THE MECHANISM OF LOW TEMPERATURE MECHANICAL RELAXATION IN DEFORMED CRYSTALS BY A. SEEGER, H. DONTH AND F. PFAFP lnstitut fur theoretische und angewaiidte Physik der Technischen Hochschule, S t ut tgart Max-Planck-Institut fur Metallforschung, Stuttgart, Germany Received 30th January, 1957 A brief discussion of the three types of mechanisms by which dislocations cause energy losses in crystals (hysteresis, resonance, relaxation) is given. The experimental evidence, in particular for the relaxation mechanism, is surveyed. A qualitative description is presented of the mechanism for the so-called Bordoni relaxation peak, which is thought to be due to dislocations overcoming the Peierls stress by thermally activated kink formation. The results of detailed calculations on the rate and the activation energy of such a process are also reported.The treatment does not employ the conventional Arrhenius equation, but derives the relaxation time from the theory of stochastic processes. It takes explicit account of the thermal stresses and the radiation losses governing the dislocation movement. The theory is compared with experimental results on metals and with dielectric measurements on quartz. 1. INTERNAL FRICTION MECHANISMS INVOLVING DISLOCATIONS In 1940, Read 1 in a series of internal friction experiments on slightly deformed single crystals of copper and zinc was able to demonstrate convincingly that dis- locations give rise to mechanical energy losses in crystals. Fig. 1 shows the ampli- tude dependence of the internal friction of copper single crystals at room tem- perature as a function of prestress.Since the number and the arrangement of the dislocations vary with prestress, these results can be interpreted in terms of energy losses caused by dislocations. Although this work and similar investigations on the dependence on crystal orientation and on the recovery of stress-induced internal friction of crystals had very early established the importance of dislocations (see Seitz’s book,2 1943), it was not until recently that a detailed picture of at least some aspects of the dis- sipating mechanism due to dislocations began to emerge. In Zener’s classical book (1948) on anelasticity of metals 3 dislocations are barely mentioned as con- tributing to energy dissipation, and even in 1952 Wert4 stated that the internal friction measurements on plastically deformed crystals have failed to tell us how the dislocations involved were arranged and anchored, and with what types of dislocations we are dealing.Two main reasons are held responsible for the slowness of progress in the field. First, the internal friction of single crystals is sensitive to static stresses even below the critical shear stress 70, as shown by fig. 1. (For copper crystals of the purity employed by Read : 1 TO rn 100 g/mm2). As reported by most experi- mental investigators internal friction measurements on metal single crystals are therefore extremely sensitive to handling damage. Secondly, the internal friction depends on the amplitude 2 of the alternating strain employed in the measurement (fig.1) well below the critical shear strain EO (in metal crystals, EO = 10-5-10-4). Since the theory of dislocations was developed mainly to account for the plastic properties on a stress level T 2 70, the internal friction experiments involved dis- location properties on which no detailed theories were available. On the other 1920 LOW TEMPERATURE MECHANICAL RELAXATION hand the damping experiments provide an excellent method of studying dis- locations on a stress level to which there is no other easy access. According to Kempe and Kroner5 the mechanical energy losses caused by the movement of dislocations which have been discussed in the literature, can be grouped as follows : (i) hysteresis losses (Nowick,6$7* 8 Weertman 9), (ii) resonance losses (Koehler lo), (iii) relaxation losses.I 2 5 Strain amplitude F FIG. 1.-Internal friction decrement A of copper single crystals (99.998 %) at room temperature after compressional prestressing (resolved shear stresses T). The loads had been applied for 1 min.1 The great majority of the energy losses by dislocation processes, including those shown in fig. 1, is of the hysteresis type. Typical hysteresis processes are, e.g., those in which dislocations are torn away irreversibly from pinning points by the applied stress. Such pinning points may be provided, e.g., by foreign atoms or other point defects,lo-l4 * and by other dislocation lines intersecting the area swept out by the moving dislocations.16-18 Tt can undergo resonance if the frequency f of an applied alternating shear stress coin- cides with the characteristic frequencyfo of the dislocation string.Plotted as a function of frequency the internal friction should go through a maximum at .fo (or near to the maximum of a distribution of resonance frequencies). The dis- * The mechanism by which impurity atoms pin dislocation lines is still under discus- sion, since-as pointed out by Weertman and Salkovitz 11-the interaction energy between individual substitutional foreign atoms and dislocation Iines is in general too small to pin the dislocation lines effectively at room temperature, at which most experiments have been made. Furthermore Baker 15 has recently shown by internal friction experiments employ- ing a small superimposed biasing static stress that some of the " break-away ¶ ¶ effects in the amplitude dependence of the decrement cannot be due to impurity pinning.A dislocation line fixed at two points behaves somewhat like a string.A. SEEGER, H. DONTH AND F. PFAFF 21 location loops which are thought to be present in crystals are so short, however, that fo is in general larger than the frequencies accessible with present megacycle techniques.19 There is, however, experimental evidence 20 that in attenuation experiments in germanium up to f ;= 300 Mc/sec one is approaching the resonance frequency fo. A typical relaxation process in crystalline solids is the Snoek process of carbon atoms in a-iron (see, e.g., ref. (4), (8)). Its characteristic features are as follows : in thermal equilibrium the physical system under consideration passes with a tem- perature-dependent mean frequency v from one configuration of minimum energy through a saddle point configuration to one of several other minimum configara- tions.An applied stress favours in general some of these minimum positions with respect to others. If we apply an alternating stress of frequency f, well- known arguments show that the energy dissipation is a maximum if the two frequencies f and v are equal. We know of only one process in crystals which involves dislocations only and which shows all characteristic features of a relaxation mechanism, namely that discovered in various f.c.c. metals by Bordoni.21. 22 We shall discuss the experi- mental evidence for this dislocation relaxation mechanism in 5 2. In passing, we mention that there are also relaxation mechanisms involving both dislocations and impurity atoms, for instance the one found by the Ke-method at about 220" C in cold-worked iron containing carbon or nitrogen.23--27 2.EXPERIMENTAL EVIDENCE FOR THE DISLOCATION RELAXATION MECHANISM The Bordoni peak, i.e. a maximum in the internal friction measured at a fixed frequency f as a function of temperature T, has been observed in various mono- crystalline and polycrystalline f.c.c. metals, namely, copper,22.28929~ 17 alu- minium,30.22 silver,22 and lead,319 329 33.349 22 For the frequency range normally employed (f- 3 x 104 c/s) the temperature of the peak ranges from about 35" K (lead) to about 100' K (aluminium). The shift of the peak with frequency cor- responds to activation energies H ranging from about 1000 cal/mole (lead) to about 4000 cal/mole (aluminium).This is of an order of magnitude which is rather unusual in solid-state physics. The frequency factor f0 in the relation f =f" exp (- H/kT) between the frequency f employed in the experiment and the absolute temperature T of the peak is also of an unusual magnitude, namely, about 109 sec-1. The height of the peak may be undetectably small in well-annealed, undeformed single crystals 153 17 but increases rapidly with small prestrains (see fig. 2, 3, 4). At larger strains the height of the peak appears to decrease somewhat with further straining.29 It does not anneal out at temperatures where point defects are known to anneal out, but only at temperatures at which the specimen presumably re- crystallizes (fig.4). As Niblett and Wilks 29 point out this shows that the peak is caused by dislocations. The experimental fact that the position of the peak is independent of prestrain (see fig. 2, 3, 4) and independent of the impurity con- tent (fig. 5) shows that we are dealing with an '' intrinsic " dislocation effect 35 which does not depend on a particular length of dislocation lines or on their interaction with impurity atoms. The measurements of Niblett and Wilks revealed that in both polycrystals 28.29 and single crystals 36 of copper the Bordoni peak is accompanied by a second, smaller peak occurring at lower temperatures (see fig. 2, 3,4, 5). The most natural explanation for this is that the two peaks are caused by two types of dislocations.* The relaxation character of the Bordoni process is borne out by the way in * Presumably both peaks are due to dislocations running along the (1 10)-direction, one of them belonging to screw dislocations and the other one to dislocations with an angle of 60" between the Burgers vector and the direction of the dislocation line.3522 LOW TEMPERATURE MECHANICAL RELAXATION which the internal friction depends on temperature for a fixed frequency (see fig.2, 3, 4), particularly in the form of eqn. (2.1), which is found to hold approxim- ately if the measurements are made in a sufficiently narrow temperature range. Further, it is in agreement with the relaxation character that both the height and the position of the peak depend only slightly, if at all, on the strain amplitude.36~ 17 0 50 100 I so 2 0 0 2 5 0 Temperature [OK] FIG. 2.4nternal friction of polycrystalline copper (99.999 %) prestrained 0.1 %, as a function of temperature, showing two Bordoni peaks (f= 1100 clsec).The resonance coefficient Q is related to the logarithmic decrement A, the energy loss per half cycle AE, and the total vibrational energy E according to l/Q = A/n = AE/2rE. Fig. 2 to 5 are due to Niblett and Wilks. 0 5 0 100 i 50 200 2SO Temperature la#] FIG. 3.-Internal friction of pure polycrystalline copper (99-999 %) after prestraining and annealing (f= 1100 c/sec). It may be seen from fig. 2, 3,4 and 5 that the relaxation peaks are superimposed on a background which also depends on the prestrain, but in a different way. The annealing behaviour and the response to neutron-irradiation 36 differ also for the peaks and the background.The background is presumably due to a disloca- tion-hysteresis mechanism the details of which, however, are not yet known. In the analysis of Nowick's 7 data on this background in copper single crystals, an activation energy of about 1500 cal/mole is found. In view of the implicationsA. SEEGER, H. DONTH AND F. PFAFF 23 for the theoretical interpretation it would be very important to check experiment- ally whether there is a close relation of this value to the activation energy of the Bordoni process as is suggested by the numerical values. $A;:/z/ 'after annealing lh at 350°C 1 0 S O 100 150 2 0 0 2 50 Temperature PK] FIG. 4.--Internal friction of pure polycrystalline copper (99.999 %) after prestraining and annealing (f = 1100 cisec).I 0 5 0 I00 I50 200 2SO Temperature ["K] FIG. 5.--Internal friction of impure copper as a function of temperature. 5.5 % ; impurity content 0.0026 % Bi, 0.032 % P; f = 1100 c/sec. Strained We close with the remark that in crystalline quartz a similar internal friction peak has been observed at about 20" K 37 and has been attributed to the same cause as in metals.34.37 Since no deformation experiments have been made on quartz this correlation is still open to experimental verification. We shall return to a short discussion on quartz in 4 5. 3. QUALITATIVE THEORY OF THE DISLOCATION RELAXATION MECHANISM A dislocation relaxation mechanism which is in accord with the experimental facts summarized in 5 2 has been described elsewhere.38~ 39.35 We may therefore confine ourselves to a brief summary. Consider a dislocation line which runs24 LOW TEMPER AT U R E ME C IH A N I C A L RELAX AT I 0 N parallel to one of the close-packed directions of its glide plane.In order to move such a dislocation line by one interatomic distance without the aid of thermal fluctuations a resolved shear stress T must act on the dislocation line which is equal to the Peierls stress 7; at 0" K. (For discussions of the Peierls stress or, as it is often called, the Peierls-Nabarro force, see ref. (40), (41), (42), (38).) At finite temperatures the dislocation line will no longer lie in only one valley of its potential energy surface but will contain kinks,43,44 i.e. it will change occasionally from one valley to a neighbouring one (fig.6). Jf a stress T < T; is applied the X b +.- -. - 'Y - - . _ I t - I - - i - - - + - + - - + + - - - , - - - + - * -3a -2a -a , a 2a 30 * FIG. 6.-The potential energy surface of a dislocation line due to the Peierls stress T;. The figure is not drawn to scale, since in reality Eo $ (Tiab/n) and w & a. dislocation line can move forward by two processes : sideways movement of the kinks (which requires very small stresses only), and formation of new pairs of kinks. The formation of a pair of kinks of opposite sign requires thermal energy and occurs therefore with a temperature dependent frequency. If the frequency f of the applied stress is large compared with the frequency v with which a pair of kinks in a dislocation line is formed, kink formation contributes nothing to the strain and to the energy dissipation. I f f is very small compared with the frequency of thermal formation of kinks the kinks are always in thermal equilib- rium and no energy losses occurs either.Only if the two frequencies are approxim- ately equal do we get a contribution to the internal friction of the crystal. This process therefore fulfils all the requirements of a relaxation process, and we believe that it is responsible for the Bordoni peaks. The quantitative treatment of this dislocation mechanism, however, is much more difficult than that of the Snoek mechanism mentioned in $2. The difficulty is due to the fact that the system passes from the initial state (dislocation line lying in a single valley) to the final state (dislocation line containing two kinks of opposite sign which are sufficiently separated in order to be stable) not throughA.SEEGER, H . DONTH AND F. P F A F F 25 a well-defined saddle point configuration. Since the dislocation line is flexible there is a variety of ways by which the final state can be reached. The standard derivation (see, e.g. ref. (45)) of the Arrhenius eqn. (21) from the Eyring-Zener theory of reaction rates is therefore not applicable. In not too complicated situations there are two ways of treating rate processes.* The first one employs Boltzmann distributions of free energies. The action of thermal stresses and the damping due to the radiation losses 46 is then summarily included.The latter statement is illustrated by the standard treatment of the Snoek process by the rate theory.45 There is, for instance, no need to introduce explicitly the radiation of waves from the carbon atom oscillating in its potential well since these waves are part of the thermal equilibrium and are therefore already allowed for in the Boltzmann expression. The second way introduces explicitly the forces exerted by the thermal stresses on the dislocation line (in the problem under discussion) and must then also include the damping mechanism, in our particular case the radiation of elastic waves from accelerated dislocation lines. Donth 47 has treated the present problem of thermally activated kink formation in this way, which in complicated cases is the only reliable one.For the details of Donth’s method which is believed to be useful also in other problems, reference must be made to Donth’s paper.47 We shall give a brief outline in 5 4. 4. SUMMARY OF THE QUANTITATIVE THEORY The shape of the dislocation line y(x, t ) (see fig. 6, t = time), lying in the xy-plane (nearly parallel to the x-direction) is to a good approximation determined by the differential equation (for more details see ref. (35)) s y 32y 2VY 3x2 3t2 a Eo- - m- = bri sin- - br. (4.11 (Eo and in = energy and mass per unit length of the dislocation, b = dislocation strength, r = resolved shear stress, n = period of the dislocation energy in the crystal, ri = “ ideal ” Peierls stress in the absence of thermal or quantum- mechanical activation). Starting from eqn.(4.1) it can be shown 35 that the energy of a kink is that the kink width w (see fig. 6) is given by and that the separation d,, of (unstable) equilibrium of the kinks under the stress ~ ( r < ri) is A study of the solution of eqn. (4.1) for7 = 0 shows 47949*50 that there are “ normal modes ” of the dislocation movement which can be characterized by their energy W (per wavelength). They are ordinary harmonic waves if W e 2W,, but are related to pairs of kinks of opposite sign if W w 2Wk. The exchange of energy between these modes can be represented by a model in which particles (labelled by the co-ordinate W ) diffuse under the influence of thermal shear stresses and * One of the authors (A. S.) gratefully acknowledges discussions on this topic at and after the Lake Placid (1956) Conference on Dislocations and Mechanical Properties of Crystals, in which particularly G.Leibfried and K. Lucke participated. Further results, particularly on higher terms in eqn. (4.4) and allowing for a second sin term in eqn. (4.1), are given elsewhere.4826 LOW TEMPERATURE MECHANICAL RELAXATION radiation daniping.47 The generalized diffusion equation representing this process is solved, subject to the boundary condition that kinks of opposite sign separated by a distance larger than d,, are independent of each other, since they are torn apart by the applied stress T. The final result is an expression (which cannot be written in closed form) for the mean frequency v with which a dislocation leaves the potential wells of the E(y)-surface shown in fig.6 . FIG. 7.-The function Fl(r ; a) of eqn. (4.5). It is related to the function n(r) employed by Donth 47 by the equation Fl(r ; E) = - In 2vn(r). The frequency v and activation energy H can be found as a function of tem- perature from the equations : In (v/B) = Fl(r; a), (4.5) and (4.5a) (4.6) Here v is the appropriate sound velocity (for screw dislocations in an isotropic medium, the velocity vt of transversal waves) ; fi(r ; a) are functions shown in fig. 7 and 8, which depend on the parameters r = 2Wk/kT, (4.7) and (Tn most cases a w 1.) (4.8)A . SEEGER, H . DONTH AND F . PFAFF 27 Using rate theory in a similar way as Mason 33 it can be shown that the decre- ment Amax at the temperature of the relaxation peak is given by 48 where (4.9a) Here NO denotes the number per unit volume of dislocation loops (of average length L) contributing to the relaxation process; A is the area swept out by one dislocation during the process, and G the shear modulus of the crystal.Putting / / employed A = La gives us a lower limit forp and (Q-l)max (provided L/a is sufficiently large), since some of the dislocations may move by more than one interatomic distance. An upper limit for (Q-l)max can be obtained from the assumption that each dislocation loop sweeps out the largest possible area which is compatible with the applied stress, the line energy Eo, and the loop length L. The upper limit is (4.10) 5. COMPARISON wm EXPERIMENTAL DATA The only unknown parameter in the theory of § 4 is the " ideal " Peierls stress T;. In principle, the precise measurement of the temperature of the Bordoni maximum at a given frequency suffices to determine 7;.There are, however, practical difficulties, since the frequency depends exponentially on temperature and since the temperature of the peak is often not well defined. It is therefore sometimes preferable to use eqn. (4.6) and to determine the activation energy H28 LOW TEMPERATURE MECHANICAL RELAXATION (which itself is weakly temperature dependent) from the slope of the usual lnf against l/kT plot, if a sufficient number of experimental points is available. in order to test the theory as critically as is possible at present we have cai- culated, in table 1, T; separately from each measurement on metals which is known TABLE 1 metal and shear modulus G (d y neslcmz) lead 0.8 x 1011 aluminium 2.7 X 1011 silver 2-9 x 1011 copper 4-5 x 1011 -EVALUATION OF EXPERIMENTAL DATA ON THE BORDONI PEAK IN METALS author Bordoni 22 Welber 32 Bommel (after Mason 33) Bordoni 22 Hutchison- Filmer 30 Bordoni 22 Niblett- Wilks 28 Bordoni 22 f = v (sec-1) 10,334 10,163 10,348 20,500 29,500 50,000 10.1 x 106 26.5 x 106 38,665 39,600 5 x 106 31,332 400 400 1,100 1,100 30,300 30,450 28,260 28,600 37,992 T (“W 36 36 36 43 47.5 50 120 140 112 97 155 62.5 61 65 68 74 82 80 91 87.5 97 r 7.3 7.3 7.3‘ 7-05 6-9 6.7 4.5 4.1 7.6 7-5 5.4 7.1 9-25 9.3 8.85 8.85 7.4 7.35 7.5 7.45 7.35 H mole (e!) 1.2 1.2 1.2 1.4 1.5 1.5 2.4 2.6 3.9 3-3 3.7 2.0 2.6 2.8 2.8 3.0 2-75 2-7 3-1 3-0 3.0 % (23 0.28 0.28 0.28 0.37 0.44 0.46 1.2 1.3 2.8 2.1 2-75 0.7 1.5 1.7 1.7 2.0 1.7 1.6 2.2 2.0 2.1 T ; ~ G x 104 3.5 3.5 3.5 4.6 5.4 5.6 14.6 16.5 10.3 7.5 10.0 2-3 3.3 3.7 3.7 4.4 3.8 3.5 4.8 4.4 4-6 to us.* It will be seen that the various measurements give about the same ri- values for both copper and aluminium.Best values for copper appear to be ri/G = 4.0 x 10-4 and w/a = 48. There appears to be a systematic deviation in the sense that the measurements at higher frequencies give larger Peierls stresses. At the moment we do not know whether this discrepancy is partly due to experimental inaccuracies. There are, however, several possibilities which might give rise to deviations from the theory as pre- sented here; 35 one of them is that higher harmonics may be required for a satis- factory representation of the potential energy surface E(y) (cp.§ 4). From the decrement at the peak information on the number of dislocation per unit volume NO and on the loop length L can be derived, in particular as a function of prestrain. If we confine ourselves to very small deformations (see fig. 2), we find that the observations can be accounted for by values of the order of No = 10+12 cm-3 and L = 2 x 10-5 cm. These are reasonable magnitudes. The magnitude of NO shows that only about 1/100 of the total number of loops in the crystal contributes to the peak. This is in agreement with our model, which assumes that only dislocations lying nearly parallel to a close-packed direction take part in the relaxation process. * We used rn = Eo/vt2 and EO = Gb2/2. The latter relation was found to hold ap- proximately for the present situation after a lengthy calculation for screw dislocations in copper carried out by H.Donth along the lines indicated by Schoeck and Seeger.51 In f.c.c. metals : (2/2/3)a = b = distance between nearest neighbours. For lead, the agreement is less good.A. SEEGER, H . DONTH AND F . PFAFF 29 We can summarize the discussion on metals by saying that the theory accounts rather well for the majority of observations. It allows one to deduce numerical values of ri from internal friction experiments. The magnitude of 7-i is found to be in rather good agreement with theoretical predictions.52~ 38 We conclude with a few remarks on quartz, which exhibits also a low-temper- ature mechanical relaxation, as already mentioned in § 2.Its interpretation as a Bordoni process 37 along the lines which have been shown to be rather successful for metals, presents some difficulties, however. Experimentally 37 eqn. (2.1) seems to be much better satisfied than our theory can account for. Since it seems unlikely that prestraining experiments will decide this problem in near future it is tempting to resort to measurements of dielectric loss. Dielectric measurements on clear, unirradiated quartz crystals do not reveal a relaxation peak corresponding to the mechanical peak.53 Natural crystals of smoky quartz and both natural and synthetic crystals irradiated with X-rays or electrons, however, do show a dielectric relaxation peak (in the kilocycle range) in the temperature interval at or below 20" K (so-called p-peak).s49 57 Both the position and the relaxation strength of the peak vary from crystal to crystal.Volger and Stevels56 suggest that the peak might be due to electrons or holes which are trapped in colour centres and which undergo electrically active transi- tions from one location in the centre to another one. Another possibility is that there exists a connection with the mechanical relaxation process mentioned above, for which eqn. (2.1) is reported 37 to hold with H = 155 cal/mole andf" = 2 x 10s sec-1. The corresponding values for the dielectric measurements are somewhat uncertain but seem to be of the same order of magnitude.55 This suggests that both processes are due to the same mechanism. Further experiments are required in order to confirm or refute this view.They might also help to clear up the open question whether the mechanical losses in quartz are of the Bordoni type and whether the Peierls stress in quartz is really so surprisingly small as would be implied by such an interpretation. The authors would like to thank Prof. U. Dehlinger for his encouragement and for his interest in their work, and Dr. P. Haasen and Dr. J. Diehl for valuable comments on the manuscript. 1 Read, Trails. Amer. Inst. Min. Met. Eng., 1941, 143, 30. 2 Seitz, The Physics of Metals (McGraw-Hill, New York, 1943). 3 Zener, Elasticity and Anelasticity of Metals (Univ. of Chicago Press, Chicago, 1948). 4 Wert, Modern Research Techniques in Physical Metallurgy (American Society for 5 Kempe and Kroner, Z. 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