Tensors for Physics

11.3 Generalized Cross Product,-Tensors 189 11.3.3 Action of the Differential Operator 13.3 Irreducible Spin Tensors Applicatio ...

190 11 Isotropic Tensors Δrrμ 1 rμ 2 ···rμ =(+ 1 )r−^2 rμ 1 rμ 2 ···rμ, andconsequentlytheresult(11.31)isrecovered.This,inci ...

11.4 Isotropic Coupling Tensors 191 11.4 Isotropic Coupling Tensors 11.4.1 Definition ofΔð‘;^2 ;‘Þ-Tensors Isotropic tensorsΔ(· ...

192 11 Isotropic Tensors Δμν,λκ,σ τΔμ′ν′,λκ,σ′τ′= 5 48 (Δμν,μ′ν′Δστ,σ′τ′+Δμν,σ′τ′Δστ,μ′ν′) − 1 24 Δμν,σ τΔμ′ν′,σ′τ′. (11.39) The ...

11.4 Isotropic Coupling Tensors 193 aμbλ aλaν ≡Δμν,λκ,σ τ aλbκ aσaτ = 1 6 a^2 aμbν+ 1 6 a·baμaν. (11.46) Puttingb=a, in these eq ...

194 11 Isotropic Tensors Now, letAbe the traceless tensor constructed from the components of the vector b,viz.Aν 1 ν 2 ···ν=bμ ...

11.6 Coupling of Second Rank Tensors with Irreducible Tensors 195 bνbλ bμ 1 bμ 2 ···bμ =bνbλbμ 1 bμ 2 ···bμ + 2  2 + 3 b^2 Δ ...

196 11 Isotropic Tensors All these triple products of three tensors are invariant under a rotation of the coordi- nate system. N ...

11.7 Scalar Product of Three Irreducible Tensors 197 =(u 1 ,u 2 ,r)= ∑  ′∑  1 ∑  2 φ( 1 , 2 ,)(r)P(^1 ,^2 ,)(u 1 ,u 2 ...

Chapter 12 Integral Formulae and Distribution Functions Abstract This chapter is devoted to integral formulae and distribution f ...

200 12 Integral Formulae and Distribution Functions 12.1.1 Integrals of Products of Two Irreducible Tensors Integrals ∫ ...d^2 ̂ ...

12.1 Integrals Over Unit Sphere 201 The orientational average of the corresponding multipole potential tensorsX···rather than th ...

202 12 Integral Formulae and Distribution Functions whereSis symmetric traceless second rank tensor, yields a product of the for ...

12.2 Orientational Distribution Function 203 whered^2 uis the surface element on the unit sphere, the average〈ψ〉of an angle depe ...

204 12 Integral Formulae and Distribution Functions The expansion tenors pertaining to= 1 , 2 ,3 and=4are φμ= √ 3 uμ,φμν= √ 15 ...

12.2 Orientational Distribution Function 205 leads to εμν=n ( εiso+ 2 3 ) 2 〈αμν〉+... , whereεisois the orientationally averaged ...

206 12 Integral Formulae and Distribution Functions Likewise, up to second order inβ 1 , the distribution function is equal to f ...

12.2 Orientational Distribution Function 207 The coefficientc(^2 |^11 )defined in (12.24) is given by c(^2 |^11 )=〈P 2 (x)〉/〈P 1 ...

208 12 Integral Formulae and Distribution Functions and similarly, from (12.10) follows FμνFλκFστFαβ〈φμνφλκφστφαβ〉 0 = 15 7 (Fμν ...

12.2 Orientational Distribution Function 209 Phenomenologically, theelectro-optic Kerr effectis described by εμν = 2 KEμEν, (12. ...