If two identical electrodes are used in the two half-cells and if the electrolyte is uni-univalent, Equation 3 reduces to EF
=
RTln
a”z ail
+ RT
J2t,d
In ag
+ RT
si2
t,d In a,
(4)
where subscript i refers to cation, q to co-ion (Le,, anion), and w to water. Integrals are extended to the membrane system on the whole. In such an Equation 4, the first term corresponds to the theoretical emf for an ideally permselective membrane, the second one is a correcting factor due to the transport of co-ion, and the third one takes into account the solvent transport. It is difficult to calculate all these terms and it is necessary to point out that the error due to water transport could increase with increasing concentrations according to the sequence Ca2+ N Mg*+ > NaS E K+. However a careful inspection of Figures 2 , 3, and 4 suggests that these contributions are negligible for the potassium stearate gel, but not for the sodium stearate coagel. There are also some effects which are common to liquid ion exchangers and must be taken into account in a more complete evaluation of the phenomenon. First of all, there is the solubility of the ion exchanger in aqueous medium and its influence on the ionic activity. This effect could be responsible for a tendency of the plot E cs. log a to acquire a less steep shape in diluted solution. The effect, due to the solubilization of the stearate inside the “membrane,” will have different influences according to the nature and concentration
of the saline solutions bathing the two sides of the membrane (negligible for the slightly soluble stearate salts, i.e., Ca*+ and Mgzi, but not for the more soluble salts, i.e., N a + and Ki). The experimental data reported in Figure 5 show how the presence of several non-ionic compounds affects the electrochemical behavior of the potassium stearate gel membrane. The recorded emf value is due to two different phenomena: the different ionic activity of KS ion in the two media, Le., aqueous and hydro alcoholic; and the asymmetry potential effect due to a different distribution of stearate molecules in the two layers in contact with the two solutions. To prove the existence of an asymmetry potential ( 6 ) , the plots obtained are compared with the ones obtained with a synthetic ion exchange membrane (Asahi CKI), dotted lines of Figure 5, in which case any asymmetry potential is excluded. In conclusion such an asymmetry effect is a function of the composition of the solution, of the nature, and of the concentration of the non-ionic molecule in solution, and of the concentration of the ionic strength in solution (7). The future development of such a research program is to check other molecules suitable to influence the structure of such membranes. In this way, it could be possible to determine by means of electrochemical measurements, the concentration of uncharged compounds. RECEIVED for review November 2, 1971. Accepted January 31, 1972. (6) A. M. Liquori and C. Botrk, J . Phys. Clzern., 71, 3765 (1967). (7) C. Botrk, M . Mascini, A. Memoli, and M. Marchetti, Farrnaco, Ed. Sci., 24, 873 (1969).
Evaluation of Precision of Quantitative Molecular Absorption Spectrometric Measurements J. D. Ingle, Jr.,’ and S . R. Crouch Department of Chemistry, Michigan State Uniuersity, East Lansing, Mich. 48823
A unique signal-to-noise ratio expression is derived for quantitative molecular absorption measurements. Under certain experimental conditions this expression can be considerably simplified, and three limiting cases are discussed in which measurements are either readout limited,. photocurrent shof noise limited, or source flicker limited. This treatment reveals that the usual assumption that optimum measurement precision occurs near 37% T can be grossly in error under certain conditions. Use of the signal-to-noise ratio expressions for evaluation of measurement precision and optimization of ex per imental conditions i s presented. Highest measurement precision is obtained when readout error is negligible and measurements are source flicker limited. Direct current and photon counting measurement systems are critically compared in terms of signal-to-noise ratio and other criteria for application to molecular absorption measurements.
NUMEROUS TREATMENTS have been presented of the precision expected from normal and expanded-scale molecular absorption spectrophotometric measurements [see for example (1-4) Present address, Department of Chemistry, Oregon State University, Corvallis, Ore. 97331, (1) N. T. Gridgeman, ANAL.CHEhi., 24,445 (1952).
and the references cited in these papers]. Many of the previous treatments assume that the limiting factor in determining spectrophotometric precision is the uncertainty in reading a linear scale. The treatment by Hughes (3) is one of the most comprehensive and considers how the source, the background, the noise, and the measurement time influence the optimum transmittance. With modern spectrophotometric instruments, capable of scale expansion or containing digital readout devices, the influence of reading error on the precision can be considerably lessened or made completely negligible with respect to the influence of noise on the readout signal. Hence, with modern instrumentation it is most important to have a complete description of all the parameters which influence the magnitude of the readout signal, the system noise, and the signal-tonoise ratio (SjN). Such a signal-to-noise ratio description of a spectrometric system can be used to optimize parameters
(2) C. M. Crawford, ANAL.CHEM., 31, 343 (1959). (3) H. K. Hughes, Appl. Opt., 2, 937 (1963). (4) “Optimum Spectrophotometer Parameters.” Cary Instruments Application Report AR 14-2, Cary Instruments, Monrovia, Calif.. 1964. ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
1375
Nx
Table I. Definition of Symbols in Equations 1 and 2 source spectral radiance, W sr-l nm-1 c1XF5 for tungsten lamp, Nx = - l~j(X,T)Tw(X)
=
c1 = 1.190 X 1016 W nm4 cm-* sr-1 cz = 1.4388 X 107nm "K q(X,T) = emissivity of tungsten filament, dimensionless, 0
5 t,(X,T) 5 1 Tu@) = transmission factor of lamp window, dimensionless, 0 5 T&) 5 1 T = color temperature of the tungsten filament, "K nx = source spectral radiance, sec-1 sr-1 cm-2 nm-1 c3x-4 for tungsten lamp, Nx = ec2,XT - 4hlT)TW(N c3 = 5.9958 X 1031nm3 cm-2 sec-1 sr-1 T&) = monochromator transmission factor, sr cm2
,
=
To,(X)DHWt(X,XrJ) T&) = transmission factor of monochromator optics, dimensionless D = solid acceptance angle of monochromator, sr H = slit height (assuming equal entrance and exit slits), cm W = slit width, cm t(A,Xo) =
slit function
=
{1 - " '0'1 for
XO
- s I x I ho + s
= 0 elsewhere
XO = monochromator wavelength setting, nm
spectral bandpass, nm reciprocal linear dispersion of monochromator, nm cm-1 (Pet)x = stray light spectral radiant power, W nm-1 ( p & = stray light spectral radiant power, sec-1 nm-1 T,(X) = sample compartment transmission factor, dimensionless s = RdW = R d =
k =
(1 - f)(1 - cuc)exp(-2.36
ct(A)C,) 1=1
f
=
fraction of radiant power lost due to reflection at sample cell surfaces, dimensionless
ac = fraction of radiant power absorbed by sample cell, dimensionless
S(X)
=
b = cell path length, cm €,(A) = molar absorptivity of ith absorbing species, 1. mole-' cm-l C, = concentration of ith absorbing species, moles 1.-l k = number of absorbing species where solvent is k = 1 photomultiplier sensitivity factor, A W-1
=
Q@)V
Q(A) = radiant cathodic sensitivity, A W-1
collection efficiency of first dynode, dimensionless average gain of tube, dimensionless photomultiplier sensitivity factor, anode pulses per incident photon 7 =
m
s(X) =
=
=
K(X)
=
quantum efficiency of photocathode, photoelectrons released per incident photon
(5) to increase the S/N. If measurement system variance is a major factor in determining the overall variance in a spectrophotometric method, rather than factors such as sampling imprecision, increasing the signal-to-noise ratio will in general increase the precision of the method. Winefordner and coworkers (6, 7)have used S/N theory to describe the precision of molecular absorption spectrometric measurements, although these treatments were only applied at the limit of detection and neglected the irduence of reading error. The treatment presented here is unique in that a S/N is defined for an absorbance measurement which incorporates both electrical noise and the influence of reading error. In addition, the signal-to-noise ratio theory described here can be applied over a very wide absorbance range, which encompasses those values obtained in routine spectrophotometric methods. The theory can then be used to calculate the optimum transmittance for a given system if the major contributors to system variance can be identified. Also the treatment is novel in that both dc measurements and (5) J. D. Winefordner, W. J. McCarthy, and P. A. St. John, J . Chem. Educ., 44,80 (1967). ( 6 ) J. J. Cetorelli, W. J. McCarthy, and J. D. Winefordner, ibid., 45, 98 (1968). (7) J. J. Cetorelli and J. D. Winefordner, Talanta, 14,705 (1967). 1376
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
photon counting are considered for detection systems, which reveals the differences and similarities of these two techniques when applied to molecular absorption spectrometry. The system to be described for dc measurements is a modern single beam spectrophotometer with a stabilized tungsten or deuterium source, a monochromator, a sample cell compartment, a photomultiplier transducer with stabilized power supply, a high quality operational amplifier current-tovoltage converter, and an expanded scale readout device such as a recorder or digital voltmeter. For photon counting, the output pulses from the photomultiplier flow through a load resistor which is the input to a fast pulse amplifier and discriminator. A high speed counter is the readout device. In the first two sections, expressions relating the readout and variance of dc and photon counting measurements to the system parameters which influence the measurement are presented. In the third section, overall signal-to-noise expressions for absorbance measurements and optimization of experimental conditions in spectrophotometry are developed. The fourth section considers simplification of S/N expressions and the effect of reading error in dc measurements. The fifth section compares the merits of dc measurement cs. photon counting as the detection system for molecular absorption spectrometry. The final section discusses the
optimization and calculation of the precision of molecular absorption measurements. RELATIONSHIP OF READOUT TO SYSTEM PARAMETERS Current and Pulse Output Expressions for Photomultiplier. For dc measurements, the photoanodic current from the photomultiplier is given by iap
=
+ lm [NJ,(V
(1)
For the remainder of this paper it will be assumed that conditions are arranged so that Equations 5 and 6 are valid (Le,, s is much smaller than the half-intensity width of the absorption band of the species being measured) and that the contribution of stray light to Equations 5 and 6 is negligible ( i a s f have been discussed in detail (10). Unless otherwise stated, it will be assumed that photon counting is used only under conditions of negligible pulse overlap so that Equations 10 and 1 1 are valid. Ro P Rod Rat,
(10) J. D. Ingle, Jr., and S. R. C r o u c h , A ~CHEW, ~ ~ . 44,777 (1972). ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
1377
A useful form for R a p taken from equations developed in the first section is Rap
=
and the readout device are negligible over the time needed to perform an ordinary analysis. The photomultiplier and current-to-voltage converter OA are assumed to be operated under conditions where errors due to nonlinearity are negligible. The errors considered are therefore random and indeuendent, and the total instrumental variance is thus the sum of the different individual variances in the system. A treatment of variance sources in a generalized spectrometric system utilizing dc measurement or photon counting has recently been presented (9). Expressions for the total variance in E,,, E r d , No,, and Nodfor a single-beam molecular absorption spectrophotometer are presented in Table 11.
n ~ , T o p ( X o ) Q H W 2 R-f) ~(l X k
( 1 - a , ) ~ ( ~ o ) ? [ e x(-2.3036 p
E i ( ~ O ) ~ i (12) )l
i=l
RELATIONSHIP OF VARIANCE TO SYSTEM PARAMETERS
If the complete spectrophotometric system is considered, many sources of error can be identified. Excellent discussions of the chemical and instrumental deviations that affect the accuracy have been presented (8, 11) and will not be
(TEId2
=
Table 11. Expressions for Variance in E,,, Eld,No,, and Nod total variance in E,,, V 2 (UEop2)p
+
(UEop2)asc
f
(bEop2)pm ~
~
~
=~ variance ~ 2 )in
+
(UEop2)f
+
6J2
f
64’
+
UR2
Eo, due to photon or quantum noise, V2 = 2mei,,R(2Af = ?.m2eqicp’R~2Af = 2m2ei,,R~2Af icp‘ = photocathodic current = iop/qm,A i,, = effective photocathodic current = icp’q,A Af = noise equivalent bandwidth of amplifier-readout system, Hz ( ~ ~ =~ variance ~ 2 in) E,,~ due ~ to ~ secondary emission noise, V2 = 2meiapaRj2Af (Y = relative variance in photomultiplier gain due to secondary emission, dimensionless ( ~ ~ =~ variance ~ 2 in ) E,, ~ due~ to photomultiplier flicker noise, V 2 ( ~ ~ =~ variance ~ 2 )in E,, ~ due to light source flicker = ( E i a S R j ) 2 , V 2 f = light source flicker factor or square root of relative variance of light source spectral radiance over measurement bandwidth, dimensionless uJ2 = variance in E,, due to Johnson noise in feedback resistor, V2 = 4kTRjAf k = Boltzmann’s constant, W sec OK-’ T = absolute temperature of Rf, “K U A = ~ variance in E,, due to amplifier noise, V 2 U R = ~ variance in E,, due to readout noise = U R ~ U~ R ~ V, 2 uxe2 = variance in E,, due to electrical noise generated in readout device, V 2 U R = ~ variance in E,, due to resolution or readability of readout device, V 2 total variance in &, V2 (
(13) (14)
+
u ~ , d 2=
= (uEod2)q
+
(UEod2)ssp (uEod2)p =
+
(qEod2)pm
+ + + uJ2
C.4’
(19)
UR2
variance in Eod due to quantum noise, Vz = 2eRf2Af(miate Tiad*) i,d* = iad - iot, iad = total dark current of pulsed nature = iad - i a l , A iol = leakage dark current at the anode, A ietc = anodic current due to thermal emission at the photocathode, A y = effective current gain received by iod*, dimensionless ( u ~ = ~variance ~ ~in Eod ) due ~ to ~ secondary ~ emission noise, v2 = 2eRj2aAf(rni,t, y i , d * ) ( u ~=~ variance ~ ~in Eod ) due ~ to ~ photomultiplier flicker noise, v2 total variance in No,
+
(20)
+
I J X ~ ,= ~
(UNop2)q
+
(USop2)f
+
UjVR2
+
(U,Vop’)ez
( L T . ~ ~=, variance ~)~ in No, due to photon or quantum noise = Nop = AIRapt
variance in No, due to light source flicker
( u . b ~ ~ ~= 2)~
=
f 2 N Z , ,= (A1R,,tf)2
u1vR2= variance in No, due to readout noise o,V,d2
= =
( ~ ~ =~ variance ~ ~ in2 No, ) due ~ to ~ excess noise ( 9 ) total variance in Nod (‘J.h”,d2)q $. udb”R2 (6A’od2)6z ( U . V , , ~ ~ )= ~ variance in N o d due to quantum noise = ( ~ ~ =~ variance ~ 2 in) Nod ~ due ~ to excess noise ( 9 )
+
considered here. Only random instrumental errors that limit the precision of making the absorbance measurement will be discussed so that imprecision due to such factors as sampling or cell positioning is not evaluated. A modern, stable, single-beam spectrophotometer is considered such that with suitable warmup time, the stability of the various components is excellent, and undirectional drifts in the light source, the photomultiplier, the amplifiers, (11) L. S.Goldring, R . C . Hawes, G . H. Hare, A. 0. Beckman, and M. E. Stickney, ANAL.CHEM., 25, 869 (1953). 1378
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
Nod
=
(AIRatc
+ AdR,d*)t
(25)
(26)
A more detailed discussion and explanation of the variance terms and parameters utilized in Table I1 is found in Reference (9). If a photodiode is used instead of a photomultiplier, a, and thus (olop2)secr equals zero. For Table I1 and for the remainder of the paper, unless otherwise stated, it is assumed that E,, = EO, and E,d = E o d . If E,, and E,, are modified before reaching the readout device, additional variance terms must be added to Equations 1 3 and 19 and the individual variance terms multiplied by suitable factors to make their contribution appropriate for the final readout voltage. For instance, if logarithmic amplification is used
after the current to voltage converter so that E,, then
=
log Eo,,
where u L 2is the variance in the readout voltage in volts squared due to noise in the log amplifier, and the log amplifier is assumed to have an output of 1 V per decade. The variance due to the readout deserves special mention. For photon counting, uNR2is unity because of =k1 count uncertainty inherent in a nonsynchronous digital frequency measurement. This will rarely be of importance in molecular absorption measurements unless extremely high absorbances are being measured. For dc measurement systems, uR2can often be a significant contributor to the total variance. Usually uRT2will determine the magnitude of u R 2 , In the case of an analog readout device (oscilloscope, meter, recorder), uR, is a fraction of the total readout scale, which represents the standard deviation in reading a value from the scale. The dead zone of a meter or recorder can also contribute to the magnitude of uR,2. For instance, a normal meter can be read to &0.5 % of full scale, which yields a readV2 if full scale is one volt. In a out variance of 2.5 X similar manner, for a digital readout device, error is caused by uncertainty in the least significant digit. Thus for a 3digit decimal readout reading 1-V full scale, the resolution is 1 mV which yields a readout variance of V2. Analog readout devices with multiple scales have a variable u R 2 . RELATIONSHIP OF SIGNAL-TO-NOISE RATIO TO SYSTEM PARAMETERS
rent readout voltage for the reference and sample solutions, the precision of making an absorbance measurement depends on the precision involved in making the sample and reference solution measurements. For this discussion, it will be assumed that the concentration of the jth species is being determined by measuring the solution absorbance. The reference photocurrent readout voltage, (E,,),, is equal to Rf(iap),,where (iap), is the photoanodic current with the reference solution in the sample cell, as given by Equation 9 with C, = 0. The sample photocurrent readout voltage, (E,&, is likewise equal to l?,(iu,)s, where (iap)s is the photoanodic current with the sample solution in the sample cell, as given by Equation 9 with C, some finite value. The refractive indices of the sample and reference solutions are assumed to be nearly equal so that the same value o f f i n Equation 9 can be used for both solutions. The only difference in calculating the group variance terms for each voltage measurement is that for the photocurrent dependent individual variance terms, iap is replaced by (iup),and (iap)sfor the reference and sample group variances, respectively. S/N FOR TRANSMITANCE MEASUREMENT. The transmittance Tis defined by Equation 32. T = (Erp)J(Erp)r = [(Ert)s - Erd]/[(Erdr - Erdl (32) where (Ert),= total readout voltage with the reference solution in sample cell, V = (E& Erd = total readout voltage with the sample solution in sample cell, V = (Erp)s -I- Era
+
The standard deviation of a transmittance measurement, uT, DC Measurement System. S/N FOR PHOTOCURRENT is found by applying propagation of error mathematics to MEASUREMENT. In a dc system, the S/N for measuring E,,, Equation 32 with the result (S/N)mdc,is given by ( 9 )
where Ert
= =
a2,
b 2 , c2
=
total readout voltage, V Erp Erd = i,,Rf group variance terms which arrange the total variance into three groups which are proportional to i c p , proportional to i e p 2 , and independent of i,, ( 9 ) , respectively
+
The group variance terms for molecular absorption measurements are determined in the following equations : a2 =
+
( r ~ ~ ~( u~ E o p)2 )q sec
+
(u~~~’)prn
(30)
b2 = (uEop2)f c’
=
2[(rJ2
+
uA2
+
UR2
+
( ~ ~ , d ’ > g
+
(uEod2)sec
(29)
+
where u(Ert)sZ and u ( ~are~ the~ variances ) ~ ~ for determining (E7OSand (E,,),, respectively. Here it is assumed that Erd in the numerator and denominator represent separate measurements and, hence, are independent measurements. The S/N for a transmittance measurement is given by Equation 34
+ bS2+ cS2 (6,)s ’
a32
(31)
The individual variance terms in the c 2 group are multiplied by 2 because two measurements ( E , , and E T d )are needed to obtain E,,. The precision with which a particular photocurrent readout voltage can be measured is given by (S/N)rndC. For absorbance measurements, the information desired is encoded in the measured value of transmittance ( T ) , reciprocal transmittance (l/T), or absorbance (A). Therefore it is useful to define S/N’s in which T , 1/T, or A become the “signal.” Since T , l/T, and A are functions of the ratio of the photocur-
+ br2+ c,’ (Erp)r2
+
-1 (34)
where
(2)
=
UT
( ~ ~ ~ d ’ ) p r n l
ur2
S/N or the reciprocal of the relative standard deviation of a transmittance measurement, dimensionless
=
T
+ bs2 + cS2
+ bT2+ c,2]1/2
as2
aT2
(Erp)r2 values of group variance terms with sample solution in sample cell, V 2 ar2,bT2,c r 2 = values of group variance terms with reference solution in sample cell, V2 +
a s 2 bs2, , cs2 =
Note that measurement of either ( E , P ) Sor requires two measurements because of the presence of dark current. The transmittance may be found by making separate measure, Erd or by electronically setting ments of (E,,),, ( E T f ) ,and Erd to 0 % T and (E,,),, to 100% T on the readout device and measuring ( E r t ) ,as the transmittance. In either case, A N A L Y T I C A L CHEMISTRY, VOL. 44, NO. 8, J U L Y 1972
1379
~~
~~
Table 111. Equations and Optimum Conditions When One Group Variance Term Predominates Case I Case I1 Case I11 Dominant group variance cr2 ar bPZ Dependence of variance on (icp)? independent a(iep)r diCp Sources of variance Shot noise in dark current, Shot noise in photocurrent Source flicker noise Johnson and amplifier noise, readout variance dc -((Erp)rlnT -(E,,)Jn T -(E&ln T (limiting form of Equation 36) (47) (44) (41) &(I +T-l)1/2 e,( 1 +T-2)1/2 1.4146, Limiting form of Equation 40 In T = -(T* 1) (42) In T = -2(T 1) (45) None" Optimum transmittance, Top 0.331 0.109 0" 0.480 0.962 ma Optimum absorbance, A,,
(k)
+
+
do
(k)d.ap a
None@
(43)
Until limited by stray light and/or dark current noise.
Equation 34 is valid since the operation of setting 0 % or 100% T is a measurement. The S/N for a 1/T measurement, (S/N)i/TdC,equals (S/N)Tdcsince the standard deviation of a 1/T measurement, ul/T, equals CTT/T'. In some instances, only one measurement of dark current is made, and Erd is assumed to be the same for the sample and reference measurements. This case is treated in the Appendix. S/N FOR ABSORBANCE MEASUREMENT. The S/N for an absorbance measurement, (S/N)Adc,is given by
);(
= A
A
-
=* -
-In T ar2
+ bs2 + cs2
us2
[
(E,,),'
+
+ br2+cT2 (Erp)r2
entering the log circuitry to yield an accurate value of A . If it is assumed that the dark current is nulled by using a very sensitive meter to make the readout variance negligible, Equation 36 can still be used with the group variance terms for the reference solution being redefined as a r t 2 ,b r z 2and , crlZas follows: u r 1 2= aT2 b,l' = br2
CTZ2 =
1
(35)
dimensionless 0.4343 T U I / T= 0.4343U~/T absorbance = -log T
The relationship uA = (0.4343)uT/Tis only an approximation since differentials are replaced by finite errors (variances). However, this relationship is a good approximation (within 1%) if uT/T < 0.02. Therefore, for reasonably small uT, this approximation is valid except at extremely small transmittances. Equation 35 is the most useful S/N form, because the S/N is directly proportional to the absorbance and hence the concentration of the sought-for species. Thus (S/N)Adc gives a direct estimate of the relative precision of a concentration determination. Because of the different photocurrent dependencies of the group variance terms, it is easily shown that us2= Tar2,bX2 = bT2T2, and c S 2= cr2. If these relationships are substituted in Equation 3 5 , Equation 36 results.
The value of (S/N)Adcis thus dependent on the magnitude of the group variance terms for the reference solution and on the value of transmittance. If the log is taken electronically with a log amplifier, and log (Erp)ris set equal to zero, the readout variance term enters differently into the total variance as was indicated in Equation 27. Usually for direct absorbance readout, the dark current is considered negligible compared to the photocurrent. If the dark current is significant, it must be suppressed before 1380
+
(38)
uL']
- 2URZ
(39)
To find the optimum value of (S/N)AdCwith respect to transmittance, Equation 36 is differentiated with respect to T and set equal to zero, which results in Equation 40.
+ + 2b,'T2 + cT2(Tz+ 1)
ur2(T2 T )
ud = standard deviation of an absorbance measurement,
A
c,'
[uR'
lI2
where
= =
+ (2,303)' (E,,'),
(37)
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, J U L Y 1972
In T
=
(40)
/aT2T -(T +Cry
This is a transcendental equation which can be solved numerically for the optimum value of transmittance, Top,if values of a r 2 ,b r z ,and cr2can be calculated or measured. LIMITINGCASES. Under certain conditions one of the three group variance terms may predominate, which allows considerable simplification of Equations 36 and 40. Table I11 presents these simplified expressions, values for the optimum transmittance (Top),the optimum absorbance ( A o p ) ,and the S/N at T o p ,(S/N)A,opdc, for the three limiting cases. Note that these limiting cases hold for a specific absorbance range within a given accuracy which can be calculated from Equation 36. As the transmittance approaches zero, the cr2(1 T2) term in Equation 36 becomes dominant, and in the limit, (S/N)Adc always approaches zero. Case I occurs when cr2 >> a,2 b r 2 ,so that (S/N)Adcis limited by factors independent of the photoelectron current such as amplifier-readout noise, dark current shot noise, or readout variance. From Equation 41, (S/N)Adcis directly proportional to ( i c p ) , , where ( i C p )is , the effective photocathode current with reference solution in the sample cell [(i,,), = (iap),/m].If T2 in Equation 42 is considered negligible compared to unity, which would result if there were no variance in determining ( E r P ) ,in Equation 33, the optimum transmittance is 0.368, which is the value usually quoted (8). Case I is the least favorable of the three limiting cases because the S/N can always be improved by varying parameters until one of the other limiting cases holds. The most common reason that a spectrophotometer might conform to Case I
+
+
is the lack of readout resolution. If readability is limiting, the precision can always be increased by scale expansion or by using a higher resolution readout device until noise can actually be observed on the readout device. Once the readout variance is negligible compared to other noise sources, increasing the readability will not result in further improvement in (S/N)Adc. Case Ia occurs if only one dark current measurement is made and is treated in the Appendix. Case I1 occurs when ur2 >> cr2 br2,so that (S/N)AdC is limited by variances proportional to ( i c p ) r such as photocurrent shot noise or photomultiplier flicker noise, and (S/N)AdCincreases with the square root of (&Jr. Although operation under conditions where Case I1 holds provides higher S/N than operation under Case I, the S/N can still be improved by increasing ( i e J r until the br2 variance terms 1) predominate (Case 111). At high absorbances, cr2(T+ will no longer be negligible in Equation 3 6 , and a combination of Cases I and I1 results. If b r 2>> a r 2 c T 2 ,Case I11 occurs, and (S/N)Adcis limited by variances proportional to ( i c J r 2 . Hence Equation 47 indicates that (S/N)AdCbecomes independent of ( i e p ) rwhen Case I11 holds. Equation 40 cannot be solved for the optimum transmittance since T o p approaches zero. Thus (S/N).4dCincreases linearly with absorbance until stray light, T I ) which has been neglected, becomes limiting or a r 2 ( 1 T-') are no longer negligible in Equation 3 6 . or crz(l Operation under conditions where Case I11 applies represents the highest achievable S/N for a given spectrophotometer. The above discussion is equally valid if the log is taken electronically rather than manually, except that the readout variance is a b,* rather than a c r 2type variance. Thus under Case I11 a direct absorbance measurement can be limited by either readout variance or source flicker. As for Case I, the readout resolution should be increased until u R 2 is negligible compared to ( g E , , * ) , . SIGNAL-TO-NOISE RATIOPLOTS. For a particular instrument and analysis, a plot of (S/N)AdCus. transmittance can be made to determine the absorbance range giving optimum S," and thus the best precision. From such a plot the expected measurement precision for any concentration analyzed can be evaluated. Figure 1 shows plots of the normalized S/N for an absorbance measurement, (S/N)Adc/ (S/N),,!Tdc, us. transmittance, where (S/N)mrdC is the S/N for measuring the reference intensity. Curves 1-3 are for limiting Cases 1-111, respectively. Curves 4 and 5 are for two intermediate cases in which two of the group variance terms must be considered. The normalized (S/N)adCfor any instrument will lie in the area between curves 1 and 3 . The absolute magnitude of (S;").4dC at a particular transmittance is found by multiplying the normalized (S/N).4dCat the transmittance value of interest by the (S/N)mrdC. For instance, if Case I1 is valid, the normalized S/N at Top = 0.109 is 0.695. If the SIN),,^^ is 1000 (the rms shot noise is l / l O O O of the reference signal), (S/N)AdC= 695, and the relative precision of an absorbance measurement at 10.9% T is 0.149x. Photon Counting System. S/N EXPRESSIONS. Analogous S/N expressions can be developed for photon counting which has recently been used for molecular absorbance measure-
+
1.5
-
421.0
-
F
%
0 2 ? A 0,5
-
+
+
+
ments (12, 13). The measured S/N, , ' :):(
a9
1.0
I
I
I
I
I
I
I
0.8
0.7
0.6
0.5 T
0.4
0.3
0.2
0.0
0.1
Figure 1. Normalized signal-to-noise ratio us. transmittance X Case1 0 Case I1 0 Case I11 c2 = (0.1)u2 >> l? 0 u2 = (0.1) l? >> c*
A
+
Equation 48,
where total observed pulse count = Nop -/- Nod = [Al(Rap Rare) AdRad*lt x z , y z , z 2 = group variance terms for photon counting which are proportional to R,,, proportional to R o p 2and independent of R,, (9), respectively. No t
=
+
+
For molecular absorption spectrometry, they are defined as : x2 = Y' = 2'
(49)
(UNO,')*
+
(wop2)1. (~NopZ)ex
= 2[(uNod2)ex
+
(~N,d2)9
+
(50)
(51)
UNE2I
The SIN expressions for transmittance and absorbance measurements,
(i)Tp (2)
P
and
, are given by Equations
52 and 53,
T
(i)Tp [ =
+ y S 2+ z s z
xrz
xS2
=
(Nop2)s
+
+ y r 2 + z r 2 -u2 (Nop2)r
1
(52)
is given by
(12) H. H. Ross. ANAL.CHEM., 38,414 (1966).
(13) M . L. Franklin, G. Horlick, and H. V. Malrnstadt, ibid., 41, 2 (1969). ANALYTICAL C H E M I S T R Y , VOL. 44, N O . 8, J U L Y 1972
1381
4 4 -t
,o/-o-o
4
/SI:=::
jf
+
3 -
*
/
F
-
I X x-x-x-x-x
8
readout variance is only 1 count and usually (N& > Nod> 1 . Hence, in photon counting, x r 2 is usually much larger than y7'. Case 111 (source flicker noise limit) is also unlikely for photon counting because y r 2 terms are dependent on and will only dominate at high pulse rates, which are unachievable because of pulse overlap. Case I1 is most common in photon counting since usually xr2 >> y r 2 2,'. Thus (S/N)Apis limited by quantum noise and is directly proportional to (No&1'2. If pulse overlap is significant, the (S/N)Apincreases faster than predictea by Poisson statistics as has been discussed in detail in a previous paper ( I o ) . If Nod is not negligible compared to (No&, a mixture of Cases I and I1 is likely. (S/N)Apcan be plotted us. T from Equation 53 for known values of y r 2 , xr2, and z r 2 . Plots of the normalized S/N for an absorbance measurement, (S/N)Ap/(S/N),rp, are analogous to those shown in Figure 1 .
1
PRACTICAL CONSIDERATIONS
-
0 14
13
12
11
IO
-Log
(iw,
9
8
Figure 2. Effect of readout variance and photocathodic current on signal-tonoise ratio
x A
c
o
x x x = 1 x
uR = 5 aR = 5 aR = 1 uli
10-3v 10-4 v 10-4 v 10-5 v
where
Simplification of Group Variance Terms. The magnitudes of the group variance terms determine the magnitude of (S/N)Adcor (S/N)Apand indicate if one of the limiting cases discussed in the last section applies. In molecular absorption spectrometry, light levels are usually much higher than encountered in other types of spectrometry. Because of this, photoelectron shot noise is expected to be a significant contributor, and many of the individual variance terms will be negligible compared to shot noise variance A previous treatment (9) has revealed that u J 2 , u A 2 , ( u ~ , , , ~and ) ~ ~(uBod2)pm , are usually negligible compared to other individual variance terms, particularly in molecular absorption spectrometry. With these assumptions, the group variance terms for dc measurement of the reference solution can be simplified to a,'
= (
u
~
1382
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
2m ~ 4 i a p ) , R ~ Z A f ( l a ) (54)
( g E o~ , 2 ) s e~ c =)
br2 = All other terms are as previously defined except that subscripts r and s indicate the reference solution in the sample cell (C, = 0 in Equations 10 and 12) and the sample solution in the sample cell ( C j finite), respectively, and the reference and the sample group variances are related by xs2 = Txr2, y s 2 = T2yr2and zs2 = z r 2 . If (S/N)AP is maximized with respect to T, an equation identical to Equation 40 results except that a,', b r 2 , and c r 2 are replaced by xr2, y r 2 , and z r 2 ,respectively. If pulse overlap is significant, the definition T = (No& (No& may cause difficulty at high photoelectron pulse rates and large absorbances. If pulse overlap in the reference beam is significant and T i s small so that ( R a p ) sis sufficiently reduced from ( R a p ) ,to make pulse overlap in the sample beam negligible, there will be a positive error in the measured transmittance. At transmittances near one, this error should be negligible. Transmittance could be more accurately defined as ( N a p ) J ( N a p ) ,since , pulse overlap in the photomultiplier has been assumed to be negligible. However, this ratio is not measurable. LIMITING CASES. For photon counting, limiting Cases I, 11, and I11 can result if one of the group variances terms ( z r 2 ,x r 2 or y r 2 ) dominate. Expressions and optimum conditions shown in Table 111 are the same if (E,,),, a r 2 ,br2, and c r 2 ,are replaced by (No&, xr2,y r 2 ,and z r 2 ,respectively. Case I is unlikely when using photon counting because the
+
+~
cr2 = 2[(gEod2)Qf
(QE,,')~
("E,d')sec
f
(55)
[E(iap)rRJ2
=
gR21 =
+ +
4meiadRf2Af(l a)
~UR'
(56)
Here it is assumed that y = m in Equations 20 and 2 1 . In photon counting, ( B ~ and ~ (u.l',d2)e2 ~ ~ ) will ~ usually be negligible (9), which allows the group variance terms to be simplified to Xr'
y,' Zr2 =
=
(57)
( u N , , ) ~= AiRapt
= (g.,r,,2)f=
(A1RaDrW
2(U.vod2)p= 2t[A1Ratc
AdRad*I
(58) (59)
Effect of Reading Error. Most traditional S/N treatments for dc measurement systems can be somewhat misleading because they assume the reading error can always be made negligible by scale expansion techniques or high resolution readout devices. However for most practical situations, a particular instrument has a readout device with limited resolution and input voltage range capabilities. Thus for high photocathodic currents, either R , or m must be reduced to keep the reference solution signal within the range of the readout device. Also the maximum voltage output of the current-to-voltage converter may impose a similar restriction. The measured S/N for the reference solution from Equations 28, 54, 5 5 , and 56 is given by Equation 60 after division by m R f :
~
bT2rather than cT2 group variance. Thus if the readout =
(:)mrdc
variance is dominant,
where icd
=
effective cathodic dark current (9) = i a d / m ,A
The limitation of the readout device is easily incorporated into Equation 61 by requiring that the reference readout voltage equal the full scale reading of the readout device, If m R , = is subEO,or EO = ( E r p ) r = (iC&R,. stituted into Equation 60, Equation 61 results.
(:)mrdc
[2eAf(l
=
+
(icp>r
cy>((icp)r
+ 2icd) + (icp)r2(t2+ 2 ~ R 2 / E 0 2 ) 1 ' / 2(61)
Equation 61 indicates that as (iC& is increased, the readout variance becomes relatively more important compared to the shot noise terms. At the limit of large (i& (S/N),7dC approaches (EZ 2flR2/E02)-112.The fact that readout variance becomes more important with (iC& seemingly contradicts the fact that uR2is an cr2 type variance. However, the relative contribution of shot noise variance to the full scale reading decreases with (i& while the relative contribution of readout variance remains constant with (iC& Figure 2 presents log-log plots of (S/N)mrdecs. (iC& for different values of uR2. It is assumed that uR2is determined only by the resolution of the readout device and that Af = 1 Hz, 01 = 0.275, icd = A, E o = 1 V and ( = lo-'. Curve 1 corresponds to an instrument having a standard meter or recorder for a readout device. Curve 2 corresponds to a meter or recorder with 10-fold scale expansion capabilities. Curves 3 and 4 could correspond to digital readout devices with 4 and 5 digit readouts. For photocathodic currents greater than A, curve 1 shows that with poor readout resolution the signal-to-noise ratio is essentially limited by readout variance. For (iC& < 10-13 A, the measurement is partially shot noise and readout variance limited. As the readout variance decreases, (S/N)m$c is shot noise limited to higher photocurrents. Curve 4 is obtained if u R 2is 10-lO V 2 or zero so that over the photocurrent range represented in Figure 2 , the readout variance is negligible if u R 25 10-lO V2. Note that scale expansion techniques or current suppression at the summing point of the current-to-voltage converter do not reduce u R 2but increase the maximum voltage accepted by the readout device, which decreases the relative contribution of u R 2 . Reducing the signal voltage with a voltage divider before the readout device increases the relative contribution of u R 2to the full scale readout variance as does reducing R,. Reducing Af decreases the magnitude of all variance terms except u R 2 and hence increases its relative contribution. Since (S/N)AdCis a fraction of (S/N)mrdc, Figure 2 clearly illustrates that high resolution readout devices can greatly improve the precision of molecular absorbance measurements, The above discussion also applies to instruments in which the readout is in absorbance. If log circuitry is utilized (S/N)mrdC will be exactly the same (if n~~is negligible) unless the readout variance is significant. Readout variance affects the direct absorbance case differently because u R 2enters as a
+
(
will follow the curve for Case
I11 rather than for Case I. Under conditions where readout variance dominates, the effect of reading error on the precision of an instrument reading in transmittance can be compared with that of an instrument with direct absorbance readout by assuming that the same readout device with readout variance, u R 2 ,is used for both instruments and the only difference is a log circuit yielding one volt per decade inserted between the OA and the readout device for the direct absorbance instrument. For the transmittance reading instrument, c r 2 = 2aR2,and if readout variance predominates, a r 2 and b r 2 can be assumed negligible. For the direct absorbance instrument, br2 = (2.303)2u R 2 ,and ur2and cr2can be assumed negligible. It is also assumed for the latter case that there is no readout error in suppressing out the dark current before entering the log circuitry. The equivalent readout variance thus makes a larger contribution in a direct absorbance readout instrument. Setting Equation 41 equal to Equation 47 with the above values of br2 and cr2,shows that transmittance readout yields a higher (S/N)AdCup to an absorbance of 0.32, while direct absorbance yields the higher (S/N)Adcif the absorbance is greater than 0.32 because, for direct absorbance readout, the readout variance is a b r 2term and its contribution decreases with absorbance. If only one dark current measurement is made (see Appendix), the crossover point is moved to an absorbance of 0.42. Because there is negligible reading error in photon counting, the S/N always increases with increasing (R,& until limited by pulse overlap or source flicker. The measured signal-tonoise ratio for the reference solution is thus given by combining Equations 46, 57, 58, and 59.
COMPARISON OF DC MEASUREMENT AND PHOTON COUNTING
Recently a critical comparison of dc measurement and photon counting for a general single beam spectrometric system has revealed the advantages and disadvantages of both techniques (9). Only the conclusions relevant to molecular absorption spectrometry will be presented here. Using Equations 36, 54, 55, and 56, (S/N)Adeis given by r EO. Equation 63 with the restriction of m R f ( i C p )=
To compare the S/N advantage of photon counting measurements to dc measurements under equivalent conditions, ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
1383
Equation 64 is divided by Equation 63 where the relationships Af = 1/(2t) and (i& = e(R& are used, and it is assumed that A1 = A d = 1, i c d = eRQd,and uR*is negligible (9). The ratio indicates that photon counting has an advantage of dl a, if the shot noise terms are dominant compared to source flicker variance. This amounts to an advantage of 5 to 22% because a usually varies from 0.1 to 0.5. Since pulse overlap will usually occur in photon counting systems at light levels below the source flicker limit, under equivalent noise bandwidth conditions, the maximum achievable S/N with linearity may be much lower than can be obtained with dc techniques. For example, even with a very fast photon counting system possessing a 10 nsec dead time, pulse overlap limits the maximum pulse rate of 105 sec-l for 0.1 linearity, In dc operation, photomultipliers can be operated with excellent linearity up to currents corresponding to pulse rates of at least 1O'O sec-I. For even higher light levels, the photomultiplier can be replaced by a photodiode (14) to extend the linear range for dc techniques even further. To reduce the light level impingent on the photocathode to bring the pulse rate within the linear range of the photon counting system would have serious disadvantages. If the light level is reduced by even a factor of two, the S/N will be decreased by approximately 30 %, and thus any inherent S/N advantage in photon counting will be negated. Discrimination against dark current pulses originating down the dynode chain is expected to provide little or no advantage for photon counting because the photocurrent will normally be much greater than the dark current except possibly at very high absorbances. Photon counting is not as subject to reading errors, domain conversion errors, or nonlinearities as are dc techniques. However scale expansion methods and high quality, high resolution A-D converters can offset these advantages. Although photon counting is inherently more stable than dc measurements, which permit the use of long counting periods to reduce the noise bandwidth, usually no distinct advantage results because light levels are such that adequate S/N's can be obtained in the dc mode with short measurement times. Another advantage of dc detection is that the analog signal can be easily processed by logarithmic amplifiers to give a direct absorbance readout. With photon counting either a D-A converter must be used before logarithmic amplification or an on-line computer must be employed to calculate the absorbance. The former approach makes the photon counting system more susceptible to drift and l/f noise, while the latter adds considerable expense. Photon counting methods appear to be most advantageous in molecular absorption methods where small spectral bandpasses are necessary or where extremely low or high absorbances must be measured. In most applications spectral bandpasses of 1-10 nm are used in analysis so that photocathodic currents are usually much larger than can be measured without serious pulse overlap nonlinearity in photon counting. In a few cases, such as in the analyses of rare earths ( 1 9 , spectral bandpasses must be less than 1 nm for Beer's law to hold, and the reduced light level might make photon counting the advantageous detection technique. In the normal range of absorbances encountered in analysis (0.1 to 1.5 A), photon counting offers little advantage, and may give less precise results compared to dc measurement,
+
(14) J. D. Ingle, Jr., and S. R. Crouch, ANAL.CHEM., 43, 1331 (1971). (15) C. V . Banks and P. W. Klingman, A t i d . Chim.Acta, 15, 356 (1956). 1384
e
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, J U L Y 1972
if the light level must be reduced in order to avoid pulse pileup. For cases in which (S/N)AdC is very small (small or large A ) , it may be advantageous to reduce the light level (i.e., slit width) and use photon counting with long integration times, Which detection method gives the higher S/N depends upon the photocathodic current or pulse rate, which is a function of the combined response characteristics of the light source, the monochromator, and the photomultiplier at the particular wavelength and slit width used for the analysis as was indicated in Equations 9 and 12. If the pulse rate for the reference beam is out of the linear range of the photon counting system used because of pulse overlap, dc measurement should be employed. On the other hand, photon counting will give the higher S/N if pulse overlap is not significant for the reference pulse rate. Near the dividing line between photon counting and dc measurement ((R&T = it may be advantageous to reduce the pulse rate slightly to utilize the S/N and other advantages of photon counting. EVALUATION AND OPTIMIZATION OF SIGNAL-TO-NOISE RATIOS DC Measurement. EVALUATION OF (S/N)AdC.The instrumental precision of a molecular absorption measurement is inversely proportional to the magnitude of the signal-tonoise ratio. The value of (S/N)Adcat a particular absorbance depends on the magnitude of the group variance terms. The magnitude of the group variance terms depends on the specific instrument used and on the particular conditions ( i e . , wavelength, slit width) used for the analysis. Under certain conditions, when one of the group variance terms predominates, the expression for (S/N)AdCis considerably simplified as was shown in Table 111. For a particular analysis, the magnitudes of the group variance terms for the reference signal voltage can be calculated from Equations 54, 55, and 56. The calculated group variance terms are substituted into Equation 36, as was shown in Equation 63. From a plot of (S/N)Adc us. absorbance or transmittance, the instrumental precision of measuring a given concentration can be estimated from the value of (S/N)Adcat the measured absorbance. Calculation of the group variance terms requires that the following parameters be known or measured: (i&, icd, m, R,, a,Af, [, and u R 2 . The degree to which these parameters can be adjusted depends on the particular instrument. All instruments provide the capability of adjusting the reference readout voltage to meet the requirement that ( E r p ) ,= EO= (icJrmR,. The photocathodic current, (i&, varies over several orders of magnitude because of the great change in the response functions of the light source, the monochromator, and the photomultiplier with respect to the wavelength and slit width used for different analyses. The noise bandwidth, Af, and readout variance, u R 2 ,are usually fixed for simple inexpensive instruments. For more sophisticated instruments, Afand u R 2may be variable, and the flicker factor, E, must be measured for each noise bandwidth. The equivalent cathodic dark current, icd,and secondary emission factor, a, are somewhat dependent on the photomultiplier gain, m. Equation 6 3 can OPTIMIZATION OF SYSTEM PARAMETERS. also be used to optimize the precision by maximizing (S/N)Adc. For a particular instrument only the parameters (i,,),, Af, and g R 2 are usually variable. The readout variance should always be made insignificant. The exact relationship between and noise bandwidth is dependent on the nature of the noise power spectrum of the source flicker noise. If the flicker noise is white or negligible, (S/N)AdC is directly proportional to (Af)-l'*. Usually bandwidths of 1.0-0.1 Hz are used to keep measurement times relatively
short, Finally, the photocathodic current, (i&, should be maximized within the limits set by resolution, Beer’s law considerations, and photodecomposition. Equations 1 and 9 reveal that the following parameters can be changed to inand improve the S/N: NAocan be increased by crease (i& using a more intense light source; T,(Xo) can be increased through using a large slit width; and S(Xo) can be increased by using a photomultiplier with high cathode sensitivity in the wavelength range of interest and high collection efficiency. Equation 63 indicates that for a particular instrument and analysis, (iC& should be increased if possible, until the absorbance measurement is flicker noise limited over the absorbance range utilized (Case 111). Of course for many situations this may not be possible without obtaining significant Beer’s law deviations. Photon Counting. If the light level has been maximized for a particular analysis and the pulse rate is in the linear dynamic range of photon counting as established by the criterion in the last section, calculation of the signal-to-noise ratio, (S/N).41’,is straightforward. Calculation of the group variance terms from Equations 57, 58, and 59 requires knowledge of only t , ( & J T (or (Rap>, and A d , Rod (or Rat,, Rad*, A , , and A d ) , and (. If the group variance terms are substituted into Equation 52, as was shown in Equation 64, and (SiN).41’is plotted cs. T , the instrumental precision expected for a particular absorbance can be found graphically.
of readout used (transmittance or absorbance). This illustrates that the usual assumption that all absorbance measurements are readout limited (Case I) can cause serious error in estimating the precision and in choosing the optimum transmittance range for absorbance measurements. For instruments or analyses in which spectral bandpasses much larger than 1 nm are utilized, (Qr may be large enough over the entire spectral region that absorbance measurements will be readout or flicker noise limited (see Figure 2) and only Case I or Case I11 will apply. The magnitude of the readout variance, the flicker noise, and the mode of readout determines which case applies. Although the discussion has been directed to quantitative single-beam spectrometry, the same basic theory and conclusions can be applied to double beam scanning instruments. In double beam systems, as well as for quantitative applications involving quite small absorption bandwidths, spectral or less are often desirable. Since the bandpasses of 1 photocathodic current, (i& is directly proportional to the square of the slit width, W , measured signals will be 2 orders of magnitude or more lower than values obtained for a 1-nm bandpass. In these cases, photon counting may be applicable and the more desirable detection system. With either photon counting or dc techniques under low light level conditions, the shot noise limit is most likely, assuming insignificant readout variance, and Case I1 will apply.
DISCUSSION
APPENDIX
To evaluate the magnitude of typical photocathodic reference currents to be expected under analytical conditions, a Heath EU-701 single beam spectrophotometer was used. With distilled water in the sample cell, the wavelength region from 200 to 800 nm was scanned using a 1-nm bandpass (500-pm slit width) and either the deuterium or tungsten lamp in its appropriate wavelength region. An RCA 935 vacuum photodiode, which has the same spectral response (S-5) as the commonly used RCA 1P28 photomultiplier, It was found that (i& varied was used to measure (i&. from a maximum value of approximately 1.5 X A to a minimum of about 3.0 X 10-l2 A. The 1-nm spectral bandpass is smaller than used in many commercial instruments, and thus the photocurrents measured should represent minimum values for typical spectrophotometers. Also, a 1-nm spectral bandpass is sufficiently small to prevent errors due to Beer’s law deviations in the great majority of molecular absorption analyses. From the range of photocurrents found, a number of conclusions can be drawn. First, the corresponding photoelectron pulse rate for a photon counting system using a 1-nm spectral bandpass over the 200-800 nm wavelength region varies from about 1 X 1Oloto 2 X 107 sec-l. Thus photoelectron pulse rates are out of the linear range for state of the art photon counting systems over the entire UV-visible range. Second, since for the major part of the wavelength region ( i C J Texceeded 10-lO A, Figure 2 illustrates that absorbance measurements will generally be readout or flicker noise limited. However, in wavelength regions near the minimum photocurrent measured, measurements will be shot noise or readout limited depending on the magnitude of the readout variance. If a normal meter or recorder with 0.5 % resolution is utilized, measurements will be readout limited over the entire spectral range. Thus it is possible for any of the 3 limiting signal-to-noise ratio cases to be valid depending on the magnitude of the photocathodic current, the readout variance, the flicker factor, and the mode
S/N for Absorbance Measurement with One Measurement of Dark Current. If only one dark current measurement is performed, an equation similar to Equation 36 can be readily derived. For this case, E r d in the numerator and denominator of Equation 32 are no longer independent so that only 3 variables, ( E r t ) S(, E r & and Erd determine the transmittance. If propagation of error mathematics are applied to Equation 32 with this restriction, uT is given by:
A
(‘4-1) The last term in Equation A-1 can also be written as Since uA
=
(0.43)uT/T,the S/N for an
absorbance measurement when only one dark current measurement is made is given by: A A
‘J.4
The final equation, Equation A-3, which is analogous to Equation 36, is obtained if the following substitutions are made in Equation A-2:
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
1385
In T
[ar2(1
+ T-’) + 26, + cr2(1+ T-2 - T-1)]1/2
(‘4-3)
Equation A-3 is the same as Equation 36 except for the additional term contained in the multiplier for c r z . Thus limiting Cases I1 and 111, where cr2is negligible, are unaffected if only dark current measurement is performed. Only limiting Case I is influenced by the number of dark current measurements. If cr2variances predominate, and one dark current measurement is made, Case Ia results and the (S/N)A is given by Equation A-4. (A-4)
The only differences between Case Ia and Case I, where two independent dark current measurements are made, are that the optimum transmittance shifts to 38.8% and the T-’ term in the denominator of Equation A-4 tends to increase the signal-to-noise ratio. For transmittances between 1 and 0.1, the for Case la is 5-41 % better than for Case I. Case Ia gives better precision only if dark current drift is negligible between measurements. At low transmittances, Cases Ia and I become identical because T-2begins to predominate over the T-1 term in Equation A-4.
RECEIVED for review September 27, 1971. Accepted March 30, 1972. Work partially supported by NSF Grant No. GP-181123 and an American Chemical Society, Analytical Division Fellowship sponsored by Perkin-Elmer Corporation.
Nondestructive Charged Particle Activation Analysis Using Short-Lived Nuclides Jean-Luc Debrun,’ David C. Riddle, and Emile A . SchweikerV Activation Analysis Research Laboratory and Department of Chemistry, Texas A&M UniEersity, College Station, Texas 77843 The objective of this study was to combine the features of nondestructive determination with speed of analysis. Accordingly, this study was restricted to radioisotopes with half-lives ranging from 1 second to -1 minute. Furthermore, only those species were considered which emit one or several ?-rays besides the 511 keV annihilation peak. Thirty elements were irradiated with protons or 3He particles. Activation with 3He proved to be of little interest in the cases studied. Proton activation was found suitable for the trace determination of the following elements: Se, Br, Y, Zr, La, Pr, Dy, and Nd. Data on their respective specific activities, activation curves, and pertinent ?-ray energies ( 1 0 . 3 keV) are given. These elements can be determined in 13 matrix elements which yield little or no activity under the experimental conditions described. Among these are matrices where neutron activation analysis cannot be applied easily because of high neutron absorption or activation cross sections.
FEWREPORTS have appeared so far on the use of charged particle activation in conjunction with short-lived nuclides (t1’2 < 1 min). Markowitz et af.( 1 , 2 ) have reported on oxygen-18 and fluorine determination methods based on the detection of 2 0 F (t”2 = 11 sec). Several authors have considered the use of I7F( P ’ 2 = 66 sec) for detecting oxygen (3-7). Ricci et al. 1 On leave of absence from Laboratoire d’Analyse par Activation Pierre Sue, Saclay, France. 2 To whom correspondence should be addressed.
(1) J. F. Lamb, D. M. Lee, and S. S. Markowitz, Proc. 2nd Conf. on Practical Aspects of Activation Analysis with Charged Particles, Euratom Report Eur-3896 d-f-e, 225 (1968). (2) D. M. Lee, J. F. Lamb, and S. S. Markowitz, ANAL.Cmhf., 43, 542 (1971). (3) P. Sue, C.R. Acad. Sci.Paris, 242,770 (1956). (4) R. R. Sippel and E. Glover, Nucl. Znsfrum. Methods, 9, 37 (1960). ( 5 ) S. n i i i , ~ ;and M. Peisach, ANAL.CHEM.,34, 1305 (1962). . 28, 2111 (6) L. Hammar and S. Forsen, J . Ztrorg. N L ~Clzem., (1966). 1386
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
(8) have measured the specific activities of several radioisotopes in the 1- to 20-min range produced by 3Heactivation on boron, carbon, and oxygen. In this laboratory, we have recently studied the determination of sulfur by measuring 32Cl of 300 msec half-life obtained by proton activation (9). With the exception of the investigations dealing with 20Fand 32Cl, the applications proposed so far involve the measurement of the 511 keV annihilation peak. The procedures based on the detection of this y-ray appear, however, for reasons outlined below, to be only of limited usefulness. The present study was motivated by the intrinsic features of an approach combining nondestructive determinations with the speed of analysis associated with the use of shortlived nuclides. The objective was to obtain a survey on the analytical possibilities offered by such an approach on a wide range of elements. To evaluate these possibilities from a practical standpoint, a selection was made among the large number of possible activation reactions and potentially suitable radioisotopes according to the following three requirements :
(a) minimal or no interfering reactions yielding the nuclides of interest (b) half-lives ranging from -1 sec to -1 min (one exception was made as noted below) (c) y-ray spectra consisting of one or several y-rays in addition to the 511 keV annihilation peak. The reason for this last condition was that a majority of nuclides produced by charged particles are p+ emitters. The resulting 511-keV peaks are thus often too complex to be (7) M. J. Lacroix, M. D. Tran, and J. Tousset, Proc. 2nd Conf. on Practical Aspects of Activation Analysis with Charged Particles, Euratom Report Eur-3896 d-f-e, 351 (1968). (8) E. Ricci and R. L. Hahn, ANAL.CHEM., 39, 794 (1967). (9) J. P. Thomas and E. A. Schweikert, Nucl. Zmtrurn. Merhods, 99, 461 (1972).
