Basic Relaxation Function Formalism - Revision history

Jess at 19:29, 13 September 2022

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Jess at 19:26, 13 September 2022

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Jess at 19:26, 13 September 2022

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WikiSysop: Created page with "Relaxonomy --> here ---- If the muon spin polarization $\vec{P}$ is a function of time $t$ , then the '''''autocorrelation function''''' between the..."

2022-08-20T01:18:14Z

Created page with "Relaxonomy --> here ---- If the muon spin polarization <math>\vec{P}</math> is a function of time <math>t</math>, then the '''''autocorrelation function''''' between the..."

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[[Relaxonomy]] --> here
----

If the muon spin polarization <math>\vec{P}</math> is a function of time <math>t</math>, then the '''''autocorrelation function''''' between the initial <math>i^{\rm th}</math> component and the <math>j^{\rm th}</math> component at time <math>t</math> is given by

<center><math> G_{ij}(t) \; \equiv \; \langle P_i(0) \, P_j(t) \rangle \; . </math></center>

Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[http://musr.ca/lit/litNum.php?j=1314 Turner 1985]</sup> most formalism in ''µSR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}</math>, where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any. For the case of '''''zero magnetic field''''' ('''ZF'''), <math>z</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>x</math>.

For TF the muon polarization is often expressed as a complex quantity, <math>\tilde{P}(t) = P_x(t) + i P_y(t)</math> so that simple precession without relaxation can be written <math>\tilde{P}(t) = P \exp[i (\omega t + \phi)]</math> where <math>P = |\tilde{P}(o)|</math>, <math>\omega</math> is the precession frequency and <math>\phi</math> is the initial phase of the precession.

← Older revision		Revision as of 19:29, 13 September 2022
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	Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''µSR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}</math> , where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any. For the case of '''''zero magnetic field''''' ('''ZF'''), <math>\hat{z}</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>\hat{x}</math>.		Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''µSR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}</math> , where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any. For the case of '''''zero magnetic field''''' ('''ZF'''), <math>\hat{z}</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>\hat{x}</math>.

	For TF the muon polarization is often expressed as a complex quantity, <math>\tilde{P}(t) = P_x(t) + i P_y(t)</math> so that simple precession without relaxation can be written <math>\tilde{P}(t) = P \exp[i (\omega t + \phi)]</math> where <math>P = \|\tilde{P}(o)\|</math>, <math>\omega</math> is the precession frequency and <math>\phi</math> is the initial phase of the precession.		For TF the muon polarization is often expressed as a complex quantity, <math>\tilde{P}(t) = P_x(t) + i P_y(t)</math> so that simple precession without relaxation can be written <math>\tilde{P}(t) = P \exp[i (\omega t + \phi)]</math> where <math>P = \|\tilde{P}(0)\|</math>, <math>\omega</math> is the precession frequency and <math>\phi</math> is the initial phase of the precession.

@@ Line 6: / Line 6: @@
 <center><math> G_{ij}(t) \; \equiv \; \langle P_i(0) \, P_j(t) \rangle \; . </math></center>
-Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''&micro;SR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}</math> , where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any.  For the case of '''''zero magnetic field''''' ('''ZF'''), <math>z</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>x</math>.
+Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''&micro;SR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}</math> , where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any.  For the case of '''''zero magnetic field''''' ('''ZF'''), <math>\hat{z}</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>\hat{x}</math>.
 For TF the muon polarization is often expressed as a complex quantity, <math>\tilde{P}(t) = P_x(t) + i P_y(t)</math> so that simple precession without relaxation can be written <math>\tilde{P}(t) = P \exp[i (\omega t + \phi)]</math> where <math>P = |\tilde{P}(o)|</math>, <math>\omega</math> is the precession frequency and <math>\phi</math> is the initial phase of the precession.

@@ Line 6: / Line 6: @@
 <center><math> G_{ij}(t) \; \equiv \; \langle P_i(0) \, P_j(t) \rangle \; . </math></center>
-Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''&micro;SR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz }</math>, where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any.  For the case of '''''zero magnetic field''''' ('''ZF'''), <math>z</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>x</math>.
+Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''&micro;SR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}</math> , where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any.  For the case of '''''zero magnetic field''''' ('''ZF'''), <math>z</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>x</math>.
 For TF the muon polarization is often expressed as a complex quantity, <math>\tilde{P}(t) = P_x(t) + i P_y(t)</math> so that simple precession without relaxation can be written <math>\tilde{P}(t) = P \exp[i (\omega t + \phi)]</math> where <math>P = |\tilde{P}(o)|</math>, <math>\omega</math> is the precession frequency and <math>\phi</math> is the initial phase of the precession.

@@ Line 6: / Line 6: @@
 <center><math> G_{ij}(t) \; \equiv \; \langle P_i(0) \, P_j(t) \rangle \; . </math></center>
-Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''&micro;SR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}  </math>, where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any.  For the case of '''''zero magnetic field''''' ('''ZF'''), <math>z</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>x</math>.
+Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''&micro;SR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz }</math>, where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any.  For the case of '''''zero magnetic field''''' ('''ZF'''), <math>z</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>x</math>.
 For TF the muon polarization is often expressed as a complex quantity, <math>\tilde{P}(t) = P_x(t) + i P_y(t)</math> so that simple precession without relaxation can be written <math>\tilde{P}(t) = P \exp[i (\omega t + \phi)]</math> where <math>P = |\tilde{P}(o)|</math>, <math>\omega</math> is the precession frequency and <math>\phi</math> is the initial phase of the precession.

@@ Line 6: / Line 6: @@
 <center><math> G_{ij}(t) \; \equiv \; \langle P_i(0) \, P_j(t) \rangle \; . </math></center>
-Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''&micro;SR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}\!</math>, where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any.  For the case of '''''zero magnetic field''''' ('''ZF'''), <math>z</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>x</math>.
+Although there are interesting cases where <math>G_{ij}</math> is nonzero for <math>i \ne j</math>,<sup>[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.112 Turner 1985]</sup> most formalism in ''&micro;SR'' is focussed on the '''''transverse''''' relaxation function <math>G_{xx}</math> or the '''''longitudinal''''' relaxation function <math>G_{zz}  </math>, where the <math>x</math> and <math>z</math> directions are defined by the direction of the applied magnetic field <math>\vec{B} = B \hat{z}</math>, if any.  For the case of '''''zero magnetic field''''' ('''ZF'''), <math>z</math> is defined by the initial direction of the muon spin polarization; for '''''transverse magnetic field''''' ('''TF'''), <math>\vec{P}</math> is taken to be initially in the <math>x-y</math> plane, or usually along <math>x</math>.
 For TF the muon polarization is often expressed as a complex quantity, <math>\tilde{P}(t) = P_x(t) + i P_y(t)</math> so that simple precession without relaxation can be written <math>\tilde{P}(t) = P \exp[i (\omega t + \phi)]</math> where <math>P = |\tilde{P}(o)|</math>, <math>\omega</math> is the precession frequency and <math>\phi</math> is the initial phase of the precession.

Basic Relaxation Function Formalism - Revision history

Jess at 19:29, 13 September 2022

Jess at 19:28, 13 September 2022

Jess at 19:26, 13 September 2022

Jess at 19:26, 13 September 2022

Jess at 19:25, 13 September 2022

Jess at 19:24, 13 September 2022

Jess at 19:23, 13 September 2022

WikiSysop: Created page with "Relaxonomy --> here ---- If the muon spin polarization \vec{P} is a function of time t, then the '''''autocorrelation function''''' between the..."

WikiSysop: Created page with "Relaxonomy --> here ---- If the muon spin polarization $\vec{P}$ is a function of time $t$ , then the '''''autocorrelation function''''' between the..."