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Anomalous specific heat and nuclear spin relaxation in Al-transition metal alloys
Volume 70A, number 2
PHYSICS LETTERS
19 February 1979
ANOMALOUS SPECIFIC HEAT AND NUCLEAR SPIN RELAXATION IN Al—TRANSITION METAL ALLOYS * E. ~IMANEKand S. YOKSAN Department of Physics, University of California, Riverside, CA 92521, USA Received 17 August 1978
A mechanism involving dislocation pinning by impurities is proposed to explain the anomalous lattice specific heat and the nuclear spin—lattice relaxation in Al—transition metal alloys.
In a dilute alloy the impurity induced change of the lattice specific heat is expected to grow linearly with the impurity concentration [1,2] The dilute solid solutions of V, Cr and Mn in aluminum prepared by rapid quenching represent a drastic departure from this prediction. Aoki and Ohtsuka [3] studied the low temperature lattice specific heat of these alloys in the concentration range of 0.1 to 0.4 at% and found an enhance. ment of the Debye temperature 0D which is independent of the impurity species and saturates at the very low concentration of about 0.2 at%. Recently Parvin and MacLaughlin [4] studied the temperature dependence of the 27A1 nuclear spin—lattice relaxation in A1V and A1Mn alloys prepared by the method of ref. [3]. Their data indicate the presence of an appreciable broadening of the quasiparticle density of states also independent of the impurity species. Moreover, this broadening is found to be insensitive to the impurity concentration in the range of 2 to 5 X 10—2 at%. However, there is no anomalous broadening found on rapidly quenching pure Al samples, indicating that the broadening mechanism saturates at a very low concentration of the transition metal impurities. These similarities suggest that both the specific heat anomaly [3] and the unusual nuclear spin relaxation [4] in dilute solid solutions of 3d metals in aluminum stem from a common origin. The purpose of this note is to propose a possible .
~ Work supported by the Energy Research and Development Administration under Contract No. EY76S030034
122
explanation of these anomalies. First, let us point out that the size of the observed increase of eD is very large; it amounts to ca. ~eD leD 0.2 for c = 0.2 at%. It is instructive to compare this value with one calculated using the phase shift method [1] in a simple lattice model of Montroll and Potts [5]. In this model atoms of mass M are coupled to their nearest neighbors through equal central and non-central forces with force constant ‘y. The impurity of mass M + ~.Mis coupled to its neighbors through force constant ~ + fry. Following ref. [1] we have for small concentrations c 1
~~D1eD= —~c(a 2131(1 + K13)) (1) where a = I.VfrI/M, 13 = fry!7 and K is a numerical constant of the order of 0.1. For Mn impurity a 1 and eq. (1) implies that a large value of j3 is required to explain the observed ~0D/®D. However, even in the extreme case of 13 oe, eq. (1) yields (with K = 0.157) ~9DI®D -÷ 5.8c which is still an order of magnitude below the experimental value [3] Next we calculate /~eD/eDcaused by the lattice contraction [2] observed by Bletry [6] on a series of solid solutions of transition metals in aluminum. Using his experimental value L~V/V=—10~for c = 0.2 at% of Mn we fmd, using the Grflneisen constant ~ = 2.06 for Al, a value of = 2 X 10~,which is two orders of magnitude below the experimental value [31.These estimates show that the alloys of ref. [3] exhibit an anomalous “softening” which cannot be understood in terms of standard simple impurity effects [1,2]. A clue to the understanding of this anomaly is sug—
-~
.
Volume 70A, number 2
PHYSICS LETTERS
gested by the value of the concentration (c l0—~) at which the saturation of eD takes place. We note that at a characteristic solute concentration c = b/i, where b is the lattice constant, dislocations of average length 1 are pinned by their interactions with the impurities [7] The characteristic value of c l0~ then corresponds to 1 l0~ cm, which is a typical expected size of a dislocation network. For c ~ b/i the elastic modulus is lowered by the movement of dislocation arcs [7]. As c increases towards the value of b/i the dislocation pinning leads to an enhancement of the apparent elastic modulus towards the perfect crystal value. l’his idea is supported by the value of = 0) = 400 K observed in ref. [3] for the quenched sample of pure aluminum. This should be compared with the previously reported value of ®D(c = 0) = 431 K observed for annealed samples [8]. At low temperatures (T~0D) only the long wavelength sound waves are excited and the coefficient of the T3-specific heat depends only on the macroscopic elastic constants of the sample. Thus we use the expression relating 8D to the compressibility x and Poisson s ratio a as follows [91: -
—1/2
—
const X ~
3/2
—
L3(1
1 2a)J ‘~
—
3/2 +
[[3(1~+ °a)J1
—
1/3
(2)
—
Expressing a in terms of the shear modulus ~zand differentiating eq. (2) with respect to a we find (using a = 0.34 for Al) ~e / (3) -
D/
D
=
l2,~U
To explain the depression of 8D of pure aluminum from 431 K to 400 K via eq. (3) a value of i~p/ji= 0.15 is required. This is about three times the value which can be obtained from the Frank dislocation network [7]. A much bigger anomaly is, however, expected for fine polygonization due to higher dislocation density [7] The actual values of ®D tend to saturate at somewhat higher values than 431 K, which is difficult to understand especially in view of the fact that the effect of the strain field of immobile dislocations is known to reduce 8~[10]. The fact that the specific heat anomaly is observed in MV, AICr and A1Mn but not in AlCu poses an interesting question as to the nature of the dislocation— impurity interactions. Resonant scattering of the con.
19 February 1979
duction electrons by the transition metal impurity could be responsible for this difference. It gives rise to a strong perturbation of the forces between the impurity and its neighbors [6,11] and the surrounding stresses should be responsible for the interaction with the dislocations [7]. As for the broadening of the density of states, found in ref. [4] ,we believe that it is due to spatial fluctuations of the superconducting order parameter due to the presence of dislocations pinned by randomly distributed transition metal impurities. As shown in ref. [10] the strain field of a dislocation is accompanied by an anharmonic shift of local lattice vibration frequencies. This will produce a change of the local coupling constant of the attractive electron—electron interaction leading to spatial variations of the order parameter A rough estimate of the spatial order parameter variations can be made by relating the change of the coupling constant g to the change of local phonon frequency p caused by the local volume change ~V(r) ~.
6g(r)lg = —26v(r)/v = 2’y~V(r)/V.
(4)
The quantity ~V(r)/ V, where r is the distance from the dislocation line, is the dilation due to the anharmonic response of the “good crystal” to the dislocation strain [12] and it is of the order ofb2lr2. Using the latter value in eq. (4) and noting that near the superconducting transition temperature ~~(r)~ g(r) we obtain (r)/~ of the order of b2/r2. Recently we have applied the Anderson s—d scattering model to calculate the quantity ~~(r)/z~ near the transition metal impurity in aluminum [13] We have found that for A1Mn the resonant scattering gives a contribution of the order of (21 + 1) (kFr)2 cos 2kFr. The amplitude of the effect is of the same order as b2/r2 but due to the presence of the cos 2kFr factor we expect that the fluctuations average out to a negligible effect. Moreover the spherical symmetry of these variations should affect the bulk density of states to a considerably lesser extent than the spatially extended variations around dislocation lines. This indicates that the proposed dislocation mechanism offers a more effective source of the broadening than the single-ion resonant scattering [13]. .
The authors are grateful for helpful discussions with D.E. MacLaughlin, K. Parvin, R. Gaupsas and M. Dotson. 123
Volume 70A, number 2
PHYSICS LETTERS
References [1] B.K. Agrawal, J. Phys. C2 (1969) 252. [2] B.D. Indu, M.D. Tiwari and B.K. Agrawal, J. Phys. F8 (1978) 755. [3] R. Aoki and T. Ohtsuka, J. Phys. Soc. Japan 26 (1969) 651. [4] K. Parvin and D.E. MacLaughlin, Proc. 15th Intern. Conf. ofL,ow temperature physics (Grenoble, 1978), to be published. [5] E.W. Montroll and R.B. Potts, Phys. Rev. 100 (1955) 525. [6] J. Bletry, J. Phys. Chem. Solids 31(1970)1263.
124
19 February 1979
[7] J. Friedel, Dislocations (Pergamon Press, 1964). [8] W.M. Hartmann, H.V. Culbert and R.P. Huebener, Phys. Rev. B1 (1970) 1486. [91N.F. Mott and H. Jones, The theory of the properties of metals and alloys (Dover Publ., New York, 1936). [10] Y. Hiki, T. Maruyama and Y. Kogure, J. Phys. Soc. Japan 34 (1973) 725. [111 A. Blandin and J.L. Déplanté, J. Phys. Rad. 23 (1962) 41. [12] H. Stehle and A. Seeger, Z. Phys. 146 (1956) 217. [13] E. ~im~nek and S. Yoksan, Proc. 15th Intern. Conf. of Low temperature physics (Grenoble, 1978), to be published.
PHYSICS LETTERS
19 February 1979
ANOMALOUS SPECIFIC HEAT AND NUCLEAR SPIN RELAXATION IN Al—TRANSITION METAL ALLOYS * E. ~IMANEKand S. YOKSAN Department of Physics, University of California, Riverside, CA 92521, USA Received 17 August 1978
A mechanism involving dislocation pinning by impurities is proposed to explain the anomalous lattice specific heat and the nuclear spin—lattice relaxation in Al—transition metal alloys.
In a dilute alloy the impurity induced change of the lattice specific heat is expected to grow linearly with the impurity concentration [1,2] The dilute solid solutions of V, Cr and Mn in aluminum prepared by rapid quenching represent a drastic departure from this prediction. Aoki and Ohtsuka [3] studied the low temperature lattice specific heat of these alloys in the concentration range of 0.1 to 0.4 at% and found an enhance. ment of the Debye temperature 0D which is independent of the impurity species and saturates at the very low concentration of about 0.2 at%. Recently Parvin and MacLaughlin [4] studied the temperature dependence of the 27A1 nuclear spin—lattice relaxation in A1V and A1Mn alloys prepared by the method of ref. [3]. Their data indicate the presence of an appreciable broadening of the quasiparticle density of states also independent of the impurity species. Moreover, this broadening is found to be insensitive to the impurity concentration in the range of 2 to 5 X 10—2 at%. However, there is no anomalous broadening found on rapidly quenching pure Al samples, indicating that the broadening mechanism saturates at a very low concentration of the transition metal impurities. These similarities suggest that both the specific heat anomaly [3] and the unusual nuclear spin relaxation [4] in dilute solid solutions of 3d metals in aluminum stem from a common origin. The purpose of this note is to propose a possible .
~ Work supported by the Energy Research and Development Administration under Contract No. EY76S030034
122
explanation of these anomalies. First, let us point out that the size of the observed increase of eD is very large; it amounts to ca. ~eD leD 0.2 for c = 0.2 at%. It is instructive to compare this value with one calculated using the phase shift method [1] in a simple lattice model of Montroll and Potts [5]. In this model atoms of mass M are coupled to their nearest neighbors through equal central and non-central forces with force constant ‘y. The impurity of mass M + ~.Mis coupled to its neighbors through force constant ~ + fry. Following ref. [1] we have for small concentrations c 1
~~D1eD= —~c(a 2131(1 + K13)) (1) where a = I.VfrI/M, 13 = fry!7 and K is a numerical constant of the order of 0.1. For Mn impurity a 1 and eq. (1) implies that a large value of j3 is required to explain the observed ~0D/®D. However, even in the extreme case of 13 oe, eq. (1) yields (with K = 0.157) ~9DI®D -÷ 5.8c which is still an order of magnitude below the experimental value [3] Next we calculate /~eD/eDcaused by the lattice contraction [2] observed by Bletry [6] on a series of solid solutions of transition metals in aluminum. Using his experimental value L~V/V=—10~for c = 0.2 at% of Mn we fmd, using the Grflneisen constant ~ = 2.06 for Al, a value of = 2 X 10~,which is two orders of magnitude below the experimental value [31.These estimates show that the alloys of ref. [3] exhibit an anomalous “softening” which cannot be understood in terms of standard simple impurity effects [1,2]. A clue to the understanding of this anomaly is sug—
-~
.
Volume 70A, number 2
PHYSICS LETTERS
gested by the value of the concentration (c l0—~) at which the saturation of eD takes place. We note that at a characteristic solute concentration c = b/i, where b is the lattice constant, dislocations of average length 1 are pinned by their interactions with the impurities [7] The characteristic value of c l0~ then corresponds to 1 l0~ cm, which is a typical expected size of a dislocation network. For c ~ b/i the elastic modulus is lowered by the movement of dislocation arcs [7]. As c increases towards the value of b/i the dislocation pinning leads to an enhancement of the apparent elastic modulus towards the perfect crystal value. l’his idea is supported by the value of = 0) = 400 K observed in ref. [3] for the quenched sample of pure aluminum. This should be compared with the previously reported value of ®D(c = 0) = 431 K observed for annealed samples [8]. At low temperatures (T~0D) only the long wavelength sound waves are excited and the coefficient of the T3-specific heat depends only on the macroscopic elastic constants of the sample. Thus we use the expression relating 8D to the compressibility x and Poisson s ratio a as follows [91: -
—1/2
—
const X ~
3/2
—
L3(1
1 2a)J ‘~
—
3/2 +
[[3(1~+ °a)J1
—
1/3
(2)
—
Expressing a in terms of the shear modulus ~zand differentiating eq. (2) with respect to a we find (using a = 0.34 for Al) ~e / (3) -
D/
D
=
l2,~U
To explain the depression of 8D of pure aluminum from 431 K to 400 K via eq. (3) a value of i~p/ji= 0.15 is required. This is about three times the value which can be obtained from the Frank dislocation network [7]. A much bigger anomaly is, however, expected for fine polygonization due to higher dislocation density [7] The actual values of ®D tend to saturate at somewhat higher values than 431 K, which is difficult to understand especially in view of the fact that the effect of the strain field of immobile dislocations is known to reduce 8~[10]. The fact that the specific heat anomaly is observed in MV, AICr and A1Mn but not in AlCu poses an interesting question as to the nature of the dislocation— impurity interactions. Resonant scattering of the con.
19 February 1979
duction electrons by the transition metal impurity could be responsible for this difference. It gives rise to a strong perturbation of the forces between the impurity and its neighbors [6,11] and the surrounding stresses should be responsible for the interaction with the dislocations [7]. As for the broadening of the density of states, found in ref. [4] ,we believe that it is due to spatial fluctuations of the superconducting order parameter due to the presence of dislocations pinned by randomly distributed transition metal impurities. As shown in ref. [10] the strain field of a dislocation is accompanied by an anharmonic shift of local lattice vibration frequencies. This will produce a change of the local coupling constant of the attractive electron—electron interaction leading to spatial variations of the order parameter A rough estimate of the spatial order parameter variations can be made by relating the change of the coupling constant g to the change of local phonon frequency p caused by the local volume change ~V(r) ~.
6g(r)lg = —26v(r)/v = 2’y~V(r)/V.
(4)
The quantity ~V(r)/ V, where r is the distance from the dislocation line, is the dilation due to the anharmonic response of the “good crystal” to the dislocation strain [12] and it is of the order ofb2lr2. Using the latter value in eq. (4) and noting that near the superconducting transition temperature ~~(r)~ g(r) we obtain (r)/~ of the order of b2/r2. Recently we have applied the Anderson s—d scattering model to calculate the quantity ~~(r)/z~ near the transition metal impurity in aluminum [13] We have found that for A1Mn the resonant scattering gives a contribution of the order of (21 + 1) (kFr)2 cos 2kFr. The amplitude of the effect is of the same order as b2/r2 but due to the presence of the cos 2kFr factor we expect that the fluctuations average out to a negligible effect. Moreover the spherical symmetry of these variations should affect the bulk density of states to a considerably lesser extent than the spatially extended variations around dislocation lines. This indicates that the proposed dislocation mechanism offers a more effective source of the broadening than the single-ion resonant scattering [13]. .
The authors are grateful for helpful discussions with D.E. MacLaughlin, K. Parvin, R. Gaupsas and M. Dotson. 123
Volume 70A, number 2
PHYSICS LETTERS
References [1] B.K. Agrawal, J. Phys. C2 (1969) 252. [2] B.D. Indu, M.D. Tiwari and B.K. Agrawal, J. Phys. F8 (1978) 755. [3] R. Aoki and T. Ohtsuka, J. Phys. Soc. Japan 26 (1969) 651. [4] K. Parvin and D.E. MacLaughlin, Proc. 15th Intern. Conf. ofL,ow temperature physics (Grenoble, 1978), to be published. [5] E.W. Montroll and R.B. Potts, Phys. Rev. 100 (1955) 525. [6] J. Bletry, J. Phys. Chem. Solids 31(1970)1263.
124
19 February 1979
[7] J. Friedel, Dislocations (Pergamon Press, 1964). [8] W.M. Hartmann, H.V. Culbert and R.P. Huebener, Phys. Rev. B1 (1970) 1486. [91N.F. Mott and H. Jones, The theory of the properties of metals and alloys (Dover Publ., New York, 1936). [10] Y. Hiki, T. Maruyama and Y. Kogure, J. Phys. Soc. Japan 34 (1973) 725. [111 A. Blandin and J.L. Déplanté, J. Phys. Rad. 23 (1962) 41. [12] H. Stehle and A. Seeger, Z. Phys. 146 (1956) 217. [13] E. ~im~nek and S. Yoksan, Proc. 15th Intern. Conf. of Low temperature physics (Grenoble, 1978), to be published.