Sample-shape dependence of magnetic noise

Zero-field magnetic noise, characterized by the magnetic autocorrelation function <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi>S</a:mi><a:mi>s</a:mi></a:msub><a:mrow><a:mo>(</a:mo><a:mi>t</a:mi><a:mo>)</a:mo></a:mrow></a:mrow></a:math>, has been observed, perhaps surprisingly, to depend on sample shape <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>s</b:mi></b:math>. The reasons for this are identified and general expressions are derived that relate the autocorrelation functions for systems of different shape to an underlying “intrinsic” form. Assuming the fluctuation-dissipation theorem, it is shown that, for any noise that relaxes monotonically, the effect of sample shape is to reduce both the noise amplitude and mean relaxation time by a factor of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mn>1</c:mn><c:mo>+</c:mo><c:mi>N</c:mi><c:msub><c:mi>χ</c:mi><c:mi>i</c:mi></c:msub></c:mrow></c:math>, where <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>N</d:mi></d:math> is the demagnetizing factor and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi>χ</e:mi><e:mi>i</e:mi></e:msub></e:math> the intrinsic susceptibility, but that only the exponential decay retains the same functional form for finite <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>N</f:mi></f:math>. In frequency space, where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:msub><g:mi>S</g:mi><g:mi>s</g:mi></g:msub><g:mrow><g:mo>(</g:mo><g:mi>t</g:mi><g:mo>)</g:mo></g:mrow></g:mrow></g:math> Fourier transforms into the power spectrum <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:msub><h:mi>S</h:mi><h:mi>s</h:mi></h:msub><h:mrow><h:mo>(</h:mo><h:mi>ω</h:mi><h:mo>)</h:mo></h:mrow></h:mrow></h:math>, the above two factors combine to suppress the zero frequency amplitude of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msub><i:mi>S</i:mi><i:mi>s</i:mi></i:msub><i:mrow><i:mo>(</i:mo><i:mi>ω</i:mi><i:mo>)</i:mo></i:mrow></i:mrow></i:math> by <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msup><j:mrow><j:mo>(</j:mo><j:mn>1</j:mn><j:mo>+</j:mo><j:mi>N</j:mi><j:msub><j:mi>χ</j:mi><j:mi>i</j:mi></j:msub><j:mo>)</j:mo></j:mrow><j:mn>2</j:mn></j:msup></j:math>, while, at high frequency, sample shape dependence becomes negligible. These results are applied to various magnetic systems and experiments, including to tests of the fluctuation-dissipation and noise measurements in spin ice, to spin glasses, to surface magnetism, and to ferromagnetic critical behavior. They may be used to imply a general result that, for any near to equilibrium magnetic system with monotonic relaxation, the internal <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>B</k:mi></k:math> field will relax more slowly, and with larger amplitude, than the internal <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>H</l:mi></l:math> field, both by factors of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:msub><m:mi>χ</m:mi><m:mi>i</m:mi></m:msub></m:mrow></m:math>.