Grid of Expressions Related to the Einstein Coefficients James E. Sturm Lehigh University. Bethlehem, PA 18015 Most textbooks on spectroscopy, photochemistry, or quantum chemistry, as well as several on inorganic or physical chemistry, include some discussion of the relations among the Einstein coefficients, oscillator strength, transition dipole moment, and integrated absorption coefficient for a transition between two states. Usually a given discussion will emphasize only some of the 20 possible relations among these terms. Then, as luck would have it, a problem situation arises calling on a relation not developed explicitly in the given context. Since some of these relations involve reconsideration of not only dimensional hut also units consistency, intermittent practitioners find themselves sidetracked often enough to find these occasions disruptive. It was therefore felt worthwhile to collect all of the relations in a grid or table that hopefully also enhances students' or refreshes users' perspective (I).A useful semiquantitative chart correlating some of these terms was presented hy Kasha and Rawls (2).A chapter that relates these expressions to ohservables is in a hook by Sandorfy (3). Before oresentation of the .. erid.. it is a~nronriate .. . to clarifv some terms, all of which relate to the transition between the kth (lower) and rnth (. u.. o ~ e. rsmtesof ) thesvstem.Theeneruv differencehetween these two states is & = hco where h i s Planck's constant, c is the speed of liaht in vacuum, and o = I/.\ where Xis the wavelength of the transition. One then has: Einstein coe/f;cients (41: Aamis the first-order rate coeffi-
-
cient for spontaneous emission k m, and the B coefficients are: B,, = B, = Ak,l(8shew3) The Bkm for a composite case, such as a molecular electronic transition, is the sum of the contributing rovihronic terms. A spectrum of a species in solution may exhibit only the envelope of overlapping components (5). Oscillator strength (6):f = Ak,cm,(4r€o)/(8n2q2w2). Here, me and q are the electron mass and charge, respectively. The factor (4m0), where ro is the permittivity of free space, is retained as a separate term since i t is identified with the system of units used (7). Tramition dipole moment (8): pkm = ($*IRlfik) where R is the dipole moment operator. The square, pam2, is related more directly than Pkm to the terms of interest here. For example, pam2= 3hAk,l(64s4w3) Integrated absorption coefficient, (IAC) (9): The ahsorptivity or extinction coefficient is defined in terms of the absorbance or optical density: r = log~o(Io/O/[(conc.)(path length)]. The IAC is related to Bmk as follows: IAC = Jrdw = B,,hwN&
10
where No is Avogadro's number
Grld ot Emesslons Related to the Elnstein Coefflclents Integrated Absorption C~efflcient
Einstein Coewicients
i\-
Am
E,
= B,
Oscillator Strength
I d w
All expessionr in SI units. m. = elecbon marro (kg); o = l l h (m-I); q = elscbon charge (C); and & = Avogaddo number.
32
Journal of Chemical Education
f
Transition Dipole
Moment p2xm
The grid is presented as the tahle. Entries are ratios of the items along the top row to any given item along the left-hand column; diagonal entries are therefore unity. To avoid amhiguity, each member of the grid is defined a t the top left of its rectangle. The expressions are in SI units (MKS system). In the cgs system, commonly used in spectroscopy, the term (4~60)= 1and is dimensionsless. Applications: One should illustrate use of this grid in practice. Let us say that one has ohtained an absorption spectrum over a band and wishes to calculate the corresponding oscillator strength. Figure 1gives a particular absorption spectrum, that for 7-hydroxy-4-methyl-coumarin (coumarin 4), a laser dye, as one would obtain it with a common spectrophotometer. This spectnun can he converted to a plot of r vs, wavenumber as in Figure 2, which also shows that the band center is a t w 3.1 X 104cm-' = 3.1 X 106m-I. Then the IAC can he computed as the area under this latter curve; its value in this case is
-
Jcdw = IAC z 6.30 X 10' M-'cm-%= 6.30 X 108 m mol-' where M is (mol dm-9. Then the oscillator strength for this hand is
f=
-
(4wdc2rn,in 10 (IAC) tro2NA . "
- (1.1127 X 10-"'S'
All units cancel. The corresponding has the value
A,., coefficient
1 1.17
x loL2s kg-'
in^,,,,, the transition dipole moment - 3hc In 10 (IAC) rhm
-
loz3mol-I)
he estimated:
82Ne
- 36626 X rmZW 1.87 X lo1"
Js)(2.998 X 108ms-'1 In lO(6.30 X 108mmol-I) Era(6.022 x 1tf3mol-')(3.1 X lo6 m-I) J
3
which ean be converted to (C rn)= by the factor ( 4 4 = 1.11265 X 10-lo J-I CZm-I. Then one has
rrmZ " 2.0807 X (C m)2 * 18.7 debve2 = 4.3 (since 3.3358 X lo-" C m = debye) from which ram dehye. Hopefully these examples are illustrative of a variety of applications. Estimation of Am, the maximum absorptivity over a hand, is possible if a simple profile is assumed (2).
Knowledge of Ah, estimated indirectly from spectral meaSurementsor from r 2ohtained theoretically, allows inter~reration of observed lifetimes in systems with quenching (10). Llterdtute Cited
8mw2In 10(IAC) No
- 8d2.998 X
In lO(6.30 X lo8m mal-') (6.626 X lo+' Jd(3.1 x 10' m-')(6.022 x
C2m-')(2.998 X 10' m~-')~(9.11 X 10P1kg) In lO(6.30 X 108 m mol-I) X 1tf3mol-I)~(1.6022X 10-'~)~(6.022
0.21
Ah?"=
=
lo8 msCL)(3.1X 10' m-')'
In lO(6.30 X 10' m mol-I) 6.022 X loz3mol-'
a 1.74 X 108 8-'
or
1. PW, A. E m u on Critiiiii, in The Compleff Pmtirol Wor*sof Aleuulder Pope, Cambridge ed.;Houghfon, Miinin: 1903:iin-574-575. 2. Ka8ha.M.:Rawb,H.R.Pholochem.PholobioI.1%8,7.581-5BB,eifcdinSimons, J. P. Photochemistryand Sppetrorcopy, Wiiey-Interscience: 1971; p 145. 3. Sandori~,C.EiectmnieS~~elro ondQwrntumChemi~lry,Prcntiee-Hall:1%;Chapfcr 5. 4. Einstein, A. Physik. Zeiffh.
1917,XVm.121-128 5. Henbe~,G.Mol~culnrS~eefmondMol~eulorSlrueturp.Ill.EloefmnieS~~f~~ond Eleefn?nie Structure 01 Pol~ofamieMolecules: Van Nostrand &inhold: 1966: pp 417 N. 8. Barrow.G. M. hemduction to Moleevlor S~aelmseopy; McGrseHill: 1962; Chapter Monographs for ~ i a e h e r aNo. 15. 8. Guillary, W. A. Introducfian o Molaculor Sfrvcturo and Soectrmoov. ,", AII- & Bamn: 1977: Chanter 2. 9. Hollas, J. M. ~ighdosolvfla"Speelro~eapy;Butternorth.: 1982; Chaptc. 2. 10. Huennekens,J.; Park, H. J.: Co1bort.T.:McCiain, S. C. Phya. &".A 1981.55, 2892~
Next one can calculate the Einstein B coefficient: In 10 (IAC) B= hoN,
~
.~
2901.
Volume 67
Number 1 January 1990
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