Cesar C.L.

Mendes Filho J.

Miranda L.C.M.

Vargas H.

English

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Response to ‘‘Comment on ‘Photoacoustic determination of thermal diffusivity of
solids: Application to CdS’ ’’ [Appl. Phys. Lett. 4 7, 434 (1985)]
C. L. Cesar, H. Vargas, J. Mendes Filho, and L. C. M. Miranda 
 
Citation: Applied Physics Letters 47, 434 (1985); doi: 10.1063/1.96137 
View online: http://dx.doi.org/10.1063/1.96137 
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/47/4?ver=pdfcov 
Published by the AIP Publishing 
 
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Photoacoustic determination of thermal diffusivity of solids: Application to CdS 
Appl. Phys. Lett. 43, 555 (1983); 10.1063/1.94417 
 
 
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COMMENTS 
Comment on "Photoacoustic determination of 
thermal diffusivity of solids: Application to 
CdS" 
F. Alan McDonald 
Physics Department. Southern Methodist University. Dallas. 
Texas 75275 
(Received 11 January 1985) 
A recent article l proposed a photoacoustic method for 
determining the thermal diffusivity of optically transparent 
solids, and gave experimental confirmation. However, the 
theoretical basis for the method was partially in error, was 
unnecessarily limited to a one-dimensional treatment of heat 
flow, and failed to incorporate the effect of finite beam size. 
As the method does appear to have the advantage of simpJj-
city of experimental arrangement, a proper theoretical dis-
cussion is warranted. 
The proposed method uses a periodically modulated la-
ser beam passing laterally through a transparent sample, 
viz., at constant distance from the sample-gas interface of the 
photoacoustic cell (Figs. 1, 2 of Ref. I). Optical absorption 
(coefficient /3) leads to a periodic heat source distribution 
which is approximately independent of the lateral coordi-
nate (z) if the lateral dimension of the sample (d ) is smaller 
than the optical absorption length. A Gaussian beam of(e- 2 ) 
radius R then gives a source distribution 
S (x,y) = /310 exp{ - (21 R 2}[(X - XO)2 + y2J I exp(jwt), (1) 
where/3is the optical absorption coefficient, 10 is the incident 
intensity at beam center, Xo is the distance of beam axis from 
sample-gas interface, andf( = W/21T) is the modulation fre-
quency. The variation with Xo of the photoacoustic signal 
from this source distribution will allow determination ofthe 
sample thermal diffusivity. 
The distribution of Eq. (I) will cleady not give rise to 
one-dimensional heat flow, though in the configuration of 
Ref. 1 heat flow will be primarily transverse to the beam. The 
photoacoustic signal is then given2 by 
o.o~~--~----------------------~ 
c:n 
:z 
< 
- -0.5 Cl 
< 
eo;; 
w 
(J) 
< 
-1. 0 ::I: 
IL. 
. 
W 
Cl 
-1.5 ::::l 
~ 
-...J IL. 
:x 
-2.0 ::s 
L!l 
Cl 
...J 
-2. 5 1--________________________ --1 
o 100 200 300 400 
Xo (M I CROMETERS) 
FIG. I. Photoacoustic signal variation from Eq. (4), for three frequencies; 
R = 50 p.m. Dashed curves show phase variation. 
(2) 
where r is the gas specific heat ratio, Po and To are the am-
bient pressure and temperature, Ig is the cell gas length, and 
a; = (jwl a), with a the gas thermal diffusivity; () is the sam-
ple surface temperature, averaged over the gas-sample inter-
face (sample thermal expansion is neglected, as it is generally 
negligible for solids). This expression is technically valid 
only if there is negJjgible heat flow to the cell wans (in the gas 
region); however, the corrections3 to Eq. (2) are independent 
of XO, so will not affect the signal variation discussed here. A 
unit point source at depth Xo gives an averaged surface tem-
perature ()p or 
()p = (kps)-l{(b + 1) exp( - O'sxo) - (b - 1) 
Xexp[ - O's(2/- xo)] I [Ig + l)(b + 1) 
- (g - l)(b - J.) exp( - 2O's/)] , (3) 
where ks is the sample thermal conductivity, g = kgO'gl 
kps' b = kbO'b1ksO's (with subscript b indicating backing 
parameters), and I is the sample thickness. A beam of vanish-
ing radius will thus give B = /3 'Io()p, where/3' = /3d ',d 'being 
the lateral dimension of the gas-sample interface (d ' < d, see 
Fig. 2 of Ref. 1); this result replaces Eq. (5a) of Ref. 1. A beam 
of non-negJigible radius requires that one integrate the pro-
duct S()p over the effective source region: 
() = ["" dx f: "" dy ['dZS(x,Y)Bp 
= A exp( - O'sxo) ! erfc [(O'sR 12.[2.) - (F2xolR )], 
(4) 
where A is a complex constant, erfc is the complementary 
error function, and we have assumed a thermally thick sam-
ple, so 1- 00. This result reduces to that of Ref. 1 (/- (0) in 
the limit of R -- O. 
The variation of () with Xo is shown in Fig. 1, for three 
frequencies, and R = 50/-lm. For Xo > 100 /-lm straight lines 
are obtained (for both log amplitude and phase), consistent 
with the experimental data of Ref. I; the slope is as = ('TTf 1 
a.)l/2. The present result is derived without assuming one-
dimensional heat flow, and incorporates the effect of finite 
beam size. 
IC. L. Cesar, H. Vargas,l. M. Filho, and L. C. M. Miranda, Appl. Phys. 
Lett. 43, 555 (1983). 
'F. A. McDonald, I. Photoacoustics 1,171 (1982) . 
3F. A. McDonald, I. Phys. CoIl. 44, C6, 21 (1983); F. A. McDonald, I. Appl. 
Phys. 52, 381 (1981). 
4F. A. McDonald, Appl. Phys. Lett. 36, 123 (1979). 
Response: In the preceding Comment l McDonald has criti-
cized the theoretical basis of the photoacoustic (P A) method 
described by the present authors2 for measuring the thermal 
diffusivity of solids. In his criticism it is claimed that the 
434 Appl. Phys. Lett. 47 (4), 1 5 August 1985 0003-6951/85/160434-02$01.00 @ 1 985 American Institute of Physics 434 
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model of Ref. 2 "was unnecessarily limited to a one-dimen-
sional treatment of the heat flow and failed to incorporate 
the effect of finite beam size" and that a proper treatment of 
the effects of three-dimensional heat flow is presented in his 
Comment. 
In this reply we respond to these criticisms by firstly 
commenting on his statement on the first paragraph of his 
paper that the model of Ref. 2 "was unnecessarily limited to 
a one-dimensional treatment of the heat flow and failed to 
incorporate the effect of finite beam size." Reading carefully 
the description of our experiment the reader will find that we 
were aware of the need for taking into account the finite 
radius of the heating beam and this was indeed experimen-
tally taken into account as described in page SS6 of Ref. 2 
" ... Here we note that, because of the finite waist of the laser 
beam, the illumination position is defined relative to the 
beam waist. That is suppose that we are illuminating the 
sample from the gas side. As soon as part of the beam reaches 
the sample, the P A signal begins to grow; on further moving 
the sample, the PA signal eventually saturates when the en-
tire beam is illuminating the sample. Up to this point we have 
moved the sample a distance of roughly 100 /-lm, which cor-
responds to the beam waist. From this point on, the PA sig-
nal decreases when moving the sample. Thus, when plotting 
our data we have subtracted the beam waist in the iHumina-
tion position scale." This means that care has been taken, 
through our experimental procedure, to account for the ef-
fect of a finite beam radius. The resulting experimental data 
were then verified to decay exponentially as predicted by a 
one-dimensional model. In other words, the restriction to a 
one-dimensional model was not a question of choice but 
rather dictated by the experimental results. In this sense we 
have not "failed" to incorporate the effects of the finite beam 
size. Rather on the contrary, by taking into account the finite 
beam size as outlined above was that we have found that the 
simple one-dimensional model was perfectly adequate to de-
435 Appl. Phys. Lett., Vol. 47, No.4, 15 August 1985 
scribe our data. The effect of the finite beam radius, as dis-
cussed in McDonald's Comments, is only relevant up to one 
beam diameter. This is clearly shown in Fig. 1 of the preced-
ing Comment in which, after an initial increasing up to a 
beam waist distance (100 /-lm), the PA signal decays expon-
entially as exp( - as so), where as = (1T f la.)1/2 and Xo is the 
illumination position. Since the finite beam radius effect can 
always be taken into account by adopting the experimental 
procedure outlined in Ref. 2, the more elaborate calculation 
of McDonald adds nothing. 
The second point which is worth commenting regards 
the McDonald's claim in the abstract of his Comment that a 
proper treatment of the effects of three-dimensional heat 
flow is presented in his paper. This is certainly not what is 
done as is evident by the one-dimensional Green's function 
used in his paper [cf. Eq. (3) of the preceding Comment]; () p 
is a function only of Xo' This means that all that is mathemat-
ically done in his Comment is essentially to include the effect 
of a Gaussian heat source distribution on the one-dimension-
al model for the P A signal production. The result of the 
taking into account of this Gaussian beam of radius R, as 
shown in Fig. 1 of his Comment, is that for distances greater 
than one beam diameter the P A signal should decay expon-
entially as exp( - asxoJ in agreement with the one-dimen-
sional model of Ref. 2. 
C. l. Cesar, H. Vargas, and J. Mendes Filho 
Instituto de Ffsica. Universidade Estadual de Campinas. 13100 
Campinas. SP, Brazil 
L. C. M. Miranda 
Instituto de Estudos Avanrados, Centro Technico Aeroespacial. 
12200, S. J. Campos. SF. Brazil 
(Received 9 May 1985) 
'F. Alan McDonald, preceding Comment. AppJ. Phys. Lett. 47. 434 (1985). 
'co L. Cesar. H. Vargas. J. Mendes Filho. and L. C. M. Miranda. Appl. 
Phys. Lett. 43. 555 (1983). 
Cesar etal. 435 
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Response To Comment On "photoacoustic Determination Of Thermal Diffusivity Of Solids: Application To Cds'"

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Response To Comment On "photoacoustic Determination Of Thermal Diffusivity Of Solids: Application To Cds'"

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