Chapter 6: Product operators - The James Keeler Group

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2πJ12tI1 zI2z2I I ⎯⎯⎯⎯→cosπJ t 2I I −sinπJ t 4I I I1y3z12 1y 3z 13 1x 2z 3zIn this evolution the spin 3 operator is unaffected as the coupling does notinvolve this spin. The connection with the evolution of I 1yunder a coupling canbe made more explicit by writing 2I 3zas a "constant" γ2πJ tI zIzγ I ⎯⎯⎯⎯12 1 2 →cosπJ t γ I −sinπJ t 2 γ I I1ywhich compares directly to12 1y 13 1x 2z2πJ tI zIzI ⎯⎯⎯⎯12 1 2 →cosπJ t I −sinπJ t 2 I I1y12 1y 13 1x 2z6.7 Alternative notationIn this chapter different spins have been designated with a subscript 1, 2, 3 ...Another common notation is to distinguish the spins by using a different letter torepresent their operators; commonly I and S are used for two of the symbols2I1xI2z ≡ 2IxSzNote that the order in which the operators are written is not important, althoughit is often convenient (and tidy) always to write them in the same sequence.In heteronuclear experiments a notation is sometimes used where the letterrepresents the nucleus. So, for example, operators referring to protons are giventhe letter H, carbon-13 atoms the letter C and nitrogen-15 atoms the letter N;carbonyl carbons are sometimes denoted C'. For example, 4C xH zN zdenotesmagnetization on carbon-13 which is anti-phase with respect to coupling to bothproton and nitrogen-15.6.8 ConclusionThe product operator method as described here only applies to spin-half nuclei.It can be extended to higher spins, but significant extra complexity is introduced;details can be found in the article by Sørensen et al. (Prog. NMR Spectrosc. 16,163 (1983)).The main difficulty with the product operator method is that the more pulsesand delays that are introduced the greater becomes the number of operators andthe more complex the trigonometrical expressions multiplying them. If pulsesare either 90° or 180° then there is some simplification as such pulses do notincrease the number of terms. As will be seen in chapter 7, it is important to tryto simplify the calculation as much as possible, for example by recognizingwhen offsets or couplings are refocused by spin echoes.A number of computer programs are available for machine computationusing product operators within programs such as Mathematica or Maple. Thesecan be very labour saving.6–14
6.9 Multiple -quantum coherence6.9.1 Multiple-quantum termsIn the product operator representation of multiple quantum coherences it is usualto distinguish between active and passive spins. Active spins contributetransverse operators, such as I x, I yand I +, to the product; passive spins contributeonly z-operators, I z. In a sense the spins contributing transverse operators are"involved" in the coherence, while those contributing z-operators are simplyspectators.For double- and zero-quantum coherence in which spins i and j are active it isconvenient to define the following set of operators which represent pure multiplequantum states of given order. The operators can be expressed in terms of theCartesian or raising and lowering operators.double quantum, p =± 2( ij)11DQx≡ 2 ( 2IixIjx −2IiyIjy )≡ 2 ( Ii+ Ij+ + Ii−Ij−)( ij)11DQy≡ 2 ( 2IixIjy + 2IiyIjx )≡ 2i ( Ii+ Ij+ −Ii−Ij−)zero quantum, p = 0( ij)11ZQx≡ 2 ( 2IixIjx + 2IiyIjy)≡ 2 ( Ii+ Ij− + Ii− Ij+)( ij)1 1ZQy≡ 2 ( 2IiyIjx −2IixIjy )≡ 2 i ( Ii+ Ij− −Ii− Ij+)6.9.2 Evolution of multiple -quantum termsEvolution under offsetsThe double- and zero-quantum operators evolve under offsets in a way which isentirely analogous to the evolution of I xand I yunder free precession except thatthe frequencies of evolution are (Ω i+ Ω j) and (Ω i– Ω j) respectively:( ij) ΩitIiz+ ΩjtIjz( ij) ( ij)DQ ⎯⎯⎯⎯ →cosΩ + Ω tDQ sin Ω Ω tDQi iz j jzijDQ ⎯⎯⎯⎯ →cosΩ + Ω tDQ sin Ω Ω tDQZQ( ) + ( + )xi j xi j y( ij) ΩtI + Ω tI( ) ( ij)y( i j) y− ( i+j)x( ij) ΩitIiz+ΩjtIjz( ij) ( ij)x⎯⎯⎯⎯→cos( Ωi − Ωj) tZQx+ sin( Ωi −Ωj)tZQy( ij) ΩitIiz+ ΩjtIjz( ij) ( ij)y⎯⎯⎯⎯→cosΩi −Ωj tysin Ωi Ωj tx( ) − ( − )ZQ ZQ ZQEvolution under couplingsMultiple quantum coherence between spins i and j does not evolve under theinfluence of the coupling between the two active spins, i and j.Double- and zero-quantum operators evolve under passive couplings in away which is entirely analogous to the evolution of I xand I y; the resultingmultiple quantum terms can be described as being anti-phase with respect to theeffective couplings:6–15
Page 1 and 2: 6 Product Operators †The vector m
Page 3 and 4: and z-components of the magnetizati
Page 5 and 6: ( ) −{ } ( )≡ { − }+ { }exp
Page 7 and 8: means that the two lines of the spi
Page 9 and 10: IVxVy-yzzzyzxzzz-xz-xangle = πJtFo
Page 11 and 12: 6.1.3 Spin echoes in heteronuclear
Page 13: ( I I + I I )= ( I + iI )( I + iI )

Chapter 6: Product operators - The James Keeler Group

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