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KERNFORSCHUNGSANLAGE JÜLICH GmbHInstitut für Reaktorwerkstoffe

A Model for Irradiation InducedChanges in Graphite Material Properties

by

H. Cords, R. Zimmermann

JÜI -1506Juni 1978

ISSN 0366-0885

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Institut für Reaktorwerkstoffe Jül • 1506

Zu beziehen durch: ZENTRALBIBLIOTHEK der Kernforschungsanlage Jülich GmbH,Jülich, Bundesrepublik Deutschland

 Model for Irradiation InducedChanges in Graphite Material Properties

by

H. Cords, R. Zimmermann

EIN MODELL FÜR STRAHLUNGSINDUZIERTE ÄNDERUNGEN

DER M A T E R I A L E I G E N S C H A F T E N DES GRAPHITS

von

H. Cords

R. Zimmermann

KURZFASSUNG

Das hier vorgeschlagene Modell zur Beschreibung der durch Neutronen-bestrahlung erzielten Änderungen der Materialeigenschaften des Gra-phits basiert auf drei Sättigungsprozessen, die mit der Überführungvon Kohlenstoffatomen oder Gruppen von Kohlenstoffatomen aus eineranfänglich bestehenden Konfiguration in eine energetisch oder stati-stisch begünstigte Anordnung innerhalb der atomaren Struktur ver-knüpft ist. Der Einfluß der Prozesse ist durch Produkte von jeweilsdrei Faktoren dargestellt. Der erste Faktor enthält den exponentiellabklingenden Einfluß der Bestrahlungsdosis, der zweite vorwiegenddie Temperaturabhängigkeit entsprechend dem Popularisierungszustandeines damit verknüpften Energiezustàndes und der dritte Faktor ver-mittelt die Kopplung zwischen den mikroskopischen Umlagerungsprozessenund den makroskopisch beobachtbaren Materialänderungsdaten. Die.Energiezustände werden gleichzeitig thermisch und mit Hilfe der ausder Neutroneneinstrahlung direkt abgeleiteten Energie aktiviert. Ineinigen Fällen müssen die Kopplungskonstanten durch zusätzliche tem-peratur- und dosisabhängige Funktionen modifiziert werden, um demEinfluß der Mikrorisse gerecht zu werden, die sich in einem frühenStadium der Bestrahlung schließen. Die Anpassung der Materialänderungs-daten eines nahezu isotropen Graphits wurde für die Richtung parallel•zur Kornorientierung ausgeführt. Das Modell kann die gleiche Aus-sage machen wie diejenige, die im sogenannten Konzept äquivalenterTemperaturen enthalten ist. Eine Anzahl weiterer Anwendungsmöglich-keiten wird erörtert. Die zusätzliche Analyse anderer Graphitsortenwürde das Vertrauen in die allgemeine Verwendbarkeit des Modellserhöhen.

Kernforschungs-anlage Julien JOL - 1506 June 1978GmbH IRW

A MODEL FOR IRRADIATION INDUCED CHANGES

IN GRAPHITE MATERIAL PROPERTIES

von

H. CordsR. Zimmermann

ABSTRACT

Thé presently proposed model for irradiation induced changes ingraphite material properties is based on three saturating processeswhich are related to the displacement of carbon atoms or groupsof atoms from an initial configuration to an energetically orstatistically more favourable position within the atomic structure.The influence of each of the processes is represented as theproduct of three contributing factors. They contain firstly anexponentially decaying dependence on dose, secondly a temperaturedependence representing the population of related energy statesby concurrent thermal activation and activation from neutronirradiation, and thirdly coupling constants to link themicroscopic rearrangement processes to the macroscopicallyobserved material changes. The coupling constants in some caseshave to be modified by additional functions depending on irradiationtemperature and dose to account for the closure of microcracksduring the early stage of irradiation. The model was tested as toits capability to fit the change data of one nearly isotropicgraphite in the direction parallel to the grain orientation. It

. can be interpreted to give the identical statement as the one givenby the so-called Concept of Equivalent Temperatures. A number ofsuggestions for further application of the model was made. Theadditional analysis of other types of graphite would establishmore confidence in the general applicability of the model.

A MODEL FOR IRRADIATION INDUCED CHANGES IN GRAPHITE MATERIAL PROPERTIES

Page

1. INTRODUCTION 1

2. THE MODEL 3

2.1 Functional Dependence on Dose 3

2.2 Functional Dependence on Temperature 62.2.1 Positive Temperature Characteristic 62.2.2 Negative Temperature Characteristic 102.2.3 Microcracks 15

2.2.3.1 Young's Modulus 152.2.3.2 Coefficient of Thermal Expansion 202.2.3.3 Thermal Conductivity and Dimensional Changes 20

2.2.4 Alternative Approach 22

3. VERIFICATION OF THE MODEL 24

3.1 Least Square Fits to Irradiation Induced Dimensional Changes 253.2 Least Square Fits to Changes in Young's Modulus 263.3 Least Square Fits to Changes in Coefficient of Thermal Expansion 273.4 Least Square Fits to Changes in Thermal Conductivity 27

4. APPLICATIONS 29

4.1 The Concept of Equivalent Temperatures 294.2 Irradiation Induced Changes Obtained from Different Flux Levels 324.3 Variation of Both Temperature and Dose 334.4 Classification of Graphites by Means of Coupling Constants 354.5 Correlations between Material Properties using Coupling Constants 36

5. SUMMARY 38

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1. INTRODUCTION

The properties of structural materials in the core of a nuclear reactorare substantially modified by the influence of fast neutron irradiation.Graphite is the favoured material for the core of the helium-cooled hightemperature reactor. For design calculations of components with permanentresidence in the core of such a reactor it is important to know thedetailed behaviour of graphite exposed to extraordinarily large values of

fast fluence. Therefore, nuclear graphite has been extensively studied formany years and a wealth of experimental data has been accumulated. Theliterature on this subject is correspondingly comprehensive 1) but, despitethe large amount of information, an overall description of the data by meansof a physical model is not available. In the case of a new graphite such amodel should be at least capable of interpolating and extrapolating thedata in regions of temperature and dose where there are no materialproperties previously measured. This can be of great help in producingcomplete sets of data for design calculations thereby saving expenditure.Additionally the model may provide explanations to effects which up to nowhave not been understood.

A model to describe the physical properties of a material could bepurely theoretical in which case it would be designed without furtherinspection of experimental evidence on the basis of a few assumptions. Theother extreme is the entirely phenomenological description of the experimentaldata using for instance expansion series with coefficients that do not haveany physical meaning. Although this type of mathematical description can beused for interpolation and extrapolation in a limited way it can hardly becalled a model. However, if at least some of the parameters involved can beinterpreted as or correlated to physical quantities the situation is improved.It is this kind of semi-empirical model that is presented in the followingsections.

With respect to design calculations the irradiation induced dimensionalchanges are the most important quantities to be considered becausedimensional changes can cause mechanical stresses within the graphite blocksor lead to mutual interactions of the core components. Therefore the modelis developed in close relation to the corresponding experimental data. Sofar it has been checked only against one type of graphite planned to be

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used for permanent side reflectors. The material is an extruded pitch-cokegraphite with a fairly good, isotropic behaviour. The irradiation data aredescribed in ref 2) (graphite No 11), ref 3) and for higher temperaturesin ref 4) (ASI2-F-500). After a successful application of the model to thedimensional change values it is then used to reproduce the relative changesof Young's modulus, thermal conductivity and the coefficient of thermalexpansion. However, this is done with the different objective of showingthat the irradiation induced changes of a number of mechanical propertiescan be correlated with each other.

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2. THE MODEL

2.1 Functional Dependence on Dose

In order to understand the adopted approach it is important to know that2 0 - 2an accumulated dose of only 10 cm at ambient temperatures displaces

about 1 to 3% of all atoms in the material 5). Therefore during the22 -2prolonged irradiation of dose values up to the order of 10 cm all of

the atoms have the chance to regroup in a different way. Due to collisionswith neutrons and subsequent processes including the slowing of chargedparticles from knock on reactions, interstitial atoms and vacancies areformed. The interstitials and vacancies are subjected to thermal activationsuch that they become mobile and can combine to form di-interstitials as '"'well as larger clusters or can recombine. The possibilities for theformation of interstitial configurations and hole configurations arenumerous.

To deduce a model for the behaviour of macroscopic quantities it is notalways useful to consider in detail the situation, that is, microscopicallyin a deterministic way. Instead, the processes should be described in amore global, say statistical manner:- Given time and energy, thermal energyas well as energy from neutron collisions, the carbon atoms in the graphiteare shifted from their initial positions in the crystal structure. This notonly involves transfer of atoms from one place to another; after in astatistical manner an extreme amount of energy has been concentrated on oneor a group of carbon atoms, and after this energy has been redistributedwithin the crystal the atoms now carrying an average amount of energy maybe settled in a different configuration which is energetically orstatistically more favourable. If, in the beginning the graphite is in anunfavourable state then after some irradiation the graphite can betransformed into another more favourable state. Of course, a transformationin the other direction is also possible but less likely. The result is astate of equilibrium which is reached after some time of irradiation. In thefollowing it is useful to replace the term "time of irradiation" by dose.

In particular , the graphite is looked upon as being a statisticalsystem which is in a stationary state during a relatively short span ofdose. In the stationary state the system can be described by the

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laws of statistical thermodynamics, but if additionally energies fromneutron irradiation are considered, the system is not in equilibrium.Equilibrium can however be reached by continued irradiation in a nuclearreactor. It is understood that the clear-cut separation between thermalenergies and energies from irradiation is an over-simplification, but itserves as an initial assumption which later on can be corrected for.(section 2.2.1) In the present section 2.1 the kinetic equation governingthe transfiguration into the state of equilibrium is given. The subsequentsection deals with the stationary state determined by the thermal energies.

A number of assumptions are required to develop the model for descriptionof the irradiation induced changes in graphite material properties. The aimwas to formulate assumptions which do not contain more implications thannecessary in order to design the model. Therefore, the identification ofthé basic elements of the statistical system in terms of atomic entitiesis not given. The elements could be carbon atoms at lattice sites,interstitials, di-interstitials, clusters of interstitials with varyingnumber of members, etc. This kind of identification is left for furtherresearch after the model itself yielded a successful description of thedata.

Thus, without specifying what the states a_ and J) actually are andwithout specifying in what units the transfer of these configurations intoeach other takes place we postulate that the number of units AN that aretransferred from state a to state J> are proportional to the fast dose ADelapsed and proportional to the number of units in state £ i.e. Na availablefor a transfer to the state j). The reaction a to J) is considered to be onereaction out of a sequence of reactions. Therefore, the population of statej> depends upon the population of state a which is populated by the precedingreactions. If, the preceding reactions for some reason provide less units N&,then also the number of units Nb will be reduced after some dose (time)elapsed. It requires a certain dose X to reach a new equilibrium in thechain of reactions.

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In formula we postulate

ANa(D) = -XNQ(D)AD (1)

The proportional constant X can be interpreted as the probabilityof a unit in the configuration a available for transfer (N, = 1) beingtransferred from a_ to t during the elapse of the dose AD = 1. The inverseof X= 1/X yields the dose X necessary to transfer 63.2 % of the unitsfrom state a to state b. Integration of equation (1) yields

Nb = Na(1-e-XD) (2)

with the initial condition Nb(D = 0) = 0 equivalent to Na(D = 0) = N&.

Apart from the first assumption which is formulated essentially byequation (1), it is necessary to assume further that the transfer of anumber of units from state a_to state b^causes a macroscopic quantitysuch as the dimensional changes to be changed proportionally. There is agood reason to believe that this assumption is not a particularly good one.However for the start, having no further information it is the most obviousproposal to be made. The doubts for the validity of the assumption stemfrom the observation of microcracks which in an early stage of irradiationclose up and thus can buffer part of the dimensional changes frommicroscopic processes.

A third assumption is required to fit the experimental data: -There areat least three of the described processes (eq (2)) involved in theirradiation induced changes in the material properties of graphite. Two ofthem are dominant in the dimensional changes. The combined effect of theprocesses is the same as if they proceeded independently of dose. In otherwords, the effects of the processes have to be added. The differencesbetween the processes are their mean dose of elapse I- and their individualproportional constants CL which are required for the relation betweenmicroscopic units transferred and macroscopic changes observed. If Tt isfor instance the relative change of a material property then according tothe model it can be calculated by means of equation (3): /

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" TC = Z (a, N,-) (1-e-xT) =• Z (arNf ) a,D (3)

Ni corresponds to N of eq. (2).

Strictly speaking, eq. (3) is true if there are no changes in temperatureduring the irradiation.

To summarise there are three, independent saturating processes assumed.Each of these is described by the kinetic equation (1), and is coupledlinearly to macroscopic material changes.

2.2 Functional Dependence on Temperature

With respect to changes in temperature it was mentioned in the precedingsection 2.1 that the number of units in state £ available for transfer tostate J> can be increased or decreased depending on the preceding andpossibly succeeding reactions. Initially it is assumed that the otherreactions are exclusively activated by thermal energies. This implies arelatively short time to reach a state of equilibrium. Therefore a kineticequation is not required. The population of the state ji in equilibrium withregard to the thermally activated process depends upon temperature. Thetemperature characteristic of a process is called positive if with increasingtemperature the state of equilibrium is changed in favour of theconfiguration a i.e. N, is increased. It is negative if an increasing

— a

temperature causes a deficiency in the population of a i.e. N is decreased.Both characteristics can be observed in the data.

2.2.1 Positive Température Characteristic

Thermally activated processes are for instance the well known chemicalreactions. These obey the Law of Mass Action 7). Thermally activatedprocesses in solids have been frequently compared with chemical reactions.For instance, the formation of Frenkel pairs i.e. the generation ofinterstitial s and vacancies have been treated as chemical reactions 6)employing the Law of Mass Action 7). In the present case this formulationshall be used as an example how to derive the temperature dependence of Na.

-7-

[NU + IN, - N a l = ^ [Nv ] + [ N i s - N v ] (4)

The symbols in squared brackets represent the different states taking part

in the reaction. N and N are the numbers of interstitials and vacancies,

respectively at equilibrium* N-| is the total number of lattice sites, and

N. is the sum total of interstitial sites. Since N and N are the same

the Law of Mass Action can be written as ,

(N.-NJ (Nic-NJ= G kT (5)

is

In equation (5) k is the Boltzmann constant and E the activation energy

for the production of interstitials. Since the number of interstitials N_'"'

is very small compared with both the total number of lattice sites and'-tfie'

sum total of interstitial sites, N can be neglected in the denominator of '

eq. (5), such that eq. (5) is transformed into

, EN a = Y N ( Nis e"2KT (6)

From equation (6) we deduce that N& is proportional to the exponential

exp(-E/2kT). .

Although the Law of Mass Action has proved to be a valuable tool for

chemical reactions in gases and liquids, with respect to solids more

rigorous thermodynamic calculations should be applied. These sometimes

lead to temperature dependent pre-exponential factors in equation (6) which

may also be attained by taking a somewhat different approach.

In a second example let us consider electrons evaporating from a metal

surface. They are statistically treated as Fermi-Dirac particles in a

potential well. Integration of that part of the distribution with the

kinetic energy larger than the depth of the well and using only that part

having the direction of momentum normal to the surface leads to the2

Richardson-Dushman equation with T as a pre-exponential factor 8). If,

however, they obeyed the Boltzmann Statistics the pre-exponential factor

would be "TT 8). In the present investigations a number of pre-exponential

factors between 1/T and T were considered. All of them yielded satisfactory• 2 • • ' • • '

fits to the relative dimensional change data. The T -factor gave marginally

better results.

-8-

The measured data for dimensional changes were fitted by means ofeq. (3) as a function of dose. The four parameters (Ot^^)« (ÇX3N3), T 2

and I, were used as free parameters. The contribution of the first processis negligibly small, and was initially introduced.with relatively smallnumbers ( O t ^ = 0.4 % and Tj = 0.26-1021 EDN/cm2) in orderto model acertain effect at low dose values. Fits were obtained for temperaturesbetween 300 and 800 °C in steps of 50°C and between 900 and 1400 °C insteps of 100 °C. With respect to the quantity GL-N the results- are plottedversus temperature in the two diagrams Fig 1 and with respect to X in theupper part of Fig 2.

Fig 1 clearly shows the exponential dependence expected from eq. (6).However, eq. (6) also predicts that there would be no irradiation inducedchanges at all for zero temperature. This behaviour is not verified andas a matter of fact cannot be expected. A constant contribution to theirradiation induced changes will always remain if the temperature isdecreased. Therefore, in addition to the temperature dependence whichwas derived theoretically, the description of the measured data requiresa further parameter Y such that the positive temperature characteristiccan be written as

El- (ai Nj )exP = a,- ( T 2 e~2kT+Yj ) = oti a i T ( 7 )

with i = 2,3.

The additional constants Y-j ensure that with no thermal energies availablethe irradiation induced processes still proceed. The energy from collisionsof fast neutrons with carbon atoms produces units Na in competition to thethermal activation process. These entities are generated at a constantrate i.e. independent of irradiation temperatures. Fig 1 shows that thecontribution of units Na from the term Y^ is dominant for temperatureslower than 1000 °C.

As it is not possible to determine CC^ and N^ individually the splitinto two factors on the right hand sides of eq. (7) is arbitrary. Only theproduct of the two factors is accessible to measurements. Therefore thefactor OCj loses somewhat its original meaning as a proportional constantbetween the macroscopic and microscopic changes. For simplification, in

-9-

the following the factors OC., are still designated as coupling constants.Although the factorization on the right hand side is arbitrary it shouldbe pointed out that the function (X.. a.,, being a product of the positivenumber of units and the uni-directional coupling constant, cannot changesign as a function of temperature. This implies a positive value for y..

The activation energies E.. (i = 2,3) however retain their physicalmeaning. The faster process 2 is characterized by an activation energy of0.88 eV. The energy of 0.66 eV is attributed to the slower process 3. Thesenumbers can be deduced from the numerical constants of eq. (7) given inFig 1. The conversion from the units temperature into units of energy isdone by means of the factor 0.1723 eV/103 °K.

Considering Fig 2 it becomes clear that for phenomenological descriptionthe mean dose X = 1/X also requires one more parameter. The inverse ofthe mean dose viz X obviously depends linearly on temperature (upper partof Fig 2). At 300 °C the process 2 proceeds with a mean dose X„ =19.2-1021 EDN/cm2, the much slower process 3 with X- = 1127-1021 EDN/cm2.The mean doses at 1400 C are approximately half of the values given for300 °C. Although a physical explanation for this effect could not be found,the much shorter mean dose at higher temperatures compared to that at lowertemperatures might be related to the fact that at higher temperatures thesmaller clusters such as di-interstitials or even vacancies become mobile.This would indicate that a number of different units participate in thesame process as the temperature increases and the additional temperaturedependence of the mean dose would correct for this fact.

21 2In the relevant region of doses up to about 30*10 EDN/cm the thirdprocess having a \/ery high value of mean dose is effectively linear withdose. The saturation level is extremely far in advance.1 The faster process 2saturates during that dose interval.

After inspection of the experimental results for the irradiation induceddimensional changes, the contributions C. (i = 2,3) in the semi-empiricalformula for the description of the processes 2 and 3 assume the shape

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C j a i [ T e Y i ] [ 1 e ]

= a,- • an- • a iD

with i = 2,3

Numerical values for the constants E^, Y > Ö- and £,-. (i =2,3) may betaken from the Figs 1 and 2. Numeric results for the coupling constants (X-relate to the individual quantity whose change is to be described. Thecorresponding numbers for a. are given in section 3.1 i.e. Table II.

2.2.2 Negative Temperature Characteristic

The process 1 associated with the shortest mean dose X, is not a \/erypronounced component in the dimensional change data. However, both therelative changes in Young's modulus and those of the thermal conductivityinitially show a rapid variation in their absolute value which thenreaches a saturation level. According to our data the process 1 has a mean

21 2 'dose T. = 0.26*10 EDN/cm . A temperature dependence of the mean-dose

Tj similar to the one found for Tg and X 3 is likely but could not beextracted from the data.

Compared to the other two processes the assumed functional dependenceon dose D as described by eq. (2) is more clearly displayed. Changes asrapid as Tj are also observed in data for mechanical strength, electricalresistivity and in irradiation induced primary creep data.

Due to the fact that only small dose experiments are required formeasurement, the effects have been studied extensively. The increase inYoung's modulus was attributed to the pinning of mobile dislocations withinthe crystal basal planes by small irradiation induced defects and by theclosing of microporosity with crystal strain 9). Thermal conduction ingraphite is due to propagation of phonons in layer planes. Irradiationdefects may reduce the thermal conductivity either by in-plane scatteringof the phonons or by providing a mechanism for energy transfer betweenlayer planes 10).

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Plotting the irradiation induced changes in the thermal resistivity andYoung's modulus at saturation versus temperature, in both cases a lineardecrease is obtained. Therefore, contrary to the other two processes,process 1 has a negative temperature characteristic.

Considering a negative temperature characteristic the problem of findinga physical interpretation can be solved by assuming an intermediate energystate j* which at higher temperatures is less stable than at low temperatures.The fast irradiation induced changes are ascribed to the degree of populationof state]) depending on the population of the intermediate state. Thepopulation decreases with increasing temperature because at highertemperatures there are more units N. decaying than are formed from the

ot

preceding reactions. Such an energy state which is instable with increasingtemperature, is a well known phenomenon (Wigner effect). When graphite isirradiated at temperatures lower than about 200 °C it accumulates energywhich can be released in the form of heat by increasing the temperature. Thestored energy can be removed by annealing.

Thus, there are similarities between at least three different, observablequantities viz changes in thermal conductivity and Young's modulus and thestored energy. They are the low mean dose and the negative temperaturecharacteristic. For mathematical description recourse is made of the Vandmodel of thermal annealing of disordered solids described e.g. in ref. 18).The basic assumption in this model is that the annealing rate dn(E,t)/dtof defects with activation energies in the range E to E + dE at a constantannealing temperature T is described by

a

at= -A(E)xn(E,t)xexp(-E/kTa) (9a)

where n(E,t) is the concentration of defects remaining at time t. In thepresent investigations we neglect the detailed dependence on E consideringit to be a constant E,/2. Furthermore,. we are particularly interested inthe span of time t < 0 at which the sample of graphite is being irradiatedin a nuclear reactor. We assume that the duration of irradiation has beensufficiently long such that a state of equilibrium was reached. In -particular, this means that under the influence of neutron irradiationthere was a constant rate of generation of defects equal in amount todn(E,/2 , t=O)/dt but differing in sign such that a constant number of

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defects n(E1/2 , t=0) was maintained. The process or processes of generationand annihilation of defects during irradiation excluding the process ofthermal annealing can be treated according to eq. (4) to yield a proportionalrelationship for ïï, viz. eq. (6). ïï is the concentration of defects whichwould be obtained if the possibility of annealing was discarded. Therefore,ïï is greater than n and using eq. (6) ïï is proportional to exp(-E /2kT).

The annealing process can be included in a second step as follows. As thefactor exp(-E^/2kT) is the probability for a defect to be annihilated bythermal annealing, the probability for a defect not to be annihilated bythermal annealing is 1 - exp(-E,/2kT). Therefore, to obtain the concentrationof defects n the fictious number n has to be multiplied by [l-exp(-E,/2kT)] .Furthermore, as ïï was found to be proportional to exp(-E /2kT) theconcentration of defects n at equilibrium during irradiation is proportionalto the factor exp(-E /2kT) [ 1 - exp(-E,/2kT)]. This function of temperaturewas chosen as the temperature dependence for the first process. Similarto eq. (8) it is expected that there is a constant y^ involved describingthe direct influence of the neutron irradiation bypassing the thermallyactivated processes. Therefore, the temperature characteristic of the firstprocess can be modelled by

(9)

The additional possibility of using pre-exponential factors was not furtherexplored. Another way of including the constant y^ is discussed insection 2.2.4.

The temperature function given by eq. (9) begins as exp(-E0/2kT) untila maximum value is reached. After going through its maximum the functiondecreases almost linearly for a considerable range of temperatures. It isthis part of the temperature scale that is supported by experimental datafrom changes in Young's modulus and thermal conductivity (Fig 3). Thegeneral remarks following eq. (7) in section 2.2.1 also apply to eq. (9).The meaning of OL being a coupling constant between microscopic andmacroscopic changes has to be redefined to be a product of a new type ofcoupling constant and the temperature function (XJJ. Furthermore, thefunctional dependence given by eq. (9) compared with the one given by

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eq. (7) seems to be more general, because for lower temperatures it alsoyields the positive temperature characteristic elaborated in the previoussection. The remark may become useful with respect to the two otherprocesses if experimental data beyond 1400 °C should become available(viz section 2.2.4). The temperature functions given by eq. (7) lead to asingular behaviour for extremely high temperatures. For instance theprediction of the model for dimensional changes at very high temperaturesis the formation of a pronounced minimum at low dose with extremely rapidvariations. The other material data on irradiation induced changes do notquite support this feature.

Numerical results for the activation energies EQ = 0.071 eV andE, = 0.046 eV corresponding to temperatures of 413 °K and 268 °K,respectively are given in Fig 3. For comparison, the energy of motion ofa free interstitial is 0.25 eV 1). (Note, that in the present formulationthe activation energies are half the value as in ref 1) ). However, thevalues for E and E, obtained in the present investigation are not uniquelydetermined. Best fits to the data could also be obtained by othertemperature values 0 and 0, (for definition see Fig 3) connected toeach other by the empirically found relationship 0. + 0.44 • 0 j = 0.48.

Finally, the fact that Yi (Fig 3) turns out to be negative in bothcases has to be discussed with great care. The corresponding quantitiesY 2 and Yo were introduced to allow for irradiation induced changes atzero temperature. It is not expected that the model being derived for ,high temperatures also holds at near zero temperatures. Nevertheless, thebehaviour near 0 °K has implications for the higher temperatures.Furthermore, the temperature dependent factor CUy was introduced as apositive number representing the number of units available for transferin the processes. For this reason all values for Y-j (i = 1»2,3) must bepositive.

The derived temperature characteristic for the process 1 (eq. (9)),discarding the additional parameters (Xj and Yi nas a certain slope whichcannot be altered considerably by special choice of E and E^. However, ifthe experimental data prescribe a slope with a larger inclination, then the

-14-

factor (Xj can compensate for it. On the other hand, too large a factor(X^ yields values for the function CtjCXj-j. which are too high. These largevalues can only be compensated for by subtraction of the constant y-, -which implies a negative value for y^. This situation seems to be the case.The experimental data have a slope which is greater than expected; in otherwords the temperature characteristic observed is more negative thanaccounted for by the theoretical function given by eq. (9).

According to the model described so far, the fitted-experimental dataare composed of contributions from two other processes C 2 and Cg. These,however, were recognized to have a positive temperature characteristic.Therefore, the data should tend to show a rather more positive characteristicdeviating from linearity. The opposite is however observed. The temperaturedependency of the fastest component is linear with an unexpectedly large,negative slope. The solution to this problem can be found if another effectis introduced, which should produce a negative and linear temperature"characteristic at low dose. This is superimposed on the process 1 in thelow dose region at its saturation level. The subsequent section dealswith the additional influence underlying the initial irradiation inducedchanges in material characteristics.

The final result of this section can be given by the followingequation (10) which describes the contributing term Cj of the fastestprocess in dose.

= a, x On-x a1D

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2.2.3 Microcracks

2.2.3.1 Young's Modulus

After a fit of the irradiation induced dimensional changes was obtained,the. description of the Young's modulus change data was attempted using thethree components introduced in the preceding part of section 2.2. As thedescribed processes are fundamental with respect to elemental carbon itseems reasonable to assume that when considering other materialcharacteristics the physical constants such as mean doses X. and theactivation energies EQ and E. (i = 1,2,3) are identical. Only the coupling,constants which represent the phenomenological part in this model vary fromproperty to property. In particular cases some of these can also turn outto be zero.

The first attempt to fit the change of the Young's modulus data failedin so far as the admixture of the second and third processes completelydestroyed the saturation effect which is so clearly displayed by process 1in experimental results. Instead of forming a saturation level there wasan almost continuous rise of Young's modulus change towards the maximumvalue. This was the second indication of a further influence (see alsoend of section 2.2.2) which effects the irradiation behaviour at low dosevalues.

Studies of the relevant literature on graphite under irradiation showthat a very important feature of the material has not been accounted forin the model.

Microcracks are typical for carbon structures 11). They are formedas a consequence of the heat treatment at yery high temperatures withsubsequent cooling to room temperature. Even if the graphite should happento be perfectly dense, as it cools intercrystalline pores will be formed.The shrinking in the c-direction of some crystals which are rigidly heldin position will open voids in the centre. The reversed process of closing

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cracks by heating is also possible. Micröcracks produced by cooling toroom temperature are disposed in exactly the correct positions to accomodatesome of the expansion of the crystals in a subsequent heating. The graphiteheated up to and held at about 2300 °C undergoes an annealing process suchthat the damage is completely removed. When graphite is irradiated thecrystals grow in much the same way as they do on heating. At first thisgrowth is partially accomodated in the cracks formed during cooling, butcontinued irradiation closes the cracks 1).

Therefore, visualising the procedure to measure the irradiation inducedchanges of Young's modulus the following situation is obtained. A testspecimen which contains microcracks from the production process is heatedup in a reactor to be irradiated by neutrons. After accumulating asufficiently high dose the microcracks will be closed. The closure proceedsin two steps that is, firstly, by bringing the specimen up to irradiationtemperature, and secondly by irradiation. After the irradiation experimenthas been terminated the specimen is cooled down to room temperature and asecond measurement of Young's modulus is performed in order to establishthe change. It is likely that new secondary microcracks open up during thetransition from irradiation temperature to room temperature and thereshould be a larger amount of microcracking the higher the irradiationtemperature. In the extreme case of a well graphitized specimen irradiatedat the graphitizing temperature the specimen should have the same, amountof microcracking at room temperature as it had in its virgin state. Howeverexperiments show (ref 1, plate 7) that at least for low irradiationtemperatures the expected, secondary microcracking is not observed.Presumably the thermal strains are accomodated in the form of elasticstrains and as a result of the irradiation the body becomes more dense.Assuming a simple relationship between density and Young's modulus thereshould therefore be a positive change in Young's modulus. With respectto higher irradiation temperatures, the specimen probably does showsecondary microcracking resulting in a lower density and a smaller increasein Young's modulus compared to the lower irradiation temperature. In otherwords there exists a relationship between irradiation temperature and.increase in Young's modulus with a negative temperature characteristic.

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Considering now the dependence on dose, it becomes clear that theclosure of microcracks is reached later in dose if the irradiationtemperature is chosen to be lower. This is so because the partial closureof the microcracks at low irradiation temperature leaves a comparativelylarge amount of complementary closure to be achieved by the irradiationdose alone.

The described phenomenon was included in the model by means of amathematical switch to mark the closure of the microcracks on the dosescale.

S(D,T) = {Utahh[aD. + b(T

With increasing dose D, the values of the switch-function S change from0 to 1. Depending on the parameter a the transition may or may not besmooth. The switch operates, or in other words assumes, the value 1/2 ata dose determined by the irradiation temperature T. If T assumes the valueof c the switch operates at D = 0. Theoretically, the parameter ccorresponds to the graphitizing temperature. The operating point of theswitch can be shifted to higher doses by means of the parameter b.

In the model the switch couples and decouples the second, microscopicprocess and the corresponding macroscopic change of Young's modulus. Thesecond process was chosen because it describes a densification of thematerial which is expected to correspond to a proportional increase inYoung's modulus if, and only if, the microcracks are closed. Otherwisethere may be a weak coupling which can be modelled by means of theparameter a in the switch function (eq. (11)).

There is also a net densification considering the combined effect ofprocesses 2 and 3 at low dose. However the switch operating on the sum ofthe two contributions did not give as good results when fitting theexperimental data. Therefore, the correction for microcracks is onlyrequired with respect to the process causing contraction in the dimensionalchanges. Following the suggestion of other authors 13,17) to consider the

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contraction as due to a graphitization process, it is tempting to conjecturethat microcracks only occur in the poorly graphitized parts of the material.

The final formulation of the model regarding the irradiation inducedchanges in Young's modulus therefore is

3 3

= I S(- a,- * aiT x aiD = I Sr C,- (12)t- i = 1 i=1

with S^ = Sg = 1. The contributions C. (i = 1,2,3) and the switch functionS 2 are defined in the eqs. (8), (10) and (11).

At the end of section 2.2.2 a temperature characteristic for the firstprocess was established. However for temperatures lower than about 200 °Kthe temperature factor (Fig 3) would turn the process 1 into a negativecontribution to the change in Young's modulus. As such an inversion is.incontradiction to the concept of the model, a further effect was postulatedwhich was then recognized to be associated with the microcracks in thegraphite. It also was found to be a macroscopic process which thereforeshould modulate the coupling constants rather than being an isolatedcomponent.

For further investigation the problem was to unravel the influence ofall four contributing effects. The four components are described in Fig 4.The upper part represents the switch function which is applied to thecoupling constant of the 2 process. The three processes also shown inFig 4 are to be multiplied each by its individual coupling constant andadded together to yield a description of the change data. The couplingconstants as well as the parameters of thé switch function were determinedby linear least square fits. The three parameters of the switch determineits smoothness at the point of contact and the temperature dependence ofthis point with respect to dose. Concerning the switch, the result of thefitting procedure is already included in Fig 4. For irradiation temperaturesabove 1400 °C the influence of the microcracks is small. The third processwith its ^tery large mean dose requires a negative coupling constant inorder to describe the decrease of the Young's modulus at high dose values.

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The composition of the four constituents at saturation level21 2(D ~ 1.0-10 EDN/cm ) as a function of temperature is shown in Fig 5. The

upper part of Fig 5 is identical to the upper part of Fig 3 showing the

measured points and the fitted, theoretical temperature dependence of the.

process 1. Although the fit is satisfactory the negative part of the curve

at low temperatures cannot be justified. The separation of the curve is

represented in the lower two parts of Fig 5. In the centre part of Fig 5

the combined contribution of the processes 2 and 3, including the influence

of the switch for the process 2, is shown. The curve is approximately

linear for temperatures between 300 and 1000 °C as was required at the end

of section 2.2.2. For temperatures higher than 1000 °C the switch is2 1 ? ••••-•

activated at the selected dose of approximately 1-10 EDN/cm . Thereforethe positive and exponentially growing temperature characteristics of the

processes 2 and 3 become dominant. The points in the lower diagräm are

the differences between the ones from the upper part of Fig 5 and the curve

shown in the centre part. The fitted curve again represents the

theoretically derived temperature dependence of the first process. The

fitted parameter Yi = 0 now has a permissible value.

The switch function S(D,T) depending on irradiation temperature and dose

was designed according to the theoretical considerations on secondary

microcracking given at the beginning of this section. The present data seem

to support this feature (Fig 7) i.e. the saturation level caused by the

contribution C, is not very pronounced at irradiation temperatures higher

than 1000 °C compared to lower temperatures. Accordingly, at higher

temperatures the switch was designed to operate earlier at a comparatively

low dose value (Fig 4). Considering other data e.g. changes in Young's

modulus for petroleum coke graphite or gilsonite coke graphite 13) it

becomes clear that the switch function is a purely phenomenological part of

the model. Like the coupling constants, the switch functions are peculiar

to the type of graphite. As can be seen from the subsequent section, there

may be different switch functions for the different property changes of

one and the same graphite.

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2.2.3.2 Coefficient of Thermal Expansion

In the case of thermal expansion the switch was introduced to operateon the coupling constant of the fastest process 1. The idea is that apopulation of the intermediate state leads to an increase of the coefficientof thermal expansion. However, provided the graphite was not irradiatedsufficiently, the increase is not completely obtained as long as there aremicrocracks present. In order to establish the change in the coefficientof thermal expansion, a second measurement is performed after terminationof the irradiation experiment. The thermal strains introduced by theheating-up process are accomodated by the closure of the secondary orprimary microcracks and by the removal of elastic strains. Using thepresently derived switch function it is difficult to describe the data forthe irradiation induced'changes in the coefficient of thermal expansionpresented in ref 12). In this case, for higher irradiation temperatures,a larger fast neutron dose is required to cause a characteristic change.This temperature dependence is completely opposite to the one discussed inconnection with the switch for Young's modulus. This remark is related tothe one given at the end of the preceding section, and therefore is insupport of the statement that different graphites and different propertiesrequire their individual switch functions.

The present data, however, do show a behaviour consistent with theideas developed along with the switch function for Young's modulus.Depending on temperature, the data show a kind of sinusoidal curve for

pi ponly one half wave at irradiation doses up to between 5 and 15*10 EDN/cm .It can be modelled by the processes 1 and 2 including a switch function forthe process 1.

Aa 3 3

— = .Z Si cti x a i T *a i D = .1 S,- C,- (13)

with Sg = 1 and (X3 = 0. The contributions C^ (i = 1,2) and the switchfunction S, are defined in the equations (8), (10) and (11).

2.2.3.3 Thermal Conductivity and Dimensional Changes

The influence of microcracks on the irradiation induced changes of thethermal conductivity and the dimensional changes have not yet been explored.

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Considering the negative parameter y^ in Fig 3 (lower part) there is astrong indication that the microcracks do have a similar influence on thecoupling constants as was observed in changes in Young's modulus andcoefficient of thermal expansion. However, it has to be elaborated towhich process, or possibly processes, a switch function has to be applied.

With respect to the dimensional changes an initial deviation from alinear shrinkage at low doses is generally observed. Sometimes also a

pi psmall, positive excursion can be seen at \/ery low doses (D^l.0-10 EDN/cm )At the start of this work these observations were the justification forintroducing the process 1 for the dimensional changes. However, at thatstage there was no temperature characteristic available and the influenceof microcracking was not considered. Coming back to the starting position,an improved description of the dimensional change data is required. Thisin effect would lead to a second iteration in the process of findingimproved numerical results.

Because the influence of the first process including the microcrackingis so small and because a more detailed description in terms of the modelhas not yet been elaborated, it was decided to report the results obtained.

Also, if the effect of microcracking on the changes in thermalconductivity and dimensional changes can be solved satisfactorily, it isplanned to make simultaneous fits to all changes of material data in orderto have improved values for the physical constants such as activationenergies and mean doses. In the presently described procedure their valueswere solely derived from the irradiation induced dimensional changes.Therefore although good fits were obtained for the dimensional changes, thefits obtained for the other data were not as good. This is because for thesubsequent fitting procedures, only the coupling constants and the constantsin the switch function were available as free parameters. In a simultaneousfit the mean square deviation would be more justly distributed amongst allexperimental data.

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The equation obtained so far for the description of the dimensionalchanges £ w and changes in thermal conductivity A X / X is

The contributions C.. (i = 2,3) are specified in the eqs (8) and (10).However, with respect to the dimensional changes, CtjalT was chosen t0

be a constant with the value 0.4 %.

2.2.4 Alternative Approach .

In section 2.2.2 the constant Yi was introduced in analogy to the

constants Y ? ancl Ys- Under tne assumption that Yi ïike Y? an<* Y 3represents that portion of units Na directly activated by energies fromneutron collisions, a formulation alternative to eq. (9) becomesconceivable. In eq. (9),the parameter Yi was inserted to account for apopulation and depopulation of the intermediate state completely independentof the thermal mechanism and thus independent of temperature. Alternatively,the parameter y^ can be understood as a mechanism for the population ofthe intermediate state while the depopulation is a purely thermal mechanism.In this case eq. (9) should be altered to:

Eo _Ej_( a ^ U p = a, (e 2kT+Yi) Mr e2kT) = a iXa1T ( 9 a )

Eq. (9 a) is theoretically more attractive because, depending on theactivation energy E p the second factor in 0C1T becomes effective only forhigh temperatures. With respect to low temperatures the factor[exp(-EQ/2kT) + Yi 1 dominates. Therefore, without greatly affectingthe results obtained so far the second factor contained in 0L,T i.e.[l - exp(-E1/2kT)] can be applied to OL^j

and a 3 j as w e l 1« Tne operationwould make the three assumed processes formally identical. In this case,the second and third process are also referred to intermediate stateswhich, however, can be annealed only for temperatures much greater thansay 1400 °C. Possibly the alternative approach can also account for thesingular behaviour observed at the high temperature end of the presentdata (viz section 3.1).

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In the present investigations the eq. (9a) was not at first favouredbecause no satisfactory fits similar to the ones shown in Fig 3 could beobtained":- However, after subtraction of the effect due to microcracking,good fits were available using eq. (9a). Fig 5a shows that according tothe experimental data and using eq. (9a) the intermediate state is alsopopulated' for temperatures below 100 °K. For comparison the temperaturecharacteristic described by eq. (9) gave the results shown in the lower partof: Fig 5. For further investigations it is also planned to give thealternative solution appropriate attention.

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3. VERIFICATION OF THE MODEL

The preceding paragraph 2 served to introduce the model. Although theintroduction was accomplished in a deductive manner there are a number ofassumptions involved which are not necessarily very obvious. Therefore themain justification for the model is its capability to describe theirradiation induced changes of a number of properties, which will beextensively discussed. Nevertheless reference should also be made to otherapproaches described in the literature 13,17). In ref 13) it was proposedto consider two components for the description of the dimensional changes,one of the two components being responsible for shrinkage, the othercausing expansion. The shrinkage reaches a saturation level while theexpansion.proceeds indefinitely within the dose span considered. Thesuperposition of the two components yields the dimensional changes asmeasured. Also in ref 17) the change data are considered to be the algebraicsum of the influence of two simultaneous mechanisms. The first process isthe well-established disordering of crystallites, which is a highlytemperature dependent process, i.e. anneal able, and is most prominent atlower temperatures. The second process is a radiation induced densificationof regions with poorly organized carbon atoms within the manufacturedgraphite. The latter process may be likened to the thermal graphitizationof carbon. Implying that each one of the two processes from the two sources13,17) are non-identical i.e. the expansion process and thedisordering of crystallites allows the conclusion that three processeshave relevant contributions to the irradiation induced changes. The twonon-identical processes were in one case 13) described as having a positivetemperature characteristic while in the other case 17) the temperaturecharacteristic was found to be negative. These results obtained fromstudies of various types of graphite substantiate the present model.

Furthermore, although the subsequent paragraph 4 serves to point outthe various applications, it can be interpreted in part to support themodel. Some of the subjects dealt with in that paragraph are explanationsto effects reported by other authors.

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3.1 Leaàt Square Fits to Irradiation Induced Dimensional Changes,

The experimental data subjected to the fitting procedure were obtainedfrom the refs. 3) and 4). The graphite is the nearly isotropic ATR-2E(earlier labelled ASI2-F-5OO) produced by the company SIGRI. It is assumedto show the same irradiation behaviour as the so-called graphite no 11from ref 2. The data from ref 3) cover the low temperature region withtemperatures greater than 300 °C and less than 800 °C. With respect to thehigh temperature region the data from ref 4) were used covering temperaturesbetween 400 °C and 1400 °C. Only the data referred to the direction parallelto the grain were investigated. In the overlapping temperature region, thereare deviations of the experimental data from the two sources (Fig 6), whichprovide an estimation of the errors pertaining to the experimental method.The model fits having deviations from.the data of the same order as thedata itself can therefore be called good fits (Fig 6).

A few points should be mentioned:

Firstly,

- The small, positive excursion exhibited by the modelled curve at lowdose and in the low temperature region cannot be deduced from the datain refs 3) and 4). Nevertheless the data from ref 4) cannot beextrapolated linearly to produce no change at no dose. This indicatesthat there is a further process involved not entirely due to the processes•2 and 3 of the model. Therefore the model was provided with a smalladmixture of the process 1. This was certainly not done in the correctmanner accounting for microcracking and a proper temperature dependenceof that process. The problem has to be subject to further studies withmore data in the low dose region. The effect as such is well established 2)

Secondly,

- The model curves, as well as the data, exhibit a singular behaviour athigh temperatures. With increasing temperature the amount of maximumshrinkage increases more than linearly. Also the turn around point isshifted to ^ery low values of dose. It is suspected that for temperatures

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greater than 1500 °C the model as it is presented fails. As a matter offact, with respect to the other irradiation induced changes in materialproperties of the graphite ATR-2E the model does not provide asatisfactory description at very high temperatures. A decoupling effectof the processes at high temperature is possible. It may also be .considered that the energy states populated by the processes 2 and 3start to decay for higher temperatures.

The parameters used for the description of the changes in materialproperties of the graphite ATR-2E are listed in Tables I and II. Withrespect to the dimensional changes a total of 12 free parameters werevaried to obtain the model fits in Fig 6. Specifically they are (Xi> (Xp>(X3, E2. E3> Y2» Y3» Ö p Ö2> 63, £ 2

and £3- For comparison, inref 4) a description of the data between 400 °C and 1400 °C up to a dose

21 ?value of only 6*10 EDN/cm is given by means of the functionE w = T c- D T . There, 25 free parameters were required to obtainappropriate fits. Also the data from ref 3) covering temperatures between300 °C and 800 °C for doses including the zero crossing were fitted bythe same functional dependence. In this case 14) a total of 46 freeparameters c . gave good results. The numbers demonstrate that othermathematical tools usually applied for description of data have to bestrained wery hard to arrive at least square fits as good as the model fits.

3.2 Least Square Fits to Changes in Young's Modulus

Data and fits with respect to changes in Young's modulus are shown inFig 7. The representation by the model is not as good as in the case ofdimensional changes. This partly is due to the fact that considerably lessfree parameters were available. Free parameters are the coupling constants(<Xp 0L2» 0^3)» the constants (a,b,c) of the switch function with itsinfluence limited to low dose values and the activation energies EQ and E pnewly introduced. Using all parameters as free parameters better fits cancertainly be obtained. In consequence, it has to be noted that accordingto the model the consistency between the data for the change in Young'smodulus and those for dimensional changes is not so satisfactory. For thechanges in Young's modulus the model predicts a larger decline after the

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initial fast increase to saturation. The decline is due to the thirdcomponent which displays its falling characteristic also at high dose values.However, at.high dose values the second component has reached saturationwhile at low dose values the second component is switched.off..Althoughin the present case the intermediate decline is not supported by the data,in general, this behaviour is not untypical for graphite. It is also ,observed in changes in electrical resistivity 13).

The deficiency of the model at higher temperatures was already discussedin the preceding section 3.1 and will be mentioned again in section 3.3.Numerical values.for the constants are given in Table II.

3.3 Least Square fits to Changes in the Coefficient of Thermal Expansion•• :

Compared with the model description obtained for the changes in Young'smodulus the changes in the coefficient of thermal expansion are reproducedmore satisfactorily (Fig 8). This is so although the third component wasomitted and thus only 2 coupling constants (Xp a 2 and 3.switch parameters(a,b,c) the Matter having restricted influence were used as free parameters.The.singular .behaviour at high temperatures is shown by a sharp decreasenot observed in the experiment. Numerical values for the constants aregiven in Table II. • ,

At low irradiation temperatures i.e. 300 °C and .350 °C the initialincrease in the coefficient of thermal expansion is overestimated by themodel fits. This is no serious deficiency as according to the temperaturedependence CUy, smaller peaks are theoretically expected at irradiationtemperatures lower than 300 °C.

3.4 Least Square Fits to Changes in Thermal Conductivity

21 2For low dose values up to 5*10 EDN/cm the data for changes in thermalconductivity were taken from ref 3). With respect to dose values higherthan 5-1021 EDN/cm data from ref 2) were used (Fig 9), which includethe typical decline at high dose values. Also, as the data from the twosources were not compatible with each other in an average level, the two

-28-

21 ?parts of the curves were connected at 5-10 EDN/cm by means of

normalisation leaving the data from ref 3) unchanged. Furthermore, thenumerical values for the parameters EQ, E^, Yi

ancl ôi differ unimportantlyfrom those used for Young's modulus and coefficient of thermal expansionalso given in Table I. The change in 6, only affects the dose dependenceof the initial decrease in the thermal conductivity. The value of Yi 1S

negative and this is the reason for a planned overall reconsideration ofthe approach for the changes in thermal conductivity (see also section2.2.3.3).

Finally, attention should be drawn to Figs 10 - 13. These graphs describethe irradiation induced changes of the material properties of the graphiteATR-2E as evaluated by the model using Tables I and II. The representationin the form of lines of equal changes is clearer and also more complete.Material change data are calculated for the ranges of temperature and dosebetween 300 and 1400 °C and 0 to 30-1021 EDN/cm2, respectively. In Figs11 - 13 the line of changes with the value -1 cuts off the corner for hightemperatures and high dose values containing values of changes smallerthan one. Such values are not permissible because they imply negativevalues for Young's modulus, thermal conductivity or coefficient of thermalexpansion. Correspondingly in Fig 10, unusually high positive values ofdimensional changes up to 50 % are found in that corner. Experimentalresults show that in this region of temperature and dose the graphite issubjected to drastic changes including disintegrating parts. The modelobviously cannot account for such effects and responds with erroneousnumbers.

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4. APPLICATIONS

Having established a certain degree of confidence in the model it is nowpossible to apply it in practice. There are a number of phenomena observedin relation to irradiation induced changes of material properties. Firstof all there exists a relationship between equivalent changes from differenttypes of reactors, which is called the concept of equivalent temperatures 1).Comparing equal changes from different flux levels the model can make astatement identical to the one given by this concept. Furthermore, thequestion of dependency on flux levels can be raised and discussed morebroadly to yield additional consideration to recent findings reported inthe literature 15). Another question is the following:

- In general, change data are obtained by irradiation experiments leavingthe irradiation temperature constant. In many applications both doseand temperature are varied independently. Previously these changes havebeen interpolated knowing that the interpolation is incorrect to acertain degree. How can the interpolation be performed with the help ofthe model?

Together with two other suggestions for application these topics are nowdiscussed.

4.1 The Concept of Equivalent Temperatures

By introducing additional restrictions and using an additional assumptioneq. (12) can be simplified tö provide an approximate description of theproperty changes. The simplified formula can be shown to be identical tothe one described in the literature 1,6) as the concept of equivalenttemperatures. Moreover, the numerical values of activation energies usedin applications of the concept of equivalent temperatures are the sameas derived in the present work.

21 2The contribution C^ having a smallvalue in mean dose X^O.26*10 EDN/cmcan be considered as a constant for dose values D > X,. With respect totemperature constancy has to be assumed. The functions O^y ancl a 3 T can

be made equal to each other by replacing the two activation energies E 2 andEg and the constants y ? an<* Y3 tneir respective arithmetic averages,i.e. E and y. Another simplification would be to give up

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the temperature dependence of the mean dose, £ 2 = £ 3 = 0, and discardthe switch functions, S 2 = S 3 = 1. Also, relinquishing the pre-exponentialfactor for the temperature characteristic has little effect. For dose valuesD < T 2 < To the exponentials in D can be expanded in a Taylor seriesincluding only two terms. Denoting 71 as the property changes in general. .eq. (12) assumes the following form,

• • ' _ _ ! _

TC = Const. + D x ( G 2kT + y ) ( <X2 6 2+ (X3 63 ) (15)

In order to arrive at the concept of equivalent temperatures the constantY has to be considered as being proportional to the damage flux (p .. Thisassumption is in agreement with the model as introduced in section 2. Therethe constant y was introduced as the irradiative contribution to thethermal process populating the state a_. Therefore it can be expected thatY is an isotone function of the damage flux. However, such a dependencycould not be derived from the data of ATR-2E graphite because these arereferred to doses given in units of EDN (Equivalent DIDO Nickel) whichimplies a normalisation to a constant damage flux ^ E Q N * Dose D' s t r i c t lyspeaking the damage dose D is the time integrated flux, e.g. for constantflux (J) , • . -

D = 0 x t (16)

Introducing eq. (16) into eq. (15) and using the proportional relationshipfor Y yields

E , ^Tt = const . + t x c|>EDN ( e~ 2kT + Y ^ - ) ( a ^ + a 3 53 ) ,17)

We now consider an experiment with two samples of graphite labelled 1 and 2being irradiated at two different levels of damage flux, (J>, and <J>2, forrespective time intervals t, and tp and with respective irradiationtemperatures T^ and T2- The experiment is arranged that way that bothsamples accumulate the same damage dose,

•1 fi = $2*2 (18)

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Furthermore it is requested that the two different triplets, ( <J>i »T, ,t^),and, ( <J>2»T2»t2)> cause equal changes in properties,TCj =1X0. The constantcontribution and the constant multiplier, <J>EDN ( a 2 6 p + OL363')» in eq. (17)drop out in the comparison,

With reference to eq. (18) it becomes apparent that by the specialarrangements of the experiment also the term involving the constant ydrops out in the comparison. Then, replacing the ratio W t ^ by theratio(J>2/-0p viz eq. (18), eq. (19) transforms to

(20)

Therefore, equal changes in properties are produced when eq. (18) holdsand the temperatures are related according to eq. (20), viz ref 1,2,5).The concept of equivalent temperatures is commonly used to reduce theoriginal set of three, characterizing parameters, damage flux<J>',irradiation temperature T1 and duration of irradiation t1, to. only twoparameters,,viz dose D in EDN and irradiation temperature T. The twoparameters T. and D are then referred to a standard flux <t>cnM- As mentionedearlier all data presently analysed are given in the standardized system.

A numerical value for the activation energy Q = 2E used in connectionwith the concept of equivalent temperatures is for instance given in ref 5),Q = 1.58 - 0.18 eV. For comparison the presently derived average value ofthe activation energies of the second and third process yields Q = 1.54 eV.

Thus, by application of the model a well-known relationship has beenreestablished. Considering the fact, that the units Na could not beproperly identified it is of interest to know on what basis the conceptof equivalent temperatures was derived. In ref 1) it is explained thatduring irradiation the displaced atoms pass through a sequence ofconfigurations which depend on the rate of displacement i.e. on flux andon the rate of diffusion i.e. on temperature. If the damage flux isincreased and at the same time the temperature raised so as to increasethe diffusion rate in the same ratio, the sequence of configurations isunchanged, but it is passed through more quickly.

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4.2 Irradiation Induced Changes Obtained from Different Flux Levels

In ref 15) dimensional change data and data for changes in Young'smodulus were investigated more closely. The investigations included datafor a matrix material as well as data for a pitch coke graphite. Part ofthe scatter of the data points which is normally attributed to statisticalerrors pertaining to the experimental method was found to be consistentwith corresponding changes in the flux level. During irradiation theseexperiments had been transferred from one irradiation channel to anotherthereby changing the flux level. The effect of this operation can be tracedin the material properties although the experiments have been correctlyevaluated for equivalent dose. Furthermore, the authors of ref 15)expressed their opinion that the changes of the flux level may be related

to disturbances of the degree of formation and recombination ofinterstitial s and vacancies in the graphite.

In the model the parameters that describe the irradiation inducedprocesses populating state a are the constants y.t i = 1,2,3. Y3 was

selected for arbitrary changes at certain points in dose. These tentativechanges of the set of parameters gave results similar to those reportedin the experiments transferred from one irradiation channel to another.

Fig. 14 b shows the effect for reduction of V , by an arbitrary factor 0.74,21 2The changes were introduced at doses D = 0, 1.5 and 2.5*10 EDN/cm . Thus

variation of the constant Y3 qualitatively reproduce the experimentalresults shown in Fig. 14 a. A quantitative comparison is not warrantedas the two results refer to different materials.

The same analysis was performed with respect to irradiation inducedchanges in Young's modulus, and the results are similar (Fig 15) thussupporting the experimental findings.

The calculations for the dimensional changes E w were done according tothe following equation (21)

(D,T} with D^Dj

Ew =3

with DBDj

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The irradiation temperature T was chosen to be the same as the one in theexperiment 15) viz 830 °C. Eq. (21) also applies for other irradiationinduced changes.

The résumée of studying the effect of variations in flux on changesin material properties therefore is that there exists a dependence on fluxnot accounted for by the usually applied procedure to convert the damagedoses to the equivalent doses EDN. The concept of equivalent temperaturesused to perform the conversion has been shown to agree with the model ifthe constant y was chosen to depend linearly on flux and if a number ofsimplifications are assumed. The dependence on flux can well be morecomplicated than so far envisaged in a linear dependence on y . InsteadY could be a more complex function of 0 or other constants may alsodepend on (p . The before mentioned simplifications enable a reductionto only two independent parameters, viz damage dose D and irradiationtemperature T. The use of the original eq. (12) for such a purpose wouldadditionally involve the coupling constants being specific parameters forthe type of graphite and individual property change but the transformationwould be more correct particularly at higher doses D>To- For these reasonsthe experimentally observed consistent deviations of the dimensionalchanges and Young's modulus (Fig. 5 a, 6 a) can be considered as beinginterprétable. •

Finally it should be pointed out that the arbitrary choice of theparameter Y 3 m ay have been fortuitous such that tentative variations ofother parameters may have produced similar results. It can be demonstratedthat tentative changes in T» produce a similar effect as shown inFig 5 b and 6 b. Therefore future research should be concentrated oneq. (12) to include a proper dependence on the damage flux (p.

4.3 Variations of Both Temperature and Dose

Large graphite components in the core of the high temperature reactorrarely have a constant temperature throughout the body and do not maintainthe same temperature distribution during operational life. Fuel elements,for instance, are designed to have an internal region of heat productionand an external or internal surface for heat transfer to the coolant. Inthe fuel region there may be temperatures of about 1200 C at start oflife. When the fuel is burnt up, the temperature of the fuel region as wellas the mean temperature of the body decreases by possibly 100 °C or more.

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Therefore, in applications it is desirable to use the variables irradiationtemperature and the dose as independent variables.

Neither the measured data nor the model as described thus far can beused to this extent. Small samples of graphite are irradiated at constanttemperature in order to obtain the irradiation induced changes as reportedin the literature. The model was designed to exactly reproduce themeasured data. Therefore, using the model or the experimental data, onlyinitially are both variables independent variables. During irradiation thetemperature cannot be changed arbitrarily because then the irradiationinduced changes also depend on the temperature-dose history , which is notyet included in the model. On the other hand, experimental data cannot beprovided to cover the large variety of cases ultimately required by theapplications.

Using additional assumptions the temperature-dose history can beformally introduced into the model. Specially designed irradiationexperiments become necessary to support these assumptions, otherwise theapproach will remain purely academic. A dependence on temperature-dosehistory can be obtained by assuming that after a change of temperatureduring irradiation, the individual processes proceed as if the experimenthad just started i.e. according to eq. (2). However, the process aims at asaturation level which is determined by the new temperature as if thistemperature had been maintained right from the start of the experiment.This statement implies that in the case of a temperature change, equation(2) is used such that the number of units Na still available for transferfrom state a_ to state j> is supplemented such that the process reaches thesaturation level characteristic for the new temperature.

These assumptions imply that the number Na can not only be supplementedbut also diminished. A lower number Na is required if the new saturation

a

level corresponding to T« is lower (in absolute figures) and if thesaturation level characteristic for the original temperature T., wasalready reached when the temperature was changed at D = D^. Such a changein saturation level is consistent with the model conceived as a chain ofreactions (section 2.1). If one of the preceding reactions is for somereason'not going as fast,the population of the subsequent states willadjust to another state of equilibrium. Thus, the population of the state Jb

-35-

responsible for the changes in material properties may be reduced if thefeeding rate is reduced and at the same time the rate of decay of that stateis kept constant. It should be emphasized again that these assumptions aremore or less arbitrary, and it should be considered a key question toexperimentally resolve whether or not the state of equilibrium can bechanged in both directions. Such experiments may provide some means tofurther explore the nature of the processes and also may help to identifythe so called units.

In formulae,the current state of the i-th process at D = D, can bedescribed by CLj CX^T1 CX-JQ^« The remaining portion for the saturation isgiven by a.,- O-JTI (1 - CX.^). Additionally, a change in temperature fröm •T^ to T2 corresponds to a change in saturation level by OC^( CX^jo ~ aiTl^ *The process restarted at D = D, therefore reaches a new saturation level ofa i ^ a i T 2 " aiTl + a i T l ^ " a i D l ^ ' For eva^uation tne changes inmaterial properties Tt can be calculated employing eqs (12) to (14) ifthe dose is less than D^. A continuation of the curve for values D greaterthan Dj can be obtained by eq. (22).

Ti =is i.a i{a f T[1-e-l 6'* e ! T l ) D l]+- (22)

i = 1 1

Eq.(22) was used to obtain Fig 16 which demonstrates the behaviour ofdimensional changes if the irradiation temperature was increased ordecreased during irradiation.

Sometimes irradiation experiments are planned to have a constanttemperature, but this is not easily maintained during irradiation andaccidental deviations do occur. In such cases eq. (22) can be used tocorrect the experimental results for constant temperature.

4.4 Classification of Graphites by Means of Coupling Constants •

The proposed model can be used to analyse a great number of differenttypes of graphite. Systematic studies on graphites varying in the type ofthe filler and binder as well as the binder coke content and the number

-36-

of impregnations are reported in the literature 13,16). The results arepresented in the form of graphs showing the functional dependence on doseand temperature. Additionally the dependence on production parameters, forexample, the binder content and graphitization temperature, is presented.Discussion of the results leads to qualitative statements about, forexample, the amount of maximum shrinkage as influenced by the variousparameters.

In terms of the model, maximum shrinkage or turn around points do notpresent the characteristic properties of the graphite. The model assemi-empirical model provides a good theoretical understanding of theproperty changes depending on temperature and dose as far as the basicthree processes are involved. Thus, in analysing irradiation inducedchanges in properties with the help of the model, the basic dependence ontemperature and dose can be separated. The remaining phenomenologicalpart of the information is then contained in the coupling constants andtheir multipliers which are the switch functions. The coupling constantsand the constants describing the switch functions are characteristic forthe type of graphite. Therefore the influence of production parametersshould rather be discussed in view of these constants.

4.5 Correlations between Material Properties using Coupling Constants

In the preceding section it was pointed out that the coupling constantsand the constants describing the switch functions are characteristic forthe type of graphite. Additionally they are also characteristic for theproperty whose change is to be described.

One objective of the research of property changes under fast neutronexposure always has been to find correlations between the differentproperty changes of one graphite. In the presently proposed model thecorrelations can be identified with the three basic processes which arein common of all property changes. They correlate the property changes ina non-linear way.

-37-

These findings however do not exclude the possibility that there arefurther systematic relationships between the coupling constants as well asthe switch functions of the different property changes of one graphite.Such relationships may become apparent if a number of different types ofgraphites is studied. Therefore it is proposed that when deducing thephenomenological constants of various graphites as described in the.preceding section to also search for systematic relations among theconstants of one type of graphite.

-38-

5. SUMMARY

Under the basic assumption that there are principally three saturatingprocesses involved, a model for quantitative analyses of irradiationinduced property changes was derived. While the three processes proceedon an atomic scale their influence on macroscopic properties in some casesis buffered by microcracks closing during the initial phase of irradiation.The temperature dependences of the two faster processes were found to berelated to the population of energy states populated by concurrent thermalactivation and activation from neutron irradiation. With respect to thefastest process an anneal able energy state was assumed. Also thecharacteristic time constants of the processes i.e. the mean doses werefound to depend linearly on irradiation temperature. The model can beinterpreted to reproduce the so-called concept of equivalent temperatures 1)to the extent that the numerical values for activation energies are thesame. As a result of the comparison it was concluded that apart from theestablished dependencies on dose and irradiation temperature there alsoexists a dependency on the damage flux. This information was used to studythe influence of variations in flux and these were brought in qualitativeagreement with experimental data 15).

The presentation of the model follows more or less the way the researchon the subject was performed, thus reflecting some of the problems andalternatives encountered. For proper identification of the processes asthe regrouping of atomic configurations, further investigations arerequired. The influence of the closure of microcracks was difficult toresolve in the presence of variations from the three basic processes.Especially with respect to thermal conductivity and dimensional changesthis problem has to be studied more extensively. Also, a simultaneousfitting procedure including all experimentally found property changesof one type of graphite is a more correct way to evaluate the physicalconstants in the model. The dependency on irradiation temperature can beformulated somewhat differently to be more symmetrical with respect toall the three processes. Possibly, by using the alternative temperaturecharacteristic the capability of the model to describe the data outsidethe temperature region presently considered can be improved. The questionof anisotropy has not yet been raised. It is expected that after obtaining

-39-

good fits from a nearly isotropic graphite in the direction parallel tothe grain orientation, data perpendicular to the direction of grainorientation and from more anisotropic graphites can be described withsimilar success. In particular, it is expected that no other processesthan the assumed three processes are required to account for anisotropy.

The above list of unresolved problems indicate that the studies on thesubject are not in a final stage and further improvement of the model,possibly including corrections can be expected. However, the resultsobtained so far leave no doubt that a description of the experimentallyobtained, irradiation induced property changes by means of a theoreticalmodel can be realized. The results are such that further research includingother.types .of graphite is justified. Indeed a number of applications of ••»the presented model are pointed out specifically. The question of a furtherdependency of the change data on fast neutron flux can be studiedquantitatively. The data on other types of graphite can be analyzed toseparate off the dependency on irradiation temperature and dose as far asrelated to the three basic processes. The remaining phenomenological partof the model, i.e. the coupling constants and the constants of the switchfunctions, can be subjected to further studies on correlations withproduction parameters of various graphites and on correlations betweenthe constants for different property changes.

REFERENCES : . . -

1) J.H.W. Simmons, Radiation Damage in GraphitePergamon Press, Oxford 1965

2) J.E. Brocklehurst, J.W. Harrison, B.T. Kelly and D.G. Martin,Changes in the Physical and Mechanical Properties of Graphitedue.to Irradiation,.AI/CB/178, AERE Harwell, May 1976

3) J. Budké, G. Haag, W. Hammer, A. Mittenbiihler, M.F. O'Connor und

K. Petersen, ModelIdatensatz für den Reflektorgraphit der Kugel häufen-reaktoren, Kernforschungsanlage Julien, JOL-Report 1414, April 1977

4) H. Cords, J. Mönch, R. Zimmermann, Graphitdaten für Spannungsrechnungenals Funktionen von Temperatur und Dosis, Kernforschurigsanlage Julien,IRW-IB-4/76

5) J.C. Bell, H. Bridge, A.H. Cottrell, G.B. Greenough, W.N. Reynolds and

: J.H.W. Simmons, Stored Energy in the Graphite of Power-Producing Reactors,Phil. Trans. Roy. Soc. A254(1962)361

6) L.T. Chadderton, Radiation Damage in Crystals, New York,John Willy & Söns Inc., 1965, p. 4

7) L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon Press,London 1958, p. 316

8) A. Sommerfeld, Vorlesungen über theoretische Physik, W. Klenn GmbH,Wiesbaden 1952

9) P.T. Nettley, J.E. Brocklehurst and A.W. Martin UKAEA Culcheth andJ.H.W. Simmons UKAEA Harwell, Irradiation Experience with IsotropieGraphite, Proceedings of a Symposium on Advanced and High TemperatureGas Cooled Reactors held at Jiilich 21-25/10/68, IAEA, Wien 1969,Contribution SM 111/34 p. 603

10) J.H.W. Simmons, B.T. Kelly, P.T. Nettley and W.N. Reynolds, TheIrradiation Behaviour of Graphite, Proceedings of the third UnitedNations International Conference of the peaceful Use of AtomicEnergy, Geneva 31/08/ - 9/09/64, vol 9 Reactor Materials,Contribution P 163, page 344

11) S. Mrozowski, Mechanical Strength, Thermal Expansion and Structureof Cokes and Carbons, Proceedings of the Is and 2 n Conference onCarbon held at University of Buffalo (1953, 1955),Buffalo, New York 1956, p. 31

12) H. Bridge, B.T. Kelly and P.T. Nettley, Effect of High-FluxFast-Neutron Irradiation on the Physical Properties of Graphite,Carbon 2(1964)83

13) W, Delle, Ober das Bestrahlungsverhalten von Reaktorgraphitenunterschiedlicher Zusammensetzung - JÜL-Report 747 April 1971

14) H. Cords, J. Mönch, R. Zimmermann, Spannungsrechnungen am Reflektor-block des Kugel häufenreaktors Teil 1: Material daten, Vollblock-Test-rechnungen KFÄ-IRW-IB-11/77, Mai 1977

15) H. A. Schulze, R.E. Schulze, W. Delle and A. Naoumidis, Effect of FastNeutron Flux on the Irradiation Behaviour of Graphitic Materials andPyrolytic Carbon, Journ. of Nucl. Materials,71(1977)171-172

16) W. Delle, G. Haag, H. Nickel, H.A. Schulze, R.E. Schulze, Influenceof the Irradiation Induced Graphitization of Graphitic Materials ontheir Property Behaviour under Fast Neutron Exposure at HighTemperatures, 4 London International Conference on Carbon andGraphite, London, September 1974

17) D.R. De Halas and H.H. Yoschikawa, Mechanism of Radiation Damageto Graphite at High Temperatures, Proc. 5 Conference on Carbon,1(1962)249.

18) H. Bridge, B.T. Kelly, B.S. Gray, Stored Energy and DimensionalChanges in Reactor Graphite, Proceedings of the fifth Conferenceon Carbon vol 1 (held at the Pennsylvania State University),Pergamon Press, New York 1962, p.289

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