1. Resolution of a DMA at atmospheric measurements Hannes.Tammet@ut.ee Pühajärve 2010 (a simplified approach)

2. 1. Characterization of mobility resolution Traditional parameter of the DMA resolution is the ratio of the mobility of monomobile ions to the width of the transfer function at the half height RES = Z / ΔZ 1/2 Transfer Z Z ZaZa ZbZb 1/2 1 Δ Z 1/2

3. In atmospheric research the mobility range is wide and the distribution function is presented on decimal logarithmic scale of mobility. An intelligible measure of the mobility resolution is the number of R esolved F ractions P er D ecade of mobility RFPD. Transfer Z Z ZaZa ZbZb 1/2 1 Δ Z 1/2 RFPD = 1 / log (Z b / Z a ) The number of actually recorded fractions per decade FPD is a subject of free choice independent of RFPD. However, it is reasonable to have the value of FPD a little greater than RFPD in typical measuring situations.

4. Relative error 100 × (2.3 RES – RFPD) / RFPD is illustrated below If Z = (Z b + Z a ) / 2 then Z b / Z a = (2 + 1 / RES) / (2 – 1 / RES) and A simple approximation: RFPD ≈ 2.3 RES

5. 2. Factors of resolution neglecting the noise  Geometric factor determined by ratios of air flow rates and electric fluxes,  Diffusion factors depending on the thermal and turbulent diffusion,  Smoothing factors depending on the response time of the electrometer and online data processing (present only in the continuously scanning analyzers like BSMA and SIGMA). If the factors are of comparable importance then the resultant will be close to the Gaussian function, which width is characterized by the standard deviation σ. The estimate of traditional resolution index is RES = 0.425 Z / σ and (2.3 × 0.425 ≈ 0.98)

6. 3. Geometric resolution of an equalized DMA A DMA is equalized when The geometric transfer function of an equalized DMA has a triangle shape on the linear scale of the mobility Z Example: in BSMA α = 0.25 and RFPD g = 8.004  8 (RFPD  5) Z o ×(1–α) Z o /(1–α) Transfer ZoZo 1/2 1 ZaZa ZbZb internal electrostatic collector air flow outlet A typical choice: FPD  RFPD g = –1 / log (1 – α) → α  1 – 10 –1/FPD

7. 4. Resolution and noise A noiseless record of the spectrometer could be inverted without loss of accuracy. Thus the resolution of an absolutely noiseless spectrometer is unlimited (well known in the mathematical theory of optical spectroscopy). Inversion amplifies the noise and sets a limit for increasing the resolution. The classic criterion of resolution – the image of two monomobile lines should have minimum at the central mobility. Let us add some noise: Increasing the noise makes the lines indistinguishable even if they are split much more than the noiseless resolution interval.

8. 5. Time resolution of DMA An intuitive index of time resolution of a scanning spectrometer is the time necessary to cover a decade of mobility t D. This leaves t F = t D / FPD to measure one fraction. 6. Noise limited resolution of DMA Statistical methods for resolving the lines in a noisy record are well developed in the mathematical theory of optical spectroscopy. However, the theory is hardly used in the practice due to the sophisticated procedures and necessity to know exactly the parameters of the noise. We will accept a rough and simple approach. The possibility to resolve two mobility lines will finally perish when the standard deviation of the noise approaches and surpasses the fraction signal. Thus the criterion of noise limit of resolution is written as F = cσ F where F is the fraction concentration of ions, σ F is the standard deviation of noise in the measurements of the fraction concentration, and c is a conventional allowed value of the signal-to-noise ratio, typically about 3.

9. Increase in geometric resolution brings along decrease in the ion current and the signal-to-noise ratio. This sets the limit of DMA resolution. The maximum electrometric current of an ion fraction is I F = αQeF, where αQ is the sample flow rate, Q is the full flow rate, and e is the elementary charge. A measurement of fraction concentration F is proportional to the electrometric current I F and σ F is proportional to the noise in the electrometric current. Thus the criterion of the noise limit can be written as I F = cσ IF. The electrometric noise has several simultaneous sources. A simple and relatively steady parameter of the noise is the standard deviation of the collector charge σ q. Noise of the ion current can be estimated as σ IF = σ q / t F = FPD σ q / t D and the criterion of the noise limit can be written as αQeF = c FPD cr σ q / t D.

10. If we accept the typical choice FPD = RFPD g and a mathematical approximation α = 1 – 10 –1/FPD  2.3 / (1.2 + FPD) (if FDP > 4 then error < 1%) then we get an equation Typical charge noise is a few thousands of elementary charges. An unavoidable component of σ q is the thermal noise (see http://en.wikipedia.org/wiki/Johnson-Nyquist_noise) which value at C = 100 pF is about 4000 elementary charges. Example: Let c = 3, σ q = 4000 e, Q = 10000 cm 3 /s, F = 3 cm –3 and t D = 10 s. Calculation results in an estimate FPD cr = 7 and RES  3.

11. 6. Conclusions  If the distribution function is presented on wide logarithmic scale then an intelligible measure of the mobility resolution is the number of Resolved Fractions Per Decade RFPD.  If case of Gaussian transfer function RFPD ≈ Z / σ.  The mobility resolution is limited by the electrometric noise and depends on the ion concentrations and required time resolution.  High values of the mobility resolution known in the laboratory mobility analyzers are principally not achievable in research of atmospheric nucleation.

12. Thank You!