Chapter3
1 T f)τ
Mxy(2τ^T) = -M^xp(--~-)
[3.11]
This procedure allows increasing the diffusion time while reducing the effect
of extended relaxation which will reduce the SNR. Eq. 3.8 can also be used to
calculated spin echo signal attenuation for stimulated spin-echo PGSE sequence.
3.3.2. Drop size determination from PGSE measurement
Eq. [3.9] is limited to bulk fluids. But in many cases, the fluids are confined in
small geometries such as pores or droplets and cannot diffuse freely, which is
referred as restricted diffusion. In these cases, Murday and Cotts [15] developed
the equation for the PGSE sequence. For restricted diffusion within a sphere of
radius a, the attenuation ratio Rsp is:
Rsb = exp<-2χ2g2∑ 7 7 7 ,
sp pl z s ^a2m(a2ma2-2)[a2mD
(⅛D)jj
[3.12]
Here Ψ = 2 + e~a2-D(&~S) - 2e~a^ - 2e~a-oδ
+ e-a^lD(Δ+δ)
[3.13]
am is the mth positive root of the equation:
ota ∙ Jr5z2(Qa)-Z372 (ota) = 0 [3.14]
Jk is the Bessel function of the first kind, order k.
If Δ » a2∕2D, Δ » δ, Eq. [3.8] can be simplified as1:
Rsp = exp(-y2g2δ2a2 /5)
[3.15]
Figure 3.4 shows the predicted attenuation of emulsified water droplets with
different drop diameters from 5 to 50 μm. Here the diffusion time Δ is 500 ms,
46