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%%% Primary Title: Bessel functions: applications to diffraction by helical structures
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\begin{document}
\section{Applications of Bessel functions in Physics and Engineering}
One notes also that \htmladdnormallink{Bessel's equation}{http://planetphysics.us/encyclopedia/BesselEquation.html} arises in the derivation of separable solutions to \htmladdnormallink{Laplace's equation}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html}, and also for the Helmholtz equation in either cylindrical or spherical coordinates. The \htmladdnormallink{Bessel functions}{http://planetphysics.us/encyclopedia/BesselEquation2.html} are therefore very important in many physical problems involving \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} propagation, wave diffraction phenomena--including \htmladdnormallink{X-ray diffraction}{http://planetphysics.us/encyclopedia/LaserProducedPlasma.html} by certain molecular crystals, and also \htmladdnormallink{static}{http://planetphysics.us/encyclopedia/Statics.html} potentials. The solutions to most problems in cylindrical coordinate \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} are found in terms of \htmladdnormallink{Bessel functions of integer order}{http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html} ($\alpha = n$), whereas in spherical coordinates, such solutions involve Bessel functions of half-integer orders ($\alpha = n + 1/2$).
Several examples of Bessel function solutions are:
\begin{enumerate}
\item the diffraction pattern of a helical \htmladdnormallink{molecule}{http://planetphysics.us/encyclopedia/Molecule.html} wrapped around a cylinder computed from the \htmladdnormallink{Fourier transform}{http://planetphysics.us/encyclopedia/FourierTransforms.html} of the helix in cylindrical coordinates;
\item electromagnetic waves in a cylindrical waveguide
\item diffusion problems on a lattice.
\item vibration modes of a thin circular, tubular or annular membrane (such as a drum, other membranophone, the vocal cords, etc.)
\item \htmladdnormallink{heat}{http://planetphysics.us/encyclopedia/Heat.html} \htmladdnormallink{conduction}{http://planetphysics.us/encyclopedia/Conduction.html} in a cylindrical \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \end{enumerate}
In engineering Bessel functions also have useful properties for signal processing and filtering noise as for example by using Bessel filters, or in FM synthesis and windowing signals.
\subsection{Applications of Bessel functions in Physical Crystallography}
The first example listed above was shown to be especially important in molecular
biology for the structures of helical secondary structures in certain proteins (e.g. $\alpha-helix$) or in molecular genetics for finding the double-helix
structure of Deoxyribonucleic Acid (\htmladdnormallink{DNA}{http://planetphysics.us/encyclopedia/FCS3.html}) molecular crystals with extremely important consequences for genetics, biology, mutagenesis, molecular evolution,
contemporary life sciences and medicine. This finding is further detailed in the next subsection.
\subsubsection{X-Ray Diffraction Patterns of Double-Helical Deoxyribonucleic Acid (DNA) Crystals}
Francis C. Crick (\htmladdnormallink{Nobel laureate in Physiology and Medicine in 1962}{http://nobelprize.org/nobel_prizes/medicine/laureates/1962/}) published in {\em Acta Crystallographica} (1952;1953a,b) concise papers on X-ray diffraction patterns of a helix and coiled coils, respectively \cite{Cochran-Crick-Vand52, Crick53a,Crick53b} in which he showed that such patterns can be completely described by the Bessel functions defined above. Thus, the equatorial, or 0-layer, line contained diffraction intensities whose values were computed with the $J_0$ \htmladdnormallink{Bessel function}{http://en.wikipedia.org/wiki/Bessel_function} of the first kind with $n=0$. In fact, the entire X-ray diffraction, multiple diamond-like pattern of such helices, including those of the double helical
\htmladdnormallink{DNA molecule}{http://www.britishbiophysics.org.uk/what-is/crystal/xbdna_br.gif}, could be completely computed by means of Bessel functions of different order for each layer line; note however that there have also been occasional \htmladdnormallink{contenders to this analysis}{http://www.springerlink.com/content/h267578291367716/fulltext.pdf}. In fact, these involve {\em Fourier--Bessel series} based on Bessel functions.
There are, however, marked differences between the
\htmladdnormallink{A- and B- DNA X-ray diffraction patterns}{http://commons.wikimedia.org/wiki/File:ABDNAxrgpj.jpg} as shown by this \htmladdnormallink{web link}{http://commons.wikimedia.org/wiki/User:Bci2#DNA_Structures} which makes a comparison between the images published by H.R. Wilson \cite{Wilson58}. The Bessel function and Fourier--Bessel series analysis is however only applicable to the analysis of A-DNA patterns, whereas the X-Ray diffraction/scattering pattern of the B-DNA form is much less tractable although it is the predominant hydrated form in living cells.
The following is a web link to a \htmladdnormallink{3D animation of a Watson-Crick DNA double-helix molecular model}{http://commons.wikimedia.org/wiki/File:ADN_animation.gif}
Note also that a pairing of double helices of a \htmladdnormallink{DNA G-quadruplex}{http://www.phy.cam.ac.uk/research/bss/molbiophysics.php} has also been recently discovered that might be associated with the initiation of certain cancers; the \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} of the Fourier transform of such \htmladdnormallink{DNA G-quadruplex structures}{http://www.phy.cam.ac.uk/research/bss/bsspictures/nucleicacid.jpg} would still result in diffraction patterns constructed from Bessel functions but the new quadruplex symmetry of the `mutated' DNA \htmladdnormallink{G-quadruplex}{http://planetphysics.us/encyclopedia/TheoreticalBiophysics.html} would naturally alter the overall diffraction pattern intensities.
Further details and implications for both genomic and biotechnology applications
are presented in a related entry on \htmladdnormallink{molecular models of DNA}{http://en.wikipedia.org/wiki/Molecular_models_of_DNA}.
\begin{thebibliography}{99}
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F. Bessel, ``Untersuchung des Theils der planetarischen St\"orungen'', {\em Berlin Abhandlungen} (1824), article 14.
\bibitem{FRGG53}
Franklin, R.E. and Gosling, R.G. recd.6 March 1953. Acta Cryst. (1953). 6, 673 The Structure of Sodium Thymonucleate Fibres I. The Influence of Water Content Acta Cryst. (1953). and 6, 678 The Structure of Sodium Thymonucleate Fibres II. The Cylindrically Symmetrical Patterson Function.
\bibitem{Arfken-Weber2k5}
Arfken, George B. and Hans J. Weber, {\em Mathematical Methods for Physicists}, 6th edition, Harcourt: San Diego, 2005. ISBN 0-12-059876-0.
\bibitem{Bowman58}
Bowman, Frank. {\em Introduction to Bessel Functions.}. Dover: New York, 1958). ISBN 0-486-60462-4.
\bibitem{Cochran-Crick-Vand52}
Cochran, W., Crick, F.H.C. and Vand V. 1952. The Structure of Synthetic Polypeptides. 1. The Transform of Atoms on a Helix. {\em Acta Cryst.} {\bf 5}(5):581-586.
\bibitem{Crick53a}
Crick, F.H.C. 1953a. The Fourier Transform of a Coiled-Coil., {\em Acta Crystallographica} {\bf 6}(8-9):685-689.
\bibitem{Crick53b}
Crick, F.H.C. 1953. The packing of $\alpha$-helices- Simple coiled-coils.
{\em Acta Crystallographica}, {\bf 6}(8-9):689-697.
\bibitem{WJ-CFC53a}
Watson, J.D; Crick F.H.C. 1953a. Molecular Structure of Nucleic Acids-- A Structure for Deoxyribose Nucleic Acid., {\em Nature} 171(4356):737--738.
\bibitem{WJ-CFC53b}
Watson, J.D; Crick F.H.C. 1953b. The Structure of DNA., {\em Cold Spring Harbor Symposia on Qunatitative Biology} {\bf 18}:123-131.
\bibitem{WSWH53}
M. H. F. Wilkins, A.R. Stokes A.R. and H. R. Wilson. 1953. ``Molecular Structure of Deoxypentose Nucleic Acids'' {\em Nature}, volume 171, pages 738--740. \htmladdnormallink{Download the full text in PDF format.}{http://www.nature.com/nature/dna50/wilkins.pdf}
\bibitem{NP}{\sc N. Piskunov:} {\em Diferentsiaal- ja integraalarvutus k\~{o}rgematele tehnilistele \~{o}ppeasutustele}.\, Kirjastus Valgus, Tallinn (1966).
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\bibitem{Spain-Smith70}
Spain,B., and M. G. Smith, {\em Functions of mathematical physics.}, Van Nostrand Reinhold Company, London, 1970. Chapter 9: Bessel functions.
\bibitem{Wilson58}
The X-ray patterns of A- and B- DNA forms are compared in the following linked image \htmladdnormallink{(courtesy of Dr. H.R. Wilson, F.R.S.)}{http://spdbv.vital-it.ch/TheMolecularLevel/BiochemViews/Nucleotides/Images/CMCC0904.jpg}
\bibitem{Watson95}
Watson, G. N. {\em A Treatise on the Theory of Bessel Functions.}, (1995) Cambridge University Press. ISBN 0-521-48391-3.
\end{thebibliography}
\end{document}
