Micrograph Measuring Room

Intercept or planimetric: the difference and which to use

The intercept method counts the intersections of test lines with grain boundaries and gives the mean lineal intercept length ℓ̄; the planimetric method counts the grains in a figure and gives the number of grains per mm², N_A. Both convert to the grain size number G with the ASTM E112 equations, but they count different things (boundaries along a line, grains in an area). Vander Voort notes that the two methods sense different geometric aspects, so they agree closely but not exactly.

The effort differs. In the planimetric method each grain has to be marked as it is counted, which takes time. E112 gives about 400 intersections with the intercept method and about 700 grains with the planimetric method as the counts that reach 10 % relative accuracy. Vander Voort also finds the intercept method much faster for the same precision.

For equiaxed grains and a quick result, use the intercept method, preferably with three concentric circles (orientation effects average out; E112 recommends it). Use the planimetric method when you need the number of grains itself or want to check grain shapes as you count. A rectangular figure (Saltykov’s method) has been reported to give consistent results even with few grains per field.

When rolling has elongated the grains, measure each direction separately (on this page, count with "one horizontal line" and "one vertical line" in separate sessions). E112 describes directed ℓ̄ values and their ratio as a measure of elongation. Duplex grain structures with two size populations are outside E112 and are treated by ASTM E1181.

For the grain size of one phase in a two-phase structure, combine its volume fraction (point count) with an intercept count of that phase only (E112: ℓ̄_α = V_Vα · L / N_α, where N_α is the number of α grains intercepted).

Sources: Univ. of Michigan lab module (grain size); Vander Voort, Buehler Tech-Notes; Vander Voort (2013) M&M; ASTM E112-13(2021) overview; ASTM E1181 overview

How many to count (relative accuracy and counts)

Precision is governed less by the count in one field than by the scatter between fields. Like E112 and E562, this page computes the mean x̄, the standard deviation s, the 95 % confidence interval t·s/√n and the relative accuracy %RA = 100 × CI / mean from the field values. t is Student’s t with n − 1 degrees of freedom and grows when there are few fields (2.776 for n = 5, 2.262 for n = 10, 2.093 for n = 20).

E112 guidance: for the intercept method choose a magnification giving 40 to 100 counts per placement of the three circles (at least 50 for straight lines) and count three to five or more fields; for the planimetric method put at least 50 grains (up to about 100) in the figure and count at least three fields. 10 %RA or less is considered adequate for most purposes; above that, measure more fields and recalculate.

The confidence interval shrinks roughly as 1/√n, so four times as many fields halve it. For a point count with independent points, the %RA for a phase of fraction p, a grid of P_T points and n fields is about 200 √((1 − p) / (p · P_T · n)). Minor phases need much more counting: a 5 % phase counted with a 100-point grid needs about 76 fields for 10 %RA by this formula, and 80 fields in Table 3 of E562 (which assumes the features are randomly distributed).

E562 gives a table that picks the grid size from the visually estimated area fraction (100 points for 2–5 %, 49 for 5–10 %, 25 for 10–20 %, 16 above 20 %) and a table of the fields needed for a target %RA; the %RA itself must be calculated from your own data, not read from the table. E562 also says that at least 30 fields must be measured to calculate the 95 % confidence interval (Annex A1); with fewer fields, read the interval as a rough guide.

Sources: Univ. of Michigan lab module (grain size); Univ. of Michigan lab module (point count); Vander Voort, Buehler Tech-Notes

When boundaries are not revealed, and whether to count twins

Both methods assume that every grain boundary is visible. If part of the boundaries is missing, fewer intersections make ℓ̄ larger, fewer grains make N_A smaller, and both underestimate G (too coarse). Counting rules cannot correct this; re-etch. For under-etching, over-etching and scratches, see the preparation guide.

Choose an etchant that reveals the grain boundaries well. For ferrite grains in steel use nital. Picral attacks cementite and does not reveal ferrite boundaries (see picral in the preparation guide), so it is unsuitable for ferrite grain size. For austenitic steels ISO 643 lists electrolytic etching in 60 % nitric acid, which reveals grain boundaries but not twin boundaries.

For the grain size number, annealing twin boundaries are ignored (both sides of a twin count as one grain), as in the E112 definition. Face-centered cubic metals (austenitic stainless steel, copper, brass, nickel alloys) often contain annealing twins, while they are not normally seen in body-centered cubic ferrite. Twin boundaries usually appear as straight, parallel pairs of lines; tell them apart from grain boundaries and do not mark them. Some studies use values that include twins, so state in the report which way you counted.

Sources: Univ. of Michigan lab module (grain size); ISO 643:2019 sample; Vander Voort, Buehler Tech-Notes

Mean grain diameter and the grain size number: points to watch

There are several "mean grain diameters": (1) the mean lineal intercept length ℓ̄ itself; (2) the square root of the mean grain area Ā = 1/N_A, d̄ = √Ā (the d̄ of the E112 table, which assumes square grains and, as E112 says, has no physical significance); (3) the equivalent circle diameter √(4Ā/π); (4) the spatial diameter D (D = 1.5 ℓ̄ for spheres of one size, and D = 1.571 ℓ̄ as an estimate based on tetrakaidecahedra with a size distribution).

For the same structure, the relations in the E112 table make d̄ about 12 % larger than ℓ̄, the equivalent circle diameter about 27 % larger, and D more than 50 % larger. When you write "grain size 20 µm", state which definition (ℓ̄, d̄ or equivalent circle diameter) and which method you used. The d in the Hall–Petch relation is ℓ̄ in some papers and an equivalent circle diameter in others.

The grain size number is a logarithmic scale: each step of G doubles N_A and multiplies ℓ̄ by 1/√2. The intercept and planimetric G are defined to agree for the grain shape assumed by E112; for real structures, and for the planar synthetic structures on this page, the two can differ by 0.1 to 0.3 for the same structure. Do not average G numbers; average ℓ̄ or N_A and then convert (E112).

Sources: Univ. of Michigan lab module (grain size)

Grain size numbers in JIS, ISO and ASTM

JIS G 0551 (Steels — Micrographic determination of the apparent grain size) and ISO 643 relate the number of grains per mm², m, to the grain size number by m = 8 × 2^G (m = 16 at G = 1). ASTM E112 defines 2^(G−1) grains per square inch at 100×, which is N_A = 15.50 × 2^(G−1) = 7.75 × 2^G mm⁻² at true size.

For the same N_A, G_JIS = log₂(N_A/8) and G_ASTM = log₂(N_A/7.75), so the JIS/ISO number is smaller by log₂(8/7.75) = 0.046. Vander Voort describes the difference as negligible.

Whether JIS and ASTM use the same equation to convert the intercept length ℓ̄ into a number could not be confirmed from public sources. This page gives the grain size number by the ASTM E112 equations; if you report to JIS, check the text of JIS G 0551. The JIS G 0551 intercept procedure uses three concentric circles or four straight lines with a total test length of 500 mm at 100×, and normally averages N_L over at least five fields.

Sources: Industrial Research Institute of Niigata Prefecture (2020); Terayama et al. (2022); ISO 643:2019 sample; Vander Voort (1991); Univ. of Michigan lab module (grain size)

The synthetic practice structures and their true values

The synthetic structures are Voronoi tessellations of random points (each grain is the region nearest to one point). The points are kept a minimum distance apart, so grain sizes are fairly uniform. The image wraps around top to bottom and side to side, so it contains exactly one copy of every grain. In the two-phase structure some grains are made a dark phase (striped, like pearlite); the particle structure places non-overlapping discs (like graphite nodules) in a matrix.

True values come from the geometry, not from counting pixels. N_A is the number of grains divided by the area. The mean intercept length is ℓ̄ = πA / (2 L_B) from the total boundary length L_B (for boundaries with no preferred orientation, intersections per unit test length are P_L = (2/π) L_A). Volume fractions come from the areas of the polygons and discs. The scale is chosen so that N_A equals the value for the selected G.

In the answer check, "counted ideally" is what a machine counts on exactly the lines, figures and grids you placed: every change of grain, every point on the dark phase. Its difference from the true value is mainly sampling scatter from the limited fields; your difference from it is counting error. The planimetric count (grains inside plus half the grains cut) does run high when the frame holds few grains: re-counting these structures with frames placed all over the image gave about +9 % with some 13 grains in the frame, +3 to 4 % with 25, +2 % with 50 and +1 % with 100. A magnification that puts the E112 minimum of 50 or more grains in the frame keeps this bias small.

The synthetic grains are planar polygons, not sections of three-dimensional grains, so the intercept G comes out about 0.3 larger than the planimetric G. Compare each method with its own true value. Put the random seed in the URL to share the same image.

Sources: Rollett, Stereology (2014); Univ. of Michigan lab module (grain size)