Double-quantum transitions

A single-quantum transition involves one spin only, whereas the zero- and doublequantum transitions involve two spins at the same time. The zero- and double-quantum transitions give rise to cross-relaxation pathways, which provide an efficient mechanism for dipole-dipole relaxation. 

The transitions between energy levels in an AX spin system are shown in Fig. 1.44. There are four single-quantum transitions (these are the normal transitions A, A, Xi, and X2 in which changes in quantum number of 1 occur), one double-quantum transition 1% between the aa and j8 8 states involving a change in quantum number of 2, and a zero-quantum transition 1% between the a)3 and fia states in which no change in quantum number occurs. The double-quantum and zero-quantum transitions are not allowed as excitation processes under the quantum mechanical selection rules, but their involvement may be considered in relaxation processes. 

An interesting feature of the energy diagram is that there are two transitions that involve the simultaneous flip of both spins. These are the transitions aa )3j8 (a double-quantum process, Wf) and a/3 )8a (a zero-... 

When the zero-quantum 1% transition is greater than double-quantum Wi, the nOe enhancements will be negative. Similarly, when is greater than 1%, the resultant nOe will have a positive sign. The predominance of Wi and over one another depends on the molecular motion. It is known that the Wo transition is maximal when the molecule tumbles at a rate of about 1 KHz, while the Wi transition is fastest at a tumbling rate of about 800 MHz. On this basis, a rough idea of the sign of nOe can be obtained. For example, small molecules in nonviscous... 

Coherence A condition in which nuclei precess with a given phase relationship and can exchange spin states via transitions between two eigenstates. Coherence may be zero-quantum, single-quantum, double-quantum, etc., depending on the AM of the transition corresponding to the coherence. Only single-quantum coherence can be detected directly. 

Chen, J. C., Chang, A. M. and Melloch, M. R. (2004) Transition between quantum states in a parallel-coupled double quantum dot. Phys. Rev. Lett., 92, 176801-1-176801-4. 

Double quantum transitions have been analyzed in high-field EPR studies of Ni11 complexes with N6 355,357 and N402 357 donor set. 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

D. Collison, M. Helliwell, V.M. Jones, F.E. Mabbs, E.J.L. Mclnnes, P.C. Riedi, G.M. Smith, R.G. Pritchard and W.I. Cross, Single and double quantum transitions in the multi-frequency continuous wave electron paramagnetic resonance (cwEPR) of three six-co-ordinate nickel(II) complexes ... 

Double quantum transitions 4.5 in ENDOR one or two rf fields 2.5 Simplification of the interpretation of ENDOR spectra... 

One rf field. The transition probability for a double quantum transition (DQT) between energy levels Ek and Em generated by one single pumping field of frequency v is given by41>... 

Fig. 20 a, b. Energy level diagram with single (SQT) and double quantum transitions (DQT) a) one rf field b) two rf fields... 

First, various advanced multipulse techniques have been developed since the mid-1970s, and nowadays are routinely applicable on spectrometers of the latest generation. Particularly innovative and ingenious among these methods are two-dimensional NMR techniques (506-508) and double quantum transition measurements (INADEQUATE) (507-509), which allow one to determine connectivities between carbon atoms within a molecule. 

Detailed theoretical and experimental investigations 328-330) of such coupling effects show that they are not caused entirely by these hole-burning effects, but that double quantum Raman transitions occur and that the interaction between both light fields and the molecule via the common level leads to a dynamic Stark splitting of the probe line 33D. 

Fig. 10. Energy diagram of a IS spin system (two spins 1/2). a and P are the usual spin functions. One-quantum transitions (Ji, Sy S2, physically observable) full arrows. ZQ and DQ (dashed arrows) refer to zero-quantum and double-quantum transitions.

The longitudinal cross-relaxation rate (see Eq. (13)) originates solely from the terms in the dipolar Hamiltonian involving both spins, namely those terms corresponding to zero-quantum and double-quantum transitions so that... 

INADEQUATE incredible natural abundance double quantum transition experiment... 

The nuclear magnetic resonance (NMR) spectrum of chrom-3-ene (1) has been measured13 and the sign of coupling constants between protons on the hetero ring obtained from a study of double quantum transitions.13 An inter-ring coupling (J4 8) has been detected.14-16 In... 

The appearance of a sharp central peak in the deuteron NMR spectrum for unaligned lyotropic liquid crystalline samples has been observed by several authors (30, 33, 34). This has been interpreted in terms of phase inhomogenities (35) or isotropic motion (36). However, recently Wennerstrom et al. showed for a lamellar amphiphile-water mesophase that this could be referred to double quantum transitions (37). It is expected that double quantum transition peaks appear in NMR spectra... 

For a mesophase composed of sodium octanoate, decanol, and heavy water such double quantum transitions for 23Na are easily observed, as shown in Figure 5. This figure shows that at low rf field amplitudes a typical 23Na powder spectrum is obtained, but at high rf field strengths new peaks arise in the spectrum. 

Figure 5. 23Na NMR spectra of a lamellar mesophase sample of the system sodium octanoate-decanol-water. The rf field amplitude was increased in the order a,b,c,dt Peaks arising from double quantum transitions are clearly shown (cf. Figure 1).

In several kinds of correlation spectroscopy it is customary to also exploit the evolution of either zero quantum (ZQ) or double quantum (DQ) coherences. As shown in Fig. 7.1, they correspond to the transition — + + — and... 

The relative probability of a double quantum transition depends on... 

It may be verified that for a double quantum jump the steric factor is A precise treatment must consider simultaneous rotational transitions, because the impulse during collision depends not only on 0 but also on the manner in which 0 changes. However, the major part of the overall transition probability occurs where 6 is small, and coupling with rotation is then at its lowest efficiency. A detailed discussion of this complex problem is given by Herzfeld and Litovitz32. By integration of equation (14) through the velocity distribution, equation (13) takes the form... 

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