Difference between revisions of "- Spin-Spin Interactions"
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===Spin-Spin Interactions=== | ===Spin-Spin Interactions=== | ||
| − | Previously, we discussed spin-lattice interactions and found that it led to the concept of a ''longitudinal relaxation time'' <math>T_{1}=\ | + | Previously, we discussed spin-lattice interactions and found that it led to the concept of a ''longitudinal relaxation time'' <math>T_{1}=\tfrac{1}{2W}</math>, where <math>W</math> is the average of transition rates out of the upper and lower energy levels. That is, the spin temperature <math>T_{_S}</math> will approach the lattice temperature <math>T</math> exponentially with a characteristic time <math>T_{1}</math>. |
We now concern ourselves with spin-spin interactions. | We now concern ourselves with spin-spin interactions. | ||
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Since each nucleus possesses a small magnetic dipole moment, there will be a magnetic dipole-dipole interaction between each pair of nuclei. That is, each nuclear magnet finds itself not only in the applied steady magnetic field <math>H_{_0}</math> but also in the small local magnetic field <math>H_{_{loc}}</math> produced by its neighboring nuclear magnets. The direction of the local field differs from nucleus to nucleus depending on the spatial configuration of the neighboring nuclei in the lattice and on their magnetic quantum number <math>m</math>. | Since each nucleus possesses a small magnetic dipole moment, there will be a magnetic dipole-dipole interaction between each pair of nuclei. That is, each nuclear magnet finds itself not only in the applied steady magnetic field <math>H_{_0}</math> but also in the small local magnetic field <math>H_{_{loc}}</math> produced by its neighboring nuclear magnets. The direction of the local field differs from nucleus to nucleus depending on the spatial configuration of the neighboring nuclei in the lattice and on their magnetic quantum number <math>m</math>. | ||
| − | The magnetic field of a magnetic dipole of moment <math>\mu</math> at a distance <math>r</math> is of the order <math>\ | + | The magnetic field of a magnetic dipole of moment <math>\mu</math> at a distance <math>r</math> is of the order <math>\tfrac{\mu}{r^3}</math>. The field therefore falls off rapidly with increasing <math>r</math>, so that only nearest neighbors make important contributions to <math>H_{_{loc}}</math>. For a rough estimate of the local field let's take <math>\mu=\mu_{_0}</math> and <math>r=1\AA</math>. With this we find that <math>H_{_{loc}}\approx \tfrac{\mu_{_0}}{r^3}\approx \text{5 Gauss}</math>. |
The total magnetic field <math>H_{_{tot}}=H_{_0}+H_{_{loc}}</math> will not be the same for each nucleus, but will vary over a range of several Gauss from one nucleus to the next. This implies that the resonance condition will not be perfectly sharp. Instead, the energy levels will be broadened by an amount of order <math>g\mu_{_0}H_{_{loc}}</math>. If we have a fixed transverse RF field at <math>\nu_{_0}</math> the resonance will be found to be spread about <math>H_{_0}</math> over a range of values of the order <math>H_{_{loc}}</math>. This process is referred to as ''inhomogeneous broadening''. | The total magnetic field <math>H_{_{tot}}=H_{_0}+H_{_{loc}}</math> will not be the same for each nucleus, but will vary over a range of several Gauss from one nucleus to the next. This implies that the resonance condition will not be perfectly sharp. Instead, the energy levels will be broadened by an amount of order <math>g\mu_{_0}H_{_{loc}}</math>. If we have a fixed transverse RF field at <math>\nu_{_0}</math> the resonance will be found to be spread about <math>H_{_0}</math> over a range of values of the order <math>H_{_{loc}}</math>. This process is referred to as ''inhomogeneous broadening''. | ||
| − | Since the total magnetic field differs from nucleus to nucleus, there will be a distribution of frequencies of their Larmor precessions (<math>\omega_{_0}=\gamma H_{_0}</math>) covering a range <math>\delta\omega_{_0}\approx \gamma H_{_{loc}}\approx\ | + | Since the total magnetic field differs from nucleus to nucleus, there will be a distribution of frequencies of their Larmor precessions (<math>\omega_{_0}=\gamma H_{_0}</math>) covering a range <math>\delta\omega_{_0}\approx \gamma H_{_{loc}}\approx\tfrac{\mu_{_0}^2}{\hbar r^3}\approx 10^4 s^{-1}</math>. |
| − | If two spins have precession frequencies differing by <math>\delta\omega_{_0}</math> and are initially in phase, then they will be out of phase in a time <math>\thicksim \ | + | If two spins have precession frequencies differing by <math>\delta\omega_{_0}</math> and are initially in phase, then they will be out of phase in a time <math>\thicksim \tfrac{1}{\delta\omega_{_0}}\thicksim 10^{-4} s </math>. |
We have been discussing a physical mechanism by which the nuclear spins interact with each other. Namely, nearest-neighbor-induced local magnetic fields. There is another physical mechanism that can also be important. That is, consider two spins <math>j</math> and <math>k</math>. In a steady magnetic field both spins will precess about <math>H_{_0}</math> and produce oscillating (at the Larmor frequency) magnetic fields. If spin <math>j</math> produces an oscillating field at spin <math>k\text{'s}</math> position it may induce a transition of spin <math>k</math>. The energy for this transition comes from spin <math>j</math> so that there is a mutual energy exchange in the process. Since the relative phases of the two spins change in a time of the order of <math>\tfrac{1}{\delta\omega_{_0}}</math>, the correct phasing for this spin-spin exchange process should occur after a time interval of this order and this in turn should determine the lifetime of the spin state. It follows that this spin-spin energy exchange process further broadens the resonance line by an amount of the order of <math>H_{_{loc}}</math>. | We have been discussing a physical mechanism by which the nuclear spins interact with each other. Namely, nearest-neighbor-induced local magnetic fields. There is another physical mechanism that can also be important. That is, consider two spins <math>j</math> and <math>k</math>. In a steady magnetic field both spins will precess about <math>H_{_0}</math> and produce oscillating (at the Larmor frequency) magnetic fields. If spin <math>j</math> produces an oscillating field at spin <math>k\text{'s}</math> position it may induce a transition of spin <math>k</math>. The energy for this transition comes from spin <math>j</math> so that there is a mutual energy exchange in the process. Since the relative phases of the two spins change in a time of the order of <math>\tfrac{1}{\delta\omega_{_0}}</math>, the correct phasing for this spin-spin exchange process should occur after a time interval of this order and this in turn should determine the lifetime of the spin state. It follows that this spin-spin energy exchange process further broadens the resonance line by an amount of the order of <math>H_{_{loc}}</math>. | ||
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<math>\int_0^{\infty}g(\nu)d\nu=1</math>. Since the integral is unitless the integrand must be in units of time. The maximum value of the lineshape function <math>g(\nu)_{max}</math> will be large for narrow lines and small for broad lines. The quantity <math>\tfrac{1}{g(\nu)_{max}}</math> is then a rough guide to the width of the line. Recall that the linewidth as a function of frequency is of the order <math>\delta\omega_{_0}</math>. Consequently, the spin-spin interaction time <math>T_{2}</math> is of the order <math>g(\nu)_{max}</math>. | <math>\int_0^{\infty}g(\nu)d\nu=1</math>. Since the integral is unitless the integrand must be in units of time. The maximum value of the lineshape function <math>g(\nu)_{max}</math> will be large for narrow lines and small for broad lines. The quantity <math>\tfrac{1}{g(\nu)_{max}}</math> is then a rough guide to the width of the line. Recall that the linewidth as a function of frequency is of the order <math>\delta\omega_{_0}</math>. Consequently, the spin-spin interaction time <math>T_{2}</math> is of the order <math>g(\nu)_{max}</math>. | ||
| − | We can now make a precise definition of the spin-spin transverse relaxation time <math>T_{2}=\ | + | We can now make a precise definition of the spin-spin transverse relaxation time <math>T_{2}=\tfrac{g(\nu)_{max}}{2}</math>. |
<math>T_{2}</math> is connected only with the peak value of the normalized lineshape curve. | <math>T_{2}</math> is connected only with the peak value of the normalized lineshape curve. | ||
Latest revision as of 19:33, 13 February 2019
Spin-Spin Interactions
Previously, we discussed spin-lattice interactions and found that it led to the concept of a longitudinal relaxation time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{1}=\tfrac{1}{2W}} , where is the average of transition rates out of the upper and lower energy levels. That is, the spin temperature will approach the lattice temperature exponentially with a characteristic time .
We now concern ourselves with spin-spin interactions.
Since each nucleus possesses a small magnetic dipole moment, there will be a magnetic dipole-dipole interaction between each pair of nuclei. That is, each nuclear magnet finds itself not only in the applied steady magnetic field but also in the small local magnetic field produced by its neighboring nuclear magnets. The direction of the local field differs from nucleus to nucleus depending on the spatial configuration of the neighboring nuclei in the lattice and on their magnetic quantum number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} .
The magnetic field of a magnetic dipole of moment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} at a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is of the order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{\mu}{r^3}} . The field therefore falls off rapidly with increasing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} , so that only nearest neighbors make important contributions to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_{loc}}} . For a rough estimate of the local field let's take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu=\mu_{_0}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1\AA} . With this we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_{loc}}\approx \tfrac{\mu_{_0}}{r^3}\approx \text{5 Gauss}} .
The total magnetic field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_{tot}}=H_{_0}+H_{_{loc}}} will not be the same for each nucleus, but will vary over a range of several Gauss from one nucleus to the next. This implies that the resonance condition will not be perfectly sharp. Instead, the energy levels will be broadened by an amount of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\mu_{_0}H_{_{loc}}} . If we have a fixed transverse RF field at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{_0}} the resonance will be found to be spread about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_0}} over a range of values of the order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_{loc}}} . This process is referred to as inhomogeneous broadening.
Since the total magnetic field differs from nucleus to nucleus, there will be a distribution of frequencies of their Larmor precessions (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{_0}=\gamma H_{_0}} ) covering a range Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta\omega_{_0}\approx \gamma H_{_{loc}}\approx\tfrac{\mu_{_0}^2}{\hbar r^3}\approx 10^4 s^{-1}} .
If two spins have precession frequencies differing by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta\omega_{_0}} and are initially in phase, then they will be out of phase in a time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \thicksim \tfrac{1}{\delta\omega_{_0}}\thicksim 10^{-4} s } .
We have been discussing a physical mechanism by which the nuclear spins interact with each other. Namely, nearest-neighbor-induced local magnetic fields. There is another physical mechanism that can also be important. That is, consider two spins Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} . In a steady magnetic field both spins will precess about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_0}} and produce oscillating (at the Larmor frequency) magnetic fields. If spin Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} produces an oscillating field at spin Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\text{'s}} position it may induce a transition of spin Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} . The energy for this transition comes from spin Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} so that there is a mutual energy exchange in the process. Since the relative phases of the two spins change in a time of the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{\delta\omega_{_0}}} , the correct phasing for this spin-spin exchange process should occur after a time interval of this order and this in turn should determine the lifetime of the spin state. It follows that this spin-spin energy exchange process further broadens the resonance line by an amount of the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_{loc}}} .
These two phase disturbing and line broadening processes are only both present when identical nuclei are concerned. For a system of non-identical nuclei, the local field effect is still present but the spin-exchange process is absent (Larmor precession frequencies for different nuclei are quite different).
It is convenient to introduce a spin-spin interaction time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2}} to describe the lifetime or phase-memory time of a nuclear spin state where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2}\thicksim\tfrac{1}{\delta\omega_{_0}}\thicksim 10^{-4}} sec. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2}} is called the transverse relaxation time.
Aside: The reason for naming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{1}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2}} as the longitudinal and transverse relaxation times will become apparent later on.
Note: For liquids and gases, the reorientation and diffusion of the lattice molecules is usually so rapid that the local magnetic field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_{loc}}} is smoothed out to a very small average value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{H_{_{loc}}}} , yielding a quite narrow resonance line.
Although we have discussed here only the basic broadening mechanisms due to spin-spin interactions, we should not overlook other common sources of broadening (listed below).
1) Non-uniformity of the applied magnetic field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_0}} over the assembly of spins.
2) A very short spin-lattice relaxation time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{1}} .
3) Electric quadrapole interaction if the spin number exceeds 1/2.
Finally, let's develop a better definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2}} . Consider the absorption lineshape as a function of frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(\nu)} . We would want this to be normalized such that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^{\infty}g(\nu)d\nu=1} . Since the integral is unitless the integrand must be in units of time. The maximum value of the lineshape function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(\nu)_{max}} will be large for narrow lines and small for broad lines. The quantity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{g(\nu)_{max}}} is then a rough guide to the width of the line. Recall that the linewidth as a function of frequency is of the order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta\omega_{_0}} . Consequently, the spin-spin interaction time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2}} is of the order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(\nu)_{max}} .
We can now make a precise definition of the spin-spin transverse relaxation time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2}=\tfrac{g(\nu)_{max}}{2}} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{2}} is connected only with the peak value of the normalized lineshape curve.
