QMGreensite_merged

E/Nis a constant that does not affect the NMR experiment; accordingly,
this term is normally not written explicitly and the high-temperature
approximation to the equilibrium density operator is simply written in
terms of the Zeeman Hamiltonian as

eq¼
H=ðNkBTÞ¼

XM

k¼ 1

h! 0 k

NkBT

Ikz, ½ 2 : 126 Š

in whichhhas been included explicitly.
By convention, the complex magnetization recorded during the
acquisition period of an NMR experiment is given by ( 2 )

Mþ

ðÞ¼t N hTrðÞtFþ



, ½ 2 : 127 Š

in whichNis the number of spins per unit volume,

Fþ¼

XM

k¼ 1

Ikþ¼

XM

k¼ 1

IkxþiIky



: ½ 2 : 128 Š

The operatorF–could have been chosen as the observation operator
equally well.

2.4.2 THEONE-PULSEEXPERIMENT
The simplest NMR experiment consists of a single pulse followed by
acquisition of the FID. For a single spin, [2.126] indicates thateq/Iz.
The effect of a pulse withx-phase and a rotation angle pulse applied to
Izis calculated using [2.115] as

Rxð ÞIzR
x^1 ð Þ¼

1

2

c 
is


is c

"#

10

0 
 1

"#

cis

is c

"#

¼

1

2

cis


is 
c

"#

cis

is c

"#

¼

1

2

c^2 
s^22 ics


 2 ics s^2 
c^2

"#

¼

1

2

cos isin


isin 
cos

"#

¼Izcos 
Iysin , ½ 2 : 129 Š

56 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY