Structure factor - Revision history

Carl McBride: Defined as

2015-02-20T17:49:15Z

Defined as

← Older revision		Revision as of 19:49, 20 February 2015
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	The '''static structure factor''', <math>S(k)</math>, for a monatomic system composed of spherical scatterers is defined by (Eq. 1 in <ref>[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", Journal of Physics: Condensed Matter '''6''' pp. 8415-8427 (1994)]</ref>):		The '''static structure factor''', <math>S(k)</math>, for a monatomic system composed of spherical scatterers is defined by (Eq. 1 in <ref>[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", Journal of Physics: Condensed Matter '''6''' pp. 8415-8427 (1994)]</ref>):

	:<math>S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~{\mathrm {d}}r</math>		:<math>S(k) := 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~{\mathrm {d}}r</math>

	where <math>g_2(r)</math> is the [[radial distribution function]], and <math>k</math> is the scattering wave-vector modulus		where <math>g_2(r)</math> is the [[radial distribution function]], and <math>k</math> is the scattering wave-vector modulus

Carl McBride: Added a few references.

2011-11-16T14:41:00Z

Added a few references.

← Older revision		Revision as of 16:41, 16 November 2011
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	The '''structure factor''', <math>S(k)</math>, for a monatomic system is defined by:		The '''static structure factor''', <math>S(k)</math>, for a monatomic system composed of spherical scatterers is defined by (Eq. 1 in <ref>[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", Journal of Physics: Condensed Matter '''6''' pp. 8415-8427 (1994)]</ref>):

			:<math>S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~{\mathrm {d}}r</math>

	:<math>~~S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} (~~ g_2(r) ~~-1 ) r \sin (kr) ~dr~~</math>		where <math>g_2(r)</math> is the [[radial distribution function]], and <math>k</math> is the scattering wave-vector modulus

	~~where <math>k</math> is the scattering wave-vector modulus~~		:<math>k= \|\mathbf{k}\|= \frac{4 \pi }{\lambda} \sin \left( \frac{\theta}{2}\right)</math>.

	:<math>k= \|\mathbf{k}\|= \frac{4 \pi }{\lambda} \sin \left( \frac{\theta}{2}\right)</math>

	The structure factor is basically a [[Fourier analysis \| Fourier transform]] of the [[pair distribution function]] <math>{\rm g}(r)</math>,		The structure factor is basically a [[Fourier analysis \| Fourier transform]] of the [[pair distribution function]] <math>{\rm g}(r)</math>,
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	The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known		The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known
	as the collective (or coherent) intermediate scattering function.		as the collective (or coherent) intermediate scattering function.
			==Binary mixtures==
			<ref>[http://dx.doi.org/10.1080/14786436508211931 T. E. Faber and J. M. Ziman "A theory of the electrical properties of liquid metals III. the resistivity of binary alloys", Philosophical Magazine '''11''' pp. 153-173 (1965)]</ref><ref>[http://dx.doi.org/10.1103/PhysRev.156.685 N. W. Ashcroft and David C. Langreth "Structure of Binary Liquid Mixtures. I", Physical Review '''156''' pp. 685–692 (1967)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevB.2.3004 A. B. Bhatia and D. E. Thornton "Structural Aspects of the Electrical Resistivity of Binary Alloys", Physical Review B '''2''' pp. 3004-3012 (1970)]</ref>
	==References==		==References==
	<references/>		<references/>
	;Related reading		;Related reading
	*[http://dx.doi.org/10.~~1088~~/~~0953-8984/6/41/006~~ A. ~~Filipponi,~~ "~~The radial distribution function probed by X-ray absorption spectroscopy~~", J. ~~Phys~~.~~: Condens~~. ~~Matter~~, '''6''' pp. ~~8415~~-~~8427~~ (~~1994~~)]		*[http://dx.doi.org/10.1007/BF01391926 F. Zernike and J. A. Prins "Die Beugung von Röntgenstrahlen in Flüssigkeiten als Effekt der Molekülanordnung", Zeitschrift für Physik '''41''' pp. 184-194 (1920)]
			*P. Debye and H. Menke "", Physik. Zeits. '''31''' pp. 348- (1930)
			*B. E. Warren "X-Ray Diffraction", Dover Publications (1969) ISBN 0486663175 § 10.4
			*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids" (Third Edition) [http://dx.doi.org/10.1016/B978-012370535-8/50006-9 Chapter 4: "Distribution-function Theories"] § 4.1

	[[category: Statistical mechanics]]		[[category: Statistical mechanics]]

132.168.28.141 at 12:33, 16 November 2011

2011-11-16T12:33:00Z

← Older revision		Revision as of 14:33, 16 November 2011
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	where <math>k</math> is the scattering wave-vector modulus		where <math>k</math> is the scattering wave-vector modulus

	:<math>k= \|\mathbf{k}\|= \frac{4 \pi }{\lambda \sin \left( \frac{\theta}{2}\right)}</math>		:<math>k= \|\mathbf{k}\|= \frac{4 \pi }{\lambda} \sin \left( \frac{\theta}{2}\right)</math>

	The structure factor is basically a [[Fourier analysis \| Fourier transform]] of the [[pair distribution function]] <math>{\rm g}(r)</math>,		The structure factor is basically a [[Fourier analysis \| Fourier transform]] of the [[pair distribution function]] <math>{\rm g}(r)</math>,

Nice and Tidy: Slight tidy

2011-09-16T07:56:09Z

Slight tidy

← Older revision		Revision as of 09:56, 16 September 2011
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	from which one can calculate the [[Compressibility \| isothermal compressibility]].		from which one can calculate the [[Compressibility \| isothermal compressibility]].

	To calculate <math>S(k)</math> in molecular simulations one typically uses:		To calculate <math>S(k)</math> in [[Computer simulation techniques \|molecular simulations]] one typically uses:

	:<math>S(k) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m))> </math>,		:<math>S(k) = \frac{1}{N} \sum^{N}_{n,m=1} \langle\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m)) \rangle </math>,

	where <math>N</math> is the number of particles and <math>\mathbf{r}_n</math> and		where <math>N</math> is the number of particles and <math>\mathbf{r}_n</math> and
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	The dynamic, time dependent structure factor is defined as follows:		The dynamic, time dependent structure factor is defined as follows:
	:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,		:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} \langle \exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0))) \rangle </math>,

	The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known		The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known
	as the collective (or coherent) intermediate scattering function.		as the collective (or coherent) intermediate scattering function.




	==References==		==References==
	#[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, '''6''' pp. 8415-8427 (1994)]		<references/>
			;Related reading
			*[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, '''6''' pp. 8415-8427 (1994)]
	[[category: Statistical mechanics]]		[[category: Statistical mechanics]]

Esanz at 16:47, 15 September 2011

2011-09-15T16:47:41Z

← Older revision		Revision as of 18:47, 15 September 2011
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	:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,		:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,

	The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known as the collective (or		The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known
	coherent) intermediate scattering		as the collective (or coherent) intermediate scattering function.
	function.

Esanz at 16:46, 15 September 2011

2011-09-15T16:46:28Z

← Older revision		Revision as of 18:46, 15 September 2011
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	:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,		:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,

	The ratio between the ~~static~~ and the ~~dynamic~~ structure factor, <math>S(k,t)/S(k,0)</math>, is known as the collective or		The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known as the collective (or
	coherent intermediate scattering		coherent) intermediate scattering
	function.		function.

Esanz at 16:45, 15 September 2011

2011-09-15T16:45:41Z

← Older revision		Revision as of 18:45, 15 September 2011
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	:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,		:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,

	The ratio between the static and the dynamic structure factor, <math>S(k,t)/~~<math>~~S(k,0), is known as the collective or		The ratio between the static and the dynamic structure factor, <math>S(k,t)/S(k,0)</math>, is known as the collective or
	coherent intermediate scattering		coherent intermediate scattering
	function.		function.

Esanz at 16:45, 15 September 2011

2011-09-15T16:45:08Z

← Older revision		Revision as of 18:45, 15 September 2011
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	:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,		:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,

			The ratio between the static and the dynamic structure factor, <math>S(k,t)/<math>S(k,0), is known as the collective or
			coherent intermediate scattering
			function.

Esanz at 16:38, 15 September 2011

2011-09-15T16:38:58Z

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	<math>\mathbf{r}_m</math> are the coordinates of particles		<math>\mathbf{r}_m</math> are the coordinates of particles
	<math>n</math> and <math>m</math> respectively.		<math>n</math> and <math>m</math> respectively.

			The dynamic, time dependent structure factor is defined as follows:
			:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,



	==References==		==References==
	#[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, '''6''' pp. 8415-8427 (1994)]		#[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, '''6''' pp. 8415-8427 (1994)]
	[[category: Statistical mechanics]]		[[category: Statistical mechanics]]

Esanz at 16:35, 15 September 2011

2011-09-15T16:35:23Z

← Older revision		Revision as of 18:35, 15 September 2011
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	:<math>S(k) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m))> </math>,		:<math>S(k) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m))> </math>,

	where <math>N</math> is the number of particles and <math>\mathbf{r}_n</math> and <math>\mathbf{r}_m</math> are the coordinates of particles		where <math>N</math> is the number of particles and <math>\mathbf{r}_n</math> and
			<math>\mathbf{r}_m</math> are the coordinates of particles
	<math>n</math> and <math>m</math> respectively.		<math>n</math> and <math>m</math> respectively.