Charge conservation - Revision history

en>Chetvorno: /* Experimental Evidence */ Added evidence that elementary positive and negative charges must be equal

2014-12-01T08:03:30Z

Experimental Evidence: Added evidence that elementary positive and negative charges must be equal

← Older revision		Revision as of 10:03, 1 December 2014
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	~~In [[physics]] and [[mathematics]]~~, '~~''axiality and rhombicity''' are two characteristics of~~ a ~~[[symmetric tensor\|symmetric]] second-rank [[tensor]] in three-dimensional [[Euclidean space]], describing its directional asymmetry~~.		Nice to meet you, my name is Refugia. What I adore doing is performing ceramics but I haven't produced a dime with it. I am a meter reader but I strategy on changing it. North Dakota is our birth place.<br><br>Also visit my homepage - [http://nuvem.tk/altergalactica/MarissaepDonnt nuvem.tk]

	~~Let ''A'' denote~~ a second-rank tensor in '''R'''<sup>3</sup>, which can be represented by a 3-by-3 [[matrix (mathematics)\|matrix]]. We assume that ''A'' is symmetric. This implies that ''A'' has three real [[eigenvalue]]s, which we denote by <math>A_{{xx}}</math>, <math>A_{{yy}}</math> and <math>A_{{zz}}</math>. We assume that they are ordered such that
	~~:<math>A_{{xx}} \le A_{{yy}} \le A_{{zz}}.</math>~~

	~~The axiality of ''A'' is defined by~~
	~~:<math> \Delta A =2 A_{zz}-(A_{xx}+A_{yy}). \, </math>~~
	~~The rhombicity is the difference between the smallest and the second-smallest eigenvalue:~~
	~~:<math> \delta A = A_{xx}-A_{yy}. \,</math>~~

	~~Other definitions of axiality and rhombicity differ from the ones given above by constant factors which depend~~ on ~~the context~~. ~~For example, when using them as parameters in the irreducible spherical tensor expansion, it~~ is ~~most convenient to divide the above definition of axiality by <math>{\sqrt{6}}</math> and that of rhombicity by <math>{2}</math>~~.

	~~==Applications==~~
	The description of physical interactions in terms of '''axiality and rhombicity''' is frequently encountered in [[Spin (physics)\|spin]] dynamics and, in particular, in [[Spin (physics)\|spin]] relaxation theory, where many traceless bilinear interaction Hamiltonians, having the (eigenframe) form

	~~:<math> \hat H = \hat\vec\mathbf{a}\cdot\mathbf{A}\cdot\hat\vec\mathbf{b} = A_{xx} \hat a_{x} \hat b_{x} + A_{yy} \hat a_{y} \hat b_{y} + A_{zz} \hat a_{z} \hat b_{z}~~ <~~/math~~>

	~~(hats denote spin projection operators) may be conveniently rotated using rank 2 irreducible spherical tensor operators:~~

	:<~~math~~> ~~\hat\vec\mathbf{a}\cdot\mathbf{A}\cdot\hat\vec\mathbf{b} = \frac{\delta A}{2} \hat T_{2,~~-~~2} + \frac{\delta A}{2} \hat T_{2,2} + \frac{\Delta A}{\sqrt{6}} \hat T_{2,-2} </math>~~

	~~:<math> \hat \hat R_{\alpha,\beta,\gamma} (\hat T_{l,m}) = \sum_{k=-2}^2 \hat T_{l,k} \mathfrak{D}_{k,m}^{(l)}(\alpha,\beta,\gamma)</math>~~

	where <math>\mathfrak{D}_{k,m}^{(l)}(\alpha,\beta,\gamma)</math> are Wigner functions, <math>(\alpha,\beta,\gamma)</math> are Euler angles, and the expressions for the rank 2 irreducible spherical tensor operators are:

	~~:<math> \hat T_{2,2} = + \frac{1}{2} \hat a_{+} \hat b_{+} </math>~~
	~~:<math> \hat T_{2,1} = - \frac{1}{2}( \hat a_{z} \hat b_{+} + \hat a_{+} \hat b_{z} ) </math>~~
	~~:<math> \hat T_{2,0} = + \sqrt{\frac{2}{3}}( \hat a_{z} \hat b_{z} - \frac{1}{4}( \hat a_{+} \hat b_{-} + \hat a_{-} \hat b_{+}) ) </math>~~
	~~:<math> \hat T_{2,-1} = + \frac{1}{2}( \hat a_{z} \hat b_{-} + \hat a_{-} \hat b_{z} ) </math>~~
	~~:<math> \hat T_{2,-2} = + \frac{1}{2} \hat a_{-} \hat b_{-} </math>~~

	~~Defining Hamiltonian rotations in this way (axiality, rhombicity, three angles) significantly simplifies calculations, since the properties of Wigner functions are well understood.~~

	~~==References==~~

	~~: D.M. Brink and G.R. Satchler, Angular momentum, 3rd edition, 1993, Oxford: Clarendon Press.~~
	: D.A. Varshalovich, A.N. Moskalev, V.K. Khersonski, Quantum theory of angular momentum: irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols, 1988, Singapore: World Scientific Publications.
	~~: I. Kuprov, N. Wagner-Rundell, P.J. Hore, J. Magn. Reson., 2007 (184) 196-206.~~ [http://~~physchem~~.~~ox.ac.uk~~/~~~kuprov~~/~~docs/kuprov_et_al~~.~~pdf Article]~~

	~~[[Category:Tensors]~~]

en>Chetvorno: Reverted 1 edit by 94.54.247.91 (talk) to last revision by 89.177.105.146. (TW)

2013-11-05T20:17:05Z

Reverted 1 edit by 94.54.247.91 (talk) to last revision by 89.177.105.146. (TW)

← Older revision		Revision as of 22:17, 5 November 2013
Line 1:		Line 1:
	~~Andrew Simcox is the title his parents gave him~~ and ~~he totally enjoys this name~~. ~~Credit authorising~~ is ~~how he tends to make cash~~. ~~For a while I~~'~~ve been in Alaska but I will have to transfer in a yr or two~~. ~~I am truly fond~~ of ~~to go to karaoke but I~~'~~ve been using on new things lately~~.<br><br>~~Also visit my site~~ ... ~~tarot readings~~ ([~~http~~://~~www.zavodpm.ru~~/~~blogs~~/~~glennmusserrvji~~/~~14565~~-~~great~~-~~hobby~~-~~advice~~-~~assist~~-~~allow~~-~~you~~-~~get~~-~~going zavodpm~~.ru])		In [[physics]] and [[mathematics]], '''axiality and rhombicity''' are two characteristics of a [[symmetric tensor\|symmetric]] second-rank [[tensor]] in three-dimensional [[Euclidean space]], describing its directional asymmetry.

			Let ''A'' denote a second-rank tensor in '''R'''<sup>3</sup>, which can be represented by a 3-by-3 [[matrix (mathematics)\|matrix]]. We assume that ''A'' is symmetric. This implies that ''A'' has three real [[eigenvalue]]s, which we denote by <math>A_{{xx}}</math>, <math>A_{{yy}}</math> and <math>A_{{zz}}</math>. We assume that they are ordered such that
			:<math>A_{{xx}} \le A_{{yy}} \le A_{{zz}}.</math>

			The axiality of ''A'' is defined by
			:<math> \Delta A =2 A_{zz}-(A_{xx}+A_{yy}). \, </math>
			The rhombicity is the difference between the smallest and the second-smallest eigenvalue:
			:<math> \delta A = A_{xx}-A_{yy}. \,</math>

			Other definitions of axiality and rhombicity differ from the ones given above by constant factors which depend on the context. For example, when using them as parameters in the irreducible spherical tensor expansion, it is most convenient to divide the above definition of axiality by <math>{\sqrt{6}}</math> and that of rhombicity by <math>{2}</math>.

			==Applications==
			The description of physical interactions in terms of '''axiality and rhombicity''' is frequently encountered in [[Spin (physics)\|spin]] dynamics and, in particular, in [[Spin (physics)\|spin]] relaxation theory, where many traceless bilinear interaction Hamiltonians, having the (eigenframe) form

			:<math> \hat H = \hat\vec\mathbf{a}\cdot\mathbf{A}\cdot\hat\vec\mathbf{b} = A_{xx} \hat a_{x} \hat b_{x} + A_{yy} \hat a_{y} \hat b_{y} + A_{zz} \hat a_{z} \hat b_{z} </math>

			(hats denote spin projection operators) may be conveniently rotated using rank 2 irreducible spherical tensor operators:

			:<math> \hat\vec\mathbf{a}\cdot\mathbf{A}\cdot\hat\vec\mathbf{b} = \frac{\delta A}{2} \hat T_{2,-2} + \frac{\delta A}{2} \hat T_{2,2} + \frac{\Delta A}{\sqrt{6}} \hat T_{2,-2} </math>

			:<math> \hat \hat R_{\alpha,\beta,\gamma} (\hat T_{l,m}) = \sum_{k=-2}^2 \hat T_{l,k} \mathfrak{D}_{k,m}^{(l)}(\alpha,\beta,\gamma)</math>

			where <math>\mathfrak{D}_{k,m}^{(l)}(\alpha,\beta,\gamma)</math> are Wigner functions, <math>(\alpha,\beta,\gamma)</math> are Euler angles, and the expressions for the rank 2 irreducible spherical tensor operators are:

			:<math> \hat T_{2,2} = + \frac{1}{2} \hat a_{+} \hat b_{+} </math>
			:<math> \hat T_{2,1} = - \frac{1}{2}( \hat a_{z} \hat b_{+} + \hat a_{+} \hat b_{z} ) </math>
			:<math> \hat T_{2,0} = + \sqrt{\frac{2}{3}}( \hat a_{z} \hat b_{z} - \frac{1}{4}( \hat a_{+} \hat b_{-} + \hat a_{-} \hat b_{+}) ) </math>
			:<math> \hat T_{2,-1} = + \frac{1}{2}( \hat a_{z} \hat b_{-} + \hat a_{-} \hat b_{z} ) </math>
			:<math> \hat T_{2,-2} = + \frac{1}{2} \hat a_{-} \hat b_{-} </math>

			Defining Hamiltonian rotations in this way (axiality, rhombicity, three angles) significantly simplifies calculations, since the properties of Wigner functions are well understood.

			==References==

			: D.M. Brink and G.R. Satchler, Angular momentum, 3rd edition, 1993, Oxford: Clarendon Press.
			: D.A. Varshalovich, A.N. Moskalev, V.K. Khersonski, Quantum theory of angular momentum: irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols, 1988, Singapore: World Scientific Publications.
			: I. Kuprov, N. Wagner-Rundell, P.J. Hore, J. Magn. Reson., 2007 (184) 196-206. [http://physchem.ox.ac.uk/~kuprov/docs/kuprov_et_al.pdf Article]

			[[Category:Tensors]]

70.247.165.249: avoid redirect

2012-06-10T18:52:10Z

avoid redirect

New page

Andrew Simcox is the title his parents gave him and he totally enjoys this name. Credit authorising is how he tends to make cash. For a while I've been in Alaska but I will have to transfer in a yr or two. I am truly fond of to go to karaoke but I've been using on new things lately.<br><br>Also visit my site ... tarot readings ([http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going zavodpm.ru])