The Relative Agle Distributio Fuctio i the agevi Theory of Dilute Dipoles Robert D. Nielse ExxoMobil Research ad Egieerig Co., Clito Towship, 545 Route East, Aadale, NJ 0880 robert.ielse@exxomobil.com
Abstract The agevi theory of the polarizatio of a dilute collectio of dipoles by a exteral field is ofte icluded i itroductory solid state physics ad physical chemistry curricula. The average polarizatio is calculated assumig the dipoles are i thermal equilibrium with a heat bath. The heart of the polarizatio calculatio is a derivatio of the average dipole-field projectio, whose depedece o the exteral field is give by the agevi fuctio. The agevi problem is revisited, here, ad the average projectio of ay give dipole oto ay other dipole from the collectio is derived i terms of the agevi fuctio. A simple expressio is obtaied for the uderlyig dipole-dipole agular distributio fuctio. I. Itroductio A sigle magetic dipole µ i a exteral magetic field H has a potetial eergy: V = µ H = µ Hcos( θ ). While formulatig a theory of magetism, agevi cosidered a collectio of dipoles i a exteral magetic field. The cocetratio of the dipoles was assumed to be sufficietly diluted that dipole-dipole iteractios could be eglected, leavig oly the sum over the idividual dipole-field potetial eergies for the total eergy. agevi developed the equilibrium average value of the dipole projectio o to the exteral field, cos( θ ), by assumig that the dipoles were i cotact with a heat bath. The distributio fuctio, which allows the equilibrium averages to be calculated, is the Boltzma distributio: V kt e e = Z Z F F cosθ
where F = µ H kt, ad F cosθ Z F = e si θ dθ dφ = π π φ= 0θ= 0 π e F cosθ dcosθ dφ = φ= 0cosθ= sih ( F ) 4 π. F Z F is the partitio fuctio. 3, 4 k ad T are Boltzma s costat ad the temperature of the heat bath. The equilibrium averages are: cos ( F) θ = dz Z df = ( F) Where ( F ) are the th order agevi fuctios. cos( θ ) deoted by., 3 = is ofte simply We ow ask, for the same system of dipoles, what is average projectio of ay give dipole oto ay other dipole from the collectio. We also ask, what is the distributio fuctio for this relative projectio i terms of the value of the exteral field? Sectio II develops the relative agle distributio fuctio ad averages. Sectio III shows the relative agle distributio fuctio that is geerated umerically from Mote Carlo calculatios with some trial values of F, for compariso. II. Relative agle distributio ad averages Figure shows the relative orietatio of two dipoles, labeled ad. The relative agle betwee the two dipoles is deoted byγ. The agles that defie each dipole s projectio oto the exteral Z-axis are give by θ ad θ for dipoles ad respectively. The Z-axis will be take as the directio of the exteral field H, so that θ ad θ are the agles that eter the expressio for the potetial eergy of dipoles ad. 3
The agle γ is give the subscriptγ because it is opposite the agle γ o the spherical triagle formed from the two dipoles ad the Z axis, see figure. θ ad θ are defied likewise. Because the dipoles are dilute, the average cos ( γ ) ca be expressed i terms of the dipole-field Boltzma distributios of the idividual dipoles ad. π π F cosθ F cosθ e e cos ( γ ) = cos ( γ) d cosθ dφ d cosθ dφ Z F Z F () φ = 0cosθ = φ = 0cosθ = The two itegrals over the two agles φ ad φ ca be replaced with a sigle itegral over the relative agle γ = φ φ because cos( γ ) is periodic i both variablesφ ad φ, ad the itegratio rages exted from 0 to π. Equatio () the becomes: π F cosθ F cosθ γ () 0 cos cos γ = θ = θ = e e cos ( γ) = π cos ( γ) d cosθ d cosθ d Z F Z F The agle additio formula, ( γ) ( θ ) ( θ ) ( θ ) ( θ ) cos = cos cos + si si cos γ, (3) from spherical trigoometry gives the depedece of cos( γ ) o the itegratio variables. 5 Direct calculatio of the itegral () with = ad =, for example, gives: ad ( γ) = ( θ ) ( θ ) = cos cos cos F ( ) cos ( γ) = cos ( θ ) cos ( θ ) + cos ( θ ) cos θ 4
= 3 ( ( F) ) ( F) + While ay of the average values cos ( γ ) ca be calculated i this maer, by expadig cos ( γ ) i the itegrad usig the agle additio formula, the uderlyig distributio of cos( γ ) that govers the averages is ot trasparet. A distributio fuctio, ( cos) ρ γ, is sought, such that: ( γ ) = ( γ) ρ( ( γ) ) ( γ) cos cos cos d cos. cosγ = To establish the distributio ( cos) ρ γ, a chage of variables is made i equatio (). The variables i equatio () cosist of two sides (arcs) of a spherical triagle ad the iterveig vertex agle (see figure ). The itegral () ca be re-expressed, i geeral, i terms of ay two sides of the spherical triagle ad their vertex agle. So, for example, the followig trasformatio is possible: { cos ( θ ), cos ( θ ), γ} { cos ( θ ), cos ( γ ), θ } (4) That the Jacobia is uity for this trasformatio maybe be verified aalytically by calculatig cos( γ ), cos( γ ) cos( θ ) γ, θ γ ad cos( θ ) with θ the aid of the additio formula: ( ) ( ) ( ) cos θ = cos θ cos γ + si θ si γ cos θ, =, ad the the law of sies for spherical triagles: si ( θ ) si ( θ ) si ( γ) si ( γ ) γ = γ θ,from spherical trigoometry. 5 auxillary formula: si cos cos With the trasformatio (4), equatio () becomes: θ 5
( Z( F) ) π π F ( cos( θ ) ( + cos( γ) ) + si( θ ) si( γ) cos( θ )) cos γ = cos γ e d cosθ d d cosγ cosγ= θ = 0 cosθ = Droppig the subscript ad superscript o the agles, ( cos) ( ) ( Z ( F) ) π cosθ = = 0 ρ γ is idetified as: π F( cos( θ) ( + cos( γ) ) + si( θ) si( γ) cos( ) ) ρ cos γ = e d d cosθ (5) The itegrad i equatio (5) is simplified by the followig chage of parameters: = ( cos + ), b= F si ( γ ), c a b F ( cos( γ )) a F γ si = + = +, ( α) = bc= F si ( γ) c, ( α) ( γ) The equatio (5) is trasformed to: ( ) ( Z ( F) ) cos = ac= c b c= F cos + c π π c ( cos( α) cos( θ) + si( α) si( θ) cos( ) ) ρ cos γ = e d d cosθ (6) cosθ = = 0 The cosie additio formula, (3), ad a agular trasformatio, aalogous to (4), allows the argumet of the expoetial i the itegrad to be writte as a sigle cosie. The expressio for the distributio fuctio (6) is itegrated to give: ρ ( cos( γ) ) ( F ( + ( γ ))) + ( γ ) 4 π sih cos = Z ( F) F cos ( F ( + cos( γ ))) Z ( F) Z = π (7) The ormalizatio of the distributio (7) is verified by the chage of variables: ( ) u = F + cos γ. θ 6
cosγ= cosγ= F ( F ( + ( γ ))) ( + ( γ )) 4 π sih cos ρ ( cos( γ) ) dcos( γ) = dcos γ Z ( F) F cos ( F ) 4 π 4 π cosh = sih u du Z( F) F = Z 0 ( F) F 4 π sih ( F ) = = Z ( F) F The averages, cos ( γ ), ca be expressed, likewise by a chage of variables, as: F u cos ( γ ) = sih ( u) du sih ( F) F 0 III. Mote Carlo Mote Carlo provides a way to umerically test the distributio fuctio (7). Mote Carlo umerically geerates cofiguratios of dipoles i a exteral field that are cosistet with thermal equilibrium ( Bolztma statistics ). The iput to the Mote Carlo calculatio here is a set of 000 dipoles with radomly assiged orietatios. Each Mote Carlo cycle refies the orietatios of all 000 dipoles by makig radom chages to the idividual dipole orietatios, oe at a time. If a give dipole s eergy ( V Fcos( θ )) = is decreased or remais the same as a result of the radom reorietatio, the ew orietatio is kept ad replaces the origial orietatio. If the radom re-orietatio of a dipole leads to a icrease i eergy, the ew orietatio is ot always accepted. A move that icreases the eergy is kept with a frequecy that is dictated by the Boltzma weightig of the eergy differece: Vew Vold kt e. I other words, larger chages i eergy are accepted less frequetly tha smaller eergy chages 7
i maer that is cosistet with thermal equilibrium. A overview of Mote Carlo is available i stadard texts, where sample codes are give. 6, 7 A Mote Carlo simulatio was ru o a set of 000 dipoles for each of the values: F = /5,,, ad 5. The last 400 Mote Carlo cycles out of a 5000 cycle trajectory were used to compile statistics of the dipole orietatios for each value of F. Figure shows ormalized histograms of the values of cos( θ ) (figure, left paels) ad cos( γ ) (figure, right paels) from the Mote Carlo dipole cofiguratios. The values of F icrease from top to bottom. The solid lies are the Boltzma distributio (figure, left paels), ad the relative agle distributio fuctio (7) (figure, right paels). The Mote Carlo results umerically cofirm the aalytic relative agle distributio (7) derived i sectio II. IV. Discussio A demostratio of the relative agle distributio fuctio for dipoles i the dilute limit serves two purposes. Firstly, the derivatio ca be used as a follow up exercise to the stadard agevi problem. The calculatio of the relative agle distributio is slightly more challegig tha the calculatio of the average polarizatio. Furthermore, the problem itroduces the idea of relative vs. exteral orietatioal order, which foreshadows the itroductio of a agular distributio fuctio i codesed phase statistics. The followig qualitative questio might be posed, for example, to help explore the differece betwee the relative ad exteral order: Why, i figure, does the relative agle distributio fuctio appear to lag behid the Boltzma distributio fuctio i its depedece o the exteral field? 8
Secodly, the aalytical expressio for the distributio fuctio derived i the dilute limit (equatio (7)) is useful for compariso with the agular distributio fuctio that arises whe dipole-dipole iteractios are preset at higher desities. The geeral agular distributio fuctio at high desity reflects both the ifluece of the exterally applied field, as well as dipole-dipole iteractios. The collective dipole-dipole iteractios are ot easily described by simple aalytic formulae because dipoles form phases ad exhibit log rage order that ivolves the participatio of may dipoles. 7 A commo method of graspig the structure of the dipolar phases visually is to choose a represetative dipole ad the record the agular distributio of all other dipoles that are at some fixed distace from the cetral dipole. This procedure is repeated for multiple represetative dipoles ad distaces, ad statistics of the relative agle distributio are compiled. Oe way of testig whether the field-dipole iteractio domiates the dipole order is to compare the relative agle distributio fuctio that is observed to the dilute limit give by equatio (7). Ackowledgmets The author wishes to thak REU studet Field N. Cady, ad Dr. Bruce H. Robiso for careful readig of the mauscript before submissio. 9
Refereces 3 4 5 6 7 C. Kittel, Itroductio to Solid State Physics (Joh Wiley ad Sos, Ic., 956) d ed. pp. 70-7. P. agevi, "Sur la Theorie du Magetisme," Joural de Physique 4, 678-688 (905). R. Kubo, Statistical Mechaics (North-Hollad, 999) d ed. pp. 4-5. R. H. Fowler ad E. A. Guggeheim, Statistical Thermodyamics (Cambridge Uiversity Press, 949) d ed. pp. 60-6. E. W. Weisstei, (Math World--A Wolfram Web Resource), http://mathworld.wolfram.com/sphericaltrigoometry.html D. P. adau ad K. Bider, A guide to Mote Carlo Simulatios i Statistical Physics (Cambridge Uiversity Press, 000) 348-378. D. Chadler, Itroductio to Moder Statistical Mechaics (Oxford Uiversity Press, 987) Ch. 6, 59-83. 0
Figure Captios: Figure : The geometry of two dipoles (dark arrows). γ is the relative agle betwee dipoles ad. θ ad θ are the projectio agles of dipoles ad with respect to the Z-axis., θ,ad θ are the vertex agles of the spherical triagle formed by the γ two dipoles ad the Z-axis. Figure : Histograms compiled from Mote Carlo simulatios with 000 dipoles i a exteral field with F = /5,,, ad 5 (icreasig from top to bottom). cos θ from Mote Carlo (bars) overlaid with the eft paels: Normalized histogram of Boltzma distributio (solid lies). Right paels: Normalized histogram of the relative agle distributio (equatio (7),text) cos γ from Mote Carlo (bars) overlaid with
Figure
Figure 3