- Email: [email protected]
Hydrostatic pressure derivatives of the elastic constants of dysprosium over the range 0–100 MPa
Volume 87A, number 6
PHYSICS LETTERS
18 January 1982
HYDROSTATIC PRESSURE DERIVATIVES OF THE ELASTIC CONSTANTS OF DYSPROSIUM OVER THE RANGE 0-100 MPa D.C. JILESa, S.B. PALMERa and G.A. SAUNDERSb a Department ofApplied Physics, University of Hull, Hull, UK b School of Physics, University of Bath, Bath, UK Received 27 October 1981
Measurements of the variation of the elastic modulus C
33 of single crystal dysprosium as a function of2C pressure 2)po up to have 100 MPa yielded indicate valuesthat of 8.35 the relationship ± 0.15 (dimensionless) is not linear.and Calculations (—3.8 ±0.4) of the X 1coefficients ~—8Pa’ respectively. (aC33/aP)p~and (a 33/aP
Introduction. Recent measurements of the behavjour of the single crystal elastic moduli of the heavy
rare earths, terbium, dysprosium and erbium [1] have suggested that the pressure dependence may be non-linear. However, these early results were unable to show sufficient detail in the range 0—100 MPa except to indicate that the pressure derivative over this range was greater than that obtained over the range 0—500 MPa, and hence that there were some significant fourth order effects in these materials, Careful measurements of the pressure dependence of the elastic modulus C 33 of dysprosium over the pressure range 0—100 MPa have been carried out to investigate the matter further. Measurements were carried out at 292 K, well above the Néel temperature of 180 K and hence far removed from any magnetic transitions. Elastic constant values were determined at, typically, 4 MPa intervals, allowing a more detailed investigation of the pressure derivative over this range than had previously been possible. The velocity of sound was measured by propagating 15 MHz longitudinal waves down the hexagonal c-axis of the single crystal sample. To measure the pressure-induced changes in ultrasonic wave velocity, an automatic frequency-controlled, gated-carrier pulse superposition equipment was used [2] which was capable of a resolution of better than 1 part rn l0~the absolute accuracy of ultrasonic wave transit time was measured to ±0.2%.The hydrostatic pres0 031.9163/82/0000—0000/s02.75 © 1982 North-Holland
sure dependences of the ultrasonic wave transit times were measured in a piston-cylinder apparatus [31. Results. The results of a typical experimental run are shown in fig. 1, where the fractional change in elastic constant has been measured as a function of both increasing and decreasing pressure. It is clear that the elastic constant is not simply a linear function of pressure. The pressure derivative over the range 0—20 MPa is 6.64 while over the range 50—100 MPa it has decreased to 5.47. This shows a significant 8.0
0 0
i Is e0 . 0
2.0
•0
•0
~‘ressureincreasing preSsure decreasing
• •8
0 50
100 Hydrostatic
pressure IMP.)
Fig. 1. Hydrostatic pressure dependence of the elastic constant C33 of Dy measured at 292 K. • pressure increasing, o pressure decreasing.
297
Volume 87A, number 6
PHYSICS LETTERS
a least squares straight line fit. It should be noted that the experimental errors in fig. 2 are proportional to P~and so high pressure values are more reliable than low pressure. The values of these coefficients obtained from fig. 2 were,
0 0
8.0
•
~
• •
•• •
•
8
• • •
0
-.
0
(ac’/aP~~~ = 8.35±0.15, 0 2c/aP2)~ (a 0= (—3.8 ±0.4) X 10—8 Pa~
•
0
0
00
18 January 1982
:
0 •
0
•
~
~
where (aC/aP)~5,0is dimensionless.
6.0
Hyarostat~cpressure (Mpa)
Fig. 2. Hydrostatic pressure dependence of C’(P)/F (see text). The straight line is a least squares line fit to the data.
change in slope of the graph over these pressure ranges and the results aie in good agreement with the earlier reported values [1,4]. Since the results appeared to be non-linear the next approximation was to make a quadratic fit to the values of elastic constant against pressure. This equation may be expressed in the 2Cform \ 1 /8 C(F) = Cr0) + I8C\ ~ P+~_) p2, (1)
r=o
p=o
pressure and where higher order terms in the polynomial have been neglected. By rearranging the terms and putting C’(P) C(P) — Cr0) this gives —~----=
i\Of~ /j
+
—
1 ia2c’ i
\~jp2~
—,—_--
L
p,
has allowed a study of the pressure dependence of C33 of dysprosium. It has been confirmed that there
is a non-linear dependence of C33 on hydrostatic pressure. The appearance of this non-linear behaviour in the variation C33 is an indication that in this case the fourth order terms are not negligible. The values of (aC/aP)~,00are related to the third order elastic constants as indicated2C/DP2)~ in the results for erbium [5],while the values of (lJ 5,0are relatedbytoBrugger the fourth elasticit constants as defined [6].order However, is not possible to calculate the values of the Ciiklmnpq~the fourth
order elastic constants, from the data available at present.
where C(F) is the elastic constant as a function of
C’(P) laC \
Conclusion. The use of a high sensitivity pulse superposition technique, together with an hydrostatic pressure system operating in the range 0—100 MPa
(2)
and hence if eq. (1) is a good fit to the data it would be expected that a plot of C’(P)/P against P would be a straight line. This is given in fig. 2 and allows the coefficients
References [11 D.C. Jiles and S.B. Palmer, Phios. Mag. 44 (1981)447. [21 Y.K. Yogwtzu, E.F. Lainbson, A.J. Mifier and G.A. Saunders, Ultrasonics 18 (1980) 155. [3] M.P. Brassington, W.A. Lambson, A.J. Miller, G.A. [4] E.S.Fisher,M,H.Manghnanj andR. ~42~(l~O) 127. Chem. Solids 34 (1973) 687. [5] D.C. Jiles and S.B. Palmer, J. App!. Phys. 52 (1981) 1113.
(a2C/aP2)~ 0and (aC/aP)~~0 to be extracted using
298
[6] K. Brugger, J. App!. Phys. 36 (1965) 759.
2
PHYSICS LETTERS
18 January 1982
HYDROSTATIC PRESSURE DERIVATIVES OF THE ELASTIC CONSTANTS OF DYSPROSIUM OVER THE RANGE 0-100 MPa D.C. JILESa, S.B. PALMERa and G.A. SAUNDERSb a Department ofApplied Physics, University of Hull, Hull, UK b School of Physics, University of Bath, Bath, UK Received 27 October 1981
Measurements of the variation of the elastic modulus C
33 of single crystal dysprosium as a function of2C pressure 2)po up to have 100 MPa yielded indicate valuesthat of 8.35 the relationship ± 0.15 (dimensionless) is not linear.and Calculations (—3.8 ±0.4) of the X 1coefficients ~—8Pa’ respectively. (aC33/aP)p~and (a 33/aP
Introduction. Recent measurements of the behavjour of the single crystal elastic moduli of the heavy
rare earths, terbium, dysprosium and erbium [1] have suggested that the pressure dependence may be non-linear. However, these early results were unable to show sufficient detail in the range 0—100 MPa except to indicate that the pressure derivative over this range was greater than that obtained over the range 0—500 MPa, and hence that there were some significant fourth order effects in these materials, Careful measurements of the pressure dependence of the elastic modulus C 33 of dysprosium over the pressure range 0—100 MPa have been carried out to investigate the matter further. Measurements were carried out at 292 K, well above the Néel temperature of 180 K and hence far removed from any magnetic transitions. Elastic constant values were determined at, typically, 4 MPa intervals, allowing a more detailed investigation of the pressure derivative over this range than had previously been possible. The velocity of sound was measured by propagating 15 MHz longitudinal waves down the hexagonal c-axis of the single crystal sample. To measure the pressure-induced changes in ultrasonic wave velocity, an automatic frequency-controlled, gated-carrier pulse superposition equipment was used [2] which was capable of a resolution of better than 1 part rn l0~the absolute accuracy of ultrasonic wave transit time was measured to ±0.2%.The hydrostatic pres0 031.9163/82/0000—0000/s02.75 © 1982 North-Holland
sure dependences of the ultrasonic wave transit times were measured in a piston-cylinder apparatus [31. Results. The results of a typical experimental run are shown in fig. 1, where the fractional change in elastic constant has been measured as a function of both increasing and decreasing pressure. It is clear that the elastic constant is not simply a linear function of pressure. The pressure derivative over the range 0—20 MPa is 6.64 while over the range 50—100 MPa it has decreased to 5.47. This shows a significant 8.0
0 0
i Is e0 . 0
2.0
•0
•0
~‘ressureincreasing preSsure decreasing
• •8
0 50
100 Hydrostatic
pressure IMP.)
Fig. 1. Hydrostatic pressure dependence of the elastic constant C33 of Dy measured at 292 K. • pressure increasing, o pressure decreasing.
297
Volume 87A, number 6
PHYSICS LETTERS
a least squares straight line fit. It should be noted that the experimental errors in fig. 2 are proportional to P~and so high pressure values are more reliable than low pressure. The values of these coefficients obtained from fig. 2 were,
0 0
8.0
•
~
• •
•• •
•
8
• • •
0
-.
0
(ac’/aP~~~ = 8.35±0.15, 0 2c/aP2)~ (a 0= (—3.8 ±0.4) X 10—8 Pa~
•
0
0
00
18 January 1982
:
0 •
0
•
~
~
where (aC/aP)~5,0is dimensionless.
6.0
Hyarostat~cpressure (Mpa)
Fig. 2. Hydrostatic pressure dependence of C’(P)/F (see text). The straight line is a least squares line fit to the data.
change in slope of the graph over these pressure ranges and the results aie in good agreement with the earlier reported values [1,4]. Since the results appeared to be non-linear the next approximation was to make a quadratic fit to the values of elastic constant against pressure. This equation may be expressed in the 2Cform \ 1 /8 C(F) = Cr0) + I8C\ ~ P+~_) p2, (1)
r=o
p=o
pressure and where higher order terms in the polynomial have been neglected. By rearranging the terms and putting C’(P) C(P) — Cr0) this gives —~----=
i\Of~ /j
+
—
1 ia2c’ i
\~jp2~
—,—_--
L
p,
has allowed a study of the pressure dependence of C33 of dysprosium. It has been confirmed that there
is a non-linear dependence of C33 on hydrostatic pressure. The appearance of this non-linear behaviour in the variation C33 is an indication that in this case the fourth order terms are not negligible. The values of (aC/aP)~,00are related to the third order elastic constants as indicated2C/DP2)~ in the results for erbium [5],while the values of (lJ 5,0are relatedbytoBrugger the fourth elasticit constants as defined [6].order However, is not possible to calculate the values of the Ciiklmnpq~the fourth
order elastic constants, from the data available at present.
where C(F) is the elastic constant as a function of
C’(P) laC \
Conclusion. The use of a high sensitivity pulse superposition technique, together with an hydrostatic pressure system operating in the range 0—100 MPa
(2)
and hence if eq. (1) is a good fit to the data it would be expected that a plot of C’(P)/P against P would be a straight line. This is given in fig. 2 and allows the coefficients
References [11 D.C. Jiles and S.B. Palmer, Phios. Mag. 44 (1981)447. [21 Y.K. Yogwtzu, E.F. Lainbson, A.J. Mifier and G.A. Saunders, Ultrasonics 18 (1980) 155. [3] M.P. Brassington, W.A. Lambson, A.J. Miller, G.A. [4] E.S.Fisher,M,H.Manghnanj andR. ~42~(l~O) 127. Chem. Solids 34 (1973) 687. [5] D.C. Jiles and S.B. Palmer, J. App!. Phys. 52 (1981) 1113.
(a2C/aP2)~ 0and (aC/aP)~~0 to be extracted using
298
[6] K. Brugger, J. App!. Phys. 36 (1965) 759.
2