The rate of change of a material property can be related to displacement rate of carbon atoms (dpa s— ). However, it is not possible to directly measure dpas—1 in graphite, but dpas—1 can be related to the reactor flux. The flux depends on reactor design, and varies with position in the reactor core.

Neutron flux is a measure of the neutron popula­tion and speed in a reactor. In a reactor, neutrons move at a variety of speeds in randomly orientated directions. Neutron flux is defined as the product of the number of neutrons per unit volume moving at a given speed, as given by eqn [6] below.

number number /cm

f ——— — = « ————- 3- v — [6]

cm2 s cm3 s

Table 5 Ratio of graphite damage to nickel flux as measured in PLUTO
However, as there is a spectrum of neutrons, with many velocities, this is not a useful unit for the material scientist. Therefore, integrated flux is used over a range of energies E1 to E2 as given by eqn [7].
Position Ratio fd/fNi C4 — inside fuel element stainless steel thimble 0.518 D3 — inside fuel element stainless steel thimble 0.468 C4 — inside fuel element aluminum thimble 0.507 D4 — empty fuel element 0.564
f	[7]

Modified from Bell, J.; Bridge, H.; Cottrell, A.; Greenough, G.; Reynolds, W.; Simmons, J. Philos. Trans. R. Soc. Lond. A Math. Phys. Sci. 1962, 254(1043), 361-395.

In this way, a measure of neutron damage at any position within a structural component can be defined as follows. For a material such as graphite,

the damaging power (displacement rate), f d, can be expressed as an integrated flux as given in eqn [8].	energy E1 to produce a recoil atom with energy E2, and v(E2) is a ‘damage function’ giving the number of atoms displaced from their lattice site by recoil energy E2. The carbon displacement rate, f ds, at a standard position in DIDO is 5.25 x 10-8 dpa. The derivation of the damage function (Figure 7) is on the basis of billiard ball mechanics, energy losses to the lattice due to impacts, and to forces associated with excitation of the lattice. The early Kinchin and Pease17 form of the dam­age function was found to underestimate damage in graphite. To give greater dpa, it was recommended that ‘Lc’ was artificially increased, but this was not satisfactory. The Thompson and Wright18 damage function was used in the official definition of EDNF. However, the Norgett eta/.19 damage function is used in most modern reactor physics codes and it has been recently shown that there is little difference in the calculation of graphite damage using either of these latter two functions.20,21 It is assumed that the ratio of dpa to nickel flux (fds/fs) at the standard position, which is equal to 1313 x 10-24 dpa (n cm-2 s-1)-1, can be equated to the same ratio fd/fNi in the reactor of interest as given by eqn [11]:
1 C(E)f(E)dE 0
fd	[8]
where f (E) is the neutron flux with energies from E to E + dE and C(E) is a function to describe the ability of neutrons to displace carbon atoms.
4.11.5.4.1 DIDO equivalent flux At the standard position in a hollow fuel element, the nickel flux, fs, can be defined by eqn [9].
f	[9]
0
1 where J f s(E)sNi(E)dE is the integral of neutron 0 flux multiplied by the nickel cross-section at the standard position in DIDO, and S0 is the average nickel cross-section for energies > 1 MeV, which is equal to 0.107 barn. The value of fs at this position is 4 x 1013 n cm-2 s-1. The carbon displacement rate can be calculated using eqn [10].
1313 x 10 24dpa(ncm 2s 1) 1 [11]
f(E1)s(E1, E2)v(E2)dE1dE2
[10]
where f (E1) is the flux of neutrons with energy E1, This value was derived using the Thompson and s(E1, E2) is the cross-section for a neutron with Wright damage function and an early flux spectrum
Figure 7 Comparison of various damage function models that describe the number of displaced atoms versus energy of primary knock-on atom.

Table 6 Comparison of calculated and measured graphite damage rates using the Thompson and Wright model Location Calculated Measured/ standard DIDO hollow fuel element 1.00 1.00 PLUTO empty lattice position 0.975 1.22 DR-3 empty lattice position 0.975 0.90 BR-2, Mol, hollow fuel element 1.00 0.90 HFR-Petten core 1.02 1.0 BEPO TE-10 hole 2.31 2.04 BEPO empty fuel channel 2.36 2.04 BEPO hollow fuel channel 0.98 0.87 Windscale AGR replaced fuel 2.70 2.28 stringer B Windscale AGR replaced fuel 2.71 2.03 stringer D Windscale AGR loop stringer 2.60 2.08 Windscale AGR loop control 2.60 2.51 stringer Windscale AGR fuel element — 1.18 1.06 inner ring Windscale AGR fuel element — 1.39 1.06 outer ring Calder x-hole 2.12 2.10 Dounreay fast reactor core 0.46 0.50 Modified from Marsden, B. J. Irradiation damage in graphite due to fast neutrons in fission and fusion systems; IAEA, IAEA TECDOC — 1154; 2000.

Table 7 Energies, cross-sections, and mean number of displacement for various particles Particles Energy (eV) Cross-section Mean number of (cm2) displacements per collision Electrons 1 x 106 14.5 x 10~24 1.6 2 x 106 15.0 x 10~24 1.9 3 x 106 15.5 x 10~24 2.3 4 x 106 16.0 x 10~24 2.5 Protons 1 x 106 7.8 x 10~21 4-5.5 5 x 106 1.56 x 10-21 4-5.5 10 x 106 7.8 x 10~21 4-6 20 x 106 3.9 x 10-21 4-6 Deuterons 1 x 106 1.56 x 10~2° 4-5 5 x 106 3.12 x 10-21 4-6 10 x 106 1.6 x 10~21 4-6 20 x 106 7.8 x 10~22 4-6.5 a-Particles 1 x 106 1.25 x 10~19 4-5 5 x 106 2.5 x 10~20 4-6 10 x 106 1.25 x 10~20 4-6.5 20 x 106 6.25 x 10“21 4-6.5 Neutrons 103 4.7 x 10~24 2.83 104 4.7 x 10~24 28.3 105 4.6 x 10~24 280 106 2.5 x 10~24 480 107 1.4 x 10~24 500 Source: Simmons, J. Radiation Damage in Graphite; Pergamon: London, 1965.

for the standard position in DIDO. Hence, the EDNF or f d can be calculated at the position of interest. The equivalent DIDO nickel dose (fluence) (EDND) is derived by integrating EDNF over time, as given in eqn [12]:

fd(t )dt

0

Table 6 compares the calculated and measured graphite damage rates in various systems using the Thompson and Wright model.

Finally, for those wishing to try and reproduce damage in graphite using ion beams, Table 7 gives the energies, cross-sections, and mean number of displacement for various particles.

4.11.5.3 Energy Above 0.18 MeV

Dahl and Yoshikawa22 noted that for energies above 0.065 MeV, eqn [13] was reasonably independent of reactor spectrum under consideration:

f(E )ff(E)v(E)dE

f(£>£0 = 1 [13]

f(E)dE

E1

Equation [13] is the integral of graphite displace­ment for a position in the particular reactor of interest, divided by the integral of flux from E1 (0.065 MeV in this case) to infinity at the same position. Table 8 gives this ratio for two other values of E1.

(E)f(E, t)dE

fG = 1 — 1 [14]

(E)w(E)dE/ w(E)dE

0 d 0

Equation [14] is essentially graphite dpa divided by a normalized fission flux. A similar unit is defined by Simmons4 in his book. However, the use of this unit was never taken up for general use.

4.11.5.7 Fluence Conversion Factors

Table 9 gives the conversion factor from other units to EDND. The following should be noted:

• EDND is a definition,

• Calder equivalent dose and other units relating damage to fuel ratings are approximate,

• BEPO equivalent dose is a thermal unit and should be avoided,

• Energies above En are a good approximation,

• dpa is directly proportional to EDND.

4.11.5.8 Irradiation Annealing and EDT

The reasoning behind the use of equivalent DIDO temperature (EDT) is that if two specimens are irra­diated to the same fluence over two different time periods, the specimen irradiated faster will contain the most irradiation damage. The reasoning is that the spec­imen irradiated at the slower rate would have a longer time available to allow for ‘annealing’ out of defects caused by fast neutron damage as outlined below.

The rate of accumulation of damage dC/dt can be described by eqn [15].

Table 9 Conversion factors to EDND Fluence unit Conversion factor EDND (ncm-2) 1.0 Equivalent fission dose (ncm-2) 0.547 Calder equivalent dose (MWd At-1) 1.0887 x 1017 BEPO equivalent dose (n cm-2) 0.123 En > 0.05 MeV (n cm-2) 0.5 En > 0.18 MeV (n cm-2) 0.67 En > 1.0 MeV (n cm-2) 0.9 dpa (atom/atom) 7.6162 x 1020 Modified from Marsden, B. J. Irradiation damage in graphite due to fast neutrons in fission and fusion systems; IAEA, IAEA TECDOC-1154; 2000.

where ф is the flux, E is the activation energy, T is temperature (K), and k is Boltzmann’s constant. Equating the damage rate for two identical samples at different flux levels ф1 and ф2 and different tem­peratures T1 and T2,

ф1 = ф2 4-щ) e4-щ)

Rearranging this gives the EDT relationship:

1___ 1 = k і (ф)2

01 02 E n ф )1

The term on the left is the difference in the recipro­cal of the temperatures in the two systems (tempera­ture has traditionally been given the symbol ‘0 ’ when used in this context) and the term on the right con­tains the natural log ofthe damage flux (or fluence) in the two systems divided by each other. In practice, the activation energy E is an empirical constant.

The use of EDT has recently been investigated24 at temperatures above 300 °C. The authors concluded that the use of EDT was inappropriate (Figure 8). However, below 300 °C, there was some evidence of the applicability,15 but at these lower temperatures there is little reliable data. Therefore, the use of the EDT concept is not recommended for modern graph­ite moderated reactors where the graphite is usually irradiated above 300 °C.