ ]
+ − −( )+ −( )
0 1 2
1 21
exp exp
exp exp
where
R R R R R
R R R R R R
I I S S IS
I S I S
= − + +
= + +[ ] = + −[ ]
2 2 2
1
1
2 2
1
2
2 4σ
λ λ
These definitions ensure that λ1 > λ2. If RI and RS are not too dissimilar, R is of
the order of σIS, and so the two rate constants λ1 and λ2 differ by a quantity of
the order of σIS.
As expected for these two coupled differential equations, integration gives a
time dependence which is the sum of two exponentials with different time
constants.
The figure below shows the typical behaviour predicted by these equations
(the parameters are RI = RS = 5σIS)
0
-1.0
-0.5
0.0
0.5
1.0
1.5
time
(Sz/Iz
0
)
10 × (Iz–Iz
0
)/Iz
0
(Iz/Iz
0
)
The S spin magnetization returns to its equilibrium value with what appears to
be an exponential curve; in fact it is the sum of two exponentials but their time
constants are not sufficiently different for this to be discerned. The I spin
magnetization grows towards a maximum and then drops off back towards the
equilibrium value. The NOE enhancement is more easily visualized by plotting
the difference magnetization, (Iz – Iz
0
)/Iz
0
, on an expanded scale; the plot now
shows the positive NOE enhancement reaching a maximum of about 15%.
Differentiation of the expression for Iz as a function of τm shows that the
maximum enhancement is reached at time
τ
λ λ
λ
λ
m,max =
−
1
1 2
1
2
ln
and that the maximum enhancement is
8–13
I I
I R
z z
z
IS
R Rτ σ λ
λ
λ
λ
λ λ
m,max( )−
=





 −
















− −
0
0
1
2
1
2
2
1 2
8.4.2 The DPFGSE NOE experiment
From the point of view of the relaxation behaviour the DPFGSE experiment is
essentially identical to the transient NOE experiment. The only difference is
that the I spin starts out saturated rather than at equilibrium. This does not
influence the build up of the NOE enhancement on I. It does, however, have
the advantage of reducing the size of the I spin signal which has to be removed
in the difference experiment. Further discussion of this experiment is deferred
to Chapter 9.
8.4.3 Steady state experiments
The steady-state NOE experiment involves irradiating the S spin with a
radiofrequency field which is sufficiently weak that the I spin is not affected.
The irradiation is applied for long enough that the S spin is saturated, meaning
Sz = 0, and that the steady state has been reached, which means that none of the
magnetizations are changing, i.e. d dI tz( ) = 0.
Under these conditions the first of Eqs. [15] can be written
d
d
z
SS
z,SS
I I
t
R I I S
z
I z IS z
−( ) = − −( )− −( )=
0
0 0
0 0σ
therefore
I
R
S IIS
I
z zz,SS = +
σ 0 0
As in the transient experiment, the NOE enhancement is revealed by
subtracting a reference spectrum which has equilibrium intensities. The NOE
enhancement, as defined above, will be
η
σ
SS
z,SS ref
ref
=
−
=
I I
I R
S
I
z
z
IS
I
z
z
,
,
0
0
90°
S
90°
(a)
(b)
Pulse sequence for recording
s t e a d y s t a t e N O E
enhancements. Sequence (a)
involves selective irradiation of
the S spin leading to saturation.
Sequence (b) is used to record
the reference spectrum in
which the intensities are
unperturbed.
8–14
In contrast to the transient experiment, the steady state enhancement only
depends on the relaxation of the receiving spin (here I); the relaxation rate of
the S spin does not enter into the relationship simply because this spin is held
saturated during the experiment.
It is important to realise that the value of the steady-state NOE enhancement
depends on the ratio of cross-relaxation rate constant to the self relaxation rate
constant for the spin which is receiving the enhancement. If this spin is
relaxing quickly, for example as a result of interaction with many other spins,
the size of the NOE enhancement will be reduced. So, although the size of the
enhancement does depend on the cross-relaxation rate constant the size of the
enhancement cannot be interpreted in terms of this rate constant alone. This
point is illustrated by the example in the margin.
8.4.4 Advanced topic: NOESY
The dynamics of the NOE in NOESY are very similar to those for the transient
NOE experiment. The key difference is that instead of the magnetization of the
S spin being inverted at the start of the mixing time, the magnetization has an
amplitude label which depends on the evolution during tl.
Starting with equilibrium magnetization on the I and S spins, the zmagnetizations
present at the start of the mixing time are (other magnetization
will be rejected by appropriate phase cycling)
S t S I t Iz S z z I z0 01
0
1
0
( ) = − ( ) = −cos cosΩ Ω
The equation of motion for Sz is
d
d
z
z z
S t
t
I t I R S t SIS z S z
( )
= − ( ) −( )− ( ) −( )σ 0 0
As before, the initial rate approximation will be used:
d
d
z m
init
S
t
t I I R t S S
t I R t S
IS I z z S S z z
IS I z S S z
τ
σ
σ
( ) = − − −( )− − −( )
= +( ) + +( )
cos cos
cos cos
Ω Ω
Ω Ω
1
0 0
1
0 0
1
0
1
0
1 1
Integrating gives
d dz
z m z m m
z m m
m m
S t t I R t S t
S S t I R t S
S t I
IS I z S S z
IS I z S S z
IS I z
( ) = +( ) + +( )[ ]
( )− ( ) = +( ) + +( )
( ) = +( )
∫ ∫0
1
0
1
0
0
1
0
1
0
1
1 1
0 1 1
1
τ τ
σ
τ σ τ τ
τ σ τ
cos cos
cos cos
cos
Ω Ω
Ω Ω
Ω 00
1
0
1
0
0 0
1
0
1
0
1
1
+ +( ) −
= +
+ [ ]
+ −[ ]
R t S t S
I R S
t I
t R S
S S z S z
IS z S z
I IS z
S S z
τ
σ τ τ
σ τ
τ
m
m m
m
m
{a}
{b}
{c}
cos cos
cos
cos
Ω Ω
Ω
Ω
After the end of the mixing time, this z-magnetization on spin S is rendered
HA
HB
HC
HD
X
Y
Irradiation of proton B gives a
much larger enhancement on
proton A than on C despite the
fact that the distances to the
two spins are equal. The
smaller enhancement on C is
due to the fact that it is relaxing
more quickly than A, due to the
interaction with proton D.
t1
t2τmix
Pulse sequence for NOESY.
8–15
observable by the final 90° pulse; the magnetization is on spin S, and so will
precess at ΩS during t2.
The three terms {a}, {b} and {c} all represent different peaks in the NOESY
spectrum.
Term {a} has no evolution as a function of t1 and so will appear at F1 = 0; in
t2 it evolves at ΩS. This is therefore an axial peak at {F1,F2} = {0, ΩS}. This
peak arises from z-magnetization which has recovered during the mixing time.
In this initial rate limit, it is seen that the axial peak is zero for zero mixing time
and then grows linearly depending on RS and σIS.
Term {b} evolves at ΩI during t1 and ΩS during t2; it is therefore a cross peak
at {ΩI, ΩS}. The intensity of the cross peak grows linearly with the mixing time
and also depends on σIS; this is analogous to the transient NOE experiment.
Term {c} evolves at ΩS during t1 and ΩS during t2; it is therefore a diagonal
peak at {ΩS, ΩS} and as Rsτm << 1 in the initial rate, this peak is negative. The
intensity of the peak grows back towards zero linearly with the mixing time and
at a rate depending on RS. This peak arises from S spin magnetization which
remains on S during the mixing time, decaying during that time at a rate
determined by RS.
If the calculation is repeated using the differential equation for Iz a
complimentary set of peaks at {0, ΩI}, {ΩS, ΩI} and {ΩI, ΩI} are found.
It will be seen later that whereas RI and RS are positive, σIS can be either
positive or negative. If σIS is positive, the diagonal and cross peaks will be of
opposite sign, whereas if σIS is negative all the peaks will have the same sign.
8.4.5 Sign of the NOE enhancement
We see that the time dependence and size of the NOE enhancement depends on
the relative sizes of the cross-relaxation rate constant σIS and the self relaxation
rate constants RI and RS. It turns out that these self-rates are always positive,
but the cross-relaxation rate constant can be positive or negative. The reason
for this is that σIS = (W2 – W0) and it is quite possible for W0 to be greater or less
than W2.
A positive cross-relaxation rate constant means that if spin S deviates from
equilibrium cross-relaxation will increase the magnetization on spin I. This
leads to an increase in the signal from I, and hence a positive NOE
enhancement. This situation is typical for small molecules is non-viscous
solvents.
A negative cross-relaxation rate constant means that if spin S deviates from
equilibrium cross-relaxation will decrease the magnetization on spin I. This
leads to a negative NOE enhancement, a situation typical for large molecules in
viscous solvents. Under some conditions W0 and W2 can become equal and then
the NOE enhancement goes to zero.
I
Ω
I
Ω
S
Ω
S
ΩF1
F2
{a}
{b}
{c}
0
8–16
8.5 Origins of relaxation
We now turn to the question as to what causes relaxation. Recall from section
8.1 that relaxation involves transitions between energy levels, so what we seek
is the origin of these transitions. We already know from Chapter 3 that
transitions are caused by transverse magnetic fields (i.e. in the xy-plane) which
are oscillating close to the Larmor frequency. An RF pulse gives rise to just
such a field.
However, there is an important distinction between the kind of transitions
caused by RF pulses and those which lead to relaxation. When an RF pulse is
applied all of the spins experience the same oscillating field. The kind of
transitions which lead to relaxation are different in that the transverse fields are
local, meaning that they only affect a few spins and not the whole sample. In
addition, these fields vary randomly in direction and amplitude. In fact, it is
precisely their random nature which drives the sample to equilibrium.
The fields which are responsible for relaxation are generated within the
sample, often due to interactions of spins with one another or with their
environment in some way. They are made time varying by the random motions
(rotations, in particular) which result from the thermal agitation of the
molecules and the collisions between them. Thus we will see that NMR
relaxation rate constants are particularly sensitive to molecular motion.
If the spins need to lose energy to return to equilibrium they give this up to
the motion of the molecules. Of course, the amounts of energy given up by the
spins are tiny compared to the kinetic energies that molecules have, so they are
hardly affected. Likewise, if the spins need to increase their energy to go to
equilibrium, for example if the population of the β state has to be increased, this
energy comes from the motion of the molecules.
Relaxation is essentially the process by which energy is allowed to flow
between the spins and molecular motion. This is the origin of the original name
for longitudinal relaxation: spin-lattice relaxation. The lattice does not refer to
a solid, but to the motion of the molecules with which energy can be
exchanged.
8.5.1 Factors influencing the relaxation rate constant
The detailed theory of the calculation of relaxation rate constants is beyond the
scope of this course. However, we are in a position to discuss the kinds of
factors which influence these rate constants.
Let us consider the rate constant Wij for transitions between levels i and j;
this turns out to depend on three factors:
W A Y Jij ij ij= × × ( )ω
We will consider each in turn.
The spin factor, Aij
This factor depends on the quantum mechanical details of the interaction. For
8–17
example, not all oscillating fields can cause transitions between all levels. In a
two spin system the transition between the αα and ββ cannot be brought about
by a simple oscillating field in the transverse plane; in fact it needs a more
complex interaction that is only present in the dipolar mechanism (section
8.6.2). We can think of Aij as representing a kind of selection rule for the
process – like a selection rule it may be zero for some transitions.
The size factor, Y
This is just a measure of how large the interaction causing the relaxation is. Its
size depends on the detailed origin of the random fields and often it is related to
molecular geometry.
The spectral density, J(ωij)
This is a measure of the amount of molecular motion which is at the correct
frequency, ωij, to cause the transitions. Recall that molecular motion is the
effect which makes the random fields vary with time. However, as we saw
with RF pulses, the field will only have an effect on the spins if it is oscillating
at the correct frequency. The spectral density is a measure of how much of the
motion is present at the correct frequency.
8.5.2 Spectral densities and correlation functions
The value of the spectral density, J(ω), has a large effect on relaxation rate
constants, so it is well worthwhile spending some time in understanding the
form that this function takes.
Correlation functions
To make the discussion concrete, suppose that a spin in a sample experiences
a magnetic field due to a dissolved paramagnetic species. The size of the
magnetic field will depend on the relative orientation of the spin and the
paramagnetic species, and as both are subject to random thermal motion, this
orientation will vary randomly with time (it is said to be a random function of
time), and so the magnetic field will be a random function of time. Let the field
experienced by this first spin be F1(t).
Now consider a second spin in the sample. This also experiences a random
magnetic field, F2(t), due to the interaction with the paramagnetic species. At
any instant, this random field will not be the same as that experienced by the
first spin.
For a macroscopic sample, each spin experiences a different random field,
Fi(t). There is no way that a detailed knowledge of each of these random fields
can be obtained, but in some cases it is possible to characterise the overall
behaviour of the system quite simply.
The average field experienced by the spins is found by taking the ensemble
average – that is adding up the fields for all members of the ensemble (i.e. all
spins in the system)
Paramagnetic species have
unpaired electrons. These
generate magnetic fields which
can interact with nearby nuclei.
On account of the large
gyromagnetic ratio of the
electron (when compared to the
nucleus) such paramagnetic
species are often a significant
source of relaxation.
8–18
F t F t F t F t( ) = ( ) + ( ) + ( ) +1 2 3 K
For random thermal motion, this ensemble average turns out to be independent
of the time; this is a property of stationary random functions. Typically, the
Fi(t) are signed quantities, randomly distributed about zero, so this ensemble
average will be zero.
An important property of random functions is the correlation function,
G(t,τ), defined as
G t F t F t F t F t F t F t
F t F t
, * * *
*
τ τ τ τ
τ
( ) = ( ) +( ) + ( ) +( ) + ( ) +( ) +
= ( ) +( )
1 1 2 2 3 3 K
F1(t) is the field experienced by spin 1 at time t, and F1(t+τ) is the field
experienced at a time τ later; the star indicates the complex conjugate, which
allows for the possibility that F(t) may be complex. If the time τ is short the
spins will not have moved very much and so F1(t+τ) will be very little different
from F1(t). As a result, the product F t F t1 1( ) +( )*
τ will be positive. This is
illustrated in the figure below, plot (b).
5-1.2 5-1.2 5-1.2
F(t) F(t)F(t+1) F(t)F(t+15)
(a) (b) (c)
(a) is a plot of the random function F(t) against time; there are about 100 separate time points. (b) is a plot
of the value of F multiplied by its value one data point later – i.e. one data point to the right; all possible pairs
are plotted. (c) is the same as (b) but for a time interval of 15 data points. The two arrows indicate the
spacing over which the correlation is calculated.
The same is true for all of the other members of then ensemble, so when the
F t F ti i( ) +( )*
τ are added together for a particular time, t, – that is, the ensemble
average is taken – the result will be for them to reinforce one another and hence
give a finite value for G(t,τ).
As τ gets longer, the spin will have had more chance of moving and so
F1(t+τ) will differ more and more from F1(t); the product F t F t1 1( ) +( )*
τ need
not necessarily be positive. This is illustrated in plot (c) above. The ensemble
average of all these F t F ti i( ) +( )*
τ is thus less than it was when τ was shorter.
In the limit, once τ becomes sufficiently long, the F t F ti i( ) +( )*
τ are randomly
distributed and their ensemble average, G(t,τ), goes to zero. G(t,τ) thus has its
maximum value at τ = 0 and then decays to zero at long times. For stationary
random functions, the correlation function is independent of the time t; it will
therefore be written G(τ).
The correlation function, G(τ), is thus a function which characterises the
memory that the system has of a particular arrangement of spins in the sample.
For times τ which are much less than the time it takes for the system to
rearrange itself G(τ) will be close to its maximum value. As time proceeds, the
initial arrangement becomes more and more disturbed, and G(τ) falls. For
a
b
c
d
Visualization of the different
timescales for random motion.
(a) is the starting position: the
black dots are spins and the
open circle represents a
paramagnetic species. (b) is a
snap shot a very short time
after (a); hardly any of the spins
have moved. (c) is a snapshot
at a longer time; more spins
have moved, but part of the
original pattern is still
discernible. (d) is after a long
time, all the spins have moved
and the original pattern is lost.
8–19
sufficiently long times, G(τ) tends to zero.
The simplest form for G(τ) is
G Gτ τ τ( ) = ( ) −( )0 exp c [18]
the variable τ appears as the modulus, resulting in the same value of G(τ) for
positive and negative values of τ. This means that the correlation is the same
with time τ before and time τ after the present time.
τc is called the correlation time. For times much less than the correlation
time the spins have not moved much and the correlation function is close to its
original value; when the time is of the order of τc significant rearrangements
have taken place and the correlation function has fallen to about half its initial
value. For times much longer than τc the spins have moved to completely new
positions and the correlation function has fallen close to zero.
Spectral densities
The correlation function is a function of time, just like a free induction decay.
So, it can be Fourier transformed to give a function of frequency. The resulting
frequency domain function is called the spectral density; as the name implies,
the spectral density gives a measure of the amount of motion present at
different frequencies. The spectral density is usually denoted J(ω)
G Jτ ω( ) → ( )Fourier Transform
If the spins were executing a well ordered motion, such as oscillating back
and forth about a mean position, the spectral density would show a peak at that
frequency. However, the spins are subject to random motions with a range of
different periods, so the spectral density shows a range of frequencies rather
than having peaks at discreet frequencies.
Generally, for random motion characterised by a correlation time τc,
frequencies from zero up to about 1/τc are present. The amount at frequencies
higher that 1/τc tails off quite rapidly as the frequency increases.
For a simple exponential correlation function, given in Eq. [18], the
corresponding spectral density is a Lorentzian
exp −( ) →
+
τ τ
τ
ω τ
c
Fourier Transform c
c
2
2
1 2
This function is plotted in the margin; note how it drops off significantly once
the product ωτc begins to exceed ~1.
The plot opposite compares the spectral densities for three different
correlation times; curve a is the longest, b an intermediate value and c the
shortest. Note that as the correlation time decreases the spectral density moves
out to higher frequencies. However, the area under the plot remains the same,
so the contribution at lower frequencies is decreased. In particular, at the
frequency indicated by the dashed line the contribution at correlation time b is
greater than that for either correlation times a or c.
0
0
0
G( )
G(0)
ττ
τ
c
0 1 2 3 4 5
0
J( )ω
ωτc
Plot of the spectral density as a
function of the dimensionless
variable ωτc. The curve is a
lorentzian
00
J( )ω
ω
a
b
c
8–20
For this spectral density function, the maximum contribution at frequency ω
is found when τc is 1/ω; this has important consequences which are described in
the next section.
8.5.3 The "T1 minimum"
In the case of relaxation of a single spin by a random field (such as that
generated by a paramagnetic species), the only relevant spectral density is that
at the Larmor frequency, ω0. This is hardly surprising as to cause relaxation –
that is to cause transitions – the field needs to have components oscillating at
the Larmor frequency.
We have just seen that for a given frequency, ω0, the spectral density is a
maximum when τc is 1/ω0, so to have the most rapid relaxation the correlation
time should be 1/ω0. This is illustrated in the plots below which show the
relaxation rate constant, W, and the corresponding relaxation time (T1 = 1/W)
plotted as a function of the correlation time.
τc
1/ω0
τc
W T1
1/ω0
(a) (b)
Plot (a) shows how the relaxation rate constant, W, varies with the correlation time, τc, for a given Larmor
frequency; there is a maximum in the rate constant when τc = 1/ω0. Plot (b) shows the same effect, but here
we have plotted the relaxation time constant, T1; this shows a minimum.
At very short correlation times (τc << 1/ω0) there is some spectral density at
the Larmor frequency, but not that much as the energy of the motion is spread
over a very wide frequency range. As the correlation time increases the amount
of spectral density at the Larmor frequency increases and so the relaxation rate
constant increases, reaching a maximum when τc = 1/ω0. After this point, the
spectra density at the Larmor frequency, and hence the rate constant, falls.
In terms of the relaxation time, T1, there is a minimum in T1 which
corresponds to the maximum in W. We see that, like Goldilocks and the Three
Bears, efficient relaxation requires a correlation time which is neither too fast
nor too slow.
Motion which gives rise to correlation times which are much shorter than
1/ω0 is described as being in the fast motion (or extreme narrowing) limit. Put
mathematically, fast motion means ω0τc << 1. Motion which gives rise to
correlation times which are much slower than 1/ω0 is described as being in the
slow motion (or spin diffusion) limit; mathematically this limit is ω0τc >> 1.
Clearly, which limit we are in depends on the Larmor frequency, which in turn
depends on the nucleus and the magnetic field.
For a Larmor frequency of 400 MHz we would expect the fastest relaxation
when the correlation time is 0.4 ns. Small molecules have correlation times
8–21
significantly shorter than this (say tens of ps), so such molecules are clearly in
the fast motion limit. Large molecules, such as proteins, can easily have
correlation times of the order of a few ns, and these clearly fall in the slow
motion limit.
Somewhat strangely, therefore, both very small and very large molecules
tend to relax more slowly than medium-sized molecules.
8.6 Relaxation mechanisms
So far, the source of the magnetic fields which give rise to relaxation and the
origin of their time dependence have not been considered. Each such source is
referred to as a relaxation mechanism. There are quite a range of different
mechanisms that can act, but of these only a few are really important for spin
half nuclei.
8.6.1 Paramagnetic species
We have already mentioned this source of varying fields several times. The
large magnetic moment of the electron means that paramagnetic species in
solution are particularly effective at promoting relaxation. Such species include
dissolved oxygen and certain transition metal compounds.
8.6.2 The dipolar mechanism
Each spin has associated with it a magnetic moment, and this is turn gives rise
to a magnetic field which can interact with other spins. Two spins are thus
required for this interaction, one to "create" the field and one to "experience" it.
However, their roles are reversible, in the sense that the second spin creates a
field which is experienced by the first. So, the overall interaction is a property
of the pair of nuclei.
The size of the interaction depends on the inverse cube of the distance
between the two nuclei and the direction of the vector joining the two nuclei,
measured relative to that of the applied magnetic field. As a molecule tumbles
in solution the direction of this vector changes and so the magnetic field
changes. Changes in the distance between the nuclei also result in a change in
the magnetic field. However, molecular vibrations, which do give such
changes, are generally at far too high frequencies to give significant spectral
density at the Larmor frequency. As a result, it is generally changes in
orientation which are responsible for relaxation.
The pair of interacting nuclei can be in the same or different molecules,
leading to intra- and inter-molecular relaxation. Generally, however, nuclei in
the same molecule can approach much more closely than those in different
molecules so that intra-molecular relaxation is dominant.
The relaxation induced by the dipolar coupling is proportional to the square
of the coupling. Thus it goes as
8–22
γ γ1
2
2
2
12
6
1
r
where γ1 and γ2 are the gyromagnetic ratios of the two nuclei involved and r12 is
the distance between them.
As the size of the dipolar interaction depends on the product of the
gyromagnetic ratios of the two nuclei involved, and the resulting relaxation rate
constants depends on the square of this. Thus, pairs of nuclei with high
gyromagnetic ratios are most efficient at promoting relaxation. For example,
every thing else being equal, a proton-proton pair will relax 16 times faster than
a carbon-13 proton pair.
It is important to realize that in dipolar relaxation the effect is not primarily
to distribute the energy from one of the spins to the other. This would not, on
its own, bring the spins to equilibrium. Rather, the dipolar interaction provides
a path by which energy can be transferred between the lattice and the spins. In
this case, the lattice is the molecular motion. Essentially, the dipole-dipole
interaction turns molecular motion into an oscillating magnetic field which can
cause transitions of the spins.
Relation to the NOE
The dipolar mechanism is the only common relaxation mechanism which can
cause transitions in which more than one spin flips. Specifically, with reference
to section 8.3, the dipolar mechanism gives rise to transitions between the αα
and ββ states (W2) and between the αβ and βα states (W0).
The rate constant W2 corresponds to transitions which are at the sum of the
Larmor frequencies of the two spins, (ω0,I + ω0,S) and so it is the spectral density
at this sum frequency which is relevant. In contrast, W0 corresponds to
transitions at (ω0,I – ω0,S) and so for these it is the spectral density at this
difference frequency which is relevant.
In the case where the two spins are the same (e.g. two protons) the two
relevant spectral densities are J(2ω0) and J(0). In the fast motion limit (ω0τc <<
1) J(2ω0) is somewhat less than J(0), but not by very much. A detailed
calculation shows that W2 > W 0 and so we expect to see positive NOE
enhancements (section 8.4.5). In contrast, in the slow motion limit (ω0τc >> 1)
J(2ω0) is all but zero and so J(0) >> J(2ω0); not surprisingly it follows that W0 >
W2 and a negative NOE enhancement is seen.
8.6.3 The chemical shift anisotropy mechanism
The chemical shift arises because, due to the effect of the electrons in a
molecule, the magnetic field experienced by a nucleus is different to that
applied to the sample. In liquids, all that is observable is the average chemical
shift, which results from the molecule rapidly experiencing all possible
orientations by rapid molecular tumbling.
At a more detailed level, the magnetic field experienced by the nucleus
8–23
depends on the orientation of the molecule relative to the applied magnetic
field. This is called chemical shift anisotropy (CSA). In addition, it is not only
the magnitude of the field which is altered but also its direction. The changes
are very small, but sufficient to be detectable in the spectrum and to give rise to
relaxation.
One convenient way of imagining the effect of CSA is to say that due to it
there are small additional fields created at the nucleus – in general in all three
directions. These fields vary in size as the molecule reorients, and so they have
the necessary time variation to cause relaxation. As has already been discussed,
it is the transverse fields which will give rise to changes in population.
The size of the CSA is specified by a tensor, which is a mathematical object
represented by a three by three matrix.
σ
σ σ σ
σ σ σ
σ σ σ
=








xx xy xz
yx yy yz
zx zy zz
The element σxz gives the size of the extra field in the x-direction which results
from a field being applied in the z-direction; likewise, σyz gives the extra field in
the y-direction and σzz that in the z-direction. These elements depend on the
electronic properties of the molecule and the orientation of the molecule with
respect to the magnetic field.
Detailed calculations show that the relaxation induced by CSA goes as the
square of the field strength and is also proportional to the shift anisotropy. A
rough estimate of the size of this anisotropy is that it is equal to the typical shift
range. So, CSA relaxation is expected to be significant for nuclei with large
shift ranges observed at high fields. It is usually insignificant for protons.
8.7 Transverse relaxation
Right at the start of this section we mentioned that relaxation involved two
processes: the populations returning to equilibrium and the transverse
magnetization decaying to zero. So far, we have only discussed the fist of these
two. The second, in which the transverse magnetization decays, is called
transverse (or spin-spin) relaxation.
8–24
(a)
(b)
(c)
Depiction of how the individual contributions from different spins (shown on the left) add up to give the net
transverse magnetization (on the right). See text for details.
Each spin in the sample can be thought of as giving rise to a small
contribution to the magnetization; these contributions can be in any direction,
and in general have a component along x, y and z. The individual contributions
along z add up to give the net z-magnetization of the sample.
The transverse contributions behave in a more complex way as, just like the
net transverse magnetization, these contributions are precessing at the Larmor
frequency in the transverse plane. We can represent each of these contributions
by a vector precessing in the transverse plane.
The direction in which these vectors point can be specified by giving each a
phase – arbitrarily the angle measured around from the x-axis. It is
immediately clear that if these phases are random the net transverse
magnetization of the sample will be zero as all the individual contributions will
cancel. This is the situation that pertains at equilibrium and is shown in (a) in
the figure above.
For there to be net magnetization, the phases must not be random, rather
there has to be a preference for one direction; this is shown in (b) in the figure
above. In quantum mechanics this is described as a coherence. An RF pulse
applied to equilibrium magnetization generates transverse magnetization, or in
other words the pulse generates a coherence. Transverse relaxation destroys
this coherence by destroying the alignment of the individual contributions, as
8–25
shown in (c) above.
Our picture indicates that there are two ways in which the coherence could
be destroyed. The first is to make the vectors jump to new positions, at
random. Drawing on our analogy between these vectors and the behaviour of
the bulk magnetization, we can see that these jumps could be brought about by
local oscillating fields which have the same effect as pulses.
This is exactly what causes longitudinal relaxation, in which we imagine the
local fields causing the spins to flip. So, anything that causes longitudinal
relaxation will also cause transverse relaxation.
The second way of destroying the coherence is to make the vectors get out of
step with one another as a result of them precessing at different Larmor
frequencies. Again, a local field plays the part we need but this time we do not
need it to oscillate; rather, all we need for it to do is to be different at different
locations in the sample.
This latter contribution is called the secular part of transverse relaxation; the
part which has the same origin as longitudinal relaxation is called the nonsecular
part.
It turns out that the secular part depends on the spectral density at zero
frequency, J(0). We can see that this makes sense as this part of transverse
relaxation requires no transitions, just a field to cause a local variation in the
magnetic field. Looking at the result from section 8.5.2 we see that J(0) = 2τc,
and so as the correlation time gets longer and longer, so too does the relaxation
rate constant. Thus large molecules in the slow motion limit are characterised
by very rapid transverse relaxation; this is in contrast to longitudinal relaxation
is most rapid for a particular value of the correlation time.
The plot below compares the behaviour of the longitudinal and transverse
relaxation rate constants. As the correlation time increases the longitudinal rate
constant goes through a maximum. However, the transverse rate constant
carries on increasing and shows no such maximum. We can attribute this to the
secular part of transverse relaxation which depends on J(0) and which simply
goes on increasing as the correlation time increases. Detailed calculations show
that in the fast motion limit the two relaxation rate constants are equal.
τc
1/ω0
W
longitudinal
transverse
Comparison of the longitudinal and transverse relaxation rate constants as a function of the correlation time
for the fixed Larmor frequency. The longitudinal rate constant shows a maximum, but the transverse rate
constant simply goes on increasing.