3 2 Gradient s and spin motion
3.2. GRADIENTS AND SPIN MOTION 25 and the phase shift, Llcp, acquired by the spin after a time t = nr5, using eqn (3 14)
will be[12] Llcp( t) m=l i=l n
1Grse 2:) n
+ 1 -i)a;.
i=lThe ensem ble average
(3.18)
When dealing with a real sample with an Avogadro's number of spins we need some technique for calculating the final signal. Typically there is an echo sequence involved and so, with the exception of the steady gradient case which follows, the echo signal is labelled E. Because the diffusion process is random each spin will acquire a different phase shift. To calculate the final result of adding all the spin vectors, ei!:J.4>, together, each with a phase shift .6.cp, an ensemble average over the whole sample is used.
In order to find the coefficient by which the ensemble-averaged transverse mag netization will be attenuated we need to calculate exp(illcp ) , additional attenuation due to T2 relaxation effects will be ignored. Therefore
exp(illcp) =
j_:
P( Ll<P) exp(illcp )d( Llcp) (3. 19)where P(Ll<P) is the distribution of phase shifts Ll<P(t). This integral is most simply done by assuming that the distribution of phase shifts P(.6.cp) will be Gaussian [14] which makes eqn (3. 19) yield
exp(i.6.cp) = exp( -flcp2 /2)
By squaring eqn (3. 18) and taking an ensemble average one obtains
n =
,2G2rs2e 2:)n
+ 1 -i)2
i=l n,2G2rs2e L J2
j=l 13'2G2r;en3
(3.20) (3.21 )where
'Lj=1 p = �n3
i s evaluated by assuming thatn
is large. Substituting this result and eqn (3. 16) into eqn (3.20) one finds the signal attenuation due to diffusion in the presence of a steady gradient is[12](3.22) Often in NMR there are i nhomogeneities in the local field throughout the sample, as discussed in Section 2.3 . 1 , and these can be refocused though the use of a spin echo. If a 180° pulse is applied at a time t after the initial 90° excitation pulse t hen an echo will form at 2t.
90x center echo
�
0 r 2-r time 11 � & + gradient signalFigure 3 . 2 : The pulsed gradient spin echo sequence (PGSE) with the r.f. and gradient pulses
shown on separate time lines. The attenuation of the echo acquired at time 2r due to the presence of the gradient pulses is given by eqn (3.25).
The signal attenuation of the echo occurring at 2t in the presence of a steady gradient is then[12]
S(2t) = exp(i�4>) exp(
-�--/G2
Dt3 ) 1exp( - 12
'"·/G2
D(2t)3)3 .2.2 P ulsed field gradient s
(3.23)
A major inconvenience of the steady gradient method[15, 16] is that the gradient is applied to the sample for the whole experiment. In particular for the times when the r.f. pulses are being applied, and when the signal is being acquired. The resultant spread of the Larmor spectrum and the effective bandwidth of the receiver and transmitter will limit the maximum strength of gradient that can be applied to the sample. It is however easy to modify the hardware so that the gradients can be applied as pulses at the appropriate time, and a spin echo sequence can be used to form an echo at an appropriate time when the gradients are off. This modification, called Pulsed Gradient Spin Echo (PGSE) , first suggested by McCall, Douglas and Anderson[16] in 1963, was demonstrated by Stejskal and Tanner(17] in 1 965, and is shown in Figure 3.2. If the gradient pulses have a duration
8
and a separation .6.then the mean square phase shift is given by[18, 17, 19]
(3.24)
The echo amplitude at 2r will be attenuated by both relaxation and the gradient. The effects of any
T2
relaxation can be removed by normalising the echo signal to3. 3. PGSE, SCATTERING AND q-SPACE 27
its value with no gradient applied. The ratio S(g)
/
S(O) is often labelled E (g). The signal strength of the echo at 2r is therefore given by(3.25)
which is the well known Stejskal-Tanner equation[l 7] . A plot of the natural loga rithm of the echo against
12g282(!:l - 8/3)
will give a straight line with a slope of-
D for a sample exhibiting unrestricted Brownian diffusion. This type of graph is often referred to as a S tejskal-Tanner plot and has been used extensively to measure self-diffusion coefficients from PGSE experiments[20, 21 , 22] . The effect of any relax ation can be removed by normalising the echo signal to the value when no gradient is applied.Eqn (3. 25) has also been derived by Stepisnik[23] using a density matrix formal ism that is especially useful for finding expressions for E( q) for oscillating rather than pulsed gradients.
3.2.3 Stimulated echo
The PGSE experiment can be altered to take advantage of the stimulated echo sequence[24] of Section 2.4.4 as shown in Figure 3.3. The echo attenuation is still given by eqn (3.25) but the normal reduction of signal by a factor of two is still present. However the spins now only suffer
T2
relaxation during the two periods r1 •During the longer period
r2
the spins suffer the normally less severeT1
relaxation. However the gradient pulses can only be applied during the intervals where the magnetization is in the transverse plane which often limits the duration of r1 . Insome recent experiments, utilising the fringe field of superconducting magnets[25] to obtain large field gradients, this pulse sequence has been used to define narrow gradient pulses. Five pulse variations[26] of the stimulated echo sequence are used in the fringe field experiments to remove relaxation dependance of the echo signal.