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Deprecated: Return type of MediaWiki\Session\Session::current() should either be compatible with Iterator::current(): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /usr/share/mediawiki/includes/session/Session.php on line 599
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Deprecated: Return type of MediaWiki\Session\Session::rewind() should either be compatible with Iterator::rewind(): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /usr/share/mediawiki/includes/session/Session.php on line 617
Deprecated: Return type of MediaWiki\Session\Session::offsetExists($offset) should either be compatible with ArrayAccess::offsetExists(mixed $offset): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /usr/share/mediawiki/includes/session/Session.php on line 633
Deprecated: Return type of & MediaWiki\Session\Session::offsetGet($offset) should either be compatible with ArrayAccess::offsetGet(mixed $offset): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /usr/share/mediawiki/includes/session/Session.php on line 646
Deprecated: Return type of MediaWiki\Session\Session::offsetSet($offset, $value) should either be compatible with ArrayAccess::offsetSet(mixed $offset, mixed $value): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /usr/share/mediawiki/includes/session/Session.php on line 656
Deprecated: Return type of MediaWiki\Session\Session::offsetUnset($offset) should either be compatible with ArrayAccess::offsetUnset(mixed $offset): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in /usr/share/mediawiki/includes/session/Session.php on line 661
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Warning: Cannot modify header information - headers already sent by (output started at /usr/share/mediawiki/includes/session/Session.php:48) in /usr/share/mediawiki/includes/changes/ChannelFeed.php on line 95
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Warning: Cannot modify header information - headers already sent by (output started at /usr/share/mediawiki/includes/session/Session.php:48) in /usr/share/mediawiki/includes/WebResponse.php on line 74 http://localhost/mediawiki/index.php?action=history&feed=atom&title=Dynamic_RelaxationDynamic Relaxation - Revision history2025-12-15T10:53:38ZRevision history for this page on the wikiMediaWiki 1.35.6 Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=836&oldid=prevJess at 18:27, 17 September 20222022-09-17T18:27:34Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:27, 17 September 2022</td>
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=825&oldid=prevJess at 17:10, 15 September 20222022-09-15T17:10:29Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:10, 15 September 2022</td>
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<td class="diff-context diff-side-deleted"><div>Of course, the assumptions that go into calculating <math>g(\Delta, t)</math> are idealizations of the real field distributions at lattice sites and/or the actual processes involved in muon "hopping" and field "fluctuations". They also omit all effects of ''quantum entanglement'' between the muon spin and the nearby nuclear or electronic spins, which need not remain static during their dipolar or hyperfine interactions with the muon. As has been demonstrated in copper crystals and (more dramatically) in fluoride crystals, such coupled spin systems often evolve together to produce "static" <math>g(\Delta, t)</math> functions bearing little resemblance to that described above. </div></td>
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<td class="diff-context diff-side-added"><div>Of course, the assumptions that go into calculating <math>g(\Delta, t)</math> are idealizations of the real field distributions at lattice sites and/or the actual processes involved in muon "hopping" and field "fluctuations". They also omit all effects of ''quantum entanglement'' between the muon spin and the nearby nuclear or electronic spins, which need not remain static during their dipolar or hyperfine interactions with the muon. As has been demonstrated in copper crystals and (more dramatically) in fluoride crystals, such coupled spin systems often evolve together to produce "static" <math>g(\Delta, t)</math> functions bearing little resemblance to that described above. </div></td>
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<td class="diff-deletedline diff-side-deleted"><div>Fortunately, Kehr's recursion relation usually works (numerically) on those "static" functions as well, although of course it is still an approximation.</div></td>
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<td class="diff-addedline diff-side-added"><div>Fortunately, Kehr's recursion relation usually works (numerically) on those "static" functions as well, although of course it is still an approximation.<ins class="diffchange diffchange-inline"> </ins></div></td>
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<td class="diff-addedline diff-side-added"><div>Note that, for hop or fluctuation rates <math>\nu</math> much larger than the static width <math>\Delta</math>, the dynamic relaxation looks more and more ''exponential'', <math>G^{\rm dyn}(t) [\nu \gg \Delta] \to e^{-\lambda t}</math> where <math>\lambda</math> decreases as <math>\nu</math> increases. This corresponds to the phenomenon of "motional narrowing" in NMR.</div></td>
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</table>Jess Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=824&oldid=prevJess at 17:03, 15 September 20222022-09-15T17:03:13Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:03, 15 September 2022</td>
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<td class="diff-context diff-side-deleted"><div>[[Image:KT-DGZF.png|600px|inline image (click to see full size)]]</div></td>
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<td class="diff-context diff-side-deleted"><div>{| width="80%" </div></td>
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<td class="diff-context diff-side-deleted"><div>Dynamic Gaussian Kubo-Toyabe function for various hop or fluctuation rates. </div></td>
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<td class="diff-context diff-side-added"><div>Dynamic Gaussian Kubo-Toyabe function for various hop or fluctuation rates. </div></td>
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=820&oldid=prevJess at 16:50, 15 September 20222022-09-15T16:50:27Z<p></p>
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<td class="diff-context diff-side-deleted"><div>and the corresponding dynamic relaxation functions look like this: </div></td>
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<td class="diff-context diff-side-added"><div>and the corresponding dynamic relaxation functions look like this: </div></td>
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<td class="diff-context diff-side-deleted"><div>Dynamic Gaussian Kubo-Toyabe function for various hop or fluctuation rates. </div></td>
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<td class="diff-context diff-side-added"><div>Dynamic Gaussian Kubo-Toyabe function for various hop or fluctuation rates. </div></td>
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<td class="diff-addedline diff-side-added"><div>Of course, the assumptions that go into calculating <math>g(\Delta, t)</math> are idealizations of the real field distributions at lattice sites and/or the actual processes involved in muon "hopping" and field "fluctuations". They also omit all effects of ''quantum entanglement'' between the muon spin and the nearby nuclear or electronic spins, which need not remain static during their dipolar or hyperfine interactions with the muon. As has been demonstrated in copper crystals and (more dramatically) in fluoride crystals, such coupled spin systems often evolve together to produce "static" <math>g(\Delta, t)</math> functions bearing little resemblance to that described above. </div></td>
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<td class="diff-addedline diff-side-added"><div>Fortunately, Kehr's recursion relation usually works (numerically) on those "static" functions as well, although of course it is still an approximation.</div></td>
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</table>Jess Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=817&oldid=prevJess at 23:22, 14 September 20222022-09-14T23:22:03Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:22, 14 September 2022</td>
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<td class="diff-context diff-side-deleted"><div>which is sometimes solvable using Laplace transforms. (Numerical methods work too.)</div></td>
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<td class="diff-context diff-side-added"><div>which is sometimes solvable using Laplace transforms. (Numerical methods work too.)</div></td>
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<td class="diff-addedline diff-side-added"><div>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above<ins class="diffchange diffchange-inline">, centered on <math>B_0 = 0</math>,</ins> gives a static function </div></td>
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<td class="diff-context diff-side-deleted"><div><math>g_G(\Delta, t) = {1\over3} + {2\over3} \exp\left[-{1\over2}\Delta^2 t^2\right] \left(1 - \Delta^2 t^2\right)</math></div></td>
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<td class="diff-context diff-side-added"><div><math>g_G(\Delta, t) = {1\over3} + {2\over3} \exp\left[-{1\over2}\Delta^2 t^2\right] \left(1 - \Delta^2 t^2\right)</math></div></td>
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</table>Jess Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=816&oldid=prevJess at 23:20, 14 September 20222022-09-14T23:20:44Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:20, 14 September 2022</td>
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<td class="diff-context diff-side-deleted"><div>and the corresponding dynamic relaxation functions look like this: </div></td>
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<td class="diff-context diff-side-added"><div>and the corresponding dynamic relaxation functions look like this: </div></td>
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<td class="diff-deletedline diff-side-deleted"><div>[<del class="diffchange diffchange-inline">File</del>:<del class="diffchange diffchange-inline"> </del>DGZF.png <del class="diffchange diffchange-inline">Dynamic</del> <del class="diffchange diffchange-inline">Gaussian</del> <del class="diffchange diffchange-inline">ZF</del> <del class="diffchange diffchange-inline">Kubo-Toyabe</del> <del class="diffchange diffchange-inline">function</del>]</div></td>
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<td class="diff-addedline diff-side-added"><div>[<ins class="diffchange diffchange-inline">[Image</ins>:<ins class="diffchange diffchange-inline">KT-</ins>DGZF.png<ins class="diffchange diffchange-inline">|600px|inline</ins> <ins class="diffchange diffchange-inline">image</ins> <ins class="diffchange diffchange-inline">(click</ins> <ins class="diffchange diffchange-inline">to</ins> <ins class="diffchange diffchange-inline">see full</ins> <ins class="diffchange diffchange-inline">size)]</ins>]</div></td>
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<td class="diff-addedline diff-side-added"><div>{| width="80%" </div></td>
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<td class="diff-addedline diff-side-added"><div>! align="left" |</div></td>
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<td class="diff-addedline diff-side-added"><div>Dynamic Gaussian Kubo-Toyabe function for various hop or fluctuation rates. </div></td>
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<td class="diff-addedline diff-side-added"><div>|}</div></td>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:06, 14 September 2022</td>
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<td class="diff-context diff-side-deleted"><div>So far we have treated only the behavior of the muon polarization at ''fixed locations'' where the local magnetic fields are ''static''. This is often not the case, and situations in which the muons are '''hopping''' between different sites and/or the local fields at those sites are '''fluctuating''' comprise many of the most interesting &micro;SR studies. </div></td>
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<td class="diff-context diff-side-added"><div>So far we have treated only the behavior of the muon polarization at ''fixed locations'' where the local magnetic fields are ''static''. This is often not the case, and situations in which the muons are '''hopping''' between different sites and/or the local fields at those sites are '''fluctuating''' comprise many of the most interesting &micro;SR studies. </div></td>
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</table>Jess Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=812&oldid=prevJess at 23:05, 14 September 20222022-09-14T23:05:29Z<p></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:05, 14 September 2022</td>
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<td class="diff-context diff-side-deleted"><div>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<td class="diff-context diff-side-added"><div>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<td class="diff-deletedline diff-side-deleted"><div><math>g_G(\Delta, t) = {1\over3} + {2\over3} <del class="diffchange diffchange-inline">e^{</del>-{1\over2}\Delta^2 t^2<del class="diffchange diffchange-inline">}}</del> \left(1 - \Delta^2 t^2\right)</math></div></td>
<td class="diff-marker" data-marker="+"></td>
<td class="diff-addedline diff-side-added"><div><math>g_G(\Delta, t) = {1\over3} + {2\over3} <ins class="diffchange diffchange-inline">\exp\left[</ins>-{1\over2}\Delta^2 t^2<ins class="diffchange diffchange-inline">\right]</ins> \left(1 - \Delta^2 t^2\right)</math></div></td>
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<td class="diff-addedline diff-side-added"><div>and the corresponding dynamic relaxation functions look like this: </div></td>
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<td class="diff-addedline diff-side-added"><div>[File: DGZF.png Dynamic Gaussian ZF Kubo-Toyabe function]</div></td>
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</table>Jess Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=811&oldid=prevJess at 23:00, 14 September 20222022-09-14T23:00:37Z<p></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
<col class="diff-marker" />
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:00, 14 September 2022</td>
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<td colspan="2" class="diff-lineno">Line 21:</td>
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<td class="diff-context diff-side-deleted"><div>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
<td class="diff-marker"></td>
<td class="diff-context diff-side-added"><div>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<td class="diff-marker"></td>
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<td class="diff-deletedline diff-side-deleted"><div><math>g_G(\Delta, t) = {1\over3} + {2\over3} e^{-{1\over2}\Delta^2 t^2}} \left(1 - \Delta^2 t^2\right)</div></td>
<td class="diff-marker" data-marker="+"></td>
<td class="diff-addedline diff-side-added"><div><math>g_G(\Delta, t) = {1\over3} + {2\over3} e^{-{1\over2}\Delta^2 t^2}} \left(1 - \Delta^2 t^2\right)<ins class="diffchange diffchange-inline"></math></ins></div></td>
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</table>Jess Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579
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Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /usr/share/mediawiki/includes/libs/rdbms/database/Database.php on line 579 http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=810&oldid=prevJess at 23:00, 14 September 20222022-09-14T23:00:08Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:00, 14 September 2022</td>
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<td class="diff-context diff-side-deleted"><div>One approach, at least in ZF, is to assume that the ''static'' relaxation function <math>G_{zz}^{\rm stat}(t) = g(\Delta, t)</math> simply '''''restarts''''' at every hop/fluctuation, using the muon polarization at the time of the hop as a new starting point. This is called the '''''strong collision model''''' and can be modeled using Kehr's recursion relation: </div></td>
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<td class="diff-context diff-side-added"><div>One approach, at least in ZF, is to assume that the ''static'' relaxation function <math>G_{zz}^{\rm stat}(t) = g(\Delta, t)</math> simply '''''restarts''''' at every hop/fluctuation, using the muon polarization at the time of the hop as a new starting point. This is called the '''''strong collision model''''' and can be modeled using Kehr's recursion relation: </div></td>
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<td class="diff-deletedline diff-side-deleted"><div><math>G^{\rm dyn}(\Delta,\nu,t) = g(\Delta,t) e^{-\nu t} + \nu \int_0^t G^{\rm dyn}(\Delta,\nu, t-\tau) g(\Delta, \tau) e^{-\nu \tau} d\tau </math> ,</div></td>
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<td class="diff-addedline diff-side-added"><div><math>G^{\rm dyn}(\Delta,\nu,t) = g(\Delta,t)<ins class="diffchange diffchange-inline"> \;</ins> e^{-\nu t} + \nu \int_0^t G^{\rm dyn}(\Delta,\nu, t-\tau)<ins class="diffchange diffchange-inline"> \;</ins> g(\Delta, \tau)<ins class="diffchange diffchange-inline"> \;</ins> e^{-\nu \tau} d\tau </math> ,</div></td>
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<td class="diff-context diff-side-deleted"><div>which is sometimes solvable using Laplace transforms. (Numerical methods work too.)</div></td>
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<td class="diff-context diff-side-added"><div>which is sometimes solvable using Laplace transforms. (Numerical methods work too.)</div></td>
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<td class="diff-addedline diff-side-added"><div>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<td class="diff-addedline diff-side-added"><div><math>g_G(\Delta, t) = {1\over3} + {2\over3} e^{-{1\over2}\Delta^2 t^2}} \left(1 - \Delta^2 t^2\right)</div></td>
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