Irreducible Cartesian andspherical tensors

Tadeusz Bancewiczhttp://zon8.physd.amu.edu.pl/~tbancewi

Angers POMA lab., june 2002

Outline

• Previous work

• Irreducible tensors in Cartesian form

• The dual picture; Cartesian and spherical tensors

• Linear light scattering

• Hyper-Rayleigh scattering

• Polarization states of HR scattering

• references• F. G. Fumi, Il Nuovo Cimento, vol.IX, N. 9, 739 (1952).

• J. M. Normand and J. Raynal, J. Phys. A, Math. Gen. 15, 1437 (1982).

• J. Jerphagon, D. Chemla and R. Bonneville, Adv. Phys.,27, 609 (1978).

• U. Fano and G Racach, Irreducible tensorial sets, (New York 1959).

• S. J. Cyvin and J. E. Rauch and J. C. Decius J. Chem. Phys. 43, 4083 (1965).

• P. D. Maker, Phys. Rev. A 1, 923 (1970).

• J. H. Christie and D. J. Lookwood J. Chem. Phys. 54, 1141 (1971).

• K. Altman and G. Strey, J. Raman Spectrosc. 12, 1 (1982).

• S. Kielich and T. Bancewicz, J. Raman Spectr. 21, 791 (1990).

Physical properties of matter represent the coupling action ofmatter between influences exerted and effects obtained(VOIGT–1928). However not all quantities entering ineffect–influence relations are physical properties of matter. In thisrespect one distinguishes between field quantities and matterquantities. While matter quantities are in themselvesconsequences of the intimate structure and cohesion of matter,field quantities simply represent field induced in matter by externalcauses: typical examples are susceptibility (matter quantity) andLorentz field (field quantity).

Generally more than one number is required to specify a physicalproperty at a point in a material system: when this occurs, theproperty is called anisotropic since the property depends ondirection. When only one number is necessary in any materialsystem, the property does not depend on direction and is calledscalar. The numbers specifying a scalar property at a given pointtransforms as a tensor of order zero of polar or axial nature. Thenumbers specifying anisotropic properties frequently transforms asof polar and axial tensors of order greater then one.

Macroscopic space symmetry is responsible for the reduction ofindependent matter tensor components in symmetrical material systems.Matter tensors component must be invariant under the existingsymmetry: The number of independent components of a mattertensor in a system of a given symmetry is invariant undercoordinate transformation.

k =1

N

∑g∈G

χm

(g) χ(g) (1)

Moreover it is convenient to decompose a tensor into sets thattransform irreducibly, both with respect to the rotation group O(3) andthe permutation group Si of its indices.

Gp = G ⊗ P (2)

With respect to the three–dimensional rotation group, a tensorT = T 1...1 of order l and with each of its ranks equal to one is definedby 3l components. The Cartesian components (Ti1...il ; ik = x, y, z) andthe reducible spherical components (T 1...1

m1...ml; mk = 1, 0 − 1 are

characterized by their law of transformation:

Ti1...il =∑

i′1...i

′l

Ti′1...i

′l

l∏k=1

Ri′k

ik(3)

T 1...1m1...ml

=∑

m′1...m

′l

T 1......1m

′1...ml

′

l∏k=1

D1m

′k mk

(R) (4)

where D1m

′k mk

(R) is the irreducible rotation matrix.

A rank–l Cartesian tensor T(l) has 3l components; an irreduciblerank–l tensor is labeled by its weight j. It has (2j+1) componentswhich subtend a weight–j irreducible representation of the rotationgroup O(3). If several linearly independent irreducible tensors ofthe same weight j appear in the reduction of T(l) they can bedistinguished by an additional superscript τ , called the seniorityindex, so that the most general form of a tensor reduction is

T(l) =∑⊕τ,j

T(τ,j)(l) (5)

This equation is often refered to as the reduction spectrum of T(l).

Any tensor T 1...1 of order l greater than one can be decomposedinto irreducible tensors T = T (1...1,λ α) j of integer rank j(0 ≤ j ≤ l), and characterized by a coupling scheme specified byl-2 intermediate couplings λ = (λk; k = 1, ..., l − 2) and theirorder symbolically denoted by α. One then defines atransformation

T (1...1,λ α) jm =

∑i1...il

Ti1...il < i1...il|λ α , j m > (6)

Attention is now focused on the coupling to the highest value l of j. By definition, an irreducible Cartesian tensor,

also said to in natural form (Coope et al 1965), is a tensor of order l with l ranks one, such that all its irreducible

tensors vanish except the one with the highest rank l, denoted by (Tl)l. Such tensor is completely symmetric and

traceless.

αβl1

m1

1=

∑j1φ

C(l1 1 j1; β m1 φ) α(l1 1)j1φ (7)

∆A(1 , 1)K M =

∑m1 m2

C(1 1 K; m1 m2 M) ∆Am11

m2

1 (8)

βmim1m2li 1 1 =

∑a,ξ,Ji,Mi

C(11a; m2m1ξ) C(aliJi; ξmiMi)

βJiMi [(11)ali] (9)

Dipole moment

Q01 = µz (10)

Q±11 = ∓

(µx ± i µy

)√

2(11)

Quadrupole moment

Q02 = Θzz (12)

Q±12 = ∓

√2

3

(Θzx ∓ i Θyz

)(13)

Q±22 =

1√6

(Θxx −Θyy ± i Θxy

)(14)

Octopole moment

Q03 = Ωzzz (15)

Q±13 = ∓

√3

4

(Ωzzx ± i Ωzzy

)(16)

Q±23 =

√3

10

(Ωzxx −Ωzyy ± i Ωxyz

)(17)

Q±33 = ∓

1√20

(Ωxxx ± i Ωxxy − Ωxyy ∓ i Ωyyy

)(18)

For nitrogen ( a D∞h symmetry molecule) in its principal axesreference frame, reducible spherical dipole–octopole αm1 m2

1 3 andthe dipole–dipole–quadrupole βm1 m2 m3

1 1 2 tensors have the followingnon-zero components which are related with their Cartesiancounterparts as follows:

a) for the dipole–octopole tensor, assuming the Cartesiancomponents Ez,zzz and Ex,xxx as mutually independent

α0 01 3 = Ez,zzz (19)

α1−11 3 = α−1 1

1 3 =

√8

3Ex,xxx (20)

b) for the dipole–dipole–quadrupole tensor, assuming the Cartesiancomponents Bxx,xx, Bzz,zz, Bxz,xz and Bxx,zz as mutuallyindependent

β0 0 01 1 2 = Bzz,zz (21)

β±1∓1 01 1 2 = −Bxx,zz (22)

β0±1∓11 1 2 = β±1 0∓1

1 1 2 = −2√3

Bxz,xz (23)

β±1±1∓21 1 2 =

2√6

(2 Bxx,xx + Bxx,zz

)(24)

The irreducible polarizability tensors are constructed from — by

standard coupling methods:

A(13)j 0 =

∑m

Cj 01 m 3−m αm−m

1 3 (25)

Bj 0((11)l, 2) =∑

m1,m2

Cl−(m1+m2)1 m1 1 m2

Cj 0l−(m1+m2) 2 m1+m2

βm1 m2−(m1+m2)1 1 2 (26)

where Cj m3j1 m1 j2 m2

stands for the Clebsch-Gordan coefficient. Thelinear combinations (25) and (26) give the mutually independentcomponents and of, respectively, the irreducible sphericalmultipolar polarizability dipole-octopole (E) and dipole–dipole–quadrupole (B) tensors.

For second–rank tensors the reduction reads

Tij = T(0)ij + T

(1)ij + T

(2)ij (27)

where

T(0)ij =

(1

3Tkk

)δij; T

(1)ij =

1

2(Tij − Tji)

T(2)ij =

1

2(Tij + Tji) −

(1

3Tkk

)δij

(28)

A tensor of rank 3 symmetric in the permutation of two of itsindices can be formed by multiplication of a symmetric rank–2tensor and a rank–1 tensor and its reduction spectrum is

1 ⊗ (0 ⊕ 2) = (1 ⊗ 0) ⊕ (1 ⊗ 2) = (1) ⊕ (1 ⊕ 2 ⊕ 3)

(29)

For a fully symmetric rank–3 tensor the reduction spectrum is1 ⊕ 3. In the general case a third–rank tensor has 33 = 27

independent components. It is a sum of one pseudo–scalar (j=0),three vectors (j=1), two pseudo–deviators (j=2) and one septor(j=3) (27 = 1 + 3 x 3 + 2 x 5 + 7).

A rank–4 tensor has 34 = 81 components. According to table 1 it isthe direct sum of three scalars (j = 0 , 2 J+1 = 1), sixpseudo–vectors (J = 1, 2 J + 1 = 3), six deviators (J =2 ,2 J+1 = 5), three pseudo–septors (J=3 , 2 J+ 1 = 7) and one nonor(J=4 , 2 J+ 1 =9)

3 × 1 + 6 × 3 + 6 × 5 + 3 × 7 + 1 × 9 = 81 (30)

A number of physical properties are described by rank–4 tensors;some are listed in table 1 with their intrinsic symmetry andreduction spectrum.

THE DUAL PICTURE–CARTESIAN AND SPHERICALTENSORS

For low–rank tensors (l=0, 1) the transformation to sphericaltensors is straightforward and unambiguous, since the tensors areirreducible (3l=2 j+1). In the case of scalars l=j=0 the connectionis just identity. Vector quantities l=j=1 we can expand either inthe Cartesian basis |i > with i=x, y, z or in a spherical basis |1 µ >

with µ=+1, 0, -1. The unitary transformation < i |1 µ > betweenthe two basis is

− 1√

20 1√

2

− i√2

0 − i√2

0 1 0

(31)

i.e. for the component of a tensor T

T1 = −(Tx + i Ty)√

2T0 = Tz T−1 =

(Tx − i Ty)√2

(32)

In the case of higher-rank tensors, the correspondence is lessobvious. Starting from a rank–n tensor Ti1,i2...in one first changesthe Cartesian coordinates into spherical coordinates through aproduct of n unitary transformations < i |1 µ >. This results in areducible tensor Tµ1,µ2...µn , which is reduced by use of thewell–known properties of the Clebsch–Gordan and Racah 3n–jcoefficients to give the irreducible tensors T

(τ j)m (the seniority index

τ distinguishes among various spherical tensors with identicalweight appearing in the reduction of T.)

The transformation from Cartesian tensors T(i) to irreduciblespherical tensors T

(τ j)m is a unitary transformation

T(i) = < (i)|τ j m > T (τ j)m ; T (τ j)

m = < τ j m|(i) > T(i) (33)

where (i) stands for a whole set of Cartesian indices. Thetransformation satisfies the usual unitary properties

< τ j m|(i) > < (i)|τ ′j

′m

′> = δττ

′ δj j′ δm m

′

< (i)|τ j m > < τ j m|(i′) > = δ(i) (i

′)

< τ j m|(i) > = < (i)|τ j m >∗ (34)

The < τ j m|(i) > coefficient is build up by a product oftransformations from the Cartesian basis to a spherical one

Tµ1,µ2...µn = < 1 µ1|i1 >

< 1 µ2|i2 > ... < 1 µn|in > Ti1,i2...in (35)

and then by reducing the spherical tensor T(µ) by use of< τ j m|µ1 µ2...µn > coefficients expressible in terms of 3n− j

symbols, so that the overall transformation is, in a condensed form

Tµ1,µ2...µn =< τ j m|µ1 µ2...µn > < 1 µ1|i1 >

< 1 µ2|i2 > ... < 1 µn|in > Ti1,i2...in (36)

LINEAR LIGHT SCATTERING

Iωij = Aω N

⟨mim

∗j

⟩Iωn = Aω N

⟨n∗i nj mi m∗

j

⟩= Aω N

⟨(n∗i Aik ek

) (nj Ajp e∗p

)⟩E2

Iωn = Aω N

⟨(W : A) (W : A)

⟩E2 (37)

Iωn = AωN

1

2j + 1

(W(j) • W(j)

) ⟨(A(j) • A(j)

)⟩E2

(38)

Wij = n∗i ej (39)

isotropic scattering

φ00 =1

3Wkk δij

1

3W ∗

pp δij =Wkk W ∗

pp

3=

(n∗ · e) (n · e∗)3

(40)

F00 =1

3Akk δij

1

3A∗

pp δij =Akk A∗

pp

3= 3 A2 (41)

antisymmetric scattering

φ11 =1

3

1

2(Wij − Wji)

1

2(W ∗

ij − W ∗ji)

=1

6

[1− (n∗ · e∗) (n · e)

](42)

F11 =1

2(Aij − Aji)

1

2(A∗

ij − A∗ji)

=1

2

(Aij A∗

ij − Aij A∗ji

)(43)

anisotropic (depolarized) scattering

φ22 =1

5

[1

2(Wij + Wji) −

1

3W ∗

kk δij

][1

2(W ∗

ij + W ∗ji)−

1

3W ∗

kk δij

]

=1

10

[1 + (n∗ · e∗) (n · e)

]−

(n∗ · e) (n · e∗)15

(44)

F22 =3

(Aij A∗

ij + Aij Aji

)− 2Aii A∗

jj

6(45)

for real n and e

φ22 =1

30

[3 + (n · e)2

](46)

for real and symmetric Aij

F22 =3 Aij Aij − Aii Ajj

3(47)

Within principal axis of a molecule only diagonal elements Axx,Ayy and Azz of A are different from zero, then

F22 =(Axx −Ayy)2 + (Axx −Azz)2 + (Ayy −Azz)2

3STM︷︸︸︷=

2 (Azz − Axx)2

3(48)

Iiso =1

33 A2 = A2 (49)

Idep (V H)(n ⊥ e) =1

10

2β2

3=

1

15β2 (50)

THE POLARIZATION STATE OF HYPER-RAYLEIGHSCATTERED LIGHT

(W(s1,j1) W(s2,j2)∗

)= W

(s,J)IJK W

(s,J)∗IJK (51)

(β(s,J) β(s,J)∗

)= β

(s,J)ijk β

(s,J)∗ijk (52)

When decomposing a Cartesian tensor Tijk, symmetric in its lasttwo indices, into its Cartesian irreducible terms T

(s,J)ijk one readily

finds that one has two linearly independent terms of the first rank(J=1), a single term of the second rank (J=2) and a single term of

the third rank (J=3). Obviously, the third rank term T(3)ijk equal in

rank to the tensor Tijk itself, is completely symmetric. Separationof two linearly independent first–rank terms is not unique. Here,we shall separate its the completely symmetric part

T(S,1)ijk =

1

15

[δij (2Tnnk + Tknn)

+ δik

(2 Tnnj + Tjnn

)+ δjk (2 Tnni + Tinn)

](53)

and the non–symmetric ‘remainder’:

T(N,1)ijk =

1

6

[δij (Tnnk − Tknn)

+ δik

(Tnnj − Tjnn

)− 2 δjk (Tnni − Tinn)

](54)

Clearly, the tensor T(N,1)ijk is still symmetric in its last two indices,

and so is the irreducible Cartesian second rank tensor T(s,2)ijk . Thus

among the irreducible Cartesian tensors we are confronted with twopossibilities with regard to their permutational symmetry:(1) complete permutational symmetry–we denote this case by s=1(2) symmetry in the last two indices–to be denoted by s=2

The autocorrelation function of the scattered radiation can now bewritten in the following form:

In(t) = A2 ω∑

s1,s2,J1,J2

φ0s1J1s2J2

F 0s1J1s2J2

(t) (55)

as the sum of six terms, four of which are‘quadratic’,J1 = J2 = 1, s1 = s2 = 1; J1 = J2 = 1, s1 = s2 =

2; J1 = J2 = 2, s1 = s2 = 2; J1 = J2 = 3, s1 = s2 = 1 and twomixed terms J1 = J2 = 1, s1 = 1, s2 = 2 andJ1 = J2 = 1, s1 = 2, s2 = 1.

φ01111 =

1

45

4 (n∗ · e) (n · e∗) + (e · e) (e∗ · e∗)

+ 2 (n∗ · e) (n · e) (e∗ · e∗) + 2 (n∗ · e∗) (n · e∗) (e · e)

(56)

F 01111 =

1

15

⟨4βiij βkkj + 2βiij βjkk + 2 βijj βkki + βijj βikk

⟩(57)

φ01212 =

1

9

(n∗ · e) (n · e∗) + (e · e) (e∗ · e∗)

− (n∗ · e) (n · e) (e∗ · e∗)− (n · e∗) (n∗ · e∗) (e · e)

(58)

F 02121 =

1

3

⟨βiij βkkj − βiijβjkk − βijj βkki + βijj βikk

⟩(59)

φ02111 =

√5

45

(e · e) (e∗ · e∗) − 2 (n∗ · e) (n · e∗)

+ 2 (n∗ · e) (n · e) (e∗ · e∗)− (n∗ · e∗) (n · e∗) (e · e)(60)

F 02111 =

√5

15

⟨βijj βikk − βiijβjkk + 2 βijj βkki

− 2 βiij βkkj

⟩(61)

φ01121 =

√5

45

(e · e) (e∗ · e∗)

− 2 (n∗ · e) (n · e∗) + 2 (n∗ · e∗) (n · e∗) (e · e)

− (n∗ · e) (n · e) (e∗ · e∗)∗ ∗ (62)

F 02111 =

√5

15

⟨βijj βikk − βijjβkki + 2 βiij βjkk

− 2 βiij βkkj

⟩(63)

We note that φ02111 = (φ0

1121)∗ and F 0

2111 = (F 01121)

∗.

φ02222 =

1

15

2 − (n∗ · e) (n · e∗) − (e · e) (e∗ · e∗)

− 2 (n∗ · e∗) (n · e) + (n∗ · e) (n · e) (e∗ · e∗)

+ (n∗ · e∗) (n · e∗) (e · e)

(64)

F 02222 =

1

3

⟨2 βijk βijk + βijj βkki + βiijβjkk

− 2 βijk βjik − βijj βikk − βiij βkkj

⟩(65)

φ01313 =

1

105

5 − (e · e) (e∗ · e∗) + 10 (n∗ · e∗) (n · e)

− 4 (n∗ · e) (n · e∗) − 2 (n∗ · e) (n · e) (e∗ · e∗)

− 2 (n∗ · e∗) (n · e∗) (e · e)

(66)

F 01313 =

1

15

⟨5 βijk βijk + 10 βijk βjik

− 4 βiij βkkj − 2 βiijβjkk − 2 βijj βkki − βijj βikk

⟩(67)

On the assumption that the hyperpolarizability tensor βijk is completelysymmetric, the only non–zero molecular parameters will be F 0

1111(t) andF 0

1313(t).

F 01111 =

3

5βiij βjkk (68)

F 01313 = βijk βijk −

3

5βiij βjkk (69)

F 01111 =

∑M

|B1M |2 (70)

F 01313 =

∑M

|B3M |2 (71)

In particular, for the symmetry groups C4...C∞ and C4V ...C∞V ,the totally symmetric tensor has the formb333 = a; b311 = b131 = b113 = b322 = b232 = b223 = b. Then forthe molecular parameters we obtain:

F 01111 =

3

5(a + 2 b) = |B1

0 |2 (72)

F 01313 =

2

5(a − 3 b) = |B3

0 |2 (73)