Theory on the effect of applied stress on luminescent spectra based on a stress Hamiltonian

Journal of Lemiacsceace 24/25 0981) 329—332 North-Holland Publishing Company

329

THEORY ON TOE EFFECT OF APFL1ED STRESS OR LONINESCERT SPECTRA EASED ON A SiRESS HANTLTONIAR° David E. Barry and Ford Williams Physics

Department

University of Delaware Newark, Delaware, USA 19711

Using a mechanical modal to represent a luminescent system including the effects of an applied stress, we derive the inrm of the Hamiltnnian. We distinguish between the strain and stress Hamiltonians. The strain Hamiltonian is prnper fnr rho case nf internal strains and the stress Hamiltonisn is correct fnr on applied external stress. The Hsmiltnnisn derived from the mechanical model is the same as that developed earlier hy Curie, Berry and Williams (CEW) for hydrostatic pressure applied to cubic systems. We apply this theory to beth non—erbitally degenerate luminescent centers in cubic crystals under hydrostatic pressure and John—Teller (J—T) systems with applied uniaxial stress. We show that CBW theory predicts a linear stress term for the splitting of J—T systems.

T.

Stress vs. Strain ttamiltonians

The equilibrium configuration for the crystal with no external force is defined es Q 0. With the application of an external force, F, the crystal is deformed and acquires a new equilibrium specified by Qe~ The difference between the stress mod strain Hamiltonians becomes evident by considering the displacement from ~e to some arbitrary configurmtion,Q, maintaining F constant. In addition to doing work against the crystal forces, work must be done against the applied force,F. We define the following HamiltonisnsH0 for the crystal with no applied stress mod the equilibrium Q0, H5~-~~for the crystal with configuration Q other than because of a deformation but with no externally applied stress, and Hstress for a crystal with an equilibrium Q5 because of an applied stress. The last two are related as follows: ttsrrain + FQ + C

(1)

Hstress = where C is independent of Q, but not of F. The second term represents the work done against the constant applied force, when the crystal changes its configuration. The corresponding ~straln is the Hamiltooian if there were no external force applied to the crystal but with the equilibrium configuration being t/e, the stress equilibrium which depends on F. We will show later that the difference between N 0 and Hstraio does not depend on

Q.

We must also consider the interconnection between the adiabatic potentials, the orbital electronic energies and the total potential energy of a crystal. Each adiabatic potential represents the total potential energy for the crystal when the given electronic orbital state is occupied, whereas the orbital electronic energy does not include the ion—ion interaction energy. When we deal with the dynamics of the crystal, we must use the total potential energy.

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DI. Brim 1. It/I/lasso /

330

/9)/lcd sD:))

OH

/um/ecsccnl spce/rs

Olsen an external iorce , F, ts applied to the crystal, a srraia is developed 0e — c In crystal. l’hi s strain corresponds ta follaws a configurational change The the strain tiamiltouiao is related to li~ as 0straio

=

flu + wad’

wisero ~ad is the adiabatic wurk done when the Force P is appi led ruvern ihia the crystal. Ey reversible appli cation of F, vu aean that at all times the crystal ierces just balance the appl icd force. lv~~is

Hd

~

F~df

=

Vc(Qe)

~c0o),

tn

(I)

schere ~c is tlw crystal force nod UcOl) is the crystal poteotisl energy without an applied external Force. Cxpunding V ((/~) in terms of the strain, — P~, we Find the usual result, that Wad has no linear strain dependent tern hecause P~ is the equilibrium configuration. Therefore, the leading term in Wad is quadratic in the strain. To the present literature on laho—Teller (i—i) systems the usual assumptions made are that the s Irain Hamiitenian is the correct gamiltooian for the case of an applied external stress and that the strain Hamil tonian contains a linear strain term. See, for example, the work of gam(T) where he assumes io the case of an applied stress that the splitting of the electron (orbital) state in a uniform strain is described by a linear strain term which he gives explicitly in his equation 19 and that this electronic apI itting descrihes the splitting of degenerate i—T states. Since 0~ is not so equilibrium configuration of the electronic orbital energy, it is correct that this orbital energy is split by a linear strain term, however, it is the total potential energy that Oescrihes the transition energy spectrum and it is not split hy a linear strain term, therefore, the linear splitting with respect to the applied stress F hetween the degenerate i—f states which is seen experimentally mnst he a consequeoce of the difference he— tween t1stress and ~atrain IT.

Wechanical

Nodel

We use a mechanical model which is a linear mite chain of springs having twe different spring constants, with an applied external force having a prescribed nature. The springs are arranged, so that, the inner springs howe one spring constant and the outer the other, which allows for representation of both local and crystal modes. Using this mechanical model shows that the external force acts~ ~)O only some of the crystal modes and some of the local modes. When there is a configurational change which involves a change in a local mode on which the external force acts, then energy is expended to do work against this external force in addition to the work done against the crystal forces. We have determined the stress hlamil tonian using the normal mode representation for this model. gor those normal modes whIch do not interact with the external force, there are no added strain or stress terms to the Hamil.tonian. For the other modes, Curie, Berry and Wllliams~~ (CNW) have given a procedure for finding the added terms due to the applied stress. there we describe a derivation directly connected with the method used to apply the external force. Within the framework of this model, we consider rise following problem: We assume that the applied external force depends on time and is given by f(t), where f(t) satisfies the conditions: f(t) = 0 if t~T_, f(t) = F if t>f5 and is continuous. Letting H(t) he the Hamiltoniau for this problem, such that, for rT_, H(t) Is equal to the Hamiltonien for the situation without an applied force, Ih~, then Hatress is given by H(t) for tT+. We note: (A) that by this procedure, the added arbitrary constant is the some for both H0 and histress. Normally, we do not consider H0 and Hstress simultaneously, and, therefore, we can, for convenience, have different values of their arbitrary constants.

f(r)

DE. Bern, 1’. Will/ama / Applied n/tess on luminescent spectra

331

However, in the framework of the adiabatic approximation, the arbitrary constants interrelate the Hamiltonians of the different electronic states and, therefore, cannot be chosen independently. (B) H(t) cannot be found in general, slocewemust use the explicit form of the motion. And (C) H(t) and the energy, E(t), as indicated, are not constants of the motion nor equal, since the time variation of the force, f(t), is a way of representing the exchange of energy with the surrounding. We have considered several forms of f(t) satisfying the conditions giveo above. For example, we considered the case where f(t) = 0, for — = t T— = 0, f(r) = Pt/i, for T_ t S T+ = t; and f(t) = F, for T+ S t S . We find that

g(r) is

2

E(t) p2 and

E(t)

+

=

p =~+f(Q -

2

t(~

+ F)2 +

-

—

/3:(A sin or

—

p 2 0)

+ E(D)

fort S T_

(4)

for t 1 T+

(~)

+ E(D) +

B cos or

—

B)

where ~2 = k/N and where the motion of the system for t I T_ is given by Q(t) = A cos or + B sin ci + Q 0. We note that the lest term of equation (5)goes to zero in the adiabatic limit, that is, as i goes to Infinity. Using several other forms of f(t), we have found the followins results with respect to equation (5): (A) that rhe energy always contains the first four terms, (B) that the last term depends on the form of f(t), but is always zero in the adiabatic limit, and (C) that if the variation of f(t) is matched correctly to the motion Q(t) then the last term is identically zero. Since the Hamiltonlan is independent of the way in which we get to a particular physical situation, If there is no hysteresis, then H(t) Is Independent of r and the form of f(t). Therefore, H(t) = E(t) for t S T_ and

2 H(t)

=

Hstress

=

+ ~(Q

-

+

~)

2 +

+ E(0) for t > T~.

(6)

and that(3), thewe constant C of equation is _FQe_F2/2K. From the form of H(t) 2/2K, and equation find for this problem(1) that We can recognize that F(t), is varied, so that the amplitude of the H5t~0~0 — H0 5 Wad = F if the force, motion changes, then energy has been exchanged with the surrounding. The r dependent term represents this exchmoge. However, if the force is changed slowly enough, then the disruption of the motion is negligible, since a small Increase im the amplitude during one part of the cycle of the motion is cancelled during another part, and then the energy is equal to the Hemiltonian. The stress Hamlltonien, given in equation (6), is the same as that found by CBW. TTI.

Jaho—Teller System

We consider the application of the CBW analysis to a J—T system.a For simplicity, we treat an electronic doublet coupled to a single vibrational mode having equal harmonic potentials with force constants E, see the solid curves of Figure 1A. If we assume there is no interaction between these J—T states, then the solid curves represent the adiabatic potentials to which we apply the CBW analysis, which implies that there is no splitting of the zero phonon lines. Do the other hand, if we assume an interaction between these states, even one of zero magnitude, then the dash: curves of figure 1A represent the adiabatic potentials to which the CBW analysis is to he applied. We will consider the lower of these The linear chain model was used in the previous section only to find the relationship between Hemiltonians — not In any sense implying J—T characteristics.

Di urns, I. i’i///t~i~/ ipp/irt/ Osro oil /tutiitsrsta’tt/ apr/rn

331

two stares

in the

limit

of

a zero

magnitude

interaction.

~

~_~Z___

E(O)~——

~

>Q A

—‘f-

L—

—~‘

~ieisii

>Q

B I

lA is a schematic representatIon of Fist’ harmonic potent Lii vi thorit interaction, aol Id curves, and with interact ion • dan/i curves. lB is a scisematir representatioo of tile lower iltteras ring potent/al curve of f issure lA, without external stress, s/ash curve, and with external stress 1, sal id curve, where we have ueglecred cisnoges in the sepsrat lots of the sin La due to sue in te rae tiost. list

usuah

niuimulss

CE/i analysis

and,

s,’,ss devt-iiissed

tiierelssre,

we

must

assusane

decide

ssb

odi,sisntir

iris os

tile

is:]

stilts

minima

is

ti

0 linviss’ ise sssed

,i iii

01I5~

L

/ ire

present aaais’sis . for multi—minima curves, we use the loilslwissa riO e~ I correct nriflinsum, rise stress adiabatic poteuiial [or si] misses i’l tilt’ atteos and all configurations will iso greater titan sir equal is’ i/to oisselsstr minimssm e/ rise zero stress adiabatic usutentissl . lit/s is evident from tilt fstci that 1’etentio/ etserula sssust increane on npplicnt]oss of the ntress. Using i/sin direction:

expanded C/lw ataalvsiss , ‘‘c f issd for ss stress tlirected its tile negative 0 (A) titer rise correct sssinimusss is rite ninirsum with the sniallesi I since using rise oriser lssississsuns, titere ssould iso a discontinuity sc/sen rile S5~/il ied force I’ is equal to tue sstssxisssusss of rite crystal force for Ifs 53 1 o I ~,/K, intl (B) titat the adiabatic potential for an applied stress I is as shown itt fissure l// isv rite solid curve to first oct/er in F. /‘,‘e nsste tbsat tisere is t linear spli it/iSri /setweess rise two sstmninsa v/sic/i is~ (2~/K)l’. 1/sit; linear splitting sirises from rite ternt in use stress ilasnhltonian which takes account isf rise work dosse agafisst use stress, when rise coofigssratiois is citangeti. This tersis sssust he cons idereti for i—I systessss , since rite i—i’ spl irring is i/to ttleasure of i/to trans;it iois energy at two different configurnticsns and titero fore sreasssres this 000rssv term. Acknowledgement:

iiitpported

in

part

isv a

rant

[russ

‘11/0.

tisvs . Rev.

B20,

geferences: (1)

F.

5.

(2)

I). Citric,

Hans,

i’isvs. I).

Rev.

i/errv

166,

p.3D7

nod I’. Will muss

(1968).

,

i

p.

2323, (1979)