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Premartensitic phenomena in InTl alloys and the role of the fermi surface
ScrJpta
MIITALLUR(]ICA
Vol.
18,
Printed
pp.
575-578,
in t h e
1984
Pergamon
U.S.A.
..\11 r i g h t s
Press
Ltd.
reserved
PREMARTENSITIC PHENOMENA IN In-Tl ALLOYS AND THE ROLE OF THE FERMI SURFACE P. D. Normantt, S. W. Wilkins**, T. R. Finlayson*, P. Goodman**, and A. Olsent * Department of Physics, Monash University, Melbourne, Australia ** Division of Chemical Physics, C.S.I.R.O., Melbourne, Australia t Department of Physics, University of Oslo, Norway tt Chisholm Institute of Technology, Caulfield, Australia. (Received December 8, 1983) (Revised A p r i l 12, 1 9 8 4 ) Introduction We seek to relate various observed premartensitic phenomena in In-Tl alloys, elastic constant softening, severe acoustic attenuation of some phonon modes configurations of diffuse scattering (e.g. in electron diffraction patterns), features of the Fermi surface and to the occurrence of Kohn-like anomalies. the present study is the observation that some of the ascribed Kohn anomalies reciprocal lattice points and so can be manifested in anomalous behaviour of length properties such as elastic constants and acoustic attenuation.
such as drastic and complicated to particular A novel feature in lie very close to very long-wave-
In-Tl alloys are claimed to be random disordered alloys and, in the composition range 15.5-31 at.% TI, are found to undergo a martensitic transformation at T from face-centred cubic (fcc) to face-centred tetragonal (fct) as the temperature is lowered, m T is a sensitive function of alloy composition and decreases from 425 K for 15.5 at.% TI to 0 Kmfor ~ 31 at.% TI. The transformation produces a distinctive lamellar microstructure due to {Ii0} < IT0 > transformation twinning and is called martensitic because it is diffusionless and displaclve involving homogeneous deformation. Various anomalies in physical properties of such transforming In-Tl alloys are known to occur as the temperature is lowered toward T m. For example: (i)
Novotny and Smith (i) used the pulse-echo technique to measure sonic velocities in single crystals of fcc In-Tl alloys. Three independent elastic constants were determined from these results and they found that the shear modulus, i/2(ell - c12), becomes extremely small as T m is approached. This result was taken to provide evidence of a dramatic softening of the [ ~ O ] polarised [ ~ 0 ] transverse acoustic phonon branch near the Brillouin zone centre. More recent ultrasonic velocity measurements by Pace and Saunders (2) confirmed the presence of this severe softening.
(ii) Recent x-ray and electron diffraction observations of transforming In-Tl alloys have been made by Koyama and Nittono (3) who noted the appearance of intensive thermal diffuse scattering (TDS) near the transformation temperature. This TDS, taken in conjunction with the findings of Pace and Saunders (2), was also taken to indicate the ~resence of a low frequency acoustic phonon mode along [ii0] direction and polarized [II0], in such In-Tl alloys, The major purpose of this paper is to report additional information regarding electron diffuse scattering produced by In-Tl alloys and to seek to relate the geometrical features of this diffuse scattering to an underlying physical mechanism involving a subtle coupling between lattice modulations (such as phonons and static displacement waves) and the energy of the conduction electrons in the alloy. In simple terms, reduction in the total energy of the conduction electrons can occur when lattice modulations of spatial frequency, qLM' match that for spanning vectors, 2kF , between antiparallel sections of the alloy Fermi surface as shown in Figure I. The effect is well known and the mathematical details as to how it arises can be found, for example, in Harrison (4). The effect apparently was first considered by Peierls (5) in relation to static (Peierls) modulations of a model l-dimensional structure. Subsequently Kohn (6) considered the effect in relation to phonons and the term "Kohn anomaly" is commonly used to
0036-9748/84 Copyright (c) 1 9 8 4
575 $ 3 . 0 0 + .00 Pergamon Press
l,td.
576
PREMARTENSITIC
PHENOMENA
IN In-Tl
Vol.
18, No. 6
describe the resulting dips in phonon spectra which often occur at ~ values corresponding 2k F spanning vectors. Similarly driven anomalies in short-range order diffuse scattering substltutional alloys have been discussed by Krlvoglaz and Tu Hao (7), Moss (8) and Cowley Wilklns (9). In yet another form, spatial modulations of the electron gas are understood occur due to a similar driving force and are termed charge density waves (CDW's).
to in and to
Although Kohn-llke anomalies may occur in diffuse scattering for any 2k F vector spanning antiparallel sections of the Fermi surface, according to Roth, Zelger and ~ p l a n (I0) the strongest anomalies tend to occur between sections where the curvature is lowest . Thus, for example, strong Kohn-llke anomalies tend to occur between flattened sections of the Fermi surface such as often occur in pseudo l-dlmenslonal systems. However, one should note that it is the local dlmenslonallty in reciprocal space which is important and not in direct space, so that strong anomalies may also occur in 3-dlmenslonal systems. For a given Fermi surface, the locl of Kohn anomalies may be plotted by folding the 2k_ spanning vectors with the reciprocal lattice and this is called a Kohn construction (see H a r r ~ n (4)). The strength of Kohn anomalies in the diffuse scattering is not only related to the curvature of the Fermi surface but also to the multiplicity of the overlap of the locl of these Kohn anomaly surfaces. Furthermore, regions bounded by Kohn anomaly surfaces can yield strong diffuse scattering throughout such regions. Nonetheless, the shape of the diffuse scattering contours reflects the form of the 2k F spanning vectors. It is interesting to note that recently Gyorffy and Stocks (ii) established a strong correlation between diffuse scattering maxima observed experimentally for random alloys of Cu-Pd and the 2k F vectors spanning flat and antl-parallel sheets of the alloy Fermi surface which had been calculated for the same system using the coherent potential approximation.
I[oo]
Ifool
Fig. 1 - 2k_ ~F spanning vector for Fermi surface regions with antl-parallel group velocities.
Fig. 3 - The third zones of the Fermi surface of indium (fct).
Fig. 2 - The third zones of the Fermi surface of alumlnlum (fcc). Kohn Constructions
Unfortunately there is no Fermi surface data for cubic In-Tl available in the literature. However, aluminium and In-Tl may be assumed to have roughly comparable electronic structures within the nearly free electron picture since they beth have nearly three conduction electrons per atom,so we have chosen to use the Fermi surface of aluminlum shown in Figure 2 as the basis of our discussions concerning In-Tl. This choice is made in preference to using the Fermi surface of indium, which is illustrated in Figure 3, because both aluminium and pre-martensitic In-Tl have fcc structures whereas indium has a fct structure similar to that of the martensite phase of In-Tl. o i~ '
The Fermi surface of aluminium as illustrated in Figures 2 and 4 was determined by the 4-OPW (orthogonallzed-planewave) calculation of Ashcroft (12),whlch he compared favourably with the de Haas-van Alphen data of Gunnerson (13) and of Larson and Gordon (14). Basically, the model surface is similar to a free-electron surface for three conduction electrons with a filled first-zone, a secondzone hole surface centred at F, and a simply connected third-zone electron surface consisting of four-sided rings around the edges of the square faces of the zone. In this model there are no fourth-zone pockets. In Figure 4, the cross section of the second-zone hole surface is shaded and the cross-section of the third-zone electron surface is hatched.
III
,/ ~///
U (
0
\
I
i Ol
o 4 - A [110] section of t h e a l u m i n i u m Fermi s u r f a c e t h r o u g h the bounding face of the Brillouln zone.
Fi$.
Vol
18,
No.
6
PREMARTENSITIC PHENOblENA IN I n - T 1
577
Using this as the model Fermi surface, Kohn constructions for In-24 at.% TI have been made in the manner of Harrison (4) by drawing Kohn anomaly surfaces of radius 2k_ about each of the ~F eight nearest-neighbour reciprocal lattice points surrounding the given reciprocal lattice point (Bragg peak). The value of 2kF used in the [001], [112] and [110] sections in reciprocal space was taken to be slightly-less than the 2k F value for aluminium which is (2.24) 2~/a. Possible explanations for this rescaling are that the conduction electron/atom (e/a) ratio for In-Tl alloys is slightly lower than that for aluminium due to the presence of d-band effects and that the band gaps between the second and third zones in In-Tl are significantly different from those for aluminium. This difference is also suggested by comparing Figure 2 with Figure 3. The Kohn constructions of the [001] and [T12] sections shown in Figures 5 and 7 compare favourably with the corresponding diffuse scattering images obtained for the In-24 at.% TI alloy as shown in Figures 6 and 8. An independent analysis by Finlayson et al.(15) has indicated that the diffuse intensity distribution in reciprocal space is a maximum along directions Joining Bragg reflections. The cubic symmetry of the f.c.c, phase enabled the construction of a 3-dimensional model of this diffuse intensity from relatively few zone-axis patterns. This model is shown in Figure I0 as a [TIO] section and compares very well with the corresponding Kohn construction shown in Figure 9. Conclusion Comparisons of Figures 5, 7 and 9 with Figures 6, 8 and i0 respectively indicate that the Kohn constructions of In-24 at.% TI relate very closely to the electron diffuse scattering patterns obtained from this alloy. Accordingly, it appears that the diffuse scattering may be viewed as arising from conduction-electron energy effects on the structure of the In-Tl alloy. However, a quantitative confirmation of this source of the diffuse scattering would require detailed bandstructure calculations for cubic In-Tl and also further diffraction experiments to observe the effect on the diffuse scattering of changing the e/a ratio in In-Tl alloys by doping (see Sato and Toth (16)). The ideas which we have outlined here for relating the observed diffuse scattering to conduction-electron energy effects are also consistent with current ideas on the driving mechanism for many phase changes involving small amplitude atomic displacements. For example, Friedel (17) has suggested how conduction-electron energy effects may produce such phase changes by the softening of a phonon mode through strong electron-phonon coupling. From our present observations of In-Tl this would appear to be the likely origin of the driving mechanism of the m~rtensitic phase transformation from fcc to fct.
/', i,o; >
•
s ,~ < ,o] .-
2oo
o~o
"(9
s 't
~L_s_
020 ', /
206 , /
Fi$. 5 - Kohn construction the [001] section crosses represent reciprocal lattice
for In-24 at.% TI for where circles with out of plane points.
Fig. 6 - Electron diffraction pattern for In-24 at.% TI for [001] zone axis.
PREMARTENSITIC
578
171
. o _ 220 L • ;'
,' "
-~ "-
-
~J /-h / kJ , ,' ", "4'
13T
".
,
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IN
In-Tl
Vol.
18,
No.
6
.- . o
~ --_.'% r
'
PHENOMENA
/ ~ /---~ )
220 ; /el
'~ ',
,i%,
llI
311
Fig, 7 - Kohn construction for In-24 at.% TI for the IT12] section where circles with crosses represent out of plane reciprocal lattice points.
Fig. 8 - Electron diffraction pattern for In-24 at.% TI for IT12] zone axis.
O
002
o
IIool
t+
lql
0~
Fi@. 9 - Kohn construction for In-24 at.% T1 for the [TI0] section where circles with crosses represent out of plane reciprocal lattice points i. 2. 3. 4. 5. 6. 7. 8. 9. I0. ii. 12. 13. 14. 15. 16. 17.
Fig. i0 - Single contour representation of the diffuse intensity [TI0] projection.
References D. Novotony and J. F. Smith, Acta Met., 13, 881 (1965). N. Pace and G. Saunders, Proc. R. Soe., 326, 521, (1972). Y. Koyama and O. Nittono, J. Jpn. Inst. Met., 45, 869, (1981). W . A . Harrison, Phonons, Edit. R.W.H. Stevenson, p. 103, Oliver and Boyd, Edinburgh, (1966). R . E . Peierls, Quantum Theory of Solids, p. 108, Clarendon Press, Oxford (1955). W. Kohn, Phys. Rev. Lett. 2, 393 (1959). M . A . Krivoglaz and Tu'Hao, Defects and Properties of the Crystal Lattice, Nankova dumka, Kiev, (1968). S . C . Moss, Phys. Rev. Lett. 22, 1108 (1969). J . M . Cowley and S. W. Wilkins, Interatomic Potentials and Simulation of Lattice Defects, Edit. P. Gehlen, J. Beeler, R. Jaffee. p. 265, Plenum Press, N.Y., (1972). L. Roth, H. Zeiger and T. Kaplan, Phys. Rev. 149, 519, (1966). B. L. Gyorffy and G. M. Stocks, Phys. Rev. Lett. 50, 374, (1983). N. W. Ashcroft, Phil. Mag. 8, 2055, (1963). E. M. Gunnerson, Phil. Trans. Roy. Soc. London, A249, 299, (1957). C. O. Larson and W. L. Gordon, Phys. Rev. 156, 703, (1967). T. R. Finlayson, P. Goodman, P. D. Norman, A. Olsen and S. W. Wilkins, Acta Crystallogr. Sect. B,in press (1983). H. Sato and R. S. Toth, Phys. Rev. 124, 1833, (1961). J. Friedel, Electron-Phonon Interactions and Phase Transitions, Edit. T. Riste, p. i, Plenum, N.Y., (1977).
MIITALLUR(]ICA
Vol.
18,
Printed
pp.
575-578,
in t h e
1984
Pergamon
U.S.A.
..\11 r i g h t s
Press
Ltd.
reserved
PREMARTENSITIC PHENOMENA IN In-Tl ALLOYS AND THE ROLE OF THE FERMI SURFACE P. D. Normantt, S. W. Wilkins**, T. R. Finlayson*, P. Goodman**, and A. Olsent * Department of Physics, Monash University, Melbourne, Australia ** Division of Chemical Physics, C.S.I.R.O., Melbourne, Australia t Department of Physics, University of Oslo, Norway tt Chisholm Institute of Technology, Caulfield, Australia. (Received December 8, 1983) (Revised A p r i l 12, 1 9 8 4 ) Introduction We seek to relate various observed premartensitic phenomena in In-Tl alloys, elastic constant softening, severe acoustic attenuation of some phonon modes configurations of diffuse scattering (e.g. in electron diffraction patterns), features of the Fermi surface and to the occurrence of Kohn-like anomalies. the present study is the observation that some of the ascribed Kohn anomalies reciprocal lattice points and so can be manifested in anomalous behaviour of length properties such as elastic constants and acoustic attenuation.
such as drastic and complicated to particular A novel feature in lie very close to very long-wave-
In-Tl alloys are claimed to be random disordered alloys and, in the composition range 15.5-31 at.% TI, are found to undergo a martensitic transformation at T from face-centred cubic (fcc) to face-centred tetragonal (fct) as the temperature is lowered, m T is a sensitive function of alloy composition and decreases from 425 K for 15.5 at.% TI to 0 Kmfor ~ 31 at.% TI. The transformation produces a distinctive lamellar microstructure due to {Ii0} < IT0 > transformation twinning and is called martensitic because it is diffusionless and displaclve involving homogeneous deformation. Various anomalies in physical properties of such transforming In-Tl alloys are known to occur as the temperature is lowered toward T m. For example: (i)
Novotny and Smith (i) used the pulse-echo technique to measure sonic velocities in single crystals of fcc In-Tl alloys. Three independent elastic constants were determined from these results and they found that the shear modulus, i/2(ell - c12), becomes extremely small as T m is approached. This result was taken to provide evidence of a dramatic softening of the [ ~ O ] polarised [ ~ 0 ] transverse acoustic phonon branch near the Brillouin zone centre. More recent ultrasonic velocity measurements by Pace and Saunders (2) confirmed the presence of this severe softening.
(ii) Recent x-ray and electron diffraction observations of transforming In-Tl alloys have been made by Koyama and Nittono (3) who noted the appearance of intensive thermal diffuse scattering (TDS) near the transformation temperature. This TDS, taken in conjunction with the findings of Pace and Saunders (2), was also taken to indicate the ~resence of a low frequency acoustic phonon mode along [ii0] direction and polarized [II0], in such In-Tl alloys, The major purpose of this paper is to report additional information regarding electron diffuse scattering produced by In-Tl alloys and to seek to relate the geometrical features of this diffuse scattering to an underlying physical mechanism involving a subtle coupling between lattice modulations (such as phonons and static displacement waves) and the energy of the conduction electrons in the alloy. In simple terms, reduction in the total energy of the conduction electrons can occur when lattice modulations of spatial frequency, qLM' match that for spanning vectors, 2kF , between antiparallel sections of the alloy Fermi surface as shown in Figure I. The effect is well known and the mathematical details as to how it arises can be found, for example, in Harrison (4). The effect apparently was first considered by Peierls (5) in relation to static (Peierls) modulations of a model l-dimensional structure. Subsequently Kohn (6) considered the effect in relation to phonons and the term "Kohn anomaly" is commonly used to
0036-9748/84 Copyright (c) 1 9 8 4
575 $ 3 . 0 0 + .00 Pergamon Press
l,td.
576
PREMARTENSITIC
PHENOMENA
IN In-Tl
Vol.
18, No. 6
describe the resulting dips in phonon spectra which often occur at ~ values corresponding 2k F spanning vectors. Similarly driven anomalies in short-range order diffuse scattering substltutional alloys have been discussed by Krlvoglaz and Tu Hao (7), Moss (8) and Cowley Wilklns (9). In yet another form, spatial modulations of the electron gas are understood occur due to a similar driving force and are termed charge density waves (CDW's).
to in and to
Although Kohn-llke anomalies may occur in diffuse scattering for any 2k F vector spanning antiparallel sections of the Fermi surface, according to Roth, Zelger and ~ p l a n (I0) the strongest anomalies tend to occur between sections where the curvature is lowest . Thus, for example, strong Kohn-llke anomalies tend to occur between flattened sections of the Fermi surface such as often occur in pseudo l-dlmenslonal systems. However, one should note that it is the local dlmenslonallty in reciprocal space which is important and not in direct space, so that strong anomalies may also occur in 3-dlmenslonal systems. For a given Fermi surface, the locl of Kohn anomalies may be plotted by folding the 2k_ spanning vectors with the reciprocal lattice and this is called a Kohn construction (see H a r r ~ n (4)). The strength of Kohn anomalies in the diffuse scattering is not only related to the curvature of the Fermi surface but also to the multiplicity of the overlap of the locl of these Kohn anomaly surfaces. Furthermore, regions bounded by Kohn anomaly surfaces can yield strong diffuse scattering throughout such regions. Nonetheless, the shape of the diffuse scattering contours reflects the form of the 2k F spanning vectors. It is interesting to note that recently Gyorffy and Stocks (ii) established a strong correlation between diffuse scattering maxima observed experimentally for random alloys of Cu-Pd and the 2k F vectors spanning flat and antl-parallel sheets of the alloy Fermi surface which had been calculated for the same system using the coherent potential approximation.
I[oo]
Ifool
Fig. 1 - 2k_ ~F spanning vector for Fermi surface regions with antl-parallel group velocities.
Fig. 3 - The third zones of the Fermi surface of indium (fct).
Fig. 2 - The third zones of the Fermi surface of alumlnlum (fcc). Kohn Constructions
Unfortunately there is no Fermi surface data for cubic In-Tl available in the literature. However, aluminium and In-Tl may be assumed to have roughly comparable electronic structures within the nearly free electron picture since they beth have nearly three conduction electrons per atom,so we have chosen to use the Fermi surface of aluminlum shown in Figure 2 as the basis of our discussions concerning In-Tl. This choice is made in preference to using the Fermi surface of indium, which is illustrated in Figure 3, because both aluminium and pre-martensitic In-Tl have fcc structures whereas indium has a fct structure similar to that of the martensite phase of In-Tl. o i~ '
The Fermi surface of aluminium as illustrated in Figures 2 and 4 was determined by the 4-OPW (orthogonallzed-planewave) calculation of Ashcroft (12),whlch he compared favourably with the de Haas-van Alphen data of Gunnerson (13) and of Larson and Gordon (14). Basically, the model surface is similar to a free-electron surface for three conduction electrons with a filled first-zone, a secondzone hole surface centred at F, and a simply connected third-zone electron surface consisting of four-sided rings around the edges of the square faces of the zone. In this model there are no fourth-zone pockets. In Figure 4, the cross section of the second-zone hole surface is shaded and the cross-section of the third-zone electron surface is hatched.
III
,/ ~///
U (
0
\
I
i Ol
o 4 - A [110] section of t h e a l u m i n i u m Fermi s u r f a c e t h r o u g h the bounding face of the Brillouln zone.
Fi$.
Vol
18,
No.
6
PREMARTENSITIC PHENOblENA IN I n - T 1
577
Using this as the model Fermi surface, Kohn constructions for In-24 at.% TI have been made in the manner of Harrison (4) by drawing Kohn anomaly surfaces of radius 2k_ about each of the ~F eight nearest-neighbour reciprocal lattice points surrounding the given reciprocal lattice point (Bragg peak). The value of 2kF used in the [001], [112] and [110] sections in reciprocal space was taken to be slightly-less than the 2k F value for aluminium which is (2.24) 2~/a. Possible explanations for this rescaling are that the conduction electron/atom (e/a) ratio for In-Tl alloys is slightly lower than that for aluminium due to the presence of d-band effects and that the band gaps between the second and third zones in In-Tl are significantly different from those for aluminium. This difference is also suggested by comparing Figure 2 with Figure 3. The Kohn constructions of the [001] and [T12] sections shown in Figures 5 and 7 compare favourably with the corresponding diffuse scattering images obtained for the In-24 at.% TI alloy as shown in Figures 6 and 8. An independent analysis by Finlayson et al.(15) has indicated that the diffuse intensity distribution in reciprocal space is a maximum along
/', i,o; >
•
s ,~ < ,o] .-
2oo
o~o
"(9
s 't
~L_s_
020 ', /
206 , /
Fi$. 5 - Kohn construction the [001] section crosses represent reciprocal lattice
for In-24 at.% TI for where circles with out of plane points.
Fig. 6 - Electron diffraction pattern for In-24 at.% TI for [001] zone axis.
PREMARTENSITIC
578
171
. o _ 220 L • ;'
,' "
-~ "-
-
~J /-h / kJ , ,' ", "4'
13T
".
,
~e 2~X
IN
In-Tl
Vol.
18,
No.
6
.- . o
~ --_.'% r
'
PHENOMENA
/ ~ /---~ )
220 ; /el
'~ ',
,i%,
llI
311
Fig, 7 - Kohn construction for In-24 at.% TI for the IT12] section where circles with crosses represent out of plane reciprocal lattice points.
Fig. 8 - Electron diffraction pattern for In-24 at.% TI for IT12] zone axis.
O
002
o
IIool
t+
lql
0~
Fi@. 9 - Kohn construction for In-24 at.% T1 for the [TI0] section where circles with crosses represent out of plane reciprocal lattice points i. 2. 3. 4. 5. 6. 7. 8. 9. I0. ii. 12. 13. 14. 15. 16. 17.
Fig. i0 - Single contour representation of the diffuse intensity [TI0] projection.
References D. Novotony and J. F. Smith, Acta Met., 13, 881 (1965). N. Pace and G. Saunders, Proc. R. Soe., 326, 521, (1972). Y. Koyama and O. Nittono, J. Jpn. Inst. Met., 45, 869, (1981). W . A . Harrison, Phonons, Edit. R.W.H. Stevenson, p. 103, Oliver and Boyd, Edinburgh, (1966). R . E . Peierls, Quantum Theory of Solids, p. 108, Clarendon Press, Oxford (1955). W. Kohn, Phys. Rev. Lett. 2, 393 (1959). M . A . Krivoglaz and Tu'Hao, Defects and Properties of the Crystal Lattice, Nankova dumka, Kiev, (1968). S . C . Moss, Phys. Rev. Lett. 22, 1108 (1969). J . M . Cowley and S. W. Wilkins, Interatomic Potentials and Simulation of Lattice Defects, Edit. P. Gehlen, J. Beeler, R. Jaffee. p. 265, Plenum Press, N.Y., (1972). L. Roth, H. Zeiger and T. Kaplan, Phys. Rev. 149, 519, (1966). B. L. Gyorffy and G. M. Stocks, Phys. Rev. Lett. 50, 374, (1983). N. W. Ashcroft, Phil. Mag. 8, 2055, (1963). E. M. Gunnerson, Phil. Trans. Roy. Soc. London, A249, 299, (1957). C. O. Larson and W. L. Gordon, Phys. Rev. 156, 703, (1967). T. R. Finlayson, P. Goodman, P. D. Norman, A. Olsen and S. W. Wilkins, Acta Crystallogr. Sect. B,in press (1983). H. Sato and R. S. Toth, Phys. Rev. 124, 1833, (1961). J. Friedel, Electron-Phonon Interactions and Phase Transitions, Edit. T. Riste, p. i, Plenum, N.Y., (1977).