A Reassessment of the Quasi-Simultaneous Arrival Effect in Secondary Ion Mass Spectrometry
Ryan Ogliore, Kazuhide Nagashima, Gary Huss, Pierre Haenecour
AA Reassessment of the Quasi-Simultaneous ArrivalEffect in Secondary Ion Mass Spectrometry
Ryan Ogliore
Department of Physics, Washington University in St. Louis, St. Louis, MO 63130, USA
Kazuhide Nagashima, Gary Huss
Hawai‘i Institute of Geophysics and Planetology, University of Hawai‘i at M¯anoa,Honolulu, HI 96822, USA
Pierre Haenecour
Lunar & Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA
Abstract
Quasi-simultaneous arrival (QSA) effects in secondary ion mass spectrometrycan create mass-indepedent inaccuracies in isotope measurements when usingelectron multiplier detectors (EMs). The simple Poisson statistical model ofQSA does not explain most experimental data. We present pulse-height distri-butions (PHDs) and time-series measurements to better study QSA. Our datashow that PHDs and the distribution of multiple arrivals on the EM are notconsistent with the Poisson model. Multiple arrivals are over-dispersed com-pared to Poisson and are closer to a negative binomial distribution. Throughan emission-transmission-detection model we show that the QSA correction de-pends on the non-Poisson emission of multiple secondary ions, the secondaryion energy distribution, and other factors, making an analytical correction im-practical. A standards-based correction for QSA is the best approach, and weshow the proper way to calculate standards-normalized δ values to minimize theeffect of QSA. Keywords:
SIMS
Preprint submitted to Journal of L A TEX Templates February 9, 2021 a r X i v : . [ phy s i c s . i n s - d e t ] F e b . Introduction Measurements of the isotopic composition of micrometer-sized and smallergrains in-situ by secondary-ion mass spectrometry (SIMS) has revolutionizedcosmochemistry, geochemistry, biology, and other fields. Modern SIMS instru-ments employ two types of detectors: current- (or integrated charge) measuringFaraday cups and ion-counting electron multipliers. To achieve the smallest pri-mary beam, and therefore, the best spatial resolution, it is necessary to decreasethe primary beam current to less than one nanoamp. Secondary ion currents forthese measurement conditions typically are small compared to the Johnson noiseof the Faraday cup preamplifier, and so must be measured by electron multipliers(EMs). Additionally, secondary-ion raster imaging requires fast collection andprocessing of the secondary ion signal, which is possible with electron multipliersand not possible with Faraday cups. The accuracy of a measurement is affectedby myriad factors such as deadtime correction, variable electron multiplier gain,and the quasi-simultaneous arrival (QSA) effect [1]. In this paper we present areassessment of the QSA effect in SIMS with the goal of improving accuracy formeasurements of small samples, such as cometary material returned by NASA’sStardust mission [2], and in the near future, asteroid regolith samples returnedby NASA’s OSIRIS-REx and JAXA’s Hayabusa2 missions.
A simplified model of the QSA effect is described in [3]. In this model,primary ions impact the sample surface. The number of secondary ions createdby each primary ion impact is modeled as a Poisson process with mean equalto K . The value of K can be determined by the secondary ion count ratedivided by the primary ion count rate, the secondary-to-primary ratio. Thetransmission of secondary ions from the sample surface to the detector is ignoredin this simplified model—quasi-simultaneous arrival results directly from quasi-simultaneous emission.The true number of emitted secondary ions per primary ion impact, N true ,is the expectation value of the secondary ion emission process, assumed to be2oisson: N true = ∞ (cid:88) n =1 ( n ) P ( K, x = n ) = ∞ (cid:88) n =1 ( n ) K n e − K n ! = K (1)After emission from the sample, the secondary ions will be accelerated, travelthrough the mass spectrometer, and hit the first dynode of the EM. Electronscreated by the secondary ion impact are accelerated and impact the next EMdynode, where more electrons are created. A series of dynodes creates an elec-tron cascade that is amplified after the last dynode by a pre-amp. The signalfrom the pre-amp is processed by a discriminator, then these pulses are digitizedas an output count rate.Pulses above the user-defined threshold voltage will be counted by the dis-criminator and trigger the user-defined deadtime. If a second pulse arriveswithin this deadtime window, it will not be counted. If the arrival time dif-ference is comparable to the width of the pulse at the output of the preamp( ∼
10 ns), or smaller, the two pulses will appear as one at the preamp output,with a height equal to the convolution of the two individual pulses.These “quasi-simultaneous” ions will be measured as a single event by thepulse-counting electron-multiplier detector. The measured counts per primaryion impact N measured can be calculated similar to Equation 1, except one countis measured for all multiple emissions: N measured = ∞ (cid:88) n =1 (1) P ( K, x = n ) = 1 − P ( K, x = 0) = 1 − e − K (2)The ratio N true /N measured is then: N true N measured = K − e − K (3)This is an easy quantity to calculate and does not need to be simplifiedfurther. In the literature however, this quantity is simplified using a seriesexpansion in K , the secondary to primary count rate ratio. However, K canreach values up to ∼
1, making the higher order terms significant (up to 5%),so this simplification seems both unnecessary and inaccurate. Nonetheless, the3eries expansion yields: N true N measured = K − e − K = 1 + K K − K
720 + O ( K ) (4)The first two terms, 1 + K/
2, are typically retained and higher order termsare dropped [3].The justification for the above series expansion is that while SIMS isotoperatios (with the more abundant isotope in denominator) are often measuredto be lower with larger K , the effect does not follow the predictions of thismodel [4, and references therein]. To account for this discrepency, instead of1 + K/
2, a correction of the form 1 + βK is applied where β is determined frommeasurements of standards. The values of β vary widely—from 0.19 [5] to 1.0[6]. A β value not equal to 1 / β (cid:54) = 1 / Slodzian et al. [3] proposed that the cause of the deviation of measuredQSA effects from the above-described model may be due to “inadequacy ofPoisson statistics to describe the phenomenon or to other effects such as frac-tionations due to differences in ion selection generated by the change in K ”. Asmentioned above, the model of [3] assumes that nothing in the mass spectrom-eter modifies the time distribution of secondary ions emitted from the sample4quasi-simultaneous emission equals quasi-simultaneous arrival). A more com-plete model that accurately predicts experimental data should take into accountthe true statistical nature of the quasi-simultaneous emission process, and thephysics between emission and detection of the secondary ions at the EM.To investigate these two phenomena, we present two different measurementsin this paper: 1) high-precision measurements of pulse-height distributions usingthe Cameca NanoSIMS 50 at Washington University in St. Louis (Section 2),and 2) time-series measurements of pulses measured at the output of the mono-collector EM preamp on the UH Cameca ims 1280 (Sections 3).
2. Pulse-Height Distributions
If two pulses arrive simultaneously at the first dynode of the electron multi-plier, they combine to make a single pulse height, which would be twice as largeas a single-count event on average. A measured histogram (a pulse-height dis-tribution, or PHD) of these pulse heights would show QSA events as an excessof pulses in the high-voltage tail of the histogram.
The probability for producing n secondary electrons for one EM dynodeis best modeled by the P´olya distribution [9], a special case of the negativebinomial distribution: P ( n ) = µ n n ! (1 + bµ ) − n − /b n − (cid:89) i =1 (1 + bi ) (5)where µ is the average gain of each stage of the EM, and λ and b are non-negative constants. With b = 0, the distribution is Poisson. For a sequence of k dynodes: P k ( n ) = µn ( P k (0)) b (cid:32) n − (cid:88) i =0 ( n + ib − i ) P k ( i ) P k − ( n − i ) (cid:33) (6)5here P k (0) is the probability that the total number of secondaries is zeroat the k th dynode, and is calculated: P k (0) = (1 + bµ (1 − P k − (0))) − /b , if b > e µP k − (0) − µ , if b = 0 (7)To start the calculation, we must specify the number of electrons per incidention on the first dynode: P ( n ) = δ ( n = N p ) (8)This is the Dirac delta function at N p , the number of electrons per iongenerated at the first dynode. All subsequent dynodes after the first are assumedto have the same mean number of electrons per incident electron: N e . Therefore,the total gain of the EM with k − µ = N p ( N e ) k − (9) We collected pulse-height distributions of Si and Si on the CamecaNanoSIMS 50 at Washington University in St. Louis, using a Cs + primarybeam and 10 × µ m raster. We collected PHDs for ∼
12 hrs at count rates of ∼ K from < ∼ N p and/or N e .We checked the overall gain of the EM before and after the measurement bycomparing the pulse-height distributions. We confirmed that the EM gain didnot change significantly over the course of each individual measurement (Low K and High K ).The PHDs shown in the left panel of Figure 1 have different characteristicwidths and peak voltages. The widths and peak locations may change withspecies and the entrance slit. In the context of the above model, the causeof this difference is due to changes in N p , the number of electrons created per6 V c oun t s / m V ( a ll c u r v e s sc a l ed t o m a t c h pea k he i gh t ) Si: slits open Si: slit Si: slits open Si: slit Si: slit mV (scaled) c oun t s / m V ( sc a l ed ) Si: slits open Si: slit Si: slit
Figure 1: Left) Pulse-height distributions for entrance slits Si and Si. Right) Pulse-height distributions for Si with three different entrance slits(corresponding to K ≈ . , . , . N p . incident ion on the first dynode of the EM. The first dynode of the EM is knownto age, effectively decreasing N p with time. (To increase the lifetime of the EMs(which are expensive to replace), the EM bias voltage is periodically increasedto offset the aging effect). Because the focused secondary ion beam impacts thedetector in different places for different analyses, N p may also vary spatially. Thelocation on the EM where the secondary ion beam hits is a sensitive functionof instrument tuning (e.g., B field value, EM position, deflector values, etc.).Consequently, if the secondary ion beam strikes a slightly different place on theEM, due to changes in instrument tuning, N p will change. The right panel ofFigure 1 shows the three PHDs for Si with K ≈ . , . , . N p varied so that the distributions align.The change in pulse-height distributions reported by [3] were interpreted as aQSA effect. We simulated these PHDs to investigate this claim. We employed aPoisson cascade model, as in [3], with five total dynodes (total computation timegrows rapidly with the number of dynodes) with no multi-hit events due to QSA.Fixing N p = 10 . N e = 3, as in [3], we changed the total simulated gainto match the peak for S. (The technical document from the EM anufacturer,Hamamatsu photonics, says N p should be close to 20 for 10 keV impact energy.However, N p = 20 gives a distribution that is much narrower than the data.)Then we used these same conditions, still with no QSA effect, and changed only7
100 200 300 400 500 600050010001500200025003000350040004500
Figure 2: Pulse-height distributions from Figures 4 and 5 of [3] for S (red circles) and S(blue squares). Blue solid line is a Poisson cascade model, without QSA events, of the electronmultiplier with N p = 10 . N e = 3. Red dashed line is the same model with N p = 11 . N p to match the PHD for S. With N p = 11 . S data without QSA as well as the model by [3] with QSA (Figure 2). Weconclude that variable gain in the EM’s first dynode between measurements of S and S by [3] are more likely to be responsible for their different PHDsthan the effect from QSA.We investigated the effect of QSA in our measured PHDs with two K (sec-ondary to primary count rate ratio) values: 0 .
01 and 0 . K = 0 . N p ), to whichwe will add QSA counts (see below) to fit the PHD for K = 0 . K = 0 .
01 PHD( f ) has no pulse contributions from QSA. To estimate the distribution of the K = 0 .
083 PHD ( f H ), we first calculate the PHD of two ions impacting theEM simultaneously so that the voltage pulses at the pre-amp output combine.We assume that the pulse-height distribution of these two pulses are identicallydistributed. As shown in Section 3.2 and Figure 11, our time-series measure-ments of QSA pulses supports this assumption. The probability distribution of8
100 200 300 400 500 600 700 800 900 1000
Threshold (mV) -0.00500.0050.010.0150.020.0250.030.035 c p s / m V Single Double Triple Quadruple
Figure 3: PHDs for simultaneous pulses at the first dynode of the EM, normalized to havethe same area under the curve. a pulse of height z is given by the conditional probability: f double ( z ) = (cid:90) ∞−∞ f ( z ) f ( z − x ) dx = ( f ∗ f )( z ) (10)where ∗ is the convolution operator.Similarly, for three pulses arriving simultaneously, the probability distribu-tion would be ( f ∗ f ∗ f )( z ). These multi-hit PHDs are shown in Figure 3,which shows that the multi-hit pulses will substantially affect the high-voltagetail of the PHD.The pulse-height distribution f H for a measurement with a secondary/primaryratio of K can be modeled: f H =( f ) P oiss (1 , K ) + ( f ∗ f ) P oiss (2 , K ) (11)+ ( f ∗ f ∗ f ) P oiss (3 , K ) + · · · We scale the N p of f to match the peak of f H . This is equivalent tosimply multiplying the voltage value for each histogram bin in the pulse-heightdistribution by some factor. Since we averaged many individual pulse-heightdistributions, factors liek a decreasing secondary ion signal, or aging of the EM,will not affect the shape of our measured pulse-height distribution.9
100 200 300 400 500 600 700 800020040060080010001200
Figure 4: PHD for the low- K measurement scaled to the high- K measurement. We focus on the high-voltage tails of the PHDs because this is where the QSAeffect is most apparent. The data and model for f and f H are shown in Figure4. We found that the K = 0 .
01 scaled template (blue line) under-predictsthe number of large pulses at the EM. This implies that there is some QSAcontribution that is unaccounted for by the template. However, the K = 0 . K value of 0 .
017 (black line), five times smaller than the actual secondary toprimary ion ratio, describes the PHD well.We conclude from this analysis that the effect of QSA is significant andmeasurable, but does not follow the simple Poisson model with the measuredsecondary-to-primary ratio K . In this case of Si measured on the NanoSIMS,the Poisson QSA model of [3] overestimates the QSA effect.A possible explanation is that because secondary ions are not arriving ex-actly simultaneously at the first EM dynode, this simplified model does notaccurately model the effect on the PHD. If the QSA pulses are arriving atslightly different times, say 5 ns apart, the convolution of these two pulses willnot be approximately equal to the sum, and the PHD will not change as we havemodeled it. If there is a distribution of arrival times of the pulses, some closeenough together so that the convolution of the pulses is approximately equalto the sum of the pulses, and some far enough apart so that this is not true,10
100 200 300 400 500 600 700 80010 -4 -3 -2 -1 Figure 5: PHD for the low- K measurement scaled to the high- K measurement, comparedwith simulations for K = 0 .
083 and K = 0 . y -axis is logarithmic to show clearly theeffect of QSA in the larger recorded pulses. we will observe a QSA effect less than what is predicted by this model. In thefollowing section, we show an example of three such pulses (Figure 9), and showthat differences in the arrival times of quasi-simultaneously emitted secondaryions is expected because of the energy distribution of the secondary ions.11 . Time Series of Measured Pulses The previous section described a measurement of the energy deposited at theEM by secondary ions. The fundamental quantity of interest was energy/ion,and we used this quantity to constrain the QSA effect. In this section, we willlook at a different quantity, the time interval ∆ t between secondary ion arrivalsat the EM. This quantity will allow us to more precisely constrain the statisticsof quasi-simultaneous emission and quasi-simultaneous arrival. Ions emitted from the sample are accelerated and travel down the massspectrometer before they are detected at the EM. During this travel, they maybe blocked by the energy slits after the ESA, or at other apertures and slits inthe mass spectrometer. To build a useful model, we must link ions measured atthe EM (ion arrivals) with ions emitted from the sample (ion emissions).Jones et al. [4] proposed a two-stage model where the Poisson-distributedquasi-simultaneous emission was followed by a binomial process to create ad-ditional multiple ions. The authors provided tabulated values for this “ K ∗ s ”conditional distribution in a spreadsheet, but it can be calculated analytically(finite sum) as a binomial and Poisson conditional probability distribution: p ( x ) = (cid:88) y p ( x | y ) p ( y ) = ∞ (cid:88) y =0 (cid:18) yx − y (cid:19) p x − y (1 − p ) y − x (cid:18) e − K K y y ! (cid:19) (12)= (1 − p ) − x p x e − K x (cid:88) y = ceil ( x/ (cid:0) K (1 − p ) /p (cid:1) y ( x − y )!(2 y − x )! (13)This equation reproduces the values tabulated in the supplementary infor-mation of [4]. This probability distribution is well-approximated by a negativebinomial distribution as shown in Figure 6. However, the distribution in Equa-tion 13 suffers from instabilities when p > K and has discontinuities when x iseven or odd, so the negative binomial distribution is a more robust statisticalmodel. 12 x p ( x ) K s =0.8, P2=0.3Neg Bin: k=1.2, r=0.5K s =1.0 , P2=0.2Neg Bin: k=2.4, r=0.7K s =1.5 , P2=0.1Neg Bin: k=7.7, r=0.8 Figure 6: The probability distribution of x multiple arrivals. Solid circles show the probabilitydistribution proposed by [4] for various values of K s ( K in Equation 13) and P ( p in Equation13). Lines show the negative binomial approximation to the three different distributions. In our model, we assume that the initial emission of secondary ions from thesample is described by a negative binomial distribution instead of a Poisson dis-tribution. The negative binomial models the number of failures in a sequence ofindependent, identically distributed Bernoulli trials (with probability of success p ) before r successes occur: P ( k ) = (cid:18) k + r − k (cid:19) p r (1 − p ) k (14)The variance of the negative binomial distribution is µ (1 + µ/r ). In thelimit where r → ∞ , the negative binomial distribution reduces to the Poissondistribution with mean µ . The negative binomial allows for a mean that isdifferent from its variance, which allows for a more accurate model of the QSAeffect in the case that it is over- or under-dispersed compared to the Poissondistribution. In this paper, we will use the negative binomial distribution, whichis consistent with the distribution proposed by [4].If r can take on non-integer positive values, we replace the binomial coeffi-cient by the gamma function: P ( k ) = Γ( k + r )Γ( k + 1) Γ( r ) p r (1 − p ) k (15)The probability p ( y ) that y ions arrive at the detector (after emission from13he sample surface and passage through the apertures) is given by the marginalprobability distribution: p ( x ) = (cid:88) y p ( x | y ) p ( y ) (16)That is, the distribution of y is equal to the product of the probabilitydistribution of x given y , multiplied by the probability distribution of y , summedover all possible values of y . For example, the probability that zero ions make itto the detector ( x = 0) is equal to: the probability that zero ions are emitted bythe initial negative binomial process ( y = 0) multiplied by the probability thatzero of these ions make it through the apertures, plus the probability that one ionis emitted ( y = 1) multiplied by the probability that it does not make it throughthe apertures, plus the probability that two ions are emitted ( y = 2) multipliedby the probability that neither of these makes it through the apertures, etc.The distribution of p ( y ) is the negative binomial distribution parameterizedby p and r . The conditional distribution p ( x | y ) is the binomial distribution(with probability q ) where the number of independent experiments is equal to y , the number of ions emitted from the sample by the negative binomial process.Plugging these two distributions into Equation 16: N BB ( x ) ≡ p ( x ) = (cid:88) y p ( x | y ) p ( y ) (17)= ∞ (cid:88) y =0 Γ( y + r )Γ( y + 1) Γ( r ) p r (1 − p ) y (cid:18) yx (cid:19) q x (1 − q ) y − x (18)= p r q x Γ( r ) ∞ (cid:88) y = x Γ( y + r )(1 − p ) y Γ( y + 1) (cid:18) yx (cid:19) (1 − q ) y − x (19)We calculate N BB ( x ) by summing terms until the last term changes the sumby no more than some threshhold value (typically 10 − ). Since the distributionproposed by [4] (Equation 13) can be approximated by the negative binomialdistribution (Figure 6), it can also be approximated by N BB ( x ). Therefore,the model we propose here is consistent with the mechanism proposed by [4].14n our model we assume that q is the probability that the ion passes throughthe ESA and other apertures. Since the ion’s energy affects the arrival times wewill model, we express q as the product of two probabilities: q = q e q a , where q e is the probability that the ion has the right energy to pass through the ESA,and q a is the probability that the secondary ion will pass through all of theother slits, apertures, and energy slit to arrive at the detector. This probability q e can be calculated from the energy scan ( F ( E ) vs. E ) shown in Figure 7. q e = (cid:82) E max E min F ( E ) dE (cid:82) + ∞−∞ F ( E ) dE (20)As we have no way to calculate q a a priori , we will vary q a so that our modelfits the measured time differences in the arrival pulses. We model the probability distribution of the kinetic energy of secondary ions P (∆ E ) following [10]: P (∆ E ) = 2(∆ E ) E s (1 + ∆ E/E s ) (cid:18) erf (cid:18) E + 0 . ω √ σ (cid:19)(cid:19) (cid:18) − erf (cid:18) E + 0 . ω √ σ (cid:19)(cid:19) (21)where E s is the surface binding energy, and ω and σ are characteristic widthsof the rectangular box and Gaussian which are convolved to make the shape ofthe energy spectrum.The measured energy spectrum for the “High- K -M” magnetite standardmeasured in our analysis is shown in Figure 7. We measured the energy spec-trum by scanning a narrow energy slit and recording the counts of O − atthe EM. We fit this energy spectrum using E s = 15 . ω = 29 . σ = 45 .
20 40 60 80 100 12000.20.40.60.81
Figure 7: Energy distribution and model fit (Equation 21) of secondary ions of O − inmagnetite (High- K -M dataset) measured on the UH Cameca ims 1280. significant source of error in modeling the QSA phenomenon, as the importantfeatures of the secondary ion energy distribution are modeled accurately. We model the emission and detection of ions in a secondary ion mass spec-trometer as follows:1. Primary ions impact the sample surface.2. A number of secondary ions are emitted from the sample, described by aPoisson or Negative Binomial distribution.3. The kinetic energy ∆ E of the emitted ions before they are accelerated arerandomly drawn from the energy scan (Figure 7).4. The secondary ions are accelerated by the extraction voltage E ext andtravel the distance of the secondary flight path of the mass spectrometer D , with total kinetic energy given by E ext + ∆ E . The travel time of the16nonrelativistic) ions is given by:∆ t = D (cid:112) E ext + ∆ E ) /m (22)5. The secondary ions pass through the electrostatic analyzer which acts asa bandpass filter, where only energies from E min to E max are permitted.6. The secondary ions have some probability q a of being stopped by theapertures.7. The secondary ions arrive at the detector.An analytical solution for the arrival times at the detector is not possiblegiven the form of the probability density functions and energy dependence ofthe secondary ions, so we use numerical techniques to simulate 5 × ions foreach measurement condition. We have included our Matlab code to performthese simulations in the supplementary information. The effect of quasi-simultaneous ions for Steps 1, 2, and 7 above, has beencalculated previously (e.g., [1]). Equivalently, if one sets the length of the sec-ondary flight tube D equal to zero, an analytical solution is easy to calculate—itis the formalism described previously [3].Using the numerical procedure outlined above, we set the length of the sec-ondary ion flight tube equal to zero to reproduce this treatment of the QSA ef-fect. As shown in Figure 8, the numerical procedure reproduces N true /N measured for various values of K .Arrival times between pulses without the QSA effect will be distributedaccording to the exponential probability density function with rate parameter λ . We estimate λ from the data (assuming the QSA pulses will not affect λ significantly). This expected distribution is shown by the blue dashed curves inFigure 10. We directly measured arrival times of secondary ions ( O − from chromiteand magnetite, except O − was measured in the “Low- K ” dataset) on the17 Figure 8: Numerical simulation of the QSA correction N true /N measured vs. K , the secondaryto primary count rate ratio, compared to the true value for zero secondary ion flight distance.Table 1: Parameters for time series measurements of arrival pulses. Name K Phase CPS Energy SlitLow-
K < ×
75 eVMid- K ×
75 eVHigh-
K > × OpenHigh- K -M > × OpenUniversity of Hawaii Cameca ims 1280 ion probe. Some important measurementconditions were as follows—primary current: 1.7 nA, primary high voltage:10 kV, exit slit width: 172 µ m, mass resolving power: 7076, entrance slit:69 µ m, field aperture: 5000 µ m.We recorded output voltage from the preamplifier as a seamless data logusing a digital oscilloscope (Tektronix TDS5104B). Voltage of the output wassampled every 0.4 nanoseconds for 500,000 points. We recorded ∼ findpeaks then fit them with Gaussians, which was a good approximation ofthe peak shape. Peak widths are ∼ ±
40 -30 -20 -10 0 10 20 30 40 50-0.0500.050.10.150.20.250.30.350.4
Figure 9: Three close pulses as measured by a digital oscilloscope on the first dynode of theEM (from the High- K measurement). preamp, and the response of the oscilloscope. We are able to reliably find peakswith centers as close together as ∼ t . Then we calculated histograms of the ∆ t values with logarthmic-spaced bins to highlight the QSA counts at small ∆ t .Histograms for the four conditions described in Table 3.2 are shown in Figure10, compared to the expected distribution of arrival time differences in a Poissonprocess (exponential distribution of ∆ t ) without QSA ions.The pulse-height distribution of QSA ions can tell us if the EM has timeto fully recover before the next pulse in a QSA pair arrives. We calculated thepeak heights of QSA pairs and calculated the PHDs of the first-arriving andsecond-arriving ions (Figure 11). There is no significant difference between thePHDs of the first and second arriving ions, so we conclude that the EM fullyrecovers even between closely arriving ions. We calculate the number of multiple arrivals in the Mid- K , High- K , andHigh- K -M data sets to see how well these pulses follow a Poisson process.For the time series we identify consecutive pulses that are more closely placed19 -9 -8 -7 -6 -5 -9 -8 -7 -6 -5 -9 -8 -7 -6 -5 -9 -8 -7 -6 -5 Figure 10: Histograms of the time between pulses at the first dynode of the EM. The measuredpulse separation histograms are shown by filled black circles. Top-left: low- K , top-right:Mid- K , lower-left: High- K , lower-right: High- K -M. Error bars are ± the square root of thenumber of counts in each bin. The blue dashed curve is the exponential distribution withrate parameter λ equal to the mean count rate for that measurement. Vertical solid grey lineindicates the EM deadtime (30 ns). Vertical dashed grey line indicates the largest ∆ t whenquasi-simultaneous arrivals are significant (∆ t thresh ). Pulses arriving at smaller ∆ t than theEM deadtime (left of the solid grey line) will be counted as a single pulse. mV C oun t s Poisson Model: Np=10, Ne=2First IonSecond Ion mV C oun t s Poisson Model: Np=10, Ne=2First IonSecond Ion
Figure 11: Pulse-height distributions for first and second ions in double arrivals for the High- K (left) and High- K -M (right) measurements. t thresh . Quasi-simultaneous arrival does not necessarily mean quasi-simultaneous emission (secondary ions that were emitted simultaneously from asingle incident primary ion). Some pulses will arrive closer than ∆ t thresh becauseof the exponentially distributed time between emissions given by the Poissonprocess. This is the blue dashed-line curve in Figure 10, and the correctionfor these missed counts is just the normal deadtime correction calculation. Wecorrect for the expected double arrivals from normal Poisson emission using thecumulative exponential distribution function: 1 − e − λ ∆ t thresh ≈
10% of the totalpulses, and assume that if one ion is emitted with low energy (∆ E = 0 eV),and the next is emitted with high energy (∆ E = 100 eV), it might be possiblefor the fast ion to catch up with the slow ion and arrive at the detector closerthan ∆ t thresh . We calculate that the difference in time between two such ions is ∼
100 ns which is much smaller than the characteristic time between secondaryion emissions ( ∼ µ s), so we conclude that velocity dispersion of non-quasi-simultaneous emitted ions is not a major source of quasi-simultaneous arrivalson the EM.The number of excess double, triple, etc. arrivals was calculated for theMid- K , High- K , and High- K -M data sets, and the expected number from theexponential distribution (blue curve in Figure 10) was subtracted. We comparethese measured multiple arrivals with what is expected from a Poisson distribu-tion with a mean value K . We calculated the best fit K for each dataset as: K = 2 P (2) /P (1), where P (2) is the measured fraction of double arrivals and P (1) is the measured fraction of single arrivals.The primary to secondary ion count rate ratio was determined to be 0.152for the Mid- K measurement. The primary beam currents for the high- K andhigh- K -M measurements were below detection limit of the primary Faraday cup,so only a lower bound on this ratio was determined for these two datasets.. Thelow- K measurement had too few double counts to be useful for evaluation ofthe statistical distribution of QSA ions. For the Mid- K dataset, we can test theaccuracy of the Slodzian et al. [1] model, where multiple arrivals are Poissondistributed with mean equal to the secondary to primary ratio. The Poisson21 -4 -3 -2 -1 Poisson, K=0.092Poisson, K=0.152Neg Bin+Bin, R=0.96, p1=0.77, p2=0.32 -4 -3 -2 -1 Poisson, K=0.158Neg Bin+Bin, R=0.65, p1=0.62, p2=0.40 -4 -3 -2 -1 Poisson, K=0.156Neg Bin+Bin, R=0.71, p1=0.63, p2=0.41
Figure 12: Probability distribution of excess multiple arrivals for the Mid- K (top-left), High- K (top-right), and High- K -M (bottom) measurements. Solid red line is a Poisson fit usingonly the single and double arrivals. Red dashed line in Mid- K measurement is the Poissondistribution with mean equal to the measured secondary-to-primary ratio (0.152). Green solidline is the NBB probability distribution fit. probability distribution, with λ = 0 . K . A fit with a N BB distribution underestimates double arrivals but is more accurate for triple+arrivals. The true distribution of multiple arrivals does not appear to follow ananalytical probability distribution.
The position and width of the energy slit, a bandpass filter, will affect theenergy distribution of secondary ions. This, in turn, will affect the secondary-ion22ight time, and how close in time the secondary ions reach the EM. A change inthe binding energy will also affect the secondary ion kinetic energy, flight times,and distribution of arrival times of secondary ions. We modeled the dependenceof the pulse separation distributions on energy slit and secondary ion surfacebinding energy (results are shown in Figure 13). For these simulations we usedthe
N BB fit to multiple arrivals in the High- K -M measurement (Figure 12).For our measurements, we recorded the energy slit setting for each of the fourmeasurements shown in Table 3.2. We only measured the energy distribution ofsecondary ions for the Mid- K setting, but this distribution may change as themeasurement conditions change. To account for these changes, we adjusted thesurface binding energy E s so that the simulation best matches the data. Theprobability that the secondary ion makes it through the apertures, q a , was setequal to one as a smaller number did not significantly improve the model fit tothe data. -9 -8 -7 -6 -5
10 eV20 eV30 eV50 eV100 eVOpen -9 -8 -7 -6 -5 Figure 13: Simulations of time between pulses showing the effect of changing the energywindow (left) and changing E s in the energy distribution for secondary ions (right). Multiplearrivals follow the NBB fit to the High- K -M data and the secondary ion energy distributionis initially assumed to be equal to that measured for the High- K -M dataset. The results of the simulation for the Mid- K , High- K , and High- K -M datasets are shown in Figure 14. The surface binding energy E s (Equation 21)was adjusted for the Mid- K and High- K data to better fit the QSA peak. Weassume that counts that arrive closer than the EM deadtime are counted as a23ingle pulse, so that the final ion counts need to be multiplied by a correctionfactor greater than one. These correction factors are given in Table 2.The important observations of Figure 14 are:1. Quasi-simultaneously emitted ions may be emitted with different ener-gies (Equation 21) and drift apart from each other during travel throughthe instrument, so that their arrival time separation is larger than thedeadtime. That is, quasi-simultaneous emission does not always resultin quasi-simultaneous arrival. For smaller geometery ion probes like theCameca 7f, secondary ions may not have time to drift apart enough to bedetected separately.2. The N BB probability distribution is a better fit to the data but overes-timates pulses closer than ten nanoseconds. However, this may be a biasin the data analysis—it is more difficult to detect close pulses in our data(Figure 9) and some may be counted as one pulse.3. Chromite (High- K and Mid- K ) required a higher surface binding energy E s than magnetite (High- K -M), by factors of 1.3 and 2.1 for High- K andMid- K , respectively. This is likely because these two materials have verydifferent surface sputtering properties under the primary Cs + beam, andthat the electron flood gun was used for chromite but was not used formagnetite. Changes in the secondary ion energy distribution may causethe quasi-simultaneously emitted ions to drift apart outside the deadtimewindow, and change the correction factor.4. Opening the energy slit allows more secondary ions through, increasing thefraction of counts lost to QSA, but some of these ions of different energiesmay drift apart outside the deadtime window, decreasing the fraction ofcounts lost to QSA.5. In the Poisson model of [3], the correction factor estimated by the mea-sured K (0.152) is too large ( K = 0 .
123 yields the appropriate correction,Table 2). 24 -9 -8 -7 -6 -5 ExponentialPoissonNeg Bin -9 -8 -7 -6 -5 ExponentialPoissonNeg Bin -9 -8 -7 -6 -5 ExponentialPoissonNeg Bin
Figure 14: Histograms of the time between pulses at the first dynode of the EM. Blue dashedlines are the same as Figure 10 and represent the no-QSA case. a) Mid- K ( K =0.152) withthe energy distribution measured at the High- K -M setting but with E s multiplied by 2.1,energy window = 75 eV, and multiple-arrival threshhold = 50 ns. b) High- K with the energydistribution measured at the Huge- K -M setting but with E s multiplied by 1.3, energy windowopen, and multiple-arrival threshhold = 83 ns. c) High- K -M with energy window open, andmultiple-arrival threshhold = 122 ns.Table 2: Correction factors ( N true /N measured ) for the simulations and data shown in Figure14. True is the correction from the data (black filled circles in Figure 14),
Neg Bin Model isthe green curve,
Poisson Model is the red curve,
Poisson Expr is Equation 3 with K given inFigure 12, K fit is the K value in Equation 3 that gives the True correction factor.
Name True Neg Bin Model Poisson Model Poisson Expr K fit
Mid- K K K -M . Discussion Our measurements show that the QSA effect depends in a complicated fash-ion on the secondary ion energy distribution, the secondary ion flight length, thesecondary ion acceleration voltage, and the electron multiplier deadtime. TheNanoSIMS PHDs (Figure 5) and the Mid- K pulse-separation measurementsshow that the oversimplified Poisson model of [3] overcorrects the QSA effect.However, other SIMS measurements summarized in [4, and references therein]show that often the Poisson model undercorrects the QSA effect. The statisticaldistribution of multiple arrivals and emissions (Figure 12) may vary with sec-ondary ion species ( O − from chromite and magnetite was measured here for allbut the low- K measurement, where O − was measured). Under the conditionsmeasured here, the distribution of secondary arrivals is signficantly overdis-persed compared to Poisson. An overdispersed distribution results in more lostions to QSA [4]. The energy distribution of secondary ions and ion probe ana-lytical conditions may result in a different fraction of ions drifting apart enoughto be detected individually. A shorter EM deadtime, longer secondary flightpath, and broader secondary ion energy spectrum results in fewer counts lost toQSA. These combined effects may result in either an overprediction of the Pois-son QSA correction for some measurements and an underprediction in others.This makes it impossible to accurately correct for the QSA effect analytically.Even a semi-empirical approach, where the QSA correction is determined over arange of analytical conditions, will be inaccurate because these combined effectsdo not smoothly vary.The best protocol is to measure appropriate standards, acquired with similaranalytical protocols and with similar secondary ion count rates, and normalizethe unknown measurements to these standards.In geochemistry and cosmochemistry, isotope ratios are usually expressedwith the most abundant isotope in the denomiator. If both isotopes are mea-sured with EMs, the major isotope is more likely to be affected by QSA, sincethe QSA effect increases with the secondary/primary ion ratio. For example, in26he ratio O/ O, the QSA effect on O is ∼
500 times larger than on O. Wewill assume that the QSA effect can be ignored on the numerator isotope.Cosmo/geochemists are often interested in deviations of isotope ratios fromknown terrestrial values. The fundamental quantity of interest is R u — theisotope ratio of the unknown sample to the same ratio in some standard. Forexample, oxygen isotopes are often expressed as parts-per-thousand (per mil) de-viations from a standard ocean water value (Vienna standard mean ocean wateror VSMOW). We will call the normalizing ratio R . For O/ O, R =0.0020052for VSMOW. However, since measured SIMS ratios will differ from the true ratiodue to instrumental fraction effects [11], we cannot just calculate R u = r u /R .We measure our unknown isotope ratio ( r u ) and the same isotope ratio measuredin a standard ( r s ) of similar composition as our unknown, but with known iso-tope ratio ( R s ). Various processes in the instrument causes the measured valueof the standard r s to be different from its true value R s . We assume that therelative error in the unknown is the same as the sample, because the standardand unknown have similar compositions. The true ratio of the unknown to R u is then [12]: R u = r u R (cid:18) r s R s (cid:19) = r u R s r s R (23)This ratio, expressed in δ units (per mil), is: δ = (cid:18) r u R s r s R − (cid:19) δ valuesis often used [13, 14]: δ (cid:48) = δ unknown + ( δ standard,measured − δ standard,true ) (25)This expression for δ (cid:48) is described as an approximation to Equation 24 (e.g.a “first order” approximation [14] for δ without stating what quantity is used forsuch a series expansion, and why it is justified to drop higher order terms). In27he following we will calculate exactly what is assumed in this approximation,using the definitions of r u , r s , R t , and R defined previously: δ (cid:48) = δ unknown + ( δ standard,measured − δ standard,true ) (26)= (cid:16) r u R − (cid:17) (cid:16) r s R − (cid:17) − (cid:18) R s R − (cid:19) (cid:18) r u + r s − R s R − (cid:19) (cid:18) r u r s + r s − R s r s r s R − (cid:19) r u (cid:18) r s + r s r u − R s r s r u (cid:19) r s R − (cid:18) r u R s r s R (cid:18) r s R s + r s r u R s − r s r u (cid:19) − (cid:19) δ ’ to be equal to δ , we must have: r s R s + r s r u R s − r s r u = 1 (28) r s R s (cid:18) r s r u − r s r u R s r s (cid:19) = 1 (29)If the measured standard ratio r s is equal to the true standard ratio R s thenthis equation holds true. However, if it is not true, then δ ’ will differ from δ .The other benefit of using the expression for δ instead of δ ’ is that themeasured ratios r u and r s are only used in the ratio r u /r s . This means thatsystematic errors that are equal multiplicative factors on both r u and r s divideout, whereas they do not in the calculation of δ ’. If the standard and unknownare measured under the same conditions (count rates, energy slits, etc.) andhave similar secondary ion energy distributions, the QSA effect will be the samemultiplicative factor on r u and r s . When calculating δ , QSA will divide out.However if δ ’ is calculated, QSA will not directly divide out. The differencebetween δ , δ ’, and δ ’ with QSA is shown in Figure 15.28 .99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 1.01-30 -20-1001020304050 Figure 15: Values of δ , δ ’, and δ ’ with a QSA factor of 1.01 for a range of values of themeasured-to-true isotope ratio in a standard.
5. Conclusions
Previous assessments of the QSA effect have relied on measured isotope ra-tios [e.g. 3]. These measurements can be affected by instrumental fractionationeffects that can mask the QSA effect. To understand QSA better, it is necess-sary to directly study the arrival of ions at the EM. In this paper, we havereassessed the QSA effect using measured pulse-height and arrival-time distri-butions. These measurements have allowed us to estimate the distribution ofmultiply emitted ions. We found that multiply emitted ions are overdispersedcompared to a Poisson distribution, making the traditional Poisson model ofQSA, and its correction, invalid. The combined effects of the secondary ion en-ergy distribution, the secondary ion flight length, the secondary ion accelerationvoltage, and the electron multiplier deadtime make it impractical to correct QSAby either an analytical or semi-empirical methodology. The best approach is tonormalize unknown measurements with the appropriate standard, and calculate δ values appropriately. 29 . Acknowledgements We thank Clive Jones (Washington University in St. Louis) for many invig-orating discussions. This work was supported by NASA grant NNX14AF22Gto RCO.
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