10.2 Exposure Geometry, Inverse Square Law, and Geometric Unsharpness

The rest of official RT topic 1, Review of Basic Radiographic Principles, is geometry. A perfect energy choice still produces a useless radiograph if the source is a large blob an inch from a thick part with the film a foot away. The general exam asks whether you can name the distances, apply the inverse square law to intensity and time, compute geometric unsharpness, and explain why an IQI belongs on the source side.

ASTM E94, ASTM E1032, and ASME Section V, Article 2 all treat source-to-film distance, geometric unsharpness limits, and IQI placement as technique requirements, not as optional art. The written exam will hand you the numbers. Your job is the formula and the reason.

Name the distances before you compute anything

Draw the beam as a straight line: source → entrance surface → exit surface → film.

On a flat plate with film taped to the back face:

SFD = SOD + OFD

and OFD is essentially the part thickness plus any cassette stand-off. On a double-wall shot of a pipe, or a source inside a vessel with film on the outside, you must sketch which surface is "the object" the formula is using. ASME's geometric-unsharpness paragraph uses D for source-to-object and d for object-to-film (source side of the object to the film). This guide writes the same physics as the prompt the exam authors use in training notes:

Ug = f × t / d

with f = physical source size (focal-spot width or gamma-source diameter), t = object-to-film distance, d = source-to-object distance. If a stem uses ASME letters, map them: F → f, d_ASME → t, D_ASME → d. Do not freeze and invent a third formula.

Inverse square law

Photons leave a small source into a sphere. The same number of photons is spread over an area that grows as the square of the radius. Therefore intensity (exposure rate) obeys:

I₁ / I₂ = (D₂ / D₁)²

or, rearranged,

I₂ = I₁ × (D₁ / D₂)²

Exposure time needed for a stated film density, if nothing else changes, moves the opposite way:

T₂ / T₁ = (D₂ / D₁)²

T₂ = T₁ × (D₂ / D₁)²

Distance here is source-to-film (or source-to-detector, or source-to-survey-meter) — the distance to the plane whose intensity you care about. Do not plug OFD into the inverse-square formula and call it a day.

Worked example: intensity

A survey meter reads 80 mR/h at 20 ft from an unshielded calculation point (the stem gives a point-source idealization).

What is the intensity at 10 ft, same line of sight?

I₂ = 80 × (20 / 10)² = 80 × 4 = 320 mR/h.

What is the intensity at 40 ft?

I₂ = 80 × (20 / 40)² = 80 × 0.25 = 20 mR/h.

Halving distance quadruples intensity. Doubling distance quarters it. That is the entire law. A candidate who answers "double distance, half the dose" has applied a linear rule that radiation geometry does not use.

Worked example: exposure time

A technique produces the required density in 3.0 minutes at 20 in SFD. The Level II must move the film to 40 in SFD (same source, same screens, same kV or same isotope, same cassette).

T₂ = 3.0 × (40 / 20)² = 3.0 × 4 = 12 minutes.

A second move: the original shot was 2.0 minutes at 24 in. New SFD is 36 in.

(36 / 24)² = (1.5)² = 2.25

T₂ = 2.0 × 2.25 = 4.5 minutes.

A third move: original 5.0 mA at 30 in for 2.0 min (10 mA·min) at the required density. New SFD 45 in, same mA.

T₂ = 2.0 × (45 / 30)² = 2.0 × 2.25 = 4.5 minutes (22.5 mA·min).

If instead you hold time at 2.0 min and raise mA:

mA₂ = 5.0 × 2.25 = 11.25 mA.

For a gamma source the same arithmetic applies to Ci·min or GBq·min. If 40 Ci·min produced density 2.0 at 14 in, then at 21 in you need 40 × (21/14)² = 40 × 2.25 = 90 Ci·min. With a 30 Ci source that is a 3.0-minute shot, after you have decay-corrected the 30 Ci figure.

Inverse square is also why a source sitting almost on the part (a panoramic shot with a tiny SOD) is brutally intense at the entrance surface and why a long SFD is kinder to definition and to the exposure-rate map — at the cost of time.

Geometric unsharpness

A real source is not a point. It has a width f — the focal-spot projection of a tube, or the pellet diameter of a gamma source. Each point in the object is therefore painted by a bundle of rays, and the shadow of an edge is a penumbra. That penumbra width at the film is geometric unsharpness, Ug.

Ug = f × t / d

ASME Section V states the same ratio as Ug = F d / D with their letters. Codes then impose a maximum Ug that depends on material thickness (Article 2 publishes a small table). This guide does not treat those millimetre limits as unpublished ASNT secrets. If the stem gives a limit, compare your calculated Ug with that limit. If it does not, the principle is: smaller f, smaller t, larger d.

Worked example: Ug

A gamma source has a physical size f = 2.0 mm. The source-to-object distance d = 400 mm. The object-to-film distance t = 30 mm.

Ug = 2.0 × 30 / 400 = 60 / 400 = 0.15 mm.

A second setup on a thicker vessel: f = 3.0 mm, d = 250 mm, t = 50 mm.

Ug = 3.0 × 50 / 250 = 150 / 250 = 0.60 mm.

That second setup is four times less sharp. Fixes the Level II actually has:

1. Increase SOD / SFD (walk the projector back, or use a longer pole). Time will rise by inverse square — budget it.
2. Put the film closer to the exit surface. Eliminate cassette stand-off, use a flexible cassette on a pipe, do not let the film hang in space.
3. Use a smaller source if the projector and the procedure allow a smaller focal spot or a smaller pellet. Output may drop, so time rises again.
4. Do not "fix" Ug by raising kV. Energy does not appear in the formula.

A discontinuity on the source side of a thick part has a large t (almost the full thickness plus any gap). A discontinuity on the film side has a tiny t. That is why a source-side crack can look fuzzier than a film-side crack of the same opening, and why IQI placement matters.

Units must match. If f is in millimetres, t and d must be in millimetres. A candidate who mixes 2.0 mm with 16 in SOD without converting will invent a Ug no code recognizes. Convert everything to millimetres or everything to inches first.

Geometric magnification versus definition

Geometric magnification of a feature in a plane at SOD is

M = SFD / SOD

If the film is against the back of a thin plate, SOD ≈ SFD and M ≈ 1. If you deliberately stand the film off, or the part is thick, M > 1 and the image of a pore is larger than the pore.

That enlargement is not free sharpness:

• Magnification enlarges the indication, which can help you see a small opening.
• The same geometry increases t relative to a contact shot, so Ug grows. The enlarged image is a larger, softer blob.
• Definition (how abruptly density changes at an edge) is a sharpness idea. Definition improves when Ug is small: small source, large SOD, film in contact with the part.
• A stem that says "increase magnification to improve definition" is mixing two words. You can magnify or you can sharpen. Doing both at once requires a much smaller source (as in microfocus X-ray), which is a special technique, not the default Ir-192 weld shot.

Contact radiography (film on the part, long SFD) is the usual industrial compromise: M near 1, Ug small enough to meet the code table, IQI holes readable.

Source-side versus film-side IQI placement

An image quality indicator — hole-type plaque (ASTM E1025) or wire set (ASTM E747) — is the shim the viewer uses to judge whether the radiograph is sensitive enough. ASME Section V, Article 2 and the ASTM radiographic practices normally require the IQI on the source side of the part, in the area of interest, unless geometry makes that physically impossible.

Why source side is the conservative location:

• The IQI then suffers the full object-to-film distance, so it is imaged with the same or worse Ug as a source-side discontinuity.
• The IQI radiation also travels through the full thickness, so it is the harder absorption test.
• A film-side IQI sits almost on the emulsion. Its holes or wires are sharp and only the film-side ligament attenuates them. You can "pass" a film-side IQI on a setup that would fail the same IQI on the source side.

When the procedure or code allows film-side placement (a source inside a closed vessel, a double-wall exposure where you cannot reach the source side of the weld), the radiograph is usually identified as film-side — a lead letter F, or the technique sheet's own mark — and the required IQI may change. The Level II does not silently flip the plaque to the cassette because the wires look prettier there.

Conceptual exam traps:

• "Put the IQI on the film side so magnification improves definition." Wrong on both counts: film-side is less severe, and magnification is not a definition free lunch.
• "Source-side IQIs are illegal because they cast a shadow." They are required because they cast the honest shadow.
• "Any visible IQI outline means sensitivity is adequate." Sensitivity is the specified hole or wire, at the specified density, in the specified location — not the brass shim outline.

Putting inverse square and Ug on the same setup

The two formulas pull SFD in opposite directions for the clock and for sharpness:

• Long SFD: Ug falls (d is larger), intensity falls (time rises by D²).
• Short SFD: time is short, Ug rises, and the inverse-square gradient across a thick part is steeper (entrance surface much hotter than the film).

A technique sheet picks an SFD that meets the Ug limit and the density requirement in a time the source and the work permit. The Level II who shortens SFD to "make production" without checking Ug is the person whose IQI 2T hole disappears into a blur.

Worked combined sketch. Required Ug ≤ 0.50 mm. Source f = 2.5 mm. Film is in contact with a 20 mm plate, so t ≈ 20 mm if the formula uses source-side object to film.

0.50 ≥ 2.5 × 20 / d → d ≥ 50 / 0.50 = 100 mm SOD.

SFD ≈ 120 mm. That is a minimum distance from unsharpness, not a suggestion. If the sheet already says 600 mm SFD, you do not creep in to 120 mm because the arithmetic still "passes." The sheet wins. The formula tells you why the sheet is not written at 50 mm.

Realistic exam scenarios

A panoramic Ir-192 shot places a 3 mm source on the centerline of a 200 mm ID pipe, film on the outside, wall 10 mm. SOD ≈ 100 mm, t ≈ 10 mm, Ug = 3 × 10 / 100 = 0.30 mm. Move the same source to a contact outside shot of one weld with SFD = 400 mm and t = 10 mm: Ug = 3 × 10 / 390 ≈ 0.077 mm. The contact long-SFD shot is sharper; the panoramic shot buys coverage, not definition.

A Level II doubles SFD "to get more magnification" on a plate with film still in contact. Magnification stays about 1 (film still on the back). Time quadruples. Ug falls slightly because SOD rose. The candidate who expected a bigger image misunderstood M = SFD / SOD on a contact shot.

A hole-type IQI is taped to the cassette. The 2T hole is gorgeous. The procedure required source-side placement and the weld fails a later audit. Classification of that radiograph: noncompliant technique, not "better than required."

A stem gives f = 2.0 mm, t = 30 mm, d = 400 mm and four numbers. Only 0.15 mm matches Ug = f t / d. 6.0 mm is f d / t. Those are the distractors.

What geometry items are really testing

If the stem changes distance and asks about intensity or time, write I₁/I₂ = (D₂/D₁)² and move time the same way as D². If it names blur, write Ug = f t / d and ask which of f, t, or d moved. If it names IQI side, source side is the severe check. If it names magnification, ask whether the film left the part and whether definition paid for the enlargement. Those four questions are the geometry half of Review of Basic Radiographic Principles.