Tis known as the inversion operator and has the effect of changing the
spin quantum number fromþ1/2 to1/2 andvice versa. This leads to
Rx^1 ð Þ¼Ecosð = 2 ÞþiTsinð = 2 Þ: ½ 2 : 111 By similar reasoning,
Rxð Þ¼Ecosð = 2 ÞiTsinð = 2 Þ: ½ 2 : 112 The rotation matrix corresponding to a pulse of flip angle, , applied
along thex-axis can now be calculated. The elements of the matrix
representations of the pulse rotation operatorsRx^1 ð Þ andRx( ) are
constructed from the basis eigenfunctions using the expressions
½Rx^1 ð Þrs¼hjfr Ecosð = 2 ÞþiTsinð = 2 Þgjis,
½Rxð Þrs¼hjfr Ecosð = 2 ÞiTsinð = 2 Þgjis:½ 2 : 113 For example, ifh 1 j¼h jandj 2 i¼j i, then matrix element½Rx^1 ð Þ 12 is
½Rx^1 ð Þ 12 ¼hjf Ecosð = 2 ÞþiTsinð = 2 Þg(^)
¼isinð = 2 Þ: ½ 2 : 114
The matrix representations of the pulse operators are
Rx^1 ð Þ¼ is ccis
and Rxð Þ¼ cis cis
, ½ 2 : 115
wherec¼cos( /2) ands¼sin( /2).
Similar analysis for a pulse with y-phase ( ¼/2) generates a
rotation matrix of the form
Ry^1 ð Þ¼ cs
sc
and Ryð Þ¼ c s
sc
: ½ 2 : 116
Finally a rotation about thez-axis (which in practice is difficult to
achieve experimentally with rf pulses) has the matrix representation
Rz^1 ð Þ¼
cþis 0
0 cis
and Rzð Þ¼
cis 0
0 cþis
:
½ 2 : 117
The rotation induced by the general Hamiltonian given by [2.97],
which includes off-resonance effects and arbitrary pulse phases, can be
written as
R ð ,Þ¼expði nIÞ¼Ecosð = 2 Þi 2 nIsinð = 2 Þ, ½ 2 : 118
2.3 PULSES ANDROTATIONOPERATORS 53