Anomalous diffusion in beta zirconium, beta titanium and vanadium

2330

TECHNICAL

and Cd2+ differ in radius and this fact results in decreasing of the force constants coupling ions in MnS,+ in comparison with Cd&-. (The Cd f, S distance in CdS is 2.61 A and Mn * S distance in MnS is 2.43 A)[6]. The mode, arising in consequence of the sulphur ion’s motion and motionless manganese ions, is able to cause the straight line dependence of A (T) because its frequency is below the upper limit of the acoustic band of CdS. The second straight line dependence of A(T) at higher frequencies is most probably caused by above mentioned T,-local mode. From this analysis it seems to be reasonable to suppose that the mechanism of s-d dynamic mixing is effective even in this case and the measured temperature dependence of A(T) of Mn2+ in CdS caused by two vibrational modes localized on the Mn2+ impurity ion.

Acknowledgements-The authors are highly indebted to D. Nohavica for preparing the CdS-powder used in these measurements.

Institute of Radio Engineering and Electronics, CzechoslovakAcademy of Sciences Prague, Czechoslovakia

K. Z~~ANSKY F. KUBEC

REFERENCES 1. WALSH W. M., JEENER J. J. and BLOEMBERGEN N., Phvs. Rev. 139A. 1338 (1965): ROSENTHAL J., YkRMUS L. z&d BAkTRAM R. H., Phys. Rev. 153, 407 (1967); ROSENTHAL J. and Y+RMUS, L., J. them. Phys. 46, 1217 (1967); ZDANSKY K., Phys. Lett. 24A, 337 (1967); CALVO R. and ORBACH R., Phys. Rev. 164, 284 (1967); CALVO R., Phvs. Lett. 27A. 713 (1968). 2. SlMANEk E.ind ORBACH R:, Phy;. Rev. 145,191 (1966). 3. AbANSKq K., Phys. Status Solidi 28, 181 (1968). 4. WALSH W. M., Jr., JEENER J. and BLOEMBERGEN N., Phvs. Rev. 139A, 1338 (1965). 5. LANDOLY H. and BERNSTEIN R., Phys.-them. Tabellen. Suringer, Berlin (1958). 6. REMY H, Lehrbuch der anorganischen Chemie (Edited by L. Gees and K.-G. Portig). Leipzig (1957).

NOTES J. Phys. Chem. Solids

Anomalous

Vol. 30, pp. 2330-2334.

diffusion in beta zirconium, titanium and vanadium

(Received

beta

15 October 1968; in revised form 6 January 1969)

A NUMBER of attempts [l-3] have been made to explain the non-linearity in the Arrhenius plots of 1ogD vs. l/T in the diffusion studies of p-Zr, /3-Ti and vanadium. The first general interpretation [l] assumed the temperature dependence of diffusivity based on a single mechanism over the entire temperature range. Another explanation is on the basis of dual mechanisms [2,3] operating simultaneously in the diffusion process and thus the diffusivity is expressed as D = DoI exp

(

-$

>

+Do2 exp -& (

>

(1)

where D,, and Q, correspond to the mechanism operating at higher temperatures while Do2 and Q2 are the values for the mechanism at lower temperatures. The values of DoI and QI for the self and impurity diffusion in p-Zr, P-Ti were normal and thus the vacancy mechanism seemed to be operative in these cases while the values of D,, and Qz for the low temperature region (below 1400°C) were found to be low compared to the respective self diffusion values. This led to different speculation for the mechanism of diffusion. According to Kidson[2], the low Do2 and Q2 values could arise as a result of enhancement of diffusivity through extrinsic vacancies by the presence of interstitially dissolved oxygen impurity (especially in p-Zr). LeClaire [4] has shown that a high binding energy for oxygen vacancy pairs is necessary if this model is applicable. The second proposed model [3] assumed the enhancement of lattice diffusivity (especially in P-Zr) due to the presence of randomly oriented dislocations where Kidson

TECHNICAL

[3] used Hart-Mortlock[5, 61 relations and showed qualitatively that if Do2 and Q2 represent the dislocation diffusion parameters, its dislocation density should be at least of the order of 1Og- 1012lines/cm2. Using random walk analysis and HartMortlock [5,6] relation similar to the previous work[7] an attempt has been made to extend it for the quantitative estimation of the enhancement of lattice diffusivity due to the presence of randomly oriented dislocations and to explain the anomalous behaviour in the self and impurity diffusion in /3-Zr, /I-Ti and vanadium. Hart-Mortlock [5,6] relation for self and impurity diffusion in metals is given by (W&J

(self) = g (DJ&)

+I1

(D/D,) (solute) = g(D,/D,,)K+

(2) 1

(3)

where D is the apparent bulk diffusivity, D, is the bulk diffusion coefficient in a crystal free from dislocations, Dd is the diffusion coefficient along dislocation, g is the fraction of the total time that the impurity atom spends in the dislocation and K is the segregation coefficient. Evidently, to calculate quantitatively the enhancement terms, the values of DJD,, K and g should be known. In absence of any data of (D,/D,) for p-Zr, P-Ti and V, the value of (DJD,) as discussed earlier [7] was calculated from the known value of (DJD,) for silver on the assumption that (D,/D,) should be the same for all the other metals at the same ratio of TIT, where T and T, are the absolute temperature for the diffusion anneal and melting points of the solvent. Cottrell has shown that for a small concentration of impurity in a solvent, the segregation coefficient K = exp (-WIRT) where W is the interaction energy of the impurity with dislocation core. The energy of interaction ‘ W’ arises as a result of the elastic and electrostatic interaction of impurity with dislocation and can be written as

NOTES

2331

w=

wl+wp+w3

where WI, W2 are the elastic interaction terms of the impurity atom with dislocation core due to the difference in the size and elastic constants of the solute and solvent atoms respectively while W, takes into account the electrostatic interaction term. Ignoring the term W,, (which is very small and difficult to evaluate), W, and W, have been evaluated according to Friedel[8]. In absence of any data on the dislocation density in these metals, the value of ‘g’ was calculated taking dislocation density of the order of log lines per cm2. The value is definitely high for a well annealed single crystal. In view of the phase transformation of martensitic nature as reported by Fisher and Renken[9] and the possibility of the interlocking of the dislocations due to the presence of the dissolved impurities in p-Zr, p-Ti, a high dislocation density is possible. Even in vanadium an evidence of a polymorphic transformation [ lo] is available which also gives rise to high dislocation density. The enhancement terms (DJD,)g and (Dd/ D,)gK for the self and impurity diffusion in p-Zr, P-Ti and V have been calculated at different temperatures and a few values are given in Table 1. It is observed that the enhancement terms for the self and impurity diffusion in p-Zr, P-Ti and V is always greater than 1 below 1400”, 1300” and 1450°C respectively (Table 1). At higher temperatures (greater than 1450°C up to the melting point of the solvent) the contribution of the enhancement term due to dislocation becomes less pronounced and only lattice diffusion through normal vacancies seem to be operative. Hence it may be concluded that below the temperature of 1400°C or so, the diffusivity in P-Zr, p-Ti and V is enhanced by the presence of the randomly oriented dislocation and the experimental values of Do and Q obtained in the lower temperature region would thus represent the results along dislocations. Moreover, if diffusion along

2332

TECHNICAL

NOTES

Table 1. Enhunceme~t terms~or the self and impurity disunion in beta zirconium, beta titanium and vanadium at few temperatures TemperaSolvent

+!

ture (“C)

v zrr141

Cb[14]

Cr

(“C)

P-Zr

p-Ti

1600 1650

Zr[14] 4.31 x 102 1.94 x lo* 1.96 1-29 0.87 0.30 0.22

900 950 1350 1400 1600 1650

Ti [20] 1.00 x 102 4764 0.68 0.46 0.12 0.09

900 950 1350 g

Cb[14] 3.03 x 103 1.26 x IO3 8.04 S-08 3.28 1.02 0.72 Cb [20] 80.25 38-61 0.58 0.40 0.11 0.08

v t211

V

800 850 1450 1500 1850

4.09 x 103 1.56 x 103 1.16 0.78 0.09

Fe1211 2.25 x to* 2.09 x 104

Cr[15] 7.19 x 104 2.70 X 103 79.28

Mo[l6] 7.45 x 10s 2.98 x lo3 15.39 9.82 6.18 1.83 1.28

Cr 1201 4.05 x I03 166x 18 990 6.21 1-22 0.86

MO [20] 2.64 x 102 1.20 x 102 1.37 0.91 0.22 0.16

Cr[Zl] I.83 x lo4 l-72 X 10’

_-. Sn[17] 1-60x 103 6.85 x lo2 5.08

1.80 x 104 7.03 x 103 29.17

Sn [20] 1.51 x 104 5.88 x 103 2565 15-57 2.78 1.91

V 1201 8.85 x 102 3.85 X 102 3.30 2.10 0.47 0.34

VI181

&D91

Ce[16] l-33 x 109 1-27x 105 5.21 x 103 4.17 x lo5 23.40 6.33 x lo” 3.52 x 102 2.01 x 102 45.00 28.30 Mn [20] 3.35 x 103 1.37 X 103 8.24 540 1.os 0.76

Fe [20] 3.55 x 103 1.47 x lo” 8.97 5.67 1.13 0.79

co [20] 5.14 x I03 2.09 x lo3 11.74 7.07 1.42 0.99

Ni 1201 5.04 x 103 199 x 103 11.63 7.21 140 0.98

TECHNICAL

NOTES

2333

Table 2. Self difusion parameters in beta zirconium, beta titanium and vanadium DO Lattice p-Zr

Q

(cm2/sec) (kcal/mole) (below 1400°C) 8.50 x 10-S 3.58X 10-4

0.36

21.70 31.20 73.65

DO

Q

(cm*/sec) (kcal/mole) (above 1400°C) 1.34 1.09 2.14x 102

dislocation is predominant, the plots of log D vs. l/T in the lower temperature region should give straight lines. It is interesting to note that the self and impurity diffusion in p-Zr, /3-Ti and V in the lower temperature region follow a simple Arrhenius relation. Furthermore the present results of quantitative analysis on the self and impurity diffusion in these systems also agree well with the proposed dual mechanism model of Kidson [3]. It was considered desirable to evaluate D,, & and Qv (diffusion parameters in dislocation free single crystals). D, was calculated using equations (2) and (3), and & and Q, were evaluated from the slopes of the plots of log D, vs. l/T. In all these plots, fairly good straight lines are observed. A representative graph for the impurity diffusion in p-Zr is given in Fig. 1. For comparison, the least square values of & and Qv for the self diffusion in p-Zr, /3-Ti and V along with the experimental values are given in Table 2. It is interesting to note that the least square values of QUfor the self and impurity diffusion in p-Zr, P-Ti and V satisfy the empirical relations of the activation energy with melting point within 20 per cent. Therefore Qv in general represent the activation energy corresponding to diffusion through normal lattice vacancies in dislocation free single crystals. The calculated value of D,, in the case of vanadium (Table 2) is pretty high. Normally the values of D for face centred cubic systems, where vacancy mechanism is operative, lie within 0.1-10 cm2/sec, still a number of cases [I 1,121 have been reported where D,, is as high as 6.3 X lo7 cm2/sec [ 131.

65.20 60.00 94.14

4,

Q,

(cm*/sec) (kcal/mole) (calculated) 0.73 3.80 1.42 x lo4

61.23 62.15 114.25

[I41

WI Pll

-;b

Cb

Q

I%

Fig. 1. Temperature dependence of ‘D,’ for impurity diffusion in@zirconium.

In spite of some of the assumptions made in the above discussion, because of nonavailability of the experimental data, it seems reasonable to assume that the apparent volume diffusivity is enhanced due to the existence of randomly oriented dislocations. Bhabha Atomic Research Chemistry Division, Trombay, Bombay, India

Centre,

M. C. NAIK R. P. AGARWALA

2334

TECHNICAL REFERENCES

I. GIBBS G. B.,Acta Metall. 12,1303 (1964). 2. PEART R. F. and ASKILL J., Phys. Status Solidi

23,263 (1967). 3. KIDSON

V., Proc. Int. Co& Gatfinbura. Am. Sot. Met. (1964). ” LECLAIRE. A. D.. Proc. ilnt. Co&. Gatlinbure.LI * Term., p. 1. Am. Sot. ‘Met. (1964). HART E., Acta Metall. 5,597 (1957). MORTLOCK A., Acta Metall. 8, 132 (1960). NAIK M. C. and AGARWALA R. P., Acta Metall. 15,1521(1967). FRIEDEL J., Dis/ocarions. Wesley, London (1964). FISHER E. S. and RENKEN C. J., Phy. Rev. 135A, 482 (1964). SEYBOLT A. U. and SUMSION H. T., J. Mets/s 5, 292 (1953). ANDELIN R. L., KNIGHT J. D. and KAHN M., Trans. Am. Inst. Min. Engrs 233, 19 (1965). SPARK B., JAMES 13. W. and LEAK G. M., J. Iron Steel fnst. 203,152 ( 1965). VASILEV V. P. and CHERNOMORCHENKO S. G., Zav. Lab. 22,688 (1956). FEDERER J. 1. and LUNDY T. S., Trans. Am.

Term.,

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

G.

p. 329.

Inst. Min. Engrs 227,592 (1963). R. P., MURARKA 15. AGARWALA ANAND M. S., Trans. metall. Sot.

S.

P.

AIME

and

233,

986(1965). M. S., NAIK M. C. and 16. PAUL A. R., ANAND AGARWALA R. P., Intl Conf. Vacancies Interstitials in Metals. Julich (I 968). V. S., RYABOVA 17. GRUZIN P. L., EMELYANOV G. G. and FEDOROV G. B., 2nd lntl Co& Peacefui Uses ofAtomic Energ_y, Geneva 19,187 (1958). R. P.. MURARKA S. P. and 18. AGARWALA ANAND M. S.,Acfa Metall. l&61 (1968). 19. NAlK M. C. and AGARWALA R. P., Proc. Nucl. Radiation Chem. Symp.. p. 282. Waltair, India (1966). D., Proc. lntl Conf. Gatlinburg, Term., 20. GRAHAM p. 27. Am. Sot. Met. (1964). 21. PEART R. F.. J. Phys. Chem. So/ids 26, 1853 (1965).

J. Phys. Chem. Solids

Vol. 30, pp. 2334-2336.

Compositional ~rn~~ation garnets

points in magnetic

(Received 12 November 1968)

VERY RECENT Mossbauer-effect spectroscopy measurements performed by Czerlinsky[l] on several Ga-substituted yttrium-iron garnets determined the distribution of the Ga3+ ions over the tetrahedral and the octahedral sites.

NOTES

The experimental points of the fractional tetrahedral Ga ions, ft = x/t, as a function of the total Ga substitution, 1, in the formula are in excellent agreement Y,Ga,Fe,_,Q, with the theoretical curve derived from a previous treatment on the thermodynamics of solid solutions [2], t = [2 f 3C -(C - l)x]x/ [3C - (C - 1)x], with the value of the energy factor, C = 6.5. Since the energy factor C is related by the formula C = exp {- [be,-- A+,]/kT) to AQ,, and Acre, respectively the difference of the stabilization energy between tetrahedral and octahedral sites of the diamagnetic M cation and of Fe3+, it gives a measure of the tetrahedral preference of M in iron-substituted garnets. The chosen value is interesting if compared with C = 3-O for A13+ substitution[2], which has a weaker tetrahedral preference than Ga3+[3-63. We wish to point out in this note that also the magnetic moment compensation compositions of such garnets give a measure of the tetrahedral preference of diamagnetic substitutions in YIG and that this can easily be predicted by the theory. En fact, the magnetization compensation condition in a mono-substituted garnet {Y,} (M,Fe,_,) x = (1 + t)/2, once subW--sFeZ--t+s1012r stituted in the distribution equation provides the two roots: t,,,v. = (C+ I)/(C- 1) and t = 5; the first root is the compensation composition, the second one represents obviously the total diamagnetic substitution. The minimum value of t,,,,,. is 1 and is obtained for C = ~0 tetrahedral substitution); the (exchtsive maximum value, t = 5, is obtained for C = 312 but large t values require further discussion, as pointed out previously [2]. The value C = 6.5 leads to fcomP.= 1.36 for Ga-substituted garnets in very good agreement with t = 1-3 as reported by Geller et al. f7] and by Li.ithi@]. The value C = 3 leads to t camp.= 240 for Al-substituted garnets in good agreement with t = 2.1 as measured by Geller et al.[3]. In Fig. 1 the curves fi vs. t