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Measurements of the radiative lifetimes of MgO(B 1Σ+, d 3Δ, D 1Δ) states
Volume 178, number 2,3
CHEMICAL PHYSICS LETTERS
22 March 199 1
Measurements of the radiative lifetimes of MgO(B ‘C+, d 3A,D ‘A) states Christian
Naulin,
Michel Costes, Zakkaria Moudden
VRA 348 CNRS: Photophysique et Photochimie MoltWake
and GCrard Dorthe
‘, Universik! Bordeaux I, 33405 Take
Cedex. France
Received 16 November 1990; in tinal form 8 January 1991
MgO molecules have been produced in X ‘Z+, a ‘II, and A ‘II dark states by the Mg( ‘SD)+NZO( X ‘Z+ ) reaction in an experiment performed with pulsed, crossed, supersonic molecular beams. Analysis of fluorescence decays followingpulsed dye-laser excitation ofthe MgO reaction product has resulted in radiative lifetime determinations of the B ‘Z+ v=O-3, d ‘A and D ‘A states. The following values have been found at 95% confidence level: B ‘Z+ state, r,,0=21.5 i 1.8 ns, r,,,=,=21.9? 2. I m, 7,,~=21.7? 2.0 ns,r,~,=2l.5~3.2ns;d3Astate,7,,,=7.8~1.8ns;D’Astate,7,,,=4.3+l.0ns.
1. Introduction Mg(‘&) tN,O(X There have been few published determinations of radiative lifetimes of MgO excited states. The experimental values, 32.7 & 1.7 ns for the B ‘Cf v=O level and 11.8 f 0.5 ns for the d ‘A state, were reported by Diffenderfer et al. in 1983 [ 11. In their experiments, MgO molecules in the X ‘C- and a 311 states were both generated by the reaction of excited metastable Mg(3PJ) atoms with O2 or N20 scattering gas at pressures of 1 Pa or less. The radiative lifetimes were deduced from the temporal decay of MgO fluorescence following pulsed dye-laser excitation of MgO dark states. Values calculated ab initio for B ‘E+ v=O, 21 ns [ 21 and 24 ns [ 11, were also obtained by the same group. More recently, in 1987, another experimental value for the B ‘Cf u=O state, 22.5 + 1.5 ns, was reported by Biisener et al. [ 31. Their experimental arrangement also used pulsed dye laser excitation of MgO (X ‘C+ ) molecules, which were produced by the Mg( ‘So) +N,O reaction at 1000 K in a beam-gas experiment with typical scattering N20 gas pressures of 0.1 Pa. In this work, MgO molecules were produced in a single-collision regime by the following exoergic gasphase reaction:
’ And GDR 87 CNRS: Dynamique des Riactions Mol&ulaires. 0009-2614/91/$
‘C+)
~MgO(X’C+,a3H,A’H)+N,(X1c~).
(1)
Such reactive collisions were achieved when crossing pulsed, supersonic beams of ground-state Mg and N,O species at a relative translational energy of reactants of 0.9 eV [4]. Pulsed dye-laser excitation of the MgO products in the collision zone resulted in the determination of radiative lifetimes of the B ‘C+ ~0-3, d 3A v=O and D ‘A v=O states.
2. Experimental A complete description of the apparatus has been previously given [ 51. The Mg( ‘So) beam, generated by entraining atoms, laser ablated from a Mg rod, with H2 carrier gas emitted by a first pulsed nozzle, had the following characteristics: 3300 m s-’ velocity at the peak of the velocity distribution, 10% velocity spread full-width at half-maximum (fwhm), 3.5 ps pulse duration fwhm at the crossing point and 6.5” divergence fwhm. The pulsed N20 beam, obtained by expansion of a 22% N,O-He mixture from a second pulsed nozzle, had 1140 m s-’ velocity, 22% velocity spread fwhm, 30 ps pulse duration fwhm at the crossing point and 4” divergence fwhm. These
03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
325
Volume 178, number 2,3
beams crossed at right angles with an attenuation factor not exceeding 10%. The MgO molecules produced in the dark states were excited by pulsed laser light at the beam crossing point using a dye laser operated with 532 nm pumping of a Rh640-DCM dye mixture. Radiations at shorter wavelengths were generated by frequency doubling and stimulated Raman scattering in Hz, used separately or in conjunction: (i) MgO(X ‘Cc ~0-3) molecules were excited to the B ‘Z+ state using the B-X transitions of the Azl=O sequence between 501 and 499 nm (first antiStokes line). The fluorescence of the B-X or B-A transitions was filtered through 3 mm of Schott BG 18 or Schott OG 550 coloured glass. (ii) MgO(a’rI u=O) and MgO(A’II Y=O) molecules were excited to the d 3A v=O state using the d-a transitions at 372 nm and to the D ‘A v=O state using the D-A transitions at 380 nm (first Stokes line of the doubled dye). The fluorescence was filtered through 1 mm Schott BG3 coloured glass. The fluorescence imaged onto the photocathode of a Hamamatsu R955 photomultiplier tube (2.5 ns rise time, 50 Q loaded) was fed directly into a transient digitiser (Tektronix 7912, 7B80 time-base. and 7A 19 50 Q vertical amplifier, 0.3 ns rise time) to measure radiative lifetimes. In a typical run, 64 individual fluorescence decay traces were accumulated and then transferred to a computer for data treatment.
1
1
4
3. Results and discussion 3.1. The B’C+ state An excitation spectrum can be found in ref. [ 41. Decay measurements were performed at the wavelengths of the prominent bandheads for vibrational levels u=O-3 (la 70). The dense rotational structure near the centre of the bands precluded individual measurements for low J levels, The analysis procedure is similar to that described in a previous paper on C2 (d 3II,) radiative lifetime [6]. The observed decays have been fitted using a least-square method to an exponential function (fig. 11,
326
tA,exp(-t/z),
(2)
,
I
0
I
I
time
I,
100
1
Ins
Fig. 1. Fluorescence decay of MgO(B ‘C+ u=O). Solid line: experimental; dashed line: calculated for r= 21.5 ns; deviation of experimental values from the exponential fit is displayed on a magnified scale.
with an upper limit of dispersion error, (Azdi,/7)‘x3s2(7/nst)*,
(3)
where s2 is the variance estimate for the considered set of n data points of increment 6t. Systematic errors are mainly due to the transient digitiser time-base sweep error (2% specified). Indeed, overestimation of the time-constant due to the convolution of PMT and electronics rise times, 7’ z
y(t)=A,
22 March 1991
CHEMICAL PHYSICS LETTERS
[7* t 1
(rise times)*] “* ,
(4)
is an almost negligible factor as (7'-7)/7zO.6%. Underestimation of the time-constant resulting from the loss of laser-excited radicals moving out of the detector field of view during the acquisition of the decay curves has been shown to be negligible for laboratory velocities in the l-2 km s -’ range and timeconstants in the 100 ns range [ 61. In the present case, the higher MgO laboratory velocities (up to 4 km s-’ ) are compensated by the much lower time-constants ( ~22 ns). Loss of laser-excited radicals by collisional quenching is also insignificant in these experiments performed with a total attenuation of the Mg beam < lo%, including all elastic, inelastic and reactive events which take place during the time-offlight of the Mg atoms through the N20 beam, z 23 ps. The uncertainty due to experimental systematic errors, Az,,,/r, can therefore be set to an over-
Volume 178,number 2,3
CHEMICALPHYSICSLETTERS
Table 1 B ‘Z v=O-3 radiative lifetimes v
7 (ns)
n =’
s b’ (%)
AhI, (ns)
AT (ns)
0
21.5 21.9 21.7 21.5
185 193 173 195
5.2 6.4 5.7 12
0.58 0.71 0.69 1.26
1.8 2.1 2.0 3.2
1 2 3
a) Number of points in the (70-2%) range of signal maximum intensity; after complete decay of the excitation light. b, Standard deviation of the sample.
estimated value of 3%. The total uncertainty, at 95% confidence interval, is thus defined as A7= 2A7,, + 7(A7~,,p/~)
(5)
Our results, summarisccl in table 1, are in good agreement with the experimental determination of Biisener et al. [3], 22.5f 1.5 ns for v=O, J= 1, and the theoretical determinations of Diffenderfer and Yarkony [ 21, 2 1 ns, and Diffendetfer et al. [ 11, 24 ns. As pointed out by Biisener et al., the striking difference when compared to the experimental value of Diffenderfer et al. [I], 32.7 & 1.7 ns, remains unexplained. 3.2. The d A ‘ and D A ‘ states
.
in the same spectral region to avoid significant differences in laser-beam spatial profile and pulse duration, and under comparable saturation conditions. Let yO(t) and y( 1) be the fluorescence signals observed for the reference and the species studied, respectively. They can be expressed, using eq. (6)) as the convolution of the unknown apparatus function, and the exponential decay function, either known as reference, go(t), or to be determined, g(t). The latter functions read ga(t)=exp(-a,t)
and
-at),
g(t)=exp(
(7)
where l/aO=r,,, l/a=7, 7. and zrefer to the relevant radiative lifetimes. Let z(t) be a function relating both measurements, y,(t) and y(t), such as (8)
JJ(t)=yo(t)@Jz(t). The Laplace transform of this equation reads W)=Wo)
2
x9(z)
(9)
which can be rearranged, using eq. (6) as Y(z) =~(g)/~(go)
.
(10)
Introducing the Laplace transform expression of an exponential function yields Y(z)=(s+ao)/(s+a)=l+(ao-a)/(s+a),
When the fluorescence decay time-constant approaches the temporal width of the laser pulse, the number of data points left after complete decay of the excitation light with sufftcient signal-to-noise ratio becomes too restricted. It is then impossible to derive radiative lifetimes from a simple exponential lit. The actual apparatus function, f(t), has to be taken into account, since the observed signal, y(t), now appears as the convolution of an exponential decay function, g(t), with f(t), y(t) =f(WWt)
22 March 1991
(11) where s is the operational variable. Taking the inverse Laplace transform, z(t)=P’(l)+Y-‘[(a,-a)/(s+a)], leads to z(f)=8+(t)t(ao-a)exp(-at),
(13)
where S+(t) stands for the Dirac function, 6+(t)=lim,,,
s(t-u) 00
(6)
The apparatus function cannot be simply estimated by some scattered light signal: it only yields poor fits to experimental data, probably because such a procedure does not allow for saturation effects generally occurring in the absorption process. An alternative procedure, which takes account of absorption effects, is to record as a reference the fluorescence signal due to a species of known lifetime, preferably
(12)
with u > 0 (
and
I
0
.
S+(t)dt=l >
(14)
If 7. is known, measuring the yo( t) and v(t) functions allows one to derive 7 value with a simple fitting procedure of v( t) to the convolution product of ye(t) and z(t). The best-fit value is obtained by minimising the variance estimate, s2, of the sample, 321
Volume 178,number 2,3
Au< [ ((dy)‘>“‘+Aa,(
( >
s2= CRf
22 March 1991
CHEMICALPHYSICSLETTERS
/(n-l),
($Y/a~)2>“21
(15) (24)
where the weighted residuals, R;, are defined as Ri=[~;-_~oi@Z(t,)l~~i.
(16)
Here, yi is the fluorescence intensity measured at time t,, with standard deviation Ui taken as weight. Since the photoelectron emission of a PMT photocathode follows Poisson statistics, fl, can be approximated by (y,)“*. Dispersion errors can be evaluated by differentiating eq. ( 8),
(wwdh
dy= (wwd~f
,
(17)
where the reference parameter, a,, is also considered to be subject to error, duo. The uncertainty on a, Aa, which is simply derived by overestimating da in eq. ( 17), A~G LAY+(+iaa,)Aa,ii(ayiaa)
,
(18)
can be estimated provided the partial differential terms in eq. ( 18) be evaluated. The partial differentials can be expressed as (a~/a+~,(tp3az/aa =-~o(t)~[ltt(ao-a)]exp(-at),
(19)
and ayiaa,=y,(t)~aziaa,=Y,(t)~exp(-at).
(20)
yielding (25)
A7~is= FRAU_
Again, experimental errors are mainly due to the transient digitiser (Ar,,,/7<2%) and the total uncertainty at 95W confidence level is defined by eq. (5). The fluorescence signal of Mg( 3p ‘PI ) (r=2 &0.2 ns [ 7 ] ) obtained by laser excitation of ground-state Mg( 3s ‘So) atoms at 285.21 nm (first anti-Stokes line of the doubled dye) was recorded as the reference function, y. (t ) (fig. 2). Special attention was taken to avoid the apparent lifetime increase due to radiation trapping. This problem was solved by smoothly varying the delay between the triggering of the pulsed valve and the ablation laser, which yielded decreasing Mg( IS,,) concentrations, and consequently decreasing the apparent Mg (3p ‘PI ) lifetime. Eventually, a point was reached where the fwhm pulse duration of Mg( 3p ‘P, ) fluorescence no longer exhibited a dependence upon the Mg( ‘So) concentration, thus allowing the reference function to be recorded. The observed decays are displayed in figs. 3 (d 3A) and 4 (D ‘A). The results from the tit are listed in table 2. The dispersion error computed with AT,= 0 are also included: the very small A7odisvalues ob-
Quadratic average values of these expressions are computed from experimental data:
1 1
A~2=((dv)2>=
((WWV
(
C [Y~-YO+WC)I~
=( C (k/W2)/r:,
c Waa,)‘~=(~
Wi!ad’)h,
>
/n,
(21)
(22)
(23)
where the sums extend over the IZexperimental data points. The uncertainty of the parameter a due to dispersion errors can thus be estimated using eqs. ( 18) and (21)-(23) 328
.5
0 Li
4
0
20
40 time Ins
Fig. 2. Fluorescencedecay of Mg(‘P, ),
60
I
Volume 178,number 2,3
CHEMICALPHYSICSLETTERS
22 March 1991
I
0
40
20
time
Ins
60
0
Fig. 3. @orescence decay of MgO(d ‘A a= 0). Solid line: experimental; dashed line: convoluted for rc=2.0 ns and r=7.8 ns; deviation of experimental values from the tit is displayed on a magnified scale.
20
40
time
I ns
60
Fig. 4. Fluorescencedecay of MgO(D ‘Av=O). Solidline: experimental; dashed line: convoluted for +2.0 ns and 7=4.3 ns; deviation of experimental values from the fit is displayed on a magnified scale.
Table 2 d 3Au=O and D ‘Av=O radiative lifetimes State
7 (ns)
,a’
Sb) (%)
Ar4iso‘) (ns)
Ahs d’ (ns)
A7 (ns)
d3A D ‘A
7.8 4.3
109 74
0.67 1.7s
0.05 0.07
0.80 0.46
1.8 1.0
‘) Number of points in the (90-10%) range of signalmaximum intensity. b, Standard deviation minimised in the fitting procedure (see text). ‘) For A&,=0. d, For Are=O.2ns.
tained ( < 0.1 ns) emphasise the good quality of the tit. The main uncertainty source thus lies in the reference time-constant determination uncertainty, ArO. Our value for the ‘A v=O state, 7.8f 1.8 ns, is shorter than the value obtained by Diffenderfer et al. [2], 11.8-t 0.5 ns, but the latter experiment was performed with a boxcar integrator set with a 10ns gate, which is considerably higher than the 0.4 ns interval increment of the transient digitiser we have used.
References [ 1 ] R.N. Diffenderfer, D.R. Yarkony and P.J. Dagdigian, J. Quant. Spectry. Radiative Transfer 29 (1983) 329. [2] R.N. Diffenderfer and D.R. Yarkony, J. Phys. Chem. 86 (1982) 5098. [3] H. Biisener, F. Heinrich and A. Hese, Chem. Phys. 112 (1987) 139. [4] M. Costes, C. Naulin, Z. Moudden and G. Dorthe, Laser Chem. 10 (1990) 367. [5] M. Costes, C. Naulin, G. Dorthe, G. Daleau, J. JoussotDubien, C. Lalaude, M. Vinckert, A. Destor, C. Vaucamps andG. Nouchi, J. Phys. E. 22 (1989) 1017. [6] C. Naulin, M. Costes and G. Dorthe, Chem. Phys. Letters 143 ( 1988) 496. [ 7 ] R. Mocciaand P. Spizzo, J. Phys. 6.2 I i,1988) 1133.
329
CHEMICAL PHYSICS LETTERS
22 March 199 1
Measurements of the radiative lifetimes of MgO(B ‘C+, d 3A,D ‘A) states Christian
Naulin,
Michel Costes, Zakkaria Moudden
VRA 348 CNRS: Photophysique et Photochimie MoltWake
and GCrard Dorthe
‘, Universik! Bordeaux I, 33405 Take
Cedex. France
Received 16 November 1990; in tinal form 8 January 1991
MgO molecules have been produced in X ‘Z+, a ‘II, and A ‘II dark states by the Mg( ‘SD)+NZO( X ‘Z+ ) reaction in an experiment performed with pulsed, crossed, supersonic molecular beams. Analysis of fluorescence decays followingpulsed dye-laser excitation ofthe MgO reaction product has resulted in radiative lifetime determinations of the B ‘Z+ v=O-3, d ‘A and D ‘A states. The following values have been found at 95% confidence level: B ‘Z+ state, r,,0=21.5 i 1.8 ns, r,,,=,=21.9? 2. I m, 7,,~=21.7? 2.0 ns,r,~,=2l.5~3.2ns;d3Astate,7,,,=7.8~1.8ns;D’Astate,7,,,=4.3+l.0ns.
1. Introduction Mg(‘&) tN,O(X There have been few published determinations of radiative lifetimes of MgO excited states. The experimental values, 32.7 & 1.7 ns for the B ‘Cf v=O level and 11.8 f 0.5 ns for the d ‘A state, were reported by Diffenderfer et al. in 1983 [ 11. In their experiments, MgO molecules in the X ‘C- and a 311 states were both generated by the reaction of excited metastable Mg(3PJ) atoms with O2 or N20 scattering gas at pressures of 1 Pa or less. The radiative lifetimes were deduced from the temporal decay of MgO fluorescence following pulsed dye-laser excitation of MgO dark states. Values calculated ab initio for B ‘E+ v=O, 21 ns [ 21 and 24 ns [ 11, were also obtained by the same group. More recently, in 1987, another experimental value for the B ‘Cf u=O state, 22.5 + 1.5 ns, was reported by Biisener et al. [ 31. Their experimental arrangement also used pulsed dye laser excitation of MgO (X ‘C+ ) molecules, which were produced by the Mg( ‘So) +N,O reaction at 1000 K in a beam-gas experiment with typical scattering N20 gas pressures of 0.1 Pa. In this work, MgO molecules were produced in a single-collision regime by the following exoergic gasphase reaction:
’ And GDR 87 CNRS: Dynamique des Riactions Mol&ulaires. 0009-2614/91/$
‘C+)
~MgO(X’C+,a3H,A’H)+N,(X1c~).
(1)
Such reactive collisions were achieved when crossing pulsed, supersonic beams of ground-state Mg and N,O species at a relative translational energy of reactants of 0.9 eV [4]. Pulsed dye-laser excitation of the MgO products in the collision zone resulted in the determination of radiative lifetimes of the B ‘C+ ~0-3, d 3A v=O and D ‘A v=O states.
2. Experimental A complete description of the apparatus has been previously given [ 51. The Mg( ‘So) beam, generated by entraining atoms, laser ablated from a Mg rod, with H2 carrier gas emitted by a first pulsed nozzle, had the following characteristics: 3300 m s-’ velocity at the peak of the velocity distribution, 10% velocity spread full-width at half-maximum (fwhm), 3.5 ps pulse duration fwhm at the crossing point and 6.5” divergence fwhm. The pulsed N20 beam, obtained by expansion of a 22% N,O-He mixture from a second pulsed nozzle, had 1140 m s-’ velocity, 22% velocity spread fwhm, 30 ps pulse duration fwhm at the crossing point and 4” divergence fwhm. These
03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
325
Volume 178, number 2,3
beams crossed at right angles with an attenuation factor not exceeding 10%. The MgO molecules produced in the dark states were excited by pulsed laser light at the beam crossing point using a dye laser operated with 532 nm pumping of a Rh640-DCM dye mixture. Radiations at shorter wavelengths were generated by frequency doubling and stimulated Raman scattering in Hz, used separately or in conjunction: (i) MgO(X ‘Cc ~0-3) molecules were excited to the B ‘Z+ state using the B-X transitions of the Azl=O sequence between 501 and 499 nm (first antiStokes line). The fluorescence of the B-X or B-A transitions was filtered through 3 mm of Schott BG 18 or Schott OG 550 coloured glass. (ii) MgO(a’rI u=O) and MgO(A’II Y=O) molecules were excited to the d 3A v=O state using the d-a transitions at 372 nm and to the D ‘A v=O state using the D-A transitions at 380 nm (first Stokes line of the doubled dye). The fluorescence was filtered through 1 mm Schott BG3 coloured glass. The fluorescence imaged onto the photocathode of a Hamamatsu R955 photomultiplier tube (2.5 ns rise time, 50 Q loaded) was fed directly into a transient digitiser (Tektronix 7912, 7B80 time-base. and 7A 19 50 Q vertical amplifier, 0.3 ns rise time) to measure radiative lifetimes. In a typical run, 64 individual fluorescence decay traces were accumulated and then transferred to a computer for data treatment.
1
1
4
3. Results and discussion 3.1. The B’C+ state An excitation spectrum can be found in ref. [ 41. Decay measurements were performed at the wavelengths of the prominent bandheads for vibrational levels u=O-3 (la 70). The dense rotational structure near the centre of the bands precluded individual measurements for low J levels, The analysis procedure is similar to that described in a previous paper on C2 (d 3II,) radiative lifetime [6]. The observed decays have been fitted using a least-square method to an exponential function (fig. 11,
326
tA,exp(-t/z),
(2)
,
I
0
I
I
time
I,
100
1
Ins
Fig. 1. Fluorescence decay of MgO(B ‘C+ u=O). Solid line: experimental; dashed line: calculated for r= 21.5 ns; deviation of experimental values from the exponential fit is displayed on a magnified scale.
with an upper limit of dispersion error, (Azdi,/7)‘x3s2(7/nst)*,
(3)
where s2 is the variance estimate for the considered set of n data points of increment 6t. Systematic errors are mainly due to the transient digitiser time-base sweep error (2% specified). Indeed, overestimation of the time-constant due to the convolution of PMT and electronics rise times, 7’ z
y(t)=A,
22 March 1991
CHEMICAL PHYSICS LETTERS
[7* t 1
(rise times)*] “* ,
(4)
is an almost negligible factor as (7'-7)/7zO.6%. Underestimation of the time-constant resulting from the loss of laser-excited radicals moving out of the detector field of view during the acquisition of the decay curves has been shown to be negligible for laboratory velocities in the l-2 km s -’ range and timeconstants in the 100 ns range [ 61. In the present case, the higher MgO laboratory velocities (up to 4 km s-’ ) are compensated by the much lower time-constants ( ~22 ns). Loss of laser-excited radicals by collisional quenching is also insignificant in these experiments performed with a total attenuation of the Mg beam < lo%, including all elastic, inelastic and reactive events which take place during the time-offlight of the Mg atoms through the N20 beam, z 23 ps. The uncertainty due to experimental systematic errors, Az,,,/r, can therefore be set to an over-
Volume 178,number 2,3
CHEMICALPHYSICSLETTERS
Table 1 B ‘Z v=O-3 radiative lifetimes v
7 (ns)
n =’
s b’ (%)
AhI, (ns)
AT (ns)
0
21.5 21.9 21.7 21.5
185 193 173 195
5.2 6.4 5.7 12
0.58 0.71 0.69 1.26
1.8 2.1 2.0 3.2
1 2 3
a) Number of points in the (70-2%) range of signal maximum intensity; after complete decay of the excitation light. b, Standard deviation of the sample.
estimated value of 3%. The total uncertainty, at 95% confidence interval, is thus defined as A7= 2A7,, + 7(A7~,,p/~)
(5)
Our results, summarisccl in table 1, are in good agreement with the experimental determination of Biisener et al. [3], 22.5f 1.5 ns for v=O, J= 1, and the theoretical determinations of Diffenderfer and Yarkony [ 21, 2 1 ns, and Diffendetfer et al. [ 11, 24 ns. As pointed out by Biisener et al., the striking difference when compared to the experimental value of Diffenderfer et al. [I], 32.7 & 1.7 ns, remains unexplained. 3.2. The d A ‘ and D A ‘ states
.
in the same spectral region to avoid significant differences in laser-beam spatial profile and pulse duration, and under comparable saturation conditions. Let yO(t) and y( 1) be the fluorescence signals observed for the reference and the species studied, respectively. They can be expressed, using eq. (6)) as the convolution of the unknown apparatus function, and the exponential decay function, either known as reference, go(t), or to be determined, g(t). The latter functions read ga(t)=exp(-a,t)
and
-at),
g(t)=exp(
(7)
where l/aO=r,,, l/a=7, 7. and zrefer to the relevant radiative lifetimes. Let z(t) be a function relating both measurements, y,(t) and y(t), such as (8)
JJ(t)=yo(t)@Jz(t). The Laplace transform of this equation reads W)=Wo)
2
x9(z)
(9)
which can be rearranged, using eq. (6) as Y(z) =~(g)/~(go)
.
(10)
Introducing the Laplace transform expression of an exponential function yields Y(z)=(s+ao)/(s+a)=l+(ao-a)/(s+a),
When the fluorescence decay time-constant approaches the temporal width of the laser pulse, the number of data points left after complete decay of the excitation light with sufftcient signal-to-noise ratio becomes too restricted. It is then impossible to derive radiative lifetimes from a simple exponential lit. The actual apparatus function, f(t), has to be taken into account, since the observed signal, y(t), now appears as the convolution of an exponential decay function, g(t), with f(t), y(t) =f(WWt)
22 March 1991
(11) where s is the operational variable. Taking the inverse Laplace transform, z(t)=P’(l)+Y-‘[(a,-a)/(s+a)], leads to z(f)=8+(t)t(ao-a)exp(-at),
(13)
where S+(t) stands for the Dirac function, 6+(t)=lim,,,
s(t-u) 00
(6)
The apparatus function cannot be simply estimated by some scattered light signal: it only yields poor fits to experimental data, probably because such a procedure does not allow for saturation effects generally occurring in the absorption process. An alternative procedure, which takes account of absorption effects, is to record as a reference the fluorescence signal due to a species of known lifetime, preferably
(12)
with u > 0 (
and
I
0
.
S+(t)dt=l >
(14)
If 7. is known, measuring the yo( t) and v(t) functions allows one to derive 7 value with a simple fitting procedure of v( t) to the convolution product of ye(t) and z(t). The best-fit value is obtained by minimising the variance estimate, s2, of the sample, 321
Volume 178,number 2,3
Au< [ ((dy)‘>“‘+Aa,(
( >
s2= CRf
22 March 1991
CHEMICALPHYSICSLETTERS
/(n-l),
($Y/a~)2>“21
(15) (24)
where the weighted residuals, R;, are defined as Ri=[~;-_~oi@Z(t,)l~~i.
(16)
Here, yi is the fluorescence intensity measured at time t,, with standard deviation Ui taken as weight. Since the photoelectron emission of a PMT photocathode follows Poisson statistics, fl, can be approximated by (y,)“*. Dispersion errors can be evaluated by differentiating eq. ( 8),
(wwdh
dy= (wwd~f
,
(17)
where the reference parameter, a,, is also considered to be subject to error, duo. The uncertainty on a, Aa, which is simply derived by overestimating da in eq. ( 17), A~G LAY+(+iaa,)Aa,ii(ayiaa)
,
(18)
can be estimated provided the partial differential terms in eq. ( 18) be evaluated. The partial differentials can be expressed as (a~/a+~,(tp3az/aa =-~o(t)~[ltt(ao-a)]exp(-at),
(19)
and ayiaa,=y,(t)~aziaa,=Y,(t)~exp(-at).
(20)
yielding (25)
A7~is= FRAU_
Again, experimental errors are mainly due to the transient digitiser (Ar,,,/7<2%) and the total uncertainty at 95W confidence level is defined by eq. (5). The fluorescence signal of Mg( 3p ‘PI ) (r=2 &0.2 ns [ 7 ] ) obtained by laser excitation of ground-state Mg( 3s ‘So) atoms at 285.21 nm (first anti-Stokes line of the doubled dye) was recorded as the reference function, y. (t ) (fig. 2). Special attention was taken to avoid the apparent lifetime increase due to radiation trapping. This problem was solved by smoothly varying the delay between the triggering of the pulsed valve and the ablation laser, which yielded decreasing Mg( IS,,) concentrations, and consequently decreasing the apparent Mg (3p ‘PI ) lifetime. Eventually, a point was reached where the fwhm pulse duration of Mg( 3p ‘P, ) fluorescence no longer exhibited a dependence upon the Mg( ‘So) concentration, thus allowing the reference function to be recorded. The observed decays are displayed in figs. 3 (d 3A) and 4 (D ‘A). The results from the tit are listed in table 2. The dispersion error computed with AT,= 0 are also included: the very small A7odisvalues ob-
Quadratic average values of these expressions are computed from experimental data:
1 1
A~2=((dv)2>=
((WWV
(
C [Y~-YO+WC)I~
=( C (k/W2)/r:,
c Waa,)‘~=(~
Wi!ad’)h,
>
/n,
(21)
(22)
(23)
where the sums extend over the IZexperimental data points. The uncertainty of the parameter a due to dispersion errors can thus be estimated using eqs. ( 18) and (21)-(23) 328
.5
0 Li
4
0
20
40 time Ins
Fig. 2. Fluorescencedecay of Mg(‘P, ),
60
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Volume 178,number 2,3
CHEMICALPHYSICSLETTERS
22 March 1991
I
0
40
20
time
Ins
60
0
Fig. 3. @orescence decay of MgO(d ‘A a= 0). Solid line: experimental; dashed line: convoluted for rc=2.0 ns and r=7.8 ns; deviation of experimental values from the tit is displayed on a magnified scale.
20
40
time
I ns
60
Fig. 4. Fluorescencedecay of MgO(D ‘Av=O). Solidline: experimental; dashed line: convoluted for +2.0 ns and 7=4.3 ns; deviation of experimental values from the fit is displayed on a magnified scale.
Table 2 d 3Au=O and D ‘Av=O radiative lifetimes State
7 (ns)
,a’
Sb) (%)
Ar4iso‘) (ns)
Ahs d’ (ns)
A7 (ns)
d3A D ‘A
7.8 4.3
109 74
0.67 1.7s
0.05 0.07
0.80 0.46
1.8 1.0
‘) Number of points in the (90-10%) range of signalmaximum intensity. b, Standard deviation minimised in the fitting procedure (see text). ‘) For A&,=0. d, For Are=O.2ns.
tained ( < 0.1 ns) emphasise the good quality of the tit. The main uncertainty source thus lies in the reference time-constant determination uncertainty, ArO. Our value for the ‘A v=O state, 7.8f 1.8 ns, is shorter than the value obtained by Diffenderfer et al. [2], 11.8-t 0.5 ns, but the latter experiment was performed with a boxcar integrator set with a 10ns gate, which is considerably higher than the 0.4 ns interval increment of the transient digitiser we have used.
References [ 1 ] R.N. Diffenderfer, D.R. Yarkony and P.J. Dagdigian, J. Quant. Spectry. Radiative Transfer 29 (1983) 329. [2] R.N. Diffenderfer and D.R. Yarkony, J. Phys. Chem. 86 (1982) 5098. [3] H. Biisener, F. Heinrich and A. Hese, Chem. Phys. 112 (1987) 139. [4] M. Costes, C. Naulin, Z. Moudden and G. Dorthe, Laser Chem. 10 (1990) 367. [5] M. Costes, C. Naulin, G. Dorthe, G. Daleau, J. JoussotDubien, C. Lalaude, M. Vinckert, A. Destor, C. Vaucamps andG. Nouchi, J. Phys. E. 22 (1989) 1017. [6] C. Naulin, M. Costes and G. Dorthe, Chem. Phys. Letters 143 ( 1988) 496. [ 7 ] R. Mocciaand P. Spizzo, J. Phys. 6.2 I i,1988) 1133.
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