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The temperature dependence of the isotope effect for the diffusion of Na+ in NaCl
J. Phys.Chem.Solids, 1972,Vol.33, pp. 106I- 1069. PergamonPress. Printedin Great Britain
T H E T E M P E R A T U R E D E P E N D E N C E OF T H E ISOTOPE EFFECT FOR THE DIFFUSION OF N a ÷ I N NaCI* S. J. ROTHMAN, N. L. PETERSON, A. L. LASKAR'~ and L. C. ROBINSON Materials Science Division, Argonne National Laboratory, Argonne, II!. 60439, U.S.A. (Received 5 October 1971)
Abstract--The isotope effect for the diffusion of N a ÷ in Harshaw NaC1 crystals has been measured in the temperature range 589-796°C. The diffusion coefficients o f N a + in NaCI are given by D = 7 7 e x p ( - 2 " 0 4 - - 0 . 0 2 e V \) cm/sec. z, \ The isotope effect, f A K , decreases as the temperature increases, indicating a vacancy pair contribution to the self-diffusion of N a + in NaCI. This contribution reaches 30-45 per cent near the melting point. The temperature dependence o f f A K and D suggestsfAK is ~< 0.2 for N a + diffusion by means of vacancy pairs. This implies that, if AK for diffusion by vacancy pairs is greater than 0.8, f pairs is ~< 0.25, and hence the ratio of C1-jumps by pairs to N a ÷ jumps by pairs must be less than 0-i. Also, a large entropy for N a + diffusion by vacancy pairs is indicated. 1. INTRODUCTION
THE OBJECTIVE of the experiments described below was to find limiting values for the contribution of cation-anion vacancy pairs to the diffusion of N a ÷ in NaCI. The extent of this contribution is at present controversial. Nelson and Friauf[1] obtained a pair contribution of about 50 per cent near the melting point from their combined drift and diffusion measurements. On the other h a n d , the measurements of Beniere et al.[2] combined with the calculations of Tharmalingam and Lidiard [3] indicated that only a few per cent of the diffusion of N a ÷ in NaCl near the melting point is by means of vacancy pairs. The vacancy pair contribution is related to the isotope effect in the following way. For two simultaneously operating mechanisms, the observed value of the isotope effect, E, is given by [4] *Work performed under the auspices of the U.S. Atomic Energy Commission. t Visiting Scientist, A N L. Permanent Address: Physics Department, Clemson University, Clemson, South Carolina.
E -- E~D1 + EzD2 Dl+D2 '
(1)
where the subscripts denote the two mechanisms, in the present case single vacancies, 1, and vacancy pairs, 2. The E's are the values of the isotope effect for each mechanism, and the D's are the diffusion coefficients of the atomic species, in this case N a ÷, by means of each mechanism. Thus if E1 and E2 are known, the pair contribution, D J ( D I + D 2 ) , can be obtained from the measured value of E. Unfortunately, E1 is known only within 10 per cent, and E2 not at all. That is, for any mechanism of diffusion, Ei = fiAKi [5], where f~ is the Bardeen-Herring correlation factor for the mechanism[6], and AK is the factor that accounts for the preexponential in the Arrhenius expression for the atomic jump frequency not being proportional to (m) -112 [7, 8]. F o r NaCI, fl = 0.782[9] and 1-0 AK~ ~> 0.8[10, 11], so E1 is known to ___10 per cent. E2 is more difficult to evaluate. First, AK2 has not been measured or calculated. One can only infer from LeClaire's correla-
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S . J . R O T H M A N e t al.
tion between AK and the relaxation around a defect[12] that AK2 is not likely to be larger than AK1. Second, f2 can vary from 0 to 0.782. This is because jumps of both the cation and the anion vacancies in the pair are involved in the diffusion of N a ÷ by a vacancy pair, and thus f~=fs(v'dv~), (see Fig. 1 of Ref. [1])[1,9,13]. Here v~ and v~ are the jump frequencies of the anion and cation vacancies in the pair, respectively. The calculations [3] S. indicate that vat > vc, the data[l], which can be faulted on experimental grounds (the contact between the two crystals in the drift experiments was far from perfec0, give v~ -> v~. It was with the hope of obtaining more data on these questions that the isotopeeffect experiments were undertaken, even though the number of unknown parameters is obviously so great that only limiting values of E2 andf~ could be hoped for. 2. EXPERIMENTAL TECHNIQUES The thin layer sectioning technique was used for the diffusion measurements, and the half-life separation technique[11, 14, 15] for the isotope-effect measurements.
Materials and sample preparation Harshaw single crystals of NaCI were used in all experiments. A flat surface a few degrees off (100) was prepared on the crystals immediately before deposition of the radioisotope. The crystals were not preannealed* because we feared that impurity pickup during the preanneal would introduce more severe imperfections than the anneal would remove. Conductivity measurements on crystals cleaved from the same boule showed that the 'knee' was at about 500°C, as expected for Harshaw crystals[2, 17]. Ultraviolet spectroscopy showed about 10 ppm of O H - . All the 2SNa used in these experiments *The crystal diffused at 796.60C was preannealed at 750°C for 24 hr in an NH4CI atmosphere. The resulting pickup of PbS+[16] caused only a small deviation from linearity in the penetration plot because the diffusion annealing temperature was so high.
was shown to be radiochemicaUy pure by high-resolution gamma-ray spectroscopy. However, the first lots of this radioisotope, obtained from International Chemical and Nuclear (ICN), contained about 300tzg/ml of nonradioactive Ca 2÷. We do not know if the contaminant came from the supplier or was introduced during our processing. Specially purified 22Na was used in subsequent experiments. The 24Na was either bought from ICN, or made by irradiating Harshaw NaCI in the CP-5 reactor; neither contained harmful impurities. The absence of radioactive impurities was verified by high-resolution gamma-ray spectroscopy and by half-life measurements. The radioisotopes were deposited as NaCI on the crystals by evaporation from a platinum filament in vacuum.
Diffusion annealing The crystals were laid with their active faces on a quartz fiat, wrapped in platinum foil, sealed under a vacuum of 2 × 10-~ Torr in previously baked quartz tubes, annealed in resistance-wound furnaces, and air cooled. Corrections were made for heating up and cooling down, and the furnace thermocouples were calibrated as described earlier[18]. We estimate that the quoted temperatures are accurate to ---+I°C.
Sectioning and counting A f t e r cooling, 1-2 mm was cleaved from the lateral sides of the crystals, which were then sectioned on a Leitz sledge-base microtome. Weight losses were ~< 1 per cent during the summer when the indoor relative humidity was high. In the winter, when the indoor relative humidity was low, a humidifier had to be used to keep chip losses down; two samples that were sectioned without use of the humidifier showed chip losses of - 15 per cent. In the few other cases in which the chip losses exceeded 1 per cent, the appropriate corrections were applied. The samples were weighed, dissolved in 1 ml HsO, and counted in 3 by 3 in. well-type
THE ISOTOPE
scintillation counters with lower level discriminators [ 14, 15].
EFFECT
k ~-.'~
I
1063
I I I r I I 683.8-c EACHDIVISIONONABSCISSA=Ix 10-4¢mz CISSA=O.5 x 10"4cm:;:'
Isotope-effect measurements The separation of the radiation from 2~Na and 24Na was accomplished by counting each section at least six times in the 75 hr following sectioning, and fitting the total gamma activity above the lower level discriminator, which was set in the valley below the 0.511 M e V positron annihilation peak of 22Na, to the time elapsed from an arbitrary time zero according to the equation C = c22e -xzzt -4- c24 e-x"--4t.
3. RESULTS
The results on the diffusion of N a + and on the isotope effect are given in Table 1.
Diffusion o f N a + in NaC1 The diffusion coefficients are obtained from the penetration plots, plots of log specific activity (I) vs. (distance X) 2, according to the solution of the diffusion equation for the thin-layer initial condition ~
.
~
> ~ ~, ~o%
"~.,~ "-" ~
~ ~- "-683.8-c ~
2160C !
(2)
Here h~2=5"ll17x10-Tmin-l[19], h~4= 7-708x 10-4min-l[15], and the quantity of interest is c22/c24. H o t sections were diluted to initial counting rates ~< 104/sec, and the usual corrections[I;4] were made for background and dead time. One million counts were taken in most cases. Null effects (aliquots of different activity from the same bottle), run together with the 621.6 and 751.2°C runs, indicated that the isotopic ratio was independent of count rate and also that the errors in counting were close to the errors expected from the counting statistics.
/=10exp--
~_
2"
(3)
A number of the penetration plots (see Fig. 1, 589.3°C line; also Table 1) obtained experimentally were Gaussian as predicted by
I
i
]
I 1 X 2,10"4cmz
I
I
Fig. 1. Penetration plots (log ! vs. x 2) for the diffusion of N a ÷ in NaC1.
equation (3). Samples on which the Ca 2+contaminated 22Na was used (and the 796.6°C run, contaminated with Pb 2+) showed penetration plots concave toward the abscissa (Fig. 1, 683.8 ° line; see also Table 1). When the curvature extended for only the first few points, valid values of both D and E could be obtained from the run. In other cases, when fewer than 10 points were on the straight line, only the D value was used in the analysis; justification of the use of such plots has been given by Rothman et al.[17]. The runs at 621-6 (see Fig. 1) and 648-1°C gave penetration plots slightly curved in the opposite sense. These were the runs with the 15 per cent weight losses mentioned previously; even when the weight-loss correction was applied, the values of D obtained from these runs were about 20 per cent low. We have therefore felt justified in not including these values of D in our least-squares Arrhenius analysis; however, since the log-ratio plots were excellent, and since loss of material during sectioning does not enter into the isotope effect[11], we have used the values of E obtained from these runs.
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S. J. R O T H M A N
et al.
Table 1. Diffusion of Na +and the isotope effect Temp. (°C)
Annealing time (see)
D (cm2/sec)
D~2 D2---~--1
E
796-6
3840
1-97 x 10 -s
0-0216___0-000201
0-484
767.1
7086
9.85 x 10 -9
751.2 719.7
10416 23820
5-83 x 10 -9 3-33 x l0 -9
718-8
7200
3.49 × 10-9
683"8
14400
1 - 2 5 × 10 -9
648.1
58200
4 . 3 6 × 10 -1°
633.7 627.4
77400 61200
3 . 3 2 × 10 -19 2-90 × 10 -1°
621.6
70740
1.93 x 10-1o
603-8
66780
1 "37 × 10 -1°
602-7
67320
1.39 × 10 -1°
589-3 587-4
144360 67500
8-75 x 10-11 8-50 x 10-1~
Remarks
F i r s t 6 p o i n t s low o n p.p.* d u e to Pb z+ c o n t a m i n a t i o n . . . . . . . G o o d p.p. Log-ratio plot curved. Value ore not used. 0.0232_--_0.000290 0.522 G a u s s i a n p.p. . . . . . . L a r g e tails o n p.p. a n d logratio plot. V a l u e o f E n o t u s e d . 0 . 0 2 5 5 _ 0-000285 0.574 F i r s t 6 p o i n t s low o n p.p. d u e to C a ~+ c o n t a m i n a t i o n . 0.0281--+0.000398 0-631 F i r s t 9 points low o n p.p. d u e to C a z+ c o n t a m i n a t i o n . 0.0299-+0.000448 0.674 C o n c a v e u p p.p., large w e i g h t loss. V a l u e o l D n o t used. 0.0269-+0.000384 0.604 G a u s s i a n p.p. . . . . . . F i r s t 11 points low o n p.p. d u e to C a ~+ c o n t a m i n a t i o n . V a l u e o r E n o t used. 0-0309___0.000456 0.695 C o n c a v e u p p.p., large w e i g h t loss. V a l u e o l D n o t u s e d . . . . . . . F i r s t 15 p o i n t s low o n p.p. d u e to C a 2÷ c o n t a m i n a t i o n . Value ore not used. . . . . . . Diffusion r u n only. F i r s t s e v e n p o i n t s low on p.p. 0 . 0 2 7 7 - 0.000368 0-624 G a u s s i a n p.p. . . . . . . G a u s s i a n p.p. D i f f u s i o n r u n
only. * A b b r e v i a t i o n pp. indicates p e n e t r a t i o n plot.
The plot of these values of D vs. lIT (Fig. 2) shows excellent agreement with the carefully taken data of Beni~re et al.[2]. This agreement indicates that moisture pickup in the sections is either negligible or is the same in Paris, France and Lemont, Ill. (probably the only thing Paris and Lemont have in common). The agreement with the data of Friauf and co-workers[I,20] is no worse than the agreement between themselves. The parameters are given in Table 2.
Isotope effect Values of E are obtained from plots of In c22/c24 vs. X2/4Dt[1 1, 23] according to the equations E =- (O22/D2,) - - 1
~/24/22- 1
(4)
and
lnCZ2=const+4~--z22t[D22--1).c24 \D24
(5)
The plots of In C22/C24 VS. X2/4Dt (log-ratio plots) shown in Fig. 3 are all linear, as predicted by equation (5), except for a few poiiats at low activity on the low temperature runs, which we attribute to dislocation effects [24, 25]. The scatter in the values of E (Fig. 4) is larger than the error bars, which represent the standard deviations of the slope obtained by a least-squares fit to the lines of Fig. 3. The range of the scatter in E, - 0.08, is about.the same as we found for self-diffusion in zinc [24], and is about the same as the laboratoryto-laboratory reproducibility of the isotope
THE ISOTOPE EFFECT 10-7
~
~
I
I
I 0.96
I 1.00
.
10"8
1065
I I I =,: :PRESENT WORK ----NELSON AND FRIAUF . . . . DOWNING AND FRIAUF 0----0BENIERE
I
.= ~'~u ~ 10.9 Q
o
io-lO
io-iI 0.92
I I 1.04 1.08 103/T, K-I
I 1.12
I 1.16
1,20
Fig. 2. Arrhenius plot (log D vs. 1/T) for the diffusion of Na + in NaCI. The squares denote our 621-6 and 648.1°C points that were not used in the determination of Q and Do.
Table 2. Values o f Do and Q for the diffusion o f N a 4 in NaCl in the intrinsic range Q (eV)
Do (cm2/sec)
Ref.
1.78 ± 0.03 2.23 ___0.08 1 "975 1.80 1.60 2.04 _--_0.02
3.2 790 33.2 3-1 0.5 76.9
20 1 2 21 22 Present work
effect [24, 26]. We cannot decide whether the scatter indicates the presence of experimental errors or whether the isotope effect differs from crystal to crystal because of some physical effect. One possible reason for the large scatter is that the points at 621.6 and 648.1°C may not be valid because the penetration plots are not Gaussian. In this case, the data are best represented by the lowest line of Fig. 4. On the other hand, the curvature of the penetration plots may be in some way connected with the large weight losses in these samples, which should not enter into the isotope effect. Also,
the log-ratio plots for these samples are quite good, and further, these points agree with the value of E obtained by Barr and LeClaire [11]. On the other hand, the two low points .of Fig. 4, for which the penetration plots and the log-ratio plots are excellent, may not represent the isotope effect for volume diffusion in pure NaC1, either because of the presence of divalent cationic impurities, or because of enhanced diffusion along dislocations [27]. The first effect is unlikely inasmuch as a minimum of 10 per cent of the diffusion would have had to take place by vacancies associated with divalent cationic impurities, which is impossible in pure crystals 90°C or more above the conductivity knee.* A lowering of the isotope effect due to Hart-type diffusion along dislocations[27] is somewhat less unlikely. Since the dislocations in NaC1 have a positive charge at high temperatures[28], cation vacancies will be strongly bound to the *Barr and LeClalre(Ref. [11]) found that doping NaCI with about 200 ppm Zn2+ lowered the isotope effect for self-diffusion from 0"72 to 0"56. As our crystals were much purer, we shouldsee a much smallerdecrease.
1066
S . J . R O T H M A N et al. 0.8
I
I
I
! t/~11
0,7 -
<~ o.~
I I
-
_ ~ ~ ~ ' ~
_
0.~ -
o.4
I00
i~
I
750
EXPERIMENTAL PRESENT WORK EXPERIMENTAL, BARR
I
I
700
650
I
600
--
550
T, *C
Fig. 4. Plot of E vs. T. T h e triangle represents the point from Ref.[ 11]. T h e three lines are discussed in the text.
/
/
/ /
I
/ i /
I
¢1 I J I I I I I I I I I I I I I I t I-X2/4Dzz t Fig. 3. Plots o f In czJc24 vs. xnl4Dzzt for the simultaneous diffusion of 2ZNa and Z4Na in NaCI. Each division on the ordinate is 0.01; on the abscissa it is 0-5.
dislocation core, and fdis~oo in NaC1 may be smaller than fa~sloc in silver, 0.47 [25]. If E for diffusion along dislocations in NaC1 is indeed small, equation (1) yields a 10 per cent dislocation contribution to the diffusion co-
efficient in the 589.3 ° sample, which is not inconsistent with the scatter around our Arrhenius plot. In this case, the data are best represented by the top line of Fig. 4. As the above discussion indicates, there is no compelling reason for preferring the points with either high or low values of E. We therefore consider the middle line of the three in Fig. 4 as best representing the data. 4. DISCUSSION
S o m e limiting values The minimum value of the vacancy pair contribution near the melting point, obtained by taking E1 = 0.624, E~ ----0, is 22.5 per cent. This value, because it is a minimum, is in better agreement with the vacancy pair contribution of 40 per cent predicted by Nelson and Friauf[1] than with the I ! per cent predicted by Beniere et al.[2]. The maximum possible value of E_~, obtained by assuming that all diffusion at 796.6°C is by pairs, is 0.485. Unfortunately, we cannot obtain a maximum value of f2 and therefore of (v'alv'c), because we do not know AK2. We estimate that AK2 is not less than 0.8, so < 0.6 and u'a/u'c < 0.5. Using this value of E,, we obtain 53 per cent for the maximum value of the vacancy pair contribution at 589.3°C.
THE ISOTOPE EFFECT
Temperature dependence o f the isotope effect We can place further limits on the vacancy pair contribution. From each of the three lines in Fig. 4, we can calculate a set of values of DJD~, as a function of temperature, for an assumed pair of values of E~ (i.e., AK1) and E2. A least-squares fit of log (D2/DI) to 1/T then gives a value ofAQ = Q2 - Q~, the difference between the activation energies for the diffusion of N a ÷ by pairs and singles, respectively, and a value of Do2/Doa, the ratio of the Do's for the two mechanisms. Using this ratio, AQ and the experimental values of D at the ends of the temperature range, we can obtain the values of Do~ and Q~ corresponding to the values of El and E2 assumed above. We accept only those values of E1 and E2 that (a) give an Arrhenius dependence on temperature of D.,/D~, and (b) yield a plot of log D, calculated from the above values of D01, D0.~, Q1, and Q2 vs. I/T, which is not much more curved than the experimentally determined Arrhenius plot. The values of the parameters acceptable under these conditions are shown in Table 3. N o matter which line of Fig. 4 we choose, only Ez ~< 0.2 yields a good fit according to the above criteria, as well as reasonable values of D2/DI. This means that f2 ~< 0.25, and ~/Jv'c <<-0.1. If we accept the middle line as best representing the data, we get AKI = 0.9, and AQ = 0-82-0-9 eV. The vacancy pair contribution near the melting point is then between 30 and 45 per cent.
1067
Since AQ = Hfl2 + H2m-- H,,-- HB, AQ = 0-85 eV means that HB + H,,,1 -- H,,,2 -~ 0-23 eV. (HB is the binding energy of the pair, the Hm's are the motion energies of singles and pairs, respectively, and we have used 2-17 eV for H~, the energy of formation of a Schottky pair[29].) Thus for Ha to be close to the calculated value, for example, 0-6eV[30], H,n2 > Hml, which is in agreement with the calculations [3]. The large value of Do2/Dol ( ~ 5000) requires some explanation. DoJDol may be written as D°2 = ~a2~u26 exp [(SI--Sb+S""-)/k] O01 4a2f~vl exp [ ( Sfl2 + Sml)/k] '
where the v's are the appropriate vibration frequencies, and the S~'s are entropies corresponding to the enthalpies defined previously. The factor of one-third in D02 is introduced because the cation vacancy can jump to only one-third of its nearest-neighbor positions without separating the vacancy pair; the factor of 6 arises because the vacancy pair may assume six possible orientations. Since f, ~ 6A, 9o2
D01 = const, exp [(Sfl2--S~
Top
Middle
Bottom
AK1
E2
AQ
where the constant is of order unity. Using the experimental quantities ( S r - - S b ) = 10.3k
DozlDo~
(eV) 1.0 1.0 1-0 0.9 0.9 0.9 0"9 0.9
0.1 0-2 0.3 0.0 0-1 0.2 0-0 0.1
0.805 0-861 0-961 0-826 0.853 0.896 0.487 0.508
5x 1.2 × 5-4 x 3.5 × 5.9 × 1.2 × 8-5 x 1.3 ×
(7)
+ S,,2 -- Sin1)/k],
Table 3. Values o f the parameters for the diffusion o f N a + by means o f single vacancies and vacancy pairs in N a C l Line (Fig. 4)
(6)
103 104 104 10a 103 104 101 102
QJ
Dox
DJD~at
(eV)
(cm~/sec)
796-6°C
1.862 1.817 1-714 1.955 1.928 1.887 1-986 1"965
6.4 3-5 0-89 21 "2 14.7 8"38 30.1 22-4
0.771 1.04 1.60 0.454 0.572 0.774 0.454 0-572
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S . J . ROTHMAN et al.
[31, 32], Sin2 = 2"6k[31, 32], S f = 2"44k[29], v'/v" = 0.1 ). Also, if a pair contribution of this and Sml=3.17k[29], we obtain Do2/Dol-~ magnitude were used in the Beniere analysis, 5000, which is in good agreement with our fl would be closer to 0.78 than to 1.0. In any results. case, the combination of a small pair contribuIn this analysis we have ignored the tion and a value of v'~/v"/> 1, suggested by temperature dependence of E2, i.e., we have Beniere et al., is not consistent with our data, assumed v'/v" and Ak2 are temperature unless AK2 ~< 0.3, which seems highly unindependent. If Ez is indeed small, this likely. assumption should not introduce large errors, 5. CONCLUSIONS as we expect V'a/V'c and, hence, E2 to decrease 1. Diffusion of N a ÷ in NaC1 in the intrinsic with a decrease in temperature. F o r example, changing E2 from 0-1 to 0.01 does not affect region is described by Do = 77 cm2/sec, Q = Dz/DI (equation (1)) significantly. The vacancy 2.04 eV. pair contribution near the melting point would 2. The isotope effect for the diffusion o f N a ÷ also remain unchanged. in NaCI decreases with an increase in temThe above analysis also assumes that the perature, indicating the presence of at least N a ÷ diffuses only by means of single vacancies two mechanisms of diffusion; the 'high-temand vacancy pairs. If a third mechanism, such perature' mechanism has the smaller value of the isotope effect. as Frenkel defects[29] or trivacancies[33], 3. Assuming that the diffusion of N a ÷ in makes a significant contribution to diffusion at high temperature, the number of disposable NaC1 takes place by means of single vacancies parameters increases to six, which does not and vacancy pairs only, the best fit to the data allow estimates to be made. Our results ex- is given by clude neither of these mechanisms because a AK (singles) = 0.9, low value of Es is expected both for Frenkel defects if the interstitial diffuses as an interf A K (pairs) ~< 0-2, stitialcy[34], and for vacancy triplets, for which the correlation factor is probably small. 0.82 eV ~< [Q (pairs) - Q (singles)] ~< 0.9 eV, Comparison with the literature Our analysis of the diffusion o f N a ÷ in NaC1 and disagrees with that of Beniere et al.[2], even though the values of D are in agreement. Our 30% ~< D ( N a + due to pairs) D ( N a +) data do not support their proposed value of f ~ = 1-0; this would require ~ ~< 0.7 to agree with the isotope-effect measurements at ~< 45% at the melting point. low temperature. We think that the analysis of Beniere et al. is in error in assuming that Acknowledgements--We thank Sherman Sussman for DNa+ ( p a i r s ) = D c l - (pairs). According to the spectroscopic studies and many useful discussions on impurities in NaC1, Y. Haven for useful discussion, Nelson and Friauf[1], DN~+ (pairs)/Dcl- Larry Nowicki for help with the experiments, and F. (pairs) can vary between 5-6 (v" >> ~,h) and Beniere for a copy of his thesis. 0.179 (z,~ ~ u'). If we multiply the Beniere et REFERENCES al. value of D o - (pairs) by five to obtain DNa+ (pairs), the pair contribution is - 5 0 1. NELSON V. C. and F R I A U F R. J., J. Phys. Chem. Solids 31, 825 (1970). per cent at 796-6°C, in excellent agreement 2. BENIERE F., BENIERE M. and CHEMLA M., with our estimate from the isotope effect. J. Phys. Chem. Solids 31, 1205 (1970); BENIERE (DNa+ (pairs)/Dcl_ (pairs) = 5 corresponds to F., Thesis, Orsay (1970).
T H E ISOTOPE E F F E C T 3. T H A R M A L I N G A M K. and L I D I A R D A. B., Phil. Mag. 6, 1157 (1961). 4. BAKKER H., Phys. Status. Solidi 31, 271 (1969). 5. BAKKER H., Phys. Status. Solidi b44, 369 (1971). 6. BARDEEN J. and H E R R I N G C., In Atom Movements, p. 87, Am. Soe. Metals, Cleveland, Ohio (1952). 7. V I N E Y A R D G. H., J. Phys. Chem. Solids 3, 121 (1957). 8. MULLEN J. G., Phys. Rev. 121, 1649 (1961). 9. COMPAAN K. and H A V E N Y., Trans. Faraday Soc. 52, 786 (1956). 10. BROWN R. C., WORSTER J., MARCH N . H., PERRIN R. C. and B U L L O U G H R., Phil. Mag. 23, 555 (1971). 11. BARR L. W. and LECLAIRE A. D., Proc. Brit. Ceram. Soc. 1, 109 (1964). 12. LECLAIRE A. D., Phil. Mag. 14, 1271 (1966). 13. HOWARD R. E., Phys. Rev. 144, 650 (1966). 14. ROTHMAN S. J. and PETERSON N. L., Phys. Rev. 154, 552 (1967). 15. PETERSON N. L. and ROTHMAN S. J., Phys. Rev. 177, 1329 (1969). 16. ROLFE J., Can.J. Phys. 42, 2195 (1964). 17. ROTHMAN S. J., BARR L. W., ROWE A. H. and SELWOOD P. G., Phil. Mag. 14, 501 (1966). 18. PETERSON N. L., Phys. Rev. 132, 2471 (1963). 19. LEDERER C. M., H O L L A N D E R J . M. and PERLMAN I., Table of Isotopes, John Wiley, New York (1967).
J l ~ $ Vol. 33 No. 5 - - H
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20. D O W N I N G H. L. and F R I A U F R. J., J. Phys. Chem. Solids 31, 845 (1970). 21. MAPOTHER D., CROOKS H. N. and M A U R E R R.,J. chem. Phys. 18, 1231 (1950). 22. L A U R E N T J.-F. and BENARD J., J. Phys. Chem. Solids 3, 7 (1957). 23. SCHOEN A. H., Phys. Rev. Lett. 1, I38 (1958). 24. PETERSON N. L. and ROTHMAN S. J., Phys. Rev. 163, 645 (1967). 25. ROBINSON J. T. and PETERSON N. L., to be published in Proceedings of the International Conference on the Structure and Properties of Grain Boundaries and Interfaces. 26. M U N D Y J. N., Phys. Rev. B 3, 2431 (1971). 27. HART E. W.,Acta Metall. 5,597 (1957). 28. D A V I D G E R. W., Phys. Status Solidi 3, 1851 (1963). 29. A L L N A T T A. R. and PANTELIS P., Solid State Commun. 6, 309 (1968). 30. TOSI M. P. and F U M I F. G., Nuovo Cim. 8, 524 (1958). 31. ECONOMU N. A. and SASTRY P. V., Phys. Status Solidi 6, 135 (1964). 32. ROY D. and GHOSH A. K., J. Phys. C 4, 941 (1971). 33. F U L L E R R. G. and REILLY M. H.,J. Phys. Chem. Solids 30, 457 (1969). 34. PETERSON N. L. and BARR L. W., (to be published).
T H E T E M P E R A T U R E D E P E N D E N C E OF T H E ISOTOPE EFFECT FOR THE DIFFUSION OF N a ÷ I N NaCI* S. J. ROTHMAN, N. L. PETERSON, A. L. LASKAR'~ and L. C. ROBINSON Materials Science Division, Argonne National Laboratory, Argonne, II!. 60439, U.S.A. (Received 5 October 1971)
Abstract--The isotope effect for the diffusion of N a ÷ in Harshaw NaC1 crystals has been measured in the temperature range 589-796°C. The diffusion coefficients o f N a + in NaCI are given by D = 7 7 e x p ( - 2 " 0 4 - - 0 . 0 2 e V \) cm/sec. z, \ The isotope effect, f A K , decreases as the temperature increases, indicating a vacancy pair contribution to the self-diffusion of N a + in NaCI. This contribution reaches 30-45 per cent near the melting point. The temperature dependence o f f A K and D suggestsfAK is ~< 0.2 for N a + diffusion by means of vacancy pairs. This implies that, if AK for diffusion by vacancy pairs is greater than 0.8, f pairs is ~< 0.25, and hence the ratio of C1-jumps by pairs to N a ÷ jumps by pairs must be less than 0-i. Also, a large entropy for N a + diffusion by vacancy pairs is indicated. 1. INTRODUCTION
THE OBJECTIVE of the experiments described below was to find limiting values for the contribution of cation-anion vacancy pairs to the diffusion of N a ÷ in NaCI. The extent of this contribution is at present controversial. Nelson and Friauf[1] obtained a pair contribution of about 50 per cent near the melting point from their combined drift and diffusion measurements. On the other h a n d , the measurements of Beniere et al.[2] combined with the calculations of Tharmalingam and Lidiard [3] indicated that only a few per cent of the diffusion of N a ÷ in NaCl near the melting point is by means of vacancy pairs. The vacancy pair contribution is related to the isotope effect in the following way. For two simultaneously operating mechanisms, the observed value of the isotope effect, E, is given by [4] *Work performed under the auspices of the U.S. Atomic Energy Commission. t Visiting Scientist, A N L. Permanent Address: Physics Department, Clemson University, Clemson, South Carolina.
E -- E~D1 + EzD2 Dl+D2 '
(1)
where the subscripts denote the two mechanisms, in the present case single vacancies, 1, and vacancy pairs, 2. The E's are the values of the isotope effect for each mechanism, and the D's are the diffusion coefficients of the atomic species, in this case N a ÷, by means of each mechanism. Thus if E1 and E2 are known, the pair contribution, D J ( D I + D 2 ) , can be obtained from the measured value of E. Unfortunately, E1 is known only within 10 per cent, and E2 not at all. That is, for any mechanism of diffusion, Ei = fiAKi [5], where f~ is the Bardeen-Herring correlation factor for the mechanism[6], and AK is the factor that accounts for the preexponential in the Arrhenius expression for the atomic jump frequency not being proportional to (m) -112 [7, 8]. F o r NaCI, fl = 0.782[9] and 1-0 AK~ ~> 0.8[10, 11], so E1 is known to ___10 per cent. E2 is more difficult to evaluate. First, AK2 has not been measured or calculated. One can only infer from LeClaire's correla-
1061
1062
S . J . R O T H M A N e t al.
tion between AK and the relaxation around a defect[12] that AK2 is not likely to be larger than AK1. Second, f2 can vary from 0 to 0.782. This is because jumps of both the cation and the anion vacancies in the pair are involved in the diffusion of N a ÷ by a vacancy pair, and thus f~=fs(v'dv~), (see Fig. 1 of Ref. [1])[1,9,13]. Here v~ and v~ are the jump frequencies of the anion and cation vacancies in the pair, respectively. The calculations [3] S. indicate that vat > vc, the data[l], which can be faulted on experimental grounds (the contact between the two crystals in the drift experiments was far from perfec0, give v~ -> v~. It was with the hope of obtaining more data on these questions that the isotopeeffect experiments were undertaken, even though the number of unknown parameters is obviously so great that only limiting values of E2 andf~ could be hoped for. 2. EXPERIMENTAL TECHNIQUES The thin layer sectioning technique was used for the diffusion measurements, and the half-life separation technique[11, 14, 15] for the isotope-effect measurements.
Materials and sample preparation Harshaw single crystals of NaCI were used in all experiments. A flat surface a few degrees off (100) was prepared on the crystals immediately before deposition of the radioisotope. The crystals were not preannealed* because we feared that impurity pickup during the preanneal would introduce more severe imperfections than the anneal would remove. Conductivity measurements on crystals cleaved from the same boule showed that the 'knee' was at about 500°C, as expected for Harshaw crystals[2, 17]. Ultraviolet spectroscopy showed about 10 ppm of O H - . All the 2SNa used in these experiments *The crystal diffused at 796.60C was preannealed at 750°C for 24 hr in an NH4CI atmosphere. The resulting pickup of PbS+[16] caused only a small deviation from linearity in the penetration plot because the diffusion annealing temperature was so high.
was shown to be radiochemicaUy pure by high-resolution gamma-ray spectroscopy. However, the first lots of this radioisotope, obtained from International Chemical and Nuclear (ICN), contained about 300tzg/ml of nonradioactive Ca 2÷. We do not know if the contaminant came from the supplier or was introduced during our processing. Specially purified 22Na was used in subsequent experiments. The 24Na was either bought from ICN, or made by irradiating Harshaw NaCI in the CP-5 reactor; neither contained harmful impurities. The absence of radioactive impurities was verified by high-resolution gamma-ray spectroscopy and by half-life measurements. The radioisotopes were deposited as NaCI on the crystals by evaporation from a platinum filament in vacuum.
Diffusion annealing The crystals were laid with their active faces on a quartz fiat, wrapped in platinum foil, sealed under a vacuum of 2 × 10-~ Torr in previously baked quartz tubes, annealed in resistance-wound furnaces, and air cooled. Corrections were made for heating up and cooling down, and the furnace thermocouples were calibrated as described earlier[18]. We estimate that the quoted temperatures are accurate to ---+I°C.
Sectioning and counting A f t e r cooling, 1-2 mm was cleaved from the lateral sides of the crystals, which were then sectioned on a Leitz sledge-base microtome. Weight losses were ~< 1 per cent during the summer when the indoor relative humidity was high. In the winter, when the indoor relative humidity was low, a humidifier had to be used to keep chip losses down; two samples that were sectioned without use of the humidifier showed chip losses of - 15 per cent. In the few other cases in which the chip losses exceeded 1 per cent, the appropriate corrections were applied. The samples were weighed, dissolved in 1 ml HsO, and counted in 3 by 3 in. well-type
THE ISOTOPE
scintillation counters with lower level discriminators [ 14, 15].
EFFECT
k ~-.'~
I
1063
I I I r I I 683.8-c EACHDIVISIONONABSCISSA=Ix 10-4¢mz CISSA=O.5 x 10"4cm:;:'
Isotope-effect measurements The separation of the radiation from 2~Na and 24Na was accomplished by counting each section at least six times in the 75 hr following sectioning, and fitting the total gamma activity above the lower level discriminator, which was set in the valley below the 0.511 M e V positron annihilation peak of 22Na, to the time elapsed from an arbitrary time zero according to the equation C = c22e -xzzt -4- c24 e-x"--4t.
3. RESULTS
The results on the diffusion of N a + and on the isotope effect are given in Table 1.
Diffusion o f N a + in NaC1 The diffusion coefficients are obtained from the penetration plots, plots of log specific activity (I) vs. (distance X) 2, according to the solution of the diffusion equation for the thin-layer initial condition ~
.
~
> ~ ~, ~o%
"~.,~ "-" ~
~ ~- "-683.8-c ~
2160C !
(2)
Here h~2=5"ll17x10-Tmin-l[19], h~4= 7-708x 10-4min-l[15], and the quantity of interest is c22/c24. H o t sections were diluted to initial counting rates ~< 104/sec, and the usual corrections[I;4] were made for background and dead time. One million counts were taken in most cases. Null effects (aliquots of different activity from the same bottle), run together with the 621.6 and 751.2°C runs, indicated that the isotopic ratio was independent of count rate and also that the errors in counting were close to the errors expected from the counting statistics.
/=10exp--
~_
2"
(3)
A number of the penetration plots (see Fig. 1, 589.3°C line; also Table 1) obtained experimentally were Gaussian as predicted by
I
i
]
I 1 X 2,10"4cmz
I
I
Fig. 1. Penetration plots (log ! vs. x 2) for the diffusion of N a ÷ in NaC1.
equation (3). Samples on which the Ca 2+contaminated 22Na was used (and the 796.6°C run, contaminated with Pb 2+) showed penetration plots concave toward the abscissa (Fig. 1, 683.8 ° line; see also Table 1). When the curvature extended for only the first few points, valid values of both D and E could be obtained from the run. In other cases, when fewer than 10 points were on the straight line, only the D value was used in the analysis; justification of the use of such plots has been given by Rothman et al.[17]. The runs at 621-6 (see Fig. 1) and 648-1°C gave penetration plots slightly curved in the opposite sense. These were the runs with the 15 per cent weight losses mentioned previously; even when the weight-loss correction was applied, the values of D obtained from these runs were about 20 per cent low. We have therefore felt justified in not including these values of D in our least-squares Arrhenius analysis; however, since the log-ratio plots were excellent, and since loss of material during sectioning does not enter into the isotope effect[11], we have used the values of E obtained from these runs.
1064
S. J. R O T H M A N
et al.
Table 1. Diffusion of Na +and the isotope effect Temp. (°C)
Annealing time (see)
D (cm2/sec)
D~2 D2---~--1
E
796-6
3840
1-97 x 10 -s
0-0216___0-000201
0-484
767.1
7086
9.85 x 10 -9
751.2 719.7
10416 23820
5-83 x 10 -9 3-33 x l0 -9
718-8
7200
3.49 × 10-9
683"8
14400
1 - 2 5 × 10 -9
648.1
58200
4 . 3 6 × 10 -1°
633.7 627.4
77400 61200
3 . 3 2 × 10 -19 2-90 × 10 -1°
621.6
70740
1.93 x 10-1o
603-8
66780
1 "37 × 10 -1°
602-7
67320
1.39 × 10 -1°
589-3 587-4
144360 67500
8-75 x 10-11 8-50 x 10-1~
Remarks
F i r s t 6 p o i n t s low o n p.p.* d u e to Pb z+ c o n t a m i n a t i o n . . . . . . . G o o d p.p. Log-ratio plot curved. Value ore not used. 0.0232_--_0.000290 0.522 G a u s s i a n p.p. . . . . . . L a r g e tails o n p.p. a n d logratio plot. V a l u e o f E n o t u s e d . 0 . 0 2 5 5 _ 0-000285 0.574 F i r s t 6 p o i n t s low o n p.p. d u e to C a ~+ c o n t a m i n a t i o n . 0.0281--+0.000398 0-631 F i r s t 9 points low o n p.p. d u e to C a z+ c o n t a m i n a t i o n . 0.0299-+0.000448 0.674 C o n c a v e u p p.p., large w e i g h t loss. V a l u e o l D n o t used. 0.0269-+0.000384 0.604 G a u s s i a n p.p. . . . . . . F i r s t 11 points low o n p.p. d u e to C a ~+ c o n t a m i n a t i o n . V a l u e o r E n o t used. 0-0309___0.000456 0.695 C o n c a v e u p p.p., large w e i g h t loss. V a l u e o l D n o t u s e d . . . . . . . F i r s t 15 p o i n t s low o n p.p. d u e to C a 2÷ c o n t a m i n a t i o n . Value ore not used. . . . . . . Diffusion r u n only. F i r s t s e v e n p o i n t s low on p.p. 0 . 0 2 7 7 - 0.000368 0-624 G a u s s i a n p.p. . . . . . . G a u s s i a n p.p. D i f f u s i o n r u n
only. * A b b r e v i a t i o n pp. indicates p e n e t r a t i o n plot.
The plot of these values of D vs. lIT (Fig. 2) shows excellent agreement with the carefully taken data of Beni~re et al.[2]. This agreement indicates that moisture pickup in the sections is either negligible or is the same in Paris, France and Lemont, Ill. (probably the only thing Paris and Lemont have in common). The agreement with the data of Friauf and co-workers[I,20] is no worse than the agreement between themselves. The parameters are given in Table 2.
Isotope effect Values of E are obtained from plots of In c22/c24 vs. X2/4Dt[1 1, 23] according to the equations E =- (O22/D2,) - - 1
~/24/22- 1
(4)
and
lnCZ2=const+4~--z22t[D22--1).c24 \D24
(5)
The plots of In C22/C24 VS. X2/4Dt (log-ratio plots) shown in Fig. 3 are all linear, as predicted by equation (5), except for a few poiiats at low activity on the low temperature runs, which we attribute to dislocation effects [24, 25]. The scatter in the values of E (Fig. 4) is larger than the error bars, which represent the standard deviations of the slope obtained by a least-squares fit to the lines of Fig. 3. The range of the scatter in E, - 0.08, is about.the same as we found for self-diffusion in zinc [24], and is about the same as the laboratoryto-laboratory reproducibility of the isotope
THE ISOTOPE EFFECT 10-7
~
~
I
I
I 0.96
I 1.00
.
10"8
1065
I I I =,: :PRESENT WORK ----NELSON AND FRIAUF . . . . DOWNING AND FRIAUF 0----0BENIERE
I
.= ~'~u ~ 10.9 Q
o
io-lO
io-iI 0.92
I I 1.04 1.08 103/T, K-I
I 1.12
I 1.16
1,20
Fig. 2. Arrhenius plot (log D vs. 1/T) for the diffusion of Na + in NaCI. The squares denote our 621-6 and 648.1°C points that were not used in the determination of Q and Do.
Table 2. Values o f Do and Q for the diffusion o f N a 4 in NaCl in the intrinsic range Q (eV)
Do (cm2/sec)
Ref.
1.78 ± 0.03 2.23 ___0.08 1 "975 1.80 1.60 2.04 _--_0.02
3.2 790 33.2 3-1 0.5 76.9
20 1 2 21 22 Present work
effect [24, 26]. We cannot decide whether the scatter indicates the presence of experimental errors or whether the isotope effect differs from crystal to crystal because of some physical effect. One possible reason for the large scatter is that the points at 621.6 and 648.1°C may not be valid because the penetration plots are not Gaussian. In this case, the data are best represented by the lowest line of Fig. 4. On the other hand, the curvature of the penetration plots may be in some way connected with the large weight losses in these samples, which should not enter into the isotope effect. Also,
the log-ratio plots for these samples are quite good, and further, these points agree with the value of E obtained by Barr and LeClaire [11]. On the other hand, the two low points .of Fig. 4, for which the penetration plots and the log-ratio plots are excellent, may not represent the isotope effect for volume diffusion in pure NaC1, either because of the presence of divalent cationic impurities, or because of enhanced diffusion along dislocations [27]. The first effect is unlikely inasmuch as a minimum of 10 per cent of the diffusion would have had to take place by vacancies associated with divalent cationic impurities, which is impossible in pure crystals 90°C or more above the conductivity knee.* A lowering of the isotope effect due to Hart-type diffusion along dislocations[27] is somewhat less unlikely. Since the dislocations in NaC1 have a positive charge at high temperatures[28], cation vacancies will be strongly bound to the *Barr and LeClalre(Ref. [11]) found that doping NaCI with about 200 ppm Zn2+ lowered the isotope effect for self-diffusion from 0"72 to 0"56. As our crystals were much purer, we shouldsee a much smallerdecrease.
1066
S . J . R O T H M A N et al. 0.8
I
I
I
! t/~11
0,7 -
<~ o.~
I I
-
_ ~ ~ ~ ' ~
_
0.~ -
o.4
I00
i~
I
750
EXPERIMENTAL PRESENT WORK EXPERIMENTAL, BARR
I
I
700
650
I
600
--
550
T, *C
Fig. 4. Plot of E vs. T. T h e triangle represents the point from Ref.[ 11]. T h e three lines are discussed in the text.
/
/
/ /
I
/ i /
I
¢1 I J I I I I I I I I I I I I I I t I-X2/4Dzz t Fig. 3. Plots o f In czJc24 vs. xnl4Dzzt for the simultaneous diffusion of 2ZNa and Z4Na in NaCI. Each division on the ordinate is 0.01; on the abscissa it is 0-5.
dislocation core, and fdis~oo in NaC1 may be smaller than fa~sloc in silver, 0.47 [25]. If E for diffusion along dislocations in NaC1 is indeed small, equation (1) yields a 10 per cent dislocation contribution to the diffusion co-
efficient in the 589.3 ° sample, which is not inconsistent with the scatter around our Arrhenius plot. In this case, the data are best represented by the top line of Fig. 4. As the above discussion indicates, there is no compelling reason for preferring the points with either high or low values of E. We therefore consider the middle line of the three in Fig. 4 as best representing the data. 4. DISCUSSION
S o m e limiting values The minimum value of the vacancy pair contribution near the melting point, obtained by taking E1 = 0.624, E~ ----0, is 22.5 per cent. This value, because it is a minimum, is in better agreement with the vacancy pair contribution of 40 per cent predicted by Nelson and Friauf[1] than with the I ! per cent predicted by Beniere et al.[2]. The maximum possible value of E_~, obtained by assuming that all diffusion at 796.6°C is by pairs, is 0.485. Unfortunately, we cannot obtain a maximum value of f2 and therefore of (v'alv'c), because we do not know AK2. We estimate that AK2 is not less than 0.8, so < 0.6 and u'a/u'c < 0.5. Using this value of E,, we obtain 53 per cent for the maximum value of the vacancy pair contribution at 589.3°C.
THE ISOTOPE EFFECT
Temperature dependence o f the isotope effect We can place further limits on the vacancy pair contribution. From each of the three lines in Fig. 4, we can calculate a set of values of DJD~, as a function of temperature, for an assumed pair of values of E~ (i.e., AK1) and E2. A least-squares fit of log (D2/DI) to 1/T then gives a value ofAQ = Q2 - Q~, the difference between the activation energies for the diffusion of N a ÷ by pairs and singles, respectively, and a value of Do2/Doa, the ratio of the Do's for the two mechanisms. Using this ratio, AQ and the experimental values of D at the ends of the temperature range, we can obtain the values of Do~ and Q~ corresponding to the values of El and E2 assumed above. We accept only those values of E1 and E2 that (a) give an Arrhenius dependence on temperature of D.,/D~, and (b) yield a plot of log D, calculated from the above values of D01, D0.~, Q1, and Q2 vs. I/T, which is not much more curved than the experimentally determined Arrhenius plot. The values of the parameters acceptable under these conditions are shown in Table 3. N o matter which line of Fig. 4 we choose, only Ez ~< 0.2 yields a good fit according to the above criteria, as well as reasonable values of D2/DI. This means that f2 ~< 0.25, and ~/Jv'c <<-0.1. If we accept the middle line as best representing the data, we get AKI = 0.9, and AQ = 0-82-0-9 eV. The vacancy pair contribution near the melting point is then between 30 and 45 per cent.
1067
Since AQ = Hfl2 + H2m-- H,,-- HB, AQ = 0-85 eV means that HB + H,,,1 -- H,,,2 -~ 0-23 eV. (HB is the binding energy of the pair, the Hm's are the motion energies of singles and pairs, respectively, and we have used 2-17 eV for H~, the energy of formation of a Schottky pair[29].) Thus for Ha to be close to the calculated value, for example, 0-6eV[30], H,n2 > Hml, which is in agreement with the calculations [3]. The large value of Do2/Dol ( ~ 5000) requires some explanation. DoJDol may be written as D°2 = ~a2~u26 exp [(SI--Sb+S""-)/k] O01 4a2f~vl exp [ ( Sfl2 + Sml)/k] '
where the v's are the appropriate vibration frequencies, and the S~'s are entropies corresponding to the enthalpies defined previously. The factor of one-third in D02 is introduced because the cation vacancy can jump to only one-third of its nearest-neighbor positions without separating the vacancy pair; the factor of 6 arises because the vacancy pair may assume six possible orientations. Since f, ~ 6A, 9o2
D01 = const, exp [(Sfl2--S~
Top
Middle
Bottom
AK1
E2
AQ
where the constant is of order unity. Using the experimental quantities ( S r - - S b ) = 10.3k
DozlDo~
(eV) 1.0 1.0 1-0 0.9 0.9 0.9 0"9 0.9
0.1 0-2 0.3 0.0 0-1 0.2 0-0 0.1
0.805 0-861 0-961 0-826 0.853 0.896 0.487 0.508
5x 1.2 × 5-4 x 3.5 × 5.9 × 1.2 × 8-5 x 1.3 ×
(7)
+ S,,2 -- Sin1)/k],
Table 3. Values o f the parameters for the diffusion o f N a + by means o f single vacancies and vacancy pairs in N a C l Line (Fig. 4)
(6)
103 104 104 10a 103 104 101 102
QJ
Dox
DJD~at
(eV)
(cm~/sec)
796-6°C
1.862 1.817 1-714 1.955 1.928 1.887 1-986 1"965
6.4 3-5 0-89 21 "2 14.7 8"38 30.1 22-4
0.771 1.04 1.60 0.454 0.572 0.774 0.454 0-572
1068
S . J . ROTHMAN et al.
[31, 32], Sin2 = 2"6k[31, 32], S f = 2"44k[29], v'/v" = 0.1 ). Also, if a pair contribution of this and Sml=3.17k[29], we obtain Do2/Dol-~ magnitude were used in the Beniere analysis, 5000, which is in good agreement with our fl would be closer to 0.78 than to 1.0. In any results. case, the combination of a small pair contribuIn this analysis we have ignored the tion and a value of v'~/v"/> 1, suggested by temperature dependence of E2, i.e., we have Beniere et al., is not consistent with our data, assumed v'/v" and Ak2 are temperature unless AK2 ~< 0.3, which seems highly unindependent. If Ez is indeed small, this likely. assumption should not introduce large errors, 5. CONCLUSIONS as we expect V'a/V'c and, hence, E2 to decrease 1. Diffusion of N a ÷ in NaC1 in the intrinsic with a decrease in temperature. F o r example, changing E2 from 0-1 to 0.01 does not affect region is described by Do = 77 cm2/sec, Q = Dz/DI (equation (1)) significantly. The vacancy 2.04 eV. pair contribution near the melting point would 2. The isotope effect for the diffusion o f N a ÷ also remain unchanged. in NaCI decreases with an increase in temThe above analysis also assumes that the perature, indicating the presence of at least N a ÷ diffuses only by means of single vacancies two mechanisms of diffusion; the 'high-temand vacancy pairs. If a third mechanism, such perature' mechanism has the smaller value of the isotope effect. as Frenkel defects[29] or trivacancies[33], 3. Assuming that the diffusion of N a ÷ in makes a significant contribution to diffusion at high temperature, the number of disposable NaC1 takes place by means of single vacancies parameters increases to six, which does not and vacancy pairs only, the best fit to the data allow estimates to be made. Our results ex- is given by clude neither of these mechanisms because a AK (singles) = 0.9, low value of Es is expected both for Frenkel defects if the interstitial diffuses as an interf A K (pairs) ~< 0-2, stitialcy[34], and for vacancy triplets, for which the correlation factor is probably small. 0.82 eV ~< [Q (pairs) - Q (singles)] ~< 0.9 eV, Comparison with the literature Our analysis of the diffusion o f N a ÷ in NaC1 and disagrees with that of Beniere et al.[2], even though the values of D are in agreement. Our 30% ~< D ( N a + due to pairs) D ( N a +) data do not support their proposed value of f ~ = 1-0; this would require ~ ~< 0.7 to agree with the isotope-effect measurements at ~< 45% at the melting point. low temperature. We think that the analysis of Beniere et al. is in error in assuming that Acknowledgements--We thank Sherman Sussman for DNa+ ( p a i r s ) = D c l - (pairs). According to the spectroscopic studies and many useful discussions on impurities in NaC1, Y. Haven for useful discussion, Nelson and Friauf[1], DN~+ (pairs)/Dcl- Larry Nowicki for help with the experiments, and F. (pairs) can vary between 5-6 (v" >> ~,h) and Beniere for a copy of his thesis. 0.179 (z,~ ~ u'). If we multiply the Beniere et REFERENCES al. value of D o - (pairs) by five to obtain DNa+ (pairs), the pair contribution is - 5 0 1. NELSON V. C. and F R I A U F R. J., J. Phys. Chem. Solids 31, 825 (1970). per cent at 796-6°C, in excellent agreement 2. BENIERE F., BENIERE M. and CHEMLA M., with our estimate from the isotope effect. J. Phys. Chem. Solids 31, 1205 (1970); BENIERE (DNa+ (pairs)/Dcl_ (pairs) = 5 corresponds to F., Thesis, Orsay (1970).
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