ECEN 5005 Crystals, Nanocrystals and Devce Applcatons Class 9 Group Theory For Crystals Dee Dagram Radatve Transton Probablty Wgner-Ecart Theorem Selecton Rule
Dee Dagram Expermentally determned energy levels o rare earth ons. - Compled by Dee, 968 Snce the 4 energy levels o rare earth ons are not strongly dependent on crystal eld, Dee dagram provdes a good qualtatve descrpton or the energy levels o rare earth ons n most solds.
Radatve Transtons Consder an electron mang a transton between two states, and. The transton probablty s gven by W π =, V = π V ρ ( ω) where ω = E /. E - Ferm s golden rule Electronc transtons can be nduced by several derent mechansms whch are determned by the actual orm o V, the nteracton Hamltonan that nduces the transton. The strongest transton s electrc dpole (ED) transton ( V = er E ), e the second largest s magnetc dpole (MD) ( V = (l + s) B ), and m the thrd s electrc quadrupole (EQ) ( V ( r)( E r) = ), etc. The relatve strengths o these three transtons are approxmately ED : MD : EQ µ c ea π λ B o 5 6 ( ea ) : : :0 :0 o where a o and µ B are Bohr radus and Bohr magneton, respectvely. Thereore, when all transtons are allowed, ED process always domnates. However, the crystal symmetry produces many stuatons n whch ED process s orbdden. In those cases, transtons occur manly by MD or EQ processes. 3
Strength o Radatve Transton The strength o a transton s oten represented by a dmensonless quantty called oscllator strength,. - The term oscllator strength orgnated rom the classcal descrpton o radaton due to oscllatng dpole. Quantum mechancal denton o oscllator strength s πω = 3 e g where ħω s the energy derence between the ntal and nal levels, g s the degeneracy o the ntal level and µ e s the electrc dpole moment. Sometmes, the strength o electrc dpole transton, S (ED), s used nstead o the oscllator strength,. S, ( ED) = µ, Because o the sgn o ω, the oscllator strength,, s postve or absorpton and negatve or emsson process. However, n practce, the absolute value o ω s usually used or calculatng and thus s always regarded as postve quantty. For an allowed electrc dpole transton, the dpole matrx element, µ e e µ, has a magntude ~ ea o, where a o s the Bohr radus. Ths e gves a value or the oscllator strength o the order o unty, ~. 4
Strength o Radatve Transton For magnetc dpole transtons, the oscllator strength s smlarly dened except that the electrc dpole moment µ e s replaced by the magnetc dpole moment dvded by speed o lght, µ m /c. The oscllator strength s related to the spontaneous emsson rate, A(ED), whch s nverse o the radatve letme τ R. nω e Eloc A( ED) = = 3 τ 4πε mc E R o - E loc /E s the local eld correcton term, whch accounts or the derence between the macroscopc eld and the actual eld near the lumnescence center. - For crystals wth hgh symmetry (e.g. cubc), Eloc E = n + 3 (n = reractve ndex) Substtutng approprate values or the physcal constants, we obtan τ R =.5 0 4 λ n / 9 where λ o s the vacuum wavelength n µm. o ( n + ) The analogous relatonshp or magnetc dpole process s τ R =.5 0 For λ o = 0.5 µm, n =.5 and = (allowed dpole transton), we obtan τ R ~ 0 8 s. 4 λ n o 3 5
Strength o Radatve Transton Although the theoretcally estmated radatve letme or an allowed ED transton s ~0 ns, many lumnescent ons n crystals exhbt much longer letmes on the order o µs or even ms. Ths ndcates the oscllator strength s much smaller than unty. These transtons are due to ether wealy allowed (or partally orbdden) ED process or MD (EQ) process. What maes a transton allowed or orbdden? Recall the transton probablty s proportonal to the square o the matrx element, µ. - For electrc dpole transton, µ = er, where the summaton s over all electrons nvolved n the transton. - For magnetc dpole process, µ = ( l + s ) Whether or not a transton s allowed and, allowed, how strong t s e m depends on the value o ths matrx element. When the matrx element s zero, the transton probablty s zero thus the transton s orbdden. Snce derent processes nvolve derent operators n the matrx element, havng a zero matrx element or ED process does not mean all matrx elements are zero. They may have non-zero matrx elements or MD or EQ processes.. 6
Wgner-Ecart Theorem In order to evaluate the matrx elements, we need to nvoe the Wgner-Ecart theorem. Consder a matrx element between two states, Γ and Γ, whch belong to the row γ (γ ) o the rreducble representaton Γ (Γ ). Also, suppose the operator X transorms accordng to the row γ o the rreducble representaton Γ. To clearly specy ths property, we denote the operator as ( Γ) X. The Wgner-Ecart theorem states Γ X X l ( Γ, γ) Γ = Γ ( Γ) Γ Γ, γ Γ, Γ, γ where l Γ s the dmensonalty o Γ. Γ X ( Γ) Γ s called the reduced matrx element. Γ - Whle we wll not attempt to evaluate ths quantty n ths course, we nevertheless note that t s ndependent o γ. - Thus, the reduced matrx element s evaluated once or partcular values o Γ s and γ s, then we can nd all other related matrx elements smply by loong up the tabulated values o the Clebsch- Gordan coecents, Γ Γ, Γ, γ. 7
Selecton Rule Wthout evaluatng the reduced matrx element, we can stll nd that the matrx element s zero when the Clebsch-Gordan coecent s zero. The transton between two states whose rreducble representatons and the row numbers result n zero Clebsch-Gordan coecents s orbdden. Selecton rule. Although we can loo up the extensvely tabulated Clebsch-Gordan coecents or the system o our nterest, we can extract a general rule rom the denton o Clebsch-Gordan coecents. Recall rom Class 9 that the drect-product representaton Γ Γ can be reduced to a sum o rreducble representatons and the coecents can be determned by usng the usual reducton ormula. Γ Γ = Γ a ) ( Γ Γ ) ( a = Nχ ( C ) χ ( C ) h Snce Γ Γ s reducble, the smple product unctons, Γ Γ, are not true bass unctons or any rreducble representatons. Instead, they can be expressed as a sum o the true bass unctons. The coecents n ths summaton are Clebsch-Gordan coecents. Γ γ Γ = Γ Γ Γ, Γ, Γ 8
Clebsch-Gordan Coecents Compare the reducton o the drect-product representaton and the equaton or ther bass unctons. We can see that the product uncton, Γ Γ, would be expressed n terms o only the bass unctons belongng to rreducble representatons Γ s who appear n the reducton o the drect-product representaton. In other words, a partcular Γ does not appear n the reducton o Γ Γ, then the correspondng Clebsch-Gordan coecents are zero. ) ( Γ Γ ) ( Γ Γ, Γ, γ = 0 a = Nχ ( C ) χ ( C ) = 0 h Another way to loo at ths problem s usng a transormaton matrx. The reducton o drect-product representaton nto a sum o rreducble representatons means that we convert all matrces n the drect-product representaton nto a bloc-dagonal orm. And each bloc (submatrx) s the matrx n the correspondng rreducble representaton. Ths transormaton process s done by a smlarty transormaton, S - AS, or all matrces n the drect-product representaton. - The row ndex or S, the untary transormaton matrx, s the drect-product representaton, Γ, Γ, γ, and the column ndex s rreducble representatons, Γ. - The elements o S are, n act, the Clebsch-Gordan coecents. 9
Clebsch-Gordan Coecents Suppose we construct a matrx A n the drect-product representaton. All elements are zero except the one correspondng to a partcular bass uncton, Γ Γ. The only non-zero element has a value o. The matrx A wll loo le: γ =,γ = 0 γ =,γ = γ =,γ = 0 A = 0 0 The row and column o the sole non-zero element correspond to the values o o the product uncton we chose, Γ, Γ Γ Γ. Apply to A the same transormaton S that was used to reduce other matrces nto bloc-dagonal orm. The smlarty transormaton by S wll transorm A nto a blocdagonal orm ust as t dd to other matrces. The resultng matrx may have more than one non-zero elements and the exact orm o the transormed matrx wll depend on S. 0
Clebsch-Gordan Coecents However, one thng we now or sure s that the transormed matrx wll have the same bloc-dagonal orm as the other transormed matrces. That s, the submatrces lned up along the dagonal wll correspond to the same rreducble representatons as they dd or the other matrces. Thereore, the non-zero coecents can only appear wthn the blocs along the dagonal. Γ Γ Γ 3 0 A' = 0 Rewrtng the nverse transormaton matrx equaton, A = SA S -, n terms o matrx elements, we obtan Γ γ Γ = Γ Γ Γ, Γ, Γ The act that non-zero elements o A appear only n the dagonal blocs means that the Clebsch-Gordan coecents Γ Γ, Γ, γ are always zero Γ does not appear n the reducton o Γ Γ.
General Selecton Rule Summarzng our prevous dscusson, the transton probablty between ( Γ) and ( γ ) Γ X( Γ) Γ where ( Γ) Γ, states depends on the matrx element X s the nteracton Hamltonan that nduces the transton and transorms as the γ row o the Γ rreducble representaton. The Wgner-Ecart theorem states the matrx element s determned by the Clebsch-Gordan coecents. Γ X X l ( Γ, γ) Γ = Γ ( Γ) Γ Γ, γ Γ, Γ, γ The Clebsch-Gordan coecents are zero (thereore the matrx Γ element s zero) unless the product representaton Γ Γ contans Γ. Another way to put t s that the transton matrx element s always zero Γ does not appear n the reducton o Γ Γ. Ths s the most general selecton rule or electronc transtons.