As L increases, the central cumulative dose DL(z 0) tends to approach the equilibrium
value Deq first, i.e., before any other value DL(z 0) of cumulative dose would (cf. Fig. 2). To
distinguish the asymptotic behavior of the central cumulative dose from that of DL(z 0), and for
purposes of dose evaluation precluding measurements over infinitely long ranges, it is convenient
to introduce a parameter Leq called the “equilibrium scanning length.” Leq is defined as a finite
scanning length for which DLeq(z 0) may be deemed to be close enough to the value Deq to represent
it practically; section 2.A.2 of this report describes the convention adopted for evaluating
Leq. For the particular phantom represented in Fig. 2, Leq ≈ 400 mm (and Leq ≈ 450 mm for the
32-cm diameter right-circular cylinder PMMA body phantom). As L increases further (L > Leq),
the region over which scatter equilibrium exists expands beyond z 0, producing a flat central
region about z 0. The dose DL(z ±L/2) at the endpoints of a scanning length approaches
½ Deq as the scanning length L > Leq approaches the full-width at half-maximum of the distribution
DL(z). The average dose over (–L/2, L/2) is less than the central cumulative dose DL(z 0)
for any value of L and approaches DL(z 0) ≈ Deq only for scanning lengths L >> Leq. To a good
approximation[23] the dependence of Deq on a/b (Eq. 3) applies likewise to the cumulative doses
expressed in Eqs. (1) and (2) even for sub-equilibrium scanning lengths L < Leq, where the
dependence on a is approximate and on 1/b exact.
[23,27