Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S10 / 3

Take room temperature as $T=300\mathrm{K}$. With the supplied gyromagnetic ratio, reduced Planck constant and Boltzmann constant,soAt $293\mathrm{K}$ the ratio is $1.05\times 10^{-5}$; the room-temperature convention makes only a small difference. The corresponding fractional nuclear spin polarization has magnitude about five parts per million. The source's negative sign convention assigns the label up to the higher-energy state; it should not be confused with the lower-energy magnetic moment aligned with a positive proton magnetic field.

Write $x=\Delta E/(k_{\mathrm{B}}T)$ and $r=N_{\uparrow }/N_{\downarrow }=e^{-x}$. Since $N=N_{\uparrow }+N_{\downarrow }$, eliminate the individual populations:The Taylor expansion $e^{-x}=1-x+O(x^2)$ gives the same result, and the odd expansion of the hyperbolic tangent yieldsEquivalently, $N_{\uparrow }\simeq (N/2)(1-x/2)$ and $N_{\downarrow }\simeq (N/2)(1+x/2)$. Their sum is still $N$. The negative sign follows directly from the stated higher-energy up-state population, and does not mean a negative magnitude of the equilibrium magnetization.

There are three physical controls: alter the energy splitting, alter the temperature, or drive the spin populations away from thermal equilibrium.

Increasing the static magnetic flux density increases $\Delta E=\hslash \gamma B$, and hence the equilibrium nuclear spin polarization approximately in proportion to $B/T$. High-field magnetic resonance imaging exploits this for greater available magnetization and potentially finer spatial or temporal resolution. The disadvantages include more demanding magnets, stronger susceptibility-related distortions, and harder radio frequency field uniformity and power management. A direct experimental comparison of head imaging at 7 T and 4 T measured both the signal gain and the RF penalties.^[ref]

Lowering the spin sample's temperature increases its equilibrium polarization at fixed $B$. Cryogenic preparation of an imaging agent can exploit this, often assisted by dynamic nuclear polarization, followed by rapid dissolution and transfer for hyperpolarized magnetic resonance imaging. This offers a much larger signal from dilute nuclei than room-temperature equilibrium, but requires preparation hardware, rapid transfer and acquisition before spin-lattice relaxation erases the polarization. Cooling an RF receiver reduces electronic noise, but does not itself change the nuclei's population ratio; nor does this argument propose cooling living tissue to cryogenic temperatures. The demonstrated dissolution method combines low temperature, magnetic field and microwave-assisted polarization transfer, so its gain must not be attributed to cooling alone.^[ref]

Driving transitions with a resonant radio frequency field changes the longitudinal populations without changing $B$ or the bath temperature. A $\pi$ pulse interchanges the populations; saturation equalizes them. Inversion recovery then measures or exploits their return toward equilibrium, $M_z(t)=M_0(1-2e^{-t/T_1})$ after an ideal inversion, providing tissue contrast and signal nulling. Advantages are selective contrast and control within an ordinary scanner; costs include reduced signal, extra acquisition time, RF energy deposition and sensitivity to pulse errors. Such RF pulses alone cannot create a polarization magnitude greater than the available equilibrium value. Genuine nonthermal enhancement instead transfers polarization from another spin reservoir, for example electron spins in dynamic nuclear polarization or optically polarized noble gases in lung imaging; its transient and preparation constraints differ from ordinary RF inversion.^[ref]