Phase Cycling and Gradient Pulses - The James Keeler Group

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of the gradient pulse the vectors representing transverse magnetization in all these discs are aligned, but after some time each vector has precessed through a different angle because of the variation in Larmor frequency. After sufficient time the vectors are disposed in such a way that the net magnetization of the sample (obtained by adding together all the vectors) is zero. The gradient pulse is said to have dephased the magnetization. It is most convenient to view this dephasing process as being due to the generation by the gradient pulse of a spatially dependent phase. Suppose that the magnetic field produced by the gradient pulse, B g , varies linearly along the z-axis according to B Gz g = where G is the gradient strength expressed in, for example, T m –1 or G cm –1 ; the origin of the z-axis is taken to be in the centre of the sample. At any particular position in the sample the Larmor frequency, ω L (z), depends on the applied magnetic field, B 0 , and B g ωL = γ( B0 + Bg)= γ( B0 + Gz ) , where γ is the gyromagnetic ratio. After the gradient has been applied for time t, the phase at any position in the sample, Φ(z), is given by Φ()= z γ ( B0 + Gz) t . The first part of this phase is just that due to the usual Larmor precession in the absence of a field gradient. Since this is constant across the sample it will be ignored from now on (which is formally the same result as viewing the magnetization in a frame of reference rotating at γB 0 ). The remaining term γGzt is the spatially dependent phase induced by the gradient pulse. If a gradient pulse is applied to pure x-magnetization, the following evolution takes place at a particular position in the sample γGztIz I ⎯ →cos γGzt I sin γGzt I . x ⎯⎯ ( ) + ( ) The total x-magnetization in the sample, M x , is found by adding up the magnetization from each of the thin discs, which is equivalent to the integral M x ()= t 1 r max x 1 2 r max ∫ – 1 2 r max cos( γGzt)dz where it has been assumed that the sample extends over a region ± 1 2 r max . Evaluating the integral gives an expression for the decay of x-magnetization during a gradient pulse ( ) sin Gr t Mx( 1 2 γ max t)= 1 2 γGrmaxt The plot below shows M x (t) as a function of time y [11] 9–34
1.0 M x 0.5 0.0 10 20 30 40 50 γGr max t -0.5 The black line shows the decay of magnetization due to the action of a gradient pulse. The grey line is an approximation, valid at long times, for the envelope of the decay. Note that the oscillations in the decaying magnetization are imposed on an overall decay which, for long times, is given by 2/(γGtr max ). Equation [11] embodies the obvious points that the stronger the gradient (the larger G) the faster the magnetization decays and that magnetization from nuclei with higher gyromagnetic ratios decays faster. It also allows a quantitative assessment of the gradient strengths required: the magnetization will have decayed to a fraction α of its initial value after a time of the order of 2 ( γGαr max ) (the relation is strictly valid for α
Page 1 and 2: 9 Coherence Selection: Phase Cyclin
Page 3 and 4: We can go further with this interpr
Page 5 and 6: that the cosine and sine modulated
Page 7 and 8: y -x x -y pulse x y -x -y receiver
Page 9 and 10: phase cycling of many two- and thre
Page 11 and 12: ( ) ( )= ( ) exp −iΩtIz I± exp
Page 13 and 14: t 1 t 2 p 2 1 0 -1 -2 ∆p=±1 ±1,
Page 15 and 16: simply because cos(Ωt 1 ) = cos(-
Page 17 and 18: A A + A A 1 1 4 1+ 2 4 1− 2 which
Page 19 and 20: [ ] 1 S+ ( F1, F2)= 2 A1+ + iD1+ A2
Page 21 and 22: pulse goes x, -x and the receiver g
Page 23 and 24: step pulse phase phase shift experi
Page 25 and 26: As the cycle has four steps, a path
Page 27 and 28: dimensional spectra. This is consid
Page 29 and 30: doubles the length of the phase cyc
Page 31 and 32: t 1 τ mix t 2 1 0 -1 If we group t
Page 33: magnetic field is made spatially in
Page 37 and 38: ∑ sp i iγ iBg,iτi = 0 . i With
Page 39 and 40: We start out the discussion by cons
Page 41 and 42: The sequence below shows the gradie
Page 43 and 44: stronger the gradient the more rapi
Page 45 and 46: discrimination in the F 1 dimension
Page 47 and 48: I ∆ 2 ∆ 2 ∆ 2 ∆ 2 t 2 S t 1
Page 49: y the gradient at the edges of the

Phase Cycling and Gradient Pulses - The James Keeler Group

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