Detectors for Atomic Spectroscopy

A photomultiplier tube converts a single photon into a burst of roughly a million electrons. This happens in under ten nanoseconds. That single fact — enormous internal gain achieved before any external noise can be added — explains a lot. It explains why PMTs dominated atomic spectroscopy for sixty years. It also explains why replacing them with solid-state arrays took a genuine reinvention of detector electronics. It was not just a change of parts. The detector is the quiet partner in every atomic spectroscopy measurement. The plasma or flame does the atomizing and exciting. The monochromator or echelle does the wavelength sorting. But the detector decides how many wavelengths you can watch at once. It decides how low a concentration you can see above the noise floor. And it decides how fast you can finish a run of forty elements on a hundred samples. This matters because the choice between a scanning photomultiplier and a multichannel array is important. It is really a choice between two philosophies of measurement. Sequential detection with a single PMT measures one wavelength at a time. It moves a grating or a slit to hop from line to line. This approach is simple, cheap, and photometrically clean. But it is slow when the sample list and the element list both get long. Simultaneous detection with a charge-transfer array (CCD or CID) captures many wavelengths at once. Sometimes it captures the whole 165–800 nm range in one integration. This comes at the cost of a more complex readout chain. It also brings a differently shaped noise budget. Modern inductively coupled plasma optical emission spectrometers (ICP-OES) embody both philosophies. So do time-of-flight or scanning ICP mass spectrometers. A working analytical chemist needs to know which one a given problem demands. The physics underneath is a story of two very different approaches. Both turn photons into a countable, storable signal. A PMT is a vacuum-tube current amplifier. It consists of a photocathode, dynode chain, and anode, all riding on a high-voltage divider. It produces an analog current or a train of discrete pulses. External electronics then integrate or count this output. A CCD or CID is a semiconductor charge-integration device. Photons generate electron–hole pairs in a silicon lattice. The charge simply accumulates in a potential well — a tiny capacitor. It is read out once, at the end of the exposure. In the CID case, it can be read non-destructively many times during the exposure. One device amplifies as it goes; the other stores and waits. That distinction drives everything downstream. It shapes dynamic range, dark current, linearity, cost. It also shapes the entire architecture of a "simultaneous" versus "sequential" instrument. This article works through several key topics. It covers the physical structure of the photomultiplier and of charge-transfer arrays. It covers the governing equations for gain, quantum efficiency, dynode statistics, and shot-noise-limited signal-to-noise ratio. It includes three fully worked numerical examples. These span a textbook PMT gain calculation, a real ICP-OES detection-limit comparison, and a competition-level noise-budget problem. Finally, it surveys where each detector type is deployed today. This ranges from trace-metal environmental labs to semiconductor-fab process control.

Key takeaways

A PMT multiplies a single photoelectron through N dynodes, each with gain δ, giving overall gain $G = \delta^N$ , typically $10^5$ – $10^7$ .
CCDs and CIDs integrate photogenerated charge in silicon potential wells; the CID's non-destructive readout of each pixel is what enables true random-access, simultaneous multi-element operation.
Sequential (scanning PMT) instruments trade speed for photometric simplicity and superb dynamic range; simultaneous (array) instruments trade some per-line optimization for enormous throughput.
Dark current, read noise, and full-well capacity — not just quantum efficiency — set the practical detection limit of an array detector.
Photon shot noise is the fundamental floor for both detector families once amplifier and readout noise are minimized; $\mathrm{SNR} = \sqrt{n_p}$ for a Poisson-limited photon count $n_p$ .
Blooming, charge-transfer efficiency, and anti-blooming drains are CCD-specific artifacts that a CID's pixel architecture largely avoids.
Key idea: the detector's architecture — amplify-then-read (PMT) versus integrate-then-read (array) — dictates whether an instrument scans one line at a time or captures a whole spectrum simultaneously, and that single design choice cascades into every figure of merit that follows.

Where this lives today

Modern ICP-OES instruments split cleanly along this article's central axis. High-throughput environmental and industrial labs running EPA Method 200.7/6010 panels on hundreds of water or soil digests per day almost universally use simultaneous CID- or CCD-based (often now CMOS-hybrid) spectrometers capable of measuring 20–70 elements from a single 15–60 s plasma exposure, because the economics of sample throughput dominate. Ultratrace geochemistry and semiconductor-grade reagent qualification, where a handful of elements must be measured at parts-per-trillion levels, still sometimes favor dedicated sequential scanning instruments (or high-resolution sector-field ICP-MS, whose detector electronics — electron multipliers and Faraday cups — echo the same amplify-then-read versus integrate-then-read dichotomy) for their superior per-line dynamic range and lower stray-light contamination. Photomultiplier tubes persist essentially unchanged in flame atomic absorption spectroscopy (FAAS) and in graphite furnace AAS, in scintillation counting for radiochemistry, and in the reference channels of many double-beam UV-Vis spectrophotometers, precisely because a single, well-characterized wavelength with excellent dynamic range is exactly what those techniques need. CCD arrays dominate Raman spectroscopy, laser-induced breakdown spectroscopy (LIBS), and echelle-based simultaneous ICP-OES, where imaging an entire two-dimensional order-sorted spectrum onto one chip is the whole point of the optical design. CIDs, though a smaller commercial niche today, remain valued in process-monitoring and metallurgical ICP-OES applications where a few very bright matrix lines (Fe, major alloy constituents) must be measured alongside trace dopants in the same transient plasma burst — the non-destructive, variable-integration readout is tailor-made for exactly that combination. In biological and clinical mass spectrometry, related charge-integration and electron-multiplier hybrids (e.g., discrete-dynode electron multipliers in ICP-MS, and time-to-digital converters in time-of-flight instruments) inherit the same Poisson-statistics and gain-variance considerations developed here, linking detector physics from optical atomic spectroscopy directly to trace elemental mass spectrometry used in Inductively Coupled Plasma Spectrometry and beyond.

Ask the tutor: How would the noise budget and detection-limit formulas change for a hybrid CMOS active-pixel sensor used in newer ICP-OES instruments, compared with the classic CCD and CID architectures covered here?

The physical picture

A photomultiplier tube begins with a photocathode: a thin semiconductor film (bialkali Sb-Rb-Cs, or Ag-O-Cs for red-extended response) deposited on the inside of an evacuated glass envelope. An incoming photon of energy $h\nu$ ejects a photoelectron via the photoelectric effect, provided $h\nu$ exceeds the material's work function. That single electron is then accelerated by a potential difference of roughly 100 V toward the first dynode, a curved electrode coated with a secondary-emissive material (BeO, GaP, or Cs $_3$ Sb). On impact, the primary electron liberates several secondary electrons — the dynode gain $\delta$ , typically 3–6 per stage. This process repeats through 9–12 dynodes, each held at successively higher potential by a resistive voltage-divider network, so that the single photoelectron becomes an avalanche of $10^5$ – $10^7$ electrons by the time it reaches the anode, all within about 5–20 ns. The anode current pulse is either integrated as an analog photocurrent (the traditional mode) or, in dim-light applications, counted as discrete pulses in photon-counting mode, where each detected photon produces a resolvable pulse above a discriminator threshold. A charge-coupled device (CCD) instead uses a two-dimensional array of metal-oxide-semiconductor (MOS) capacitors fabricated on a doped silicon substrate. A photon absorbed in the depletion region promotes an electron from the valence to the conduction band, creating an electron–hole pair; the electron is trapped in the local potential well beneath a gate electrode, and holes are swept away. Charge accumulates linearly with exposure time and photon flux until the well nears its full-well capacity (typically $10^5$ – $10^6$ electrons per pixel for scientific-grade CCDs). Readout is destructive and sequential: at the end of integration, charge packets are shifted row by row into a serial register and then clocked out pixel by pixel to an on-chip amplifier and analog-to-digital converter (ADC). Because readout empties the whole array, a CCD cannot be queried mid-exposure without losing the accumulated signal — a serious limitation for the very different dynamic ranges of major and trace emission lines in atomic spectra. A charge-injection device (CID) solves exactly that problem. Each CID pixel has two capacitively coupled electrodes so the stored charge can be sensed non-destructively — by measuring the change in voltage induced when charge is shifted between the two electrodes within the pixel — without removing it from the well. This allows random-access readout: any individual pixel, or any subset of pixels, can be interrogated at any time during the integration, as many times as desired, and the accumulated charge is only actually cleared ("injected" into the substrate) when the experimenter chooses. In atomic emission work this means a bright ionic line and a faint trace-element line on the same array can each be read out with an integration time individually matched to their intensity — a capability with no CCD equivalent, since a CCD's single global readout forces one compromise integration time for the whole chip (unless multiple non-destructive reads are averaged, a technique CCDs also use but less flexibly).

Figure

Photomultiplier tube: photon in, amplified electron pulse out

Figure

CCD vs CID pixel architecture and readout timing

The core ideas

Photomultiplier gain. For a dynode chain of $N$ stages, each with secondary-emission ratio $\delta$ (the average number of secondary electrons ejected per incident electron), the overall current gain is

G = \delta^{N}

where:

$G$ = overall electron multiplication factor (dimensionless), the ratio of anode current pulse charge to photocathode photoelectron charge
$\delta$ = secondary emission coefficient per dynode stage, itself a function of interstage voltage $V_d$ roughly as $\delta \propto V_d^{\,k}$ with $k \approx 0.7$ – $0.8$
$N$ = number of dynode stages (commonly 9–12)

Reading: because $\delta$ itself grows with the interstage voltage, the total PMT gain is an extremely steep function of the applied high voltage — doubling $V_d$ can increase $G$ by an order of magnitude or more, which is why PMT power supplies must be stabilized to millivolt-level precision.

Anode signal and shot-noise-limited SNR. The mean anode current from a steady photon flux is

i_a = G\, e\, \Phi\, \eta

where:

$i_a$ = mean anode current (A)
$e$ = electron charge, $1.602\times10^{-19}$ C
$\Phi$ = photon arrival rate at the photocathode (photons s $^{-1}$ )
$\eta$ = photocathode quantum efficiency (photoelectrons per incident photon, dimensionless, typically 0.1–0.3 in the visible)

The number of photoelectrons collected in an integration time $t$ is $n_{pe} = \eta \Phi t$ , a Poisson-distributed random variable, so the photon-shot-noise-limited signal-to-noise ratio is

\mathrm{SNR} = \frac{n_{pe}}{\sqrt{n_{pe}}} = \sqrt{n_{pe}} = \sqrt{\eta \Phi t}

where:

$\mathrm{SNR}$ = ratio of mean signal to its standard deviation, assuming Poisson (shot-noise) statistics dominate over dark current and readout noise
the square-root dependence means doubling SNR requires quadrupling either the flux or the integration time

Reading: this equation is the ultimate floor for any photon-counting detector, PMT or array; everything else — dark counts, dynode statistics, read noise — only adds noise on top of this irreducible Poisson term, never removes it.

Array detector signal-to-noise ratio. For a CCD or CID pixel collecting $S$ photoelectrons of signal, with dark current contributing $D$ electrons and read noise $R$ electrons (rms) per readout, the total noise combines in quadrature (each independent noise source is uncorrelated, so variances add):

\mathrm{SNR} = \frac{S}{\sqrt{S + D + R^{2}}}

where:

$S$ = signal electrons accumulated during integration (Poisson variance $=S$ )
$D$ = dark-current electrons accumulated over the same integration time (Poisson variance $=D$ )
$R$ = read noise per readout event, root-mean-square electrons (Gaussian, added as $R^2$ since it is already a standard deviation, not a Poisson mean)

Reading: at high signal levels the $S$ term dominates and $\mathrm{SNR}\to\sqrt{S}$ (photon-shot-noise limited, matching the PMT expression above); at very low signal the $R^2$ term dominates, and read noise — not photon statistics — sets the practical detection limit, which is exactly why cooled, low-read-noise CCDs matter for trace analysis.

Derivation ADynode-chain gain and its variance — why PMT pulse-height distributions are broad

Goal: show that the overall PMT gain is $G=\delta^N$ and that the relative variance of the output pulse charge is dominated by fluctuations at the first dynode, not the later ones — the key fact behind PMT pulse-height spectra and the choice of high first-dynode voltage.

Step 1 — single-stage multiplication. A single photoelectron striking dynode 1 produces a random number $k_1$ of secondary electrons, with mean $\langle k_1\rangle = \delta$ and variance $\sigma_1^2$ (secondary emission is well modeled as a Poisson-like process, so to first approximation $\sigma_1^2 \approx \delta$ , though real dynodes show slightly super-Poissonian statistics; a common empirical factor is $\sigma_1^2 = b\,\delta$ with $b\approx 1$ – $1.5$ ). Each of those $k_1$ electrons then independently produces a further generation at dynode 2, and so on.

Step 2 — mean output after N stages. By induction, the mean number of electrons after stage $n$ is $\langle k_n\rangle = \delta^{\,n}$ , because each stage multiplies the expected population by $\delta$ regardless of the details of the distribution:

\langle k_N \rangle = \delta \cdot \delta \cdots \delta \;(N \text{ times}) = \delta^{N} = G

This confirms the gain formula stated above purely from the multiplicative structure of cascaded amplification.

Step 3 — propagation of variance through the cascade. For a branching (cascade multiplication) process, the relative variance of the final output, $\sigma_N^2/\langle k_N\rangle^2$ , obeys the recursive relation (a standard result for cascaded random multiplication, sometimes called the Burgess variance theorem):

\frac{\sigma_N^2}{\langle k_N\rangle^2} = \frac{\sigma_1^2}{\delta^2} + \frac{1}{\delta}\left(\frac{\sigma_2^2}{\delta^2} + \frac{1}{\delta}\Big(\cdots\Big)\right)

Expanding this geometric-like series with $\sigma_n^2 \approx \delta$ (Poisson single-stage statistics) for every stage gives

\frac{\sigma_N^2}{\langle k_N\rangle^2} \approx \frac{1}{\delta}\left(1 + \frac{1}{\delta} + \frac{1}{\delta^2} + \cdots\right) = \frac{1}{\delta}\cdot\frac{1}{1-1/\delta} = \frac{1}{\delta - 1}

Step 4 — interpretation. The relative variance of the fully amplified pulse converges to $1/(\delta-1)$ , a constant set almost entirely by the first dynode's gain $\delta$ , because each subsequent stage's fractional contribution is suppressed by an extra factor of $1/\delta$ . For a typical $\delta \approx 4$ –5, the relative standard deviation of the single-photoelectron pulse-height distribution is $\sqrt{1/(\delta-1)} \approx 0.5$ –0.6 — a broad, roughly 50% spread — which is exactly the characteristic single-photoelectron peak shape seen in PMT pulse-height spectra, and the reason instrument designers run the first dynode at unusually high voltage (to push $\delta_1$ as high as possible and narrow this dominant term).

Go deeper · Derivation B · 1 minDetection limit from the Poisson-plus-read-noise budget

Derivation BDetection limit from the Poisson-plus-read-noise budget

Goal: derive an expression for the minimum detectable analyte concentration $c_L$ in terms of blank signal, blank noise, and calibration sensitivity, then show how the array SNR expression feeds directly into it — connecting the microscopic noise sources to the macroscopic figure of merit reported on every spectrometer datasheet.

Step 1 — the IUPAC/analytical convention. The limit of detection is conventionally defined as the analyte concentration giving a net signal equal to $k$ (commonly $k=3$ ) times the standard deviation of the blank signal, $s_{bl}$ :

S_L - \bar{S}_{bl} = k\, s_{bl}

where $S_L$ is the signal at the detection limit and $\bar S_{bl}$ the mean blank signal.

Step 2 — introduce calibration sensitivity. If the calibration curve is linear over the relevant range with slope $m$ (sensitivity, signal units per unit concentration), then

S_L - \bar{S}_{bl} = m\,c_L

Combining with Step 1,

c_L = \frac{k\, s_{bl}}{m}

Step 3 — express $s_{bl}$ using the detector noise budget. The blank signal itself is a small photoelectron count $S_{bl}$ sitting on dark current $D$ and read noise $R$ , so by the array SNR expression derived above, the noise (not signal) on the blank is

s_{bl} = \sqrt{S_{bl} + D + R^{2}}

For a PMT operated in analog mode, replace $D$ with the dark-current-equivalent photoelectron count and $R$ with amplifier/electrometer noise; the same algebra applies.

Step 4 — assemble and interpret. Substituting,

c_L = \frac{k\sqrt{S_{bl} + D + R^{2}}}{m}

Two limiting regimes fall directly out of this expression. When $R^2 \gg S_{bl}+D$ (read-noise-limited, typical of a warm, fast-readout array), $c_L \propto R/m$ : the detection limit is fixed by electronics and can only be improved by better electronics or higher sensitivity $m$ , not by integrating longer. When $S_{bl}+D \gg R^2$ (shot-noise or dark-current limited, typical of cooled CCDs or PMTs at long integration), $c_L \propto \sqrt{S_{bl}+D}\,/m$ , and because $S_{bl}$ and $D$ both grow linearly with integration time $t$ while $m$ (proportional to accumulated analyte signal) also grows linearly with $t$ , $c_L \propto 1/\sqrt{t}$ — the familiar rule that detection limits improve with the square root of integration time, but only once you are past the read-noise floor.

Worked examples

Example 1 — Foundational: PMT anode current and photon-counting rate. A PMT with photocathode quantum efficiency $\eta = 0.22$ at 589 nm (sodium D line) has $N = 10$ dynodes, each with secondary emission coefficient $\delta = 4.2$ . The photocathode receives a steady photon flux $\Phi = 5.0\times10^{6}$ photons s $^{-1}$ from a flame emission source. Overall gain:

G = \delta^{N} = 4.2^{10}

Compute stepwise: $4.2^2 = 17.64$ ; $4.2^4 = 17.64^2 = 311.2$ ; $4.2^5 = 311.2\times4.2 = 1307$ ; $4.2^{10} = 1307^2 = 1.708\times10^{6}$ .

G \approx 1.71\times10^{6}

Mean anode current:

i_a = G\,e\,\eta\,\Phi = (1.71\times10^{6})(1.602\times10^{-19}\ \text{C})(0.22)(5.0\times10^{6}\ \text{s}^{-1})

Multiply progressively: $1.71\times10^6 \times 1.602\times10^{-19} = 2.740\times10^{-13}$ C. Then $2.740\times10^{-13}\times 0.22 = 6.027\times10^{-14}$ . Finally $6.027\times10^{-14}\times 5.0\times10^{6} = 3.01\times10^{-7}$ .

i_a \approx 3.0\times10^{-7}\ \text{A} = 300\ \text{nA}

This is a comfortably measurable current with a simple current-to-voltage (transimpedance) amplifier — no photon counting needed at this flux. Photon-shot-noise SNR for a 1.0 s integration: photoelectron count $n_{pe} = \eta\Phi t = 0.22\times5.0\times10^{6}\times1.0 = 1.10\times10^{6}$ photoelectrons.

\mathrm{SNR} = \sqrt{n_{pe}} = \sqrt{1.10\times10^{6}} \approx 1.05\times10^{3}

A shot-noise-limited relative standard deviation of about $1/1049 = 0.095\%$ — excellent precision, consistent with why flame and ICP emission measurements with adequate photon flux routinely achieve sub-percent RSD. Limiting-case check: if the dynode voltage were reduced so that $\delta \to 1$ (no multiplication), $G\to 1^{10}=1$ and $i_a$ collapses to the bare photocathode current of $\eta e \Phi \approx 1.76\times10^{-13}$ A — far below what any practical electrometer can resolve without excessive averaging, which is precisely why PMTs, not bare photodiodes, are required for weak-line atomic emission and absorption work.

Example 2 — Applied: comparing PMT-sequential and CID-simultaneous ICP-OES detection limits. An ICP-OES method for manganese at 257.61 nm is run two ways on the same plasma and optics train, differing only in the detector. Configuration A (scanning PMT): the monochromator dwells on the Mn line for $t=2.0$ s. The blank (pure matrix, no Mn) gives a mean photoelectron count $S_{bl} = 4.0\times10^{4}$ with negligible dark counts and negligible amplifier noise (PMT dark counts at this wavelength and cooled housing are $D\approx 50$ electron-equivalents, negligible here). Sensitivity from a calibration curve is $m = 8.0\times10^{4}$ counts per (mg L $^{-1}$ ). Blank noise (shot-noise dominated since $D, R^2 \ll S_{bl}$ ):

s_{bl} = \sqrt{S_{bl}} = \sqrt{4.0\times10^{4}} = 2.00\times10^{2}\ \text{counts}

Detection limit at $k=3$ :

c_L = \frac{k\,s_{bl}}{m} = \frac{3\times2.00\times10^{2}}{8.0\times10^{4}\ \text{counts}/(\text{mg L}^{-1})} = 7.5\times10^{-3}\ \text{mg L}^{-1}

Configuration B (CID array, same integration time, but read noise now matters): the same optical signal falls on a CID pixel, but the shorter effective single-pixel light-gathering geometry (an echelle image is spread over more resolution elements) gives a smaller $S_{bl} = 1.6\times10^{4}$ electrons, plus dark current $D = 2.0\times10^{2}$ electrons and read noise $R = 25$ electrons (rms) per non-destructive read, with sensitivity accordingly scaled to $m = 3.2\times10^{4}$ counts per (mg L $^{-1}$ ) (same fraction of total light, smaller absolute pixel signal).

s_{bl} = \sqrt{S_{bl} + D + R^{2}} = \sqrt{1.6\times10^{4} + 2.0\times10^{2} + 625}

s_{bl} = \sqrt{1.6825\times10^{4}} = 1.297\times10^{2}\ \text{electrons}

c_L = \frac{3\times1.297\times10^{2}}{3.2\times10^{4}} = 1.216\times10^{-2}\ \text{mg L}^{-1} \approx 1.2\times10^{-2}\ \text{mg L}^{-1}

Interpretation: the PMT configuration gives a somewhat better single-line detection limit (7.5 µg L $^{-1}$ vs. 12 µg L $^{-1}$ ) because it collects more total light onto one large-area photocathode and adds essentially no electronic noise, whereas the array pixel, though only modestly noisier, splits the same optical throughput across many resolution elements and adds a genuine read-noise term. In practice this per-line penalty of roughly a factor of 1.6 in detection limit is precisely why simultaneous CID/CCD instruments are called "compromise" detectors — but the array configuration measures all 40+ other elements in the same 2.0 s exposure, while the PMT would need 40 separate 2.0 s dwells (80 s total, plus grating-move overhead), so total analysis throughput favors the array by well over an order of magnitude for multi-element work. Limiting-case check: if read noise $R\to 0$ and dark current $D\to 0$ for the array (an idealized, perfectly cooled, zero-noise sensor), $s_{bl}\to\sqrt{S_{bl}} = \sqrt{1.6\times10^4}=126.5$ , giving $c_L = 3\times126.5/3.2\times10^4 = 1.19\times10^{-2}$ mg L $^{-1}$ — barely different from the computed 1.2×10⁻² mg L⁻¹ value above, confirming that in this example the array is already close to shot-noise-limited and the modest detection-limit penalty relative to the PMT is mostly a light-collection-per-pixel effect, not a read-noise artifact.

Example 3 — Exam/competition level: choosing integration strategy under a fixed photon budget, and the crossover between read-noise-limited and shot-noise-limited regimes. A CCD pixel monitoring a weak emission line has dark current $D_r = 15$ electrons s $^{-1}$ (already dominated by thermal generation despite cooling) and a fixed read noise $R = 8$ electrons rms per readout (independent of exposure time, since it is generated once per readout event, not once per unit time). The photon signal arrives at a rate $S_r = 40$ electrons s $^{-1}$ at the pixel (post quantum efficiency). Full-well capacity is $W = 1.2\times10^{5}$ electrons. (a) Find the exposure time $t^{*}$ at which shot noise from signal-plus-dark equals the read noise contribution — the crossover from read-noise-limited to shot-noise-limited operation. Signal and dark electrons accumulated after time $t$ : $S(t) = S_r t$ , $D(t)=D_r t$ . Shot-noise variance from these Poisson sources is $S(t)+D(t) = (S_r+D_r)t$ . Set this equal to $R^2$ :

(S_r + D_r)\,t^{*} = R^{2}

t^{*} = \frac{R^{2}}{S_r+D_r} = \frac{8^2}{40+15} = \frac{64}{55} = 1.164\ \text{s}

So for exposures shorter than about 1.16 s, read noise dominates the noise budget; for longer exposures, Poisson (shot + dark) noise dominates. (b) Find the maximum single-exposure time before the pixel saturates (reaches full well), and hence the maximum SNR obtainable from a single, unsaturated exposure.

t_{sat} = \frac{W}{S_r+D_r} = \frac{1.2\times10^{5}}{55} = 2.182\times10^{3}\ \text{s}

At this exposure, signal electrons $S(t_{sat}) = 40\times2182 = 8.727\times10^4$ and dark electrons $D(t_{sat}) = 15\times2182 = 3.273\times10^4$ ; check the sum $8.727\times10^4+3.273\times10^4 = 1.200\times10^5 = W$ ✓ (consistent by construction).

\mathrm{SNR}_{max} = \frac{S(t_{sat})}{\sqrt{S(t_{sat})+D(t_{sat})+R^2}} = \frac{8.727\times10^{4}}{\sqrt{1.200\times10^{5}+64}}

\mathrm{SNR}_{max} = \frac{8.727\times10^{4}}{\sqrt{1.20006\times10^{5}}} = \frac{8.727\times10^{4}}{346.4} = 251.9

(c) Now suppose total available observing time is fixed at $T = 6.0\times10^{3}$ s and the pixel must be read out (incurring a fresh $R=8$ e⁻ hit each time) every time it is emptied. Compare one long exposure of duration $t_{sat}=2182$ s repeated $n = T/t_{sat} \approx 2.75$ times (round to $n=3$ separate full exposures, each shorter than $t_{sat}$ to fit exactly, i.e. $t = T/3 = 2000$ s) against co-adding many short, read-noise-dominated exposures of $t=1.0$ s each ( $n=6000$ reads). *Three long exposures, $t=2000$ s each:* per-exposure signal $S=40\times2000=8.0\times10^4$ , dark $D=15\times2000=3.0\times10^4$ . Total after co-adding 3 frames: $S_{tot}=2.4\times10^5$ , $D_{tot}=9.0\times10^4$ , and total read-noise variance $= n R^2 = 3\times64=192$ (read noise adds in quadrature once per readout, and there are 3 readouts).

\mathrm{SNR} = \frac{2.4\times10^{5}}{\sqrt{2.4\times10^{5}+9.0\times10^{4}+192}} = \frac{2.4\times10^{5}}{\sqrt{3.3019\times10^{5}}} = \frac{2.4\times10^{5}}{574.6} = 417.7

Six thousand 1.0 s exposures co-added: per-exposure signal $S=40$ , dark $D=15$ ; totals $S_{tot}=40\times6000=2.4\times10^5$ electrons, $D_{tot}=15\times6000=9.0\times10^4$ electrons (identical totals, since total exposure time is the same $6.0\times10^3$ s either way), but now read-noise variance accumulates $n=6000$ times:

n R^{2} = 6000\times 64 = 3.84\times10^{5}

\mathrm{SNR} = \frac{2.4\times10^{5}}{\sqrt{2.4\times10^{5}+9.0\times10^{4}+3.84\times10^{5}}} = \frac{2.4\times10^{5}}{\sqrt{7.14\times10^{5}}} = \frac{2.4\times10^{5}}{845.0} = 284.0

Interpretation: for identical total signal and dark-current accumulation, coadding many short reads (284.0) gives markedly worse SNR than a few long exposures (417.7), purely because each readout re-injects a full $R^2=64$ e⁻² of variance. This is the rigorous, numerical justification for the standard practice in low-light array spectroscopy (and imaging) of using the longest single exposure the full well and background will allow, rather than many short ones — and it is exactly the argument used to justify a CID's or CCD's minimum practical read count in trace ICP-OES work. Limiting-case check: if read noise $R\to 0$ , both strategies give identical SNR — $\mathrm{SNR}=S_{tot}/\sqrt{S_{tot}+D_{tot}} = 2.4\times10^5/\sqrt{3.3\times10^5}=417.7$ regardless of how the total time is divided — confirming that the entire penalty for many short exposures traces to the $nR^2$ term, exactly as the algebra shows, and vanishes when $R=0$ .

Limiting cases, scope & common traps

Regime / case	Behavior	Why
$\delta \to 1$ (no dynode gain)	$G\to1$ ; PMT signal collapses to bare photocathode current, unusable at low flux	multiplication is the entire reason PMTs beat photodiodes for dim atomic lines
Read-noise-limited array ( $R^2 \gg S,D$ )	$c_L \propto R/m$ ; longer integration does not help	must reduce $R$ (better ADC/preamp) or increase sensitivity $m$ , not integration time
Shot-noise-limited array or PMT ( $S,D \gg R^2$ )	$c_L \propto \sqrt{t}/t = 1/\sqrt t$	classic "detection limit improves with square root of time" rule, valid only past the read-noise floor
CCD near full well	risk of blooming — charge overflows into neighboring pixels/columns, corrupting adjacent wavelengths	anti-blooming drains sacrifice some full well and linearity to prevent this
CID at high signal	pixel individually reset/read at shorter time before saturating, no blooming into neighbors	non-destructive per-pixel access is architecturally immune to blooming
Sequential PMT, many elements	throughput poor (each element needs its own dwell + grating move) but each measurement gets full photometric attention and superb dynamic range per line	best when element list is short or when the very lowest detection limits on a handful of lines are the priority
Simultaneous array, many elements	one exposure captures the whole spectral window; huge throughput gain for multi-element work	best for routine multi-element screening (environmental, industrial QC) even at a modest per-line detection-limit cost
PMT in photon-counting mode	best sensitivity at very low flux, since pulse discrimination rejects sub-threshold noise pulses	requires flux low enough that pulses don't pile up (dead-time correction needed)

The chemist behind it

The photomultiplier tube's essential architecture — a photoemissive cathode followed by cascaded secondary-emission stages — was developed at RCA in the 1930s, principally through the work of Vladimir Zworykin along with colleagues including Joseph Rajchman and George Morton, building on Zworykin's earlier iconoscope and photocell research for television. Zworykin, a Russian-American engineer trained under Boris Rosing in St. Petersburg before emigrating to the United States, was primarily driven by the demands of practical television camera and low-light imaging technology, not analytical chemistry — yet the PMT his group perfected became, within two decades, the single most important detector in flame and atomic absorption spectroscopy, and it remains in routine use in modern instruments to this day, more than eighty years after its invention. The later shift toward multichannel solid-state detection owes its analytical-chemistry translation largely to workers such as James Winefordner and collaborators at the University of Florida, who through the 1970s–1990s systematically evaluated and championed CID and CCD arrays for atomic emission spectrometry, publishing the noise-budget framework (signal, dark current, read noise combining in quadrature) that underlies essentially every modern array detector specification sheet. Winefordner's group's work on detector-limited figures of merit — bridging solid-state physics and quantitative trace analysis — established the vocabulary ("read-noise-limited," "shot-noise-limited," "charge-transfer efficiency") that analytical spectroscopists still use to specify and compare instruments.

Origins · 1 minVladimir K. Zworykin and how this was discovered

Vladimir K. Zworykin

1888–1982 · RCA Laboratories, Princeton, New Jersey, developer of the modern photomultiplier and iconoscope television camera tube

History

- 1887 — Heinrich Hertz observes the photoelectric effect, the physical basis of every photocathode.

- 1905 — Einstein's photon quantization explanation of the photoelectric effect,

E=h\nu-\phi

, underlying photocathode response.

- 1934–1936 — Zworykin, Rajchman, and RCA colleagues develop the first practical multi-dynode photomultiplier tube.

- 1969 — Boyle and Smith at Bell Labs invent the charge-coupled device, initially conceived as a memory element.

- 1970s–1990s — CID and CCD arrays adapted and characterized for simultaneous atomic emission spectrometry, notably by Winefordner and coworkers, establishing the modern noise-budget framework for array detectors.

Why does a PMT need such a high, stable voltage supply?

Because gain scales as $G=\delta^N$ and the per-stage secondary emission coefficient $\delta$ itself increases with interstage voltage (roughly $\delta\propto V_d^{0.7\text{–}0.8}$ ), the overall gain is an extremely steep, nonlinear function of the total applied high voltage, often scaling as $V_{total}^{6}$ – $V_{total}^{8}$ across a typical 9–12 stage tube. A mere 1% drift in supply voltage can shift gain by several percent, directly corrupting quantitative absorbance or emission-intensity measurements. This is why PMT high-voltage supplies for analytical instruments are regulated far more tightly (often to millivolt precision) than supplies for other lab equipment, and why manufacturers specify gain stability, not just absolute gain, as a key performance parameter.

What exactly is "blooming" and why don't CIDs suffer from it?

Blooming occurs when a CCD pixel's potential well fills past its full-well capacity during integration; excess photogenerated charge has nowhere to go and spills into neighboring pixels along the charge-transfer direction, corrupting the signal at nearby wavelengths — a serious problem when a strong matrix or major-element line sits near a faint trace line in an emission spectrum. CCDs mitigate this with anti-blooming drain structures that shunt excess charge to the substrate, at some cost to fill factor and linearity. CIDs largely avoid the problem because each pixel can be individually and non-destructively read and reset well before saturation, and because the two-electrode CID pixel architecture does not require charge to be physically shifted through neighboring pixels during readout, removing the principal blooming pathway.

Is photon-counting mode better than analog current mode for a PMT?

Photon-counting mode — discriminating and counting individual anode pulses above a threshold — gives superior sensitivity at very low light levels because it rejects sub-threshold electronic noise pulses that would otherwise be integrated into an analog current measurement, effectively achieving a cleaner Poisson-statistics signal. Its limitation is pulse pile-up: at high photon flux, individual pulses begin to overlap in time and the counter undercounts, requiring a dead-time correction, and eventually analog current mode becomes both simpler and more accurate. Most modern instruments switch modes (or use logarithmic/dual-mode electrometers) depending on measured light level, choosing counting for trace work and analog integration for major-constituent lines.

How do detection limits reported for simultaneous versus sequential ICP-OES actually compare in practice?

Manufacturer-published detection limits for well-optimized simultaneous CID/CCD-based instruments are typically within a factor of 1.5–3 of dedicated scanning-PMT instruments for the same element and line, consistent with the noise-budget comparison worked through in this article's examples. The gap has narrowed substantially since the 1990s as CCD/CMOS read noise and dark current have improved and as echelle optics have made better use of the full 2-D detector area. For routine multi-element panels, the modest per-line detection-limit penalty of simultaneous detection is almost always outweighed by the throughput gain, which is why simultaneous instruments now dominate commercial ICP-OES sales despite sequential instruments retaining a niche for ultratrace, few-element applications.

Why is the relative variance of a PMT's single-photoelectron pulse dominated by the first dynode?

Because each dynode stage contributes a fractional variance term that is suppressed by an additional factor of $1/\delta$ relative to the previous stage in the cascade, the sum of contributions from the second, third, and later dynodes forms a rapidly converging geometric series, while the first dynode's contribution enters unsuppressed. The closed-form result, relative variance $\approx 1/(\delta-1)$ , is set almost entirely by the first stage's own gain and statistics. This is why photomultiplier designers deliberately run the first dynode at a higher-than-average interstage voltage — a larger $\delta_1$ narrows the single-photoelectron pulse-height distribution, improving pulse-height discrimination in photon-counting applications.

Can a single instrument use both a PMT and an array detector?

Yes — many practical spectrometers are hybrids. Some ICP-OES systems dedicate an array to the bulk simultaneous spectral window while adding a scanning PMT channel (or a movable exit slit and secondary photomultiplier) for a small number of especially demanding trace lines needing the best possible detection limit or the widest linear dynamic range. Double-monochromator and dual-detector AAS/ICP-OES combination instruments similarly pair PMTs for absorption channels with arrays for simultaneous emission background monitoring. The underlying principle, amplify-then-read versus integrate-then-read, is chosen independently for each measurement channel based on which figure of merit — dynamic range and per-line sensitivity, or throughput and simultaneity — matters most for that specific measurement.

What role does quantum efficiency play separately from gain?

Quantum efficiency $\eta(\lambda)$ sets how many photoelectrons (PMT) or electron–hole pairs (array) are generated per incident photon; gain $G$ (PMT) or full-well capacity and readout amplification (array) then determines how faithfully and with how much added noise that initial photoelectric event is converted into a measurable electrical signal. A detector can have superb quantum efficiency but poor overall performance if gain is too low (signal lost in downstream amplifier noise) or too high (saturation, nonlinearity); conversely a modest quantum efficiency with excellent gain and readout electronics, as in a well-designed PMT, can outperform a higher-QE but noisier device at low light levels. Comparing detectors on quantum efficiency alone, without considering the full noise budget derived in this article, is a common and consequential error.

Detectors for Atomic Spectroscopy

Where this lives today

The physical picture

The core ideas

Worked examples

Limiting cases, scope & common traps

The chemist behind it

Before this