Similarly, the imaginary component varies from −2τcpNcycε1 to 2τcpNcycε1, which can be expressed as ±Trel(f00I − f11I)/2. The

two imaginary limiting values correspond to magnetisation that ‘swaps’ ensembles after each 180° pulse, spending equal time in the ground and excited state ensembles. The imaginary limiting values correspond to the least refocused magnetisation. All four frequency limits are proportional to Trel. This provides a strong justification for performing constant time CPMG experiments, as this means that the relaxation for each term, and the maximum phase that any one term can accrue will be constant for all values of Ncyc. The complete set of discrete frequencies that can potentially contribute to the signal intensity, parameterised in terms of the indices j and k: equation(59) Fk,j=k-2j+1Ncycf11R-f00R+2f00R+f11R+i-k-2j+3Ncyc+2f00I-f11ITrel4with the index k running from 1 to 1 + 2Ncyc describing the trinomial expansion

in buy Natural Product Library ε0 − ε1, and j running from 1 to 1 + Ncyc describing the binomial expansion in ε0 + ε1. The geometric distribution of these the real and imaginary components of these frequencies is illustrated in Figs. 3B and 4A, where the real component has been normalised by a factor of f11RTrel, and the imaginary terms by (f00I − f11I)Trel. Using these normalisations, the range of frequencies are independent on Ncyc and take the form of a diamond with limits in the imaginary dimension of (−0.5, 0.5) and in the real dimension of (f00R/f11R) to 1. As f00R ≪ f11R, on this scale the

first term appears to be very close to zero, and the terms ‘higher’ up the diamond on the real axis have significantly Forskolin order larger relaxation rates. In the constant time CPMG experiment, the range of the resolvable frequencies is identical. The spectral resolution is limited by the density of frequencies which increases substantially with increasing Ncyc ( Fig. 4A). The simultaneous binomial and trinomial expansions result in there being many different pathways that can lead to the selleck screening library same final net evolution frequency. The total number of individual pathways that will contribute at each frequency is given by the product of the coefficients of the two series, written here in terms of the Gamma function, a generalisation of the factorial, Γ(x+1)=x!Γ(x+1)=x!: equation(60) χk,j=χk,jbiχk,jtri=Γ(n+1)Γ(n-k+1)Γ(k+1)∑j=0nΓ(n+1)Γ(j+k+1)Γ(n-2j-k+1) The degeneracies of each frequency are strongly dependent on Ncyc. Initially, each of the six frequencies has equal degeneracy (Ncyc = 1, Fig. 3B). At successively higher values of Ncyc, there exists a strong combinatorial preference for terms to converge on the central frequency ( Fig. 4A). This combinatorial factor effectively describes the additional mixing between ground and excited ensembles that occur at increased νCPMG. It is important to note however that the frequencies emerging from the CPMG block are not equally weighted, and using Eq.