Line broadening of uranyl salts at low temperature

J. Phys. Chem. Solids

Pergamon Press 1971. Vol. 32, pp. 1087-1090.

LINE

Printed in G r e a t Britain.

BROADENING OF URANYL LOW TEMPERATURE

SALTS

AT

V. A. LODATO Gulf General Atomic Inq. P.O. Box 608, San Diego, Calif. 92115, U.S.A.

(Received 9 April 1970) Abstract--A model is developed for the energy transfer between uranyl ions for the resonant and

non-resonant cases. The periods of excitation transfer are estimated and the spectral broadening of the resonant line is examined in terms of the excitation transfer between ions. The question of whether the initial excitation is due to dipole or quadrupole radiation is resolved. 1. INTRODUCTION

ONE OF the main features [1, 2] of the fluorescence spectra of the uranyl salts is the loss of high resolution at low temperatures. The width of the spectral lines can be attributed to five basic processes: (1) the emission of light energy which is accompanied by the transition of the uranyl ion into the non-excited elcctron state; (2) the conversion of the electron energy of the uranyl ion to the energy of the oscillations of the lattice; (3) the conversion of the energy of the intramolecular oscillations to the energy of oscillations of the lattice; (4)the energy transfer of intermolecular oscillations initially located around the absorbing or the emitting uranyl ion along the whole lattice; (5) the transfer of the energy of the electron excitation from one uranyl ion onto other uranyl ions. Stepanov[3] has obtained an order of magnitude for the line widths for the first four processes from the Heisenberg uncertainty principle arguments, but he was not able to draw any conclusive evidence for the period of excitation transfer for the fifth process. According to Rabinowitch and Belford[4], the width of the fluorescent bands is due primarily to the excitation transfer between uranyl ions, but there has been no quantitative analysis for the frequency of exchange. In this paper, an order of magnitude calculation is presented to estimate the line widths due to

resonance and non-resonant broadening that may occur due to excitation transfer between neighboring UOa +2 ions in the lattice. In the non resonance case the excitation transfer is considered between different isotopic species of uranyl ions. (The isotopic splitting is of the order of 0-5 cm-l.) We conclude that the excitation transfer accounts for the line broadening in the fluorescent spectra at low temperature. 2. MODEL

Consider the system of two molecules .4 and B that are fixed at some distance. The interaction with lattice vibrations of the surrounding medium is considered to be negligible since the total system is at a very low temperature. We assume that each molecule has a single electronic state in the energy region of interest. Given an initial excitation on A, designated as A*, the question is what is the characteristic time for excitation transfer between two states. .4* + B ~-~-A + B *

(State 1)

(1)

If the period of molecular vibration is much' less than the period of excitation transfer between .4 and B, then for any probable nuclear configuration in the neighborhood of

1087

J.P.C.S. Vol. 32 No. 5--L.

(State 2).

1088

V . A . LODATO

the minimum of the potential surface of A* B a number of vibrations will be complete within a single period of excitation transfer: the potential field felt by the two ions is that obtained by applying the adiabatic approximation to the coupled system of A and B. The total Hamiltonian for the system is ~r: = H.a el (ql) + HA rit~(Q1 ) + H B el (q.o)

Defining the variables y~,B(t) = C4,~ (t) e -i''t and y~s,(t) = CAB*(t)e -i't, the equations become

ihS,~.,(t) = [(6~.~, Iv 14,~. 4,,) -

-

hel]3),e(t) + (~bA*~bn I v 14,~ 4,~.)

• (XAI•

(5a)

ih.f'AR,(t) = <•215

v(I q_o- q, + R I)

(2)

x )~,.B (t) + [( r

where the coordinates ql, Q1 and q2, Q.., are the collective electronic and nuclear coordinates of A and B respectively and ]RI is the distance between the two. In the present two state case the total wavefunction for the system may be written as

-- he2]YAB*(t).

+ H;3"i~'(Q.0)+

~(ql,q2,Q~,Q2, t) = CA*e(t)~bA*(qOqbR(q2)

C,4*B(0) =

A I and Cae.(O) = A.O,

the probability of excitation transfer is

{

+ CAs*(t) qbA(ql)qbH*(q.._) XA(QOXB.(Q2)e ie-'t

I CAB,(t) 12 = A -02cos 2 89[4/3.0 + y 2]..0

ih CA*B(t)e -i''t = ( ~bA*qSnI v 16.,.6,,) e - ' " ' x c.,.,,(t)+ (~:,r

x
ih CAB.(t)e -i'~' = (~bA~bB.[v[~b~.~bn)

• (x,,Ixo.) x e -i''' C..~ (t) + 16,d, ~.t v l

• qSa~bn,) C~n,(t)e-i'~q

'}

(3)

where the 6's and X's correspond to the electronic and vibrational wavefunctions and el and e-0 are the sums of the electronic and vibrational frequencies in states 1 and 2 respectively. The squares of the moduli of the two coefficients, ICA.B(t)I-0 and Ic~.(t)l.0, correspond to the probability of finding the system in State 1 or State 2. Using time dependent perturbation theory, the Schroedinger equation reduces to two coupled linear first order differential equations with explicit time dependent factors

x e-I ,._,CAs* (t)

(5b)

Solving these two differential equations subject to the boundary conditions

X XA* (QOxA(Q2) ei~'t

X

v 16A r

(4b)

4

(A2T-- 2A1/3) 2 4 / 3 2 + 7 .-' f

l/2L~

Z sin 2/89 4/32 + T-O] L '

hJ

(6)

where fl is the product of the electronic exchange energy (~bA.qS,[v[~bA~bB.) and the F r a n c k - C o n d o n factors (XA[XA*)(XJ~[X~*). and y = X/~b,~.~bBIv [~bA.~bn) -- (~bA68-[v [(hAq58.) + A E , where AE = h(e2--ej). In the case of exact resonance T = 0, and equation (6) reduces to the well known formula (5,6) for the probability of excitation transfer ICA,.(t)l 2 = A2 2 cos 2 {fl(t/h)}+A12 sin2{/3(t/h)} with the period of excitation transfer being t = (h/4/3). The second case of interest is that in which the excited electronic state of A is the same as B. and the energy difference AE is due to the vibrational isotope shift in the resonant state. The probability of excitation transfer for the case. (7.8) with the initial conditions AI = 1 andA2 = 0is

LINE BROADENING OF URANYL SALTS

4/3" ] CAB.(t)12 -- 4/32A_ AE 2

X sin2{1 [4/32A-AE2]t/2~}.

(7)

The period of transfer is ~ - : (h/2){4/3z+ AE2> -112. The effect of AE will clearly be

greatest if AE >>/3. As will be seen below, this is the relevant case. Under these conditions the period of excitation transfer becomes small, however the amplitude of the transition probability is also small, being given by

4/3'-' max{ I CAB, (t)1"} = l CAn, (r)I -'= 4/32 + AE.,. (8) Representing the decay of initially excited A* as a first order process (9)

n~+(t) = hA.(0) e -I~'

k can be evaluated as follows. The amount of A* at the end of successive periods, ~-, is given by the series: n A. (~-) = [ 1 - I C~B'(r) ~] n A,(0)

nA.(2T) = [1 --I CAB.(T)[21211A*(0)

,,..(t) = [1 - I c.~-(~)I'1"~,, ~.(0).

(10)

c..,.('r)I"3"

(11)

Then

tl - I

= e-"

1089

To carry out the evaluation of/3 the electronic exchange matrix element, the interaction Hamiltonian, which is the coulombic energy between electrons of A and B, can be expanded in a multipole series which is v = {e2/eR'~} {MA. Mn--3(MA. k)(MR./~)} + {e-~/eR4}{~(R. QA./~)(MB 9 --(M,,. Q,,./~)}

+ {e'/eR 4} {~(/~. Q,,~,.R)(MA./~) - - ( M A . QR./~)}

+ {~/eR '~}{(/~-Q. QB./~) + 88 x (/~. QB./~) +~(QA : Q~)} + . . . .

Q,,./~) (15)

Here MA," Mn and QA, Qn are the dipole and quadrupole transition moment operators MH------~] rI~i, Q n = 3 M t , M n - M B - M B 1 (16) i The vector R = IRI k is the internuclear separation, e is the electronic charge, and e is the static dielectric constant of the medium. In the case where the transitions A---> A* and B---> B* are optically allowed, the major interaction is dipole-dipole. However, if the transitions are strictly forbidden due to odd parity in the product of the wavefunctions of the initial and final states the major interaction arises from the quadrupole term in the interaction potential. Hence the matrix element may be approximated by

substituting from (8) and taking the logarithms e2 liog[1 ~"

4-7fl'-z o ] = - - k . 4/3" + AE-_I

(12)

Since AE 2 >>/32 1

4fl 2

-- k

(13)

r 4fl 2 + AE 2 approximating ~-on the same AE >> fl basis

k = 2A____E h

4/3 2 =--- 8/3'-' 4/3" + AE 2 hAE"

(14)

E R 3 II A IIB

Dipole-Dipole

e'e R 5 QA Q8

Quadrupole-Quadrupole. (17)

Here the u's and Q's correspond respectively to the dipole and quadrupole transition elements. With regard to the evaluation of the vibra-

1090

V.A.

LODATO

tional overlap integrals, it is sufficient to consider the transition O ---> O in the vibrational spectrum. Hence the F r a n c k - C o n d o n factors become (XAIXA,) = 28 8a 8a 2+ 8 ~ exp {_89

AQ2 • ~A2+~2

(18)

where 8A = {RaM~h2} 88 and RA is the force constant for A and M its mass. We have another corresponding expression for B. Evaluation of the force constant factors from the spectral data of Dieke and Duncan [9] gives a value of - 0.997; these terms can be assumed to be unity and /3 is g i v e n by electronic interaction matrix elements. The following is a table for the relaxation times where ~" is the energy transfer period in the resonant case and the inverse of the decay constant in the non resonant case, and 9 is the static dielectric constant CsUO2 (NO3)3 which is approximately equal to two. 3, CONCLUSION

Using the numerical values from Hall and Dieke[10] for CsUO2(NO3)a at 4.5~ one may calculate the dipole and quadrupole transition elements which are 3.66• 10-3s esu 2 cm 2 and 3-6 • 10-46 esu 2 cm 4 respectively. Since the uranyl ions are separated by approximately 10 ,~ one calculates from Table 1 that for dipole-dipole ( D - D ) interaction the relaxation time is 8.9• 10-H sec, while for quadrupole-quadrupole (Q-Q) interaction it is 1.06 • 10-~ sec. Since the resonant line width 13-9) is less than 0.3 cm -~ and much larger than the natural line width, one conTable 1. Dipole-Dipole

QuadrupoleQuadrupole

Resonance

h~R a 4e2/z,~btt~

heR s 4eQaQ~

Non-resonance

hAEeZR 6 8e4v,~2V,n2

hAEe~R to 8e~Qa2Qn 2

z(sec)

cludes that the excitation transfer can not be due to Q - Q interaction but agrees with the D - D case. In the non resonant case the two isotopic species of uranyl ion of the natural salt is separated by 50 ,~ and the energy splitting is 0.5cm -I at 20~ Using Table 1, one calculated a relaxation time of 3.9• 10-11 sec for the D - D case while for Q - Q interaction one has 5.2• 10-21. Hence from the relaxation time the interaction must be dipoledipole, if it were not, one would not observe the isotopic splitting. Since UO22§ is linear in character a dipoledipole transition would be zero. However the electron cloud around the uranylion is distorted, due to the crystal field interaction with other ligands. A relaxation of the selection rules occurs and one has a forced dipole transition occuring. This is based on the fact that the oscillator strength for most uranyl salts is 10 -8 [4]. Thus from the above discussion we conclude that the excitation transfer between uranyl ions must be dipole-dipole and the line width can be explained in terms of the energy transfer. Acknowledgement--I would like to thank Dr. John P. Porter for several helpful conversations. REFERENCES 1. UMREIK.O D. C. and L A R K I N G . N., Z. Prik. Spek. 3, 567 (1968). 2. S T E P A N O V B. I., Z. Eksp. teor. Fiz. 10, 1153 (1951). 3. S T E P A N O V B. I., Z. Eksp. tear. Fiz. 10, 1158 (1951). 4. R A B I N O W l T C H E. and B E D F O R D R., Spectroscopy and Photochemistry of Uranyl Compounds, p. 53. MacMillan, New York (1964). 5. L A N D A U L. D. and L I F S H I T Z E. M., Quantum Mechanics Nonrelativistic Theory, Chap. VI, Pergamon Press, London (1958). 6. Th. Forster, Disc. Faraday Soc. 27, 7 (1959). 7. R O B I N S O N G. and F R O S C H R. P., J. Chem. Phys. 37, 1962 (1961). 8. K A T S U U R A K., J. Chem. Phys. 43, 4149 (1965). 9. D I E K E G. H. and D U N C A N A. B. F., Spectroscopic Properties of Uranium Compounds, p. 60. McGraw-Hill, New York (1949). 10. H A L L L. A. and D I E K E G. H., J. Opt. Soc. Am., 47, 1092 (1957).