- Saturation

Saturation

Previously, we considered the interaction between the spins and lattice and found the characteristic relaxation time $T_{1}$. That is, when the spin temperature $T_{_{S}}$ is different than the lattice temperature $T$ how long does it take for the spin temperature to return to the lattice temperature? Answer: $T_{1}$.

We then discussed spin-spin interactions which led to the notion of inhomogeneous broadening by a spatially-varying local magnetic field. This led to the concept of a phase-memory time constant $T_{2}$. That is, when two nearly-resonant spins are excited then allowed to freely decay (RF field off) how long before the two spins are out of phase? Answer: $T_{2}$.

We now investigate the steady-state behavior of the spin-lattice system with an applied transverse RF field.

In the absence of the transverse RF field, the differential equation describing the time variation of $n$, the excess number of nuclei in the lower state, is ${\tfrac {dn}{dt}}={\tfrac {n_{_{0}}-n}{T_{_{1}}}}$. This makes sense because if $T_{S}>T$ then ${\tfrac {dn}{dt}}$ is greater than zero ($n<n_{_{0}}$) and there's a net flow of nuclei into the lower state with a characteristic time constant $T_{_{1}}$.

When the transverse RF field is present, another term must be added to account for the induced upward transitions corresponding to a net absorption of energy so that we get ${\tfrac {dn}{dt}}={\tfrac {n_{_{0}}-n}{T_{_{1}}}}-2nP$, where $P$ is the probability per unit time that a nuclei makes an upward transition. A steady state is reached when ${\tfrac {dn}{dt}}=0$ so that a steady-state value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{_{ss}}} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{n_{_{ss}}}{n_{_0}}=\tfrac{1}{1+2PT_{_1}}} . It can be seen that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{_{ss}}} decreases as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_1}} increases.

We must now evaluate the transition probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} . If a transverse RF field is applied with the correct sense of rotation at the resonant frequency the probability of a transition in unit time between the states is given (with the help of Fermi's golden rule) by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{_{m\rightarrow m'}}=\tfrac{1}{2}\gamma^2H_{_1}^1|<m|I|m'>|^2g(\nu)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <m|I|m'>} is the appropriate matrix element of the nuclear spin operator.

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m'=m-1} it can be shown that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |<m|I|m'>|^2=\tfrac{1}{2}(I+m)(I-m+1)} . For the case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I=\tfrac{1}{2}} this reduces to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=\tfrac{1}{4}\gamma^2H_{_1}^2g(\nu)} so that we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{n_{_{ss}}}{n_{_0}}=[1+\tfrac{1}{2}\gamma^2H_{_1}^2T_{_1}g(\nu)]^{-1}\equiv Z} .

If an RF field is applied, whose amplitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_1}} is large, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{n_{_{ss}}}{n_{_0}}} becomes quite small; the spin temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_S}} becomes very high and the spin system is said to be saturated. Remembering that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_2}=\tfrac{1}{2}g(\nu)_{_{max}}} we can write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{n_{_{ss}}}{n_{_0}}=[1+\gamma^2H_{_1}^2T_{_1}T_{_2}]^{-1}=Z_{_0}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{_0}} is the value of the saturation factor at the maximum of the lineshape function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(\nu)} .

Recall that, in thermal equilibrium, the lattice temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is defined in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{_0}} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=\tfrac{N\mu H_{_0}}{n_{_0}k}} . Similarly, when the system is not in thermal equilibrium, the spin temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_S}} is related to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_S}=\tfrac{N\mu H_{_0}}{kn}} . These two results yield

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_S}=T\tfrac{n_{_0}}{n_{_{ss}}}=\frac{T}{Z}} .

The spins can easily be heated up to extremely high temperatures. For example, at room temperature for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^1H} , a transverse field of 0.1 Gauss, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_1}=1 \text{sec}} (cold water) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_2}=10^{-4}} seconds, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z\approx 0.05} giving a spin temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_S}=T/Z\approx5400K} .

These high spin temperatures don't require the absorption of much energy from the transverse field. The amount of energy required to raise a spin system, similar to that discussed above, to infinity (the excess number of spins in the lower state equal to zero) is approximately 1 erg or 100 nJ and (from the Wikipedia link above on the Erg) is about the energy a housefly expends doing one pushup.

The rate of approach of a spin system to the steady state while experiencing a steady RF transverse field is described by a solution having the factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\tfrac{-t}{T_{_1}Z}}} . That is, the return to equilibrium of a driven spin system occurs exponentially (as seen with the undriven case) with a characteristic time of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_1}Z} rather than just Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{_1}} .