"Diffusion Studies In Uranium Carbide Systems".
A Thesis Submitted For The Degree of
Doctor of Philosophy
in. The rniverGity c; .ondon.
Hee iyAyong Lee.
Dep&rtnent of Chemical Engineering and Chemical Technology, Imperial College of Science & Technology, London, S.W.T.
March, 1965. C OWTELTS
Abstract. Acknowledgement s. Contents, Section I. Introduction. 1 Section II. Survey of the State of Knowledge. 5 II - 1, Properties of Uranium Carbide 6 1 - 1. Phase Diagram and Crystal Structure 6 1 - 2. Uranium Monocarbide 6 1 3. Nature of Chemical Bonds 10
1 7 4. Uranium Di carbide .11 1 0 5. Uranium Sesqu-Carbide 12 1 - G. Therual Conductivity and Expansion 13 1 7. Heat Capacity 13 1 - 8. Electrical Resistivity 14 1 - 9. Thermoionic Emission 15 - 10. Thermodynamical Properties 16 1 - 319 Mechanical Properties 21 1 - 12. Radiation Damage 30 1 - 13. Fission Gases Release II - 2, Diffusion in Solids 37 2 - 1. Imperfection in Solids 38' 2 - 2. Types of Defects 38 2 - 3. Frenkel Defects 39 2 - 4. Sbhottky Defects 40 2 - 5. Role of Defects in Diffusion 40 2 - 6. Diffusion Equations 42 2 - 7. EgiTations for D 44 2 - 8. Diffusion Mechanism 46 2 - 9. Emr6rical Relations 49' Section III. EtperimentaA Work 51 III - 1. Materials - Preparation and Properties 52 III - 2, Descriptions of Experimental Apparatus 57 2 - I. Vacuum Purnac 58 2 - 2. Grinding Machine 62 2 - 3. Evaporation Unit 6$- 2 - 4. Gamma-ray Spectrometry 66 2 - 5. Liquid Scintillation Counting Techniques and Counter 78 III - 3. Experimental Procedures. - Self-Diffusion of Carbon in Uranium Carbdules 88 3 - 1. Diffusion Couple 89 3 - 2. Diffusion Anneal 92 3 - 3. Grinding and Sectioning 93 3 - 4. Combustion Technique and Apparatus 96 3 - 5. Carbon-14 Activity Measurements 100 3 - 61 Results 103 III - 4. Experimental Procedures. - Self-Diffusion of Uranium in Uranium Carbide 126 4 - 1. General Conadirations 127 (A) Alpha-Ray Method 130 4 - 2. Diffusion Couple and Anneal 131 4 - 3. Sectioning 132 4 - 4. Determination of Uranium-234 Isotopes 133 4 - 5. Results. 135 (B) Gamma-Ray Method 154 4 - 6. Diffusion Couple and Anneal 156 4 - 7. Sectioning 156 4 •- 8. Determination of Uranium-235 Isotopes 156 4 -- 9. Resets 159 Section IV. Discussion. 165 1 - 1. The diffusion of carbon in uranium carbide 166 1 - 2. The diffusion of uranium inruranium carbide 173 2 - 1. The diffusion of carbon in hyperstoichio:letric uranium carbide (5.0, 51 and 5.6 IA %) 178 2 - 2. The diffusion of carbon in hypostoichiometric uranium carbide (4.7 74) 181 Section V. Applications and Suggestions for future Work. 195 Section 11. General Conclusions. Appendix - Metalio'graphical Studies of Uranium Carbides References 221 ABSTRACT.
The present work 7E0 carried out to investigate transport properties of material in uranium carbic'le in a wide range of compositions and temperatures.
Self-diffusion coefficients of carbon in stoichiometric and non-stoicnion2etric uranium carbide have been measured by means of carbon-14 and standard section technique. For stoichiometric uranium carbide, diffusion of carbon in the temperature range 1286 - 1884°C can be represented by the equation
.= 1.75 exp ( 53000 /
In non-stoichiometric uranium carbiie (carbon-rich) carbon diffusion rates are increased, and the activation energy is decreased, whereas, in the carbon-deficient side, the diffusion rates are clecreased and the activation energy is increased.
In tile carbon--rids side, the additisu of suall amount of carbon to essentially si:oichiwnetric uraniun carbide results in an increase in the carbon clifftEion coefficients and a decrease in the activation energy. Self-diffusion coefficients of uranium in stoichiometric uranium carbide were also investigated by means of alpha and gamma-ray methods. Uranium diffusion in the temperature range o 1505° - 1863 C can To,J represented by the equations
4.49 x 103 exp (- 228TP
(alpha-ray method)
104. 000 1) = 8.47 exp ;
(gamma-ray nethod).
The diffusion of uranium exhibits only one activation energy over the temperature range studied, suggesting that a single mechanism of diffusion predominates; the volume (or bulk) diffusion. In view of the close-packed arrangement of uranium atoms in the structure, Schottky type of defects seems more probable. Using thc above resnii,s, as attelcipt as been made to evaluate quantatively diffusion models for sintering and high-temperature deformation, for which 'Apprcpriae experimental data is available. ACKNOVIADGEtr:ANT.
The author is grateful to his supervisor, Mr. L.R. Barrett for his guidance and encouragement throughout the course of the investigation.
The author would also like to thank Mr. H.J. Hedger and his colleagues for their help in preparation of the material and their discussion of the work carried out.
The author is also griAeful for the helpful discussion with professor P.L. Pratt and Dr. 1.1). Le Claire.
The author is indebted to the Atomic Energy Research Establishment, Harwell, for their sponsorship of the work and facilities made available to the author through Dr. P. Murray and the Metallurgy Division. Section I. INTRODUCTION
In view of higher ecficiency and for economic reasons, nuclear technology turns to higher telliparatures. This necessitates the use of material in nuclear reactors of high melting point and high stability against thermal streas, dry and wet conditions.
So far, metallic uranium ha2 been used as a fuel element most widely. but it suffPro a nnmber of disadvantages such as low melting point, alIntropic phase-transition at practical tel and dimensional changes under irradiation. As a consequence of these undesirable properties, considerable efforts have been directed to the del-elcpment of ceramic fuel elements such as uranium oxide and carbide. The reasons for this interest lies in their high melting points, lack of phase-transformation at operation temperatures, and dimensional stability under irradiation, coupled Trith considerable higher thermal conductivity and its high uranium density.
Of the alternative materials which have been developed, uranium dioxide has received most attention and much is now known about its properties. (1 to)8 Uranium dioxide has a very high melting point (2800°C), it is monophase and is relatively inert towards many coolants such as hydrogen, water, and carbon dioxide. Under irradiation it does not change significantly. Oxygen has a very low thermal neutron cross- section (0.002 barn) and will not act, therefore, as a neutron parasite.
- 1 - Inevitably, however5 uranium dioxide has its disadvantages. It is brittle, cracks due to thermal stress and has a low uranium density. Moot of all, it has a poor thermal conductivity, especially at high-Aemperatures,
.A comparison of the properties of uranium carbide with uranium dioxide has shown that the carbide possesses many of the advantages of uranium dioxide besides having superior mechanical properties, a thermal conductivity close to metallic uranium, and a high uranium density.
TABLE I. Comparison of properties of UC, UO2 and Uranium Matal.(7)
Uranium Uranium Uranium Monocarbide Dioxide Metal
U density 12.99 10.5 (gm/cm3) M.P.(°C) 2450 2750 — 40 1,133 i 1 0 _,0 There. al 0.06 at, ion —7c, 0.018 at 100°C 0.065 at 100°C Conductivity per 5 mt % C 0.012 at 400C 0.081 at 400°C (cal/sec/en/°C) 0.008 at 700°C 0.104 at 700°C
Theoretical Cubic 13.06 X—ray gensity 13.53 10.97 tetra 18.11 (6"m/cm ) Othonon 19.12 Covalent & Covalent & Chemical Metallic Bond. metallic Ionic
—2 — Furthermore, carbi& has a low thermal neutron capture cross— section and need not complicate the subsequent chemical reprocessing, since it readily oxidises in air to give uranium oxide which is soluable in conventional solvents. Also, material—compactability studies show the carbide to be particularly suitable for sodium—, organic and high temperature gas—cooled reactors because of its higher thermal conductivity and uranium density than uranium dioxide.
However, evaluation of the properties of uranium carbide is not complete and work on the fabrication, properties and irradiation behaviour of this material is being currently (5.6.7.8 to 17) reported. A property whibh until recently has received little attention is that of the self—diffusion coefficients of carbon and uranium in uranium carbides. Besides being of theoretical importance, self—diffusion studies on uranium carbides may provide a basic understanding of phenomena such as sintering; grain growth, plastic flow, oxidation reactions and diffusion of the fission products.
however, since application of radioactive tracers is possible ii research, a great amount of work has been done to determine diffusion coefficients of metals, alkali halides and oxides. But little progress has been, made to determine diffusion coefficient in carbide systems because of its hardness and brittleness. Consequently, a series of preliminary tests have been carried out at the laboratory to examine the feasibility of determining the diffusion coefficients, and the work described in this report is a continuation of that
— 3— reported by Chang and Hilton.(Is.19)
Measurements of the diffusion coefficients of uranium and carbon in uranium carbide 17.avo been reported from the Battelle Memorial Institute. It has been shown that, by means of radioactive tracers such as C-lif and U-235, the diffusion coefficient vary from 2.0 x 10-7 cm2/sen to 7.0 x 10-11 in the temperature range 1200° to 2120°, the activation energies of diffusion being 50 kcal/nolo for carbon and 65 kcal/mole for (20,21) uranium. But their results are very much scattered, Particularly, uranium Iiiffusicn coefficients in uranium carbide, It is therefore vary difficult qA) evaluate the results accurately.
It was, therefore, thought that the diffusion study of uranium and carbon in uranium carbide was worthwhile investigating. In order to get a perfect knowledge of material transport properties, the diffusion coefficients of two constituents were simultaneously determined and conpared with each other by introducing new techniques into the erperi oroceeures. Section II
Survey of the State of
Knoviedge. To have a fuller understanding of the process of material transport in uranium carbide it is necessary to survey the properties of urani-a carbide, as these play an important part in the process.
It is also vital to have a sound knowledge of the nature of imperfections in solids because a great many of the important properties of solids strongly depend upon the types and concentrations of various imperfections which are present in solids.
II -19 Properties of Uranium Carbides.
1 - 1, The Phase Diagram and C6Aal Structures.
The latest version of the uranium-carbon phase diagml, proposed by Rough and Chubb1(22) is shown Fig. 1. The features which are fir4y established are the existence of the three compounds, their melting or decomposition temperatures and the phase transformation in UC2. The region between UC and UC above 1750°C is not very firmly established, but recent works indicate a fairly extensive solid solution region.
1 - 2, Uranium Monoccrbide.
This compound is of invariant co.lposition, i.e. has a very narrow composition range, and crystallizes in the f.c.c. Na Cl
- 6 *- structure which is by far the most common structure for mono-carbides of the transitional metals. The lattice parameter :.s1 4.1-361 t 0.00Ae, The interatomic distances are
U- 12U 3.50e C - 12C 3.50A°
U - 6C 2.48A0
(23) and the X-ray density is 13.63 gu/cn3. The melting point is 2450°C. The structure is not only isomorphous pith the transition and actinide metal monocarbide but also with U0 and UN and continuous solid solution with all of these compotnds appear to exist
Although uranium monocarbide is generally believed to be of invariant cocIpo:Ation, rect x-ray suggests that it may accommodate excess uranium atoms (or aore probably a deficit of carbon atoms) at high temperatures. Williams, Sat bell and (24) Wilkinson detemine6. the lattice parameter of UC in a series of arc-melted specimetxb oil' varing carbon content and founG a systematic variation.
(25' Buckley quenched a 47 at ';'S C alloy from temperatures ,o between 1100 and 230o C and thea. measured the lattice para- meter; again this showed a periodic variation reaching a o minimum corresponding to quenching from 2000 C.
These results are interpreted as being an indication of the ability of UC to adopt a uranium-excess structure at high
Fig . 1 . The uranium - carbon constitutional diagram.
Carbon wt % 0 1 2 3 4 5 6 7 8 9 10 1 I I i i i 2600 Liquid + C
2400 Liquid
Cubic UC-UC2 2200 Solid Solution Cubic UC2
, • 2000 -- / C / U C O
/ Cubic UC 1800 / , I i .-1 - - - - UC + Tetra UC2 < I Tetra ture / '1 /- -7------\ 1-- Ili \ UC ra %I U ‘1 2 e . o• 1600 / e U C Temp Liquid + UC 2 1400 -/ UC C U C UC 2 3 2 U2C3 1200 1- C
1000- U UC U C U C 2 3 800 0 10 20 30 40 50 60 65 Carbon at %
- 8 - Fig. 2 . Structure of Uranium Monocarbide.
0 Uranium 0 Carbon
. 9 — temperatures by omission of carbon atom from the unit cell, which causes the lattice parameter to decrease.
Accray (34) et al have studied the solubility of uranium and carbon in uranium nonocarbide and reached the same, con- clusion.
1 - 3. Nature of Chemical Bond in Carbides.
(25) Hagg made an extensive study of this class of compound and concluded that pro7i.dad the ratio of the atomic radius of the non-metal to that of metal ion was less than 0.59, a "normal" interstitial structure would be formed, i.e. the metal atoms would form a close packed cubic or hexagonal lattice and the non-metal atomo occupy the interstices. Uhbelohde (27)7 Umanoki (28) and Kiessling also postulated that the non-metallic atoms give up electrons to the metallic atoms, which in the case of the transition metals implies the filling of the d-band and in the ,ase of the actinide elements the filling of the f-band s the non-metallic atoms being converted to the metallic state.
Rundle (29) loses his arguments on the fact that the non- metal atoms in the Na Cl type of interstitial compound (e.g. C UC) are arranged in the octohedral interstices and that the metal-metal interatomic distances tend to increase considerably on formation of these compounds. This implies the formation
- 10 - of strong metal-non--metal bonds with strong directionality, at the expense of metal-metal bonds.
Griffiths (30) has recent17 considered the nature of the bonding and the band structures in UC, ThC and FuC. The metal- carbon bond may consist of an octet of electrons in a 6d/5f metal - 3p carbon covalent linkage or a 7s/6d.5f metal - 3p carbon linkage, since in the carbide the metal atoms are further apart than in the metal lattices and occupation of the 7s states may be feasible.
1 - 4, Uranium Dicarbide ((UC2 )
Uranium dicarbide has a cubic structure above 1800°C, with a maximum carbon content of about 9.2 wt %. Below 1800°C it transforms from the cubic form to a tetragonal one, with lattice parameters a .3.519A°. C = 5,9891° corresponding to an X-ray 3 density of 11.68 g/cm . At 1820°C the structure transforms to the fluorite (CO' ) structure The structure of UC is studied 2 - (31) by ttodi and Kernel and may be considered as an interrietallic compound of U and C2, where the c-c distance is a double bond and C the tetragonal phase decompose into U C and graphite. below 1500° 2 3
.This -phase has alrays been found to be sub-stoichiometric, the composition tying in the range and commonest UC1:75-1.90 the composition being UC/.8 (32'33
- 3.1 - 1 - 5, Uranium Sesqutrarbide (U2 C3)
Uranium sesoui.carbide may be formed either by a synthetic reaction, UC UC2 U2C3 or by a decomposition of uranium dicarbide, 2UC2 -= U2C3 C. The structure is body-centred o (35 36 37) cubic, with a = 8.088 - 0.001A 8 molecules/unit cell ' ' and an X-ray density of 12.88 g/cm3.
1 - 6, Thermal Conductivity and Expansion.
Extensive measurements of this important property have been made over a wide range of temperatures and compositions at the (38) Battelle Uemorial Institute and elsewhere. (39,40,41)
However, over the greater part of the temperature range of interest, the conductivity han a mean value of 0.050 cal/cm.sec°C, compared with values at rompalr.able temperatures of just under
i.0 0 and 0.08 for uf:aniuo. metai 2 'e'h thermal ex :t of (.4:e-af4it, wuposildons in the 08,39942) rklilgp 4.33 to 5,G5 1;r
expf_mr_Dir: ui ftoo 4.9 -ko 5.05%C were , .1entio.- to 2000
•- 1 . - 3F, • , 0'34 17 T 20 '
o c letvth at 91) C a"Ad T l'erveraturcl in C. elue to the reversion of UC UC to U C (43) 2 2 3 a slight increase in thermal expansion was observed on cooling.
To 400°C, the expanclon y hyposoi4hiometric UC was identical to the above; beyond terperature, expansion became slightly greater than that of UC 1.00 , the difference increasing with decreasing carboL content and increasing temperature.
Above 850°C the expansion increased rapidly) and progressively larger permanent deformation was observed.
1 - 7 ) seat Capacity.
• Few ueasurements have been made of the heat capacity of uranium carbide. BMA reported values of Cp for UC which could be described by the relationship; (44)
Cp 9.6 +2.,35 x 1
between 2980 and 2400° K.
(45) Katz and aabinowitch gave a corresponding relationship for•UC2)
Cp = 8.92 3.95 x 10-3 T
- 13-
The measurements were also made on a sintered UC pellet 4 having an initial composition of 4.88 wt.70 C.(46) The expression for capacity is
14.223 1.594 x 10-3 T — 4.118 x 105 T-2 (293 — 1495°K).
1 — 81 Electrical Resistivity.
B.M.I investigators have measured the resistivity of arc—cast 50 to 53 at 7'04 C9 60 at iL and 67 at C alloys between room temperature and 1600°C. Lt any given temperature the resistivity increases with increasing carbon content.
The measured resistivity at !loom temperatures are.
Composition Resistivity
4.8 wt 7.0 41.2/v—ohm—cm 5.0 wt 41.9 — 43.2 //—ohm—cm 6.75 wt 7:1 50.8 ,/1 —ohm—cm
Norreys et al also have measured and found that the resistivity of UC rises from 40.5 microhm. aa at room o (47) temperature to 299 microhi . cm at 2252 K.
— 14— 1 — 9, Thermoionic Emission.
The work function of UC is interesting in its possible application as the cathode of a. thermoionic diode converter(48) (49), General measurements have been (ilade, :add et el Haas and (25) Jenson (50) caul Norroys et al mansured the emission at various temperatures .radar a high accelerating potential and deriveJ the work function from the Richardson equation. Their results are listed below.
(e.U). Speemen
o 4.57 at 0 3.i Cast UC. Fidd et al. 2.94 Towder zReposite .0(; Eaas and Jenson. 4.69 at 0°K
3.9 at 1250° Cast UC Norreys et al. 3.4 at 164C°
( dopkin 51'measured :the contact potential difference between a clean tungsten reference surface and UC, deriving a value of 3.9T 0.615 e'a at about 300°K.
— 15— 1-10, Thermodynamic Properties.
Experimental data for the uranium carbides are by no means complete, but data published hes reduced considerably the speculation concerning the thermodynamical properties of these carbides.
(a) Uranium Monocarbide.
Values of formationAll have the heat of 298 been derived by combustion calorimetry by Farr, Huber, Head and Holley (52) (53) and by Tripler, Snyder and Duckworth and by an equilibrium technique by Alcock and Grieveson (54)
The values derived were as listed below.
Method Ref. 4.H298
4 - 21.0 - 1.0 Combustion calorimetry 52
- 20.0 - 5.0 It o 53 J. - 21.0 -:-. 1.0 Equilibrium technique 54
The data appears to be reasonably accurate and the results of various workers in the field are in fair agreement.
- 10 From experii2ental data of uranium pressures over UC2 - C mixtures at 2000 - 2100°K and uranium activities in the UC UC2 (54) and U - UC region at 1723 - 1823°K, Grieveson and Alcock o determined the free energy of formation of UC as CGr. f 24,400 + 3T cal/mole for the equation
u(i) + uc(s)
A comparison of free energy data obtained by three procedures is shown in Tale 2.
Table 2. Estimates ca thri 'Energy of Formation of Uranium Uenocarbide.
.M.I. 1 B 1 Huber and Alcock and Temp. B.M.I. effusion (° K) Calculation data Nolley Grieveson.
1000 -20,675 -19,500 -21,600 1500 -20,004 -19,600 -125040 -19,800 2000 -18,607 -18,100 -18,822 -18,000 2500 -17,435 -16,500 -20,412 -16,000
The B.M.I. calculations are based upon Doreyes heat of 56) formation and Krikoriants (56, technique for estimating heat capacity and entropy. The data of Grieveson and Alcock are extrapolations of the values measured in the range 1723 - 1823°K. Huber and Holley obtained their values from the equation
- 17 - Ci G .n,14 T 2 T
Over most of the temperature range quoted the values for free energy of formation do p.ct 7ar3 more than few Kcal/mole. Thus the values for UC appear to be reliable.
(b) Uranium Sesquicarbide.
The heat of formation as reported by Huber and Holley (50 is H296 =, —49.0 ± 4 Kcal/mole. Phase equilibria studies o indicate that U2C3 decomposes at about 2050 K to form UC and UC2, and at 1760°K UC is reported to decompose to U C and 2 2 3 graphite.
Huber and Holley oalcuAated the free energy of formation by taking the experimental value for the free energy of formation of UC and adding the values to obtain free energy 2 of formation of UC and obtained &G° values shorn Table 3 and compared with effusion data.
Table 3. Standard Free Energy of Formation of U2C3
Temp. Huber and Holley B.M.I. Effusion (°K) Calculation Data.
1000 —46,000 —55,200 1500 —44,500 —52,200 2000 —43,000 —45,600 The current data con3idered at present to be the most suitable is
G°U2C3 - 72000 + 13.2T 2U (1) + 3C (gr)
U C (S) (1405 - 2000°K). 2 3
(c) Uranium Dicarbide.
'Huber and Holley deterained the heat of formation to (57) be -18.0 - 4 Kcal/molc) of ukaibn et al determined UCI.86. H the heat capacity as Cp -2.93 4 4.33 x 10-2 T - 3.17 x 10-5T 2 o (373 - 673 K).
Alcock and Grieveson calculated the free energy of formation in the temperature range 2000 - 2100°K from measurements of uranium pressure over a UC2-C mixture. They obtained the equation
o G UC - 32.600 4- 3.6T. 2 (58) Leitnaker and Witter man have also studied the dissociation pressure of UC2 and estimated an entropy value of 18.80;u. Suggesting as a basis, the similarity of UC2 and CpC21
4298 (CaC2) - (Ca) sf298 (UC2) SLET sf298
- 19 - Using an estinate entropy of 19 e.u. for UC2, Huber and
Holley estimated a free energy of fomation of UC at 298°K 2 of —19000 oal/moIe and o,e,Sf298 of 4 e.u. Their esticated ::aatinn.T.TC- equation for the :reo energy of for 2 =18,000 — 3T dal/mole, rhen correlated with the data for UC, and Huber and Volley's data for U2C3, is found to be in substantial agreement with the phase equilibria studies and with the values of o o tkG computed at 2000 — 2100 K from Grieveson and Alcock's UC2 • data. This equation is found to be
eN G°UC2 - -• 18000 — 3T (1405 — 2500°K)
20 - 1 - 11, Mechanical Properties.
(a) Compressire and Transverse Rupture Strengths.
A series of measurements .11Lc,ve beer_ made at the Battelle Memorial Institute of the rocm-temperature compressive and transverse rupture strengths of arc-cast carbides in the (59). composition range 50 to 67 at % carbon The results are summarized in Table 4.
Table 4.
Moduli and Compressive Rupture Strengths of Uranium-Carbon Alloys.
Alloy Carbon Coapressive Elastic Strain ,. Rupture Strength at rupture Modulus Content (p . s.i .x10-u) (wt.%) (p.s.i) (/))
4.8 54500 0.17 31.5 45000 80000 46400 40600 0.13 26.4 35300 0.12 32.5 39600 0.24 27.6 67500 Average 57100 29.5
- 21 - 7.0 85600 67600 63400 60400 52600 0.16 32.1 64700 0.25 25.9 Average 60700 — 29.0
Several interesting features emerged; firstly, maximum strength is observed at 60 at 11, carbon, secondly the elastic strain at rupture is of the order of 0.150 and thirdly the value 6 of Youngs Modulus ic about 30 x 10 p.s.i. for 50 and 60 at % carbon alloys.
Alsc, some ner.suvements yore done at G.E.C. on the bend strength. of sintered UC of density varying from 11.36 to 12.60 glom3 and gave values of Young'c Modulus of average value 5 5.0 x 10 p.s.i. The reason for the large discrepancy between the two sets of value;;; is not c_7 ear.
(b) Hardness
Fig. 3. showing transverse rupture strengths also includes elata on the variation of room temperature hardness with composition. In agreement with the rupture strength variation, maximum. hardness is achieved at about 60 at 0 carbon. Similar measurement at G.E.C. agree sub— stantially with these findings, but a slight drop of hardness
— 22 —
40000 a
17 20000 •a- .
05 o 0 , 0 10000 , , , !, rwor2,5uvreerse': E 800 ,—eT 600 strength
40
E 2001 en
t 100' ) x "g 80' ) N ` 60 ) -- x T` Hardness xx 4,‘1 xtE 40 ) I
.r. x 20 8 0 20 40 50 60 65 Carbon conten tat/.)
2 4 • 6 ;3 1,) O Carbon content OA/I'M Fig. 3. Transverse rupture strength and hardness of arc—cast TM alloys
10000 I I 6000 CRYSTALS 500,50b,580,58b,-STOICHIOMETRIC CRYSTALS 530,53b,-HYpoSTOicHomETRIC 6000 CRYSTALS 1 2,64,27-HYPERSTOICNOVETRIC
CRYSTAL ZT 4000 CRYSTAL 64
7,- — CRYSTALS 50a, 50b CRYSTAL i2 13,
2000 cc o " tr) c 3 O (I) CRYSTAL 53o 1000 CRYSTALS 58a, 586, 800
CRYSTAL. 530 600
400 1400. 1500 1600 1700 1800 1900 2000 2100 TEMPERATURE (SC) Fig. 4. Flow Stress vs Temperature ( U—C single crystals )
— 23 — is observed between 50 and 60 at 2,1, carbon which way reflect the limits of miscibility of UC and UC2 above 1800°C. G.E.C. heat treated a CO at carbon alloy to fore U2C3 and obtained a hardness value of i550 (Vickers) compared with B.M.I. values of Knoop hardness varying between 1080 and 1260. Both investigations were made on arc—cast material.
Regan and Hedger (so)studied the variation in hardness with temperature in reaction—sintered UC, while Brown and (61) Stobo made similar measurement on arc—cast UC. Both 2 showed a steady drop from a Ilardness of about 900 Kgimm at o room temperature to about 100 -1g/mm2 at 800 C.
(c) Creep.,
(47) G.E.C. worh3rs have carried out compressive creep tests on arc—cast cylinders 0.7 cm. diameter x 0.16 — 0.8 cm. long of composition 50, 56 and 50 at 'I, carbon, at temperatures between 1100 and 1500°C at stress of about 6000 p.s.i. Some typical creep curves are shown in Fig. 3. There is a marked dependence on composition. The presence of even very small quantities of free uranium markedly accelerate the creep rate, e.g. a slightly hypostoichiometric alloy containing only 0.1 vol % free uranium crept at an average rate of 0.0/h over a period of 16 hrs. at 1400°C under a stress of 6000 p.s.i., whereas the rate for a stoichiometric alloy was 0.001 %. Similarly a 58 at carbon alloy, which transformed under showed a curve in which tertiary creep set in stress to U2C3'
—24— Fig. 5. Effect of Stoichiometry on creep.
Test Conditions 1300 0C and 28 6000 psi
24
49.5 at % C
20 WW1
/V.
20 40 60 80 100 120 140 T I M E, H
—25— after 30 h, indicating that the sesquicarbide has poor creep properties. Qualitative support for these results is given by compressive and bend creep experiments at Farwell and Dounray.
Chang () has reported a more basic study of the flow and recovery properties of near stoichiotsetric IJC in the temperature range 1400° to 2000°C at strain rates varying -5 -3 between 2 x 10 /sec and 2 A 10 /sec.
Stress-strain cur;ios at strain rates give positive evidence of strain hardening as its temperature dependence. From the horizontal shift of time with respect to temperature, an activation energy for transient creep of about 80,000 cal/ -3 mole was obtained. Data obtained at strains of 10 gave the following creep-temperature-stress relationship at steady- state;
0 14 C ,-; exp 01,T) /sec. which is valid for values of E° exp (Q/'. ,T}7 . 1. i 2 In this region, K =-7 3. » 10-40 if -r• 5 and G is in dynes/cm , or K 5.0 x 10-11 if-r =, 5 and (f is in p.c.i; and Q . 37.5 kcal/ mole. This activation energy is interpreted as the energy for migration of vacancies away from dislocation which are moving non- conservatively.
- 26 - The relaxation of stress rith time at constant strain was very closely described by the equations,
ct € E117, exp (_
KT/ In (1 -4- (0,,,a)e Ent
CT is the applied stress at ti(:se. "zerY", when strain is HereI o fixed; K is the Baltzeman Constant,(2 is the activatiun volume, E is the prior (constant) strain rate, and Em is the effective elastic aoduius of the test assembly Laeasure during loading prior to yield. These equations are similar in form to those (63). derived from theory by Cottrell and Aytekin Plots of (3 -3 v.s. flow stress at C. 10 showed that the activation volume is uniquely related tc, the floe: ,Aress, independent of temperature -21 -20 3 and strain rate. Values owe' 0 ranged Iron 10 to 10 cm . The anergyto form a vacancy (36,500 cal/mole) is obtained from the stress dependence of the activation volume.
Subsequently, plastic flow experiments were performed on single crystals, whish had been hydrogen—reduced to 4.?, 4.3 and 4.9 wt 70 carbon._ It was shown that the first two coapositions were single—phase at the test temperature, while the last two—phase (UC UC2). The results are summarized in Table 5.
— 27 — Table 5. Summary of Compressional Flow Data Flor Stress, p.s.i. at strain = 0.001 -4 and rain rate ,,.. 4 x 10 kgin.
1500o 1700°C 1800°C 2000°C Co position n. (cal/nole) C
4,8 wt % 5.2 65,000 2,460 1,700 1,440 1,090 4.8 wt ro 5.1 73,000 2,150 1,430 1,190 880 4.8 wt 7- 4.4 72,000 2,900 1,840 1,500 1,070
4:7 vt 'i, 4.3 57,000 2,300 1,530 1,270 930 4.7 wt c/r 4.8 74,500 1,900 1,280 1,070 770
4.9 wt /., 6.5 109,000 5,100 3,170 2,550 1,800 14.9 wt 7, 7.1 125,000 9,800 5,500 4,400 2,950
The flow stress for the given strain rate was corrected -3 back to a "constant structure" condition of 0.6 x 10 strain, and the results are plotted vs teuperature in Fig. 4. It is immediately seen that the single phase (UCI-x) and two-phase
- ) crystals fall into two separated flow—stress bands, (UC — UC2 that of the latter being about twice that of the former. Within the two—phase system, the flow stress appeared larger for a crystal containing 8 vol f UC, than o;• one containing 3 vol i.e. it correlated positively with the UC fraction. UC2' 2 Within the single—phase there seemeA to be no consistent or important dependency of the flow stress or its activation energy upon carbon deficiency. 1 — 12, Radiation Effect on Uranium Carbide.
Two radiation processes are possible in the carbide, whereas only a single process occurs in non—fissionable materials. The first process, the displacement of atoms from their original lattice sites in uranium carbide by elastic collision with neutrons and fission fragments, is similar to neutron or other particle collision process in non—fuel materials. however, the energy release to the lattice by fission fraguents is for greater and causes very high local temperatures, The second process, i,e" the chemical or mechanical damage wrought by the very existence of fission products within the lattice, has no counterpart in non— fissionable materials except in very small volumes immediately adjacent to fissionable material, that is, within the fission— fragment recoil distance.
(a) Basic Studies of Radiation Damage.
These have been based, to a very large extent, on the use of X—ray diffraction and electrical resistivity measurements.
The first reported X—ray measurements were by Block and (64) olorcheix who found that lattice parameter changes with 18 2 neutron dose saturated at about 10 neutrons/cm . The decrease in lattice parameter with annealing showed sharp drops at about 13009 550° and 700°C.
—30 — (65) Adam and Rogers measured the changes in unit cell size by X—ray powder diffraction techniques with both UC and UN, irradia. in the BEPO reactor at Harwell at about 60°C. The unit cell sizes increased up to a saturation value of about , (1. 0.15% at 0.5 x 1018 neutrons/ca22. An analysis of the number of atoms affected by a single fission event gave vcAues of 6 6 2.8 x 10 for UC and 3.5 x 10 for UN which may be compared 7 7 With estimates of 10 for uranium and 3.4 x 10 for U308.
The recovery of the unirradiatei cell size was measured o after annealing at a so:ieo of tomperatures up to 640 C.
Two annealing stages were observed in UC, 58% in the range 70° to 150°C and 27% at about 525°C, some 15% remaining unannealed at 640°C. The UC used in these experiments was made by reaction sintering and was of approximately stoichiometric compositions in the D.M.T.R. reactor at Dounray at a maximum specimen temperature of 80°C, in a thermal neutron flux of between 1.3 : 1012 and 1.4 x 1013 nicm2 sec., i.e. an order of magnitude higher than the BFPO flux in !dam and Rogers' experiments.
Fractional increases in lattice parameter as determined (66) by Childs et al are plotted in Fig. 6. For hypirand. hypostoichiometric specimens and for U2C3 from a two phase ) specimen. Except for this last specimen, saturation (UC + U2 C3
—31— x 3E OF VALUES 010 34 FREAS et1 . 4.7.1020 , -3 W It 10
1.5
a
I- 1- 1.0 ••• ..,tICIMEN 65
z :/ --.• SPECIMEN 44 •6 (W-CONTAINING) 0 ADAM'S DATA
LL
1076 1017 lot° 101, Ntuipom ;7XPOSURE (LOG SCALE) 6. Increase in lattice parametre for irradiated UC
1.0
LL
z rc z
I-
P
0 SPECIMEN 53 0 I- V LL PECIMEN 54 PECIMEN 62 PECIMEN 45
0 200 A00 600 BOO t000°C ANNEALING TEMPERATURE Fig. 7. Resistivity decrease on annealing irradiated UC 18n/cm2 Anneal time = 5 h ; Exposures — 45 9.7 x 1018 2 53 8.3 x 10 n/cm 19 2 54 1.6 x 10 n/cm2 62 3.5 x 1016n/cm
— 32 — 17 occurred between 10 and 1018 nicm2 dose, the curves running roughly parallel with one another with the carbon-rich material giving the larger values, Lnnealing experiLeents showed removal of cpproxieately of the coil .,Aze inerease between 100° and o 300°C and the remaining 30% between 400 end 600°C; attempts to confirm the 700°C stele founci by Block and Uarchaix having failed.
(67) At the Battelle Memorial Institute, lattice parameter measurements were made or. arc-ctst UC, 52, at carbon contentl , over the burn--up range G 1.0 at % uranium, although in these experiments the specimen temperatures were 150°C up to 0.3 at % burn-up and between 430 and 810°C at higher burn-ups. The lattice parameter burn-up curve saturated at 0.15%4- a/a at a similar dose as found by other workers.
(68) More recent studies Chi. ds and Ruckman have shown a correlation between the resistivity increase at saturation in hypostoichiometric ITC with the lattice parameter of the uni_rradiated specimens_ At the hypostoichiometric composition a 4.955972 and the resistivity increase is 157%. Childs believes that this is due either to the vacant carbon sites accommodating displaced uranium atoms with a minimum of strain or to the defect lattice reducing the range of uranium crowdions. (See Fig. 7). Griffiths (69) carried out a parallel set of experiments to determine the resistivity increase in reaction sintered UC
—33— of approximately 4.8 wt % carbon composition and observed tro recovery stages c.:t 110 — 130°C (304) and 700 — 730°C (70%). Furtheraore the high temperature stage corresponds very closely with the annealii temperature for thermally (70) inducedivakmacies and Griffiths concludes that the low temperature stage corrospona,1 with interstitial migration. — 13, Fission Gas Release.
The can of a fuel element cannot be very thick in practice, and while fission gas release might be accommodated by the provision of free space within the fuel element, it is obvious that a loy level of gas release is technologically desirable.
Sone indication o:? the probable level of gas release may be obtained by irraliating specimens to a low dose 133 85 followee by measureL,:ent of the release of Xe or Kr on heating to successiely higher temperatures. From such experiments the diffusion coefficients can be obtained. Cowan and Orth have reassured the diffuSion coefficients of (71) Xe — 133 in uranium carbide and also Luskern and Osara 72) have done similar fleasurements. Their results are summarized in Table 8 and a typical value for UO is given 2 for cof_parison.
Table 8. Surve of Fission Gas Diffusion.
Xe — 133
Ref. Do g -7 UC 1.27 x 10 42.5 71 2.1 x 10-5 85.1 72 -8 UO 2.0 x 10 48.9 73 2 -6 6.6 x 10 71.7 74 -8 1.5 x 10 46.0 75
— 36 — Uranium aonocarbide, with regard to fission gas diffusion, shows a behaviour siailar to the dioxide. There is no essential difference between these two :laterials with regard to fission gas containment at high temperatures.
rlore direct indication of fission gas release comes from the sampling of irradiated capsules or fuel eletents. The capsules containing the fuel are punctured and the fission gases removed to an evacuated glass vessel and sub— jected to mass spectrometric analysis for Xe and Kr isotopes. Insufficient data exists for a details i analysis of the release behaviour but the results obtained data are all of a low level, generally less than 11'0 for maximum centre temperature up to 1100°C.
—36-- Section 11 •- 2,
Diffusion in Solids,
3`! 2 — 1, Imperfection in Solids.
The 1:.io5t stable state of crystal at the absolute temperature zero is that of complete order. .Lt finite temperatures the structure of crystal still approaches very closely a regular 1ritticc. Lo the temperature is; further raised, the rIcan amplitude of the thermal vibrations of the atoms about their mean positions increases. Owing to these vibrations there is at any moment a certain departure from periodicity in the position of the atoms in a crystal lattice. Thus, defects were introduced into a crystal,,
Furthermore, defectn ccal he shown to arise from purely thermodynamic reasons , D) when the temperature of a crystal is raised. For a crystal to be in thermodynamic equilibrium at given temperature, its free energy must be a minimum. Although energy must be expanded to form a defect against the cohesive forces of the crystal, the inereu.2e in entropy resulting from the defects causes the free energy to be a minimum for a definite concentration of defeets at a given temperature. Thus, we see that acme of these defects will then appear as natural features of the crystal in equilibrium at this temperature.
2 — 2, Type of Defects.
Since diffusion phenomena in crystals are intimately related to defects, a preliminary description of these defects is necessary for this study.
—33— The most Laportr.mt defect? for our considerations are
(a) T7..cancieo 03) interstitials point .:10?Ifoct (C) impurities
(d) dislocation lino dofoct
One of the most important properties possessed by interstitial° and vacancies is that they x.suclly cap migrate relatively easily when assisteJ by thermal. :luctlpy'A.on and permit the diffusion of matter, (14)
() 2 - 3, Frenkel Def?cts
An atom or jot moving from noraal :Ate to an interstitial position leaves a vacany behind, This mechanism of defect formation is called the l?ri.tkel mechanm, and a pair of defects - the intert,titia7. together. with he 7acancy - is known as a Frenkel defect. The ni.:E.bor Cf Frenkei defect in equilibrium at a temperature T giros by
(2.3.1) (U N )P (- Uf/MT)
Where N and AT are, recpeedvely, the total number of sites and of possible interstitial sites, Uf is the energy of formation of such a defect. Thus, one would expect, the number of atoms in interstitial positions increases rapidly, and vacancies and
-39-- interstitials can diffuse separately until they meet again and anihilate each other.
2 - '4, Schottky De5..'ect (m)
From previously described thermodynamic reasons, vacant lattice sites in a crystal may be formed in the way which does not involve the production of interstitial atoms. The number n of Schottky defects which ate in thermal equilibrim:1 in a crystal at the temperature T, is given by
(2.4.1) n (N-n) exp (-U4T ). in which U is the s energy of formation of the vacancy, N the number of possible sites and n the number of vacancies. L salient feature of this defect is its mobility. In a compound, the nobilities as veil as the energy o± formation of vacancies corresponding to the different kind of atone or ions which are present differ considerably.
In general both tyres of defect will occur. Whether Schottky or Frenkel defects predominate in a particular crystal depends on the relative values of 116 and Uf , which, in turn are essentially determined by geometrical features.
2 - 5, Role of Defects in Diffusion.
Defects usually can migrate about the crystal relatively easily ahem assisted by thermal fluctuation and hence permit
40 - the diffusion of matter. In fact, these defects were introduced into the theory in order to explain facts concerning diffusion and electrolytic conduction. (80
Ln interstitial atom may diffuse through the crystal by jumping randomly from one interstitial position to another one. Similarly, a vacant lattice site may move through the crystal by the jump of an adjacent atom of the appropriate sign into the vacancy, Thc vacant lattice point thus =yes to a nev position in thc, ciryeetaly and this process can repeat itself. 1„ potential barrier must be overcome in the jump of an interstitiali atom into a net! position or by the jump of a normal atom into a vacancy, hence these processes require an activation energy and the movement of these defects is strongly tenperature-dependent.
- 41 - 2 — 6, Diffusion Etuations.
Ltorie diffusion in solids usually refer to a net flux of atoms of one species induced by a concentration gtadient of this particular species. if an unhomogeneous single— phased alloy is annealed, maillr will flow in a manner which will decrease the concentration gradients. If the specimen is annealed long enough, it irth become homogeneous and the net flow of !natter rill C.*C-_,E.0 r The mathematical representation of this phenomena
(2,6.1) grad. n,
—2 -1 (j- the flux of atoms m sec n the number of atoms per 3. cm ) is known az nich's law,. The constant D is known as the diffusion coefficient.
The original Fick's lay may be generalized in several directions. The diffusion coefficient D is a sealer only in isotropic materials or in crystals with cubic symmetry. In crystals with noncubic symmetry, D must be a tensor. The law must thus be generalized to
(2.6.2) J• . = D.. grad n. 13 3
The fundamental diffusion equations as written in (2.6.1) involve the concentration because the driving force is furnished by a gradient in concentration. This connotation is of course not strictly correct. The actual driving force
— 42 — is the gradient of the cbeminal potantialA-. The chemical potential may be constant even though the concentration gradient is non vanishing. The equation (2.6.2) may thus be generalized to
(2.6.3) J = K. grad
Although the diffusion coefficients defined by Equation (2.6.3) may be difficult to measure experimentally, they have thr, virtue that they have an exact physical inter— pretation. Since, by our present definitions, we have excluded all external influence on the atomic motion, any flux of atoms must originate purely in the random thermal motion of atoms through the lattice. it is therefore reasonable to assume that di fusion occurs by the periodic jumping of atoms from one lattice site to another. If this is true, then the mathematics of the random walk problem Trill allow us to iind sorrelaitdonships between the observed microscopic diffusici: ,:oefficients and the jumping frequencies and jump etiatc,nce of the diffusing atoms. 2 — 7, Eouations for D.
alpirically it is found that D can be described by the equation
(2.7.1) D D exp — ) o RT
Where D and .6.Tri; may vary with composition but are o independent of temperature. H'xperimentally Do and L are obtained by plotting la D v2rous --. The slope of this plot gives
(2.7.0 d it ;3 1
An alternate equation for T.) in the c:.,se of interstitial diffusion is also given
AT (2.7.3) \ to D =Is exp — o 11T—
Ti'nere , gerrzietri cv.1 fast,:r. a Y= jump distanc€ o juop frecuency A .9 = entropy change e-=,. If enthalpy of r2obiity
— 44. —
Comparing this with Eq. (2.7.1) we see that the first term in parenthesis is .:7.,quc..1 to Do and that & E equals the quantity ZIRLI.
For diffusion by vanancy mechanism in a pure metal D is; given (2.7.4) +ThelH 2 42ft -f n ) D ( exp r, ) exP (-- RT
where entropy change for formation of vacancies. 4kS entropy ,hange for mobility of m vacancies, Bf enthalpy for formation of vacancies. enthalpy for nobility of vacancies. hn
The tern square brLLckets is again while AE is the in D0, sun of .6, is f and Q N.
—45— 2 --E-* Diffusion Mechanisms.
Three types of mechanisms have been proposed for crystals.
(1) Vacancy Mechanism.
Some lattice sites may be vacant and an atom neighbouring a vacancy may then become displaced into the vacancy. As successive atoms move in this way the vacancy migrates through the cry.7,taln Since work must be done to create a vacancy site by removing the atom and putting it on a surface, the con- centration of vacancies in pure crystals in thermodynamic equilibkium is very low at low temperatures and only becomes appreciable in most substances at high temperatures. It is now well established that dislocations and grain boundaries can act as efficient sources and sinks of vacancies. In compounds vacancies may also be introduced by impurities or by departures from stoichiometric composition.
For this mechanism it is possible that vacancies may cluster in groups of two or more and that the groups may remain associated through many unit jumps. For a pair to retain its identity the atom jumping into one vacant site of the pair must originate from a set which is a nearest neighbour of both sites in the vacancy pair.
The vacancy mechanism is well established as the
- 46- dominant mechanism of diffusion in f.c.c. metals and alloys and has been shown to be operative in many b.c.c. and h.c.p. metals as r;:11 as in ionic compounds and oxides.
(2) Interstitial Mechanism.
The diffusing atoms move through the interstitial spaces between the atoms occupying the normal lattice sites. The interstitial defect must have a continuity over at least several iner—stomic distances, r.lthough the defect need not centre on the same atoz. th3.; distance. In a special type of interstitial meshanism, designated interstitialey by (82) Seitz the intersti.tial defect moves when the interstitial atom squeezes into a nearest lattice site, displacing an atom from that site into another interstitial position. By a repetition of such displacements the defect may move continuously over considerable distances.
The interstitial mechanism is thought to operate in alloys for those solute atom3 which normally occupy interstitial (83) positions, e.g. C in CA — iron. In solid solutions in which one component is interstitial under equilibrium conditions and appreciably smaller than theautrix atoms, it is recognized that the iieerstitia/ atom should diffuse through the interstitial space; while lattice atoms diffuse through the substitutional sites. Since the two kinds of atoms do not
—47 — compete directly for the name sites, two independent diffusion coefficient.,: should be obtained.
(3) Exchange and ling12Lechanip.,
A pair of neighbouring atoms may si,Lply exchange places or a ring of neighbouring atoms may execute a "rotation" in which each atom jumps and takes the place of the one before it. A (84) version of this mechanism due to Nachtrieb describes the fluctuations in atomic positions which cause the re—arrangement as local melting followed by freezing. It may be said fairly that there is presently no experimental evidence for a ring or rotation mechanism in any metallic system.
—48— 2 — 9 , Empirical Relations.
There exist several approximate empirical relations giving quantities of activation in terms of bulk properties of the system.
(G5) Nachtrieb and :candler found empirically that 41,E for self—diffusion correlates closely with the latent heat of melting L 9 for many cube substances, according to 119
(2. .3.2) r 16.0 7
A similar relation to this one betweenpEand the melting temperature Tm is rather analogous,
(2..9.3) LYiff 33 I'm cal/mole (Van Liempt's law)
(86) Since Hildebrand had observed that the entropy of melting, where
(2. 9.4) r G. S L/Taa was nearly constant for material which had the sane crystal structure.
A third relation of the same kind holds between entropies
— 49 — and volumes of activation.
(2.D .5) L where oe„ in the coefficient of thennai expansion and is the ) isothermal conpressibility.
The correlation with melting properties lead Nachtrieb and Handler to postulate a ralaxed vacancy mechanism for all these cubic solids. It is not clear that the vacancy described in thermodynamics differs substantially from that described by Hintington and Seitz.
— 50 — ii ~C ION III
Experimental Work.
--51 -- III — 1, Material—preparation and properties.
—52— Naterials.
The uranium carbides used in this study were prepared and supplied by the Metallurgy Departiient, L.E.11Z.L, Harwell.
Lrc-Cast Uranium Carbide.
The carbides were prepared by arc-pelting of uranium and graphite together and casting into a graphite mould followed by annealing at 1800°C in vacuo. The casting was machined to size of i" din. x i" length.
Cintered Uranium Carbide.
The carbide specimens were reaction sintered at 1100°C, followed by annealing at 2000°C in vacuo. Nominal content of carbon was 5% by weight.
C-14 labelled Uranium Carbide.
The specimens were all reaction sintered in a similar manner; the carbon composition 'consisted of 5% wt C-14 and 95 wt % Dag 621 graphite. The activity of carbon-14, measured after the preparation was 2.45 micro curie/mg of uranium carbide.
U-235 labelled Uranium Carbide.
The specimen was cold pressed at 50 t.s.i.,reacted 1130°C
- 53 - for 2 hours and annealed 1700°C for 2 hours. All heat -5 treat lent was done in vacuo of the order of 2 x 10 am pg.
4.725 gras of uranium carbide (5.5 wt `12) contained 4.466 gns of uranium metal at 92.7 V enrichment and 0.260 gms of carbon.
Impurities. Some of the analytical results for the arc-cast uranium carbide follow:-
Specimen No. Carbon w/o. Oxygen P.P.m. Nitrogen P.P.m.
175 5.1 140 115 179 5.0 130 135 183 4.7 70 140
Carbon contents were obtained grminetrically from CO2 absorption after combustion in oxygen at high temperature. Nitrogen was determined by the Kjelchahl method, and oxygen was measured by vacuum fusion method in a platinum bath at A 2000°C.
Metallic impurities were determined by spectrographic method. It normally contained ti 100 ppm tungsten, -,- 150 ppm 2 ppm cobalt and 20 ppm aluminium.
-54- Density Measurement.
The density of uranium carbide specimen, prior to any experiments, were determined with the mercury balance in the I'mboratory.
The results of measurement are listod in Table 9.
The specimens had an average density of 9'd% of the theoretical (X—ray) density for arc—cast and 78%for the sintered carbide spenimens.
-- - TABLE 9. Density of Uranium Carbide.
Carbon Density Specinen Content wt .% t,u/cc Theoritiad Dentt Melt No. 179-1 5.0 13.39 98.2 n 11 2 5.0 12.61 92.5 Melt No. 175-1 5.1 12.68 92.3 tl " -2 5.1 13.51 99.1 II " -3 5.1 1!.;.64-- 95.8 U -4 5.1 13.37
Helt Yo. 183-i 4.7' 13.49 98.1 -2 4,7 13.43 95.7 1, -3 4.'s 181 101.3 fl It -4 4.7 13.62 100.0 II U -5 4,7 13.5 99.2 n " -6 40 7. 13.85 101.6 4.82 13.03 95.8 ______.12., r.2.71 93.2 verge ,.'o Sintered No, 1 5.0 11.55 84.7 II 00. 2 5.0 11,05 73.7
tt No. 3 5.0 10.92 71.2 it No. 4 5.0 11.23 82.4
Average 78%
-56- III — 2, Description of Experi:aental Lp2c,ratus.
- 57 - 2 - 1, Vacuum Furnace.
In view of rapid oxidation of uranium carbide in air, a vacuum or inert-gas filled furnace must be used for experiments.
Diffusion annealing experiments were performed in a vacuum furnace supplied by Edward :iigh VacuumIAL(See Fig. 8), which is specified to maintain 2400°C with a tungsten heater.
The furnace comprises a mild steel water cooled cylindrical furnace chamber fitted with a detachable water-cooled lid and containing the static radiant heater assembly. It consists of a hexagonal molybden bar heater mroviding a work space of 7 in. long x ka in. diameter and surrounded by six concentric tungsten radiation shields, insides the outer shell. It is also provided with circular tungsten radiation shields at the bottom of the assembly, a set of similar shields at the top. The entire assembly is enclosed in a circular steel shield. (See Pig. 9).
The chamber lid is fitted with two pyrex observation windows for viewing the process and for the purpose of optical pryrometery. An alternative means of temperature measurement is by a thermo- couple for which purpose a blanked-off port is provided at the rear of the furnace chamber.
Power to the heater is supplied from a Brentford type REO 8/02 transformer.
The safety circuit is set to cut off the power supply to the furnace in the event of a short circuit.
- 58 -
Fig. 8. Vacuum Furnace
Control Units
Lid not :,ho
Water Sighting Hole 0 7 Cooling C O
0
0 iiolybden ILadiation neat Element Sheild
Copper Copper Conductor Conductor
Insulator . -
Vacuum Chamber
azpviinst4,-4.0 P--
Vertical Cross Section of The Vacuum Furnace
9,
- 6o - Tempera ture Fig. 10.HeatCycleofVacuumFurnace. -0-0-0--0-0-0 0 TIME
2 — 2, Grinding_plachine.
J photograph of the grinding machine used in the experiments is shown in 'Zig. 11, it was originally designed and built by Rayrlent Tools Ltd., Southampton, to the require— uent of Dr. Chang Shih. (U,K.h.:0,..E. Progress Report. Contract No. 13/5/165/1026, march, 1960).
The complete arrangement consists of a high speed grinder (Bosch LPI/IJAZ 65/51), mounted on the arm of a robust drill stand and a slowly rotating platform which is connected to the main motor.
The tight—fitting semi—flexible plastic bag can also be used to hold an inert atmosphere when grinding is in progress, if required.
When in use, the drilling head is first swung aside to permit adjustment of the specimen. It is then returned to a position slightly off -the centre of rotation of the specimen, and the platform raised until contact is just made. The direction of rotation is opposite each other.
In view of higher efficiency of grinding and the pyrophoric nature of uranium carbide a slight modification has been made. The platform has been changed to a cylindrical dish type and used as a grinding collector as well as a rotating platform. L spring loaded sample holder was attached to a high speed grinder instead of a diamond iapregneted rim cutter.
— 62— Fig. 11. Grinding Machine
-63- In the dish silicon carlkAa powder, moistened with toluene, was spread. Then in use the specimen on the holder was slowly touched by -ale rotating platform and the grinding was accomplished. The grinding chips end used silicon carbide powder remained in the dish and was collected for the pubadquent treatuent.
64- 2 - 3, Evaporation Unit.
The deposition of thin-film of metals and certain of their compounds, such as the oxides and salts, on metallic and non-metallic surface is a most popular technique in diffusion studies. In order to see the feasibility of this technique an evaporation unit was constructed and is described in the following paragraph.
Pumping System and work Chamber.
The vacuum work chamber is a hard glass bell-jar (6 inches outside diameter) and a moulded rubber "L" shaped gasket to make a vacuum seal with base--plate The chamber is evacuated by a "Speedevac" Model 203 B vapour diffusion pump, backed by "ISC150L" rotary vacuum pump. I. "Speedevac" Pirani-Penning type gauge is fitted, with the Penning gauge head in the work-chamber and the Pirani type head in the backing line to ensure correct operation of the vapour pump.
Electrical Equipment.
The equipment includes a low-tension transformer, maximum 7.5V, 20! continuous rating, 3012 intermittent rating with a variable resistance in the secondary for the adjustmenta of the filament current and the panel mounted an ammeter with all necessary switches.
The filament used was generally tantalum fail.
- 65- Fig. 12. Evaporation Unit
- 66 - 2 - 4, Gamma-ray Spectrometry.
General Considerations.
A variety of methods etist for determination of the isotopic abundance of uranium-235 in uranium-containing materials.
Methods for determination of uranium-235 concentration relative to that of uranium-234 and 238 have been based on differences in mass (88'89/6°) and on isotopic line shifts 01,92,93) in emission spectroscopy. The natural radioactivity of uranium samples has been employed, based on alpha particle (24) disintegration rate, m:,?.,asurement of the growth of beta- emitting daughters from the chemically separated uranium, (95) and a gamma ray emission of uranium-235 based on a discriminat- ing counting techniques. (96)
Several methods involve neutron activation follower!. by radioactive measurement. Thus, neutron-induced fissions have (97) been detected by using a fission counter and barium fission product has been isolated and counted after activation. (98)
The method described is rapid and non-destructive within 1% over the concentration range of 0.72 to 00 uranium-235, although it can be extended to approximately 0.00 uranium-235.
It is based on the use of gamma scintillation spectrometry to measure the photo-peak resulting from 0.184 MeV. gamma photon emitted by uranium-235 during its natural radioactive dekay.
-- 67 i4ajor Components.
The gamma-spectrometry used throughout this study consists essentially of .a single channel analyser feeding a modified Honeywell strip-chart recorder which incorporates the special scanning feature.
This scanning feature comprises a precision potentiometer which is driven from the chart driving mechanism in such a manner as to sweep through the pulse height control for each 12 indhes of chart travel.
The single-channel pulse height analyser (Type 1168B) is proceeded by a wide-band linear anplifier (Type 14301) and followed by the rateneter (Type 1037C) with meter display and recorder output.
The spectrometry also includes
k sealing unit (Type 1009~!,) and a power unit (Type 1359A)
A thallium activated sodium iodide crystal (IP x 1") was provided in connection with a photomultiplkr tube. It was a yell type crystal and used in conjunction with a k diameter test tube through the aperture in the lid of the counter.
All major components, except a Honeywell recorder, were manufactured byfii.,YnatIbn ElectronicSLtd., and supplied by A.E.R.E., Harrell.
-68- The Function of Gamma-ray Spectrometry.
The gamma scintillation spectrometer is used to analyse the gamma-spectra of radioisotopes in a 5anner similar to the analysis of the emission spectra of elements with the emission spectrograph. The sensing unit required for the detection of the emitted garsia radiation is the thaliUm-activated sodium iodide crystal.
A typical gamma-spectrometer is shown in Fig. 13 and 14.
The gamma photons from the sample, penetrating the crystal, have a high probability of interacting with the crystal. This interactinn results in the conversion of a single gamma photon, via a recoil electron, to a shower of light photons which are detecteei by a photo-multiplier tube.
The current output from the multiplier phototube anode is directly proportional to the energy lost by the incident gamma photon. These current pulses are fed into an amplifier of sufficient gain to produce voltage output pulses in the amplitude range of 0 to 100 volts.
One method of analysis of tlAe pulse spectra is by use of a single-channel pulse-height analyser. The analyser slowly scans the,pulse distribution over a 0 to 100 volt range with an acceptance slit a few volts in width. Only those pulse falling within an acceptance slit are passed on to a rate-meter and plotted.
- 69 - ks the acceptance slit completes its scan down to zero energy, the complete pulse height distribution is plotted.
The curve of pulse height distribution of a aonoenergetic gamma emitted can be divided into two characteristic curves. ee Fig. 18).
The sharp symmetrical peak is the result of total absorption of the gamma energy by the NaI(Tl) crystal, and is normally referred to as the full—energy peak. The continuous curve below the full energy peak is due to conpton interaction, whereilill the gamma—photon loses only part of its energy to the crystal. The location of the full—energy peak on tine pulse amplitude or gamma— ray axis is proportional to the gamma energy of the incident photon, and is the basic for the qualitative or quantative application of the gamma scintillation spectrometer.
The compton continuum serves no useful purpose in full— energy peak analysis.
—70 — Calibration of the Spectrometer.
Using standard sources, purchased from The ladiochemical Centre, the spectrometer wasu.librated by comparing the photo peak area of the specimen with that of the standard source (or by measuring the photopeak heights). The used standard source was a well—calibrated Cs-137 and Co-0 isotopes. An energy spectrum of Cs-137 was shown in Fig. 18. The energy spectrums of natural and ”enriched" ur.7,niuci was also shown in Pig. 15, 16 and 17,
A calibration ;..hrixt vias cIso plette0. in Fig. 19. It shows a good linear relationvhir) between the activity and weight of samples.
—73k— Fig. 13. Gamma-ray Spectrometry
- 72 - 7
8
6
Fig 14 . Schematic arrangement of a single-channel analyser.
H 1. NaI crystal ( 1 x ) 5. E. H. T.
2.photomultiplier tube 6. analyser
3.cathode follower 7. recorder 4.amplifier 8. ratemeter
:7- .73 184 KeV 94 KeV X-ray ( 16 KeV )
0 0. dil. .' .. . - .
• ...... _ _4 . . I 1 . I I 111
R IMMO IM 1 MIMMININE --,-- irm _. ...im. a 1 ili MN .'11 .1 i ....., ...MIS ..II II III i IIIIIIIE111111111 II
-- II
• ^ I ,1 _.--- __ i . ' _ I WM INISIIIIMIIMINS I I: MI i s:III _ _ -111111111111111114 _ L
- 4.
, .• !, • , .. .. _ , . • 0 ar II
Fig. 15. Energy Spectrum of "Enrichrd" Uranium ( 92.7 )
- 74 - 184 KeV 94 KeV X - ray ( 16 KeV )
Fig. 16. Energy Spectrum of Natural Uranium
- 75 - 184 KeV 94 KeV X -ray ( 16 KeV )
- --- 2 7 N, __._ • t --_;, _ . __ 0. f- _, ___.1--+---f- __ —7
- • . I , _ il 1- - -
=: i Min . _____I 1._. .= MEM=
Mill
-4 iii II=1 - --- 6 111111111MIIM' ■ NM ? NMI _ .1 _ _ II • I - 1 i , ,' 1 ... L I M - - 1 ____L 1 11111MEMNIIIM - - -
i • 4-- AIIIIIIIIIM NM --1 Mil =7E7 1 1111.=111111111111
nor _ IC — - F ' g
_ ... . _ . - - • .11.1al11111111 Ili I, IR _ =.1, ,.....,..... -t IIIIIL , "..ill t , mall:... 1 • - r ,,,,r, • ch
Fig. 17. Energy Spectrum of " Enriched " Uranium ( 7 % )
- 75' - Fig. 18. Energy Spectrum of Cs-137
Energy Fig. 19. Calibration of Gamma-ray Spectrometry
0 s.o
0 - LIN
) /mg in /m c ( ity 0 tiv 1 -- c cu a
0 0
0 5 10 15 weight of " enriched " uranium carbide ( mg.)
- 77 - 2 - 5, Liquid Scintillation Counting Technique and the Counter.
General Considerations.
The neagurement of C14 02 particularly that derived from biological system, has received considerable attention since the earliest days of carbon-14 availability.
At first the use of internally filled G-M tubes for direct 14 counting of 0 vied with precipitation of BaC 0 and counting 1402 3 on planchets as the nreferred method of 0140 determination. 2
14 (1) Direct measurement of 0 02
This requires a fairly elaborate sample handling train in order to purify the gas and ensure correct filling of the counting device. Great care must be exercised in sample preparation since tracer contathinaAes, and even moisture, can cause serious inter- ference with the count. The amount of sample which can be measured at one time is severely limited and routine low activity measurements are almost out of the question. Finally, contamination of the counting chamber is an ever-present possibility. A typical example was shown in Fig. 20. This apparatus was previously adopted for C-14 determination in the laboratory.
14 (2) Measurement of BaC 03 on planchets.
This method is fraught with :,any of the sane problems, although
-78- 14 Fig. 20. An Example of Direct 1'ieasurement of C 02 with an Internally Filled G M Thbe. ( set up by Dr. Chang Shih and used by Mr. A. D. Hilton )
-79- planchet does have an advantage of permitting automatic sample handling. Preparation of planchets reproducible surface characteristics has received much attention.
Because self-absorption is a problem which cannot be over- 14 come, BaC 0 planchets are generally made up to have "infinite 3 thickness". The amount of sample which can be measured is there- fore severely limited and, since counting efficiencies are quite low, the method nut be classified as relatively insensitive. 14 Planchet preparation including BaC 03 precipitation, filtration and drying is quite time-consuming.
In recent years, liquid scintillation counting has become the method of choice for the measurement of soft beta-activity.
(loo) Imong the early workers, Hayes, et al reported on an extensive survey of potential solvent systems, while Passmann, (101) Rodin and Cooper outlined a new concept - the use of a chemical solubilizing agent, the hydroxide of Ayamine - which dissolves in toluene.
102) Hayes, Rogers and Langham ( reported the first evidence that the liquid scintillation method could be successfully used for the measurement of material in suspension, rather than in solution. C-14 labelled substances were ground and dispersed in toluene - ppo - popop system, merely by shaking.
-H0- 14 Many workers sought to trap C 0 in a medium suitable 2 for liquid scintillation work. The first to elaborate on a ()103 method was Pa ssman, Rodin and Cooper and their approach — (104) the use of the hydroxide of Hyamine — is still the system most widely used. Since this first paper on the subject, several other trapping materials have been employed.
The advantages of using liquid scintillation techniques 14 over the methods discussed earlier for C 0 measurement are 2 several. Sample preparation is rapid and relatively easy. Trace contamination is but a minor problem. Counting efficiencies are quite high and relatively large amounts of sample can be measured, thereby, decreasing counting tire.
Liquid Scintillators.
A liquid scintillator must have two essential ingredients; solute and solvents.
A scintillator solute must be an efficient light emitter and must produce a photon spectrum which, in a conventional scintillation detector will be efficiently transmitted and reflected in the optical system and converted into electrical energy by the photomultiplier. The best light emitters have emission cpectra of two short wavelengths to satisfy the conditions and so it has become conventional to employ a two— solute combindtionic a primary to insure a large number of eventually emitted photons, and a secondary to be the actual emitter and to control the spectrums of the. photons.
For a detector which operates at room temperature, p-terphenyl and POP0(2, 2-p-phenylenelus (5-phenyloxazoleV are an excellent and favourite solute pair.
A liquid scintillation solvent must have not only good energy transfer characteristics but also a small absorption coefficient for the light that the solute emits. Early studies of scintillation solvents showed that best performance resulted from alkyl benzene structures; and the simplest of these, toluene, became the most popular.
The best performed composition of solution at room teuperature is 5g/I p-terphenyl and 0.5g/1 POPOP in triple distilled toluence.
The "scinstant" 2cintillators with the same composition was commercially available from Nuclear Enterprises Ltd.
--62— CO Trapping Agent - Hyamine. 2
Nyamine hydroxide is a high molecular weight quaternary arimonium hydroxide, I is reilciiiy soluble in toluene, xylene, water and alcohols, and has found a good absorbent for internal counting of CO2.
On) Passman, et al demonstrated that a 100 molar excess of "Byamine" in methanol-toluene absorbed CO2 quantatively.
ketual sample preparation has been even further simplified by aadin and co-workers who have shorn that it is at times possible to perform combustions in a flask with direct trapping of the evolved CO in Hyamine, thereby eliminating PICO collection. 2 ) have shown that CO in expired breath Frederickson and Ono (105 2 is quantatively trapped by bubbling through Hyamine, enabling direct breath collection.
Results of CO work using gyamine have been reproducible and 2 14 precise. Counting efficiency for C 0 as Hysmine Carbonate has 2 been found to be above 60 for a 15 ml toluene solution containing 1 al of I molar Eyamine.
The "Hyamine" hydroxide used throughout this work was supplied by Nuclear Enterprises Ltd., Edinburgh, Scotland.
-83- Liquid Scintillation Counter.
The P864A type scintillation counter, supplied by EKCO l']lectToncE, Ltd., is a '111,111) .,,;,riding; shielded instrument for "in vitro" counting of millirAnrocurie quantities of beta and gamma emitters.
The counter comprises a thirteen stage photo—multiplier tube which, together with a source coupling unit, is continued in a lead shield fitted with a counter—balanced lid. The lead shielded counter assembly is mounted on a steel case, which also contains a wide band linear amplifier.
The source coupling unit is an aluminium alloy casting containing a light proof shutter, which covers the sample to count and interlocks with the shutter to prevent accidental exposure of the photomultiplier tube to light, and above this a spring loaded carrier to hold the special quartz sample holder. To ii-Iprove the light ooliection efficiency, a liquid coupling medium (Silicone Oil .thS200. 2 centistrokes from Eduard Eigh Vacuum Ltd") was used between the sample container and tube face.
Radiation absorbed by the scintillators is converted into light pulses which are, in turn, converted into electrical pulse by the photo cathode and amplified by the photomultiplier tube.
— 84 — The signal at the collector is then amplified still further, by the built-in linear auplifier, to a level sufficient to operate a sealer recyiring a positive going input signal of five volts minimum.
A complete counting system wc.s formed using the N664A Irith a sealer, a power unit and a pulse analyser (Type 1168B) from Daynatron Ltd. gptimum Working Conditions.
It is generally accepted that the best operation conditions of any counter are those Which will determine the activity of any sample to a given accuracy, in the shortest time. It can be 2 shown that these conditions occur when the factor Rs /Rb is a (1019 107) maximum, where Rs is the counting rate from the sample alone and Rb is the background count rate. While the counting efficiency (which is proportional to Rs) increases; due to increasing either the H.V, the amplifier gain, or by reducing the discriminator bias setting a point is reached when the 2 background rises more rapidly than Rs .
Under such conditions, in the case of carbon-14 counting, efficiency of 70 - 80% was obtained with the background count rate of about 3 count per second at room temperature. The efficiency was determined by using an internal standard.
The instrument setting was as below.
Applied voltage 1500 volts.
-85-
Amplifier Gain 50
Bias Voltage 25 volts.
Saonle Container.,
For measuring the licuid samples, quartz container with a noninal capacity of 16 n1 was used. These container* have a white coating which provides a reflecting surface, thereby increasing the efficiency of the light collection. (See Fig. 21)
The sample containers retain a slight residual phosphorescence after exposure to light ane. wore allowed to "dark—adapt" by placing then in the dark plae for up to 5 pinutes when used with carbon-14.
— 86 — L
Fig. 21. Liquid Scintillation Counter anc Sample Containers.
-87- III - 3, Experimental Procedures. - Self-Diffusion Study of Carbon in Uranium Carbide. 3 - 1, Diffusion Couple.
Measurements of diffusion rates in readily achieved when the diffusion of an isotope from a labelled region to an un- labelled region is followed by conventional chemical or radio- chemical analysis.
axperimental error arises if there is a bad adhesion between the active and inactive material, and if the initial distribution is not plane and perpendicular to the X axis.
It is therefore vital to c,btain a well-attached, thin uniform layer of radioactive isotopes on the pellet surface to be measured. This has been the subject of extensive investigations by any authors. (103, 109, 110) study of the literature reveals that for the present work one of three possible methods could be used.
(1) Thin-film Vacuum Deposition. (2) Butt Joint Couple, in which material diffuse across a join between two uniform regions of different concentration. (3) Uniform Spreading Method.
In the course of this study, all three methods have been employed to see the feasibility.
-89 - Since thin-film deposition technique is the most favoured in diffusion studies ail attempt was made to make the diffusion couple by evaporation. Vozzella et al(111) have studied the thermal decomposition of uranium monocarbide and observed that the deposited layer on the cold parts of the apparatus was found to be the carbide, and interpreted it due to recombination of free uranium and carbon. nrom this fact an evaporation unit (Fig. 12) was constructed and a series of experi.:ents was perfomed to prepare the diffusion couple by evaporation of the carbide. Indeed, considerable activity of carbon-14 we.s observed from the deposited carbide film, but due to the thermal decomposition and oxidation of uranium carbide it could not succeed to accumulate enough the activity on the surface of specimen. Therefore this LLethod was rejected.
(112) Chubb et al have employed a butt joint couple by inserting a thin foil of uranium metal for diffusion studies in uranium monocarbide. Thin method gave reasonably good results, but their results here very widely scattered.
This method also pr,p.sente?, a number of experimental difficulties, because of iow melting point and high vapour pressure of uranium metal., Xn studies at high temperatures, surface diffusion and vaporization reduced the &mount of uranium in the layer between the slabs. There was also other possibility of fart ing a new phase in the couple by changing the ratio of carbon and uranium in the interface. Therefore, it was not a very favourable method. 114) Finally, Zhelankin et al (113, have measured the diffusion coefficients of carbon and niobium in niobium carbide by employing a uniform—spreading method. Also Nerrari (115) measured the diffusion coefficients of nitrogen in uranium dioxide in similar manner. This method was very si.sple and gave very satisfactory results. At higher temperatures, a perfect contact was obtained as the activity profile in Fig. 37 *ndicatea.
The labelled carbide was crushed from the cylindrical form and ground to fine powder, by a lAortar, and the powder was kept in suspension in toluene solution. This presented oxidation of the carbide powder in
The carbide powder, suspended in toluene, was dropped by a pippette on the freshly polished surface of the specimen, while still wet, the carbide powder was spread as uniformly as possible and repeated several times to accumulate enough of the labelled carbide.
After drying the couple was placed in the uranium carbide bottle (or wrapped with tantalum foil). The radioactive surface was kept upward for the purpose of temperature measurement. 3 - 2, Diffusion Anneal.
Theicothermal diffusion anneal was performed in the vacuum furnace described in III. 2 - 1. The diffusion couple was put in a uranium carbide or tantalum container to prevent evaporation and oxidation or other possible contamination, and the container was placed in the constant-temperature zone in theheat element. The anneal time ranged from few hours to several days. The temperature range was from 1200°C to 1800°C. The anneal time was measured from the time it reached tha desired temperature. Once reached the temperature there was almost no variation in the temperatures unless the power failure occurs. This was shown in Fig. 10.
-5 The vacuum during diffusion anneal was less than 10 mm Hg.
Temperature Measurements.
The temperature was measured with an optical pyrometer to ± 10°C, through an adjustable mirror and the silica-glass viewing window on top of the furnace.
The optical readings were taken on a surface of the couple in the holder, the pyrometer being calibrated beforehand for absorption loss through the window and mirror. The pyrometer (Tinsley type) had been previously calibrated at the National Physics Laboratory, Teddington.
- 92 - 3 — 3, Grinding and Sectioning.
After diffusion anneal the couple was cooled down to room temperature in the furnace. Then, the couple was sectioned by a grinding machine or hand.
Machine Grinding.
After the anneal the rough surface of the couple was smoothed with silicon carbide paste and then mounted to the sample holder of the machine, Wet silimn carbide powder was spread on the platform, and slowly the sample holL,)r was lowered and very carefully touChed on the rotating surface where silicon carbide powder was spread. From time to time toluene solvent vas dropped and redistributed the one—side collected silicone carbide powder.
After pre—determined time, the sample holder was lifted upward end the machine vas stopped. Then the platform was dismounted and collected the carbide chips vith silicon carbide powder together by a brash for subsecuent trzatment,
Hand Grinding.
Machine grinding does not give a satisfactory result because of the brittleness of carbides, and the speed of rotation. Therefore, hand grinding was employed a great deal in this study. Lfter the anneal the couple was taken from the furnace and one end of the couple was cut off to the nearest point to the interface by the grinder. Then, layers were ground from the surface with silicon carbide powder moistened with toluene until the diffusion zone was reached, and then each time about 20 — 30 microns of thickness of a layer was sectioned. The number of layers removed in the diffusion couples ranged from 7 to 13. Each time, the residual portion of the specimen was examined in several places with a micrometer to make sure the ground surface was parallel with the original surface, and the thickness removed was also checked by measuring the couple thickness before and after each layer was sectioned off with a micrometer. The corresponding weight change was approximately 20 mg.
klthough uncertainty in parallelism was involved, no wide spread in the penetration curve, as shown in Fig. 23 — 37, was observed. It is, therefore, considered that the uncertainty in parallelism from this technique does not seriously affect the measurement of diffusion rate.
From the measured density of the specimen and the weight change, both the volume of the section removed,,,ndd the distance from the interface could be calculated and used for plotting the penetration curve.
The hand grinding made it somewhat more convenient to weigh the sample during the section than to use the mechanical grinding method where the sample had to be removed from the holder, weighed, and then remounted.
— 94 — The errors due to irregularities in sectioning were considered.
016) qajda and .;luntington •- ' have arrived at an expression which gives the effect of misalignment and finite thickness of section on the measured diffusion coefficients.
If U (x) is the average value of C (X) in a section with midpoint)C, their results shell that if Z (x) .. C (x) + d2C x) - x then for the solution of the effusion eouation used here,
where k = (c12 S2) /24, gives the effect 1-4 k 2 of misalignment and finite thickness of sections. Dm is the measured values of D, d the section widthsre 2 (D.03. and S is the misalignment distance. The value of k computed for a dadel 2 of the worst example was k = 0.078d . The calculation based on this model indicates that error due to this effect would be less than 1%. Considering all other effect such as the roughness of the sectioned surface, it seems that in most of the cases the combined effects should be less than a 3% increase in the diffusion coefficients.
after sectioning, the chips and silicon carbide used were collectee. together and transferred to a small tube of the combustion apparatus. Because a mixture of uranium carbide and silicon carbide was used, loss of the powder was expected to be very small and the exposure time of the powder in air was short because, most of the tthe, it vas handled in the wet form; the loss due to the oxidation was therefore negligible.
- 95 3 — 4, Combustion Techniques and. 12paratus.
01) Buyske and cc—worker's have demoted considerable effort to the deve/cpmeot of 'tkmbustion techniques to facilitate liquid scintillatioa counting of samples which otherwise might be difficult to handle. Peets, Florini and We) Buyske have described a macroGmethod for combustion of both liquid and solid samples of biological origin. Kelly, 019) Peets, Gordon and Buyske ' have also described the design of a modifiedMhOniger flask for closed combustion of samples and subsequent trappir4; in I:yamine.
(120) Kelleener and Rutschmann • have worked out a simplified SchOniger combustion technique for use with C-14 .Sled compounds which promises to have extremely wide utility when other methods of sample preparation are not applicable.
Peakall et al( 121) ' exposed reaction—sintered uranium carbide pellet to dry oxygen,i_n a thermal balance at temperatures o 0 C. The reaction product was U 0 in the range 350 1000 3 8 every case, according to the equation;
3UC 40 U 0 2 3 8
?lost of the carbon subsequently being oxidized
C t 02 -4 C02
— 06 — 2 At 350°C the reaction rate in mg/cm hr. was in the range 3.7 to 11 and at 1000°C, 347.
Figure 22 is a schematic diagram of the combustion apparatus. It was designed and constructed applying the principle of an oxygen filled SchOniger flask, simplified by Kelly. The oxidation and trapping of CO2 gas generated was done in the same apparatus. The oxidation rate of uranium carbide was quite fast even at low temperature, whereas that of silicon carbide was very slow, particularly below 1000°C. Carbon dioxide, produced by these oxidation reactions, was sbscl:.sei in Hyanine solution.
The carbide chips to he determined were collected with silicon carbide powder used for sectioning and placed in a small tube in the side arm of the apparatus.
A solution of 1 M Uyamine in methanol (3 ml) was then delivered into the main tube by a pipette, and the apparatus was flushed with a stream of oxygen to facilitate the oxidation.
Tie amount of sample used was 20 — 30 mg. of uranium carbide powder which contained about 1.5 mg. of carbon to be oxidized. The amount of oxygen filled was considered to be sufficient. After firmly closing up the openings to prevent leakage of the gas, the main tube was codtled to reduce internal pressure and the sample in a small tube was placed inside of a furnace tube placed vertically on the bench, and then heated at about 700°C for one hour.
—97— Fig. 22. Combustion Apparatus.
/'-- Thermocouple
V V F V 4 a A A 0 0 0 4 O 0 ( UC + SiC ) powder O 0 ° O 0 6 0 0
Hyamine Solution
Iced Water
— The temperature vc,s raeasured by a thermocouple placed between the tube and furnace. The apparatus vas stood overnight, then an aliquot of }bromine solution ,as removed for counting.
The disadvantages of using Eyamine solutions are (1) it is very slow in absorbing CO2 gas and (2) colour change due to the impurities in a gas, thereby decreasing the counting efficiency. It as reported that 96% of CO2 gas was absorbed in 3 hour at room temperature. In order to make sure absorption vas :complete, the apparatus was permitted to stand overnight. Since the oxidation product was ra1ati72y simple, mainly CO2, colour change in the solution vas not observed in the course of the experiments. 3 - 6, Carbon-14 Activity Measurement.
After completion of the absorption, an aliquot of Eyamine Carbonate solution was transferred from the main tube by a pipette to a sample container (shown in :dig. 21), and mixed well with 15 ml of the prepared liquid scintillatore and left in the dark place to eliminate the light effect for several hours.
Then, it was counted with a liquid scintillation counter. The counting efficiency eras about 70%. This efficiency was dependent upon the quantity of Hyamine used and the instrument setting.
After setting the instrument on optimistic conditions, the efficiency was measured by using n-hexadecane-l-C-14 as an internal standard source, purchased from the Radiochemical Centre, Amersham, and whenever an uncertainty arose the efficiency was re-examined and kept the constant efficiency all the time.
The counting time was usually 10 minutes. This gives the standard deviation about 0.1%.
From the counting reading it was corrected with the background and the dead-time loss. Then the true counting rate was calculated from efficiency of the counter and derided by weight of the carbide powder used.
In view of the long half-life of C-14 and the high efficiency of the counter, these experimental data were uncorrected for the
- 100 -- decay and self-absorption.
The accuracy of the experiment depends on (1) the measure- ment of the annealing temperature and tine, (2) the errors in and trapping in hyaauine solution, collection, oxidation 01402 and (3) the error in .the measurement of carbon-14 activity. Apart from the annealing temperature and time, the errors in (2) and (3) procedures were estimated by using the known activity of carbon-14 as a tracer from the labelled uranium monocarbide (the activity was 2.45 micro curie/gm.) After the diffusion anneal the same experivnantal i,rocedures described earlier were followed and the activity was measured, afterward, an aliquot of internal standard was introduced into the mixture of Hyamine and scintillators used, and the counting efficiency was determined for each sample. The internal standard was n-hexadecane-l-C14 whose accuracy was less than 4% in error.
A sufficient counting time was given in each measurement, and it was determined in every case with a standard deviation of less than 1 0.1%. The experimental results are as follows:
- 101 - Weight of Calculated Total Measured Total Efficiency Activity (c/min) Activity(c/min UC14 (mg) %
1.93 10422 8940 85 4.51 24300 20005 82 2.08 11232 9800 87 4.94 26676 24285 91
Average 86%
It shows about 154 discrepancy, and it includes the errors in collection, transferring, oxidation, CO2 trapping and the activity measurenent.
— 109 — 3 - 6, Results.
The condition of the present investigations for carbon cpptoximate that of a semi-infinite cylinder extending from x=- 0 to x with an instantaneous source of diffusion media at x-“). Under such conditions, the solution to the diffusion equation is 2 (3.6.1) X ATE:CO C 4Dt. C where Co is the initial concentration per unit area of active material at the original plane, C is the concentration at a distance X after a diffusion time t, and D is the self-diffusion coefficient.
The experimental data would be expected to lie along a straight line in a plot of the logarithm of the activity as a function of the square of the distance from the interface, since taking the logarithm to the base 10 Of Equation (3.6.1) *olds
,2 const. log C - 0.4343 4Dt 2 X + const. (3.6.2) or log C - - 0.1086 :fit the slope of the linear portion in the plot of log C versus X2 was used to calculate the diffusion coefficients D since the slope is equal to
- 0.1086 (3.6.3) D.t.
--103- 2 Penetration Curve (log C v.s. X )
From the carbon-14 activity data and the penetration distance, the penetration curve (log C v.s. X2) vere plotted and showain Fig. 23-27.
The best fitting lines could be drawn through the points on all curves. Some scattering in the experiment data was found, but generally the penetration curves defined good straight lines. If these lines are straight, this indicates that volume (or bulk) diffusion is predominant. Since most of them are good straight lines one could say that the volume diffusion is predominant.
In some cases the curves were not a straight line, suggesting that short-circuit diffusion is operative. Surface effects such as a "lock-up" of the diffusing species at the initial interface and/or short-circuit paths such as dislocations (or grain boundary) in the crystal cause deviation from Fickian diffusion and may yield penetration of several segments. The surface "lock- up" yields a lower diffus!vn rate while the dislocation or grain boundary yields a higher diffusion rate than that expected for bulk diffusion. In this study both cases vere found to be existing.
In order to detect any effect of grain boundary ob aiffuzion, Fisher's analysis was applied to see if there vas any evidence. (See for details Section IV. Discussion).
- 104 - All 32 diffusion couples in sintered and arc—cast uranium carbide were analysiid. The composition of the carbide ranged 0 from 4.7 to 5.6 wt The temperature range studied was 1215 o to 1800 C.
Self—Diffusion Coefficients.
Self—diffusion coefficients for carbon were calculated from the slope of the linear portion in the penetration curves, plotted by the procedure described above. The results of these calculations are given in Table 10 — 14 for the temperature studied.
An example of calculation of the diffusion coefficients at 15050C is given; from the Nquation (3.6.3)
Anneal time = 0400 sec. Slope r_ 1.06Z -4 1086 x 10 D /50? x U4a
= 7.28 x 10 g cm2/sec.
— 105— Fig 23• Penetration Curves for 4,7 wt % Arc-cast Uranium Carbines.
103
0 1800 °C e 1764 °0 ..) 0 1690 °C bo -.; g 1618 °C ED 1573 °0 3 1505 °(.: ° \ .4 0 144Q C 0 (t ,,__ C 40 --,,,„,,,,.„
4) 41 01
,..„„„....„,,
e
.
•
10 10 20 • 30 40 50 0 0
x2 10-5 cm2
-106- Count/ min / mg 10 o 2 r '1 0 1, ' el, \ - ,
• 20 Fig. •
24. m • 40 Penetration Curvesfor4: Uranium Carbides. ;T n.•
X 2 x10 60 —107— """""
1 -5 • cm 80 ' S e 0 0 05 2 0 0
4 8wt %Arc—cast 157 136 146 159 126 168 el 100 ° ° °
C C C C Ne 120 N .
_ ,
...... 140 Fig. 25. Penetration Curves for 5.1 vt k Arc-cast Uranium Carbides.
103
401596 CG . 01505 4:0
:: Q1412 C 01287 C c'1215 6
631 C+ Cy lib,
e NIIIII
IIIIIIIIIIS \go. 1st e
10 0 10 20 30 40 50 60 2 X x 10-5 cm2
- 108 - Fig. 26, renetration Curves for 5.6 wt % Arc—cast Uranium Carbides.
103
—______0 •
o .
. g e .
c o Q'o'(B
2 10
4 1613 ° . iii, . 0 1467 °O 0 1448 ° El) 1380 °C C., ® 1293 ° E) 1256 10 G> 1215 '°'
20 40 60 80 100 120 1, 0 2 2 X x 10-5 cm
— 109 — "
Fig. 27. penetration Curves for 5.0 wt 2 Sintered Uranium Carbides. 104 0 Q 1626 0 0 .6) 1580 0
(0 1525 0 0 •
0 1448 0 CD 1399 • 0
e 1280 0 C • 3 .
/mg 10 ►120 in /m t n
Cou ----, . • D s(D---'—"-- • 1.' •------, • '..6., '....'"'
------7,-, .--. -.. 0 .
. .
10
0 20 40 60 80 100 120 140
X2 x 10-5cm2
- 110 - Table 10. Coefficients of Self-diffusion for Carbon in Arc-cast Uranium Carbide (4.7 vt%)
Temperature, Annealing Time, Diffusipn Coefficient, C sec. cm sec
2 -8 1800 45.0 x 10 1.20 x 10 2 -9 1761 73.8 x 10 9.40 x 10 2 -9 1690 84.0..x 10 3.0 x 10 2 -9 1618 469.8 x 10 1.54 x 10 ,2 -10 1573 603. m 1•,, 8.83 x 10 3 -10 1505 105.6 x 10 2.82 x 10 3 -10 1440 508 x 10 1.86 x 10
Table 11. Coefficients of Self-diffusion for Carbon in Arc-cast Uranium Carbide (4.82 w0)
Temperature, Annealing Time, Diffuspn Coefficient, vC sec. cm /sec
5.7 -8 1684 53.0 x 10- 1.70 x 10 2 -9 1596 144 x 10 7.70 x 10 2 -9 1575 144 x 10 6.0 x 10 2 -9 1465 549 x 10 2.31 x 10 2 -10 1362 552 x 10 7.0 x 10 3 -10 1266 177 x 10 2.0 x 10 Table 12. Coefficients of Self-diffusion for Carbon in Arc-cast Uranium Carbide (5.1 it %)
Temperature, Annealing Time, Diffusion Coefficient, oe Sec aa /see.
2 -8 1596 33.0 x 10 1.82 x 10 2 -9 1505 95.0 x 10 7.28 x 10 2 -9 1412 162 x 10 2.30 x 10 L) -9 1287 70.2 x 10 1.30 x 10 3 1215 108.42 x 10 3.40 x 10-10
Table 13. Coefficients of self-diffusion for carbon in Arc-cast Uranium Carbide (5.6 wt
Temperature, Annealing Time ti Diffusion Coefficient oc Sec cm sec.
-8 1613 72.0 x 102 2.26 x 10 -9 1467 216 x 102 6.07 x 10 -9 1448 288 x 102 9.74 x 10 -9 1380 313 x 10' 3.47 x 10 -9 1293 936 x 102 1.28 x 10 3 -9 1256 172.8 x 10 1.00 x 10 3 -10 1215 234.0 x 10 5.91 x 10
- 112 - Table 14. Coefficients of Self-diffusion for Carbon in Sintered Uraniun Carbide (5.0 ITO).
Teoperature, Annealing Tioe, Diffusion Coefficient, °C Sec. cm sec
2 -8 1626 76.3 x 30 1.35 x 10 2 -8 1580 72,0 x 10 1.01 x 10 2 -9 1525 282 x 16 , 3.50 x 10 2 9 1448 377. x 10 3.40 x 10 2 -9 1399 735 x 10 2.67 x 10 3 -10 1280 108 x 10 7.10 x 10 3 -10 1205 244.2 x 10 2..34 x 10
- 113 - Activation Energy and Pre-exponential Terms.
From the experimental data in Tables 10 - 14. the logarithms of the diffusion coefficients versus the absolute reciprocal temperature are plotted in Fig. 28 and 29. A straight line was drawn through the points by the method of least squares. From the slope of lines the activation energy was calculated from the equation
/ D Do exp k- 7-7 .1 ) (3.6.4) (3.6.5) log. D -2.s6-3 .0 I- log Do E since the slope is equal to - 2.303.R Therefore the activation is given as (3.6.6) 4 E = - 2.303.11.. $lope
The values of activation energy, calculated from (3.6.6) are listed in Table 15. The most notable features of this study are composition' dependence of the activation energy for diffusion process of iarbon. In the carbon-rich side the activation energies vary almost linearly as shown in Fig. 30 but in the carbon-deficient side it varies very abruptly.
It is also interesting to note that the diffusion rates of carbon increases abruptly as the carbon content is increased and reach to constant values. The variation of diffusion rates are shown in Fig. 31 as a function of carbon content.
- 114 - From the intercept of the Arrehnius line to y axis the pre—exponential term (Do) were calculated, r.nd the results are listed in Table 15 and shown in Fig. 32 as a function of carbon content. Tnis is also the composition dependence. As the carbon content increases the values of Do decrease very rapidly. Fig' 28. TEMiER;vTURE (oC ) 1800 1700 1600 1500 1400 1300 1200 I SELF-DIFFUSION OF CARBON IN URANIUM CARBIDE ( Arc-cast ) a a 0
-8 9 10 Carbon Content(wt a 0 a a ( a ) 4.7 % 4) ( b ) 4.82%
U a ( c ) 5.1 ( d ) 5.6 %
vi U 0
0 -9 10 0 0 ion ffus Di 0 O
0 a ( d )
. 1 0-10 ( b )
t A 4.5 5.0 5.5 6.o 6.5 7.0 7.5 1 x 104 °K T Fig. 29.
- SELF-DDE.PUSION OF Ahta IN 4, SINTERLD URANIUM 04.R.b1DE - ( CARBON 5.0 vt % ) . 7 _ 7
--. i ii C.) 0 . 8 . U t ien ic
f -9 f 10 e Co
ion -.- fus
f _ Di _ H 0
-10 10 5.0 5.5 6.0 .5 7.0 .5
1 104
- 117 -- Table 15. Diffusion Parameters for Carbon in Uranium Carbide.
Carbon 2i A SA (ut %) Do (cm/sec) LE(cal/mole)
4- 4.7 32.3 39000-6000 -#7.87
4.82 1.75 63000 t 1000 +4.97 arc—cast -2 -4- .5.1 2.95 x 10 54090 — 15000 +0.81
...a 5.6 2.76 x 10 45000 4-- 6000 —1.49
-2 sintered 5.0 3.21 x 10 55000 — 8000 +0.97 Fig. 30. Variation of activation energy as a function of carbon content in uranium carbide.
90 0 C) arc-cast
sintered 80
V 0 70
C.) ta V 60
tion a
tiv 50 Ac
40 1 1 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Carbon Content ( wt % ) Fig.31. Variation of diffusion rates as a function of c,1:rbon content.
1632°C b— o
0 10-8 0 ,o o 149 C
0
0 0 1394°C 0
-9 10 0 . .
0 I
10-1
4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Carbon Content ( wt % )
— 120 — Fig. 32. Variation of Do as a function of carrion content.
10 1 0 arc-cast
O sintered
1
L
0 L..
V
CV V
O -1 10
r-
-2 1 0
. 3 10 1 .1 I 1 I ;. 5.7 4.5 4.6 4.7 4.8 4.9 5.0 5.T 5.2 5.3 5.4 5.5 5.-j- Carbon Content ( wt % )
- 121 Calculation of Entropy Change (AS).
The diffusion equation can be re-written as follows:- 2 (3.6.7) D 75" exp o • ) exP ( -NE ) Hence, the D is 0 2 (oS ) (3.6.8) D 17: 1. . a o . `I) exp
The ZNS term can be evaluated from the known values of 2 D o 1 and an assumed value of v 9 which is usually o taken to be the Deby4 FrequeAcy of solids.
In the case of interstitial atos l id can be calculated by assuming that the potential energy curve of the atom varies sinusoidally along the diffusion path and the maximum value is the energy for the mobility of interstitial atoms.
For simplicity, taking 13 -1 u as 10 sec and ao as the interatomic distance of carbon in uranium carbide (3.50A°) which is the shortest jump distance, the 4, 3 was evaluated and listed in Table 15.
A sample calculation of the entropy change at stoichio- metric uranium carbide is given below. Experimentally Do was found to be 1.75 cm2/sec. and (122) 7f = 1 for f.c.c, crystal structure -8 ao 3.50 x 10 cm. 13 -1 10 sec .
- 122 - Inserting Ainto Eq. (3.6.8) this gives -8 2 13 S 1.75 = 1 x (360 x 10 ) x 10 exp ( R by rearrangenent / zsS exp t) 143 tt LIS 2.303 log 143
12-- 4.97
Therefore S = 9.9 cal/mole/degree
The oalcuiated vatues 2 decrekie very rapidly from positive to negative shown in Fig. 33. Experimentally it is found that2SS will be positive unless there is a phase change. The negative vc.iue of L at the 5.6 wt fo composition is most probably connected with the occurence of a 2nd phase — UC2 or U2C3.
gowever, the values ef65 evaluated depends on the values of -I) assumed. In view of vaguenesP as to what *alue40.fle u to be llsed, S ne cietermined with precision. As will be seen in the Di6eussion section, the evaluation of even an approximate 4N S car. be helpful in checking experimental results.
7- I r -r-
4 0 Fig. 33. Variation of entropy chan: as a function of carbon content
+ 6 r
+ 5 L.
0 arc-cast
sintered
0
- 2 t i i t t 4.5 4 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Carton Content ( wt ) Comparison with Other Work.
Experimental data on the self-diffusion of carbon in uranium carbide is 7ery scarce.
Chubb et al at B.M.I. have studied this by radioactive tracer and standard section technique and obtained an activation of :00 kcal/mob for arc-cast uranium carbide (5.0 wt%).
(123) Craven et al have also studied the reaction kinetics of uranium carbide and obtaip.ed an activation of 63 kcal/mob. Since the reaction is diffusion controlled, these results were used for comparison.
The results are listed in Table 16. The results available are good agreement with the present data.
Table 16. Comparison of Energies of Activation with Other Work,
Present work Others lgo-t. carbon activation energy activation energy method red (wt%) Kcal/rcole Kcal/mole 4.7 89 t 6 ------arc-cast 4.82 63 :t I 63 - 6 Kinetics 123 511 54 t 15 50 t 20 Diffusion 21 5.6 45 ± , ,; - sintered 5.0 55 t 8
- .25- III - 4. Experimental Procedures
- Self-diffusion of Uranium in Uranium Carbide. 4 - 1, General Considerations.
Uranium found in nature consists of three isotopes of Mass 238, 235 and 234, and abundance 99.276, 0.718 and 0.0056% respectively. These isotopes are all radioactive, 9 8 5 having half-limes of 10 10 and 10 years.
It is a well known fact that the important fission process occurs with slow neutron only in U-235. This fact developed the ways to separate large quantities of U-235 from the more abundant U-238, for use in atomic reactors. The separated U-235 is added to natural uranium to form "enriched" fuel elements. It can be mixed up to 93.2%.
It is interesting to note that, when mixed with natural uranium, the nuclear properties of uranium have been considerably changed.
The nuclear properties of "enriched" uranium are listed in Table 16.
12Y (124) Table 17. Specific Activity of Uranium Isotopes in "Enriched" Uranium.
• Specific Principal ' 1 kbu ndance Isotopes Activity Alpha-ray (ciminAmg) (9 (MeV)
234 13,500 1.08 4.76 235 4.7 93.7 4.20, 4.40,4.58 236 140 0.2 4.50 238 0.74 5 4.18
From the table one can see that, in spite of the fact that it contains only 1.00 of Uranium-234, almost all the alpha activity of uranium comes from the disintegration of Uranium- 234 isotope because of its short half-life. This fact could be employed to determine uranium isotope content for diffusion studies in uranium containing compounds.
There exists a variety of ways for the determination of uranium isotopes. Here, we deal with only two methods, which could be easily applied in the laboratory.
(1) Determination of Uranium-234 Isotope by Alpha-ray Method.
This method is based on the fact that, as described above, the major source of the gross alpha activity is contributed by uranium-234 isotope. (2) Determination of Uranium-235 Isotope by Gamma-ray Lethod.
Uranium-235 isotopes, beside being an alpha emitter, also emits a gamma ray with a maximum energy of 0.184 This fact serves as the basis for the determination of the isotope by gamma ray spectrometry. This determines only a 0.184 MeV gamma- ray and does not determine other uranium isotopes, except the isotopes of similar gamma-emitter.
It is therefore obvious that, if one uses "enriched" uranium as a radioactive source, one can employ either method to determine the uranium isotopes. (A) Alpha—'lay Method.
-- 13c — 4 - 2. Diffusion Couple and Diffusion Anneal;
(O. Diffusion Couple.
The arc-cast carbide was usez.: and contained 4.82% of carbon and was expected, according to the phase-diagram, to consist of a single phase (UC).
The highly "enriched" uranium carbide was used as a labelled compound for the preparation of diffusion couple. (See III, 3-71).
The details of the couple preparation is described in 111.3,1.
(b) Diffusion Anneal Procedure and temperature measurement are described in 111.3.a.
7.01. 4 - 3. Sectioning.
After the diffusion anneal the couple was taken from the vacuum furnace. It was found that on removal the radioactive faces of all specimen were bright and free from the oxide film.
The most important thing in the sectioning was grinding off layers parallel to the original interface. After the anneal the exposed surface of the radioactive layer was not flat. This surface had to be made flat and parallel to the interface using the backface of the originally non-labelled surface as a reference plane.
Prior to actual grinding the rough surface was ground off to parallel to the backface of the original surface with silicon carbide powder, and the parallelness was checked with a micivneter at several points on the surface. After the couple was made flat and parallel, the activity was measured from the surface -lith a counter and this was taken as the initial activity (Ailoi-Ai) for the calculation of diffusion coefficients.
After this, successive layers were sectioned by measuring with a micrometer from the original face. Each time the activity was measured from the freshly polished gurface.
The couple was sectioned on the paU, glass plate with silicon carbide powder and toluene which providedaa greasy medium in which to collect the radioactive grindings and prevented any loss of material and also oxidation of the carbide powder in air.
- 132 - The weight change of the couple vas measured by a semi- microbalance in the laboratory. From the measured density and the weight change, both the volume of the section removed and the distance from the interface could be calculated. Each section removed was 4 - 5).Athiok, and the corresponding weight change was approximately 6 - 7 mg.
4 - 4, Determination of Uranium-234 Isotope.
The counting of uraninn/-234 isotope vas accomplished with a zinc sulphide scintillation counter.
The zinc sulphide crystal was connected to an eleven stage photomultiplier tube and operated on the scintillation counter principle. The crystal was activated with cadinum and provided 100% detection efficiency for alpha-rays and has a low efficiency for beta and gamma-rays, thus, it can be tiser to detect alpha- rays in the presence of a beta or gamma background. It was usually protected with c light-tight cover of aluminized Mylar or aluminium foil.
After each successive polishing, the specimen was radio assayed by placing it in a plastic holder, which was placed on the top of a counter as shown in Fig. 34. This facilitated reproducible positioning of the specimen with respect to the counter and defined the portion of the surface of the uranium carbide to be counted. 1 516 in. hole in the centre of the plastic plate permitted only that fraction of the alpha-ray
- 13 -
specimen
ZnS:alpha-:cOunter
Fig.34. Counter Arrangement
-, 134 - emitted from a freshly polished surface to reach the detector.
With a source of known activity the counting efficiency was esti,lated.
Under these conditions, it was estimated that approximately 15% of the alpha—rays emitted from the specimen surface were detected. From the counter readings the background and dead—time loss were corrected and usd for plotting the penetration curves.
In view of the long of U-234 these experimental data were uncorrected for the decay and self—absorption.
4 — 5. Results.
Boundary Conditions.
Subject to appropriate conditions, the diffusion equation gives the distribution of a diffusion material at a given temperature after a given tie. Fig. 35 and Table 18 show three solutions to this equation for which simple forms and boundary conditions may be realized experimentally. The con— centration distributions in Fig. 35 have been calculated for a typical value of the diffusion coefficient encountered.
- 135 - Table 18. Tabulation of experimental conditions commonly used to prepare diffusion couples, and the resulting concentration distributions.
Type of Couple Experimental Conditions, Concentration Distribution
1.Butt—joint couple Pla,aar- join of block of base material to X block having diffusing 4.C — oerfC(C=.iut ) pac,teicl in uniform concentration Co
2.Sandwich couple Fixed amount of material 2 diffusing S initially in thin t X A ..4kyyt exPk ---- ) layer between two VDt— slabs of base naterial -.._
3. Vapour Deposition Planar surface of base material maintained at , X Couple CerfCko — ) constant concentration 2 C o
— 136— Fig. 35. Plots of concentration distrihution of uranium atoms as a function of -11 2 penetration distance. ( D = 3.00 x 10 cm /sec, anneal time = 2 2525 x 10 sec ) calculated, E) experimental . 1.0
-2- = erf c ( ) Co 21ET C = erf c ( X C 2 )- 0.5-r o
0 I I 1_1 I III,' I I 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 -4 1-enetration Distance ( x. 10 cm )
Concentration distribution for a typical diffusion couple, as determined by a zinc sulphide scintillation counter are shown in Fig. 36. A plot of the activity v.s. distance is also shown in Fig. 37. The observed concentration distribution curve is identical to that of the case 3 in Table 18. This boundary condition predicts a concentration distribution
C/Co erfc X (4.4.1.) g /Dt where C concentmtiqn as el function of X, t.
Co concentration at X 0 D diffusion coefficient t = time
If radioactive tracers are used, the equation (4.4.1) may be modified to read
Ax - Lb (4.4.2) ` A° - kb
where Ax =tile activity at distance from the interface. Ao ve the initial activity. Lb g the background.
The inverse coripleuentary error formation of (lxAo - AAb) should, therefore, be linear with respect to penetration. The value selected for(Ao - Ab) is somewhat arbitrary, because the slope did not vary appreciably with the choice of (Lo - Ab).
- 138 -
j'7 Activity Profile at 1708 °C.
3 000
2000
2000
1500
T-4 1 0 0 0 -
500
10 2U 30 40 50 60 penetration depth ( x 104 cm ) iig. 36. Concentration Distribution EA 170F °C.
sn 1
0
0
0 10 20 30 40 50 60
F enetn.ttion epth ( X I0 c9fl ) Therefore, from a plot of
-1 Ax - lab erf C ( v.s. X (4.4.3) loo - kb 1 it was possible to determine the diffusion coefficients from the slope as below.
(4.4.4.) Slope - 1 gra
-1 Ax lo - fib) is plotted as a function In Fig. 38 - 45 erf C (( of penetration distance (X) for all specimen measured. The deposited layer wc,s relatively thick for the high temperature sample, and-was relatively thin for the others.
There was some deviation at the beginning of the linear portion in the plot of the inverse complementary error function v.s. penetration distance. This is believed to be due to back diffusion of uranium atoms into the radioactive layer. The thickness of the layer has been found to correspond precisely with the depth at which the deviation was observed.
From the slope of the linear portion of curve, the value of the diffusion coefficients was computed.
Eight diffusion couples were analysed. The inverse complementary error function plots are given in Fig. 38 - 45. Prior to the diffusion couple preparation the background activity (kb) of the uranium in the sample were determined from
— 141 — 1.4 Fig. 38. Mot of inverse complemen— tary error function vs 1 .3 penetration depth at 1536 °C. 1.2 —
1.1
1.0 1-
0. g 0.8 0 o.7
0 0.5
0 -4 1-
0.3
0.2 O/
0.1 0
0 10 20 30 4.0 lenetration depth ( x 104cm )
—142 — 1
Fig. 39. blot of inverse complementary error function vs penetration depth at 1591 °C.
0.9
0
0 10 20 30 40 50 60 1enetration depth ( x 104 cm )
— 143 — 1.3 Fig. 40. clot of inverse complementary error function vs penetration depth at 1.2 1637 °C.
1.1
1.0
0.91-
0.8
0.7
0.6
0.5
0 0.4
0.3
0.2 1
0.1 0 i
70 80 90 100 110 120 130 ienetration depth ( x 104cm )
- 144 -
1.4 Fig. 41. Plot of inverse complementary trror fuction vs penetration depth at 1702 °C.
0.9
0.8— p; -g' 0.7.-
0.6
0 1 0 5 4-4 0 0.4-
0.3 - / 0 0 0.2--
- 0.1! I 1 . I 30 40 50 • 60 70 80 90 -4 ienetration depth ( x 10 cm )'
- 145 - 1.0 f I T I 1"--
Fig. 42. Clot of inverse complementary 0.9- error function vs penetration depth at 1708 °C.
0 10 20 30 40 50 60 70 penetration depth ( x 10-4cm )
— 146,— .3 • Fig. 43. i1ot..: iliverse complementary error function vs penetration / 1.2 D .depth at 1749 °G. ; 1.1
/ 1.0 ;-- c . 9 _ 0(i 0.8 0/
0.6 —
0.5 -
0.4
0.3
0.2-- zI 0 00 H
0.0 I 1/ t 50 60 70 80 90 100 110 120 130 140 150 160 170 ienetration depth ( x 10 cm )
— 147 — 1.2 — Fig. 44. clot of inverse complementary error function vs penetratic•n depth at 1797 °C. 1.1 I--
1.0 —
L..
0.2
0.7 0 0.6 L. sl
xc• 5 I--
7 0-4 G.) 0.3 r-
0.2
0.1
0.0 10 20 30 kenetration depth ( x 10-4 cm )
— 148— 1.4r- i 1 1 1 I 1 I 1 1 1' 1 i I i
Hz. 45. [-lot of inverse complementary error function vs penetration depth at 1863 °C. 1
<4
0 0 9 0 0.8- N 0.7
$.1
0.5H 1
0.4;- 1 • 0.31-
0.2 - o . 0 / ----Lr / __O--- • / 0 I L I I 0----10-----W---30 40 50 60 70 80 90 100 110 120 -4 Penetration depth ( x 10 cm )
- 149 - the polished surface. The initial concentration, Co ort(ko - lb)) of U-234 in the layer of the couple were determined by averaging the first 2 - 3 counting readings.
The self-diffusion coefficients for uranium calculated from the slope of the linear portion in Fig. 38 - 45. The results of these calculations are given in Table 19 for the temperature range of 1536° to 18630C. It also includes the anneal time and temperatures studied.
From the experimental data one can see some irregular scattering in the curves. I. large part of this scatter is probably caused by an itncertainty in the measurement of the thickness of the layers, by differences in the nature of the polished surfaces, and possibly by small variation in positioning of the specimen with respect to the counter.
From the diffusion coefficients in Table 19, a conventional graph of the logarithms of the diffusion coefficients versus the absolute reciprocal temperatures aas plotted and shown in Fig. 46. L. straight line was drawn through the points by the method of least squares. The activation energy was calculated from the slope of the least squared line. From the intercept in y axis the pre-exponential factor was also calculated. The temperature dependence of the self-diffusion coefficient for uranium in stoichior:etric uranium carbide was found to be
3 28 ,000 ) D 4.49 x le exp ( RT
- 150 - Table 10. Diffuzion CoefficieUts of Uranium in Uranium Carbide.
Temp (°C) Anneal Time B (cm2/oec) (sec) ,
2 -10 1863 790 x 10 2.87 x 10 2 -10 1797 522 x 10 1.20 x 10 2 -11 1749 621 x 10 5.20 x 10 2 -11 1708 2529 x 10 3.00 x 10 2 111 1702 846 x 10 2.70 x 10 2 -12 1637 2358 x 10 8.30 x 10 2 -12 1591 3456 x 10 3.18 x 10 2 -12 1536 6024 x 10 1.38 x 10 ,..... Fig. 46.
SELF-DIFFUSION OF URANIUM IN URANIUM CAlthl DE ( air ha-ray method )
1 0-1 2 4.5 5.0 5.5
1 x 104 — 152 — The activation energy for self—diffusion of uranium is 128 .1-. 8 kcal/mole.
The entropy change (AoNS) was also calculated in a similar manner, as described III. 3 — 6. The calculated value was 28.4 cal/mole. Comparing with that of the carbon diffusion, /degree the value is quite large. It is obvious that a large uranium atom will bring more local disturbance when it moved to a new site.
For this study a number of possible errors exist in the determination of the diffusivity of uranium. The error in the determination of thickness and parallelism of the layer polished from the diffusion couple was 0.5/A. This uncertainty, in addition to possible errors in the radioassay caused by slight difference in the nature of the polished surface or the positioning of the specimen, probably accounts for the scatter in the data. Therefore, the sum of possible errors in measuring activity, thickness, time and temperature was estimated to be 15 to 200.
— (B) Garma-Ray ethoth
- In order to confirm the previous results, the diffusion coefficients were measured by collecting the grinding chips from each layer and counting a gamma—ray from U-235 isotopes in the chips.
For the calculation of diffusion coefficients a different solution of the diffusion equation was employed to see if there is any significant difference between both cases.
:1 4 - 6. Diffusion Couple and inneal.
The details of diffusion couple preparation and anneal are described in III. 31-1.
4. - 7. Sectioning.
After the diffusion anneal the couple was taken from the vacuum furnace.
Prior to actual grinding and counting the rough surface was ground off to the interface of the couple.
After this, successive layers were ground from the inter- face of each couple on the flat glass plate with silicon carbide powder moistened with toluene. .fter drying the chips, together with the solicon carbide powder, they were carefully collected into a snail glass tube, which fitted to a well of NaI crystal for the purpose of assaying the activity.
4 - 6, Deterclination of U-235 Isotopes.
The quantative application of gamma-ray spectrometry depends on the measurement of the heights or areas of photo- peaks, since the area is proportional to the absolute gamma- emission rate of the corresponding isotopes.
- i53- It is therefore necessary to measure the number of counts in the photopeak. Under normal conditions this can be done in three ways;
(1)by measuring the height of the photopeak, (2)by -measuring the number of counts in the channel corresponding to the centre of the photopeak, (3)by measuring the area under the photopeak.
The reproducibility of such measurementsis about 10%, (125) 5% and 2% respectively.
In this case, the t:letbod (2) has been employed to measure the intensity of U-235 isotopes. It is possible to obtain the energy spectra of U-235entomatically or manually, as shown in Fig. 15, 18 and 17. The analyser channel was first tuned to the centre of the photopeak, which corresponds to a gamma—ray of 0.184 MeV energy fros U-235, then the intensity of the gamha— ray was counted with a sealer. This procedure is identical to measuring the height of the photopeak. During the measurement the width was kept at constant voltage.
The error introduced by this method is expected to be about 5%.
— 157 -• The spectrometer setting used for determination of U-235 was as follows;—
Lpplied Voltage R000 differential Time Const. 0.16,,v s. Integral Time Const. 0.16 jos. Amplifier Attenuation 2 db. Channel width 5 volt. Time of Rise 1 itA s. Disc Dead Time 5 Amplifier Backbias 19.5 volts.
158 -- 4 — 9, Results.
Under the conditions of the experimental procedure in this study, an infinitely thin layer can be said to have been deposited on a semi—infinite solid. For these conditions the gradient of tracer concentration C at distance X beneath the original interface is given by
a In C 1 - 4Dt (4.8.1) X2
2 Where D is the diffusion coefficient in cm /sec and t is the time in seconds.
Diffusion coefficients were obtained by plotting the logarithm of the activity against the square of the distance from the initial interface, and setting the slope of the straight 1 line so obtained equal to 4Dt— Typical plots of activity versus the square of the distance are shown in Fig. 47.
The linear portions of the curves in Fig. 47 were used to calculate the diffusion coefficients. The values of the calculated diffusion coefficients are included in Table 20. When the diffusion process conforms exactly to (4.8.1), there results a straight line when log C is plotted 2 versus X for all cases. Four diffusion couples were analysed. The penetration curves obtained defined good straight lines. The diffusion coefficients were calculated from these curves and are listed in Table 20. It also includes the annealing time and the temperatures.
— 159 -- Fig. 47. ienetration Curve
103
co 1863 cc e 1741 00 114 o 0 1650 °C * . 0, _ o 0 1505 G
O fl
! \\\ e .- as E 102 \\\\
O *
o
.- O
10
4. 1 I ' i
. .
0 20 30 40 50 60 70 -6 2 2 x 10 cm
' - 160 - , Table 20. Self—diffusion Coefficients of Uranium in Uranium Carbide. (by gamma—ray method).
Annealing Time Diffusion 2 Temp. (°C) (sec). Coefficients (cm /see)
2 -10 1863 72 x 10 1.83 x 10 2 -11 1741 287 x 10 5.00 x 10 2 -11 1650 3276 x 10 1.50 x 10 -12 1505 4908 x 102 3.13 x 10
From the above data, a conventional Arrhenius plot is given in Fig. 48. The least square method was employed to draw the line through the points. From the slope of the line, activation energy was calculated, and from the intercept the pre—exponential was also calculated. term Do
The temperature dependence of the self—diffusion coefficient for uranium in stoichiometric uranium carbide was found to be
104,000 D = 8.47 exp ( RT
The activation energy for self—diffusion of uranium was 104 t 7 kcal/moD. The entropy change was also calculated to be IT.5 cal/mole./degree.
Y61 I 0 Coefficient 10 Fig. 48. 10 -
-9 ...
. • _ • • -162- 1 O T . x 10 • • 4
SELF - DIFFUSION ( gamma-ray URANIUM CARBIDE IN 5 method ) OF URANIUM 7.0 These results are in good agreement with those of the elpha—ray aethod.
Because of the exponential form of diffusion equations a moderate error in activ&tion energy can lead to a large discrepancy in D.
Comparison with other work.
The experimental data of uranium diffusion in uranium containing compounds are very scarce.
Chubb et al at B.M.I. have studied the self—diffusion of uranium in uranium carbide and have obtained an activation energy of 65 kcal/moll Their results are so widely scattered, that it is very hard to compare them with the present results, but a comparison was made in Fig. 49.
— 163— Fig. 49. Comparison with other work
10- 9 L.
0 o b. L. I.
0 I. G. o alpha-mothoa • gamm&-fclCtLCC
▪ 1 0-1 0 I (.9 0
1 0-12 4.0 4.5 5.0 5.5 6 .0 6.5 1 4 x 10 T
- 164 - Section IV. Discusoion. 1. Discussion on the diffusion processes in stoichiometric uranium carbide.
1 - 1. ne diffusion of carboa. in uranium carbide.
Uranium monocarbide is a typical example of an interstitial alloy and assumes the f.c.c. NaC1 crystal structure.
Because of the relatively small size of carbon atoms, they fit interstitially into the f.c.c. uranium sub-lattice, forming a caroon f.c.c. sub-lattice. The carbon atoms ere isolated from each other, the c-c distance being 3.50 i.
G ialgg (126) regarded these interstitial alloys as solid solutions of the small non-metal atoms in the metal structures, his view of these alloys being apparently supported by the following generalizations,
(1). The relative positions of the metal atoms are essentially the same as in tl paro metal,
(2). The electrical conductivity is me-t,al/ic.
(3) U A :ride range of cmpositions for an interstitial alloy (-xists,
(4). The types o1 ho yes occupied is determined by ratio of the radii 3i and asa-metal atoms.
Yrou arxetens4-7 ;;carve-:y a acti carbides and nitrides, 0.27°4 Rundle shoad s tAat sane of Uggis generalizations were not valid for these ,_:or_:poundA3, beca73se (1) the arrangement of metal atoms in the compound UC or UN is different from that in the parent metal itself. (2). Llthough many of these interstitial alloys exhibit variable composition in a wide rangetUC or UN does n.,t, exhibit this behaviour (128) (or in a very short range).
When uranium carbide is formed from its parent metal, lr - uranium, the uranium atoms are placed :In the close-packed f.c.c. sub-lattice. The distance of the uranium atoms in the carbide is 3.50R, whereas 3.4742 is the distance talr- uranium. There is no variation in the meta?-metal distance by forming a carbide.
More recently, surveys of the properties of uranium carbide revealed that,
(1). The nature of chemical bonding in uranium carbide is metallic and coval entl in fact, more metallic.
(2). From a quenched-in experiant, Griffiths (70) observed that uranium carbide behaved like a metal, showing a single vacancy c,zhanism with a long life time (a). From plastic deformation studies of the carbide, Hollax (129) and Smallman showed a striking similarity of the behaviour with f.c.c. metals.
(4) • The vacancy state of carbon atoms in uranium carbide is not clear, decent Russian (154) calculations of the electronic state in transition metal carbide shored that carbon atoms wore positively charged, consistent with the observation that carbon atoms migzate t.. - ,he negative electrode when Fe—C alloys ware diffused under an electric field (155) This was also supported by the fact that the activation (156,157) volume was nil in Fe--CC alloys. The behaviour found in this system may we-Al Tce true for uranium carbide system.
Jill these factp support that, although uranium carbide classified as a cerr.,21e, it e mere wetallic in nature and carbou atoms are present as an interstitial impurity in the uranium matrix.
Therefore, in this pnper, a stoichiometric uranium carbide will be treated ae a solid zolution of carbon atoms in the uranium metal.
In solid solutions in which one conlponent in an inter— stitial atoms it is a well recognized fact that ,t,kzs inter— Atitial atoms should diffuse *T-ough the interstitial space, while lattice atoms d7iffuse t::arough the sub—lattice sites. 2ince the two kinds oi ,o1.7a, ::Vnot co:.,Ipete jirectly for the same sites, two indepen.;en'.:2, dfusior,, ccfficients should be obtained. This p‘-oblem war, thoroughly studied in the diffusion 130,131) of smail Tsese results have indicated clearly that an ia:ver: titia diffusion mechanism vas for th'D case of intt:rstitia alloys,
— 167 — Application of Zener's Theor, (132)
For interstitial diffusion! Zener has developed the first Satisfactory theory of Do, assuming that the activation energy is mainly used to strain the atomic lattice.
In the next, we shall discuGs the application of Zener's theory, into the carbon diffusion in uranium carbide. Experimentally, this procedure is useful for estimating Do and6S, and cnecking the results.
In III, 3 — 6, for simplicity, the vibration frequency was 13 assumed to be 10 sec and the entropy change calculated was 9.0 cal/mole/degree. ,Tin-;;D app icatior of Zener's theory requires a rigorous treatment of the vibration frequency, this was re— calculated by assuming that potential—energy curve of the atom varies sinusoidally along the diffusion path and its maximum value is the activation energy for the interstitial diffusion.(132) Thus, the relationship is expressed in the equation
20E )' (TV.1.1) ( c4" where GE the activation energy the shortos-6 jump distance mass of the diffusing atom.
From the experiAeatal data the numerical value of parameters were obtained as belo. 63000 cal/mol 2.33 ev/at3m and 10--J.0 2 12.26 x cm .
(133) From the mass-energy relationship the mass of carbon-14 is given in terms of the energy
E yr== 14x931x 10 ev.
Inserting these valir)s into the equation (V.1.1)
ev) - (2 x 14 931 12.25 x C (ev)(c41
9.48 x IC12 sec-1 13 -1 1 x 10 sec
Thus. ::,he assumption mc72a in IIT. 3-61 twos in right order. The accurate calculatfon 3f 4. at stoichionetric uranium carbide is also given
x x 1c12 exp (4s.S) 1.75 = 1 x 12.25 x l"16 R •
— = 5.0, GIS . 9.94 cal/mole/degree.
In 'ohe case of interstitial e.iffusion, the process consists si.mply of the jumping of a carbon atom from one interstitial site into a neighbouring interthAtial site. Zener postulates
1C9 that the activation energy or this jump is used mainly in straining apart the neighbouring uranium atoms between the carbon atom passes. This strain energy which is imparted to the surrounding lat'Ace, causes a lowering of the elastic constants and hence an ihoreGse in the vibrational entropy. Thus, making use of the fact that the elastic constants decrease linearly with increasing temperatures, he calculated the entropy change (LNS) and obtained the following relation;
E a .2) rn. (iv.3
where activc,:don enerxy To - the absolute melting temperatures the oonetant that depend on elastic constants and has a value between 0.25 and 0.45 for moot metals.
Since the experi,..1:elltoI 3atci on ela,7,tic properties of the carbide ir,; not avalabl as average vaiu& of 0.4 was taken for the caLculaon GO 4_12 40 WOOG 23 - cuA/mcie/degrei),,
This .rosuIt i2 in e!.:3eliert aFrre.Doent with that of the caY.cuatedeiS in str,icMooetr:1 1)run3AYA carbide.
ry the p.G1 pc:oer a number of very acourai.i. measuL-eentE on .1.Le di.rfusion of interstitial atoms such us oxTgen 3 an nitrogen wazi done in transition
170 (134) b.c.c. metals. The prevent. results were compared with them and summarized in Table 21. (134) Table 21. Diffusion Data For interstitial Carbon ktoms.
C in C in C in C in cx -Fe Ta Pb UC
Do(cm2/sec) 0.020 0.015 0.015 1.75
-1 13 13 13 13 1r(sec ) 1.30 x 10 1.31x10 1.31x10 0.95 x 10
(3 0.43 0.40 0.40 0.40
ZYZ(Kcal/mole) 20.1 27.0 27.0 83.0
SIR (Th) 2.38 ,1.65 2.01 4.73 -..1- 4s/a (Exp) 2.41 11.84 1.84 5.00
O course, the diffeicence cf the crystal structure was compensated in the calcuIationy takiagy -,==.1 for f.c.c. and 1 /6 for b.c.c. crystal in the Equation (3.6.8) (p.122).
Since a f.c.c. crystal structure is more closely packed than a b.c.c. structure, it requires a larger activation energy and entropy change, and a lower vibration frequency.
Hence, the experimental results are in complete agreement with Zener's predictiono. On the other hand, if one used the Langmuir—bushman equation (158} for the Do, there is also a good agreement. The equations is given by 2 ao 4.,E r. where ao is the nearest neighbour distance (3.5031, A is the Avagadro's number, h is Plank's constant and [5E is the activation 2 energy, Do calculated using this equation is 1.79 on /sec.
The Langmuir—Dushma:: theoryiiS not as complete as the Zeher's theory, but it till gives a good result.
From this discussion one can see that a carbon atom in uranium carbide behaves like an interstitial impurit:, the metal, not like an anion in ionic crystals. This fact might imply that the carbon is present as a positively charged ion.
Therefore, it can be concluded that in stoichiometric uranium carbide the carbon atoms diffuse through the interstitial Space in the structure; that is to say, an interstitial diffusion mechanism in predominant. — 2. The eldffusion df Urrnium in Uranium carbide.
Ls discussed in the previous paragraph, stoichiometric uranium carbide behaved like a simple cubic metal as a bulk.
In simple cubic metals, empirical rules for predicting the activation energy was found as described in II, 2-9. and it works very well for close4packed metals, although its applicability to all To.c,c. metals is not wholly justified.
Therefore, it is of interest to compare the present results with the values predicted from empirical rules. Since diffusion process by any mechanism involves the straining of atomic lattice, the coo only employed method is to relate the activation energy with Quantities which reflect the strength of atomic lattice.
Thus, Van Limpt first suggested that the activation energy is directly proportional to the melting temperature of metals. In an analysis of self—diffusion data for metals, (135) Le Claire proposed the Following proportionality constant,
Z2,- E 38.Tm cal/mole,
Taking Tm equal to 2723uK for uranium carbide, the activation energy should be about 103 kcal/mole. This is a good prediction.
,herby and Sinto4? (136) ::.afire also attempted an empirical correlation of the activation energy - Tith the absolute melting temperature, crystal structure, and valence of pure metals. Their relationship is P D = Do exp (—(Ko Tm/T) &El/sec.
— 173 where ;a) is a constant Alich depends only on the crystal structure9 V is a valence attributed to the metal, here 9 uraniuu. Sherby and 2irarod!), have suggested that Do should 2 be equal to 1 cm /eec for metals.
The values of Ko for f.c.c. lattice is 17. Assuming a valence of 4 for uranium metals, the equation yields an activation energy of 113 Kcal/mole. This is a very good prediction for the uranium diffusion in uranium carbide.
These predicted values are sunrarized in Table 22 and compared with the present results.
Table 22. Summary of Predicted and Measured Values of Do and,eNE.
c,E(Kcal/nole) ref
103 Le Claire redicted Values Sher by C•nd 1.13 Si Datiel ,-, 1 :.49 ; 10"' 122 8 klpha—r,lathoa AeasIlred Valuep I- t-- t 3.0 J.04 ± 'Y Gamma-methoi
ill of the activaiori energy vas generally very good thi? ut;suitab3e for the prediction cf T1c) Zener's theory worked very well in the case of interstitial carbon atoms c,,s discussed. in IV, 1-1. Hence, it is desirable to extend it to the case of uranium diffusion in the carbide. The argument leadin: to this case is exactly the same as the case of interstitial atoms. Thus, theiNS is written
s (3 Tm where ,)1 is a constant which should be equal to 1 and r.ds 0•_ _4 •
Taking tvo activatior.. .7:e:.c.gies obtained, the S was
3aiculated an be low.
x 128.009_
18.8 caVmole/degree.
}iron the experim,,iatal data, were calculated in 12 1 III, 3 — 6 and III 4 — 5, c-Lsuzling that 10 sec .. The values of calculated z.'2:_, P) were 175 and 28.4 cal/mole/degree.
All the results are summarized below and compared.
,!..8 (Tb) .4c.; (TExp) Lethod
15.3 17.5 Gamma—ray method
18.8 28.4 Alpha—ray method
All the results are in fair agreement and the agreement with that of the gamma—ray method is rather good. But Zener
115 also proposed a value of 0.55 for the constant y with the more accurate data for f.c.c. metals. The agreement is very poor with this value.
Js these results indicate, the diffusion of uranium atoms must be via the vacant uranium lattice sites, because the large uranium atoms cannot diffuse through the interstitial space. This is consistent with the vacancy diffusion mechanism. Thus, it is obvious that in uranium carbide the responsible imperfections are an interstitial carbon atom and a vacancy. The vacancy may be formed by a Schottky defect.
It is E41no of interest to note that the activation energy for carbon diffusion is about half of the activation energy for uranium diffusion, and tae diffusion rates are much greater than that of uraniula diffusioa. This is analogous to the fact that observed in many cases of iiiipur;ty difftsion in metals. , According to atomic theory of diffusion in 'clids these effects Could not be assigned merely to an incre=ased jump frequency for the impurity (MO atoms alone, so the Jc,b_nsn • ' mechanism was introduced to show that if thc,,re a h.iding ecrgy 115ctween the impu-ity atoms and vac.-)mcies, 1,-.-ipurity-gat;ancy wuld result, -vihich could cliEb thE, fl.tt.ins thus Jiving a higher diffusion rate and a iowel' activation '1nargy. if the Johnson mechanism i correct, an interstitial-vacancy complex may be forued in ura-lium narhice o There is no experimental evidence la urcniaci carbide ye, but in .1?-C- system a. carbon-vacancy compL,,y 7C,0 experimelAeai_y c,bo.rved and the binding energy was (160,161) found to n about IC Ri:al/mgolc,, In Summary; in stoichiometric uranium carbide the carbon ator diffuse through the interstitial space - by the interstitial mechanism, wherear the uranium atoms diffuse via the vacant lattice site - by a vacaney 2. Discussion On the Giffuson process In Non-stoichiometric Uraniuu aarbhde.
P - 1. The diffusion of Carbon in hyperstoichiometric graaium Carbide, (5.0, 5,1 and 5.6 wt %)
This is a carbon-rich alloy, consisting of a matrix of UC with UC lates in the form of a Tidmansatten structure (see 2 p Fig. 63). The structura relatienship between these phaoes have (13) not been studied in any detail. But Wilson confirmed the existence of a solid solution region between UC and UC2. pointing out that the uranium atone in tAe two compounds are in virtually the sane poitions, both interstitial type structure with the carbon atoms occupying in one case the octahedra) interstices and in the other case the tetrahedral interstices. (31) The neutron diffraction study reveals that the structure ofUC2 correspondstothatofCaC,and is considered to be an intermetallic compound of U and C,, where the c-c bond is a double bond. It Faso indicates that a difference in valency states between U in UC and in tha metal exists. 2 til-,,refor, (37,:ions that a hyperstoichionetric uranium carbide no longer can be treated in the same manner .as was done in IV. 1 - 1 and 2.
Tn an interstitial typo structure, there are two kinds of holes, namely, the octahedral and tetrahedral holes. In a cubic close-packed arrangement of atoms, if the octahedral holes are all occupied the arrangement of the atoms is that of NaCl structure , occupied the CaF structure results. and if all the tetrahedral ar. 2 Since re are dealing with a composition consisting of UC and UC dr a mixture of Nan Lnd CaF structure), two features 2 ( 2 follow frog the above discussion.
( 1) The poition of uranium aci.s is ac-; changed in the phase o2 UC or UC2.
(2) The carbon atoms can be placed in two positions in the structure, the octahedral and tetrahedral.
In pra,.:.tice, the matrix convicts of mainly UC, with a small portion of rC , that in, all the octahedral are occupied by carbon 2 atoms and some of the tetrahedral positions are also occupied, leaving a large portion of the tetrahedral positions to be occupied.
If the cl!.ffusion loace in the structure some kind of the defect mast be .nroe..ucee, cc t3lat tb carbon atop can riove. This car be eapily reil,e7ing the carbon from the octahedral position into M-..e totrahe_rai positions, because carbon atoms are cc snail that no aistcrtion 7311 occlzr ir the lattice, So, rz vacancy and an interctial carbon at= are formed. This corresponds to 7renkel type of dc'.fectq
Here, tLe neaning 'intarstitiai atoms" is cone- what :lifferent from that of TV, i - 2 and 2, referring now to the tetrahedral position in the structure.
If exce6s Jarboka adijied to the stoi,hiometric composition the ca..-ocu- atcms ao-v.id be placed in the interstitial space because all the octahedral positions are occupied. The carbon atoms in the interstitial space can be moved in two ways, (1)directly to the next interstitial positions, or (2)to the lattice site, by knocking out the carbon atoms in the site to the interstitial site.
Since the distortion involve:1 in the case of (2) is quite small, it can occur more easily thab (1). This fact nay be attributed to the lower activation energy in hyperstoichiometric uranium carbide. (Fig. 36). Therefore, the diffusion mechanism can be considered to he one in which an interstitial carbon atom and a lattice carbon atc,n move in combination. This is consistent with the int9rstitialoy L:echanism.
The excess carbon atoms in the interstitial positions would form a diffusion carrier and facilitate the diffusion of carbon. This is justified by the fact that the diffusion rate of carbon is strongly dependent on the carbon content (Fig. 31). As the amount of excess carbon increases, the diffusion rate increases rapidly, but not linearly, and finally reaches a constant value at about 5.6 wt % composition. This uay be due to the fact that a variation in the valency state of uranium or the effect of double bond in UC2. It was also observed that a negative value of .AS was exhibited in 5.6 wt % composition (Fig. 33). This indicates that phase becomes nore effective. the influence of UC2 In summary the diffusion of carbon in hyperstcichiometric uraniuci carbides was tackled by steric considerations. It seems that interstitial atous form a diffusion carrier and the diffusion mechanism is considered to be interstitialcy. 2 — 2, The diffusion of .,ar'Jcbn in Hkrwstoichipmetric uranium carbide (4.7 wt%)
This is-a carbon—deficient alloy, consisting of a matrix of UC with free uranium.
William et al (24) and Buckley (25) determined the lattice parameter of UC in a series of arc—melted uranium carbides and found a systematic variation. These results are interpreted as being an indication. of the ability of uranium carbide to adopt a uranium—excess at high temperatures by omission of carbon atoms from the unit cell, which causes the lattice parameter to decrease. This suggests that the vacancy concentration of carbon in this composition is very small.
According to atomic theory of diffusion, the diffusion coefficient is proportional to the concentration of vacancies. This is expressed in 2 a w. Nv
where Nv is the concentration of vacancies, and w the probability that the next neighbouring sit is vaca:at
Therefore, in thi2 composition, one would expect that the diffusion coefficient should be very small., This is verified as the experimental results shuw. 28 and 31). The diffusion coefficients in this ^omposition are smaller than any other composition studieE:.
Ai;tivatioK energy is used Eainly in straining apart the neighbouring uranium atono between whch carbon atot passes. Since, according to Wi'iLliao and Buckley, th,:= vacant carbon sites in the unit cell are occupied by the uranium atoms, the cell is almost an aggregate of uranium atoms. In order to pass through this aggregate, the carbon atoms require a large sum of energy to strain them apart to make a path. Therefore, it will require a large activation energy. The activation energy obtained was 89 Kcal/mole, which is very high value for the carbon diffusion.
There is sore evidence of grain boundary diffusion as shown in the next paragraph. Also the metallugraphical studies show grain growth in the composition (Fig. 65 and 66). Both results may be correlated, since the material migrates along the grain boundaries and forms a new grain. Therefore, the carbon atoms cr uranium atoms may migrate along the grain boundaries. If one applies Fisher's analysis in this system a plot of log c v.s. X may exhibit a good straight line. Indeed, the evidence of grain boundary diffusion was observed as discussed in the next paragraph. 3. Discussion on the effect of grain boundaries on diffusion.
The diffusion of carbon and uranium atoms may proceed through the lattice as well as along the grain boundaries.
Many experiments have shown that, (133, 139, 140, 141 ) in general, lattice diffusion is predominant at high temperatures, whereas grain boundary diffusion is predominant at low temperatures. The diffusion rates along the grain boundary may be several orders of magnitude greater than the lattice diffusion at low temperature. The :pper limit of te,uperatures at which grain boundary diffusion is predominant about Lalf of the melting temperature. (142) J.C. Fisher ' first developed a theory, which is able to separate approximately the lattice and grain boundary Liffusion (143) when both occur together. Later, Whipple extended the theory, but its applicability is limited. The theory is based on a model of a grain boundary as a narrow region having a high diffusion rate relation to the adjacent grains. The lattice and grain boundary diffusion rates are assued to be independent of concentration, Thus -t,1. mathematical :.olution of Fisher's equation leads to t X const. e where f. is the aEoant ok material in a section of thickness QX cud tis the time: ole cAffusi311
Hence, a plot of . g G activity) v. s. X should give a straight line of slupe
77 Tit,) iki:L;ttr o if grain boundary diVfu!Aoll Experimentally this can be observed if the ratio Db/De (grain boundary and lattice diffusion) is large enough, but is large Fisher's analysis gives no indication of where Db/De enough. Therefore, Pisher's analysis is only approximate.
According to Fisher's analysis, log C.w.s. X was plotted for all the experimental data obtained in the course of this study. Three types of characteristic plots were found and shown in Fig. 50, 51 and 52. Their interpretation is given below.
Type (1). In Fig. 50, a pronounced curvature was obtained, indicating that the major portinn of the diffusion took place via the lattice. This is a case where, at high temperatures, the lattice diffusion is more pronounced than the grain boundary diffusion. A plot of log C v.s. X2 was a straight line,
Type (2). In Fig. 51, a straight line was obtainer, indicating that, as Fisher predicted, the grain boundary diffusion is :2 predominant, and also a plot o.? log C v.s. X was curved.
Type (3)% In Fig. 52. This an intermediate stage where the lattice and grain boundary diffusion occur together and equally important. Therefore, the plot is a combination of straight line and curve.
Fisher's analysis was applied to all the experimental results obtained, to see if there is any evidence of grain boundary diffusion, and typical examples for each series of uranium carbide specimen were selected and shown in Fig. 53 — 58. Fig. 5. . Effect of Grain boundary on Uranium Diffusion in Arc-cast Uranium darbice ( 4.82 wt % ) at 1741 G. 0 log C vs X
AO 7 S-1
0 0
bp O
2 X 105 cml 10 0 30 40 50 .0 70
10 20 30 40 50 60 ' x 10-4 cm
- 185 - 1-- Fig. 51. Effect o Grain Boun ary on Carbon D'ffusion in Arc—cast Uranium Carbide 103 ( 4.7 wt % ) at 1690 °C. _ _ • . —
•
bo ,tF • a z log C vs X
2 10
• _ 0 log C vs X —
. —
--,
2 5 2 X x 1t cm •
10 20 30 40 4 60 70 10 0 50 100 150 200 250 . 300
X x 10— cm
- 186 -
0 an ( count / min / mg ) 10 10 10 2 3 ) - _ r _ _ F _ l- r . 5 iv O 10. log 50 0
15 C vsX X
2
x
?c) 0 2 10 ` inn " 5
cm
2 Fig. 52. 39 X
35 — 187 4, 1 log ..in x
4Q,
-- C vsX iTt Carbon DiffusioninAre Effect ofGrainBounoary Uranium Carbide(4.7 - • • T505 10 -4 -Q onn cm C s, - , 4 I
ogn wt %)
1 -cast on , ,nn _
- .... _ _ J _.. _ _ j Fig. 53 . Effect f Grain Boundary on Carbon ifftision in Arc-cast , Uranium Carbide ( 4.7 wt %•)
, _ • _
I; _ c • )
e tiv la re (
• to0
._ 0 1761 C • _ C) 1690 c - CIO 1440
O
0 50 100 150 200 250 300
X x 10-4ern
7 188 - T
Fig. 5 4. Effect o Grain Boundary on Carbon Diffusion in Arc-cast Uranium Car ide ( 4.82 % )• 103 . • _ i-- _
_ p _ •
•
a
102 U • 40 O _
— 1465 - 1362
10 100 300 400 x lo-4 cm
- 189- _...
_ - Fig. 55 .Effect of Grain Boundary on Carbon Diffusion in' .. ' rbi firs _ ( 5.1 wt % ) _ r • ' _ 1r. _ — . IP L- • _ 1- _ 1 if. • 1 _ - _ h-- 0 • _ •
,.. _ tat) 0
- r- p 1505 — _ .._. r• @ 1412 3c _ L--. 0 1215 c
0 50 100 150 200 250 300
X x 10-3 cm
- 190 - J
Fig. 6. Effect Of Grain Boundary on - Carbon Diffusion ill Arc-cast - Uranium Carbide ( 5.6 wt % ) - 0
_ • _
_ r _ • _
cd
41) 0
_ 0_ - _ _ _ _ 01467 IC - , . _ 0 1215 ' C . .
- _
C)
0 100 200 300 400. 500 600 —4 X x' 10 cm
- 191,- - Fig. 57 . Effect of G ain Bouict - 0 lit • • -dary on Ca bon Diffu -_ - • -sion in Si tered - ,-- • Uranium Car ide ( 5.0 - wt % ).,. - •
0 4,
4 1
.... .-.
_ I. • ) _
tive _ la _ . • 4 C ( re
log 0 1626 °C
o 1280 C -
_ 0 1205 °C _ -
50 100 150 200 250 300 X x 10-4 cm
- 192 - Fig. 58 Effect of Drain .boundary on Uranium Diffusion in 4c-cast .110, Uranium Carbide ( 4.82 % ) . •
.4
2 10
/mg in /m t r O ou c ( •
tuD O 0
10
GO 186 0 174
0 165 ,
0 20 40 - - 0 x 10-4cm
- 193 - (a). 7ffect o.fgrain bounda,-v on the diffusion of carbon. Three types of tendency were observed in arc—cast uranium carbide, that is (1). The evidence of grain boundary diffusion was more marked on the lower temperature side. (2). The evidence of grain boundary diffusion was more frequently found in the lower carbon composition side.
(3) • Even at high temperature thy diffusion was not entirely due to the lattice diffusion alone, because the plot of log C v. s. X contained considerable length of linear portion.
In sintered uraniuu carbide more pronounced curvations were obtained throughout the temperature range studied — a strong indication of lattice diffusion.
(b). Effect of grain boundary on the diffusion of uranium. Since the temperature range investigated was much higher than half of the melting temperature, as one expected, all the results of Fisher's analysis were pronounced curvatures. This is a strong indication of lattice diffusion dominancy.
— 11)4
Section V. Applications and
Suggeztions for Future Work.
— 195._ Introduction.
The study of diffusion provides a basis for the under- standing of activated processes such as sintering, high temperature deformation, grain growth and diffusion of fission products. Hence, the determination of diffusion coefficients of uranium and carbon in uranium carbide has provided a means to evaluate various diffusion models, proposed for sintering and deformation. In an activated process, it has generally been assumed that the clover species must 'oe rate-controlling. Since in uranium carbide the uranium LA,DLas are the slower species, this must be rate-controlling in sintering and deformation. However, the diffusion coefficients were calculated from the sintering and creep data, and compared with that of the diffusion studies. This type of comparison may throw some light on the nature of the activated process itself, or the feasibility of diffusion models proposed by many authors.
I. Ugh-Temperature Deformation in Uranium Carbide. 0:45 Nabarro (144) and Tiarring have shown that a crystal can change shape by the ,:,elf-diffusion of vacancies under an applied stress. In a polycrystalline solid, diffusion within the grain causes a viscous type of flow.
This relation ,;am be expressed as 10, r. L. V. (v.1.1) (G2)6. h..T where e is the strain rate, r is the applied stress, H is the diffusion coefficients, V is the vacancy volume, (GS) is the grain size, and kT is Boltzmann's constant and absolute temperature. It is apparent that, if appropriate numerical values are inserted into the equation (V.1.1) the diffusion coefficients could be caicula-ced, Since the uranium atoms are iate—controlling the calculated diffusion coefficients must be equalto that of the uranium atoms and should exhibit an equal activation energy, provided that a diffusion creep mechanism is operative. (146,147) Norreys and Wheeler at G.E.C. Wembley, measured the compression creep rates of uranium carbide at elevated temperatures, Their rersults are expressed in the equation exp ( 49:0) exp (B.6) o where e is the minimum creep rate under a stress of 6, R. is the gas constant and !. and Rare constants for the material. The strain rate was interpolated from the plot of log e 1 v.s. for hyperstoichionetric composition and the diffusion coefficients were calculated from the N:,barro—Herring formula (Eq. (IV.1.1) ). One of the calculations is shown, qrz 6000 7.s.i 7 • 41.4 x 10 dynes/cm2 2 (GS) = 3 x 10_ co -23 3 Vcm• 3.04 x 10 . 1.3S x 10-16 dynes cm/degree -4 4 x 4 h T 1673`K.
— 197 — therefore, -4 -6 4x 10 x 9 x 10-4 x 10-4 x 1.38x 10 x 1673 D +7 -23 3600 x 10 x 41.4 x 10 x 3.04 x 10 -10 2 / 1.9 x 10 en / sec.
The calculated diffusion coefficients are listed below.
Temp (°C) D (en /sec) -10 1400 1.9 x 10 -10 1343 1.2 x 10
1289 15.3 x 10-11 -11 1242 3.1 x 10 -11 1/98 1.7 x 10 and a conventional Arrehenius plot was drawn and compared with that of the diffusion studies in Fig. 59. Contrary to the prediction from Nabarro-Herring model, the calculated diffusion coefficients are very near to the carbon diffusion coefficients, in spite of the fact that the uranium atoms are rate-controlling. This coaparison indicate that tlhe deformation process nay not be by a Nabarro-Fierk.ing diffusion mechanism only. It is well known that ext&nsive grain-boundary sliding occurs in most polycrystalline solids during deformation. Uranium carbide was not exceptional. R. Chang has produced evidence of grain-boundary sliding in hot-pressed uranium (148) carbide
- 198 - Since the calculated diffusion coefficients are considerably higher than those whieh can be accounted for by the Nabarro-Herring uodei, ucly we conclude:.1T that the creep deformation of uranium carbide is proceed not only by vacancy migration, but also grain boundary sliding at the stress applied and tempek-ature studied.
— 199 — Fig. 59.
O 10 0 o carbon diffusion doefficients in uranium carbide ...... _ 0
10-9
0
-1 o 10 • creep diffusion 0 V 0 coefficients in uranium carbide OQ o o
r-i -1 10 o 1 uranium diffusion coefficients in uranium carbide
0
0 -1 10 ...
4.5 5.0 5.5 6.0 6.5 7.0 7.5 T x 104 - 200 - 2. Sintering Of Uranium Carbide.
The phenomenological theory of sintering, especially of its initial stage during which bonds between the adjacent particles form, has :teen fairly well established. In the case of a simple system such as two spheres in contact with each other a rather si-Tle relationship between the neck growth X, tine t and sintering temperature T has been ( found (149)
in A (T). t. (nr.2.1) M a where a is the radius .of the particle, A(T) a function of temperature only appropriate for a given flow of mass, and n, m, constants for types of sintering mechanism. These relation— ships were verified in a series of experiments, (150, 151, 152) and demonstrated the predominance of the volume diffusion mechanism. In this case the equation (IV.2.1) can be written in the form
5 7c. (IV.2.2) X. 2 R,T a where '6 is surface tension o± the solid, V its atomatic volume (or vacancy volume), R -,fie gas constant and D the self—diffusion coefficient, K is a constant depending on the geometry of the sample and also on the diffusion path. The exploration of the constant K has been the subject of many researcher on the sintering Aechan±so of scdids. Never al authors 'aa7re worked out this constant K under suitable assumptions aid some of their results follow;
— — 2/5 (1) AL (T. V. D. t Kuczynski Lo 32. a3.11.T 2/ 5 (2) AL . /30.'Y'. V. D. t Kingery & Berg. 3 2.. a .k.. n/ t 5 (a) 4L = / 10. -A . Coble. LO 3 a . k, T
These are evaluated by measuring neck-growth or shrinkagee,datay and Obtained some success in application to the initial stage of sintering in solids. The neck-growth or shrinkage are due to bulk flow from the region adjacent to the contact area to the neck. the paths Tor material flo either via the bulk or the grain boundary. (153) raclaran, Regan and hedger have investigated the kinetics of sintering in uraniun carbide and obtained an activation energy of 122 Kcal/mole, which is similar to that in v.s. log t these diffusion studies, and the slope of log 4111Lo way 2/7 cvc-, the temrature range studied. In order to throw Eloe light on the sintering process in uranium carbide? using tne thl'ee -2ode1s ciuoted ear..ier, the apparent diffusion cc,efficients were calculated from the shrinkage data, An example of the calculations is given
a 1,.; 16-5 en -23 3.05 Y 10 CQ Z.)c at i,tjt zoc 5. ;.noc.; erg/k-A.
- Inserting these valfuez, into the above equations, D „. (0.0335)5/ 2 x i.38 x .10-16 x 1573 4 -23 10 x 3.05 x 10 = 6000 -11 9.90 10 cm2/:ec.
The apparent diffsion coeificients, calculated from three models, are shown in Fig. 60. L11 these values seei: to fall fairly well cn a straight line and shor an activation energy of 95 — 7 10 Kcal/mole., but the agreement between the measured and calculated values of diffusion coefficients is very poor.
The possible reasons for this disagreement are considered to be (1)lack ox -the accuraile --afcrnation such as surface energy, particle size and shape or (2)inadequaby el the nroposed models. (3)introduction cf other r2a-:;hanisms in the sintering process such as plastic “orT,
202 Fig. 60. -13
uraniu diffusion coeffi ients in uraniu, carbide. -14 10
10-15
sinterng diffbsioil coeffi ients'in A -16 uraniu carbide 10 O Coble O Kin erY & berg O Kuc
-17 10
-18 10 5.0 5.5 6.0 6.5 7.0 7.5 8.0 1 4 T x 10 - 204 - 3. Diffusion of Fission Gases. When uranium undergoes fission, the rare gases Xenon and krypton comprise a large fraction of the total fission products atoms. The amount is so great that the diffusion of rare-gas atom becomes an important phenomenon in nuclear reactor fuels. (72)_ 133 Auokern, and Osawa have studied the diffusion of Xe produced in uranium carbide. They found a Do of 1.27 x 10-7 2 cm /sec and activation energy of 85.1 Kcal/mole. The evolution of fission gas has been measured in structure of graphite and uranium carbide by Lindner and Matzke. (73) The activation energy for ire-diffusion from this material was found to be 42.5 Kcal/mole. There is a large discrepancy between two results. The reason is not clear. The results of Xe-diffusion were compared with that of the extrapolated uranium diffusion in Fig. 01. This result suggests that Xenon may be diffuse through the bulk of uranium carbide There seems little contribution from structure-sensitive dAffzfiion processes such as grain boundary or void diffusion. Fig. 61.
1 0 1 —
uranium diffusion in uranium carbide.
in1 1,4/ -16 ---
Xe1 l 3diffusion in U iU uranium carbide.
-t
1L=-1 0 0
FISSION GAS DIFFUSION
IN
URANIUM CARBIDE 10-19
-20 i v --- 4.0 5.0 6.0 7.o 8.0 9.0
x 104 - 206 - 4. Suggestions for Future Work.
From the information obtained in the course of this study the following experiments are worthwhile to investigate. (1). determination cf the self-diffusion coefficients for uranium in non-stoichiometric uranium carbide. (2). determination of the self-diffusion coefficients for uranium and carbon in single crystal uranium carbide. (3). direct evidence of grain boundary diffusion by auto- radiographic to using C-14 and U-235 as tracers. (4). determination of rare-gas diffusion coefficients in uranium carbide. (5). investigations on plastic and steady-state deformation of uranium carbide. (6). investigation on the sintering process of uranium carbide. (7). investigatiOn of the valence state of carbon in uranium carbide by a diffusion technique under applied electric field - or electrolysis of uranium carbide. (8). determination of the self-diffusion coefficients for uranium and carbon in doped uranium carbides.
- 207 - Section VI
General Conclusions.
— P0,9 — 1. Self-diffusion of carbor in stoichiometric uranium carbide is given by the relationship
D . 1.75 exp (-63,000/RT) for the temperature range 1266 - 1684°C. The mechanism is considered to be an interstitial mechanism. The addition of a small amount of carbon to essentially stoichionetric carbide results in a large increase in the carbon diffusion coefficients and a decrease in the activation energy. The presence of excess carbon atoms favours an inter- itialcy mechanism. In hypostoichionetric uranium carbide there is a large increase in the activation energy and a decrease in the carbon diffusion coefficients, There is some evidence of grain boundary diffusion at lower temperatures.
2. Self-diffusion of uranium in stoichiometric uranium carbide is given by the relationship 128,000 4.49 x lu exp ( _. ) D RT (alpha-ray method)
D = 8.47 exp ( 1%14000) (gamma-ray nettod). o for the temperature range 1505-C - 1863 C. The possible rechanism is considered to be a vacancy mechanism. The effect of grain boundary diffusion is minor. The whole picture of self-diffusion studies in uranium carbide is shown in Fig. 62.
— 2C9 — Fig. 62. i
q., 40
• 1 0-8
O
• 10-9 •
0 •
C ( 3) ( 4 ) ( 5 ) E-1• 10 * 10 C.)
I V ( 2 ) c.) SELF—DIFFUSION OF URANIUM
• rik %,. I & CARBON IN URANIUM CARBIDE. ( 1 ) U in UG ( 4.82 % ) ' ealpha cgamma —11 10 alk W ( 2 ) C in UC ( 4.7 % ) 0 ( 3 ) C in UC ( 4.82 % ) e ( 4 ) C in UC ( 5.1 % ) 0 ( 5 ) C in UC ( 5:6 % ) 6)
-1 10 r)
( 1 )
..., 4.5 5.0 5.5 6.0 6.5 7.0 •
1 4 T x 10
—210— APPENDIX - Metallographical Studies of Uranium Carbides
- 211 - Fig. 63. Typical microstructure of arc-cast uranium carbide prior to diffusion anneal.
( a ) Microstructure of hypostoichiometric U - C alloy ( 4.7 wt °A ) . Mag x 100
( b ) Microstructure of hyperstoichiometric U - C alloy ( 5.6 wt ck. ) . Mag x 400
- 212 - Fig. 64. Microstructure of arc-cast U C alloy ( 4.7 wt ) afetr aiffusion anneal.
Conditions; Temp; 1500 0C , Anneal Time; 5 hours, Flag x 350.
* White conFtituent in grain boundaries is uranium metal.
_ 213 - Fig. 65. Grain growth in arc-cast U - C alloy ( 4.7 wt ).
Conditions;
Temp; 1500 °C , Anneal Time; 5 hours x 750.
* White constituent in old grain boundaries is uranium
metal.
- 214 - Fig. 66. Grain growth in arc-cast U-C alloy ( 4.7 wt ;10 ). Conditions;
Temp; 1500 °C , Anneal Time; 5 hours , Mag x 750.
* 1 hlte 'pots in grain are an old grain bounaary ana uranium metal.
- 215 - Fig. 67. Micro-crackes in arc-cast U - C alloy ( 4.7 wt ). Conditions;
Temp; 1500 °C , Anneal Time; 5 hours , Mag x 50.
- 216 - Fig. 68. Microstructure of arc-cast u - C alloy ( 4.82 14 ) after diffusion anneal.
Conditions;
Temp; 1750 °C , Anneal Time; 20 hours , Mag x 750.
-217- Fig. 69. Microstructure of arc-cast U - C alloy ( 5.6 wt (), )
after diffusion anne 1.
Conditions;
Temp; 1250 °C , Anneal Time; 30 hours, Mag x 750.
* needle-like structure in grain is U C . 2
-218- Fig. 70. I recipitation of U C 2 in grain boundaries. Conditions;
Temp; 1750 °C , Anneal Time; 20 hours , Mag x 3 50.
- 219 - Fig. 71. Frecil:Itation of U C 2 in grain.
Conditions;
Temp; 1750 °C , Anneal Time; hours , Nag x 750.
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