# σ& # = ' ( # %". σ. # + %- %"0 (1) Evaluating the partial derivatives: (2) %- (3) %- %"0

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Aileen Tate
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Reading: Santrock, Studley and Hayes ( ); Ricci, Merrit and Hayes ( ). 1) Introduction a) At this point we have separated masses and converted ion currents to stable, amplified analog voltages.

1 Reading: Santrock, Studley and Hayes ( ); Ricci, Merrit and Hayes ( ). 1) Introduction a) At this point we have separated masses and converted ion currents to stable, amplified analog voltages. Now what? Final job of the IRMS is to do something with that signal; generally requires conversion to a digital signal, transmission to the computer, and then data processing. b) An important aspect to understand is the precision with which we must treat these signals. To get at that, start by considering the statistics of counting things, and potential noise sources. 2) Poisson statistics a) Ions arriving at a detector are random events. They do not arrive at regular intervals. A good analogy is counting cars on the freeway, and we are interested in knowing the ratio of cars to trucks with precision of 1 part in a million. Obviously if we count only 10 cars we don t know. i) We want to know the rate of ions hitting detector (ion beam current, proportional to abundance); if we measure for 1 second, we won t know that rate as well as if we measure for 10 seconds or 100 seconds and take the average. You have intuition for this. But how to quantify? b) Counting discrete events, like cars or ions, produces a skewed (not Normal) distribution, because you cannot have negative values. Described by the Poisson distribution. Fundamental value of interest is the number of events per time interval. c) d) One property of the Poisson distribution is that the mean value of our measurement population is correlated with the variance of that mean, ie σ 2 N. In other words, as the number of ions gets bigger, the standard deviation of our measurement gets bigger but only as sqrt(n). Our precision is thus fundamentally limited by the number of ions we measure, and there are diminishing returns to counting more ions. e) Example: Imagine we have an ion current of 100 fa (10 13 A), which is about 1 million cps (1.06 million to be exact). We are measuring it at 1 msec intervals (pretty fast). i) On average (the mean) we expect 1060 ions to hit our detector in each interval, but in reality we sometimes get a few more, sometimes a few less. This is the fundamental nature of shot noise associated with discrete events. ii) The variance of our measurements is correlated with the mean, such that σ and so σ sqrt(1060) (68% of the time we would measure between 1028 and 1092 ions) 1

2 iii) If we now lengthen detection interval to 100 msec (0.1 sec), we would measure 106,000 ions and σ sqrt(106,000) 326 (68% of time we measure between 105,674 and 106,326; already you can see this is less significant variation) iv) Although measuring more ions results in a greater variance (which seems bad), remember that what we really care about is the relative standard deviation, signal/noise (N/σ). (1) For 1 msec, N/σ 1060/ (2) For 100 msec, N/σ 106,000/ (3) So, clearly the signal/noise ratio is improving as sqrt(n) v) VERY USEFUL RULE OF THUMB: In general, if we measure N ions, we can say that we have measured that ion current to one part in sqrt(n). If we want to measure an ion current to 1 part in a thousand (1 permil), need to measure ions, or For an isotope ratio, this would apply mainly to the rare isotope. Also, we need to encode and handle those measurements with a similar level of accuracy (think of precision of A to D conversion, number handling). vi) Remember: this assumes that precision is limited by counting statistics. If everything not working properly, that is not necessarily the case. With a new measurement, often one of the first things we want to do is to check that we are operating near the shotnoise limit. 3) Derivation of relationship. a) Goal is to develop quantitative relationship between number of ions we count and the limiting precision of our analysis (often called counting statistics limit or Shot noise limit ). b) Start with fundamental thing we measure, which is an ion current. i) i Ne/t (1) where N is number of ions, e is electronic charge, t is time interval of msmt. From standard propogation of errors, the variance in this ion current is given by: σ& ' ( N ii) σ " %" %& (1) where partial derivative of i with respect to N is e/t, and variance in N is equal to N comes from Poisson statistics. We can substitute (ine/t) and divide both sides by i to get the relative standard deviation of the ion current:, iii) * + " & (1) which shows that the relative variance varies inversely with number of ions counted c) Next thing we do is to compare two ion currents to calculate an (apparent) isotope ratio. Handle this the same way. i) R i m /i M (1) where m s denote minor and Major ion currents. Propogation of uncertainties then gives σ. + % σ0 ii) σ % %". %"0 (1) Evaluating the partial derivatives: (2) % 1/i %". 0 (3) % i %"0./i 0 2

3 (4) Combining terms and factoring out R 2 i m 2 /i M 2 gives + * 7 "0 iii) σ R * 6 ". (1) Dividing by R 2 then gives the relative variance of the isotope ratio iv) v) vi) * 6 ". + * 7 "0 (1) which should be intuitive, because we know that variances of independent noise sources add in quadrature. This says that the relative variance in isotope ratio is the sum of the relative variances of the two ion currents. (2) Next we substitute in our relationship for the relative variance in ion current, developed above. This gives:, +, & 7 (1) Thus the relative standard deviation of an isotope ratio is a simple function of the number of ions counted. When in doubt, this is the equation I usually return to. From the definition of R N m /N M, we can get a slightly more convenient form requiring us only to know the number of major ions (assuming R is approximately that of natural abundance),,9 & 7 d) Final task is to convert this to useful expression for uncertainty in a delta value. i) δ, (1) Assuming that errors in R1 and R2 are not correlated, the standard propogation of uncertainties equation gives ii) σ < %< %, σ + %< % σ (1) we have assumed that the variance in R1 and R2 is approximately equal (ie, σ 2 R) because they are measured at roughly the same time on the same instrument. (2) Next evaluate the partial derivatives assuming that R1 and R2 are approximately equal. (3) %<, %, (4) %< >,, >, (5) Plug these into the equation above to get iii) σ < 2 10 C (1) We ve already written the equation for the relative variance of R in terms of N. Substituting that in gives iv) σ < 2 10 C, +, & 7 (1) For most isotope systems, N M is 100fold or more greater than N m, thus 1/N M is less than 1% and so can be ignored. Dropping N M and taking the square root gives the approximate but very useful form: v) σ <,E or N.,E * 3

4 (1) So, as an example: If we want precision for δ 13 C of 1, we must count 2 million 13 C ions. For a precision of 0.1, we must count 200 million 13 C ions. To get to 0.01, it is 20 billion ions. (2) Important note. For systems in which isotope ratio is not <<1, must go back to equation iv with both N m and N M in it. Jess has written a paper in which they derive a parameter called N effective that subsumes both of these. 4) Analog to Digital Conversion a) One consequence of the preceeding discussion is that to measure a delta value to 1 permil, we not only need to count 2 million ions, but we need to convert them to digital signals with similar accuracy, and store and handle those numbers with similar accuracy. Places a very high requirement on the analogtodigital conversion process. b) A to D conversion (ADC). This is a generic problem in electronics, and many different solutions have been devised. Each have different characteristics and are designed to handle different situations. Helpful for you to have at least some idea of how different ADC converters work. Properties to consider: i) Full Range. The range of input signals our ADC can accept. For example, we ll use 10V. ii) Resolution. The smallest difference that can be resolved. I think of this as how many different output levels can be generated. Since it is digital, generally expressed in bits, ie powers of 2. Thus a 3bit converter can generate output levels, and the minimum resolution would be 10V/8 1.25V. Our output signal would thus look like 0, 1.25, 2.50, 3.75,. iii) From above, if we want to know δ (or R) to 1 part in 1000, we need to know i and R to one part in That corresponds to roughly 20bit precision ADC. When our IRMS was built, there were very few ADC s that could achieve this kind of precision (I remember paying thousands of dollars for a fancy 14bit ADC). Now they are quite common, can even buy 24bit ADC s. iv) Accuracy. Main criteria here is that we want something that is very linear. Not all methods achieve this. v) Frequency response. Not a big issue for us because signals change slowly (a few Hz), but in video capture (for example) this is a big issue. c) Types of ADC i) Directconversion ADC. (Also called Flash ADC). Essentially have a comparator for each output level, all operating in parallel. Conversion takes only 1 iteration, so maximum possible speed. Drawback is the number of components needed, ie a 16bit ADC must have 65,000 discrete comparators, so tend to be very big and expensive. ii) Successiveapproximation ADC. Use a single comparator. First compare input to midpoint of output range; next compare to midpoint of halfrange; next of quarterrange, etc. Can reach arbitrary resolution, but quite slow. iii) Pipeline ADC. Combines two approaches above, using multiple comparisons per step, plus several rounds of subranging. Represents a compromise between speed and resolution. iv) Integrating ADC. Apply input voltage to a capacitor for fixed length of time, then apply a known (negative) reference voltage to the same capacitor and measure length of time it takes to reach zero. Resolution is easily changed by adjusting length of integration time. Very flexible, used in most autoranging voltmeters, etc. 4

5 v) VFC ADC. In an intermediate step, voltage is converted to an AC sine wave, with frequency varying as a function of voltage. In second step, frequency is counted and used to generate digital output. Primary benefit is that resolution can be improved, at the expensive of response time, by counting cycles for a longer time period. This is the type of converter used in Thermo instruments. (1) Details. Each Faraday signal (voltage) is first fed into a voltagetofrequency converter operating at 2000 Hz/V. A pair of counters for each channel then counts pulses from the primary signal, and from a secondary oscillator (clock, nonius) operating at much higher frequency (4.2 MHz). To avoid biasing at low frequency, the integration interval is terminated when the signal crosses zero (ie, whole number of cycles counted) and the current calculated from the number of clock cycles counted. The system is very linear and very high precision, but slow (typically <8Hz). Side effect is that length of each integration interval is slightly different. Common mistake when exporting continuousflow data is to assume all intervals are equal width when integrating. 5) Communications a) Since about 1990, all data processing has been done by computer. Mention story about chart recorder paper. Understanding the details of digital communication protocols is way beyond our scope here, so just describe briefly. b) GPIB. Older instruments use a wired digital interface known as the General Purpose Instrumentation Bus. Developed in 1960 s by Hewlett Packard (originally HPIB) as a simple means of connecting test/measurement equipment. Is a 24wire passive connection, 8 data lines, 8 ground lines, 3 handshake, 5 bus control lines. Uses a 24pin Dshaped connector that is proprietary. Unique in having male/female on opposite sides, so can be stacked. Equipment can be connected in serial or star configuration. Up to 15 possible devices, 20m total cable length. i) Several generations of standards have been written by IEE, all designated IEE488. Go up through IEE488.2, latest version (2004) now designated IEC ii) System uses an interlocked, 3wire handshake protocol that limits throughput to ~ 1MB/second, about the same as the original USB1 standard. More recent version relax handshaking and achieve up to 8 MB/sec. Both are inadequate for transmitting fullresolution data from more than 3 channels. iii) Because of the paired signal/ground connections, groundfault loops with the computer were a common problem. Essentially requires that computer and instrument share the same ground. All modern instruments come with the computer plugged into the mass spectrometer. c) Optical. Beginning with the Delta+XP and MAT253 instruments in ~2003, all communications is now optical. Chosen mainly for improved ground performance, although throughput is also helpful. 5

Module 1: Introduction to Experimental Techniques Lecture 2: Sources of error. The Lecture Contains: Sources of Error in Measurement

Module 1: Introduction to Experimental Techniques Lecture 2: Sources of error. The Lecture Contains: Sources of Error in Measurement The Lecture Contains: Sources of Error in Measurement Signal-To-Noise Ratio Analog-to-Digital Conversion of Measurement Data A/D Conversion Digitalization Errors due to A/D Conversion file:///g /optical_measurement/lecture2/2_1.htm[5/7/2012