Crystal field effect on atomic states

Crystal field effect on atomic states Mehdi Amara, Université Joseph-Fourier et Institut Néel, C.N.R.S. BP 66X, F-3842 Grenoble, France References : Articles - H. Bethe, Annalen der Physik, 929, 3, p. 33 (Selected Works of Hans A. Bethe, World Scientific, 997) - B. Bleaney and K.H.W. Stevens, 953, Rep. Prog. Phys. 6, p 8 - Books - M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill, NY, 964

Introduction spherical symmetry permutation properties of an electrons set Energy levels and wave functions of atoms Question : What happens if the atom belongs to a crystal, at a site of well defined, lower than spherical, symmetry? "Answer" : Properties of a quantum system in an environment of defined symmetry Direct group theory application : Hans Bethe 929

Summary -Back to the free atom - The atom inserted in a crystal Expansion of the crystal field potential The various degrees of crystal field correction: * Strong crystal field * Intermediate crystal field * Weak Crystal field - Crystal-field splitting of a multiplet (for a cubic symmetry) Decomposition of the terms of odd multiplicity Decomposition of the terms of even multiplicity Related theorems : Kramers, Jahn-Teller Crystal Field Operator Equivalent: Stevens method - Conclusion

Free atom Electron in a spherical potential V( r ) nucleus other electrons Symmetries of the hamiltonian : Schrödinger equation solutions : Full rotational = any angle about any axis Inversion center = nucleus Time reversal ψ ( r ) = R n (r).y l m (θ,ϕ) Radial part (Energy in V(r)) Angular part: Spherical Harmonic Hydrogen type Wave functions Quantum numbers : n, l, m

Example l = : Spherical Harmonics Y m (2l + ) (l - m)! l (θ,ϕ) = P m l (cosθ) e imϕ 4π (l - m)! For a given l, set of 2l+ functions with m= -l, -l+,..., l set of functions Y (θ,ϕ) = 2 Y (θ,ϕ) = 2 Y (θ,ϕ) = 2 3 sinθ eiϕ 2π 3 π cosθ 3 sinθ e iϕ 2π Rotation about the z axis : R z (α )Y l m (θ,ϕ) = Y l m (θ,ϕ α ) = e imα Y l m (θ,ϕ) transforms into itself Rotation about the x axis : R x (α )Y l m (θ,ϕ) = R y ( π 2 )R z(α )R y ( π 2 )Y l m (θ,ϕ) R y ( π 2 )Y (θ,ϕ) = 2 Y (θ,ϕ) + 2 Y (θ,ϕ) + 2 Y (θ,ϕ) R y ( π 2 )Y (θ,ϕ) =) 2 Y (θ,ϕ) 2 Y (θ,ϕ) R µ (α )Y m l (θ,ϕ) = C µ (α ) Y m l (θ,ϕ) l m = l set of functions with common l = Irreducible representation of the sphere rotation group

Beyond the Hydrogen Unfilled shell with x electrons for 2(2l+) electronic states single electron Hartree-Fock generalization : ψ ( r ) = R nl (r).y m l (θ,ϕ) Antisymmetrized x electrons wave-functions Intra-shell electrostatic interactions Tideous approach! First approximation : the hamiltonian commutes with total orbital and Total orbital momentum L = l i total spin momentum: S spin momentum = s i number i Energy classification according to quantum numbers L and S : atomic terms 2S+ L i Hund's Rules letter: S, P, D, F,... ) Among all S values consistent with Pauli's principle, the largest is of the lowest energy 2) Among all L values consistent with the first rule, the largest has the lowest energy.

Multiplet Wave Functions Orbital wave-function Spin wave-function Hund's ground term Atomic state : ψ = ψ L ψ S (2L+) X (2S+) degeneracy crystalline anisotropic environment electrons positions Spin-orbit coupling : J = Partial lifting of the L + S ψ L (2L+) orbital degeneracy associated J good quantum number H s.o. = A L S = A (J(J + ) S(S + ) L(L + )) 2 Third rule (Russel-Saunders) : For L and S resulting from Hund's rules, the J value with minimal energy is: J = L-S, for a less than half filled subshell (x<2l+) (A > ) J = L+S, for a more than half filled subshell (x>2l+) (A < ) Ground state Multiplet Wave Functions ψ i = L,S,J,J z

The atom inserted in a crystal ionic solid covalent solid metal Unfilled shell Electrostatic Interactions distant ions : ionic, covalent, metal electronic exchange: metal, covalent Crystal field hamiltonian : H CEF = e V( r i ) x i= V( r ) = aspherical potential invariant under the symmetries of the crystallographic site

Expansion of the crystal field potential Spherical harmonics expansion : V( r ) = A p q r p Y p q (θ,ϕ) p p= q= p satisfies Laplace's equation Where to stop the expansion? Single electron wave-function : ψ m ( r ) = R nl (r).y m l (θ,ϕ) Matrix elements of the perturbing hamiltonian : ψ m2 H CEF ψ m = e Y l m2* (θ,ϕ) Y l m (θ,ϕ). R nl (r) 2.V( r )dv sum of spherical harmonics of order =< 2l Terms of order > 2l yield zero contribution Terms with odd parity are forbidden for an inversion center The term of order zero doesn't contribute to the splitting 2 V( r ) = V + r 2 A q 2 Y q 2 (θ,ϕ) + r 4 A q 4 Y q 4 (θ,ϕ) + r 6 A q 6 Y q 6 (θ,ϕ) +... l = q= 2 q= 4 q= 6 l = l = 2 l = 3 + symmetry relations between the remaining A p q 4 6

V 6 V 2 V 4 Expansion for the cubic case V( r ) = V + r 2 A q 2 Y q 2 (θ,ϕ) + r 4 A q 4 Y q 4 (θ,ϕ) + r 6 A q 6 Y q 6 (θ,ϕ) +... q= 2 Cartesian coordinates 2 x = r sinθ cosϕ, y = r sinθ sinϕ, z = r cosθ Order 2 ( x,y,z) = A xx 2 x 2 + A yy 2 y 2 + A zz 2 z 2 + A xy 2 xy + A yz 2 yz + A zx 2 zx = A 2 (x 2 + y 2 + z 2 ) =A 2 r 2 No crystal field splitting to be expected for l = Order 4 ( x,y,z) = A xxxx 4 x 4 + A yyyy 4 y 4 + A zzzz 4 z 4 + A xxyy 4 x 2 y 2 + A yyzz 4 y 2 z 2 + A zzxx 4 x 2 z 2 = A 4 r 4 + A 4 (x 2 y 2 + y 2 z 2 + z 2 x 2 ) q= 4 Order 6 ( x,y,z) = A xxxxxx 6 (x 6 + y 6 + z 6 )+A xxyyyy 6 (x 4 (y 2 + z 2 ) + y 4 (x 2 + z 2 ) + z 4 (x 2 + y 2 ))+A xxyyzz 6 (x 2 y 2 z 2 ) =A 6 r 6 + A 6 r 2 (x 2 y 2 + y 2 z 2 + z 2 x 2 )+A 6 2 x 2 y 2 z 2 4 6 q= 6 V(x,y,z) = V + V 2 (x,y,z) + V 4 (x,y,z) + V 6 (x,y,z) +... First crystal field splitting term V(x,y,z) = V (r) + (A 4 +A 6 2 r 2 ) (x 2 y 2 + y 2 z 2 + z 2 x 2 ) + A 6 2 x 2 y 2 z 2 +... Number of cubic crystal field parameters reduced to : for l = for l = 2 (iron group ions) 2 for l = 3 (lanthanides) polar plot

The various degrees of crystal field correction Intra-shell electrostatic interactions: H ee (Hund's rules) atomic shell: x electrons for 2(2l+) states r max Spin-Orbit coupling: H SO ("Hund's" third rule) Crystal Electric Field : H CEF Corrective hamiltonian : H = H ee + H CEF + H SO Usually, different magnitudes for H ee, H CEF and H SO Sequence of applied perturbations Strong crystal field Intermediate crystal field Weak crystal field H CEF H ee H ee H ee H CEF H SO H SO H SO H CEF

In real systems Strong crystal field : 3d Iron group exposed, nearly empty of filled 3d shell, light atoms H CEF > H ee > H SO Intermediate crystal field : 3d Iron group exposed, close to half-filled 3d shell, lighter atoms H ee >H CEF > H SO Weak crystal field: 4f Lanthanides series inner shell, heavy atoms H CEF < H SO <H ee L, S L, S L, S, Γ L, S, J L, S, Γ, Γ SO L, S, J, Γ

Crystal-field splitting of a multiplet G.T. approach : inside a given energy level, eigen functions of the hamiltonian transform according to an irreducible representation of the hamiltonian space group. free atom quantum state basis irreducible representations of the hamiltonian point group Number of levels and respective degeneracies of the perturbed hamiltonian Strong crystal field : Starting representation D l { Y l l (θ,ϕ),y l l (θ,ϕ),...,y l l (θ,ϕ)} Intermediate crystal field : Crystal Field Hamiltonian acting on the multi electronic orbital wave-functions ψ L Quantization of orbital momentum + starting hamiltonian of spherical symmetry Orbital multiplet L for x electrons { L, L z = L, L, L z = L,..., L, L z = L } Transform identically Share the same representation set of 2L+ spherical harmonic { Y L L (θ,ϕ),y L L (θ,ϕ),...,y L L (θ,ϕ)}

Weak crystal field : starting manifold associated with L,S and total angular momentum J J = integer : { J, J z = J, J, J z = J,..., J, J z = J } Y J J Transforms as J (θ,ϕ),y J (θ,ϕ),...,y J J (θ,ϕ) { } J = half integer no associated set of spherical harmonics Generalization? Crystal field acting on a manifold of odd degeneracy Starting manifold :l, L or integer J { L, L z } { J, J z } { Y m l (θ,ϕ)} Spherical harmonics representation D l Irreducible spherical representation Crystal Field Splitting Γ ι Point group irreducible representations

O h Point group of the octaedra Rotation Classes: E C 3 C 4 C 2 = (C 4 ) 2 C 2 Example : Cubic point group C 4, C 4 -, C 2 = (C 4 ) 2 C 2 C 3, C 3 - Inversion/reflection Classes: I I C 3 = C 6 /reflection I C 4 =C 4 /reflection I (C 4 ) 2 = σ h I C 2 = σ v 24 elements 24 elements O h = O I Parity (-) l for atomic states Octaedra rotation group O h has twice as many classes and I.R. as O l even : decomposition over the positive IRs of O h l odd : decomposition over the negative IRs of O h Only half of the rows/columns is useful for the decomposition one can restrict to the 24 elements of the O rotations group

Number of I.R. = number of classes 5 irreducible representations Construction of the O group characters table I.R. dimension Dimensionality theorem: n i 2 Γ, Γ 2, Γ 3, Γ 4, Γ 2 + 2 +2 2 +3 2 + 3 2 =24 5 Bethe's notations Γ Γ 2 Γ 3 Γ 4 Γ 5 i = g Group order Weighted orthogonality between rows Orthogonality between columns Class multiplication : χ Γ (R n )χ Γ (R m ) = n Γ g n g m Rotations only : O characters table E 8 C 3 3 (C 4 ) 2 6 C 2 6 C 4 Γ Γ 2 - - Γ 3 2-2 Γ 4 3 - - Γ 5 3 - - mult. constant for R p in R n X R m p C p nm g p χ Γ (R p )

Characters for rotation classes Spherical harmonics representation R z (α )Y l m (θ,ϕ) = e imα Y l m (θ,ϕ) D l e ilα... e i(l )α... D l (R z (α )) =............... e ilα Character of a rotation of angle α for D l : χ l (R z (α)) = Tr(D l (R z (α))) = l e ilα = e ilα 2l (e iα ) n m= l n= χ l (R(α )) = e ilα e i(2l+)α e iα = e ilα i(2l+)α /2 e e i(2l+)α /2 e i(2l+)α /2 e iα /2 e iα /2 iα /2 e χ l (R(α )) = sin((l + / 2)α) sin(α / 2) For a rotation of angle α about any axis character for a rotation of angle α class

Decomposition of the terms of odd multiplicity Characters of the O group classes for a spherical harmonics representation of dimension 2l+ sin((l + / 2)α) Identity : E: χ l (E) = 2l+ Rotations : χ l (R(α )) = sin(α / 2) C 3 : χ l (R( 2π sin( 2π for l =, 3, 6,... (l + / 2)) 3 )) = 3 sin( 2π = for l =, 4,... 6 ) - for l = 2, 5,... C 2 or (C 4 ) 2 : χ l (R(π )) = sin(π(l + / 2)) sin( π 2 ) = for l =, 2, 4,... - for l =, 3, 5,... C 4 : χ l (R( π sin( π (l + / 2)) 4 )) = 4 sin( π = 8 ) for l =,, 4, 5... - for l = 2, 3, 6,... Decomposition formula : class order a Γ = g group order R χ l (R). χ Γ (R) * = g i χ l (R). χ g Γ (R) * i

Application for l =,, 2, 3, 4 l = l = l = 2 l = 3 l = 4 Spherical harmonics characters 3 5 7 9 E "Normalized" O group characters table - - - (C 4 ) 2 χ( E) χ( C 3 ) χ((c 4 ) 2 ) χ(c 2 ) χ(c 4 ) - - x g /g i x g i /g x g i /g x g i /g x g i /g - - C 4 C 3 C 2 Γ /24 /3 /8 /4 /4 Γ 2 /24 /3 /8 -/4 - /4 Γ 3 /2 - /3 /4 Γ 4 /8 -/8 -/4 /4 Γ 5 /8 -/8 /4 - /4 Γ Γ 2 Γ 3 Γ 4 Γ 5 2 3 3 2 3 4 In cubic field, no splitting for l Γ 5 (t2g) Composition table nb levels degeneracy Strong crystal field 3d approach : l = 2 Γ 3 (eg)

Decomposition of the terms of even multiplicity (double group approach) Generalization : χ J (R(α)) = sin((j + / 2)α) sin(α / 2) for J half integral Double valued characters : χ J (R(α + 2π )) = sin((j + / 2)(α + 2π )) sin((α + 2π ) / 2) = sin((j + / 2)α) sin(α / 2 + π ) = χ J (α) Bethe's approach : R(2π) and the double group χ J (R(α + 2π ) 2 ) = χ J (α) Introduction of a new group element R : R = R(2π ) E and R(4π )=R 2 = E Double group (octaedral case): O = O R 48 elements in 8 classes 3 additional classes: R, RC 3, RC 4 3 additional double valued I.R.: Γ 6, Γ 7, Γ 8 dimensions 2 2 4

Double group : O' characters table E R 8 C 3 8 RC 3 3 (C 4 ) 2 + 3 R(C 4 ) 2 6 C 2 + 6 RC 2 6 C 4 6 RC 4 Γ Γ 2 - - - Γ 3 2 2 - - 2 Γ 4 3 3 - - Γ 5 3 3 - - - Γ 6 2-2 - 2 2 Γ 7 2-2 - 2 2 Γ 8 4-4 - Identity : Characters of the J manifold representation χ J (E) = 2J+ 2π rotation : χ J (R) = -(2J+) Rotations : χ J (R(α )) = sin((j + / 2)α) sin(α / 2) Composition table J = /2 J = 3/2 J = 5/2 J = 7/2 J = 9/2 degen. Γ 6 2 Γ 7 2 2 Γ 8 3 2 4 nb levels 2 3 2 2

Related theorems Kramers: (anti-unitary Time reversal for half integer spins) In presence of only electrostatic fields, the energy levels of a system containing an odd number of electrons have an even degeneracy. odd number of electrons even number of electrons Jahn-Teller: Kramers ions half integral S and J minimal degeneracy = 2 Non-Kramers ions integral S and J minimal degeneracy = double-valued representations single-valued representations Any non-linear molecular system in a degenerate electronic state will be unstable and will undergo distortion to form a system of lower symmetry and lower energy thereby removing the degeneracy crystal field ground-state degeneracy > Kramers minimum Symmetry lowering via crystal distortion