Calculations of optical properties of transition metal and ...

Calculations of optical properties

of transition metal and rare-earth ions

Mikhail G. Brik

Institute of Physics, University of Tartu, Riia 142, Tartu, 51014, Estonia

LUMINET winter school, Costa Adeje, Tenerife, December 2-6, 2013

Outline

Introduction: impurities in crystals and glasses

Transition metal and rare earth ions

Energy level schemes of free ions

Correlation between the Racah parameters, spin-orbit constant and

atomic number for isovalent ions

Basic foundations of crystal field theory

Splitting of free ion energy levels in crystal field

Tanabe-Sugano diagrams for d ions

Conclusions

3d unfilled shell

4f unfilled shell

Introduction: d and f ions

4d unfilled shell

5d unfilled shell

5f unfilled shell

Y3Al5O12:Nd3+

Phosphor powders

Quartz

Ice -

snowflake

Emerald

Amethyst

Azurite

Diamond

Introduction: crystals, glasses, phosphor powders

Where does the color come from?..

Corundum Al2O3

If pure – colorless or grey

Impurities (imperfections) in crystals

Ruby: corundum Al2O3

with a few percent of chromium

Sapphire: corundum Al2O3

with a few percent of titanium

Even a small amount of impurities (added either artificially or in a natural way) can drastically change optical properties of a crystal !!


Corundum Al2O3 Chromium Ruby

+ =

Corundum: http://www.kaycircle.com/Wha

t-Is-The-Average-Cost-Of-

Corundum-Per-Gram-Pound-

Ton-Average-Corundum-Price

~ 50 $ per pound

Math properties of addition cease to work, when it comes to gemstones …

Chromium: http://www.metalprices.com/Fre

eSite/metals/cr/cr.asp

~1.6 – 2.6 $/kg +

Ruby: http://www.ruby-sapphire.com/r-

s-bk-prices.htm “good quality”

(on a scale “poor – fair – good –

very good - exceptional”)

from 50 to 13,000 $ per carat 1 carat = 0.2 gram

≠


http://www.kaycircle.com/What-Is-The-Average-Cost-Of-Corundum-Per-Gram-Pound-Ton-Average-Corundum-Price




























http://www.metalprices.com/FreeSite/metals/cr/cr.asp

http://www.metalprices.com/FreeSite/metals/cr/cr.asp

http://www.ruby-sapphire.com/r-s-bk-prices.htm









Outline









Conclusions

Electron configurations

Impurities (3d, 4f ions) – MULTI-ELECTRON systems

Electron states of free ions

3d, 4f ions – MULTI-ELECTRON systems

d electrons quantum numbers:

Principal quantum number n=3, or 4, or 5

Orbital quantum number l=2

Magnetic quantum number ml=-2,-1,0,1,2

Spin quantum number mS=-1/2, +1/2

10 states in total

f electrons quantum numbers:

Principal quantum number n=4, or 5

Orbital quantum number l=3

Magnetic quantum number ml=-3,-2,-1,0,1,2,3

Spin quantum number mS=-1/2, +1/2

14 states in total

Pauli exclusion principle: no electrons with identical quantum numbers!

Wolfgang Ernst Pauli,

25.04.1900-15.12.1958

Nobel prize in physics 1945

http://upload.wikimedia.org/wikipedia/en/8/85/Wolfgang_Pauli2.jpg

d2 configuration (V3+, Cr4+, Mn5+, Fe6+)

OR:

OR…

!!

!

kqk

qN

Number of

permutations of k

electrons through

q orbitals

OR:

45

!210!2

!10

N

d2 configuration: q=10, k=2



Electron

configu-

ration

Number of states

Number of energy levels

p1 , p5 6 ?

p2 , p4 15 ?

p3 20 ?

Electron

configu-

ration

Number of states


d 1 , d 9 10 ?

d 2 , d 8 45 ?

d 3 , d 7 120 ?

d 4 , d 6 210 ?

d 5 252 ?

Electron

configu-

ration

Number of states


f 1 , f 13 14 ?

f 2 , f 12 91 ?

f 3 , f 11 364 ?

f 4 , f 10 1001 ?

f 5 , f 9 2002 ?

f 6 , f 8 3003 ?

f 7 3432 ?

How can we find the number of energy levels for each of these configurations?..

How can we calculate the energies of those levels?..

!!

!

kqk

qN

Number of

permutations

of k electrons

through q

orbitals

??


Wave function of a particular

state can be obtained from the

one-electron functions

using the angular momenta

addition rules and

(again!) Pauli exclusion

principle.

2121 LLLLL

1L

2L

M1

M2

L

21 MMM

In quantum mechanics

2+2 is not always 4!

The rule of addition of two

momenta L1 and L2

For example, L1=2, L2=2,

L=0, 1, 2, 3, 4

Why?.. Only one out of three components

of angular momentum operator can be

determined at the same time

,,, lmnlnlm YrRr

0

12

00

2/3

0

3

22exp

2

2!

!1)(

na

ZrL

na

Zr

na

Zr

na

Z

nln

lnrR l

ln

l

nl

General form of the

radial wave functions

One-electron wave

function

zyxkjiLiLL kji ,,,,,ˆˆ,ˆ

Spectral terms of free ions

Ψ stands for the wave function

of a particular state;

it can be obtained from the

one-electron functions

using the angular momenta

addition rules and

(again!) Pauli exclusion

principle.

Electrostatic (Coulomb) interaction between electrons of the unfilled shell

is mainly responsible for the formation of the energy level schemes

21212121

2

2

2

1

21

2* sinsin dddddrdrrr

rr

e

+Ze

-e

-e

r12

Spherical system of coordinates

2121 LLLLL

1L

2L

M1

M2

L

21 MMM

In quantum mechanics

2+2 is not always 4!

The rule of addition of two

momenta L1 and L2

For example, L1=2, L2=2,

L=0, 1, 2, 3, 4

Why?.. Only one out of three components

of angular momentum operator can be

determined at the same time

Ψ stands for the wave function

of a particular state;

it is a linear combination of

products of one-electron wave

functions

MMM

MLML

LM

MLMLLM C21

22112211

The Clebsch-Gordan coefficients

They have the following physical meaning: the

square of the absolute value of the

coefficient is equal to the probability that two

states (L1, M1) and (L2, M2) after addition

produce the state (L, M). If M ≠ M1+ M2, the

corresponding Clebsch-Gordan coefficient is zero.

LM

MLMLC2211


,,, lmnlnlm YrRr

0

12

00

2/3

0

3

22exp

2

2!

!1)(

na

ZrL

na

Zr

na

Zr

na

Z

nln

lnrR l

ln

l

nl

General form of the

radial wave functions

One-electron wave

function

Clebsch-Gordan coefficients and Wigner 3j-symbols

mmm

jjjjC

mjjjm

mjmj

21

21121 21

2211

z

mjjz

zmjjzmjjzmjzmjzjjjz

mjmjmjmjmjmj

jjj

jjjjjjjjj

mmm

jjj

!!!!!!

1

!!!!!!

!1

!!!

2131232211321

333322221111

321

321321321

321

321

321

General definition of the Wigner 3j-

symbol

Not that difficult to write a computer program to calculate the 3j-symbols

Many standard packages (MAPLE, for example) have the built-in procedures

for evaluation of the 3j-symbols

But in a “pre-computer era” one was supposed to calculate the 3j-symbols

“by hands” (special tables were created…)

It is a common practice to use the 2S+1L notation for the multi-electron states

where S is the TOTAL spin, and L is the TOTAL orbital momentum

After the wave functions of two electron configurations are built, one can repeat the

procedure by adding the third electron to get the wave functions of three electron

configuration etc … (fractional parentage coefficients, seniority quantum number, 6j-, 9j-

symbols)


L 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

S P D F G H I K L M N O Q R T

How to memorize the above-given sequence … mnemonic rules?..

Each 2S+1L spectral term is degenerated; the degree of degeneracy is

(2S+1)(2L+1). E.g. 4F term: (2S+1)(2L+1) = 4(2 x 3 +1) = 28 states

Sober Physicists Don't Find Giraffes Hiding In Kitchens Like My Nephew


Equivalent configurations for p-electrons:

p1 and p5 – 2P

p2 and p4 – 1SD, 3P

p 3 – 2PD, 4S

Equivalent configurations for d-electrons:

d 1 and d 9 – 2D

d 2 and d 8 – 1SDG, 3PF

d 3 and d 7 – 2PD(2)FGH, 4PF

d 4 and d 6 – 1S(2)D(2)FG(2)I, 3P(2)DF(2)GH, 5D

d 5 – 2SPD(3)F(2)G(2)HI, 4PDFG, 6S

Equivalent configurations for f-electrons:

f 1 and f 13 – 2F

f 2 and f 12 – 1SDGI, 3PFH

f 3 and f 11 – 2PD(2)F(2)G(2)H(2)IKL, 4SDFGI

f 4 and f10 – 1S(2)D(4)FG(4)H(2)I(3)KL(2)N, 3P(3)D(2)F(4)G(3)H(4)I(2)K(2)LM, 5SDFGI

f 5 and f 9 – 2P(4)D(5)F(7)G(6)H(7)I(5)K(5)L(3)M(2)NO, 4SP(2)D(3)F(4)G(4)H(3)I(3)K(2)LM,

6PFH

f 6 and f 8 - 1S(4)PD(6)F(4)G(8)H(4)I(7)K(3)L(4)M(2)N(2)Q, 3P(6)D(5)F(9)G(7)H(9)I(6)K(6)L(3)M(3)NO,

5SPD(3)F(2)G(3)H(2)I(2)KL, 7F

f 7 - 2S(2)P(5)D(7)F(10)G(10)H(9)I(9)K(7)L(5)M(4)N(2)OQ,

4S(2)P(2)D(6)F(5)G(7)H(5)I(5)K(3)L(3)MN, 6PDFGHI, 8S

A subscript denotes a number

of terms with the same S and L.

Hund’s rule to find the ground term: i) max spin S; ii) max orbital momentum L

d 2 and d 8 – 1SDG, 3PF - the ground term is 3F Friedrich Hund, 04.02.1896 – 31.03.1997

Electron states and spectral terms of free ions

Electron

configu-

ration

Number of states


p1 , p5 6 1

p2 , p4 15 3

p3 20 3

Electron

configu-

ration

Number of states


d 1 , d 9 10 1

d 2 , d 8 45 5

d 3 , d 7 120 8

d 4 , d 6 210 16

d 5 252 16

Electron

configu-

ration

Number of states


f 1 , f 13 14 1

f 2 , f 12 91 7

f 3 , f 11 364 17

f 4 , f 10 1001 47

f 5 , f 9 2002 73

f 6 , f 8 3003 119

f 7 3432 119



!!

!

kqk

qN

Number of

permutations

of k electrons

through q

orbitals

? ?

Coulomb interaction between electrons of the unfilled shell is

mainly responsible for the formation of the energy level schemes

21212121

2

2

2

1

21


rr

e

+Ze

-e

-e

r12


Energy of spectral terms of free ions

22

*

111

12

,,12

41

kmkmk

k

k

k

km

YYr

r

kr

0

21

2

1

0

21

1

21211

1

1

r

k

kr

k

k

k

k

drr

rdr

r

rdrdrdr

r

r

r1

r2

r1>r2

r2>r1

Not zero, if l1+k+l2 is an even number. k=0, 2, 4 for d-electrons

k=0, 2,4,6 for f-electrons

,,,~ lmnlnlm YrRr Eventually we have the integrals from

the product of three spherical functions

21

212121

0

2

0

*

0004

1212121sin 1

2211 mmm

lkllkllklddYYY

m

mlkmml

This integral is expressed in terms of the Wigner 3j-symbols:

Coulomb interaction between electrons of the unfilled shell is

mainly responsible for the formation of the energy level schemes

21212121

2

2

2

1

21


rr

e

+Ze

-e

-e

r12


R(r) – radial parts of the wave functions


The energy of the LS terms will be a combination of the Fk parameters

(Slater integrals) defined as:

21

2

2

2

1

2

22

2

111

2 )()( drdrrrrRrRr

reF

k

k

kThe integrals can be found numerically

k=0, 2, 4 for the d-electrons

k=0, 2,4,6 for the f-electrons

It is a common practice to express the

energy of the LS terms for 3d ions in the

so called Racah parameters A, B, C:

44240

63

5;

441

5

49

1;

441

49FCFFBFFA


Equivalent configurations for d-electrons:

d 1 and d 9 – 2D



d 4 and d 6 – 1S(2)D(2)FG(2)I, 3P(2)DF(2)GH, 5D

d 5 – 2SPD(3)F(2)G(2)HI, 4PDFG, 6S


1S A + 14 B + 7C

3P A + 7 B

1D A – 3 B + 2C

3F A – 8 B

1G A +4 B + 2C


2P 3A – 6 B + 3C

4P 3A

2D1, 2D2

2F 3A + 9 B + 3C

4F 3A – 15 B

2G 3A – 11 B + 3C

2H 3A – 6 B + 3C

3dn electron configurations and

corresponding ions from the periodic table

d 1 Ti3+

d 2 Ti2+, V3+, Cr4+, Mn5+, Fe6+

d 3 V2+, Cr3+, Mn4+, Fe5+

d 4 Cr2+, Mn3+, Fe4+

d 5 Mn2+, Fe3+

d 6 Fe2+, Co3+

d 7 Co2+, Ni3+

d 8 Ni2+

d 9 Cu2+

A can be omitted

(a shift of all terms)

Ion B (cm-1) C (cm-1)

Ti2+ 718 2629

V3+ 861 4165

Cr4+ 1039 4238

V2+ 766 2855

Cr3+ 918 3850

Mn3+ 965 3675

Fe4+ 1144 4459

Mn2+ 960 3325

Fe3+ 1015 4800

Co2+ 971 4366

Ni2+ 1041 4831

Cu2+ 1238 4659

Cr 4+ – 1SDG, 3PF

3F – 8312 cm-1

1D 5359 cm-1

3P 7273 cm-1

1G 12632 cm-1

1S 44212 cm-1

Typical values of the Racah parameters for some free 3d ions

Electron states and spectral terms of free ions

Electron

configu-

ration

Number

of

states

Number

of

energy

levels

p1 , p5 6 1

p2 , p4 15 3

p3 20 3

Electron

configu-

ration

Number

of

states

Number

of

energy

levels

d 1 , d 9 10 1

d 2 , d 8 45 5

d 3 , d 7 120 8

d 4 , d 6 210 16

d 5 252 16

Electron

configu-

ration

Number of states

Number

of

energy

levels

f 1 , f 13 14 1

f 2 , f 12 91 7

f 3 , f 11 364 17

f 4 , f 10 1001 47

f 5 , f 9 2002 73

f 6 , f 8 3003 119

f 7 3432 119



!!

!

kqk

qN

Number of

permutations

of k electrons

through q

orbitals

Outline









Conclusions

25

)1(14,2

)(

LLhFHN

i

ii

k

k

k

FI sl

Coulomb interaction Spin-orbit interaction Trees correction

Slater integrals

CBF 772

5634 CF Racah parameters

Electronic states:

Hamiltonian for free d-ions

C.A. Morrison, Crystal

Fields for Transition-

Metal Ions in Laser Host

Materials, Springer-

Verlag, 1992

Is there any trend

between these

parameters and a

number of d-

electrons (or atomic

number)?

Systematic trends across

the 3d ions series

Is there any trend in behavior of the

B, C parameters across these groups?..

1 2 3 4 5 6 7 8 9

800

1000

1200

3000

4000

5000

C= 2902.60 + 254.22 Z

CuNiCoFeMnCrVTi

B

C

Energ

y,

cm

-1

Number of

d electrons, N

Divalent 3d ions

Sc

B= 778.82 + 69.76 Z

1 2 3 4 5 6 7 8 91000

1200

1400

1600

4000

4500

5000

5500

6000Trivalent 3d ions

Energ

y,

cm

-1

B

C

CuNiCoFeMnCrVTi

Number of

d electrons, N

Zn

C= 3691.11 + 240.24 N

B= 978.39 + 66.22 N

1 2 3 4 5 6 7 8 91200

1400

1600

5000

5500

6000

6500

En

erg

y,

cm

-1

Tetravalent 3d ions

B

C

Zn GaV Cr Mn Fe Co Ni Cu

Number of

d electrons, N

B= 1154.26 + 64.55 N

C= 4387.62 + 233.59 N

Free di, tri-, tetravalent 3d ions: Racah parameters

B and C – linear increasing functions of

number of d-electrons (atomic number)

M.G. Brik, A.M. Srivastava,

Opt. Mater. 35 (2013) 1776

1 2 3 4 5 6 7 8 9

3.69

3.70

3.71

3.72

3.73

3.74

3.75

3.76

3.77

3.78C/B=3.79382-0.00628N

C/B=3.76464-0.00551N

C/B

Divalent 3d ions

Trivalent 3d ions

Tetravalent 3d ions

Number of d electrons, N

C/B=3.71974-0.00351N

Free di, tri-, tetravalent 3d ions: C/B ratio

C.A. Morrison, Crystal

Fields for Transition-

Metal Ions in Laser Host

Materials, Springer-

Verlag, 1992


Opt. Mater. 35 (2013) 1776

21 22 23 24 25 26 27 28 29 30 31

3

4

5

6

= 0.27778 (Z-8.571)

= 0.28437 (Z-9.444)

= 0.29695 (Z-10.619)

, [c

m]

GaZn

Divalent 3d ions

Trivalent 3d ions

Tetravalent 3d ions

CuNiCoFeMnCrVTi

Atomic

number, Z

Sc

*4/1 29695.0619.1029695.0 ZZ Divalent

*4/1 28437.0444.928437.0 ZZ Trivalent

*4/1 27778.0571.827778.0 ZZ Tetravalent

Free di, tri-, tetravalent 3d ions: SO coupling

Increase of an effective

nuclear charge

with increasing oxidation

state


Opt. Mater. 35 (2013) 1776

Free di, tri-, tetravalent 3d ions: relation

between B, C and SO constant

2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 6.4

1000

1500

4000

5000

6000

, [cm]

Energ

y,

cm

-1

B-divalent

C-divalent

B-trivalent

C-trivalent

B-tetravalent

C-tetravalent

Conclusion: if the value of any single parameter (B, C, or ζ) is known

for one ion from the group of di-, tri- or tetravalent 3d ions, all

parameters for all members in the considered isovalent ions can be

estimated.


Opt. Mater. 35 (2013) 1776

Free di, tri-, tetravalent 4d/5d ions

C.-G. Ma, M.G. Brik,

J. Lumin. 145 (2014) 402

Similar behavior:

The Racah parameters

A, B, C, and SO constant

ζ1/4 all proportional to

atomic number Z.

If one parameter is

known for one ion from

the group of di-, tri- or

tetravalent 4d/5d ions,

all parameters for all

members in the

considered isovalent

ions can be estimated.

Free di, tri-, tetravalent 4d/5d ions

C.-G. Ma, M.G. Brik,

J. Lumin. 145 (2014) 402

Similar behavior:

The Racah parameters A, B, C, and

SO constant ζ1/4 all proportional to

atomic number Z.

If one parameter is known for one

ion from the group of di-, tri- or

tetravalent 4d/5d ions, all

parameters for all members in the

considered isovalent ions can be

estimated.

SO constant

Racah parameters

Free di, tri-, tetravalent 4f/5f ions

C.-G. Ma, M.G. Brik, in preparation

Slater integrals, 4f ions

58 59 60 61 62 63 64 65 66 67 68 69 7020

30

40

50

60

70

80

90

100

110

120

130Divalent lanthanides

F6=-59672.22+1663.49 Z

F4=-82612.02+2310.41 Z

F2

F4

F6

Energ

y,

10

3 c

m-1

Atomic number, Z

F2=-130428.62+3676.62 Z

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb

59 60 61 62 63 64 65 66 67 68 69 70 7130

40

50

60

70

80

90

100

110

120

130

140

Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

F6=-41837.26+1456.23 Z

F4=-58338.49+2027.99 Z

F2=-93815.31+3248.91 Z

Energ

y,

10

3 c

m-1

Atomic number, Z

F2

F4

F6

Trivalent lanthanides

60 61 62 63 64 65 66 67 68 69 70 7140

50

60

70

80

90

100

110

120

130

140

Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

F2

F4

F6

Tetravalent lanthanides

Energ

y,

10

3 c

m-1

Atomic number, Z

F2=-72429.77+3042.09 Z

F4=-43942.72+1889.27 Z

F6=-31183.95+1353.59 Z

90 91 92 93 94 95 96 97 98 99 100 101 102

10

20

30

40

50

60

70

80

90

100

110

Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No

Divalent actinides

F6=-120996.33+1631.76 Z

F4=-161536.70+2191.03 Z

F2

F4

F6

Energ

y,

10

3 c

m-1

Atomic number, Z

F2=-238506.92+3270.85 Z

91 92 93 94 95 96 97 98 99 100 101 102 103

20

30

40

50

60

70

80

90

100

110

Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

F6=-97633.32+1420.14 Z

F4=-130266.05+1906.87 Z

F2=-193109.92+2855.93 Z

Energ

y,

10

3 c

m-1

Atomic number, Z

F2

F4

F6

Trivalent actinides

92 93 94 95 96 97 98 99 100 101 102 103

30

40

50

60

70

80

90

100

110

U Np Pu Am Cm Bk Cf Es Fm Md No Lr

F2

F4

F6

Tetravalent actinides

Energ

y,

10

3 c

m-1

Atomic number, Z

F2=-169408.29+2657.98 Z

F4=-113647.18+1769.28 Z

F6=-85101.95+1316.94 Z

Slater integrals, 5f ions

57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

5

6

7

8

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

-4.86638 + 0.17606 Z

-5.43241 + 0.18275 Z

Atomic number, Z

divalent 4f

trivalent 4f

tetravalent 4f

SO

consta

nt 1/4

, cm

-1/4

-6.3991 + 0.19556 Z

89 90 91 92 93 94 95 96 97 98 99 100 101 102 103

5

6

7

8

9

Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

-10.97840 + 0.19305 Z

-12.35019 + 0.20563 Z

Atomic number, Z

divalent 5f

trivalent 5f

tetravalent 5f

SO

consta

nt 1/4

, cm

-1/4

-14.97853 + 0.23113 Z

SO constant

Outline









Conclusions

Corundum Al2O3 crystal

lattice (one unit cell)

Al

O

Some Al ions are replaced by

Cr: imperfections (defects) of

crystal lattice

Impurity ions in crystals:

what happens to the energy levels?..

Spherical symmetry of a free ion is broken. As a result, degeneracy

of some energy levels is removed (they are split).

The splitting can be calculated using the crystal field theory.

Octahedral impurity centers

x

y

z

1

2

3

4

5

6

a

a

a

–a

–a

–a

Z

Y

X

Six surrounding ions (called ligands) have an electrical charge –Ze.

Single d-electron in an octahedral crystal field (Ti3+, V4+, Cr5+, …)

6

1

2

)(i i

ZeV

rRr

potential energy V of the single d-electron of the

central ion

Perturbation theory: V(r) much smaller than the free ion Hamiltonian. Since the 2D term

is 5-fold degenerated, the effects of the small perturbation will be revealed after

diagonalizing the following 5 by 5 matrix, which can be built using the above-

mentioned wave functions:

EVU

m)()(

2

2

rr

a free ion Hamiltonian )(2

2

r

Um

Values of M

(from -2 to

+2) ddrdrYRVYRV nnnn sin),()()(),()( 2

'232

*

232', rrr

The energetic states of a single d-electron (2D term) in a free ion are described by

the following FIVE wave functions:

),,()( 2232 YrR ),,()( 2132 YrR ),,()( 2032 YrR ),,()( 1232 YrR ),()( 2232 YrR

How to manage this integral?..

ddrdrYRZe

YRV n

i i

nnn sin),()(),()( 2

'232

6

1

2*

232', rrR

r

Single d-electron (Ti3+, V4+, Cr5+, …) in an octahedral crystal field –

calculation of the matrix elements

A solution is to expand the Coulomb potential in terms of the spherical harmonics:

k

km

iikmkm

i kk

k

i i

YYkr

rZe

ZeV

,,

12

4)(

*6

1 01

26

1

2

rRr

Here r< and r> are the smallest and the greatest of r (an electron coordinate) and a (a

distance between the central ion and surrounding point charges), (θ, φ) and (θi, φi) are the

spherical angular coordinates corresponding to the electron and ligands, respectively.

Since r < a (an electron is between the central ion and ligands) , we have:

k

km

iikmkm

i k

k

YYka

r

a

ZeV

,,

12

4)(

*6

1 0

2

r

Will it help?..

Oh yes! A use will be made of some remarkable properties of the spherical functions!!

Calculation of the matrix elements of a crystal field potential

– a further simplification

From the sum (formally an infinite one!) over index k many terms vanish. The reason is that

the matrix elements of V(r) are proportional to the following integrals from the product of

three spherical functions:

(in our case l1 = l2 = 2)

2

0 0

*

', sin),(),(),(~2211

ddYYYV mlkmmlnn

21

212121

0

2

0

*

0004

1212121sin 1

2211 mmm

lkllkllklddYYY

m

mlkmml

This integral is expressed in terms of the Wigner 3j-symbols:

)(0)2

;evenis)10

000 21

2121

llk

lkllkl

If l1 = l2 = 2 (d electrons) then k = 0, 2, 4.

If l1 = l2 = 3 (f electrons) then k = 0, 2, 4, 6.

Calculations of the matrix elements

of the crystal field potential

,,14

5,

2

6)( 4

4

4

4

4

0

4

5

22

CCCra

Ze

a

ZeV r

ddrdrYRVYRV nnnn sin),()()(),()( 2

'232

*

232', rrr

The calculated eigenvalues are:

–4Dq (three roots) and 6Dq (two roots).

Energy levels of a single d electron in an octahedral field

The group theoretical analysis: D → E + T2

Crystal field theory allows for quantitative estimation of

the energy interval between the split states

5

42

3

510

a

rZeDq The crystal field

strength

The character of this splitting can be understood

from the spatial distribution of the electron

density

Directed into the space BETWEEN

the ligands – smaller energy of the

Coulomb repulsion – minimum of

total energy

Directed into the space TOWARDS

the ligands – higher energy of the

Coulomb repulsion – maximum of

total energy

Outline









Conclusions

Tanabe-Sugano matrices for energy levels

of impurity ions in a cubic crystal field

Three papers by Y. Tanabe and S. Sugano:

1. "On the absorption spectra of complex ions I". Journal of the Physical Society of Japan 9 (5),

1954, 753–766.

2. "On the absorption spectra of complex ions II". Journal of the Physical Society of Japan 9 (5),

1954, 766–779.

3. "On the absorption spectra of complex ions III". Journal of the Physical Society of Japan 11 (8),

1956, 864–877.

Famous book:

S. Sugano, Y. Tanabe, H. Kamimura, Multiplets of Transition-Metal Ions in Crystals, Acad. Press, New

York, 1970

Splitting of all LS terms of the d-electron configurations in the cubic

crystal field as a function of the Dq, B, C parameters

Energy levels of two d-electrons in a tetrahedral field

The ground state configuration (e)2(t2)0, two excited configurations (e)1(t2)

1 and (e)0(t2)2.

Tanabe-Sugano diagram for

the d2 configuration in a

tetrahedral field. The

horizontal axis – Dq/B; the

vertical axis – energy in (E/B).

C/B=4.25

Weak field – broad

emission corresponding

to the spin-allowed

transitions

Strong field – narrow

emission corresponding to

the spin-forbidden

transitions

3F → 3A2+ 3T1+ 3T2

3P → 3T1

1D → 1E+ 1T2

1G → 1A1+ 1E+ 1T1+ 1T2

1S → 1A1

Energy levels of three d-electrons in an octahedral field

The ground state configuration (t2g)3(eg)0 (the energy is 0), and three excited configurations

(t2g)2(eg)1, (t2g)1(eg)2, (t2g)0(eg)3 with the energies 10Dq, 20Dq, and 30 Dq, respectively.

Tanabe-Sugano diagram for

the d3 configuration in an

octahedral field. The

horizontal axis – Dq/B; the

vertical axis – energy in (E/B).

C/B=4.25

Weak field – broad

emission corresponding

to the spin-allowed

transitions

Strong field – narrow

emission corresponding to

the spin-forbidden

transitions

Tanabe-Sugano diagram and absorption spectrum of Cr3+

Absorption (left) and emission

(right) spectrum of Cr3+ in MgO

Energy levels of four d-electrons in an octahedral field

Tanabe-Sugano

diagram for the d4

configuration in an

octahedral field. The

horizontal axis –

Dq/B; the vertical

axis – energy in (E/B).

C/B=4.25

Weak field – HIGH-SPIN

state

Strong field – LOW-SPIN

state

The ground state configuration (t2g)3(eg)

1 (High spin), and excited configurations (t2g)2(eg)

2, (t2g)1(eg)

3,

(t2g)0(eg)

4, with the energies 10Dq, 20Dq, 30Dq, respectively.

Or: the ground state

configuration (t2g)4(eg)0

(Low spin), and excited

configurations (t2g)3(eg)1,

(t2g)2(eg)2, (t2g)1(eg)3, and

(t2g)0(eg)4, with the energies

10Dq, 20Dq, 30Dq, and

40Dq respectively.

5E

5E

3T1

3T1

Difference between the high and low spin

states of a d4 configuration in an

octahedral field

High spin (energy of

spin-pairing is greater

than the crystal field

strength 10Dq)

S = 2 Low spin (energy of

spin-pairing is less than

the crystal field strength

10Dq)

S = 1

10Dq d orbital

t2

e

10Dq d orbital

t2

e

5E 3T1

How the energy levels of the d ions are formed…

A multielectron

ion

Coulomb

Interaction 2S+1L terms 2S+1 L

~104 cm-1

~104 cm-1

Crystal

field Spin-orbit

interaction

~102 cm-1

How the energy levels of the f ions are formed…

A multielectron

ion

Coulomb

Interaction 2S+1L terms 2S+1 L

~104 cm-1

~104 cm-1

Spin-Orbit

Interaction 2S+1LJ

Crystal

field

splitting

~102 cm-1

Outline









Conclusions

Multielectron ions energy levels:

things to remember

Multielectron ions energy levels:

things to remember

Multi-electron configurations

Electrostatic (Coulomb) and spin-orbit interactions produce the (rich)

energy level schemes

Different terms are denoted by the 2S+1L notation (spin S and orbital

momentum L; TM ions) OR 2S+1LJ notation (total angular momentum

J; RE ions)

These states are highly degenerated: (2S+1)(2L+1) number of states

The energies of different terms are expressed in terms of the Racah

parameters (Slater integrals)

Parameters of electrostatic interaction increase linearly with atomic

number Z in the isovalent series

The SO constant increase linearly with Z4

Crystal field removes degeneracy of free ion energy levels

When I see a cow in the field, it is not a cow,

but … a crazy dance of electrons.

S. Chase

Acknowledgement: financial support

Sir Winston Churchill about Christopher Columbus’ voyage to

“India”, which eventually led to the discovery of the New World:

“He left without knowing where he was going, arrived where he did not think he would, and all of it at the expense of others.”

The taxpayers should be always acknowledged!

Marie Curie Initial Training Network LUMINET, grant agreement no. 316906

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http://en.wikipedia.org/wiki/File:Columbus_Taking_Possession.jpg

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