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Radiation Effects on Electrical and Thermal Properties
CHAPTER TEN
Radiation Effects on Electrical and Thermal Properties R. E. NIGHTINGALEf
10-1 Electrical Resistance The mean free path of the conduction electrons in unirradiated electrographites is relatively large, being limited only by scattering at the crystallite boundaries. The introduction of crystal defects by neutron and heavy-particle radiation changes the concentration of the charge carriers (electrons and holes) and also the number of scattering centers. The study of these changes has aided the understanding of the nature of the radiationinduced defects. The impact of changes in electrical resistivity per se on the engineering aspects of reactor technology has been relatively minor. Such changes have introduced no problems in the operation of graphite-moderated reactors. Primak 1 has suggested the use of resistivity of graphite as a means of monitoring the damaging portion of the neutron flux in radiation-damage studies (Sec. 7-5.5). 10-1.1
PARTICLE-ACCELERATOR IRRADIATIONS
A number of studies have been conducted on the effects of chargedparticle irradiation on the electrical resistivity of graphite. The irradiation temperatures range from that of liquid helium to 250° C; the bombarding particles include protons, 2 deuterons, 3 ' 4 alpha particles, 3 ' 4 and electrons.5-7 In terms of the energy dissipated during the irradiations, the efficiency of the particles of the same energy for increasing the resistivity increases in the order: electron, proton, deuteron, and alpha particle. 4 The effects of proton bombardments have been compared with neutron irradiations at several temperatures. As shown in Fig. 10.1, the changes resulting from proton and neutron irradiations are similar. The initial slopes of the curves and the saturation values decrease with irradiation temperature, whereas the exposures where saturation begins increase with irradiation temperature. By assuming that the two-dimensional electronband model is applicable and by making certain other simplifying assumptions, Hove 8 has concluded that the number of electrons trapped per effective scatterer decreases with increasing irradiation temperature. The rapid decrease with irradiation temperature of the initial slopes of the curves in Fig. 10.1 is believed to reflect the fact that fewer free electrons are trapped t Hanford Laboratories, General Electric Company, Richland, Wash. 295
296
R. E .
NIGHTINGALE
at the higher temperatures; the change in saturation values is attributed to the decrease in the relative number of trapped electrons per scattering center. These conclusions are compatible with the damage model proposed by Hennig and Hove 9 wherein it is suggested that at low temperatures (—200°C) close interstitial-vacancy pairs are produced which trap only i.o
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ft.
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—#
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. f .
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Irradiation Temperature
10
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Neutron irradiations in range - I 9 5 ° C to - I 4 8 ° C (Brookhaven) and at about 30°C ( Hanford)
1
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20 30 40 50 Proton Exposure , / u a - h r / c m ^
1
60
—\
J_j 70
in electrical resistivity of graphite irradiated with neutrons and open symbols refer to samples irradiated at a lower temperature temperature indicated. An equivalence of 1 ^a-hr/cm2 to 1.5 (From Hove, Progress in Nuclear Energy, Series V, Metallurgy Press, Ref. 8.)
one electron. Those defects that anneal only at higher irradiation temperatures are believed to include isolated interstitials that trap one electron, isolated vacancies that trap two electrons, and clustered defects that trap several electrons. Presumably all these act as single scatterers, and therefore, when the close pairs are annealed, the ratio of trapped electrons to scatterers decreases. It will be noted from Fig. 10.1 that proton irradiations at 30°C and above, followed by annealing at a higher temperature, result in a slightly larger change in electrical resistivity than irradiations carried out at higher temperatures. The phenomenon of irradiation annealing is suggested; however, of all the physical properties studied, electrical resistivity has been found to be the least affected by annealing during neutron irradiation (Sec. 13-4). 10-1.2 REACTOR IRRADIATIONS The effects of reactor radiation on the electrical resistivity of graphite have been measured on a wide variety of materials ranging from graphitized lampblack to carefully selected single crystals. Nuclear graphites, which lie between these two extremes, have been most thoroughly studied. The data on two grades are shown in Fig. 10.2. In contrast to most other proper-
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
297
ties, the changes in electrical resistivity rapidly approach a saturation value, and the changes thereafter occur over a long period of time. This rapid approach to saturation has been explained 9 ' 12 in terms of the relative change in the number of charge carriers and scattering centers. At low doses the increase in the concentration of positive holes is almost balanced by the 3.5 CSF ( I I )
^
.C.SFJJ-) Δ/)/^=3.3
■2.5
at 11,100 Mwd/At
TSGBF(II)
• ^ - ^ ^ T S G B F
Ap/Po-- 1.87 at 8,100 M Mw wdd//A Att
U)
Δ/>//> 0 = Ι.4Ι at 8,100 Mwd/At
_ i.o
0.5
p0t ohm-cm (ID CSF 7.1 TSGBF 13.7
1000
xlO 4 Ratio U ) U)/(ll) 11.8 1.66 18.5 1.35
2000 Exposure , Mwd/At
3000
FIG. 10.2 The fractional change in the room-temperature electrical (Δρ/po) resulting from irradiations near room temperature. 10 · 11
4000
resistivity
decrease in the concentration of free electrons, and therefore the increase in resistivity is almost proportional to the number of scattering centers. After a relatively low dose (^200 Mwd/At at 30°C), the free-electron concentration is greatly reduced and does not contribute significantly to the conductivity. At this point the increasing concentration of holes counteracts the decreasing mean free path between scattering centers, and the resistivity remains relatively constant. Although there is little change in p after 200 Mwd/At, the changes that continue to occur in the graphite cause the activation energy for annealing electrical resistivity to increase (see Fig. 13.12). Apparently the scattering centers become more complex and more difficult to anneal as the structure becomes heavily damaged.
298
R. E. NIGHTINGALE
Because the initial resistivity of graphite is strongly dependent on the degree of crystallinity, the fractional increase in resistivity caused by irradiation is also sensitive to the crystallinity. Relatively large differences in the saturation value of fractional resistivities are found, even between different grades of nuclear graphite (see Fig. 10.2). The differences in the graphitization temperatures of CSF and TSGBF (2800 and 2450°C, respectively) and the graphitizability of the starting cokes largely account for the different degree of crystallinity (Table 5.2) and different values of p0. More extreme examples may be found in the literature. For example, graphitized carbon black (p0 = 60 X 10-4 ohm-cm) saturates 13 at Ap/p0 ~ 0.15 when irradiated at 30°C; whereas natural flake crystals with a very high anisotropy ratio [p 0 (l)/p 0 (ll) ^ 140] saturate 14 at Δρ/ρ 0 (||) ~ 19 and Δρ/ρ 0 (±) ~ 2. Saturation values of Δρ/ρ0 for nuclear graphites reported by other investigators 12 ' 15 are similar to those shown in Fig. 10.2. The changes in electrical resistivity of graphite irradiated above room temperature are shown in Fig. 10.3. If the changes are compared with those '
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Exposure , Mwd/At
FIG. 10.3 The fractional change in the room-temperature electrical resistivity (Δρ/ρο) of CSF (±) graphite at several irradiation temperatures.16
of stored energy, length, c spacing, and thermal conductivity, it will be seen that electrical resistivity is the least sensitive to the irradiation temperature. At 1000 Mwd/At the ratios of the fractional changes in various properties at 30°C to those at 300°C vary from about 12 to 25 for most cases; this ratio is only 1.7 for electrical resistivity. The fact that irradiation does not greatly enhance annealing of electrical resistivity (Sec. 13-4) is undoubtedly related to this temperature dependence.
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
299
10-2 Magnetoresistance The magnetoresistance decreases very rapidly as the electron mobility decreases (Sec. 6-4.4). For example, the magnetoresistance coefficient (A) definedf by ^
p
= AH2
(10.1)
where H is the magnetic-field strength, is reduced in KC (||) graphite from 5.1 X 10"10 to <0.01 X 10"10 emu even before the Hall coefficient reaches a maximum.17 At an exposure of approximately 50 Mwd/At at 30°C, the magnetoresistance coefficient becomes negative and then slowly approaches zero at higher exposures. The detailed changes following the rapid initial decrease are not well understood, particularly the anomalous negative values. 10-3 Hall Coefficient The Hall coefficient is of considerable interest because it provides direct information on the nature and number of electron carriers (Sec. 6-4.3). The effect of irradiation on the Hall coefficient has been reported by several investigators with generally similar results. The studies include short-term (<200 Mwd/At) irradiations, 18 ' 19 and longer term irradiations followed by annealing.20 The effects of exposure on two materials with widely different crystallite sizes are illustrated in Fig. 10.4. In Sec. 10-1.2 the changes in electrical resistivity were explained in terms of the scattering due to damage centers and the change in the number of charge carriers. The Hall effect can be qualitatively explained in a similar fashion.21 Since the Hall coefficient is negative in unirradiated graphite, the charge carriers are predominantly electrons [i.e., ηβμ2β > ηημ\ (see Eq. 6.8)]. As traps are produced by irradiation, the ratio of the charge carried by free electrons to the charge carried by positive holes decreases. The Hall coefficient passes through zero and attains a maximum positive value. At higher exposures when ηβμξ <
300
R.
E.
NIGHTINGALE
for KC ( | | ) . I t reaches a maximum value (not shown in the figure) of 2.1 emu at 24 Mwd/At. The difference in these two curves has been attributed 13 to additional trapping of free electrons at the crystallite boundaries of the graphitized lampblack. Changes in the Hall coefficient of irradiated graphite have been correlated with the changes produced chemically by a known concentration of bisulfate ions.23 The bisulfate ions were introduced quantitatively by the electrolysis of concentrated sulfuric acid between graphite electrodes.
0
500
1000 Exposure, M w d / A t
1500
2000
FIG. 10.4 The effects of room-temperature irradiation on the Hall coefficient for a nuclear graphite, KC (||), and a graphitized lampblack, SA-25. The crystallite size of SA-25 is approximately one-tenth that of KC.13, "
Both the chemical impurities and the radiation-induced defects act as electron traps. I t was found that 9 X 10~5 chemical traps per carbon atom produce the same change in the Hall coefficient as a reactor exposure of 1 Mwd/At. The Hall coefficient of the irradiated graphite and of the graphite containing bisulfate ions was described satisfactorily by the two-band conductor model,24 although the electrical resistance and magnetoresistance of the latter material were not. 10-4 Thermoelectric Power As does the Hall coefficient, the thermoelectric power provides information on the effects of irradiation on the electronic band structure. The
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
301
thermoelectric power is an extremely sensitive function of the dose (Fig. 10.5). If it is assumed that the number of electrons trapped from the conduction band is proportional to the exposure, the two-dimensional band theory correctly predicts the shape of the curves in Fig. 10.5 for KC (||) graphite. 25 ' 26 For graphites with relatively large crystallites, the mean free path of the conduction electrons appears to be independent of their energy. Exposures at 3 0 ° C
100
200
300
Temperature ,°K
FIG. 10.5 The thermoelectric power of irradiated KC (II) and SA-25 graphitized lampblack.18
The thermoelectric power of SA-25 is not as sensitive to irradiation effects as that of KC, and it is not until high exposures are reached that their thermoelectric behaviors become similar. This has been explained in terms of the type of scattering center.26 In graphites with large crystallites, defect scattering is also relatively important even in unirradiated samples, and irradiation only increases the number of defects. In graphites with small crystallites, boundary scattering is expected to predominate until the crystallites have suffered appreciable damage. This explanation cannot account for the different behavior of the unirradiated sample. Although it has been noted13 that the difference could be due to the fact that scattering at boundaries and scattering at internal defects probably depend differently upon the energy of the electrons, no calculations have been published. Although the Hall coefficient and magnetic susceptibility give a common value of about 9 X 10~5 traps per carbon atom per megawatt-day per adjacent ton, 23 ' 26 no such correlation exists for the thermoelectric power.2 Moreover, the correlation between neutron and proton irradiations which has been derived from other properties does not apply to thermoelectric
302
R.
E.
NIGHTINGALE
power. One must conclude that thermoelectric effects in graphite are not well understood at the present time. 10-5 Magnetic Properties The theory of electronic phenomena in graphite indicates that the magnetic susceptibility should depend only on the free-electron concentration. In contrast to other electronic properties, it should be independent of scattering effects. The magnetic susceptibility is unique in another respect. Although the susceptibility is highly anisotropic, the total susceptibility (the sum of the susceptibilities measured in any three orthogonal directions) is almost independent of direction of measurement and crystallite size for materials that have been heated to graphitization temperatures. Graphite is highly diamagnetic. Values of the total susceptibility measured in electromagnetic (cgs) units range from —21.5 X 10~6 for graphitized lampblack (SA-25) to —23.2 χ 10~6 for compressed flakes of natural Ceylon graphite. 27 A slightly lower diamagnetism of —20.8 X 10~6 emu has been measured for nuclear KC graphite. The diamagnetism is attributed to the presence of electrons near the edge of the Brillouin zone (Sec. 6-4.1). These electrons are trapped by damage centers, and therefore the susceptibility is rapidly reduced by
IOOO
2000
3000
4000
Exposure, M w d / A t
FIG. 10.6 The change in total magnetic susceptibility of KC graphite due to irradiation at 30° C. The total susceptibility is the sum of the susceptibilities measured in any three orthogonal directions.10
irradiation as illustrated in Fig. 10.6. A correlation26 with the susceptibility resulting from chemical traps indicates that over the exposure range 0 to 20 Mwd/At approximately 9 X 10~5 electron traps per atom are formed per megawatt-day per adjacent ton.
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
303
As neutron bombardment proceeds, trapping centers having an associated spin will make a paramagnetic contribution to the susceptibility. If it is assumed that the paramagnetic centers are isotropic and the Peierls type diamagnetism is anisotropic, then the ratio of susceptibilities measured parallel and perpendicular to the axis of extrusion can be used to separate the paramagnetic and diamagnetic contributions. The data 27 indicate that the paramagnetic contribution is small for exposures less than 1000 Mwd/ At, although the evidence on this point would be more conclusive on graphites having a higher anisotropy. From an analysis 28 of the available data, it was concluded that the paramagnetic contribution to the susceptibility is proportional to the square root of the neutron exposure and attains a value of only 0.5 X 1 0 6 emu at 1000 Mwd/At.
o
O
200
400
600 800 Exposure , Mwd/At
1000
FIG. 10.7 Paramagnetic resonance9 of graphite irradiated at 30°C.
The measurement of paramagnetic spin centers by magnetic susceptibility is difficult because of the large diamagnetism. A paramagnetic resonance method has been employed29 for this purpose, and the results are shown in Fig. 10.7. The number of paramagnetic spin centers formed per carbon atom (Nc) during room-temperature irradiations is given approximately by Nc = 4 X 10"5#1/2
(10.2)
where E is the room-temperature exposure in megawatt-days per adjacent ton. Thus the rate of production of electron traps is several times the rate of production of paramagnetic centers in the exposure range (0 to 20 Mwd/ At) over which the former has been measured. The concentration of paramagnetic centers appears to saturate at about 1000 Mwd/At, which suggests that the centers are simple defects, perhaps single interstitials or single vacancies. Since paramagnetic centers produced at — 196°C are mobile at — 100°C, it is more probable that the centers are predominantly interstitials.
304
R. E .
NIGHTINGALE
10-6 Thermal Conductivity 10-6.1
REACTOR IRRADIATIONS
Conduction of heat in polycrystalline graphite takes place mainly by lattice waves rather than by electronic transport. The evidence for this comes largely from the Wiedemann-Franz ratio for graphite, which is about 100 times larger than the theoretical value for purely electronic conduction (Sec. 6-2.1). The mean free path of the lattice waves, to which the thermal conductivity is roughly proportional, is of the same order of magnitude as the size of the individual crystallites, indicating that the lattice waves are scattered at crystal boundaries. The thermal conductivity (fc) is markedly reduced by radiation-induced scattering centers created within the crystallites. This reduction in thermal conductivity in the moderator structure is not necessarily undesirable. However, it is necessary to be able to anticipate changes in thermal conductivity so that the reactor lattice can be designed to accommodate the accompanying changes in heat transfer which will occur during the life of the reactor. The radiation-induced changes in the thermal resistivity measured at room temperature are shown in Figs. 10.8 and 10.9. As with electrical resisCSF (II) CSF ( 1 )
TSGBFU) TSGBF(II) k 0 ,cal/(sec)(cm)(°C) Ratio (ID U ) lll)/U) CSF 0.26 1.6 0.40 TSGBF 0.20 0.18 I.I Korite 0.015
— —
0
2000
4000
600C
8000
10000
Exposure, Mwd / A t
FIG. 10.8 Fractional change in the room-temperature thermal resistivity due to reactor irradiation near 30° C. The crystallinity of the unirradiated materials decreases in the following order: CSF, TSGBF, and Korite. The curves for exposures >3000 Mwd/At are based upon only a few points that are scattered about the curves. The uncertainty in the values above 3000 Mwd/At is ±10 per cent.10,11'30
tivity, the fractional change in thermal resistivity, (k0/k) — 1, is greater in the more crystalline materials. Prolonged neutron irradiation at 30°C reduces the thermal conductivity of each graphite to a common value, which approaches 0.005 to 0.007 cal/(sec) (cm) (°C). Data on the thermal conductivity of graphite irradiated in the temperature range 50 to 400°C are limited to low exposures (<1200 Mwd/At).
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
305
More information is available at higher temperatures; after 4500 Mwd/At the fractional change 31 in the thermal resistivity of CSF (1) is 3.4 at 400°C and 2.3 at 500°C. Of more direct interest in reactor applications is the effect of hightemperature irradiation on the thermal conductivity measured at high
O
500
1000
1500
Exposure, M w d / A t
FIG. 10.9 The fractional change in the room-temperature thermal resistivity of CSF (_L) graphite irradiated at several temperatures.16,31
temperatures. Only fragmentary results have been reported, and one must be guided by measurements made on unirradiated material and by theoretical predictions. The lattice thermal conductivity of solids is given approximately by the Debye relation k = iCp\v
(10.3)
where Cp is the lattice specific heat, λ is the mean free path of the scattered lattice waves, and v is the effective velocity of the waves (see also Sec. 6-2.1). Carter 32 has derived a family of curves of k vs. T, each curve representing a graphite characterized by a ratio λι/λθ, where Aj is the temperature-independent wave length determined by the boundary scattering and λθ is the temperature-dependent wave length assumed to be inversely proportional to the absolute temperature.f The value of λ is given by the equation
\ =f+f
(10 4)
·
t Recent work41 indicates that better agreement with experiment is obtained if λθ is assumed to be inversely proportional to the square of the absolute temperature.
306
R. E. NIGHTINGALE
The product CPX contains the temperature-dependent part of the thermal conductivity. A set of curves normalized by choosing νλ\ to give the peak thermal conductivity of CSF (1) graphite at 300°K is shown in Fig. 10.10. As \i/\e decreases, the curves tend to flatten out, and the maximum in the curves shifts to higher temperatures.
0
500
1000
1500
2000
Temperature,°K
FIG. 10.10 Theoretical curves for the thermal conductivity of graphite for various values of λ»/λθ at 300°K. The value of λί/λ θ is less than 60 for graphites that are incompletely graphitized or that have suffered radiation damage.32
The trends suggested by experimental measurements are in general accord with these predictions in those few cases where it is possible to make valid comparisons. The experimental work is summarized in Fig. 10.11. The KC graphite is a well-graphitized material having a preferred orientation somewhat greater than CSF. Figure 10.11 and additional measurements at low temperature 32 show that the maximum thermal conductivity occurs near 0°C, slightly lower than that for CSF. Other generally similar curves have been reported on unirradiated polycrystalline graphites. 32-35 The graphites MH4LM-90, R-0025, and ATL-82 are high-density (p ~ 1.90 g/cm 3 ) low-permeability materials (see Table 6.23). The high thermal conductivity of MH4LM-90 suggests that it is highly crystalline; the anisotropy is low, and little difference is expected in the thermal conductivity measured in the transverse and parallel directions. After a short irradiationf at 360 to 420° C, the thermal conductivity is reduced to 50 per cent of its original value at 30°C and is almost independent of the measuring temperature. The thermal conductivities of ATL-82 and R-0025 are lower than that of MH4LM-90. In the case of the R-0025, this may be due to the nongraphitizing carbon black that was added to the base formulation. Again f The changes in thermal conductivity suggest that the dose of 3.6 X 1019 neutrons (thermal)/cm2 in the Materials Testing Reactor is roughly equivalent to 100 Mwd/At.
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
307
^Y-V-V-H-I-H-I-M-M-I-I-H-H-I-Χ-Χ-Χ-Χ-Χ-Χ-Χ-Χ-Χ-χ-χ-χ-χ-χ-χ-χ.χ-χ-
O
200
400
600
800
1000
Temperature °C
FIG. 10.11 The change in the thermal conductivity of graphite with temperature. Where two curves are shown for a given material, the lower one pertains to the irradiated sample.
the effect of irradiation on both materials is to reduce the conductivity and displace the maximum of the curves to higher temperatures. The CSO and CSGBF (1) curves show that irradiation to much higher doses reduces the thermal conductivity further and almost completely eliminates the maximum in the curves. 10-6.2
INTERPRETATION
The radiation-induced changes in the low-temperature (<100°K) thermal conductivity have been interpreted on the basis of a two-phase structure composed of graphitic particles, each consisting of many single crystallites, and bonded by a thin isotropic layer of carbon.36 I t is suggested that the isotropic layer is formed from some of the pitch that does not graphitize completely. Since the extent of graphitization of the pitch binder is at present a controversial point and is not essential to the argument, it is perhaps best to simply assume that well-crystallized grains are separated by isotropic carbon regions whose dimensions are large compared with the wave length of the thermal wave. The equation that gives the
308
R. E. NIGHTINGALE
temperature dependence of thermal conductivity for this model has the form37 j
(10.5)
= AT + BT* + C
where A is inversely proportional to the crystalline size, B is directly proportional to the number of internal defects of the crystallite, and C is a function of the thickness of the isotropic regions. Experimental values of Ta/k are shown as a function of T2 in Fig. 10.12. The slopes of the curves
Exposure at 30°C, Mwd/At
0
2000
4000 2
2
6000
8000
FIG. 10.12 The function T*/k vs. T2 for KC (II) graphite at several exposures at 30°C. Note the change in scale for the 460 Mwd/At curve. (From Hove and Smith, Physical Review, Ref. 36.)
decrease with temperature, as predicted from Eq. 10.5, and become almost constant at the higher temperatures. Similar behavior is found with AWG graphite and the graphitized lampblack SA-25. The slope of the linear portion of each curve should be proportional to the number of defects in the crystalline phase. As expected, these slopes increase rapidly with increasing exposure. By subtracting the slope of the unirradiated sample and correcting for the type of graphite, Hove and Smith36 derived a function proportional to the number of radiation-induced defects as a function of exposure for KC, AWG, and SA-25. This function indicates that, although the crystallite size of SA-25 is of the order of one-tenth that of KC and AWG, about twice as many defects appear to be retained by SA-25 as by the other two materials. This result is contrary to expectations (Sec. 12-4.2) and has not been explained. The validity of Eq. 10.5 is supported by the dependence on temperature
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
309
of the thermal conductivity of dilute bromine-graphite residue compounds. Since the residual bromine in these compounds is thought to be at the boundary sites, the linear portion of T3/k vs. T2 curves should have the same slope until bromine begins to enter the lattice ( > 1 per cent bromine) ; then the slope should increase. This behavior has been observed experimentally. 36 A detailed interpretation of the radiation-induced changes to the thermal conductivity above about 100°K in terms of the structure of graphite has not been developed. I t is apparent that Eq. 10.5 is not valid in the temperature range of 30° C and above because it incorrectly predicts that the thermal conductivity increases monotonically with temperature. 10-7 Specific Heatt Irradiation causes a small increase in the spécifie heat (Cp) of graphite. De Sorbo and Tyler 38 have measured the effect over the temperature range 13 to 300°K on a nuclear graphite irradiated to about 2200 Mwd/At at 30°C. Goodman et al.39 have measured the increase in specific heat between 20 and 80°K for three different neutron exposures. The increase in Cp can be understood if it is assumed that the atoms which are displaced to interstitial positions have an effective Debye temperature much less than that of the average lattice atom. If the displaced atoms are assumed to be isolated interstitials, they will contribute to the specific heat at low temperatures like simple Einstein oscillators. The fraction (/) of displaced interstitial atoms has been estimated 38 by fitting the excess specific heat, ACP, to an Einstein function with a characteristic temperature of 180°K. The ACP is normalized to a saturation value at 300°K of 3Ä/, where R is the gas constant. This gives a value of / = 0.014, which is about an order of magnitude less than the average value calculated from other properties for this exposure (Sec. 7-4). The lower value may be due to the considerable amount of clustering which would be expected in a severely damaged sample. In any case, the agreement is satisfactory in view of the uncertainties in such estimates. Goodman et al.39 calculated the characteristic frequency of an atom displaced to an interplanar position directly below the center of a carbon hexagon. The characteristic frequency normal to the (0 01) planes was calculated with two different forms of the potential function and was found to be several times greater than that calculated for the two vibrational modes parallel to the planes. I t was therefore concluded that ACP is determined by the two parallel modes. The ACP VS. T curves of Goodman are represented quite well by Einstein functions with a characteristic temperature of 105 ± 5°K. The calcut Specific heat as used in this section refers to the specific heat measured at temperatures low enough that no annealing of irradiation damage takes place during measurement.
310
R. E .
NIGHTINGALE
lated frequencies of the parallel vibrational modes were equated to the frequency derived from the characteristic Einstein temperature, and an average value was calculated for the local expansion of the planes of 1.4 ± 0 . 4 A. This compares favorably with the value of e calculated by Bacon and Warren 40 (see Sec. 9-1.4). To calculate /, Goodman assumed that ACP approaches the value 2Rf since the calculations indicated that only two vibrational modes contributed to ACP. At low neutron exposures (E), the ratio f/E approached a value of (3.4 ± 0.5) X 10~22 per thermal neutron/cm 2 . Comparison with other calculations of / is difficult because E is given in terms of thermal neutrons per square centimeter in an unreported facility. However, if a reasonable equivalence of 1 Mwd/At = 1017 to 1018 neutrons/cm 2 is assumed, these results are in good agreement with other estimates of the displacement rate. The pre- and postirradiation measurements of De Sorbo and Tyler were not performed on the same sample. Goodman has reported that a scatter in Cp of up to 10 per cent may be expected on different samples. Hence the uncertainties in De Sorbo and Tyler's values of the characteristic temperature and of /, which are derived from the ACP curves, are probably large enough that they are not in serious disagreement with Goodman's values. References 1. W. Primak and L. H. Fuchs, Fast Neutron Damaging in Nuclear Reactors. I. Radiation Damage Monitoring with the Electrical Conductivity of Graphite, Nuclear Sei. and Eng., 2: 49-56 (1957). 2. G. E. Deegan, Thermal and Electncal Properties of Graphite Irradiated at Temperatures from 100 to 426°K, USAEC Report NAA-SR-1716, Atomic International, Dec. 15, 1956. 3. C. R. Malmstrom, Studies on Nuclear Reactors. 10. Experimental Method for the Determination of Resistivity Change in Graphite Bombarded with Deuterons and Alpha Particles, USAEC Report NAA-SR-10, North American Aviation, Inc., Mar. 1, 1948. 4. F. Faris, Results of Resistivity-Range Measurements on Graphite Bombarded with Charged Particles, USAEC Report NAA-SR-14 (Del.), North American Aviation, Inc., Oct. 12, 1950. 5. D. T. Eggen and W. E. Parkins, Preliminary Experiments on Radiation Damage Due to Electron Bombardment, USAEC Report NAA-SR-37, North American Aviation, Inc., Sept. 12, 1949. 6. D. T. Eggen, Energy Required for Atomic Displacements in Graphite Determined by Electron Bombardment, USAEC Report NAA-SR-69, North American Aviation, Inc., Apr. 10, 1950. 7. S. B. Austerman and J. E. Hove, Irradiation of Graphite at Liquid Helium Temperatures, Phys. Rev., 100: 1214-1215 (1955). 8. J. E. Hove, Low Temperature Irradiation and Annealing Effects in Graphite, in Progress in Nuclear Energy, Series V, Metallurgy and Fuels, H. M. Finniston and J. E. Hove (Eds.), Vol. 2, pp. 551-569, Pergamon Press, New York, 1959. 9. G. R. Hennig and J. E. Hove, Interpretation of Radiation Damage to Graphite, in
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
311
Proceedings of the First International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1955, Vol. 7, pp. 666-675, United Nations, New York, 1956. 10. W. K. Woods et al., Irradiation Damage to Artificial Graphite, in Proceedings of the First International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1955, Vol. 7, pp. 455-471, United Nations, New York, 1956. 11. E. M. Woodruff, Hanford Laboratories, General Electric Company, unpublished data, November 1960. 12. G. H. Kinchin, The Effects of Irradiation on Graphite, in Proceedings of the First International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1955, Vol. 7, pp. 472-478, United Nations, New York, 1956. 13. A. W. Smith and J. D. McClelland, Electronic Properties of a Graphitized Lampblack and Their Dependence on Neutron Irradiation, USAEC Report NAA-SR-248, North American Aviation, Inc., June 1, 1953. 14. W. Primak and L. H. Fuchs, Radiation Damage to the Electrical Conductivities of Natural Graphite Crystals, Phys. Rev., 103: 541-544 (1956). 15. V. I. Klimenkov and Yu. N. Aleksenko, Change in the Properties of Graphite When Irradiated by Neutrons, in Conference of the Academy of Sciences of the USSR on the Peaceful Uses of Atomic Energy, July 1-5, 1955, Session of the Division of Physical and Mathematical Sciences (in English translation as USAEC Report AEC-TR-2435, Pt. 1, pp. 227-237). 16. J. F. Fletcher, Controlled Temperature Irradiation of Graphite, Interim Report No. 3, PT-105-403-P, USAEC Report HW-36221, Hanford Atomic Products Operation, Sept. 5, 1956. (Classified) 17 W. P. Eatherly and J. J. Donoghue, A New Circuit for Precision Measurement of the Hall and Magneto-Resistive Effects with Results of Observations on Reactor Irradiated Graphite, USAEC Report NAA-SR-68, North American Aviation, Inc., Apr. 5, 1950. 18. R. J. Maurer and R. C. Ruder, The Hall Effect in Neutron Irradiated and Annealed Graphite, USAEC Report CP-2890, Carnegie Institute of Technology, Apr. 25, 1945. 19. G. H. Kinchin, Changes in the Electrical Properties of Graphite Due to Neutron Irradiation, J. Nuclear Energy, 1: 124-129 (1954). 20. C. J. Delbecq and G. R. Hennig, The Hall Coefficient of Pile Irradiated Graphite, USAEC Report ANL-4416, Argonne National Laboratory, Mar. 9, 1950. 21. H. Brooks, Nuclear Radiation Effects in Solids, Ann. Rev. Nuclear Sei., 6: 215-276 (1956). 22. D. F. Johnston, A Calculation of the Density of Electron-Trapping Defects in Neutron-Irradiated Graphite from Measurements of the Temperature Variation of the Hall Coefficient, J. Nuclear Energy, 1: 311-318 (1955). 23. G. R. Hennig, A Chemical Model of Radiation Damage in Graphite, Nuclear Sei. and Eng., 3: 514-528 (1956). 24. G. L. Montet, Interstitial Compounds of Irradiated Graphite, Nuclear Sei. and Eng., 4: 112-133 (1958). 25. W. P. Eatherly and N. S. Rasor, The Thermoelectric Power of Graphite Dependence on Temperature, Type and Neutron Irradiation, USAEC Report NAA-SR-196, North American Aviation, Inc., Nov. 21, 1952. 26. J. E. Hove, Radiation Damage Effects on Graphite, in Proceedings of the First and Second Conferences on Carbon Held at the University of Buffalo, pp. 125-136, Waverly Press Inc., Baltimore, Md., 1956. 27. J. J. Donoghue and J. D. McClelland, The Magnetic Susceptibility of Graphite and the Effect of Irradiation on Artificial Graphite, USAEC Report NAA-SR-153, North American Aviation, Inc., Dec. 18, 1951.
312
R. E .
NIGHTINGALE
28. J. E. Hove and J. D. McClelland, Magnetic Susceptibility of Neutron-Damaged Graphite, / . Chem. Phys., 26: 1028-1030 (1957). 29. G. R. Hennig and B. Smaller, Paramagnetism of Irradiated Graphite, USAEC Report ANL-5385, Argonne National Laboratory, Jan. 20, 1955. 30. J. F . Fletcher and W. A. Snyder, Use of Graphite in the Atomic Energy Program, Am. Ceram. Soc. Bull, 36: 101-104 (1957). 31. R. E. Nightingale et al., Damage to Graphite Irradiated up to 1000°C, in Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, Vol. 7, pp. 295-300, United Nations, New York, 1959. 32. R. L. Carter et al., Recent Developments in the Technology of Sodium-Graphite Reactor Materials, in Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, Vol. 7, pp. 72-81, United Nations, New York, 1959. 33. R. W. Powell and F . H. Schofield, The Thermal and Electrical Conductivities of Carbon and Graphite to High Temperatures, Proc. Phys. Soc. (London), 5 1 : 15ä-172 (1939). 34. W. P . Eatherly et al., Physical Properties of Graphite Materials for Special Nuclear Applications, in Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, Vol. 7, pp. 389-401, United Nations, New York, 1959. 35. D . E. Baker, Hanford Laboratories, General Electric Company, unpublished data, 1958. 36. J . . E. Hove and A. W. Smith, Interpretation of the Low-Temperature Thermal Conductivity of Graphite, Phys. Rev., 104: 892-900 (1956). 37. J. E. Hove, Thermal Properties of Graphite, in Proceedings of the Third Conference on Carbon Held at the University of Buffalo, pp. 515-527, Pergamon Press, New York, 1959. 38. W. De Sorbo and W. W. Tyler, Effect of Irradiation on the Low-Temperature Specific H e a t of Graphite, / . Chem. Phys., 26: 244-247 (1957). 39. B. B. Goodman et al., L'effet de l'irradiation neutronique sur la chaleur spécifique du graphite, Compt. rend., 248: 956-959 (1959). 40. G. E. Bacon and B. E. Warren, X-ray Diffraction Studies of Neutron-Irradiated Graphite, Acta Cryst., 9: 1029-1035 (1956). 41. R. L. Carter, Atomics International, unpublished data, 1961.
Radiation Effects on Electrical and Thermal Properties R. E. NIGHTINGALEf
10-1 Electrical Resistance The mean free path of the conduction electrons in unirradiated electrographites is relatively large, being limited only by scattering at the crystallite boundaries. The introduction of crystal defects by neutron and heavy-particle radiation changes the concentration of the charge carriers (electrons and holes) and also the number of scattering centers. The study of these changes has aided the understanding of the nature of the radiationinduced defects. The impact of changes in electrical resistivity per se on the engineering aspects of reactor technology has been relatively minor. Such changes have introduced no problems in the operation of graphite-moderated reactors. Primak 1 has suggested the use of resistivity of graphite as a means of monitoring the damaging portion of the neutron flux in radiation-damage studies (Sec. 7-5.5). 10-1.1
PARTICLE-ACCELERATOR IRRADIATIONS
A number of studies have been conducted on the effects of chargedparticle irradiation on the electrical resistivity of graphite. The irradiation temperatures range from that of liquid helium to 250° C; the bombarding particles include protons, 2 deuterons, 3 ' 4 alpha particles, 3 ' 4 and electrons.5-7 In terms of the energy dissipated during the irradiations, the efficiency of the particles of the same energy for increasing the resistivity increases in the order: electron, proton, deuteron, and alpha particle. 4 The effects of proton bombardments have been compared with neutron irradiations at several temperatures. As shown in Fig. 10.1, the changes resulting from proton and neutron irradiations are similar. The initial slopes of the curves and the saturation values decrease with irradiation temperature, whereas the exposures where saturation begins increase with irradiation temperature. By assuming that the two-dimensional electronband model is applicable and by making certain other simplifying assumptions, Hove 8 has concluded that the number of electrons trapped per effective scatterer decreases with increasing irradiation temperature. The rapid decrease with irradiation temperature of the initial slopes of the curves in Fig. 10.1 is believed to reflect the fact that fewer free electrons are trapped t Hanford Laboratories, General Electric Company, Richland, Wash. 295
296
R. E .
NIGHTINGALE
at the higher temperatures; the change in saturation values is attributed to the decrease in the relative number of trapped electrons per scattering center. These conclusions are compatible with the damage model proposed by Hennig and Hove 9 wherein it is suggested that at low temperatures (—200°C) close interstitial-vacancy pairs are produced which trap only i.o
i
! Jt1^
0.8 L
i
—■—^Το —· M
1
.-.r-. Ο
°C
Ί
1
Λ
υ
—
250°C
9
'
\I jf ff MAr s u/ S V i
FIG. 10.1 Change 8.6-Mev protons. The and annealed at the Mwd/At is assumed. and Fuels, Pergamon
ft.
o' —r-\ ru
A
—#
0
n
. f .
0.6 |
1
i
Irradiation Temperature
10
'
'
***
_ J
A
1
Neutron irradiations in range - I 9 5 ° C to - I 4 8 ° C (Brookhaven) and at about 30°C ( Hanford)
1
1
20 30 40 50 Proton Exposure , / u a - h r / c m ^
1
60
—\
J_j 70
in electrical resistivity of graphite irradiated with neutrons and open symbols refer to samples irradiated at a lower temperature temperature indicated. An equivalence of 1 ^a-hr/cm2 to 1.5 (From Hove, Progress in Nuclear Energy, Series V, Metallurgy Press, Ref. 8.)
one electron. Those defects that anneal only at higher irradiation temperatures are believed to include isolated interstitials that trap one electron, isolated vacancies that trap two electrons, and clustered defects that trap several electrons. Presumably all these act as single scatterers, and therefore, when the close pairs are annealed, the ratio of trapped electrons to scatterers decreases. It will be noted from Fig. 10.1 that proton irradiations at 30°C and above, followed by annealing at a higher temperature, result in a slightly larger change in electrical resistivity than irradiations carried out at higher temperatures. The phenomenon of irradiation annealing is suggested; however, of all the physical properties studied, electrical resistivity has been found to be the least affected by annealing during neutron irradiation (Sec. 13-4). 10-1.2 REACTOR IRRADIATIONS The effects of reactor radiation on the electrical resistivity of graphite have been measured on a wide variety of materials ranging from graphitized lampblack to carefully selected single crystals. Nuclear graphites, which lie between these two extremes, have been most thoroughly studied. The data on two grades are shown in Fig. 10.2. In contrast to most other proper-
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
297
ties, the changes in electrical resistivity rapidly approach a saturation value, and the changes thereafter occur over a long period of time. This rapid approach to saturation has been explained 9 ' 12 in terms of the relative change in the number of charge carriers and scattering centers. At low doses the increase in the concentration of positive holes is almost balanced by the 3.5 CSF ( I I )
^
.C.SFJJ-) Δ/)/^=3.3
■2.5
at 11,100 Mwd/At
TSGBF(II)
• ^ - ^ ^ T S G B F
Ap/Po-- 1.87 at 8,100 M Mw wdd//A Att
U)
Δ/>//> 0 = Ι.4Ι at 8,100 Mwd/At
_ i.o
0.5
p0t ohm-cm (ID CSF 7.1 TSGBF 13.7
1000
xlO 4 Ratio U ) U)/(ll) 11.8 1.66 18.5 1.35
2000 Exposure , Mwd/At
3000
FIG. 10.2 The fractional change in the room-temperature electrical (Δρ/po) resulting from irradiations near room temperature. 10 · 11
4000
resistivity
decrease in the concentration of free electrons, and therefore the increase in resistivity is almost proportional to the number of scattering centers. After a relatively low dose (^200 Mwd/At at 30°C), the free-electron concentration is greatly reduced and does not contribute significantly to the conductivity. At this point the increasing concentration of holes counteracts the decreasing mean free path between scattering centers, and the resistivity remains relatively constant. Although there is little change in p after 200 Mwd/At, the changes that continue to occur in the graphite cause the activation energy for annealing electrical resistivity to increase (see Fig. 13.12). Apparently the scattering centers become more complex and more difficult to anneal as the structure becomes heavily damaged.
298
R. E. NIGHTINGALE
Because the initial resistivity of graphite is strongly dependent on the degree of crystallinity, the fractional increase in resistivity caused by irradiation is also sensitive to the crystallinity. Relatively large differences in the saturation value of fractional resistivities are found, even between different grades of nuclear graphite (see Fig. 10.2). The differences in the graphitization temperatures of CSF and TSGBF (2800 and 2450°C, respectively) and the graphitizability of the starting cokes largely account for the different degree of crystallinity (Table 5.2) and different values of p0. More extreme examples may be found in the literature. For example, graphitized carbon black (p0 = 60 X 10-4 ohm-cm) saturates 13 at Ap/p0 ~ 0.15 when irradiated at 30°C; whereas natural flake crystals with a very high anisotropy ratio [p 0 (l)/p 0 (ll) ^ 140] saturate 14 at Δρ/ρ 0 (||) ~ 19 and Δρ/ρ 0 (±) ~ 2. Saturation values of Δρ/ρ0 for nuclear graphites reported by other investigators 12 ' 15 are similar to those shown in Fig. 10.2. The changes in electrical resistivity of graphite irradiated above room temperature are shown in Fig. 10.3. If the changes are compared with those '
1
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'
Irradiation °C Temperature, 30
—
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—
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LL
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i
1
500
i
1
1000
i
L
1500
Exposure , Mwd/At
FIG. 10.3 The fractional change in the room-temperature electrical resistivity (Δρ/ρο) of CSF (±) graphite at several irradiation temperatures.16
of stored energy, length, c spacing, and thermal conductivity, it will be seen that electrical resistivity is the least sensitive to the irradiation temperature. At 1000 Mwd/At the ratios of the fractional changes in various properties at 30°C to those at 300°C vary from about 12 to 25 for most cases; this ratio is only 1.7 for electrical resistivity. The fact that irradiation does not greatly enhance annealing of electrical resistivity (Sec. 13-4) is undoubtedly related to this temperature dependence.
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
299
10-2 Magnetoresistance The magnetoresistance decreases very rapidly as the electron mobility decreases (Sec. 6-4.4). For example, the magnetoresistance coefficient (A) definedf by ^
p
= AH2
(10.1)
where H is the magnetic-field strength, is reduced in KC (||) graphite from 5.1 X 10"10 to <0.01 X 10"10 emu even before the Hall coefficient reaches a maximum.17 At an exposure of approximately 50 Mwd/At at 30°C, the magnetoresistance coefficient becomes negative and then slowly approaches zero at higher exposures. The detailed changes following the rapid initial decrease are not well understood, particularly the anomalous negative values. 10-3 Hall Coefficient The Hall coefficient is of considerable interest because it provides direct information on the nature and number of electron carriers (Sec. 6-4.3). The effect of irradiation on the Hall coefficient has been reported by several investigators with generally similar results. The studies include short-term (<200 Mwd/At) irradiations, 18 ' 19 and longer term irradiations followed by annealing.20 The effects of exposure on two materials with widely different crystallite sizes are illustrated in Fig. 10.4. In Sec. 10-1.2 the changes in electrical resistivity were explained in terms of the scattering due to damage centers and the change in the number of charge carriers. The Hall effect can be qualitatively explained in a similar fashion.21 Since the Hall coefficient is negative in unirradiated graphite, the charge carriers are predominantly electrons [i.e., ηβμ2β > ηημ\ (see Eq. 6.8)]. As traps are produced by irradiation, the ratio of the charge carried by free electrons to the charge carried by positive holes decreases. The Hall coefficient passes through zero and attains a maximum positive value. At higher exposures when ηβμξ <
300
R.
E.
NIGHTINGALE
for KC ( | | ) . I t reaches a maximum value (not shown in the figure) of 2.1 emu at 24 Mwd/At. The difference in these two curves has been attributed 13 to additional trapping of free electrons at the crystallite boundaries of the graphitized lampblack. Changes in the Hall coefficient of irradiated graphite have been correlated with the changes produced chemically by a known concentration of bisulfate ions.23 The bisulfate ions were introduced quantitatively by the electrolysis of concentrated sulfuric acid between graphite electrodes.
0
500
1000 Exposure, M w d / A t
1500
2000
FIG. 10.4 The effects of room-temperature irradiation on the Hall coefficient for a nuclear graphite, KC (||), and a graphitized lampblack, SA-25. The crystallite size of SA-25 is approximately one-tenth that of KC.13, "
Both the chemical impurities and the radiation-induced defects act as electron traps. I t was found that 9 X 10~5 chemical traps per carbon atom produce the same change in the Hall coefficient as a reactor exposure of 1 Mwd/At. The Hall coefficient of the irradiated graphite and of the graphite containing bisulfate ions was described satisfactorily by the two-band conductor model,24 although the electrical resistance and magnetoresistance of the latter material were not. 10-4 Thermoelectric Power As does the Hall coefficient, the thermoelectric power provides information on the effects of irradiation on the electronic band structure. The
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
301
thermoelectric power is an extremely sensitive function of the dose (Fig. 10.5). If it is assumed that the number of electrons trapped from the conduction band is proportional to the exposure, the two-dimensional band theory correctly predicts the shape of the curves in Fig. 10.5 for KC (||) graphite. 25 ' 26 For graphites with relatively large crystallites, the mean free path of the conduction electrons appears to be independent of their energy. Exposures at 3 0 ° C
100
200
300
Temperature ,°K
FIG. 10.5 The thermoelectric power of irradiated KC (II) and SA-25 graphitized lampblack.18
The thermoelectric power of SA-25 is not as sensitive to irradiation effects as that of KC, and it is not until high exposures are reached that their thermoelectric behaviors become similar. This has been explained in terms of the type of scattering center.26 In graphites with large crystallites, defect scattering is also relatively important even in unirradiated samples, and irradiation only increases the number of defects. In graphites with small crystallites, boundary scattering is expected to predominate until the crystallites have suffered appreciable damage. This explanation cannot account for the different behavior of the unirradiated sample. Although it has been noted13 that the difference could be due to the fact that scattering at boundaries and scattering at internal defects probably depend differently upon the energy of the electrons, no calculations have been published. Although the Hall coefficient and magnetic susceptibility give a common value of about 9 X 10~5 traps per carbon atom per megawatt-day per adjacent ton, 23 ' 26 no such correlation exists for the thermoelectric power.2 Moreover, the correlation between neutron and proton irradiations which has been derived from other properties does not apply to thermoelectric
302
R.
E.
NIGHTINGALE
power. One must conclude that thermoelectric effects in graphite are not well understood at the present time. 10-5 Magnetic Properties The theory of electronic phenomena in graphite indicates that the magnetic susceptibility should depend only on the free-electron concentration. In contrast to other electronic properties, it should be independent of scattering effects. The magnetic susceptibility is unique in another respect. Although the susceptibility is highly anisotropic, the total susceptibility (the sum of the susceptibilities measured in any three orthogonal directions) is almost independent of direction of measurement and crystallite size for materials that have been heated to graphitization temperatures. Graphite is highly diamagnetic. Values of the total susceptibility measured in electromagnetic (cgs) units range from —21.5 X 10~6 for graphitized lampblack (SA-25) to —23.2 χ 10~6 for compressed flakes of natural Ceylon graphite. 27 A slightly lower diamagnetism of —20.8 X 10~6 emu has been measured for nuclear KC graphite. The diamagnetism is attributed to the presence of electrons near the edge of the Brillouin zone (Sec. 6-4.1). These electrons are trapped by damage centers, and therefore the susceptibility is rapidly reduced by
IOOO
2000
3000
4000
Exposure, M w d / A t
FIG. 10.6 The change in total magnetic susceptibility of KC graphite due to irradiation at 30° C. The total susceptibility is the sum of the susceptibilities measured in any three orthogonal directions.10
irradiation as illustrated in Fig. 10.6. A correlation26 with the susceptibility resulting from chemical traps indicates that over the exposure range 0 to 20 Mwd/At approximately 9 X 10~5 electron traps per atom are formed per megawatt-day per adjacent ton.
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
303
As neutron bombardment proceeds, trapping centers having an associated spin will make a paramagnetic contribution to the susceptibility. If it is assumed that the paramagnetic centers are isotropic and the Peierls type diamagnetism is anisotropic, then the ratio of susceptibilities measured parallel and perpendicular to the axis of extrusion can be used to separate the paramagnetic and diamagnetic contributions. The data 27 indicate that the paramagnetic contribution is small for exposures less than 1000 Mwd/ At, although the evidence on this point would be more conclusive on graphites having a higher anisotropy. From an analysis 28 of the available data, it was concluded that the paramagnetic contribution to the susceptibility is proportional to the square root of the neutron exposure and attains a value of only 0.5 X 1 0 6 emu at 1000 Mwd/At.
o
O
200
400
600 800 Exposure , Mwd/At
1000
FIG. 10.7 Paramagnetic resonance9 of graphite irradiated at 30°C.
The measurement of paramagnetic spin centers by magnetic susceptibility is difficult because of the large diamagnetism. A paramagnetic resonance method has been employed29 for this purpose, and the results are shown in Fig. 10.7. The number of paramagnetic spin centers formed per carbon atom (Nc) during room-temperature irradiations is given approximately by Nc = 4 X 10"5#1/2
(10.2)
where E is the room-temperature exposure in megawatt-days per adjacent ton. Thus the rate of production of electron traps is several times the rate of production of paramagnetic centers in the exposure range (0 to 20 Mwd/ At) over which the former has been measured. The concentration of paramagnetic centers appears to saturate at about 1000 Mwd/At, which suggests that the centers are simple defects, perhaps single interstitials or single vacancies. Since paramagnetic centers produced at — 196°C are mobile at — 100°C, it is more probable that the centers are predominantly interstitials.
304
R. E .
NIGHTINGALE
10-6 Thermal Conductivity 10-6.1
REACTOR IRRADIATIONS
Conduction of heat in polycrystalline graphite takes place mainly by lattice waves rather than by electronic transport. The evidence for this comes largely from the Wiedemann-Franz ratio for graphite, which is about 100 times larger than the theoretical value for purely electronic conduction (Sec. 6-2.1). The mean free path of the lattice waves, to which the thermal conductivity is roughly proportional, is of the same order of magnitude as the size of the individual crystallites, indicating that the lattice waves are scattered at crystal boundaries. The thermal conductivity (fc) is markedly reduced by radiation-induced scattering centers created within the crystallites. This reduction in thermal conductivity in the moderator structure is not necessarily undesirable. However, it is necessary to be able to anticipate changes in thermal conductivity so that the reactor lattice can be designed to accommodate the accompanying changes in heat transfer which will occur during the life of the reactor. The radiation-induced changes in the thermal resistivity measured at room temperature are shown in Figs. 10.8 and 10.9. As with electrical resisCSF (II) CSF ( 1 )
TSGBFU) TSGBF(II) k 0 ,cal/(sec)(cm)(°C) Ratio (ID U ) lll)/U) CSF 0.26 1.6 0.40 TSGBF 0.20 0.18 I.I Korite 0.015
— —
0
2000
4000
600C
8000
10000
Exposure, Mwd / A t
FIG. 10.8 Fractional change in the room-temperature thermal resistivity due to reactor irradiation near 30° C. The crystallinity of the unirradiated materials decreases in the following order: CSF, TSGBF, and Korite. The curves for exposures >3000 Mwd/At are based upon only a few points that are scattered about the curves. The uncertainty in the values above 3000 Mwd/At is ±10 per cent.10,11'30
tivity, the fractional change in thermal resistivity, (k0/k) — 1, is greater in the more crystalline materials. Prolonged neutron irradiation at 30°C reduces the thermal conductivity of each graphite to a common value, which approaches 0.005 to 0.007 cal/(sec) (cm) (°C). Data on the thermal conductivity of graphite irradiated in the temperature range 50 to 400°C are limited to low exposures (<1200 Mwd/At).
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
305
More information is available at higher temperatures; after 4500 Mwd/At the fractional change 31 in the thermal resistivity of CSF (1) is 3.4 at 400°C and 2.3 at 500°C. Of more direct interest in reactor applications is the effect of hightemperature irradiation on the thermal conductivity measured at high
O
500
1000
1500
Exposure, M w d / A t
FIG. 10.9 The fractional change in the room-temperature thermal resistivity of CSF (_L) graphite irradiated at several temperatures.16,31
temperatures. Only fragmentary results have been reported, and one must be guided by measurements made on unirradiated material and by theoretical predictions. The lattice thermal conductivity of solids is given approximately by the Debye relation k = iCp\v
(10.3)
where Cp is the lattice specific heat, λ is the mean free path of the scattered lattice waves, and v is the effective velocity of the waves (see also Sec. 6-2.1). Carter 32 has derived a family of curves of k vs. T, each curve representing a graphite characterized by a ratio λι/λθ, where Aj is the temperature-independent wave length determined by the boundary scattering and λθ is the temperature-dependent wave length assumed to be inversely proportional to the absolute temperature.f The value of λ is given by the equation
\ =f+f
(10 4)
·
t Recent work41 indicates that better agreement with experiment is obtained if λθ is assumed to be inversely proportional to the square of the absolute temperature.
306
R. E. NIGHTINGALE
The product CPX contains the temperature-dependent part of the thermal conductivity. A set of curves normalized by choosing νλ\ to give the peak thermal conductivity of CSF (1) graphite at 300°K is shown in Fig. 10.10. As \i/\e decreases, the curves tend to flatten out, and the maximum in the curves shifts to higher temperatures.
0
500
1000
1500
2000
Temperature,°K
FIG. 10.10 Theoretical curves for the thermal conductivity of graphite for various values of λ»/λθ at 300°K. The value of λί/λ θ is less than 60 for graphites that are incompletely graphitized or that have suffered radiation damage.32
The trends suggested by experimental measurements are in general accord with these predictions in those few cases where it is possible to make valid comparisons. The experimental work is summarized in Fig. 10.11. The KC graphite is a well-graphitized material having a preferred orientation somewhat greater than CSF. Figure 10.11 and additional measurements at low temperature 32 show that the maximum thermal conductivity occurs near 0°C, slightly lower than that for CSF. Other generally similar curves have been reported on unirradiated polycrystalline graphites. 32-35 The graphites MH4LM-90, R-0025, and ATL-82 are high-density (p ~ 1.90 g/cm 3 ) low-permeability materials (see Table 6.23). The high thermal conductivity of MH4LM-90 suggests that it is highly crystalline; the anisotropy is low, and little difference is expected in the thermal conductivity measured in the transverse and parallel directions. After a short irradiationf at 360 to 420° C, the thermal conductivity is reduced to 50 per cent of its original value at 30°C and is almost independent of the measuring temperature. The thermal conductivities of ATL-82 and R-0025 are lower than that of MH4LM-90. In the case of the R-0025, this may be due to the nongraphitizing carbon black that was added to the base formulation. Again f The changes in thermal conductivity suggest that the dose of 3.6 X 1019 neutrons (thermal)/cm2 in the Materials Testing Reactor is roughly equivalent to 100 Mwd/At.
10. EFFECTS ON ELECTRICAL AND THERMAL PROPERTIES
307
^Y-V-V-H-I-H-I-M-M-I-I-H-H-I-Χ-Χ-Χ-Χ-Χ-Χ-Χ-Χ-Χ-χ-χ-χ-χ-χ-χ-χ.χ-χ-
O
200
400
600
800
1000
Temperature °C
FIG. 10.11 The change in the thermal conductivity of graphite with temperature. Where two curves are shown for a given material, the lower one pertains to the irradiated sample.
the effect of irradiation on both materials is to reduce the conductivity and displace the maximum of the curves to higher temperatures. The CSO and CSGBF (1) curves show that irradiation to much higher doses reduces the thermal conductivity further and almost completely eliminates the maximum in the curves. 10-6.2
INTERPRETATION
The radiation-induced changes in the low-temperature (<100°K) thermal conductivity have been interpreted on the basis of a two-phase structure composed of graphitic particles, each consisting of many single crystallites, and bonded by a thin isotropic layer of carbon.36 I t is suggested that the isotropic layer is formed from some of the pitch that does not graphitize completely. Since the extent of graphitization of the pitch binder is at present a controversial point and is not essential to the argument, it is perhaps best to simply assume that well-crystallized grains are separated by isotropic carbon regions whose dimensions are large compared with the wave length of the thermal wave. The equation that gives the
308
R. E. NIGHTINGALE
temperature dependence of thermal conductivity for this model has the form37 j
(10.5)
= AT + BT* + C
where A is inversely proportional to the crystalline size, B is directly proportional to the number of internal defects of the crystallite, and C is a function of the thickness of the isotropic regions. Experimental values of Ta/k are shown as a function of T2 in Fig. 10.12. The slopes of the curves
Exposure at 30°C, Mwd/At
0
2000
4000 2
2
6000
8000
FIG. 10.12 The function T*/k vs. T2 for KC (II) graphite at several exposures at 30°C. Note the change in scale for the 460 Mwd/At curve. (From Hove and Smith, Physical Review, Ref. 36.)
decrease with temperature, as predicted from Eq. 10.5, and become almost constant at the higher temperatures. Similar behavior is found with AWG graphite and the graphitized lampblack SA-25. The slope of the linear portion of each curve should be proportional to the number of defects in the crystalline phase. As expected, these slopes increase rapidly with increasing exposure. By subtracting the slope of the unirradiated sample and correcting for the type of graphite, Hove and Smith36 derived a function proportional to the number of radiation-induced defects as a function of exposure for KC, AWG, and SA-25. This function indicates that, although the crystallite size of SA-25 is of the order of one-tenth that of KC and AWG, about twice as many defects appear to be retained by SA-25 as by the other two materials. This result is contrary to expectations (Sec. 12-4.2) and has not been explained. The validity of Eq. 10.5 is supported by the dependence on temperature
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of the thermal conductivity of dilute bromine-graphite residue compounds. Since the residual bromine in these compounds is thought to be at the boundary sites, the linear portion of T3/k vs. T2 curves should have the same slope until bromine begins to enter the lattice ( > 1 per cent bromine) ; then the slope should increase. This behavior has been observed experimentally. 36 A detailed interpretation of the radiation-induced changes to the thermal conductivity above about 100°K in terms of the structure of graphite has not been developed. I t is apparent that Eq. 10.5 is not valid in the temperature range of 30° C and above because it incorrectly predicts that the thermal conductivity increases monotonically with temperature. 10-7 Specific Heatt Irradiation causes a small increase in the spécifie heat (Cp) of graphite. De Sorbo and Tyler 38 have measured the effect over the temperature range 13 to 300°K on a nuclear graphite irradiated to about 2200 Mwd/At at 30°C. Goodman et al.39 have measured the increase in specific heat between 20 and 80°K for three different neutron exposures. The increase in Cp can be understood if it is assumed that the atoms which are displaced to interstitial positions have an effective Debye temperature much less than that of the average lattice atom. If the displaced atoms are assumed to be isolated interstitials, they will contribute to the specific heat at low temperatures like simple Einstein oscillators. The fraction (/) of displaced interstitial atoms has been estimated 38 by fitting the excess specific heat, ACP, to an Einstein function with a characteristic temperature of 180°K. The ACP is normalized to a saturation value at 300°K of 3Ä/, where R is the gas constant. This gives a value of / = 0.014, which is about an order of magnitude less than the average value calculated from other properties for this exposure (Sec. 7-4). The lower value may be due to the considerable amount of clustering which would be expected in a severely damaged sample. In any case, the agreement is satisfactory in view of the uncertainties in such estimates. Goodman et al.39 calculated the characteristic frequency of an atom displaced to an interplanar position directly below the center of a carbon hexagon. The characteristic frequency normal to the (0 01) planes was calculated with two different forms of the potential function and was found to be several times greater than that calculated for the two vibrational modes parallel to the planes. I t was therefore concluded that ACP is determined by the two parallel modes. The ACP VS. T curves of Goodman are represented quite well by Einstein functions with a characteristic temperature of 105 ± 5°K. The calcut Specific heat as used in this section refers to the specific heat measured at temperatures low enough that no annealing of irradiation damage takes place during measurement.
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lated frequencies of the parallel vibrational modes were equated to the frequency derived from the characteristic Einstein temperature, and an average value was calculated for the local expansion of the planes of 1.4 ± 0 . 4 A. This compares favorably with the value of e calculated by Bacon and Warren 40 (see Sec. 9-1.4). To calculate /, Goodman assumed that ACP approaches the value 2Rf since the calculations indicated that only two vibrational modes contributed to ACP. At low neutron exposures (E), the ratio f/E approached a value of (3.4 ± 0.5) X 10~22 per thermal neutron/cm 2 . Comparison with other calculations of / is difficult because E is given in terms of thermal neutrons per square centimeter in an unreported facility. However, if a reasonable equivalence of 1 Mwd/At = 1017 to 1018 neutrons/cm 2 is assumed, these results are in good agreement with other estimates of the displacement rate. The pre- and postirradiation measurements of De Sorbo and Tyler were not performed on the same sample. Goodman has reported that a scatter in Cp of up to 10 per cent may be expected on different samples. Hence the uncertainties in De Sorbo and Tyler's values of the characteristic temperature and of /, which are derived from the ACP curves, are probably large enough that they are not in serious disagreement with Goodman's values. References 1. W. Primak and L. H. Fuchs, Fast Neutron Damaging in Nuclear Reactors. I. Radiation Damage Monitoring with the Electrical Conductivity of Graphite, Nuclear Sei. and Eng., 2: 49-56 (1957). 2. G. E. Deegan, Thermal and Electncal Properties of Graphite Irradiated at Temperatures from 100 to 426°K, USAEC Report NAA-SR-1716, Atomic International, Dec. 15, 1956. 3. C. R. Malmstrom, Studies on Nuclear Reactors. 10. Experimental Method for the Determination of Resistivity Change in Graphite Bombarded with Deuterons and Alpha Particles, USAEC Report NAA-SR-10, North American Aviation, Inc., Mar. 1, 1948. 4. F. Faris, Results of Resistivity-Range Measurements on Graphite Bombarded with Charged Particles, USAEC Report NAA-SR-14 (Del.), North American Aviation, Inc., Oct. 12, 1950. 5. D. T. Eggen and W. E. Parkins, Preliminary Experiments on Radiation Damage Due to Electron Bombardment, USAEC Report NAA-SR-37, North American Aviation, Inc., Sept. 12, 1949. 6. D. T. Eggen, Energy Required for Atomic Displacements in Graphite Determined by Electron Bombardment, USAEC Report NAA-SR-69, North American Aviation, Inc., Apr. 10, 1950. 7. S. B. Austerman and J. E. Hove, Irradiation of Graphite at Liquid Helium Temperatures, Phys. Rev., 100: 1214-1215 (1955). 8. J. E. Hove, Low Temperature Irradiation and Annealing Effects in Graphite, in Progress in Nuclear Energy, Series V, Metallurgy and Fuels, H. M. Finniston and J. E. Hove (Eds.), Vol. 2, pp. 551-569, Pergamon Press, New York, 1959. 9. G. R. Hennig and J. E. Hove, Interpretation of Radiation Damage to Graphite, in
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Proceedings of the First International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1955, Vol. 7, pp. 666-675, United Nations, New York, 1956. 10. W. K. Woods et al., Irradiation Damage to Artificial Graphite, in Proceedings of the First International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1955, Vol. 7, pp. 455-471, United Nations, New York, 1956. 11. E. M. Woodruff, Hanford Laboratories, General Electric Company, unpublished data, November 1960. 12. G. H. Kinchin, The Effects of Irradiation on Graphite, in Proceedings of the First International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1955, Vol. 7, pp. 472-478, United Nations, New York, 1956. 13. A. W. Smith and J. D. McClelland, Electronic Properties of a Graphitized Lampblack and Their Dependence on Neutron Irradiation, USAEC Report NAA-SR-248, North American Aviation, Inc., June 1, 1953. 14. W. Primak and L. H. Fuchs, Radiation Damage to the Electrical Conductivities of Natural Graphite Crystals, Phys. Rev., 103: 541-544 (1956). 15. V. I. Klimenkov and Yu. N. Aleksenko, Change in the Properties of Graphite When Irradiated by Neutrons, in Conference of the Academy of Sciences of the USSR on the Peaceful Uses of Atomic Energy, July 1-5, 1955, Session of the Division of Physical and Mathematical Sciences (in English translation as USAEC Report AEC-TR-2435, Pt. 1, pp. 227-237). 16. J. F. Fletcher, Controlled Temperature Irradiation of Graphite, Interim Report No. 3, PT-105-403-P, USAEC Report HW-36221, Hanford Atomic Products Operation, Sept. 5, 1956. (Classified) 17 W. P. Eatherly and J. J. Donoghue, A New Circuit for Precision Measurement of the Hall and Magneto-Resistive Effects with Results of Observations on Reactor Irradiated Graphite, USAEC Report NAA-SR-68, North American Aviation, Inc., Apr. 5, 1950. 18. R. J. Maurer and R. C. Ruder, The Hall Effect in Neutron Irradiated and Annealed Graphite, USAEC Report CP-2890, Carnegie Institute of Technology, Apr. 25, 1945. 19. G. H. Kinchin, Changes in the Electrical Properties of Graphite Due to Neutron Irradiation, J. Nuclear Energy, 1: 124-129 (1954). 20. C. J. Delbecq and G. R. Hennig, The Hall Coefficient of Pile Irradiated Graphite, USAEC Report ANL-4416, Argonne National Laboratory, Mar. 9, 1950. 21. H. Brooks, Nuclear Radiation Effects in Solids, Ann. Rev. Nuclear Sei., 6: 215-276 (1956). 22. D. F. Johnston, A Calculation of the Density of Electron-Trapping Defects in Neutron-Irradiated Graphite from Measurements of the Temperature Variation of the Hall Coefficient, J. Nuclear Energy, 1: 311-318 (1955). 23. G. R. Hennig, A Chemical Model of Radiation Damage in Graphite, Nuclear Sei. and Eng., 3: 514-528 (1956). 24. G. L. Montet, Interstitial Compounds of Irradiated Graphite, Nuclear Sei. and Eng., 4: 112-133 (1958). 25. W. P. Eatherly and N. S. Rasor, The Thermoelectric Power of Graphite Dependence on Temperature, Type and Neutron Irradiation, USAEC Report NAA-SR-196, North American Aviation, Inc., Nov. 21, 1952. 26. J. E. Hove, Radiation Damage Effects on Graphite, in Proceedings of the First and Second Conferences on Carbon Held at the University of Buffalo, pp. 125-136, Waverly Press Inc., Baltimore, Md., 1956. 27. J. J. Donoghue and J. D. McClelland, The Magnetic Susceptibility of Graphite and the Effect of Irradiation on Artificial Graphite, USAEC Report NAA-SR-153, North American Aviation, Inc., Dec. 18, 1951.
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28. J. E. Hove and J. D. McClelland, Magnetic Susceptibility of Neutron-Damaged Graphite, / . Chem. Phys., 26: 1028-1030 (1957). 29. G. R. Hennig and B. Smaller, Paramagnetism of Irradiated Graphite, USAEC Report ANL-5385, Argonne National Laboratory, Jan. 20, 1955. 30. J. F . Fletcher and W. A. Snyder, Use of Graphite in the Atomic Energy Program, Am. Ceram. Soc. Bull, 36: 101-104 (1957). 31. R. E. Nightingale et al., Damage to Graphite Irradiated up to 1000°C, in Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, Vol. 7, pp. 295-300, United Nations, New York, 1959. 32. R. L. Carter et al., Recent Developments in the Technology of Sodium-Graphite Reactor Materials, in Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, Vol. 7, pp. 72-81, United Nations, New York, 1959. 33. R. W. Powell and F . H. Schofield, The Thermal and Electrical Conductivities of Carbon and Graphite to High Temperatures, Proc. Phys. Soc. (London), 5 1 : 15ä-172 (1939). 34. W. P . Eatherly et al., Physical Properties of Graphite Materials for Special Nuclear Applications, in Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, Vol. 7, pp. 389-401, United Nations, New York, 1959. 35. D . E. Baker, Hanford Laboratories, General Electric Company, unpublished data, 1958. 36. J . . E. Hove and A. W. Smith, Interpretation of the Low-Temperature Thermal Conductivity of Graphite, Phys. Rev., 104: 892-900 (1956). 37. J. E. Hove, Thermal Properties of Graphite, in Proceedings of the Third Conference on Carbon Held at the University of Buffalo, pp. 515-527, Pergamon Press, New York, 1959. 38. W. De Sorbo and W. W. Tyler, Effect of Irradiation on the Low-Temperature Specific H e a t of Graphite, / . Chem. Phys., 26: 244-247 (1957). 39. B. B. Goodman et al., L'effet de l'irradiation neutronique sur la chaleur spécifique du graphite, Compt. rend., 248: 956-959 (1959). 40. G. E. Bacon and B. E. Warren, X-ray Diffraction Studies of Neutron-Irradiated Graphite, Acta Cryst., 9: 1029-1035 (1956). 41. R. L. Carter, Atomics International, unpublished data, 1961.