Scanning mobility particle sizer (SMPS) plots become much easier to translate when the reader separates three operations: electrically classifying charged particles, counting particles that pass the classifier, and mathematically reconstructing a size distribution. The instrument does not directly photograph every nanoparticle and measure its diameter. It selects electrical mobility with a differential mobility analyser (DMA), counts through a detector such as a condensation particle counter (CPC), and converts signals into reported particle numbers.
An SMPS plot often shows dN/dlog10(Dp) on the vertical axis and particle diameter in nanometres on a logarithmic horizontal axis. The result is not a conventional histogram whose bar height is simply particles per cubic centimetre in that exact bin. The log-size density requires multiplication by the bin’s logarithmic width before estimating the particle concentration represented by that range. Misreading the denominator can create errors while leaving a chart visually attractive.
This article explains electrical mobility, charging and multiple-charge correction, aerosol and sheath flow, scanning time, CPC detection, diffusion losses, particle number versus particle mass, and the limits of source claims. Worked examples keep units and assumptions visible. The aim is to let readers translate an SMPS result accurately into plain English and know which inferences require measurements the instrument never made.
Why classify nanoparticles by electrical mobility?
Particles far smaller than a human hair can be difficult to size reliably with light-scattering methods because their response depends strongly on diameter, optical properties and instrument sensitivity. An SMPS instead charges an aerosol, uses an electrical field to select particles by how rapidly they drift through a carrier gas, then counts what passes through. It is a powerful approach for characterising number distributions in the ultrafine range under controlled instrument conditions.
Electrical mobility is related to particle charge and motion through air against viscous drag. In a simplified spherical-particle expression, mobility depends on charge, gas viscosity, a slip correction and particle diameter. This explains why a small singly charged particle can have a different mobility from a larger singly charged particle. It also explains a central ambiguity: a larger multiply charged particle can share mobility with a smaller particle carrying fewer charges.
Mobility diameter is therefore an equivalent diameter based on electrical transport under a specified model. It is not automatically the aerodynamic diameter used by an impactor, the optical diameter inferred from scattering, or the physical longest dimension of an irregular aggregate. A soot agglomerate may possess complex morphology, making equivalence especially important. Translation must keep the qualifier “mobility” rather than replacing it with an unsupported geometric claim.
The measurement typically addresses number concentration rather than mass concentration. Counting a thousand small particles need not imply more mass than counting a hundred much larger ones; volume scales approximately with diameter cubed for comparable spheres. Density, morphology and chemistry must be known or assumed before converting a number-size distribution into mass. This is why an SMPS result cannot by itself be substituted for a regulatory PM2.5 mass measurement.
The neutralizer and charge distribution
An aerosol entering a DMA has to possess charge for the electrical field to influence it. A charge-conditioning or neutralizing device is often used to establish a statistically predictable distribution of charge states. Not every particle emerges with one unit of charge. Depending on particle size and conditions, some remain neutral, some carry one charge and some carry more than one. The size-dependent charge distribution feeds directly into the data inversion.
“Neutralizer” is a potentially misleading everyday word here. It does not mean every particle becomes electrically neutral and stops responding. It means that the charge distribution approaches a prescribed state suitable for classifying and interpreting the particles. The precise arrangement, residence time, condition of the device and model assumptions matter. A report that says “all particles receive one electron” can seriously misrepresent the counting and sizing process.
Multiply charged particles are an important source of ambiguity. A DMA selects particles based on mobility, not on an independent measurement of the exact charge carried by each particle. A larger particle with two charges may travel through a selection window intended for a smaller singly charged particle. The software uses charge-probability and instrument-response models to correct the reported distribution. That correction is better understood as inversion than as a simple shift of labels.
The term “inversion” refers to solving the measurement problem backward: from observed signals through a known or modelled instrument response to an estimate of the underlying size distribution. Any inverted answer depends on response functions and assumptions. If counts are noisy, flow rates wrong or charge probabilities unsuitable, a mathematically tidy curve can still be biased. The model is part of the reported measurement, not an invisible afterthought.
The differential mobility analyser
A DMA commonly includes a controlled aerosol stream and a sheath-air stream passing through a geometry containing electrodes. An applied voltage establishes the field that deflects charged particles. At a given operating condition, particles within a narrow electrical-mobility range reach the classifier’s outlet and can be counted downstream. By scanning voltage, the instrument changes which mobilities are transmitted and therefore maps out an estimated particle-size range.
Sheath flow and aerosol flow have distinct roles. The sheath stream helps establish the classifier’s transport and resolution conditions, while the aerosol stream carries the particles to be selected. Their ratio influences how sharply mobilities are separated and how much signal arrives at the counter. Stating only the voltage without the flow configuration is insufficient to reproduce an SMPS measurement. Unstable flows can shift inferred diameters or broaden the response.
The DMA has a transfer function: a relationship describing which mobilities are transmitted, and with what efficiency, under its operating conditions. Real instruments do not select a mathematically perfect single diameter at each voltage. Finite width, transport losses and flow geometry affect the transmitted population. The reconstruction software must account for an instrument response rather than pretend that each detector count belongs unambiguously to one true size bin.
A voltage scan takes time. During a changing aerosol event, particle concentrations may evolve while the instrument is moving through the selected sizes. A distribution drawn from one scan may therefore combine different moments rather than represent an instantaneous atmosphere. Fast sources, chamber nucleation experiments and moving aircraft measurements deserve special care. Scan direction, duration and repeated scans can help diagnose whether the apparent mode shape reflects temporal change.
The condensation particle counter
The CPC is the counting stage often paired with the DMA. It enlarges sufficiently small particles through controlled vapour condensation until an optical detector can count the resulting droplets. This is a physical way to make nanoparticles visible to a detector; the CPC does not ordinarily provide the original size of each particle in the SMPS. The DMA classification supplies the size selection, and the CPC measures the number transmitted.
Counting efficiency is size-dependent near the lower detection range. Very small particles may not activate, grow or be detected with the same probability as particles comfortably within the instrument’s operating range. Coincidence or counting limitations can also appear at high concentrations. The laboratory’s calibration, CPC model and response corrections therefore matter. A numerical zero at the lower cutoff is not proof that no smaller particles existed in the sampled air.
Flow through the CPC determines how many cubic centimetres of classified aerosol were examined. Counts can be converted into a detector-side concentration by dividing by sample volume and appropriate counting efficiency, but this is not yet the complete ambient number-size distribution. Transmission through the DMA, dilution or losses, the charge state distribution and scan timing still need to be considered. Confusing the counter’s raw concentration with ambient total number is a frequent interpretive mistake.
For an illustrative detector-side check, 0.20 litres per minute for one minute corresponds to 0.20 litres, or 200 cubic centimetres. If the CPC records one thousand particles and detection efficiency is assumed to be one, the detector-side concentration is 1000/200 = 5 particles per cubic centimetre. If efficiency were independently established at eighty percent for that relevant population, a simple efficiency correction gives 6.25 per cubic centimetre, before the SMPS inversion.
What the horizontal axis really means
Many SMPS graphs use logarithmic diameter because aerosol populations can span wide size ranges. Equal visual distances on a logarithmic axis correspond to equal diameter ratios, not equal nanometre increments. A bin from 40 to 50 nm spans a ratio of 1.25; a bin from 80 to 100 nm also spans 1.25. Although their widths in nanometres differ, their base-ten logarithmic widths are equal. This matters directly to displayed number-size densities.
A label dN/dlog10(Dp) means particle concentration per unit interval in the base-ten logarithm of diameter. It is common to express its vertical unit as particles per cubic centimetre per log-diameter interval. The concentration within a small bin is approximately the plotted density times the bin width in log10 diameter. The approximation assumes the density is reasonably representative of that bin, and actual instrument outputs may already provide integrated per-bin counts.
Do not confuse dN/dln(Dp) with dN/dlog10(Dp). Natural logarithm and base-ten logarithm differ by a constant factor: dN/dlog10(Dp) equals ln(10) times dN/dln(Dp) for the same distribution. Mixing them without conversion introduces a factor of approximately 2.303. Check the caption and exported data dictionary rather than deciding the axis base from the fact that the horizontal scale looks logarithmic.
Diameter is often written Dp, Dm or another method-specific symbol. A plot may show central bin diameters, lower and upper bin edges, or a selected mobility corresponding to a voltage. Those are not interchangeable metadata. For number integration, use the real edges where available. Treating centre-to-centre spacing as the bin width can be particularly troublesome when bin spacing varies or the graph has been rebinned for publication.
Worked example: integrate a logarithmic bin
Assume a hypothetical report gives dN/dlog10(Dp) = 600 particles cm⁻³ per decade across the 40–50 nm bin. Its logarithmic width is log10(50/40) = log10(1.25) ≈ 0.09691. Multiplying yields approximately 58.1 particles per cubic centimetre in that interval. The plotted density of 600 is not itself the integrated particle number in the bin, because the denominator includes log-diameter width.
To check the result, the bin occupies about one tenth of a decade, so a density of 600 per full decade should contribute roughly sixty per cubic centimetre. That order-of-magnitude sense check catches a common error: summing displayed density values as if they were independent concentrations without multiplying by bin widths. In a carefully spaced instrument output, integration may be straightforward, but the reported data format must be confirmed first.
Now compare a 40–50 nm interval with an 80–100 nm interval. Each has log10 width approximately 0.09691, despite the second being twenty nanometres wide rather than ten. If both display density 600 per decade, they contribute roughly equal particle number concentrations. The larger particles nevertheless occupy more physical volume and can contribute much more mass under spherical-density assumptions. Number and mass emphasize different parts of a size distribution.
Suppose instead the exported column says particles per cubic centimetre per bin, and its value is 58.1 for the same interval. Multiplying again by 0.09691 would incorrectly reduce the concentration to about 5.63. The repair is simple: read the exported variable definition before calculating. A figure can be correct and a spreadsheet derived from it wrong solely because the analyst failed to distinguish number density from integrated number.
Why the physical air pathway matters
Aerosol particles can be lost on sampling tubing, bends, valves, walls and internal surfaces. Very small particles may diffuse to walls, and losses may increase as their diameter decreases and residence time grows. The data-processing chain may include a size-dependent penetration correction, but its adequacy depends on the inlet, flow, line geometry and particle behaviour. A clean software output does not prove that all airborne particles reached the classifier.
Drying or conditioning the aerosol can also change reported mobility diameter. Hygroscopic particles grow in moist air and may shrink after drying. If the instrument measures a conditioned dry distribution, it should not be described as the unmodified ambient wet size without qualification. Different humidity conditions and drying systems complicate comparisons across experiments. A report can be valuable precisely because it measured dry mobility diameters consistently, but that scope must be stated.
Volatility is another concern. Heated or low-pressure sample paths can change semi-volatile particle material and even the number of particles, depending on the chemistry and time scale. Dilution used for a concentrated source can change particle concentration and sometimes gas–particle partitioning. Multiplying by a simple dilution factor restores a counting denominator only if other physical effects are understood. It does not magically recreate particles that evaporated before detection.
In aircraft or mobile observations, inlet transmission can depend on airspeed, orientation, pressure and sampled particle size. The measurement volume and pressure reference should be recorded. An airborne SMPS mode seen during one scan cannot be transferred into a stationary street-level exposure estimate without accounting for location, ambient conditions and instrument configuration. The term “ambient” refers to the sample environment under study, not automatically to the general population’s breathing zone.
Particle diameter is not a synonym for health risk
The mobility-diameter spectrum helps study nanoparticle populations, nucleation, growth, coagulation and changes associated with emissions or atmospheric processing. It can reveal a mode at tens of nanometres, but it does not identify molecular composition, toxicity or deposited dose in a person’s airway by itself. A health or safety conclusion requires exposure timing, chemistry, deposition modelling and relevant toxicological evidence beyond the SMPS measurement.
Two aerosols with similar number-size spectra can contain different chemical mixtures. Conversely, the same material may appear in different size distributions after coagulation, condensation or atmospheric dilution. The instrument’s strength is a controlled description of particle counts by mobility-related size. A clear article should celebrate that strength while keeping other questions open. Readers gain more by knowing what further evidence to seek than by being offered false certainty.
Mass inference needs additional assumptions. For ideal spheres of diameter D and effective density rho, single-particle mass is rho times pi D cubed divided by six. Aggregates depart from ideal spheres and may have complicated effective densities. Applying a mass model to an SMPS spectrum can be useful when validated, but it is not a direct gravimetric measurement. A plot of particle number should not be relabelled PM2.5 mass to make it more familiar.
A very large concentration of small particles may dominate total number while larger particles dominate mass. This can happen because particle mass increases roughly with the cube of diameter for similar spheres. Therefore an increase in the number mode at 20 nm may change total counts dramatically while barely changing a mass monitor reading. Neither instrument is necessarily wrong: they answer different questions about the same air.
Common errors and immediate repairs
Error one is describing the DMA as a size-selective camera. It selects electrical mobilities under a flow-and-voltage configuration; inferred diameters require charge and drag models. Error two is describing the CPC as a device that independently sizes the original nanoparticle. It counts droplets grown from transmitted particles. Naming the contribution of each stage prevents readers from assigning a measurement to the wrong component.
Error three is treating every selected particle as singly charged. The multiple-charge population can contaminate nominal mobility bins and requires an inversion correction informed by charge statistics. Error four is assuming a perfect sharp mobility window. Real transfer functions have finite widths and losses. The method’s reported resolution, flow stability and correction framework should be retained when comparing narrow modes.
Error five is summing dN/dlog10(Dp) heights without integrating bin widths. The cure is to determine the logarithmic bin edges and multiply each density by its own log width. Error six is confusing natural-log and base-ten densities. A factor of ln(10) matters even when curves look identical by shape. Dimensional labels should be copied and checked rather than abbreviated away.
Error seven is comparing wet ambient particle diameters with conditioned dry mobility diameters as though humidity had no effect. Error eight is inferring chemical identity or human toxicity from number-size curves alone. The repair is to state drying, inlet and instrument limitations and cite additional chemistry or exposure data where relevant. Honest scope makes technical numbers transferable without pretending they are omniscient.
How to read a scan without mistaking a model for a direct observation
An SMPS spectrum is a reconstruction from a changing selection window. During a voltage scan, the DMA transmits a mobility range that changes with time, while the CPC counts particles reaching the detector. The software must associate counts with the appropriate voltage, flow, residence time and classifier response. Consequently a smooth curve is evidence of the instrument and inversion working together, not a photograph of a stable aerosol suspended in space. Record the scan duration and the aerosol conditions before comparing two distributions.
A scanning instrument assumes enough stability over a sweep for sequential measurements to be combined into one meaningful distribution. Suppose an emission plume doubles in intensity halfway through a scan. The later diameter bins then experience a different aerosol concentration from the earlier bins. An artificial spectral slope or peak can appear even when particle sizes were unchanged. A valuable cross-check is comparing successive scans, including up-scan and down-scan directions when available, and asking whether a faster instrument independently corroborates the change.
Software inversion uses a transfer function linking possible true particle diameters and charge states to measured mobility channels. It can incorporate charging probabilities, CPC response, penetration and instrument-specific corrections. The calculation is not equivalent to subtracting a fixed background and smoothing a line. An altered charging model or an unnoticed change in sheath flow can move inferred distributions. Readers should retain the model version and settings whenever they report a narrow mode, a small shift in geometric mean diameter or a trend close to the instrument resolution.
The distinction between mobility and particle shape matters especially for aggregates. A cluster of soot primary particles can have a different frictional response from a compact sphere containing the same material volume. A conventional spherical inversion still produces a valid operational mobility diameter under its assumptions, but converting that output into surface area, volume or mass may be badly biased. If morphology matters, electron microscopy, effective density experiments or an aggregate-specific model can provide constraints; the SMPS trace alone cannot choose among those geometries.
Read the counting denominator before reading the headline
The CPC reports counts associated with air processed through its sampling path, and the software converts those events into concentration. The relevant gas volume depends on actual aerosol sample flow and sampling duration, not on the nominal sheath-air flow circulating through the DMA. If a report lists 12,000 counted particles in 60 seconds and an aerosol sample flow of 0.30 litres per minute, the volume is 0.30 litre, or 300 cubic centimetres. An idealised concentration before additional corrections is 12,000 divided by 300, equal to 40 particles per cubic centimetre.
That numerical example deliberately separates ideal counting from detection efficiency. If the CPC counts only 80 percent of a suitable test population under stated conditions, a simplified corrected estimate is 40 divided by 0.80, or 50 particles per cubic centimetre. Do not apply this single factor indiscriminately across all diameters. Detection efficiency often varies at the lower cut-off, and losses along the transfer path can be size dependent. The instrument-specific response curve, not a convenient constant, controls a proper correction.
Particle counts fluctuate even in an ideally steady sample. Under the simplifying Poisson model, the standard deviation associated with 100 independent counts is approximately the square root of 100, or 10 counts, a relative standard uncertainty of ten percent. With 25 counts the corresponding relative uncertainty is about twenty percent. These are counting-statistics examples, not complete instrument uncertainties. Flow calibration, sampling inefficiency, coincidence, background and temporal variability introduce further terms that may not be independent.
If a figure reports one-second values beside a sixty-second average, treating both as equally precise is unsafe. A short interval may contain too few events for a tight estimate, while averaging can conceal rapid plume changes. State the integration time and sample volume along with concentration. When comparing neighbouring periods, check whether identical averaging and quality-control filters were applied. A change in the appearance of a trace can come from a display setting rather than a real environmental shift.
Worked example: converting a logarithmic spectrum into particle number
Imagine a hypothetical exported table with dN/dlog10(Dp) equal to 12,000 particles per cubic centimetre in a 50–60 nanometre bin and 9,000 in a 70–80 nanometre bin. These numbers are densities per base-ten logarithmic diameter, not direct bin totals. For the first bin, log10(60 divided by 50) is approximately 0.079181. Multiplication gives 950.17 particles per cubic centimetre represented by that bin under the constant-density approximation.
The second bin has logarithmic width log10(80 divided by 70), approximately 0.057992. Multiplying by 9,000 yields about 521.93 particles per cubic centimetre. The combined concentration for those two separated bins is 1,472.10 particles per cubic centimetre. It would be incorrect to claim that this is the total spectrum: the 60–70 nanometre interval and every other diameter range have not been included. The example teaches summation over bins that are actually present, not invention of missing intervals.
Now consider an exported table giving three adjacent bins: 40–50 nanometres at 12,000, 50–63 nanometres at 8,000 and 63–80 nanometres at 4,000, all expressed as dN/dlog10(Dp) in particles per cubic centimetre. Their integrated contributions are about 1,162.92, 802.96 and 415.00 particles per cubic centimetre respectively. The sum is approximately 2,380.88 particles per cubic centimetre over 40–80 nanometres. A student can reproduce each term with a calculator before trusting software totals.
If instead the density is written as dN/dln(Dp), the natural-log convention changes the numerical value of a density even for the identical physical distribution. Because ln(Dp) equals ln(10) times log10(Dp), dN/dln(Dp) equals dN/dlog10(Dp) divided by ln(10). Thus a base-ten density of 5,000 corresponds to approximately 2,171.47 per natural-log unit. Simply copying a y-axis number between these two conventions would introduce a factor of about 2.303.
Checking the dimensional story prevents arithmetic that happens to look plausible. A density of particles per cubic centimetre per logarithmic diameter interval becomes particles per cubic centimetre when multiplied by its dimensionless log width. The nanometre unit sets the position of the bin, while the ratio of its upper and lower edges is dimensionless. Do not multiply a log-density by the linear width in nanometres unless the report instead defines dN/dDp in units per nanometre.
Multiple charging and model-conditioned recovery
A mobility channel can accept particles of different physical diameters when their charges differ. For example, a larger particle carrying two elementary charges can travel with a similar mobility to a smaller singly charged particle, subject to the drag correction. A basic plot may therefore overstate one nominal size region unless the inversion accounts for the charge population. The proper correction requires information from neighbouring channels and the charge distribution; it is not a universal subtraction of twenty percent.
Consider a purely illustrative diagnostic: a channel records 1,000 events, and a charge-state model estimates that 180 of the detected events were produced by multiply charged particles assigned to another diameter range. The remaining 820 events are only an intermediate allocation, not a fully corrected concentration. Detection efficiency and transport loss still require adjustment, and uncertainty in the model affects both groups. A readable report should distinguish measured events, model allocations and the final inverted spectrum.
A false sense of certainty arises when authors treat an inversion curve’s decimal places as independent experimental evidence. An inversion may be mathematically stable yet depend on imperfect charge equilibrium or an assumed spherical shape. Reporting a mode at 61.37 nanometres suggests unwarranted resolution when the stated transfer-function width, scan conditions and calibration are looser. A more faithful explanation gives a rounded estimate, its fitting method, the supported interval and the reasons the exact numerical peak may move.
Size-dependent penetration introduces another layer. Very small particles move rapidly by diffusion and are more likely to deposit on tubing walls, bends, surfaces and internal pathways. If upstream penetration is 0.72 and detection efficiency is 0.90 for a chosen test size, their product is 0.648. An observed concentration of 648 particles per cubic centimetre would correspond to 1,000 upstream under this simple independent-factor model. Actual correction requires size-specific measured functions and uncertainty, not an unexamined blanket multiplier.
Flow checks, tubing and environmental conditioning
Aerosol sample flow and DMA sheath flow do different jobs. Sample flow governs the aerosol admitted to the classifier and contributes to sampled volume; sheath flow establishes a controlled carrier stream and affects electrical-mobility selection and resolution. Their settings are coupled to a particular DMA geometry and scan programme. Switching nominal flows without updating classification calibration can distort the diameter axis as well as the concentration estimate. Always copy the actual calibrated values, not only factory defaults printed in a brochure.
A longer tube is not simply a delayed copy of the air outside. Its length, internal diameter, conductive character, bends, residence time and electrostatic state can cause losses. Diffusion dominates many losses for very small particles, while inertia or gravitational settling becomes more important for other size regimes. Particle concentrations measured at a detector can therefore differ from those at the inlet. Before blaming the atmosphere for a spectral change, verify whether the sampling pathway changed.
Humidity changes the interpretation of size. Hygroscopic particles can absorb water and grow in diameter, while dryers or conditioning stages change the ambient mixture before measurement. A dry mobility diameter and a humid ambient particle’s effective diameter are not interchangeable. Record the relative humidity, dryer configuration, conditioning temperature and any known volatile losses. When a spectrum shifts after a change of conditions, it may reflect water uptake or evaporation rather than fresh emissions.
Electrical equipment also needs disciplined quality checks. A leak can change actual flow or introduce clean dilution air; a mis-set neutralizer can alter charge conditioning; a CPC working fluid issue can change detection response. Routine checks may include flow verification, zero or filtered-air backgrounds, timing checks, known-size challenge aerosols and scan repeatability. The appropriate tests depend on the particular configuration. A pretty graph is not an alternative to a quality-assurance record.
From particle counts to surface and mass: what remains unknown
Suppose one particle is a sphere of diameter 200 nanometres and another is a sphere of diameter 20 nanometres. At equal density, the larger sphere has a diameter ten times as great and volume one thousand times as great because spherical volume scales with the third power of diameter. Therefore one 200-nanometre sphere can carry the same material volume as one thousand 20-nanometre spheres. Counting more objects does not automatically mean more particulate mass.
For a geometric sphere, volume is pi times diameter cubed divided by six. Turning number into mass additionally requires density, physical diameter and assumptions about morphology; the mobility diameter is not always equal to the geometric diameter. For fractal aggregates, voids and drag complicate any naive cubic scaling. A scientifically careful result can report particle number confidently while stating that mass is undetermined. This is a legitimate conclusion rather than a missing calculation to hide.
Surface-area approximations face a similar problem. Geometric surface area of compact spheres scales with diameter squared, but the accessible surface of branched aggregates may be different. Moreover, surface area alone does not determine dose or health effect: chemistry, deposition, solubility, oxidative activity and exposure conditions matter. The right use of an SMPS curve is to help formulate the next measurement question, not to turn an equivalent diameter into a complete toxicology assessment.
Diagnose a changing distribution before naming a source
Imagine a monitoring station showing a new mode near 30 nanometres in the morning. One hypothesis is a nearby fresh combustion plume; others include new particle formation, a changed inlet condition, scan artefacts or a different air mass. An SMPS can support the statement that a mobility-size number mode was observed at that time under the reported conditions. It does not independently prove which source created that mode. Wind, time-resolved gases, chemistry, meteorology and repeated sampling are needed for source attribution.
An apparent fall in the total number can likewise have several explanations. The true aerosol concentration may have decreased, but a restricted CPC response near its detection limit, a new tubing run, reduced neutralizer output or unusual multiple-charging populations can create an apparent change. An effective diagnostic sequence first checks valid data flags and flows, then background and detector response, then scan consistency, then environmental context. It is more economical to eliminate known instrumental causes before building an elaborate atmospheric story.
A common comparison error is joining size spectra from instruments with different lower cut-offs. One analyser may count many particles below 10 nanometres while another begins only at 20 nanometres. Their reported integrated totals are not directly comparable because they integrate over different domains. The same principle applies to scans with different upper ranges. First harmonise the diameter bounds and integration convention; then discuss efficiency and calibration differences with honest uncertainty.
Independent practice problems
Practice one: a CPC records 4,800 countable events over two minutes with aerosol sample flow of 0.40 litre per minute. Ignore all efficiencies for this question. Find the sampled volume in cubic centimetres and the count-derived concentration. State why using a separate sheath-flow figure would change the denominator incorrectly. Work it before reading the solution; this is a unit and mechanism exercise rather than a mere division test.
Practice two: a plotted base-ten size density is 10,000 particles per cubic centimetre per log10 diameter in a bin from 60 to 75 nanometres. Estimate the integrated bin number. Would multiplying by 15 nanometres give the right answer? Identify the logarithmic width in your working. The difficult step is not arithmetic but recognizing what the vertical-axis denominator means.
Practice three: two spherical particles of equal density have diameters of 40 nanometres and 120 nanometres. Compare their individual volumes and explain whether a mobility-based measurement necessarily proves this same geometric ratio for two irregular soot aggregates. Practice four: a low-size bin contains 36 counts in the specified measurement volume under a Poisson approximation. Estimate the relative counting standard uncertainty, then name two additional uncertainty sources not described by that square-root rule.
Practice five: the instrument detects an apparent shift toward smaller mobility diameters after a room humidifier is switched off and a dryer is inserted. Suggest at least three checks before claiming a new atmospheric source. Practice six: two researchers report integrated particle numbers of 5,000 and 6,000 particles per cubic centimetre, but the first integrates 10–500 nanometres and the second 5–500 nanometres. Decide whether their totals alone demonstrate a disagreement.
Practice answers and reasoning
Answer one: 0.40 litre per minute multiplied by two minutes is 0.80 litre, or 800 cubic centimetres. Divide 4,800 by 800 to obtain 6 particles per cubic centimetre under the stated idealization. Sheath air circulates to classify aerosol mobility and must not be substituted for the aerosol sample volume. The explanation is more important than the final 6, because using the wrong flow could yield a numerically reasonable but scientifically incorrect concentration.
Answer two: log10(75 divided by 60) is log10(1.25), approximately 0.09691. Multiply by 10,000 to obtain approximately 969.1 particles per cubic centimetre. The linear difference of 15 nanometres does not match the plotted density unit; that multiplication would produce a spurious quantity. A distribution labelled dN/dDp, with an explicit per-nanometre denominator, would require a different integration operation.
Answer three: the diameter ratio is 120 divided by 40, or three, so the volume ratio for equal-density spheres is three cubed, or 27. The mobility diameters of irregular aggregates do not prove their actual geometric volume ratio; morphology and drag need separate constraints. Answer four: the relative Poisson uncertainty is the square root of 36 divided by 36, equal to one sixth, or approximately 16.7 percent. Examples of extra uncertainty include flow calibration, diffusion loss, charging assumptions and CPC counting efficiency.
Answer five: verify the humidity and drying configurations, inspect sample and sheath flows, compare repeated scans, examine inlet losses and check whether charge conditioning or detection settings changed. Environmental aerosol composition may also have changed, but instrument-induced size changes must be evaluated first. Answer six: no. The second interval includes particles between 5 and 10 nanometres that the first deliberately excludes. Recalculate both totals over a common size range, then consider differences in penetration and lower-cutoff efficiency.
Frequently asked questions from careful readers
Is an SMPS simply a very small particle counter? Not exactly. A CPC provides a number response for particles it activates and detects. The SMPS adds electrical classification, voltage scanning and a mathematical inversion to produce a mobility-size distribution. Counting alone does not reveal the diameter of each particle. Conversely, a diameter label produced by the combined system does not imply direct imaging or chemical identification. Keeping the instrument components separate clarifies the meaning of the final curve.
Can an SMPS be trusted below ten nanometres? It depends on the specified classifier, charger, CPC, flow settings, sampling losses and calibration. Some configurations measure in that regime, but charging fraction, diffusion and detection efficiency can become difficult. A claim that an instrument covers a particular lower diameter must be grounded in that exact setup rather than in the general acronym. Near a detection edge, values may need larger uncertainty or explicit qualification.
Does the highest peak mark the largest number of particles? A peak in dN/dlog diameter indicates the highest density per logarithmic diameter at that position. A broader, lower part of the curve can contribute more integrated particles over its interval than a narrow tall peak. For actual totals, integrate each relevant bin with its proper width. If the graph is normalized to total count, check what normalizing convention has been used before comparing peak heights across samples.
Can the instrument tell whether particles are dangerous or from a particular engine? It supplies size-resolved particle-number evidence, not a toxicological verdict or forensic source fingerprint. Combustion particles, secondary atmospheric aerosols and engineered nanoparticles may overlap in mobility diameter. Additional chemical, morphological and contextual measurements are needed. Readers can responsibly say that a distribution is consistent with a proposed source under certain conditions, but should distinguish that from proof.
How should a translated report handle qualifiers such as corrected, equivalent, apparent and inferred? Keep them in the sentence. “Corrected number concentration” identifies an adjustment based on instrument response; “electrical-mobility equivalent diameter” tells the reader what kind of size the calculation produces; “inferred distribution” acknowledges inversion. Replacing these terms with unqualified “real particle diameter” erases the evidence chain. Good technical translation transfers the conditions and limitations as carefully as the numbers.
Primary method references and further reading
TSI Incorporated, Scanning Mobility Particle Sizer (SMPS) Spectrometer: Nanoparticle Aggregate Mobility Analysis Software Module, application note SMPS-002: manufacturer technical note. The document describes bipolar charging, electrical classification and the limits of spherical assumptions for aggregate geometry.
TSI technical software documentation and equipment manuals should be consulted for the precise DMA transfer function, charging correction, flow specifications, CPC detection function and software setting on the actual instrument. General guidance cannot replace an instrument-specific operating manual or calibration certificate; different SMPS configurations have different useful size ranges and response characteristics.
University of Erlangen–Nuremberg, Scanning Mobility Particle Sizer facility description: facility instrumentation overview. It distinguishes the electrostatic classifier and DMA from the ultrafine CPC detector in one installed configuration.
Google Search Central, Creating Helpful, Reliable, People-First Content, recommends original useful analysis, transparent limitations and reliable supporting evidence rather than writing to an imagined ranking word threshold.
Related reading in the eduKate library
For the CPC detection mechanism and counting efficiency, see How to Read Condensation Particle Counters. For why particulate mass is a distinct physical quantity, compare How to Read TEOM PM2.5 Mass. The worked-problem teaching structure follows the read-only Clementi Mathematics teaching reference, without altering it.