Theory of self-transducer photoacoustic spectroscopy

Solid State Communications, Vol.78, No. 1,pp. l-4,1991.

0038-1098/91$3.00+.00 Pergamon Pressplc

PrintedinGreatBritain.

THEORY R

OF SELF-TRANSDUCKR

P. Vardapetyan.

PHOTOACWSTIC

A. P. Hovhanesyan

SPECTROSCOPY

and A. A. Araratyan

In&itute of Applied Problems of Phyeics, Armenian Academy of Sciencea. 375014, Yerevan, Armenia, USSR_ (Received 12 December

1990 by G.

S.

Zhdanov)

A

aelf-transducer method of photoacoustic.signal detection ia propoeed, baeed on piexo- or pyroelectric properties of the sample. An expression for the potential difference between Surface electrode8 ie found , which depend8 on optical, acouetic and thermal properties of the 88mple and intensity and modulation frequency of the light.

The theory of photoacoustic apectroscopy with piezoelectric detection Is

Periodical heating and expansion of the crystal, due to the light with harmonically modulated intensity W=Wo( l+coowt)/Z excite acoustic oscilla-

given in Cl] for low (below 10'Hz) modulation frequencies and in C21 for high frequences in the case of opaque and thermally thick samples. Here we preeent the theory of selftransducer photoacoustic 8peCtrO8COpy (ST PAS).when the intrineic piezo- or pyroelectric propertiee of the sample are used to detect the photoacouatic signal. The geometry of the problem ie shown in Fig. 1. The lateral side8 of the dielectric crystal plate are much larger than the thickneae L. A longitudinal acoustic wave can propagate along the X-direction (X-cut a-quartz, Y-cut lithium sulfate, Z-cut lithium niobate, piezoceramice, etc.). Since the edge effects are neglected, the problem is one dimensional. The crystal plate is coated with infinitely thin and transparent metallic electrode8 and is in contact with ieotropic nonpiezoelectric medium.

I

tions in the sample. Then on it8 lateral aurfacee there appears a potential difference whose value depend8 on thermoelastic, optical and acouetic properties of the sample and medium. If W(l-r) i8 the intenaity of the light entering the crystal ( r is the reflection coefficiis the energy ent), then pW(l-r)oxp(+x) abaorbed in the unit volume within the crystal during a unit of time. Heat generation ie assumed to occur instantly just in the point of the sample where the light ie absorbed, and thus. the heat 8ource power decreases a8 OXP(+X), where P ie the optical absorption coefficient. Heating lead8 to the temperature increment T=@a,,. where 8>0 is the real temperature

and 8o - the natural

state temperature, is taken a8 zero. The natural state ia characterized by the and tensiona. absence of deformation8 The thermal conductivity equation ha8 the form

I

1aT U2T Wcl(l+) -- - --~--= _ (je-/-?x ax2 2n xat with

l-

boundary

eiwt

, (1)

condition8

I

L

X

where

x is the thermal

is the 8urface is the thermal

conductivity,

x*

conductivity, x = ~/PC" diffusivity, p is the

denaity, c" is the specific heat at constant volume _ The constant component of the heat source is not taken into account in equation (11, a8 it doe8 not contribute to the piezoresponse. 1

2

THEORY OF SELF-TRANSDUCERPHOTOACOUSTIC SPECTROSCOPY In the wave

equation

bu

&Y

P--iat

=-ax

where

the tension o(x,t)

Q ie a'u dU = c" - - eE - yT + n--__6X f9xat

u(x)=t4e-ikx+Ne

, ,%(l-eg/~)

of the

Es.

'

(7) fs

ikx+pe-Ax+gexx+Re-13x

ui(x)=Ui~x~(ikox)

I

,

(6)

,

u2(x)=U2~xp(-ik,,(x-L)) where

Ui and U2 are the wave

in front of and behind

(5)

where g is the pyroelectric coefficient and E is the dielectric constant, must be added to these equations. As the crystal is in non-conductive medium, the Poisson's equation is hold all other the range of the values of x from -cot0 +aD _ At the same tiae there are no electric fields outside the crystal and D = 0 , therefore, taking info account the boundary conditions for continuity of the normal components of the electric displacement, then D ia equal to zero within the crystal as well. All the values of T, E. u. 0 and V appear due to the intensity-modulated light and, therefore, depend on time as

amplitudes

the eample

and ke

is the accouatic wave vector in the isotropic medium. Substituting now Eqs. (6) and (8) into Eq. (7) and grouping the terms, we obtain

-Ax+(pw2Q+&.2Q-~AB)eAx+

+(po'P+&'P+AyA)e

+(pw2R+&2R+&C)e-'x=0

_

(10)

From the linear independence of exponential functions it follows for the amplitudes P, Q and R: ;A/;: $3 = _ ---A k' +A2 GA/;:

Q=-=---El

I G-3/;:

,

R

=

_

k"+A*

1x.i-__.

c

-

k2+f3'

exp(iwt)_

Eqs. (4),(5) and Poisson's equation form simultaneous differential equations of thermal piezoelectricity with the unique solution C31. The general solution of Eq. (1) is + Re xx + CeepX)eiot

In this case we use the dispersion

(6)

u*(O)=u(O)

*

u(L)=u2 (L)

9

Q2(0)=0(0)

9

g(LW2(L)

-

Using

(5) we get from Bq.

(11)

C A = -C(T+J3)(T~)eAL+(,-13)(r-A)e-13Ll, P

c

=

.

*

-C(T-13)(T+A)e-~L-(T~)(T--X)e , aJ_

,

r=Il*/x , p=(r-A)

, C="(lWr) 2w(hZ-f32)' 2

e

-AL

_

(4)

(12)

Similarly for elastic tensions in the medium in front of and behind the crystal o1 and u2 we have

- (T+A)'d"

Using Eqs. (4) and (5) the wave Rq_ (3) may be reduced to the following form d2u po2u+c.----y--=0 dX*

Eq.

du o(x)=%-- - ;T dX

-AL,

P

A=(l+i)a

rela-

tion po2="ck2 which also results from Eq. (10). Rxpreesions for the amplitudes Id and N are obtained using the boundary conditions for displacements and tensions on the crystal faces:

with

B

, (6)

where k is the acoustic wave vector. The elastic waves escape the plate outside the crystal, so

module, c" is the rigidity, 7 is the thermoelastic coefficient, equal to the volume expansion coefficient multiplied by the volume rigidity module, and n is the phonon viscosity_ Poisson's equation SD/&t = 0 for the electric displacement

T(x,t)=(AevAX

,

(4)

where u is the acoustic wave amplitude, E is the electric field, e is the piezo-

&I D=e-+gT++EE bX

&=cx(l+?+i~~/c~)

K7 (e /EC ) _ The solution given by

(3)

Vol.78, No. 1

u1

(x)=$

"2(x)=:o

dT

d”, & -

FoTi

_d!!_. dX 7,T,

,

(13) ,

(7) dx

where

io= co+itivo _ Here c,, , I),,and Y,

Vol. 78, No. 1

are the elasticity, viacoaity mal elasticity of the medium, lyThe medium temperature6 and behind the cry&al T, and

and theraccording-

is the piezoelectric

Eq.

GpT&*=-~pT~,,, (14) z@T I &=x,pT*

I:x=L

where T1=Biexp(h,.,x)

(fi0)

T2*,exP(-~o(x-L))

(tiL) ,

* (15)

X,=(l+i)a 0' medium thermal It yields

(w/2x0)

and x, is the

eiot,

V,2iZo

conductivity.

3

- ReAL)+g

Ce"L]

0

9

3

ia 0(1-r)

zg= 2Ex where

md_

pois

the density

v. is the velocity

dz_eBikL- feikL nd+- md_e

9 of the medium

and

of sound.

The reaulta of our theory are shown in Figs. Z-6. As a model object a 0.2 cm

-ikL

thick xe

,iwt

2”“WoPo

0

md+eiw-

-ikL_

2

2*=

.

Condition6 of continuity (11) for displacement8 and tenaiona lead to the syetem of algebraic equations for M. N. U, and U,, whose solution has the form

N=

-

XWo(l-r)

” [;(Ae-AL

H=

and

is the pyroelectric voltage. The final expression (16) for V have no limits on 13. a, 0 and ie valid if only the boundary conditions are fulfilied. Eq (16) could be easily eimpll.fied in the case of optically and thermally thick eample (i?Ls>l, aL>>l. T~,u) at the resonant frequencies f=v(2v+l)/ZL, where v ia the velocity of the longitudinale wave in the crystal, v=O,l,Z... We get

VgtzgA

0

voltage

in front of Tz are so-

lutiona of the thermal conductivity (1) at W=O with boundary conditions

ez=

3

THEORY OF SELF-TRANSDUCERPHOTOACOUSTIC SPECTROSCOPY

d~eikL +

where m=l+P+l_Q+b+R-y,e*+;(A+B+C)

,

n=l_Pe -hL+l + eA %I+b_e-%+yo8,-;T(L) T(L)=Ae-XL+BehL+Ce-(3L d+=iko;02ik;?

,

b,=iko&o%fi&

,

,

,

Z-cut

LiNbC3

platelet,

vibrating

in air at normal conditions, wae chosen. From Es. (16) and Pigs. (Z-6). the following predictions are made. (1) ST PAS signal is proportional to the reflection corrected incident power. (2) For small values of P, the signal in directly proportional to 13. (3) The signal ie inverely proportional to modulation frequency.

(a) 'l,=iko~o&

.

(bl

The

potential difference between L. electrodes V= -SE dx we get by integration

of Es.

(cl

(50) at D=O:

v=vo+vg*

(16)

where cl

V =" . c

10-7

I

I

I

10-5

[t4(e-ikL-1)+N(eikL-I)+

Optical abkption

Fig-Z_ +P(e -AL_l)+B(exL-l)+R(e-'L

-l)Jeiwt

I05

ST PAS niennl 1OHz (h) and

km“)

VR 3. w/%n-:ltlz 1OOtlz (cl.

(.=I),

4

THEORY OF SELF-TRANSDUCERPHOTOACOUSTIC SPECTROSCOPY

Vol. 78, No. 1

,

Z

(a) (cl

10-8

’

10-4 Optical

Fig.3.

I

I

I

IO

absorption

102

IO

6

I03

I04

Frequency Uiz)

(cm-‘)

ST PAS

signal va I? in high frequv=O (a) , 1 (b) and ency case. 5 Cc)-

Fig-S.

ST PAS signal P=O.Ol

cmvL

100 cm-*

vs frequency

(a), 1 cm-'

w/%-r.

(b) and

(a).

b_

6_

(a) 0.2

, -

Pig.6.

I

I Parameter

Fig.4.

008

0.6

I.0

(cm)

v8 L. w/%=100

(a), 3 cm-'

Hz.

(b) and

Cc).

0.12

u

The frequency response of ST PAS signal Mod V and phaee Arg V near the "cut-off" frequency at v=O . p=10

cm-'

10 cm-'

I

I

0.04

06

thickness

ST PAS signal PzO.1

0

0.4

Sample

Cp?

(5) Thick samplea signals.

tend to yield

higher

ST PAS might be applied in practice because of its simplicity and large number of piezo-, and pyroelectric materials.

(4) At high frequencies the signal amplitude can be enhanced using the resonant properties of the mmple. In this case the photoacoustic quality of the cry&al could be measured by a standart remonanceantiresonance method-

Acknowlegment-We would like to thank Dr. Kh.V.Kotandjian for kind aeeietance. Reference8

1. W.Jackson and N.H.Amer. J.Appl- Phys. 3343 (198015l, A.H.Horozov and 2. Yu.V.Gulyaev. V.Yu.Raevakii.Soviet Phya. Acouetice. 278 (1985). 3.L.

3. W.Nowacki. Efekty w stalych cialach Warszawa (1983).

elektromagnetyczne odksztalcalnych.