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Electron microscope observations of the formation of Θ from Θ' in A1–4% Cu
LETTERS
TO
states. Equally electrons in band states with wave functions and approximated as linear combinations of atomic d orbitals would give the same results. On either interpretation, the description of the magnetic electrons is in terms of little modified and little hybridised atomic d functions; the non-spherical nature of the spin density results from selective population of the 5 cl states available. Shull and Yamada(Q (also Shullc5)) have shown by neutron scattering methods that interaction between 49 and 32 electrons will give rise to a negative and not to a positive magnetisation. This means, therefore, that the second Pauling view of a positive contribution 3d interaction is of 0.22,s for conductivity/atomic incorrect, and it is unlikely that any model is correct which gives Fe atoms in a state with a moment of exactly 2pB. Of the Pauling approaches, the first is, therefore, now the more probable. As has been shown elsewhere (Hume-Rothery’7)) the assumption of the same constant valency for all the elements from Group VIA to VIIIC in all three Transition Series is not in agreement with the facts, and modified schemes have been suggested. The exact electronic configurations of these transition metals are still in dispute, but it is satisfactory to know that the magnetic electrons are exclusively d like, and that atomic electronic structure may still be significant in the solid. The author acknowledges helpful discussion with Dr. W. M. Lomer and Dr. S. L. Altmann. W. HUME-ROTHERY Department of Metallurgy University of Oxford References L. PAULINO, Phys. Rev. 54, 899 (1938). L. PAULINQ, PTOC.Nut. Acud. Sci. 89. 551 (1953). C. ZENER, Phys. Rev. 81, 440 (1951); 62, 463 (1951); 83, 299 (1951). L. PAULINO, The Nature of the Chemical Bond. Cornell, New York (1960). C. G. SHULL, Electronic Structure and Alloy Chemistry of the Transition Metals. , D. 69. Interscience. New York (1963). 6. C. G. SHULL and Y. YAMADA, J. Phys. Sot. Japan, Suppl. B, BIII, 17, 1 (1961). 7. W. Hum-ROTHREY, Atomic Theory for Student8 of Metallurgy, p. 378. Inst. of Metals (1962); also W. HUMEROTHERYand B. R. COLES,Advanc. Phye. 3, 149 (1954). &
* Received July, 2 1964. i Jn Fig. 1, the upper left end lower right hand corners are the positions of two atoms which are nearest neighbours.
Electron microscope observations of the formation of 0 from 0’ in A14% Cu* In the early stages of an investigation of precipitation at grain boundaries, an Al4% Cu alloy has
THE
EDITOR
1463
given incidental information on the formation of 0 phase from the transitional 0’ phase. A sheet specimen 0.005 in. thick, was solution treated at 500°C and transferred to an ageing furnace at 380% for 15 min. Electron microscope examination of thin foils revealed plates of 0’ containing two types of rod-like 0 particles. Figures 1 and 2 are from foils oriented respectively with [510],, and [3Ol],i directions approximately parallel to the electron beam. The 0’ phase forms as (OOl), platelets on the {lOO},i matrix planes. Two orientations of the 0’ plates, A and B, are parallel or almost parallel to the electron beam in the Figs. 1 and 2. The third 0’ plate orientation is almost perpendicular to the electron beam, that is at a shallow angle to the foil surface. Such plates contain 0 rods (v, w, x, y and z in Figs. 1 and 2). Rods v, w, x and y are at angles of 12” to 13’ to one of the two (loo),, directions in the plane of the 0’ plate (this is near to the value 11’ 18’ for a (510),, direction). It can be seen from the edge-on view of the 0’ plates A and B that the rod axes lie in the plane of the 0’ plates. The orientation relationship between rods such as v, w, x, y and the matrix was obtained by electron diffraction from five different 0’ plates, and is: [lOOI, 11[lOOI*,; [Oil],
11[001],1 (orientation 1)
The [lOOI, direction is perpendicular to the 0’ plate, It follows from symmetry considerations that there are four possible 0 orientations for each 0’ plate, all of which are present in Fig. 2 (i.e. v, w, x and y). The orientation relationship is the same as that observed by Heimendahl and Wassermann(l) when 0 formed from 0’ at 45O”C, but the details of the morphology of the @ particles were not given. The orientation of one rod axis is shown on the right hand side of Fig. 3; all rods form in (032)e directions. The rods are clearly defined and have very straight sides so that a simple explanation of their shape might be expected. If the morphology of the @ phase arose entirely from a highly preferred growth direction then it might be expected to be the same as that in the melt, in which case the growth direction is [OOl],. Since the rods form in this case with an (032 )e axis and only that (032)e axis which is almost parallel to an (051),i direction, it is apparent that the growth direction is dependent on the orientation of 0 with respect to the matrix. It might be expected that the preferred direction of growth would correspond to a zone axis of planes of easy fit with the matrix and high atomic density. (The rods are growing in the Al matrix at the stage observed
1464
ACTA
METALLURGICA,
VOL.
12,
1964
FIG. 1. Formation of 0 rods from 0’ plates in an Al-4 y0 Cu alloy.
and not in direct contact with the 0’ plates.) Figure 3 shows the projected atomic position of 0 and the aluminium matrix on a common (100) plane (that is in the plane of the 0’ plate). It is immediately
apparent that there is no obvious matching of atomic positions in the [OOl], direction. In the [Oil], direction, a contraction of only 5% is required for the matrix atom at A’ to coincide with the 0 atom at A. It is surprising that the rods do not grow in this direction but a possible explanation is that the contraction is greater than the average linear contraction (2.4 %) for 0 formation. Considering the plane containing the rod axis which is also perpendicular to the (100) plane in Fig. 3 (in fact the (023)e plane) there are groups of atoms in 0 such as B, D, E; F, G, H; J, K, L; which require relatively small displacements from the matrix positions (B’, D’, E’, etc.) to 0 positions. This group of three atoms re-occurs regularly in this plane but in all cases either one atom position is a good fit (e.g. D) and the other two require about 1 A movement (e.g. B and F) within the plane, or two atom positions fit well (F and H) and the centre atom requires about 1 A movement (e.g. G). The movements are small and are within the (023) plane so that this repetition of a matching group of atoms gives reason for the easy growth of the (023) plane. Within this plane the [032], rod direction may result from : (a) the good matching every 20.6 A in this direction or (b) the development of additional planes with an [032], zone axis (such as the (loo), plane). Ol-
Fro. 2. Formation of 0 rods from @‘plates in an Al-4 % Cu alloy.
(c)
restriction of the growth of 0 to the plane of the 0’ platelet which is providing the Cu atoms.
LETTERS
TO
THE
1465
EDITOR
ALUMINIUM
MATRIX
X AL ATOMS IN PLANE +
OF PROJECTION
A( ATOM5 2 021 ABOVE AND BELOW PLANE Of PROJECTION
0 PHASE .
Cu ATOMS IN PLANE OF PROJECTION
0
AND BELOW
0
Cu ATOMS ,.O,R ABOVE PLANE OF PROJECTION At ATOMS 0.97x ABOVE PROJECTION . A6 ATOMS 0,971 BELOW PROJECTION h At ATOMS 2 06x ABOVE PROJECTION A At ATOMS 2.Ob8 BELOW PROJECTION
PLANE OF PLANE OF PLANE OF PLANE OF
FIG. 3. Atomic positions for the aluminium matrix and the @ phase projected on a common (100) plane. The unit cells of the aluminium and 0 structures are outlined respectively by squares and rectangles. The figure on the right relates the axis of the 0 rod (broken line) to the structures of 0 and aluminium.
Since the rods show no clearly developed prism faces it is probable that (a) and (c) are the most important factors. No other explanation has been found for this unusual rod axis. The 0 particles marked Z in Figs. 1 and 2, bear a different orientation relationship to the matrix:
[lOO], 11[lOOlal; [OOl], 11[Oil],,
(orientation 2)
This is the orientation given by Guinier(s) as that which nucleates from 0’ but not from the matrix since there is good lattice matching perpendicular to the 0’ platelet for the [lOO], direction with the [OOl],, direction but not with the [lOO],, directions. The same argument also applies to orientation 1 for this direction is common to both orientations and they differ only by a swing of 6’ about the common direction (i.e. about an axis perpendicular to the 0’ plate). The rods with orientation 2 have a tendency to be elongated in the [OOI], direction and are frequently more sharply defined than the examples in Figs. 1 and 2. This swing of 6” thus completely alters the direction of growth of 0. The groups of atoms discussed above for orientation 1 no longer have matching matrix atoms. In the direction of the rod axis a contraction of 15% is required to make the matrix atoms coincide with those of the 0 structure. It seems probable that in orientation 2, since the growth direction is the same as from the melt, that the rod direction is governed by the 0 lattice and
not 0 to Al lattice matching. 0 phase precipitates have also been observed with no apparent association with 0’ plates and having the following orientation with respect to the matrix: [OOl], 11[OOl],,; [130], 11[IOO],, (orientation 3) This is similar to the Group IV and V orientations observed by Guinier and expressed as [OOl], II [OOl],,, [lOO]@ making an angle of -&SOwith [lOO],,. Orientation relationship 3 was predicted by Mehl et aE.c3) by considering the matching of Al atom positions in the 0 structure with the matrix atomic positions. Their model appears to fit the experimental observations more closely than the model used by Guinier to explain the Group IV and V orientations. It is obvious from the shapes of the 0 particles and also from the irregular gaps between the 0 particles and the 0’ plates that the growth mechanism from 0’ is not a shear transformation, but involves nucleation of 0 within the 0’ plate and subsequent growth by dissolution of 0’. This is in agreement with the observations of Heimendahl and Wassermann.(l) This work was carried out at C.E.R.L., and we would like to thank C.E.G.B. for permission to publish. D. VAUGHAN J. M. SILCOCK Central Electricity Research Laboratories Leatherhead, Surrey
1466
ACTA
METALLURGICA,
References 1. M. v. HEIMENDAHLund G. WASSERMANN,2. Metallk. 59, 275 (1962). 2. A. GUINIER.J. Phvs. Radium 3. 124 0942). 3. R. F. ME&, C. S: BARRETTkd F.‘ N. WHINES, Trans. Anter. Inst. Min. (Met&.) Engrs. 99, 203 (1932). * Received June 4, 1964.
The activation
energy of high internal friction*
In a wide range of materials there is a rapid rise in the internal friction at high temperatures which is independent of strain amplitude and appears to follow a law of the type (1)
A review of the data has been given by Mason(l) and Niblett and Wilks.c2) This intern-al friction occurs in single crystals as well as in polycrystals and is almost certainly due to some thermally activated dislocation relaxation. This can be concluded from the magnitude of the effect which is too large to be explained by any reasonable mechanism involving point defects only. The value of U in equation (1) is usually called the activation energy of the internal friction and it is obtained by plotting In Q-l vs. l/T which gives a straight line if equation (1) is obeyed. The values of U range in different metals under different conditions from 0.3 eV to 2 eV(l12) and it is difficult to associate these values with a specific dislocation mechanism. It is the purpose of this note to point out that the value of U is not necessarily identical with the activation energy U, of the controlling dislocation mechanism, and usually will be smaller than U,. Let us assume that the internal friction is caused by the motion of dislocations which are in some way interacting with point defects and move at high temperatures in a viscous way. This mechanism is not necessarily invoked in the following discussion; however, it is a convenient model to give a definite meaning to the otherwise abstract parameters occurring in the equations. Under a wide range of conditions at high temperatures and low stresses the dislocation velocity is given by dx/dt = p exp
(- U,IkT) * 3
12,
1964
Under internal friction conditions with a periodic small stress, o = a, exp (&~t), the acting stress 3 can generally be represented by ~7= a, exp (id)
-
9%
(3)
where q represents a restoring force on dislocations due to the line tension or internal stresses. Usually, there exists a wide distribution of values of q according to the geometrical arrangement and in special cases q can be vanishing. The dislocation movement is then described by
temperature
Q-l = A exp (- U/kT)
VOL.
(2)
where ~7is the stress acting on the dislocation, U,, the activation energy of the rate controlling dislocation mechanism and p a proportionality factor which may depend on the geometrical arrangement but not on temperature.
exp (U,/kT) dx P
z
+ qx = 0, exp (iwt)
(4)
The internal friction can be easily calculated by observing that equation (4) results in a dislocation movement which is out of phase with the applied stress, and by remembering that the movement of dislocations of total length A over a distance x gives rise to an anelastic deformation of the magnitude Ed = yAxb
(5)
where y is a geometrical orientation factor of the order of magnitude 0.1, and b the magnitude of the Burgers vector. For small values of the internal friction its magnitude is given by the imaginary part of .sd divided by the elastic deformation E, = a# where (2 is the shear modulus.(3) In the case of q = 0 this results in an internal friction of the magnitude Q-l = yGAb
P e=p (-
UJkT)
w
We see here that even if we have a distribution of the values of p the activation energy U measured in internal friction by equation (1) is identical with the value of U, which is the one of the rate controlling process. In this case however, the value of Q-l must also be proportional to w-l and under any experimental condition the frequency dependence of Q-r should be checked to assure if equation (6) is applicable or not. In addition any sample stressed under these conditions should show a permanent deformation and not return to its original state upon unloading. If q # 0 and the values of the parameters p and q are constant it is easily seen that the internal friction is given by
Q-l=
yAGb 1 ;rL2T2
(7)
where r = exp (u,lkt)l~n
(8)
TO
states. Equally electrons in band states with wave functions and approximated as linear combinations of atomic d orbitals would give the same results. On either interpretation, the description of the magnetic electrons is in terms of little modified and little hybridised atomic d functions; the non-spherical nature of the spin density results from selective population of the 5 cl states available. Shull and Yamada(Q (also Shullc5)) have shown by neutron scattering methods that interaction between 49 and 32 electrons will give rise to a negative and not to a positive magnetisation. This means, therefore, that the second Pauling view of a positive contribution 3d interaction is of 0.22,s for conductivity/atomic incorrect, and it is unlikely that any model is correct which gives Fe atoms in a state with a moment of exactly 2pB. Of the Pauling approaches, the first is, therefore, now the more probable. As has been shown elsewhere (Hume-Rothery’7)) the assumption of the same constant valency for all the elements from Group VIA to VIIIC in all three Transition Series is not in agreement with the facts, and modified schemes have been suggested. The exact electronic configurations of these transition metals are still in dispute, but it is satisfactory to know that the magnetic electrons are exclusively d like, and that atomic electronic structure may still be significant in the solid. The author acknowledges helpful discussion with Dr. W. M. Lomer and Dr. S. L. Altmann. W. HUME-ROTHERY Department of Metallurgy University of Oxford References L. PAULINO, Phys. Rev. 54, 899 (1938). L. PAULINQ, PTOC.Nut. Acud. Sci. 89. 551 (1953). C. ZENER, Phys. Rev. 81, 440 (1951); 62, 463 (1951); 83, 299 (1951). L. PAULINO, The Nature of the Chemical Bond. Cornell, New York (1960). C. G. SHULL, Electronic Structure and Alloy Chemistry of the Transition Metals. , D. 69. Interscience. New York (1963). 6. C. G. SHULL and Y. YAMADA, J. Phys. Sot. Japan, Suppl. B, BIII, 17, 1 (1961). 7. W. Hum-ROTHREY, Atomic Theory for Student8 of Metallurgy, p. 378. Inst. of Metals (1962); also W. HUMEROTHERYand B. R. COLES,Advanc. Phye. 3, 149 (1954). &
* Received July, 2 1964. i Jn Fig. 1, the upper left end lower right hand corners are the positions of two atoms which are nearest neighbours.
Electron microscope observations of the formation of 0 from 0’ in A14% Cu* In the early stages of an investigation of precipitation at grain boundaries, an Al4% Cu alloy has
THE
EDITOR
1463
given incidental information on the formation of 0 phase from the transitional 0’ phase. A sheet specimen 0.005 in. thick, was solution treated at 500°C and transferred to an ageing furnace at 380% for 15 min. Electron microscope examination of thin foils revealed plates of 0’ containing two types of rod-like 0 particles. Figures 1 and 2 are from foils oriented respectively with [510],, and [3Ol],i directions approximately parallel to the electron beam. The 0’ phase forms as (OOl), platelets on the {lOO},i matrix planes. Two orientations of the 0’ plates, A and B, are parallel or almost parallel to the electron beam in the Figs. 1 and 2. The third 0’ plate orientation is almost perpendicular to the electron beam, that is at a shallow angle to the foil surface. Such plates contain 0 rods (v, w, x, y and z in Figs. 1 and 2). Rods v, w, x and y are at angles of 12” to 13’ to one of the two (loo),, directions in the plane of the 0’ plate (this is near to the value 11’ 18’ for a (510),, direction). It can be seen from the edge-on view of the 0’ plates A and B that the rod axes lie in the plane of the 0’ plates. The orientation relationship between rods such as v, w, x, y and the matrix was obtained by electron diffraction from five different 0’ plates, and is: [lOOI, 11[lOOI*,; [Oil],
11[001],1 (orientation 1)
The [lOOI, direction is perpendicular to the 0’ plate, It follows from symmetry considerations that there are four possible 0 orientations for each 0’ plate, all of which are present in Fig. 2 (i.e. v, w, x and y). The orientation relationship is the same as that observed by Heimendahl and Wassermann(l) when 0 formed from 0’ at 45O”C, but the details of the morphology of the @ particles were not given. The orientation of one rod axis is shown on the right hand side of Fig. 3; all rods form in (032)e directions. The rods are clearly defined and have very straight sides so that a simple explanation of their shape might be expected. If the morphology of the @ phase arose entirely from a highly preferred growth direction then it might be expected to be the same as that in the melt, in which case the growth direction is [OOl],. Since the rods form in this case with an (032 )e axis and only that (032)e axis which is almost parallel to an (051),i direction, it is apparent that the growth direction is dependent on the orientation of 0 with respect to the matrix. It might be expected that the preferred direction of growth would correspond to a zone axis of planes of easy fit with the matrix and high atomic density. (The rods are growing in the Al matrix at the stage observed
1464
ACTA
METALLURGICA,
VOL.
12,
1964
FIG. 1. Formation of 0 rods from 0’ plates in an Al-4 y0 Cu alloy.
and not in direct contact with the 0’ plates.) Figure 3 shows the projected atomic position of 0 and the aluminium matrix on a common (100) plane (that is in the plane of the 0’ plate). It is immediately
apparent that there is no obvious matching of atomic positions in the [OOl], direction. In the [Oil], direction, a contraction of only 5% is required for the matrix atom at A’ to coincide with the 0 atom at A. It is surprising that the rods do not grow in this direction but a possible explanation is that the contraction is greater than the average linear contraction (2.4 %) for 0 formation. Considering the plane containing the rod axis which is also perpendicular to the (100) plane in Fig. 3 (in fact the (023)e plane) there are groups of atoms in 0 such as B, D, E; F, G, H; J, K, L; which require relatively small displacements from the matrix positions (B’, D’, E’, etc.) to 0 positions. This group of three atoms re-occurs regularly in this plane but in all cases either one atom position is a good fit (e.g. D) and the other two require about 1 A movement (e.g. B and F) within the plane, or two atom positions fit well (F and H) and the centre atom requires about 1 A movement (e.g. G). The movements are small and are within the (023) plane so that this repetition of a matching group of atoms gives reason for the easy growth of the (023) plane. Within this plane the [032], rod direction may result from : (a) the good matching every 20.6 A in this direction or (b) the development of additional planes with an [032], zone axis (such as the (loo), plane). Ol-
Fro. 2. Formation of 0 rods from @‘plates in an Al-4 % Cu alloy.
(c)
restriction of the growth of 0 to the plane of the 0’ platelet which is providing the Cu atoms.
LETTERS
TO
THE
1465
EDITOR
ALUMINIUM
MATRIX
X AL ATOMS IN PLANE +
OF PROJECTION
A( ATOM5 2 021 ABOVE AND BELOW PLANE Of PROJECTION
0 PHASE .
Cu ATOMS IN PLANE OF PROJECTION
0
AND BELOW
0
Cu ATOMS ,.O,R ABOVE PLANE OF PROJECTION At ATOMS 0.97x ABOVE PROJECTION . A6 ATOMS 0,971 BELOW PROJECTION h At ATOMS 2 06x ABOVE PROJECTION A At ATOMS 2.Ob8 BELOW PROJECTION
PLANE OF PLANE OF PLANE OF PLANE OF
FIG. 3. Atomic positions for the aluminium matrix and the @ phase projected on a common (100) plane. The unit cells of the aluminium and 0 structures are outlined respectively by squares and rectangles. The figure on the right relates the axis of the 0 rod (broken line) to the structures of 0 and aluminium.
Since the rods show no clearly developed prism faces it is probable that (a) and (c) are the most important factors. No other explanation has been found for this unusual rod axis. The 0 particles marked Z in Figs. 1 and 2, bear a different orientation relationship to the matrix:
[lOO], 11[lOOlal; [OOl], 11[Oil],,
(orientation 2)
This is the orientation given by Guinier(s) as that which nucleates from 0’ but not from the matrix since there is good lattice matching perpendicular to the 0’ platelet for the [lOO], direction with the [OOl],, direction but not with the [lOO],, directions. The same argument also applies to orientation 1 for this direction is common to both orientations and they differ only by a swing of 6’ about the common direction (i.e. about an axis perpendicular to the 0’ plate). The rods with orientation 2 have a tendency to be elongated in the [OOI], direction and are frequently more sharply defined than the examples in Figs. 1 and 2. This swing of 6” thus completely alters the direction of growth of 0. The groups of atoms discussed above for orientation 1 no longer have matching matrix atoms. In the direction of the rod axis a contraction of 15% is required to make the matrix atoms coincide with those of the 0 structure. It seems probable that in orientation 2, since the growth direction is the same as from the melt, that the rod direction is governed by the 0 lattice and
not 0 to Al lattice matching. 0 phase precipitates have also been observed with no apparent association with 0’ plates and having the following orientation with respect to the matrix: [OOl], 11[OOl],,; [130], 11[IOO],, (orientation 3) This is similar to the Group IV and V orientations observed by Guinier and expressed as [OOl], II [OOl],,, [lOO]@ making an angle of -&SOwith [lOO],,. Orientation relationship 3 was predicted by Mehl et aE.c3) by considering the matching of Al atom positions in the 0 structure with the matrix atomic positions. Their model appears to fit the experimental observations more closely than the model used by Guinier to explain the Group IV and V orientations. It is obvious from the shapes of the 0 particles and also from the irregular gaps between the 0 particles and the 0’ plates that the growth mechanism from 0’ is not a shear transformation, but involves nucleation of 0 within the 0’ plate and subsequent growth by dissolution of 0’. This is in agreement with the observations of Heimendahl and Wassermann.(l) This work was carried out at C.E.R.L., and we would like to thank C.E.G.B. for permission to publish. D. VAUGHAN J. M. SILCOCK Central Electricity Research Laboratories Leatherhead, Surrey
1466
ACTA
METALLURGICA,
References 1. M. v. HEIMENDAHLund G. WASSERMANN,2. Metallk. 59, 275 (1962). 2. A. GUINIER.J. Phvs. Radium 3. 124 0942). 3. R. F. ME&, C. S: BARRETTkd F.‘ N. WHINES, Trans. Anter. Inst. Min. (Met&.) Engrs. 99, 203 (1932). * Received June 4, 1964.
The activation
energy of high internal friction*
In a wide range of materials there is a rapid rise in the internal friction at high temperatures which is independent of strain amplitude and appears to follow a law of the type (1)
A review of the data has been given by Mason(l) and Niblett and Wilks.c2) This intern-al friction occurs in single crystals as well as in polycrystals and is almost certainly due to some thermally activated dislocation relaxation. This can be concluded from the magnitude of the effect which is too large to be explained by any reasonable mechanism involving point defects only. The value of U in equation (1) is usually called the activation energy of the internal friction and it is obtained by plotting In Q-l vs. l/T which gives a straight line if equation (1) is obeyed. The values of U range in different metals under different conditions from 0.3 eV to 2 eV(l12) and it is difficult to associate these values with a specific dislocation mechanism. It is the purpose of this note to point out that the value of U is not necessarily identical with the activation energy U, of the controlling dislocation mechanism, and usually will be smaller than U,. Let us assume that the internal friction is caused by the motion of dislocations which are in some way interacting with point defects and move at high temperatures in a viscous way. This mechanism is not necessarily invoked in the following discussion; however, it is a convenient model to give a definite meaning to the otherwise abstract parameters occurring in the equations. Under a wide range of conditions at high temperatures and low stresses the dislocation velocity is given by dx/dt = p exp
(- U,IkT) * 3
12,
1964
Under internal friction conditions with a periodic small stress, o = a, exp (&~t), the acting stress 3 can generally be represented by ~7= a, exp (id)
-
9%
(3)
where q represents a restoring force on dislocations due to the line tension or internal stresses. Usually, there exists a wide distribution of values of q according to the geometrical arrangement and in special cases q can be vanishing. The dislocation movement is then described by
temperature
Q-l = A exp (- U/kT)
VOL.
(2)
where ~7is the stress acting on the dislocation, U,, the activation energy of the rate controlling dislocation mechanism and p a proportionality factor which may depend on the geometrical arrangement but not on temperature.
exp (U,/kT) dx P
z
+ qx = 0, exp (iwt)
(4)
The internal friction can be easily calculated by observing that equation (4) results in a dislocation movement which is out of phase with the applied stress, and by remembering that the movement of dislocations of total length A over a distance x gives rise to an anelastic deformation of the magnitude Ed = yAxb
(5)
where y is a geometrical orientation factor of the order of magnitude 0.1, and b the magnitude of the Burgers vector. For small values of the internal friction its magnitude is given by the imaginary part of .sd divided by the elastic deformation E, = a# where (2 is the shear modulus.(3) In the case of q = 0 this results in an internal friction of the magnitude Q-l = yGAb
P e=p (-
UJkT)
w
We see here that even if we have a distribution of the values of p the activation energy U measured in internal friction by equation (1) is identical with the value of U, which is the one of the rate controlling process. In this case however, the value of Q-l must also be proportional to w-l and under any experimental condition the frequency dependence of Q-r should be checked to assure if equation (6) is applicable or not. In addition any sample stressed under these conditions should show a permanent deformation and not return to its original state upon unloading. If q # 0 and the values of the parameters p and q are constant it is easily seen that the internal friction is given by
Q-l=
yAGb 1 ;rL2T2
(7)
where r = exp (u,lkt)l~n
(8)