Experiment: Calibrating the Densitometer in Both Modes

Purpose

An object becomes an instrument on the day somebody measures what it does wrong. This session does that, twice — once for the transmission path and once for the reflection head — and ends with a certificate that says, band by band, which of its own readings the instrument may be trusted for.

The hypothesis. That the densitometer built over the last two pages reports diffuse transmission densities that agree with a calibrated reference to within a stated tolerance across 0 to 3.0 D, that the departures from it have the shapes the arithmetic predicts, and that its repeatability is limited by mechanics rather than by electronics.

The control. The calibrated step wedge — a reference wedge — read step by step: an object whose densities were established by somebody else’s instrument and are not in question here. Every arm of the experiment holds the wedge fixed and varies exactly one thing about the way it is read. In reflection there is no equivalent control and the page says so plainly rather than inventing one — the white tile is a transfer standard, not a reference, and what it fixes is repeatability rather than truth.

The variable. One per arm, and never two. Arm 1 varies the density presented, holding everything else still, and produces linearity. Arm 2 varies nothing at all and produces repeatability. Arm 3 varies time from switch-on and produces drift. Arm 4 varies room temperature. Arm 5 varies the position of the aperture within a step. Arm 6 varies which of two light sources is switched on, and produces the one figure no datasheet in this course’s corpus can supply: the linearity of your own detector chain over the range you actually use.

You finish holding an uncertainty statement and an instrument certificate, and every quantitative claim in the seven later parts that read this instrument cites them.

!Every number on this page is a template, and the course says so once

The course has no measurements of its own reference build to publish. Nothing here is a result. The figures in the worked examples are arithmetic on stated assumptions, labelled as such wherever they appear, and they exist so that you can check your own working rather than so that you can copy an answer.

That is not a gap to be apologised for; it is what the page is for. A calibration you did not perform is worth nothing, because the quantity being measured is your instrument. Where the course could source a figure it gives it and cites it. Where it could not — the linearity of the detector chain, whether a wedge ages, the absolute reflectance of an uncertified tile — the page says so and turns the gap into something you measure.

Learning objectives

By the end of this session you should be able to:

• distinguish a dark reading from a zero and say when each is taken and how often;
• perform a two-point calibration in the form a commercial instrument uses, and say which of the two points drifts and why that decides the working routine;
• plot measured against reference density, read the residual rather than the correlation, and name the fault behind each of three residual shapes;
• separate the electronic component of repeatability from the mechanical one with two ten-reading runs;
• derive a warm-up rule from your own drift curve instead of copying one;
• predict the temperature sensitivity of your own chain from three datasheet lines, then measure it and say which prediction was wrong;
• measure detector linearity by superposition, using two light sources and no reference standard at all;
• anchor a reflection scale on a tile and a trap, quote a separate repeatability figure for it, and state in one sentence what your reflection densities may and may not be compared with;
• write an uncertainty statement that keeps Type A and Type B terms apart and states a figure per density band rather than one figure for the instrument.

Prerequisites

The densitometer electronics and firmware, built and passing its own tests E1 to E6, with the noise figure in counts recorded at air and at the top of the range.

The densitometer optical head, with its stray-light fraction and the ceiling that fraction sets. That ceiling is not measured again here; it is used, and half the results below are meaningless without it.

Calibrating the sensitometer, which established the method this page mirrors: a budget with its terms separated by axis, a stated rule for combining them, and a certificate that refuses four specific claims. Read its error-budget section again before you start. This page uses the same rule and does not re-argue it.

Measurement and uncertainty from Part II, which is where the course’s combination rule was ruled on.

Safety classification

Level A, throughout, with no raised step. The rubric’s criteria are met without argument: no chemistry is handled, nothing is heated, nothing is built or modified, the whole session runs from a certified USB supply at 5 V, and the only waste is the paper you write on.

What is not a hazard here, and why. This page produces no waste stream at all and handles no substance. There is no developer to splash, no fixer to collect, no dust to breathe, no vapour to extract, and no glove-selection problem, because there is nothing for a glove to keep off you. Saying so plainly matters more here than on a page with real hazards, because a session of reading numbers off a screen invites the assumption that a safety section is a formality — and the assessment that produced this paragraph is the same assessment that would have found a hazard if there had been one.

Nor is the electricity a hazard in the ordinary sense. Nothing exceeds 5 V; the analogue side runs at 3.3 V. Five volts across dry skin drives a current far below the threshold of sensation. Nothing mains is opened, built or modified anywhere in this part.

The only genuine risk in the session is to the equipment, and specifically to one object.

Hazards

Damage to the calibrated wedge, which is the experiment’s control. If the wedge changes, everything this page produces changes with it and you will not know. X-Rite’s instructions for their own transmission reference are the model, and they are specific: handle it at the edges only, because fingerprints or other foreign substances on the measurement area cause errors; do not attempt to dust or clean the surface with anything other than a soft camel-hair brush, because anything else may change the densities; and minimise change by storing it in a dark, cool, dry place. Treat your own wedge to the same three rules from today onwards.

Eye strain and a warm room, over three hours. This is a long session of small numbers. The instrument’s own specification depends on your not making arithmetic slips at hour three, so break between arms, and write each result down as it appears rather than at the end.

Handling the archived negative and the control strip. Both are irreplaceable in the sense that matters: they were made under conditions you cannot exactly repeat. Cotton-free handling by the edges, back into their sleeves between arms, and never on a bench where anything is being drunk.

The instrument’s own lamp, briefly. The emitter runs at a few milliamps behind a diffuser, and the head is closed for every reading. Do not look into the aperture with the head open and the lamp on out of curiosity; there is nothing to see and it is a habit worth not forming.

Required PPE

No personal protective equipment is specified for this session, and the reason is that no hazard here is addressed by wearing anything. There is no chemical, no projectile, no heat and no dust. Writing “none needed” would be a failure to do the assessment; what the assessment actually returns is that the controls for this page are procedural rather than protective — handling the wedge by its edges, closing the head before every reading, and writing numbers down when they appear.

Clean, dry hands are the one thing that matters, and they are a control for the wedge and the tile rather than for you. Skin oil on a measured face is a density change you will spend an evening chasing.

Ventilation

Ventilation is not among the controls for this session, because nothing in it produces a vapour, a dust or a fume. The instrument is solid-state, the samples are dry, and nothing is heated. An ordinary room is what the page assumes.

There is one environmental control that does matter and it is not ventilation but stability. A draught across the head changes its temperature, and arm 4 will show you what a few degrees is worth. Close the window during the drift and temperature arms, and note in the log if you cannot.

Materials

Item	Quantity	What it is for
Calibrated 21-step transmission wedge, with its certificate	1	The control. Stouffer’s T2120CC and T1530CC are the calibrated parts, compared with NIST Standard Reference Material 38120C on a densitometer conforming to ANSI PH2.19-1986; the T5100C is sold specifically for densitometer calibration
Uncalibrated wedge, if that is what you own	1	Everything on this page still works and produces a relative scale. See Alternative route
The archived Part XIII step-wedge negative	1	The cross-check. It has been waiting since Part XIII for an instrument to read it
A Part XIV control strip	1	The second cross-check, and the link between the two instruments
White reference tile	1	The reflection scale’s top anchor
Black trap	1	The reflection scale’s zero
Two prints, glossy and matt, of similar visual density	1 pair	The reflection geometry check, from the optical head page
One print carrying a full tonal scale, dry	1	The reflection repeatability and superposition arms
Squared paper, or a spreadsheet	—	The residual plots. Draw at least one by hand: a residual you have plotted yourself is one you will recognise on sight for years

Chemicals

This page handles no chemical at all, and the section says so rather than inventing a row for the sake of a table. There is no solution to mix, none to dilute, none to store and none to discard.

What is present, and stays on the bench without being handled, is worth inventorying because it is the subject of the measurement rather than a reagent in it: three pieces of processed photographic film — the wedge, the archived negative and the control strip — each carrying an image of developed silver in hardened gelatin on a plastic base, and two or three sheets of processed photographic paper. All are dry, all are fully fixed and washed, and none of them presents a route of exposure while it is being read. The silver in them is metallic and locked in a gelatin layer; it is the reason for the page rather than a hazard on it.

Equipment

Item	For	Notes
The densitometer, both heads	Everything	With its label complete: aperture, plate thickness, gap, LED wavelength, drive current, tile serial, build date
Computer with a serial terminal, logging to a file	Every reading	Capture the whole session to one file. Do not retype numbers; retyping is where the errors come from
Room thermometer, reading to 0.5 °C	Arms 3 and 4	The one from Part II’s commissioning session, with its own calibration record
A second light source for arm 6	The superposition test	You already own it: the blue LED the optical head’s parts list told you to buy and fit later. Today is later
Spreadsheet	The residuals and the statistics	Anything that opens comma-separated values
Stopwatch or phone timer	The drift arm	Two-minute intervals over thirty minutes

Estimated cost

Cost band £, and it is the lowest band in the part because the session buys nothing. Everything on the Materials list was bought for an earlier page or made on one.

The one item that can cost real money is the calibrated wedge, and the decision about it was made before this page: an uncalibrated wedge produces a repeatable instrument with a relative scale, and a calibrated one produces a repeatable instrument with an absolute one. The planner records that it could confirm no United Kingdom price for a 21-step transmission wedge at all, calibrated or not, and that gap is the largest single unknown in the cost of this part.

Estimated consumables cost

Nothing in this session is consumed except paper and electricity. There is no chemistry, no film is exposed, no print is made, and every object on the Materials list goes back in its sleeve at the end in the state it arrived in.

Consumed	This session	Sourced price	Cost this session
Squared or plain paper for the residual plots	4 to 6 sheets	None. The planner carries no stationery line	—
Electricity, for a USB-powered instrument over three hours	under 5 Wh	None. The planner prices no utilities	—

No row carries a dated price, so no total is given and none should be inferred. This is as close to a free session as the course contains, and the reason is worth noticing: calibration is cheap and skipping it is expensive. Every figure the next seven parts quote costs one afternoon here, once.

Equipment stays out of the table on purpose. The wedge, the tile, the trap, the thermometer and the instrument are not consumed by being used, and a page that counted them would be reporting capital as though it were a running cost.

Waste streams

This session produces no chemical waste of any kind, because it handles no chemical. The only things leaving the bench are your notes, which go into the notebook rather than into a bin.

The one disposal question worth naming is the one that arrives years later: at end of life the instrument’s LED, photodiode, converter and microcontroller are waste electrical and electronic equipment and belong in that stream rather than in a household bin, whatever their size. Local regulation governs where that stream goes and it differs by jurisdiction — check your local regulations.

Alternative route

Nothing here needs a darkroom, a UV source or mains equipment, so the facility question does not arise. The question that does is what a reader without a calibrated wedge should do, and there is a real answer rather than an apology.

Run the whole page on an uncalibrated wedge and change one word in the certificate. Everything except the absolute scale survives: repeatability, drift, temperature sensitivity, the stray-light ceiling, the range trim, the linearity shape, the placement tolerance and both reflection figures are all internal measurements that need no reference at all. What you lose is the right to say that your 2.00 is anybody else’s 2.00.

That loss is smaller than it sounds and larger than it looks. Smaller, because Stouffer are explicit that calibrated and uncalibrated guides are the same product from the same batches, and that calibration consists of reading each step and recording the readings — so an uncalibrated wedge is not a worse object, it is the same object without a certificate. Larger, because the missing certificate is exactly what an absolute claim rests on, and a nominal 0.15 increment is a manufacturing intention rather than a measured value, as Part IX established and every page in this part inherits.

The honest form of the uncalibrated certificate is therefore: “Densities relative to a nominal 0.15 increment on wedge serial XXXX; absolute scale not established.” Write that, and every comparison you make between two of your own negatives is still valid, because a common scale factor cancels out of a difference. Only the comparisons with somebody else’s numbers are lost.

Preparation

The day before. Read the certificate that came with your wedge, if it has one, and type its twenty-one densities into a spreadsheet column. Check two of them against each other for a typing error by looking at the differences: they should all be near the nominal increment, and one that is not is either a real feature of the wedge or a slip of the fingers, and you want to know which before the session rather than during it.

One hour before. Put the instrument, the wedge, the tile and the prints in the room where the session will happen, and leave them. Everything in arm 4 is about temperature, and starting with a cold wedge from a cold cupboard puts a transient into arm 1 that you will spend an hour chasing.

At the bench. Room temperature written down. Instrument label read aloud and checked against the build record — aperture, gap, drive current, firmware version. Terminal open and logging to a file before the first reading, not after the third. A blank page headed with the date, the instrument’s serial and the wedge’s serial.

Then, and only then, switch the instrument on — because arm 3 measures the thirty minutes from switch-on and it cannot be measured retrospectively.

Procedure

Twelve arms, in this order. Arms 1 to 8 are transmission; 9 to 11 are reflection; arm 12 is the cross-check that ties the whole part back to Part XIII. Nothing later depends on your finishing all of them in one sitting, but arms 3 and 4 must not be interrupted once started.

The calibration in the order that makes each arm interpretable

1. 11 — Dark, then zeroLamp off for the dark, lamp on and stage empty for the zero. They are different readings taken at different moments and the difference is the whole of the next three hours.
2. 22 — Two-point calibrationCal Lo is the zero you just took; Cal Hi is one certified step. The slope that makes them agree is the instrument's scale, and it is one number.
3. 33 — Linearity, all 21 stepsRead every step as an unknown. Plot measured against certified, then plot the residual, which is where the information is.
4. 44 — Repeatability, twiceTen readings untouched, then ten with the film lifted and replaced. The difference between the two spreads is the mechanics.
5. 55 — Drift, thirty minutesOne step every two minutes from switch-on, zero taken once and never retaken. Produces the warm-up rule.
6. 66 — TemperatureThe same dense step at two room temperatures, against a prediction made from three datasheet lines before the measurement.
7. 77 — PlacementThe aperture walked across a step boundary in half-millimetre moves. Produces a placement tolerance rather than an anecdote.
8. 88 — SuperpositionTwo LEDs, three readings, no reference standard: the linearity of your own chain, which no datasheet in this course can give you.
9. 99 — Reflection anchorsWhite tile first, black trap second, in that order and re-done whenever the head is refitted.
10. 1010 — Reflection repeatability, three waysUntouched, sample replaced, head refitted. Three numbers, and the third is usually the largest.
11. 1111 — Reflection linearity, honestlySuperposition again, because there is no certified reflection scale at a home-lab price and the page will not pretend there is.
12. 1212 — The cross-checkRe-read Part XIII's archived negative and a Part XIV control strip. The first tells you what the interim method was worth; the second links the two instruments.

Each arm holds every variable fixed except one. That is the only reason a result from one of them can be read as a statement about the instrument rather than about the afternoon.

Arm 1 — Dark and zero, which are not the same reading

Run zero(). It takes a dark reading with the lamp off on both ranges, then switches the lamp on and takes an air reading with nothing on the stage. Record both, with the spread of each.

The two are answers to different questions. The dark is everything the chain reports when no light reaches it: the detector’s 7.5 mV single-supply pedestal, the op-amp’s input bias current through the feedback resistor, the photodiode’s own dark current, and the converter’s offset. It is a property of the electronics and it moves with temperature. The zero is what the lamp delivers through an empty stage, and it is a property of the lamp and the optics: it moves with drive current, with lamp temperature, with dust on the diffuser and with anything that changes the geometry.

Take the dark before every zero, and take the zero at the start of every session and again whenever anything at all has been touched. X-Rite’s manual is unambiguous about which of the two points a commercial instrument re-establishes most often, and why: their Quick CAL procedure exists to re-take the zero alone, because the zero is the major factor of drift over a period of time. Yours will behave the same way and arm 3 will show it.

Arm 2 — The two-point calibration, and which step to use

A commercial instrument does exactly two things to establish its scale. It takes Cal LO with all film removed — the zero you have just taken. Then it takes Cal HI on the marked step of a calibrated reference and enters that step’s certified value. Everything between the two is arithmetic.

Yours works the same way. Put the chosen certified step on the stage, close the head, and run calibrate(d_ref) with the certificate’s value for that step. The firmware reads the step, computes the raw density from the ratio of counts, and prints the slope that makes the two agree:

slope = D_certificate ÷ log₁₀(V_zero ÷ V_step)

The scale, from one certified step

D_certificate is the reference step’s certified density, V_zero the dark-corrected air reading and V_step the dark-corrected reading through the step. The slope should be within a per cent or two of 1.000, because a correctly built instrument computing a logarithm of a ratio has almost nothing left to be wrong about. A slope more than about ten per cent from unity is a fault, not a calibration, and the firmware says so — the usual causes are a stale zero, a wrongly entered certificate value, or a step read at the wrong place on the wedge.

Which step to calibrate on. The choice matters because a slope error grows in proportion to density, so the higher the calibration point, the tighter the whole scale is pinned. Against that, a point too close to your stray-light ceiling is a point contaminated by the floor. The commercial reference this page is modelled on runs from about 0.06 to 4.0 in five steps and uses the fourth — high, but not the highest. For a 21-step wedge with a 3.0 D ceiling, a step near 2.0 D is the equivalent choice, which on a nominal T2115 scale is step 14. Record which step you used; the certificate names it.

The check, and how often. From now on, every session starts by taking a zero and reading the calibration step. If it reads within 0.02 D of its certified value, the instrument is calibrated and you may work. If not, recalibrate before anything else. That tolerance is not the course’s invention: it is the check tolerance X-Rite specify for their own instrument against their own reference, and it is adopted here as a defensible starting point rather than derived. Their recommended interval under normal use is once a week; a home instrument that lives in a warmer, dustier, more mobile place than a printing works should not stretch that, and re-reading one step takes twenty seconds.

Arm 3 — Linearity, and reading the residual rather than the plot

Read all twenty-one steps as unknowns, with series(), pressing the read button at each. Take them in order from clear to dense and then once more from dense to clear; a systematic difference between the two passes is drift, and you have just measured it for free.

Now make two plots. The first is measured density against certified density, and it will look magnificent, because any instrument that is not broken produces a beautiful straight line on that plot and it hides everything. The information is in the second plot: the residual, measured minus certified, against certified.

Three residual shapes, and the fault behind each

• Stray light, f = 0.4 % — flat, then falls away
• Slope error, scale 0.90 — a straight line through the origin
• Offset, zero taken through base — a horizontal line off zero

Show the numbers behind this plot

Three traces of residual against certified density on a horizontal axis from 0 to 3.0 and a vertical axis from minus 0.35 to plus 0.15. The first, for a stray-light fraction of 0.4 per cent, sits almost exactly on zero up to about 1.0 density and then curves downwards ever more steeply, reaching minus 0.028 at 1.25, minus 0.107 at 1.85, minus 0.193 at 2.15 and minus 0.51 at 2.75; its distinguishing feature is that it is indistinguishable from a perfect instrument over the first third of the range. The second, for a scale error of ten per cent, is a straight line from minus 0.005 at 0.05 to minus 0.275 at 2.75, passing through the origin extended, so its error is proportional to the density and is already visible at low densities. The third, for a zero taken through film base of 0.10 rather than through air, is a horizontal line at plus 0.100 across the whole range, identical at every density. Two guide lines mark plus and minus 0.02, the linearity a commercial instrument specifies from 0 to 5.0 D. The three shapes are the point: a curve that only bends at the top is additive, a straight line through the origin is multiplicative, and a horizontal offset is a zero error, and each has a different cure.

Series	Certified density of the step	Residual: measured − certified
Stray light, f = 0.4 % — flat, then falls away	0.05	0.00
Stray light, f = 0.4 % — flat, then falls away	0.35	-0.00
Stray light, f = 0.4 % — flat, then falls away	0.65	-0.01
Stray light, f = 0.4 % — flat, then falls away	0.95	-0.01
Stray light, f = 0.4 % — flat, then falls away	1.25	-0.03
Stray light, f = 0.4 % — flat, then falls away	1.55	-0.06
Stray light, f = 0.4 % — flat, then falls away	1.85	-0.11
Stray light, f = 0.4 % — flat, then falls away	2.15	-0.19
Stray light, f = 0.4 % — flat, then falls away	2.45	-0.33
Stray light, f = 0.4 % — flat, then falls away	2.75	-0.51
Slope error, scale 0.90 — a straight line through the origin	0.05	-0.01
Slope error, scale 0.90 — a straight line through the origin	0.35	-0.04
Slope error, scale 0.90 — a straight line through the origin	0.65	-0.07
Slope error, scale 0.90 — a straight line through the origin	0.95	-0.10
Slope error, scale 0.90 — a straight line through the origin	1.25	-0.13
Slope error, scale 0.90 — a straight line through the origin	1.55	-0.15
Slope error, scale 0.90 — a straight line through the origin	1.85	-0.18
Slope error, scale 0.90 — a straight line through the origin	2.15	-0.21
Slope error, scale 0.90 — a straight line through the origin	2.45	-0.24
Slope error, scale 0.90 — a straight line through the origin	2.75	-0.28
Offset, zero taken through base — a horizontal line off zero	0.05	0.10
Offset, zero taken through base — a horizontal line off zero	0.35	0.10
Offset, zero taken through base — a horizontal line off zero	0.65	0.10
Offset, zero taken through base — a horizontal line off zero	0.95	0.10
Offset, zero taken through base — a horizontal line off zero	1.25	0.10
Offset, zero taken through base — a horizontal line off zero	1.55	0.10
Offset, zero taken through base — a horizontal line off zero	1.85	0.10
Offset, zero taken through base — a horizontal line off zero	2.15	0.10
Offset, zero taken through base — a horizontal line off zero	2.45	0.10
Offset, zero taken through base — a horizontal line off zero	2.75	0.10

Computed from the arithmetic of an additive floor, a multiplicative scale and an offset — not measured from any instrument. Plot your own residuals on these axes and the shape will tell you which of the three you have before you have run a single further test. The curve is drawn to show the shape, not measured from a real material. Your own materials will differ, and measuring them is what the sensitometry part of the course is for.

Three shapes, three completely different faults, and the whole diagnostic method of the break/fix page in one drawing. A residual that is flat and then falls away at the top is additive — stray light, ambient leakage, or a dark reading that was taken and then went stale. A residual that is a straight line through the origin is multiplicative — a wrong certificate value, a mis-set slope, a lamp at a different level than when the zero was taken. A residual that is a horizontal line away from zero is an offset — a zero taken through something.

Record the density above which your residual leaves the ±0.02 band. That density, not your ceiling, is where your certificate stops making a strong claim.

Arm 4 — Repeatability, twice, because there are two of them

Choose a mid-scale step. Run repeat(10) without touching anything: ten readings of an undisturbed sample. Then run repeat(10, lift=True), which waits for the read button between readings so that you can lift the arm, slide the wedge away, put it back against its fences and close the arm again.

Two standard deviations come out. The first is the electronics, and it should be close to the noise figure you measured as test E6. The second is the electronics plus the mechanics, and on almost every home-built instrument it is several times larger. The difference between them is the number that tells you where the next hour of work should go, and on a taped-together head that hour is always spent on the arm and the fences rather than on the amplifier.

Then repeat both at a dense step, near the top of your usable range. The electronic spread will grow — the density-per-count equation says it must — and the mechanical spread will not, because a millimetre of misplacement is worth the same fraction of the signal wherever you are on the scale. Which of the two dominates therefore changes with density, and your certificate has to say at which density each figure was taken.

Arm 5 — Drift, and writing your own warm-up rule

Switch everything off, wait until the head is at room temperature, and switch on. Immediately run drift(30, 120): it takes a zero once, then reads one mid-scale step every two minutes for thirty minutes and never re-zeroes.

The refusal to re-zero is the design of the experiment. Re-zeroing removes exactly the multiplicative drift being measured, and an arm that re-zeroed would report a magnificently stable instrument and tell you nothing at all.

Plot density against minutes, and separately plot the logged monitor column against minutes. Then read three things off them and write them in the certificate:

• the total excursion over the first ten minutes, in density;
• the residual slope after it settles, per ten minutes;
• whether the monitor moved with the sample or stayed flat, which says whether the drift is in the lamp or in everything else.

Your warm-up rule is then one sentence — “readings are taken no sooner than N minutes after switch-on” — and N is where your own curve goes flat. For scale rather than for comparison: a commercial instrument in a metal case specifies a two-minute warm-up and a zero stability of ±0.02 D per eight hours. A home-built head in a plastic box will not match either, and the number you need is not theirs but yours.

What a warm-up curve looks like, and what you read off it

• Reported density, zero taken once at t = 0
• The same step re-zeroed at each reading

Show the numbers behind this plot

Two traces against minutes from switch-on, on a vertical scale of reported density from 1.48 to 1.53. The first, taken with the zero established once at switch-on and never retaken, starts at 1.500 and climbs steadily, reaching 1.510 at ten minutes and then flattening, creeping only from 1.512 to 1.5143 over the remaining eighteen minutes. The excursion over the first ten minutes is 0.011 density and the residual slope afterwards is about 0.001 per ten minutes. The second trace, of the same step re-zeroed immediately before every reading, is flat at 1.500 within a thousandth throughout. A vertical guide at twelve minutes marks where the first curve flattens, which is the warm-up rule the instrument's certificate would state. The drawing's point is that the whole of the drift is multiplicative and lives in the zero, so it disappears entirely when the zero is retaken, and that an instrument which re-zeroes before every reading needs no warm-up rule for this term at all.

Series	Minutes from switch-on	Density reported for one unchanged step
Reported density, zero taken once at t = 0	0.00	1.500
Reported density, zero taken once at t = 0	2.00	1.503
Reported density, zero taken once at t = 0	4.00	1.506
Reported density, zero taken once at t = 0	6.00	1.508
Reported density, zero taken once at t = 0	8.00	1.510
Reported density, zero taken once at t = 0	10.00	1.511
Reported density, zero taken once at t = 0	12.00	1.512
Reported density, zero taken once at t = 0	14.00	1.512
Reported density, zero taken once at t = 0	16.00	1.513
Reported density, zero taken once at t = 0	18.00	1.513
Reported density, zero taken once at t = 0	20.00	1.514
Reported density, zero taken once at t = 0	24.00	1.514
Reported density, zero taken once at t = 0	28.00	1.514
The same step re-zeroed at each reading	0.00	1.500
The same step re-zeroed at each reading	4.00	1.500
The same step re-zeroed at each reading	8.00	1.500
The same step re-zeroed at each reading	12.00	1.500
The same step re-zeroed at each reading	16.00	1.500
The same step re-zeroed at each reading	20.00	1.500
The same step re-zeroed at each reading	24.00	1.500
The same step re-zeroed at each reading	28.00	1.500

Illustrative arithmetic on a lamp falling 2.6 per cent over twenty minutes, not a measurement of any instrument. What is real in it is the pair of shapes: a rising curve that flattens, and a flat line, from the same instrument on the same step. The curve is drawn to show the shape, not measured from a real material. Your own materials will differ, and measuring them is what the sensitometry part of the course is for.

✦The cheapest warm-up rule is a re-zero

Look at the two traces above. The drift is entirely in the zero, so re-zeroing before every reading removes it completely, and a session that re-zeroes every ten minutes needs a warm-up rule of about zero.

That is not a reason to skip the arm. You need to know the magnitude, because there will be sessions — reading twenty-one steps in a row, say — where re-zeroing between every reading is not practical, and because a drift that does not vanish on re-zeroing is a different fault entirely and this is how you find out you have one.

Arm 6 — Temperature, with the prediction made first

Predict before you measure. Three datasheet lines are all you need, and the naive prediction is wrong, which is the point of the arm.

The obvious candidate is dark current, and it is negligible. The detector’s photodiode dark current is 2.5 pA, doubling every 7 °C. Through the 1 MΩ feedback resistor that is 2.5 µV at 25 °C, and even twenty degrees warmer it is about 18 µV — against a 7.5 mV pedestal it is not measurable.

The bigger term is on the same datasheet page and is easy to miss. The amplifier’s own input bias current is 165 pA, doubling every 10 °C — sixty-six times the photodiode’s dark current. Through the same megohm that is 165 µV at 25 °C and about 660 µV twenty degrees warmer: on the fine range, 84 counts, which is half the entire signal at 3.0 D.

And both are removed by re-taking the dark, which is why the firmware retakes it rather than storing it. What survives is the pedestal’s drift at ±10 µV/°C between the dark and the reading, and the lamp, whose output moves with junction temperature by a fraction no datasheet in the course’s corpus quantifies for the emitter you happen to have bought.

So the prediction is: the dominant temperature term is the lamp, it is multiplicative, and it will cancel if you re-zero. Now test it. Read a dense step at your normal room temperature, then change the room by four or five degrees — a cool morning against a warm evening is an entirely legitimate way to do this at home — let everything equilibrate for an hour, and read the same step again, once with a fresh zero and once against the old one. Record the room temperature both times.

If the reading against a fresh zero barely moves and the reading against the old zero moves a lot, the prediction held. If the fresh-zero reading also moves, something in the optics is moving with temperature — a diffuser lifting off its seat, an arm relaxing — and you have found a mechanical fault by thermal means. For scale, a commercial instrument specifies an operating range of 10 to 40 °C without saying what happens across it.

Arm 7 — Placement, and how big a mistake a millimetre is

Put a mid-scale step under the aperture and read it. Now move the wedge along its fence in half-millimetre steps, reading at each, until you have crossed a step boundary and are fully inside the neighbouring step.

Plot density against position. You will get a plateau, a transition and another plateau, and the width of the plateau is your placement tolerance: how far the wedge may be out before the reading changes by more than your repeatability. On a T2115 the steps are about 6 mm long and the aperture is 2 mm, so the plateau ought to be around 4 mm — but ought is not a measurement, and a bore drilled off centre or an aperture plate not square to the fence eats that margin from one end.

Record the plateau width. It is the number that tells you whether an index mark on the fence is worth adding, and it is the honest answer to “how carefully do I have to place the film?”

Arm 8 — Linearity by superposition, which needs no reference at all

This is the arm that fills the hole the datasheets left. The electronics page established that the OPT101’s published nonlinearity is quoted at a full-scale output of 24 V and cannot be transferred to a 3.3 V circuit, and that TI publish no figure for the conditions this instrument uses. Hamamatsu’s note tells you the photodiode is linear over more than nine orders of magnitude. Nobody tells you about your amplifier and your converter together, over your three decades.

You can measure it, and the method needs nothing you do not own.

🧪The two-source method, in four readings

A detector is linear if the response to two sources together equals the sum of the responses to each alone. That is the definition, and it is testable without any calibrated anything.

1. Switch on source A alone. Read. Call it a.
2. Switch on source B alone. Read. Call it b.
3. Switch on both. Read. Call it c.
4. The non-linearity at that light level is c − (a + b), expressed as a fraction of c.

Your two sources are already fitted: the green LED and the blue one the optical head’s parts list told you to buy and leave for later. Their wavelengths differ, which does not matter — this is a test of linearity in flux, not of spectral response.

Now walk it down the range. Put wedge step 1 in the light path and repeat the four readings; then step 5, step 9, step 13, step 17. At each level you get a fractional non-linearity, and plotting it against the density at which it was measured is a linearity curve for your own chain, obtained without a reference standard.

Convert the worst fraction to a density with log₁₀(1 + ε) and put it in the budget as a Type B term. If it comes out below 0.001 D across the range, the chain is linear as far as anything else on this page can tell, and you may stop worrying about it. If it grows towards the dense end, you have found a real limit and you know its size, which is the whole difference between an instrument and an object.

Two cautions, because a superposition test can be fooled. The two sources must not interact: if switching one on warms the other, or loads the same supply enough to dim it, you are measuring the supply rather than the detector, so check that source A alone reads the same whether B has recently been on. And the sum must be inside the range: if c saturates, the test reports a nonlinearity that is simply the top of the scale, so reduce both currents until c sits comfortably below full scale.

Arm 9 — The reflection anchors, in the order they must go in

Fit the 45/0 head. Then, and in this order:

1. The white tile. Set the LED current so that the tile reads near two-thirds of the coarse range’s full scale, and record both the current and the reading. This is the reflection scale’s zero in the densitometric sense — the top of the scale, against which everything is a ratio.
2. The black trap. Read it. Compute the stray-light fraction as trap over tile, exactly as in transmission, and the reflection ceiling as its negative logarithm.
3. The black card, for comparison. The gap between card and trap is what the trap is worth, in your own numbers, and it is usually the difference between a ceiling of 1.5 and one above 2.5.

The order matters because step 1 sets the gain that steps 2 and 3 are measured on. Reverse it and the trap is measured at one current and the tile at another, and the ratio is meaningless.

Re-anchor whenever the head is removed and refitted, without exception. A head that has been off and on is a different geometry until proved otherwise, and arm 10 measures exactly how different.

Arm 10 — Reflection repeatability, in three flavours

Take a print with a full tonal scale and choose a mid-tone patch. Then three runs of ten readings:

• Untouched. The print and the head stay where they are. This is the electronics, again, and it should be close to the transmission figure because it is the same chain.
• Lifted and replaced. Take the print off the port and put it back against the same register. This is sample presentation, and it is where flatness against the port, the backing behind the print and the pressure you apply all show up.
• Head refitted. Take the reflection head off the instrument and put it back, re-anchoring on the tile each time. This is usually the largest of the three, and it is the reason the certificate carries a separate reflection repeatability figure rather than borrowing the transmission one.

Quote all three. A single reflection repeatability figure is a fiction: which of the three applies depends on whether the reader is comparing patches on one print, prints in one session, or prints made a month apart, and those are three different measurements with three different uncertainties.

Arm 11 — Reflection linearity, and the gap the course will not paper over

The course could not establish that a printed reflection step scale with certified densities is obtainable at a home-lab price, and it will not assert that the reflection scale is linear on the grounds that the transmission scale is. They share a detector, an amplifier and a converter; they do not share a geometry, and geometry is where a reflection measurement’s errors live.

So the page states what is known and what is not.

Known, because it was measured in arm 8: the electronic chain is linear to the figure you obtained, and that figure carries over, because it is the same chain.

Not known, and honestly not knowable here: whether the optical relationship between a print’s reflectance and your head’s reading is linear across the range from paper white to maximum black. The sources of departure are real and nameable — inter-reflection between the print surface and the head’s interior, a specular component that varies with surface and with flatness, and show-through from the backing — and none of them has a published magnitude for a home-built head.

What you can do about it, today: run the superposition test of arm 8 in reflection. The 45/0 head has two opposed sources by construction, which is exactly the pair the method needs: read a patch with the left arm alone, the right arm alone, and both. Repeat on a light patch, a mid-tone and the deepest black. A linear chain gives c = a + b at every one of them; a departure that grows towards the dark end is telling you that the dark end is where your reflection densities stop being trustworthy, and it does so without any certified scale at all.

And what to write: “Reflection densities are relative to white tile serial XXXX and internally consistent to ±N; the reflection scale’s absolute linearity has not been established.” That is a complete and honest specification. It supports every comparison a printer actually makes — this paper against that one, this print against yesterday’s, base white against maximum black — and it supports nothing that involves somebody else’s numbers, which is correct, because it cannot.

Arm 12 — The cross-check, and what Part XIII was worth

Two readings, and they close a loop that has been open for two parts.

Re-read the archived Part XIII step-wedge negative, every step, on the calibrated instrument. Overlay your new curve on the interim one you obtained there with a lux meter, a spot meter and a set of matched grey patches. Two things come out: the slope of new against interim, which is the calibration factor that method had all along and did not know; and the scatter about that line, which is what that method’s precision actually was. Part XIII promised you this measurement and today is the day it can be taken.

Re-read a Part XIV control strip. It was exposed by one instrument you built and is now read by another. The process-control page told you what a control strip is for; this is the first time the reading of one has had an uncertainty attached. Record the density of every step with your new uncertainty figures beside them, because from here on that strip is the reference against which process drift is judged.

Expected observations

Arm 1. The dark reading will not be zero and should not be; on the fine range it will be several hundred counts. Its spread should be a handful of counts. A dark reading that wanders by tens of counts is a fault, not a measurement.

Arm 2. A slope within one or two per cent of unity. Anything further is a mistake somewhere upstream.

Arm 3. A residual within ±0.02 across the lower two thirds of the range and departing downwards near the top. That downward departure is the stray-light floor arriving, exactly where the head page’s ceiling said it would.

Arm 4. The lifted-and-replaced spread several times the undisturbed one. If they are equal, either your mechanics are extremely good or you did not really replace the sample.

Arm 5. A rise of a few thousandths to a few hundredths over the first ten minutes, flattening. A monitor column that rises or falls in step with it.

Arm 6. Very little change against a fresh zero; a visible change against a stale one.

Arm 7. A flat plateau of a few millimetres with sharp shoulders. A plateau with a slope on it means the light through the aperture is not uniform, which is a diffuser problem.

Arm 8. A fractional non-linearity in the small fractions of a per cent at low densities. If it is large everywhere, suspect the two sources are interacting before you suspect the amplifier.

Arms 9 to 11. A reflection ceiling well below the transmission one — reflected light is scarce and the geometry is open — and a head-refitted repeatability several times the untouched one.

Arm 12. A slope near but not equal to 1.00 between the interim and instrument curves, and a scatter that makes plain why this part exists.

What is happening chemically

No chemistry happens on this page. Nothing is mixed, nothing reacts, nothing is consumed, and the section says so rather than manufacturing a mechanism to fill a heading.

What is worth saying under it is what the numbers are of, because the object being measured is a chemical one even though the measurement is not. Every density you read today is a layer of metallic silver — filaments grown by development from exposed halide crystals and then fixed and washed until nothing light-sensitive remains — suspended in hardened gelatin on a plastic base. The instrument reports the logarithm of how much light that layer removes, by absorption and by scattering together, which is why the geometry lesson had to come before the build.

That has one practical consequence for this page, and it is a real open question rather than a rhetorical one. A calibrated wedge is a silver image, and a silver image is a chemical object that can change. Residual thiosulfate, atmospheric sulfur, humidity and heat are the mechanisms Part XII catalogued for every other silver image in this course, and there is no reason a step wedge should be exempt from physics that applies to a negative.

!Whether a wedge ages: what the course could and could not find

Could not find. No supplier statement in this course’s corpus gives a shelf life, an ageing rate or a recalibration interval for a transmission step wedge. Stouffer’s own material distinguishes calibrated from uncalibrated guides and says nothing about how long a calibration remains good.

Could find, and adopts by analogy rather than by authority. X-Rite’s handling instructions for their own calibrated transmission reference: edges only, because foreign substances on the measurement area cause errors; a soft camel-hair brush and nothing else, because anything else may change the densities; and storage in a dark, cool, dry place to minimise change. A manufacturer who writes “minimise change” is telling you that change is possible.

What to do instead of guessing. Nominate one step — the calibration step is the obvious choice — and read it at the start of every session for the rest of the instrument’s life, logging the value and the date in the calibration log of the calibration records. Any drift in that number is either your instrument or your wedge, and the two are separated the first time you borrow a second wedge. A three-year log answers a question no datasheet in this course’s corpus could.

Data to record

Everything goes in the lab notebook with the instrument’s own line in the calibration records, whose sheets carry the reading-reference- deviation-action shape this page’s arms produce and which should be used rather than duplicated. The worksheet set records honestly that the densitometer is one of the instruments it does not yet carry a sheet for; until it does, the balance and thermometer sheets are the right shape to copy.

Per session. Date; room temperature at start and end; instrument serial; firmware version; drive current; aperture and gap from the label; wedge serial and whether calibrated; the certificate’s own reference if it has one; warm-up given before the first reading.

Per arm. The raw log file name, so a number on a plot can be traced back to the row that produced it. Never transcribe a reading you could keep.

The nine numbers that go on the certificate, each with the arm that produced it:

Number	Arm	Units
Calibration slope, and the step it came from	2	dimensionless
Density at which the residual leaves ±0.02	3	D
Repeatability, undisturbed, at a stated density	4	D
Repeatability, sample replaced, at a stated density	4	D
Warm-up excursion and the rule derived from it	5	D, and minutes
Temperature sensitivity against a fresh zero	6	D per °C
Placement plateau width	7	mm
Chain non-linearity, worst case across the range	8	D
Reflection repeatability, all three flavours	10	D

Analysis

Three products: a set of residuals with a shape you can name, an uncertainty statement per density band, and a certificate.

The residuals

Fit a straight line to measured against certified over the range where the residual is flat, and report its slope and intercept. The slope should be 1.000 if arm 2 did its job, and a slope that is not 1.000 after calibration means the calibration step was misread. The intercept should be 0.000, and an intercept that is not means the zero was taken through something.

Then report where the residual leaves ±0.02 and where it leaves ±0.05, and note both. The first is where the instrument stops matching a commercial specification; the second is where it stops being able to tell one wedge step from its neighbour.

The uncertainty statement

The course keeps two kinds of term apart, and defines them here because it must.

!Type A and Type B, as this course uses them, and the source it could not get

The course could not obtain a plain-language, GUM-based beginner’s guide to uncertainty from a source that meets its sourcing standard. Rather than paraphrase one from memory — which is exactly what Rule 1 forbids — it states its own definitions and labels them as its own:

Type A, in this course, is an uncertainty evaluated from the statistics of repeated observations you made. Arms 4, 5 and 10 produce Type A terms.

Type B is an uncertainty evaluated by any other means: a manufacturer’s specification, a certificate, a bound you can argue for. The wedge certificate, the converter’s gain match and the non-linearity from arm 8 are Type B terms.

The distinction earns its keep because the two behave differently when the instrument changes. A Type A term improves if you take more readings; a Type B term does not, and no amount of patience will make an uncertified wedge into a certified one.

The combination rule is Part II’s and is not re-argued here: the terms are added, because that is arithmetic anyone can check and because a proper treatment of how independent uncertainties combine belongs to a metrology text this course has not read. As on Part XIV’s certificate, you may compute the root-sum-square figure as well — it will be roughly half — but you must label which rule produced which number, and never print the smaller one bare.

Then build the statement per density band, because a single figure for an instrument whose resolution changes by a factor of a hundred across its range is not a statement, it is an average of incompatible things.

Band	Type A terms	Type B terms	Combined, Part II’s bound	What may be claimed in this band
0.0 to 1.0	Repeatability replaced, at 0.5	Wedge certificate; chain non-linearity
1.0 to 2.0	Repeatability replaced, at 1.5	Wedge certificate; chain non-linearity; range trim
2.0 to the residual limit	Repeatability replaced, at 2.5; drift if the warm-up rule is not obeyed	All of the above, plus the stray-light residual
Above the residual limit, up to the ceiling	—	—	—	Nothing. Report the reading and mark it as at or near the instrument’s floor
Above the ceiling	—	—	—	Nothing at all. The instrument is reporting its own stray light

Fill it in for your own instrument. The two rows at the bottom are not padding: the most valuable thing a certificate does is say where the instrument stops, and an instrument whose certificate has no such line is one whose owner will eventually quote a number it never measured.

🧮A worked band, on numbers that are not yours

Take the 2.0-to-2.5 band. Suppose repeatability with the sample replaced is 0.008 D, the wedge certificate states 0.02, the chain non-linearity from arm 8 came out at 0.004, the range trim residual is 0.0004, and the stray-light residual at 2.5 is 0.010.

Part II’s bound: 0.008 + 0.02 + 0.004 + 0.0004 + 0.010 = 0.042 D.

Root-sum-square, labelled as such: √(0.000064 + 0.0004 + 0.000016 + 0.00000016 + 0.0001) = 0.024 D.

Which to quote: the bound, per Part II’s ruling, with the smaller figure named and labelled if you want it.

Which term to attack first: the wedge, at 0.02, which is nearly half the total and is the one term you cannot improve by building anything — it is bought. After it, the stray-light residual at 0.010, which is black paint and an afternoon. Repeatability at 0.008 is the arm and the fences. The range trim, at 0.0004, is a hundredth of the total and no work on it will move the answer, which is the entire reason for writing a budget instead of improving whatever is nearest to hand.

Every figure in this callout is invented for the arithmetic. None of them is a measurement of anything.

The instrument certificate

One page, and it is the deliverable of the whole part. Every later page that quotes a density from this instrument cites it.

Field	What goes in it
Instrument and serial	The name you gave it and its number, from the build record
Date of calibration, and by whom	And the date the next one is due
Reference	Wedge serial; calibrated or not; the certificate’s own reference if it has one; the step used for Cal HI
Geometry	Diffuse, opal in contact, aperture diameter, plate thickness, detector gap. “Modelled on ISO 5-2; conformance not claimed and not tested”
Spectral condition	The LED’s dominant wavelength and half-width from its datasheet. “A narrow green band, not a spectral product. Modelled on nothing; ISO 5-3 is cited by number as what a spectral condition is”
Working range	From 0.00 to the density at which the residual leaves ±0.02
Stray-light ceiling	From the head page’s T2, with its date
Repeatability, transmission	Undisturbed and replaced, each at a stated density, with the number of readings
Warm-up rule	One sentence, from your own drift curve
Temperature sensitivity	D per °C against a fresh zero, and the two temperatures it came from
Chain non-linearity	From arm 8, with the method named as superposition
Reflection mode	Tile serial; trap construction; the three repeatability figures; the reflection ceiling
Reflection scale	“Relative to tile serial XXXX. Absolute reflectance unknown; absolute linearity not established”
Known limitations	Every gap on this page that applies to you, written out
Signature and notebook page	So the certificate points back at the raw data

✕Five claims this certificate must not make

“Conforms to ISO 5-2” or “ISO 5-4 reflection density”. The course does not buy the standards, cites them by number as what its conventions are modelled on, and quotes nothing from them. ISO 5-4 in particular has not been read by this course at all — only its title is known, from the foreword of another part — so no statement of the form “meets these requirements and not those” can honestly be made about the reflection head, and none is made. What the certificate says instead is what the head is: illumination at 45°, collection on the normal, and a stated aperture.

“Accurate to 0.01 D.” Accuracy and repeatability are different words and Part II keeps them apart. A repeatable instrument with a wrong reference is precisely wrong every time.

“Traceable.” Traceability is an unbroken chain of comparisons each with a stated uncertainty, ending at a national standard. A calibrated wedge is one link of it — the supplier’s own words are that their calibrated parts compare with a NIST standard reference material by means of a densitometer conforming to a named standard — and your certificate may say exactly that much and no more.

“Reads to 3.05 D.” Only if your residual is still inside your stated band there. A reading taken above the density where the residual left the band is a reading of your own stray light.

“Reflection density 2.10.” Not without the tile. Write “2.10 relative to tile XXXX”, every time, because the day you compare that number with somebody else’s is the day the omission costs you.

Troubleshooting

What you see	Likely cause	What to do
The calibration slope comes out at 1.4	The zero was stale, the certificate value was mistyped, or the aperture is not on the step you think	Re-zero and re-run. If it persists, read the step either side; a slope that jumps to 0.9 on the neighbour means you were on the wrong step
Residual is a straight line through the origin	A scale error: wrong reference value, or the lamp at a different level than when the zero was taken	Recalibrate. Do not reach for the black paint; this is not stray light
Residual is flat and offset by a constant	The zero was taken through something — film base, a fingerprint on the diffuser, the wedge’s own clear leader	Take the zero on an empty stage and repeat
Residual is flat and then dives at the top	Stray light, as predicted. This is the expected shape rather than a fault	Compare where it dives with the head page’s ceiling. If they agree, the model is right and the cure is optical
Undisturbed and replaced spreads are identical	The sample was not really moved, or the fences are so tight the film cannot go back wrong	Check by deliberately misplacing it a millimetre and reading again. If nothing changes, arm 7’s plateau is very wide and that is a result
Drift does not settle in thirty minutes	Something with a long thermal time constant — a metal head, a heatsink, or a room that is itself warming	Run it for an hour. Then write the rule you actually measured rather than the one you hoped for
The dense steps read differently on the two passes of arm 3	Drift within the session, which is what the two passes were for	Average the passes only if the difference is inside your repeatability; otherwise re-zero between passes and say so
Reflection readings differ every time the head is refitted	Expected, and it is why the third repeatability figure exists	Add a positive register — a pin and a slot rather than a pencil line — and re-measure. Whatever remains goes on the certificate
The black trap reads the same as the black card	The trap is not deep enough, or its baffle is missing	Five times the opening diameter, with a sloping matt baffle. A shallow trap is a card with ambitions
Superposition gives a large non-linearity everywhere	The two sources are interacting through the supply	Read A alone, then A alone again immediately after B has been on. If the two differ, fix the supply before believing the test

Clean-up

Wedge, negative and control strip back into their sleeves, handled by the edges. Tile face down in its own sleeve, in the dark. Trap somewhere it cannot collect dust, which for a black cavity means covered. Instrument off, and the head closed rather than left open with the aperture facing the room.

The log file is the clean-up that matters. Save it, name it with the date and the instrument serial, and put a copy somewhere other than the machine it was written on. A calibration you cannot produce the raw data for is an assertion.

Storage

The wedge goes back in its own sleeve, flat, away from heat and light, on the three rules borrowed from X-Rite’s reference: edges only, camel-hair brush only, dark and cool and dry. It is now the most precisely known object in your darkroom and it is a piece of film.

The white tile is stored in the dark, in a sleeve, never face down on a bench. Skin oil and yellowing are both real and both slow, which is exactly why you will not notice them happening. Its serial number is on the certificate, so a replacement tile is a new calibration and not a spare part.

The certificate goes in the notebook and a copy in the calibration records. Both, because one of them will be lost.

The instrument lives with its head closed and its label legible. If it moves house, or is dropped, or sits in a car in August, the certificate is void until arm 2’s check has been run again.

Disposal considerations

This session produces no waste stream, so there is nothing to dispose of at its end. The disposal question for this page arrives at end of life rather than at end of session, and it has two parts.

The electronics — LED, photodiode, converter, microcontroller, display — are waste electrical and electronic equipment. The chemistry that makes that a separate stream is that they contain metals which are worth recovering and, in trace quantities, some that are not wanted in landfill leachate; the general practice everywhere is separate collection.

The film and paper in the wedge, the negative, the strip and the prints are fully processed silver gelatin on a plastic or paper base. The silver in them is metallic, bound in gelatin, and present in milligram quantities; it is the same material Part XII’s silver-recovery page treats as worth recovering in bulk and not worth chasing in ones and twos.

Local regulation governs both streams and differs by jurisdiction — check your local regulations. The course describes the chemistry and the general practice and gives no jurisdiction-specific instruction as though it were universal.

Questions

1. Your residual plot is a horizontal line at +0.09 across all twenty-one steps. Name the fault, name the one reading you would take to confirm it in under a minute, and say why recalibrating the slope would make the plot look better without fixing anything.
2. Repeatability with the sample undisturbed is 0.002 D at 1.0 and 0.006 D at 2.5. With the sample replaced it is 0.009 D at both. Explain why the first pair grows with density and the second does not, and say which figure belongs on the certificate for a reader who will compare two negatives.
3. Your drift arm shows the reported density rising by 0.012 over ten minutes and then flat, and the monitor column falling by 2.7 per cent over the same period. Show that the two are the same observation, and give the one change to your working routine that removes this term entirely.
4. You are offered a certified white tile for a sum you can just afford. State precisely which line of your certificate would change, which lines would not, and what new claim you would then be entitled to make about a paper Dmax measured next year.
5. Arm 8 gives a = 0.812 V, b = 0.655 V and c = 1.459 V. Compute the fractional non-linearity and express it as a density. Then say whether that term belongs in the Type A or the Type B column, and why.
6. A friend’s certificate reads: “Diffuse visual density, ISO 5-2, accurate to ±0.01 D, traceable to NIST, range 0 to 3.05 D.” Identify every claim it cannot support, and write the honest version of each line.

Further experiments

Borrow a second wedge and separate the instrument from its ruler. Every figure on your certificate that involves the wedge cannot distinguish an error in your instrument from an error in your wedge. Read somebody else’s wedge on your instrument and yours on theirs, and the two errors separate immediately. It is an afternoon and a stamp, and it is the only way to break the circularity that a single reference imposes.

Read the same wedge every month for a year. Log the calibration step at the start of every session anyway; once a year, read all twenty-one. A slow, monotonic change at the dense end is the wedge; a change everywhere is the instrument; noise is noise. This is the experiment that answers the ageing question the course could not source, and the only cost is a habit.

Find the aperture at which your repeatability stops improving. Arm 4 with three aperture plates instead of one. The replaced-sample spread should fall as the aperture grows, because a larger patch is less sensitive to placement — until the aperture starts to overlap the neighbouring step, at which point it rises sharply. The minimum is your instrument’s best aperture for that wedge, and it is a different number from the one the geometry lesson chose on other grounds.

Compare the two ranges as though they were two instruments. Read all twenty-one steps forcing the coarse range, then again forcing the fine range wherever it is on scale, and plot the difference against density. A flat difference is the range trim, already known. A difference that grows is something the trim does not describe, and finding out what would be a genuinely original measurement on your own hardware.

Check your understanding

Question 1. Your residual plot sits within 0.005 of zero up to 1.2 D and then falls steadily away, reaching minus 0.34 at 2.6 D. What have you found, and what is the first thing to try?

• A scale error; recalibrate the slope
• An additive term; take a fresh dark reading before touching anything optical
• A zero taken through film base; re-zero on an empty stage
• A wedge that has aged; buy a new one

Show the answer and why

Answer: An additive term; take a fresh dark reading before touching anything optical

A residual that is flat and then curves away is the signature of an error that adds light rather than scaling it, because an additive term is negligible while the signal is large and dominant once the signal is small. Three things produce it: stray light inside the head, ambient leakage through the shroud, and a dark reading taken cold and then left to go stale as the head warms. The last is free to eliminate and takes forty seconds, so it is tried first. A scale error would give a straight line through the origin and a base-zero error a horizontal line, so options 1 and 3 have the wrong shape. Option 4 diagnoses the most expensive cause first on no evidence.

Question 2. Which of these belongs in the Type B column of your uncertainty statement, as this page defines the terms?

• The spread of ten readings with the sample lifted and replaced
• The spread of ten readings with the sample untouched
• The wedge certificate quoted uncertainty
• The spread of the drift arm over thirty minutes

Show the answer and why

Answer: The wedge certificate quoted uncertainty

Type A, in this course, is evaluated from the statistics of repeated observations you made; Type B is evaluated by any other means, chiefly from a specification, a certificate or an argued bound. Options 1, 2 and 4 are all standard deviations of your own repeated readings and are therefore Type A. The wedge certificate is somebody else own measurement handed to you as a number, which makes it Type B. The distinction earns its keep because they behave differently under effort: taking more readings shrinks a Type A term, and no amount of patience turns an uncertified wedge into a certified one.

Question 3. Why does the two-point calibration use a step near 2.0 D rather than the clearest step or the densest one?

• Because the clearest step is easiest to misplace under the aperture
• Because a slope error grows in proportion to density, so a high calibration point pins the scale more tightly, while a point near the stray-light ceiling would be contaminated by the floor
• Because the densest step is the one the certificate is most accurate about
• Because a mid-scale step is where the converter changes range

Show the answer and why

Answer: Because a slope error grows in proportion to density, so a high calibration point pins the scale more tightly, while a point near the stray-light ceiling would be contaminated by the floor

The scale is one multiplicative constant, so the error it leaves behind is proportional to the density: calibrating on a step of 0.2 D and then reading 2.5 D magnifies any error in that one reading more than tenfold. That argues for calibrating as high as possible. The counter-argument is the stray-light floor, which contaminates the top of the range, so a point taken there would bake the floor into the slope. Near 2.0 D is the compromise, and it is the same compromise a commercial reference makes by marking the fourth of its five steps rather than the fifth. Option 4 describes something to avoid rather than to seek.

Question 4. In the superposition test, source A alone reads 0.940 V, source B alone reads 0.720 V, and both together read 1.652 V. What is the fractional nonlinearity, and what is it worth as a density?

• 0.5 per cent, about 0.0021 D
• 0.5 per cent, about 0.05 D
• 5 per cent, about 0.021 D
• There is no nonlinearity, because 1.652 is between 0.940 and 0.720 doubled

Show the answer and why

Answer: 0.5 per cent, about 0.0021 D

A linear chain gives the reading for both sources as the sum of the two single readings, which here is 1.660 V. It gave 1.652, so the shortfall is 0.008 V, which as a fraction of 1.652 is 0.48 per cent, near enough half a per cent. As a density that is the logarithm of 1.0048, which is 0.0021, and it goes into the budget as a Type B term. The virtue of the method is that it needs no reference standard at all: the two sources need not be known, matched or calibrated, only independent and stable, because the test is of an addition rather than of a value.

Question 5. Which sentence may honestly appear on your reflection certificate?

• Reflection density measured to ISO 5-4, 45/0 geometry
• Maximum black 2.08, traceable to a national standard
• Maximum black 2.08 relative to tile serial 0041; absolute reflectance unknown and absolute linearity not established
• Reflection densities accurate to plus or minus 0.01 D

Show the answer and why

Answer: Maximum black 2.08 relative to tile serial 0041; absolute reflectance unknown and absolute linearity not established

Only the third states what was actually established. The course has not read ISO 5-4 at all, knowing only its title from the foreword of another part, so no statement about meeting or missing its requirements can be made and option 1 is a claim of conformity nobody has tested. Traceability is an unbroken chain of comparisons each with a stated uncertainty ending at a national standard, and an uncertified tile provides no link of it, so option 2 is false. Option 4 confuses repeatability with accuracy. The third sentence is unglamorous and complete: it supports every comparison a printer actually makes and refuses the one it cannot.

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