- Saturation - Revision history

Bsboggs at 19:27, 18 February 2019

2019-02-18T19:27:36Z

← Older revision		Revision as of 19:27, 18 February 2019
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	<math>T_{_S}=T\tfrac{n_{_0}}{n_{_{ss}}}=\frac{T}{Z}</math>.		<math>T_{_S}=T\tfrac{n_{_0}}{n_{_{ss}}}=\frac{T}{Z}</math>.

	The spins can easily be heated up to extremely high temperatures. For example, at ~~just below~~ room temperature for <math>^1H</math>, a transverse field of 0.1 Gauss, <math>T_{_1}=1 \text{sec}</math> (cold water) and <math>T_{_2}=10^{-4}</math> seconds, <math>Z\approx 0.05</math> giving a spin temperature <math>T_{_S}=T/Z\approx5400K</math>.		The spins can easily be heated up to extremely high temperatures. For example, at room temperature for <math>^1H</math>, a transverse field of 0.1 Gauss, <math>T_{_1}=1 \text{sec}</math> (cold water) and <math>T_{_2}=10^{-4}</math> seconds, <math>Z\approx 0.05</math> giving a spin temperature <math>T_{_S}=T/Z\approx5400K</math>.

			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 [https://en.wikipedia.org/wiki/Erg 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 <math>e^{\tfrac{-t}{T_{_1}Z}}</math>. That is, the return to equilibrium of a driven spin system occurs exponentially (as seen with the undriven case) with a characteristic time of <math>T_{_1}Z</math> rather than just <math>T_{_1}</math>.

Bsboggs at 19:07, 18 February 2019

2019-02-18T19:07:05Z

← Older revision		Revision as of 19:07, 18 February 2019
Line 28:		Line 28:
	<math>T_{_S}=T\tfrac{n_{_0}}{n_{_{ss}}}=\frac{T}{Z}</math>.		<math>T_{_S}=T\tfrac{n_{_0}}{n_{_{ss}}}=\frac{T}{Z}</math>.

	The spins can easily be heated up to extremely high temperatures. For example, at room temperature for <math>^1H</math>, a transverse field of 0.1 Gauss, <math>T_{_1}=1 \text{sec}</math> (cold water) and <math>T_{_2}=10^{-4}</math> seconds, <math>Z\approx 0.05</math> giving a spin temperature <math>T_{_S}=T/Z\approx5400K</math>.		The spins can easily be heated up to extremely high temperatures. For example, at just below room temperature for <math>^1H</math>, a transverse field of 0.1 Gauss, <math>T_{_1}=1 \text{sec}</math> (cold water) and <math>T_{_2}=10^{-4}</math> seconds, <math>Z\approx 0.05</math> giving a spin temperature <math>T_{_S}=T/Z\approx5400K</math>.

Bsboggs at 23:59, 15 February 2019

2019-02-15T23:59:16Z

← Older revision		Revision as of 23:59, 15 February 2019
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	===Saturation===		===Saturation===
	Previously, we considered the interaction between the spins and lattice and found the characteristic relaxation time <math>T_1</math>. That is, when the spin temperature <math>T_{_S}</math> is ~~greater~~ than the lattice temperature <math>T</math> how long does it take for the spin temperature to return to the lattice temperature? Answer: <math>T_1</math>.		Previously, we considered the interaction between the spins and lattice and found the characteristic relaxation time <math>T_1</math>. That is, when the spin temperature <math>T_{_S}</math> is different than the lattice temperature <math>T</math> how long does it take for the spin temperature to return to the lattice temperature? Answer: <math>T_1</math>.

	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 <math>T_2</math>. 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: <math>T_2</math>.		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 <math>T_2</math>. 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: <math>T_2</math>.
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	We now investigate the ''steady-state behavior'' of the spin-lattice system with an applied transverse RF field.		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 <math>n</math>, the excess number of nuclei in the lower state is <math>\tfrac{dn}{dt}=\tfrac{n_{_0}-n}{T_{_1}}</math>. This makes sense because if <math>T_S>T</math> then <math>\tfrac{dn}{dt}</math> is greater than zero (<math>n<n_{_0}</math>) and there's a net flow of nuclei into the lower state with a characteristic time constant <math>T_{_1}</math>.		In the absence of the transverse RF field, the differential equation describing the time variation of <math>n</math>, the excess number of nuclei in the lower state, is <math>\tfrac{dn}{dt}=\tfrac{n_{_0}-n}{T_{_1}}</math>. This makes sense because if <math>T_S>T</math> then <math>\tfrac{dn}{dt}</math> is greater than zero (<math>n<n_{_0}</math>) and there's a net flow of nuclei into the lower state with a characteristic time constant <math>T_{_1}</math>.

	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 <math>\tfrac{dn}{dt}=\tfrac{n_{_0}-n}{T_{_1}}-2nP</math>, where <math>P</math> is the probability per unit time that a nuclei makes an upward transition. A steady state is reached when <math>\tfrac{dn}{dt}=0</math> so that a steady-state value of <math>n_{_{ss}}</math> is given by <math>\tfrac{n_{_{ss}}}{n_{_0}}=\tfrac{1}{1+2PT_{_1}}</math>.		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 <math>\tfrac{dn}{dt}=\tfrac{n_{_0}-n}{T_{_1}}-2nP</math>, where <math>P</math> is the probability per unit time that a nuclei makes an upward transition. A steady state is reached when <math>\tfrac{dn}{dt}=0</math> so that a steady-state value of <math>n_{_{ss}}</math> is given by <math>\tfrac{n_{_{ss}}}{n_{_0}}=\tfrac{1}{1+2PT_{_1}}</math>. It can be seen that <math>n_{_{ss}}</math> decreases as <math>P</math> or <math>T_{_1}</math> increases.

	we must now evaluate the transition probability <math>P</math>. 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		We must now evaluate the transition probability <math>P</math>. 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

	<math>P_{_{m\rightarrow m'}}=\tfrac{1}{2}\gamma^2H_{_1}^1\|<m\|I\|m'>\|^2g(\nu)</math>, where <math><m\|I\|m'></math> is the appropriate matrix element of the nuclear spin operator.		<math>P_{_{m\rightarrow m'}}=\tfrac{1}{2}\gamma^2H_{_1}^1\|<m\|I\|m'>\|^2g(\nu)</math>, where <math><m\|I\|m'></math> is the appropriate matrix element of the nuclear spin operator.
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	<math>T_{_S}=T\tfrac{n_{_0}}{n_{_{ss}}}=\frac{T}{Z}</math>.		<math>T_{_S}=T\tfrac{n_{_0}}{n_{_{ss}}}=\frac{T}{Z}</math>.

			The spins can easily be heated up to extremely high temperatures. For example, at room temperature for <math>^1H</math>, a transverse field of 0.1 Gauss, <math>T_{_1}=1 \text{sec}</math> (cold water) and <math>T_{_2}=10^{-4}</math> seconds, <math>Z\approx 0.05</math> giving a spin temperature <math>T_{_S}=T/Z\approx5400K</math>.

Bsboggs: /* Saturation */

2019-02-15T20:19:29Z

Saturation

← Older revision		Revision as of 20:19, 15 February 2019
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	We now investigate the ''steady-state behavior'' of the spin-lattice system with an applied transverse RF field.		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 <math>n</math>, the excess number of nuclei in the lower state is <math>\tfrac{dn}{dt}=\tfrac{n_{_0}-n}{T_{_1}}</math>. This makes sense because if <math>T_S>T</math> then <math>\tfrac{dn}{dt}</math> is greater than zero (<math>n<n_{_0}</math>) and there's a net flow of nuclei into the lower state with a characteristic time constant <math>T_{_1}</math>.

			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 <math>\tfrac{dn}{dt}=\tfrac{n_{_0}-n}{T_{_1}}-2nP</math>, where <math>P</math> is the probability per unit time that a nuclei makes an upward transition. A steady state is reached when <math>\tfrac{dn}{dt}=0</math> so that a steady-state value of <math>n_{_{ss}}</math> is given by <math>\tfrac{n_{_{ss}}}{n_{_0}}=\tfrac{1}{1+2PT_{_1}}</math>.

			we must now evaluate the transition probability <math>P</math>. 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

			<math>P_{_{m\rightarrow m'}}=\tfrac{1}{2}\gamma^2H_{_1}^1\|<m\|I\|m'>\|^2g(\nu)</math>, where <math><m\|I\|m'></math> is the appropriate matrix element of the nuclear spin operator.

			For <math>m'=m-1</math> it can be shown that <math>\|<m\|I\|m'>\|^2=\tfrac{1}{2}(I+m)(I-m+1)</math>. For the case of <math>I=\tfrac{1}{2}</math> this reduces to

			<math>P=\tfrac{1}{4}\gamma^2H_{_1}^2g(\nu)</math> so that we have

			<math>\tfrac{n_{_{ss}}}{n_{_0}}=[1+\tfrac{1}{2}\gamma^2H_{_1}^2T_{_1}g(\nu)]^{-1}\equiv Z</math>.

			If an RF field is applied, whose amplitude <math>H_{_1}</math> is large, <math>\tfrac{n_{_{ss}}}{n_{_0}}</math> becomes quite small; the spin temperature <math>T_{_S}</math> becomes very high and the spin system is said to be ''saturated''. Remembering that <math>T_{_2}=\tfrac{1}{2}g(\nu)_{_{max}}</math> we can write

			<math>\tfrac{n_{_{ss}}}{n_{_0}}=[1+\gamma^2H_{_1}^2T_{_1}T_{_2}]^{-1}=Z_{_0}</math>, where <math>Z_{_0}</math> is the value of the saturation factor at the maximum of the lineshape function <math>g(\nu)</math>.

			Recall that, in thermal equilibrium, the lattice temperature <math>T</math> is defined in terms of <math>n_{_0}</math> as <math>T=\tfrac{N\mu H_{_0}}{n_{_0}k}</math>. Similarly, when the system is not in thermal equilibrium, the spin temperature <math>T_{_S}</math> is related to <math>n</math> as <math>T_{_S}=\tfrac{N\mu H_{_0}}{kn}</math>. These two results yield

			<math>T_{_S}=T\tfrac{n_{_0}}{n_{_{ss}}}=\frac{T}{Z}</math>.

Bsboggs: Created page with "===Saturation=== Previously, we considered the interaction between the spins and lattice and found the characteristic relaxation time $T_1$ . That is, when the spin..."

2019-02-14T20:11:52Z

Created page with "===Saturation=== Previously, we considered the interaction between the spins and lattice and found the characteristic relaxation time <math>T_1</math>. That is, when the spin..."

New page

===Saturation===
Previously, we considered the interaction between the spins and lattice and found the characteristic relaxation time <math>T_1</math>. That is, when the spin temperature <math>T_{_S}</math> is greater than the lattice temperature <math>T</math> how long does it take for the spin temperature to return to the lattice temperature? Answer: <math>T_1</math>.

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 <math>T_2</math>. 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: <math>T_2</math>.

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

- Saturation - Revision history

Bsboggs at 19:27, 18 February 2019

Bsboggs at 19:07, 18 February 2019

Bsboggs at 23:59, 15 February 2019

Bsboggs: /* Saturation */

Bsboggs: Created page with "===Saturation=== Previously, we considered the interaction between the spins and lattice and found the characteristic relaxation time T_1. That is, when the spin..."

Bsboggs: Created page with "===Saturation=== Previously, we considered the interaction between the spins and lattice and found the characteristic relaxation time $T_1$ . That is, when the spin..."