MATERIALES Y MÉTODOS
CARCINOGÉNESIS DE PIEL DE RATÓN.
2.2. Unión de factores nucleares al promotor de CD-E.
2.2.3. Análisis de los factores que interaccionan con la caja CCAAT.
Thus far, the arguments put forward have involved only a single, isolated nuclear spin. In a real system, one would expect around 1022 individual spins and in any NMR experiment, one would expect to observe the macroscopic magnetic properties of the sample caused by the superposition of individual magnetic moments from all of these nuclear spins. This macroscopic magnetisation is known as the bulk magnetisation of a sample and it turns out that bulk magnetisation possesses very similar properties to those of the classical individual nuclear spin. Let us consider an ensemble of spin-1/2 nuclei each of which obeys the semi-classical model. We will also assume that any interactions between these nuclei are negligible.
Recall that for spin-1/2 nuclei there are two accessible eigenstates of
I.
la) and I fi). Onapplication of a magnetic field, these two degenerate states undergo Zeeman splitting such that the energy difference between them is given by AE =
hyB
z as per Equation2.10. Assuming a state of thermal equilibrium, the relative populations of the two states are given by Boltzmann statistics, i.e.
N
f tA E )
[2.24]Na =
exp —k BT
Where
N
a is the number of spins in the la) state,N
I,
is the number of spins in the 1,3)state,
T is
the temperature andk
is the Boltzmann constant. Equation 2.24 clearly shows that an excess number of nuclear spins lie in the la) state i.e. precessing about the z-axis and aligned parallel to it. In the limit wherek
BT
>> AE , we can expand the exponentialkeeping only the first term such that:
yB, N hg3,
[2.25]N
a–N
f l=N
c r— —
k
BT
– 2k
BT
Where
N
represents the total number of spins in the spin system. This is known as the high temperature approximation and holds for most typical laboratory magnetic fields.The total bulk magnetisation
M
is given by the vector sum of magnetic moments of allthe individual spins in the system. Therefore the magnitude of the magnetisation parallel
to the static field (known as longitudinal magnetisation) is given by /14,
= N
ap,"
+ N
flpf ,
and sincep," = -4 = IA
, we can write:N
(hy)213,M, =
[2.26]4
kBT
This shows that the longitudinal magnetisation induced by placing nuclear spins in a magnetic field is directly proportional to the field strength. Magnetisation perpendicular to the applied field (known as transverse magnetisation) is dependent on phase coherence between precessing spins. Whilst each individual precessing nuclear magnetic moment encompasses a rotating transverse component of magnetisation, unless a significant fraction of these transverse components rotate in phase, then their vector sum will equal zero, and hence no observable transverse component of bulk magnetisation will exist. This is the case in the equilibrium state, and so if a sample is placed in a magnetic field and it remains unperturbed by rf radiation, no transverse component of magnetisation exists and the induced bulk magnetisation is given by
M = M
Zi .In applying the oscillating rf field, a change in the relative population of the
la)
and I fi)states, and thus a change in the longitudinal magnetisation
Af,
is induced. A secondaryeffect of the rf field is to introduce phase coherence between precessing spins. Whereas in the equilibrium state, no transverse magnetisation exists because all of the individual nuclear magnetic moments will have a random phase, following the application of an rf pulse, the induced phase coherence produces a finite component of transverse magnetisation that will rotate in the transverse plane at the Larrnor frequency (in the laboratory frame).
Mathematically, it can be shown that the expectation value of the bulk magnetisation
M
obeys the same classical equation as that used in section 2.3.1 to describe a single spin, i.e.
dM ,
=
x B)
dt
[2.27]Given an initial state of thermal equilibrium such that M(0)= Mzi, we can use the
equation of motion to derive the subsequent time evolution of bulk magnetisation. From the above calculations, we deduce that if an rf-field is applied at the Larmor frequency then M will be tipped away from the z-axis and a transverse component of magnetisation exists. Again the angle through which M is tipped is given by 0= Atp
where tp is the duration of the rf pulse. (Notice that this is equivalent to Equation 2.23,
except in this case, no restrictions on the angle are introduced by quantum mechanics since M is a macroscopic property and so does not obey the same quantum mechanical rules.) By the use of if-pulses of appropriate lengths, the angle through which M rotates can be controlled, and of particular interest are the special cases in which 0= 90°, and 0= 180°. In the case of the 90-degree pulse, the relative populations of the la) and 1/3) states are equalised such that no longitudinal magnetisation exists and the transverse component of M is a maximum. In the case of the 180-degree pulse, the equilibrium
magnetisation is inverted such that M = —Mzi and the transverse component is reduced
to zero. The effect of an rf pulse is depicted in Figure 2.6 for the special case where 0= 90°.
Figure 2.6: Motion of bulk magnetisation before, during and immediately after the application of a 0= 90° RF pulse. A) The equilibrium magnetisation aligned parallel to the z-direction and hence the applied static field. B) Motion of bulk magnetisation during application of the RF pulse. Magnetisation is shown in the rotating frame and the pulse causes a rotation about the x'-axis into the transverse plane. C) Bulk magnetisation immediately after application of an RF pulse shown in the laboratory frame.
x