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The X-ray debye temperature of tungsten
Mat. Res. Bull. Vol. 4, pp. in the United States.
137-14Z,
1969.
Pergamon
Press,
Inc.
Printed
THE X-RAY DEBYE TEMPERATURE OF TUNGSTEN
L. K. Walford Physics Department, Southern lllinois University Edwardsville, Illinois 62025 {Received D e c e m b e r
Z7, 1968; Refereed)
ABSTRACT The x-ray Debye temperature (0M) of tungsten has been measured to be 377 i II°K. The method used was the measurement of the integrated intensity of the {310} and {321} reflections at two temperatures.
Introduction There is a scarcity of adequate data on x-ray Debye temperatures in the literature.
This is evidenced by the large ranges given in the International
Tables (i) and from review articles, e.g. Herbstein (2).
A recent paper by
Salter (3) shows that in general Debye temperatures determined by one method cannot usually be used for description of other properties.
Barron et al (4)
show for example that 9 C values derived from heat capacity data cannot be related to the @M values used in diffraction applications in the simple way suggested by Zener and Bilinsky (5).
Therefore it would seem advisable to measure
@M by a diffraction experiment. The range of values for tungsten quoted in the International Tables is 270-384°K,, Theory The integrated intensity of a Bragg reflection depends on the specimen temperature.
The intensity is reduced relative to the absolute zero value by
the Debye-Waller factor, exp(-2M).
The derivation of the equations quoted here
is given by James (6) pp. 216-239. 2M(T) = 12h2sin2O
[~(x)/x + i/4]/mk@M(T)% 2
2M(T I) - 2M(T 2) = £n[l(T 2)/I(T I)] 137
(i) (2)
138
TUNGSTEN
Vol. 4, No.
2
x
1 ~(x) = ~ 0
f
~d~ (e£-l)
(3)
where m is the mass of the atom h is Planck's constant k is Boltzmann's constant T is the absolute temperature x = @M/T I(T) is the integrated intensity at temperature T.
The function
(~(x) + x/4) is tabulated by James who also gives it as a power series in x, valid for x
< 27.
Barron et al (4) also indicate that the typical variation of GM(T) is likely to be much less than
that of @C(T).
They give a high temperature
(T > hmm/2~k) expansion for @M(T), viz.:
@C being the Debye temperature deduced from heat capacity data. @C and @M in the literature is about 300-400 °K. (@ C / @ M )
The range of
Taking the extreme case of
= 3/4, we find that for T = 77°K the value of @M(T) is only 4% less
than that of @ M.
This is less than twice the expected standard deviation of
the final result.
Thus, although Waite et al (7) quote a range of 8 C of 305°K
to 378°K for the temperature range 300°K to 0°K, it is very unlikely will vary that much.
The analysis of equations 1 and 2 may be made much sim-
pler by assuming @M(T) to be a constant. (@M) 2
=
that @M(T)
12h2[Tl~(X I) sin201 - T2~(x 2)
The final equation is then:
sin282]/mkX2£n[l(T2)/l(Tl)]
The calculation of the function ~(x) requires knowledge of @M.
(5)
However,
~(x) does not change much with @M, so a guessed value of @M (such as that predicted from the Lindemann equation)
is used to calculate ~(x).
If the value
of @M so derived differs from the guessed 0M, the process is repeated with the new 0M in the expression for ~(x).
The iteration is continued until the cal-
culation is internally consistent.
Convergence is rapid and two cycles usually
suffice. If we assume
that the dominant errors are in the integrated intensity
Vol. 4, No.
Z
TUNGSTEN
139
TABLE i Comparison of Results for @ for Tungsten
0
Temperature
(°K)
Method
Reference
359 • 26
-
Resistivity
310
-
Thermal Expansion
Nix and MacNair
384
0
Elastic constants
Featherston and Neighbours
f Gruneisen (9) Meissner and Voigt (I0) (Ii)
(12) 370 ± 4
293
Elastic constants
Neighbours and Alers (13)
371
300
Elastic constants
364
300
Young's modulus
Koster (16)
/ Bridgeman (14) Wright (15)
388 ± 17
0
Specific heat
Clausius and Franzosini (17)
378 ± 7
0
Specific heat
Waite, Craig, and Wallace
(7) 305
300
Specific heat
Waite, Craig, and Wallace
(7) 312 ± 3
293
291
-
377 ~ ii
measurements,
Specific heat
J. de Launay (18)
Lindemann equation
77-293
X-ray diffraction
Present work
then the error 40 in @ using equation (4) is given to a first
approximation by
=
1 2 £n (12/II)
&12/12
+
AII/II
(6)
Experimental Procedure The integrated intensities of the {310} and
{321} reflections were
measured by the balanced filter technique using copper radiation, pulse height selection and 2@ scans.
The powdered tungsten (99.95% pure) was sieved
140
TUNGSTEN
Vol. 4, No. Z
through a 325 mesh sieve and mounted on the cold finger of a Materials Research Corporation low temperature attachment for the General Electric XRD6 Diffractometer.
Liquid nitrogen was used in the dewar, the sample temperature being
measured with an iron-constantan thermocouple.
The reference junction was an
ice-water mixture and the thermo-e.m.f, was measured using a Leeds and Northrup K3 potentiometer. to be ~ I°K.
The precision of the temperature measurements was estimated
The thermal diffuse scattering correction was calculated using
the method of Chipman and Paskin (8). Results and Discussion Equation 6 shows that when £n (I2/I I) is small, very large errors in @M occur even though the precision of the individual integrated intensity measurements might be of the order of 2% (on a relative scale). high angle reflections should be used.
For this reason
The reflections used were the {310}
and {321} reflections. The measured value of @M is 377 i II°K.
This result is compared to pre-
vious work by other methods in Table i. The present result is appropriate in x-ray diffraction experiments in which the Debye-Waller factor must be used to correct the intensity of a Bragg peak for temperature effects. If the x-ray scattering were measured as a function of temperature down to 4°K then some knowledge of the phonon frequency distribution could be gained. The values of @o M and @ M could be used with O's measured by other means (@ C, @oC, @ E and 0 oE) to determine the moments of the frequency distribution.
This
in turn can yield information about how closely the solid approximates to a Debye Solid. References
i.
I n t e r n a t i o n a l T a b l e s f o r X - R a y C r y s t a l l o g r a p h y , Vol. ILl, e d i t e d by K. Lonsdale,
K y n o c k Press,
Birmingham
(1962).
2. F. H. Herbstein, A d v a n c e s in Physics I0, 313 (1961). 3. L. S. Salter, A d v a n c e s in P~ysics 14, 1 (1965). 4. T. H. K. Barron, A. J. Leadbetter, Acta Cryst. Z.0, 125 (1966). 5. C. Z e n e r and S. Bilinsky, Phys. Rev. 6. R. W. J a m e s , (1962). 7. T. R.
(1956).
J. A. M o r r i s o n and L. S. Salter, 50, I01 (1936).
T h e Crystalline State, Vol. II, G. Bell and Sons, L o n d o n
Waite, R. S. Craig and W. E. Wallace, Phys. Rev.
104, 1240
Vol. 4, No. Z
TUNGSTEN
141
8
D. R. C h i p m a n and A. Paskin, J. Appl. Phys. 3__O0,1998 (1959).
9
E. Gruneisen, Ann. Physik.
I._6_6,530 (1933).
i0
W. Meissner and B. Voigt, Ann. Physik 7, 893 (1930).
ii
F. C. Nix and D. MacNair,
IZ
F. H. Featherston and J. R. Neighbours,
13
J. R. Neighbours and G. A. Alers, Phys. Rev.
14. P. W. Bridgeman,
Phys. Rev. 6__O0,597 (1941); 6_.J_l,74 (194Z).
Proc. A m e r .
Phys. Rev.
130, 13Z4 (1963).
l__!_l,707 (1958).
Acad. Arts Sci. 6__O0,305 (1925).
15. S. J. Wright, Proc. Roy. Soc. AIZ6, 313 (1930). 16. W. Eoster, AppI. Sci. Res. A4, 3Z9 (1954). 17. K. Clausius and P. Franzosini,
Z. Naturforsch.
14a, 99 (1959).
137-14Z,
1969.
Pergamon
Press,
Inc.
Printed
THE X-RAY DEBYE TEMPERATURE OF TUNGSTEN
L. K. Walford Physics Department, Southern lllinois University Edwardsville, Illinois 62025 {Received D e c e m b e r
Z7, 1968; Refereed)
ABSTRACT The x-ray Debye temperature (0M) of tungsten has been measured to be 377 i II°K. The method used was the measurement of the integrated intensity of the {310} and {321} reflections at two temperatures.
Introduction There is a scarcity of adequate data on x-ray Debye temperatures in the literature.
This is evidenced by the large ranges given in the International
Tables (i) and from review articles, e.g. Herbstein (2).
A recent paper by
Salter (3) shows that in general Debye temperatures determined by one method cannot usually be used for description of other properties.
Barron et al (4)
show for example that 9 C values derived from heat capacity data cannot be related to the @M values used in diffraction applications in the simple way suggested by Zener and Bilinsky (5).
Therefore it would seem advisable to measure
@M by a diffraction experiment. The range of values for tungsten quoted in the International Tables is 270-384°K,, Theory The integrated intensity of a Bragg reflection depends on the specimen temperature.
The intensity is reduced relative to the absolute zero value by
the Debye-Waller factor, exp(-2M).
The derivation of the equations quoted here
is given by James (6) pp. 216-239. 2M(T) = 12h2sin2O
[~(x)/x + i/4]/mk@M(T)% 2
2M(T I) - 2M(T 2) = £n[l(T 2)/I(T I)] 137
(i) (2)
138
TUNGSTEN
Vol. 4, No.
2
x
1 ~(x) = ~ 0
f
~d~ (e£-l)
(3)
where m is the mass of the atom h is Planck's constant k is Boltzmann's constant T is the absolute temperature x = @M/T I(T) is the integrated intensity at temperature T.
The function
(~(x) + x/4) is tabulated by James who also gives it as a power series in x, valid for x
< 27.
Barron et al (4) also indicate that the typical variation of GM(T) is likely to be much less than
that of @C(T).
They give a high temperature
(T > hmm/2~k) expansion for @M(T), viz.:
@C being the Debye temperature deduced from heat capacity data. @C and @M in the literature is about 300-400 °K. (@ C / @ M )
The range of
Taking the extreme case of
= 3/4, we find that for T = 77°K the value of @M(T) is only 4% less
than that of @ M.
This is less than twice the expected standard deviation of
the final result.
Thus, although Waite et al (7) quote a range of 8 C of 305°K
to 378°K for the temperature range 300°K to 0°K, it is very unlikely will vary that much.
The analysis of equations 1 and 2 may be made much sim-
pler by assuming @M(T) to be a constant. (@M) 2
=
that @M(T)
12h2[Tl~(X I) sin201 - T2~(x 2)
The final equation is then:
sin282]/mkX2£n[l(T2)/l(Tl)]
The calculation of the function ~(x) requires knowledge of @M.
(5)
However,
~(x) does not change much with @M, so a guessed value of @M (such as that predicted from the Lindemann equation)
is used to calculate ~(x).
If the value
of @M so derived differs from the guessed 0M, the process is repeated with the new 0M in the expression for ~(x).
The iteration is continued until the cal-
culation is internally consistent.
Convergence is rapid and two cycles usually
suffice. If we assume
that the dominant errors are in the integrated intensity
Vol. 4, No.
Z
TUNGSTEN
139
TABLE i Comparison of Results for @ for Tungsten
0
Temperature
(°K)
Method
Reference
359 • 26
-
Resistivity
310
-
Thermal Expansion
Nix and MacNair
384
0
Elastic constants
Featherston and Neighbours
f Gruneisen (9) Meissner and Voigt (I0) (Ii)
(12) 370 ± 4
293
Elastic constants
Neighbours and Alers (13)
371
300
Elastic constants
364
300
Young's modulus
Koster (16)
/ Bridgeman (14) Wright (15)
388 ± 17
0
Specific heat
Clausius and Franzosini (17)
378 ± 7
0
Specific heat
Waite, Craig, and Wallace
(7) 305
300
Specific heat
Waite, Craig, and Wallace
(7) 312 ± 3
293
291
-
377 ~ ii
measurements,
Specific heat
J. de Launay (18)
Lindemann equation
77-293
X-ray diffraction
Present work
then the error 40 in @ using equation (4) is given to a first
approximation by
=
1 2 £n (12/II)
&12/12
+
AII/II
(6)
Experimental Procedure The integrated intensities of the {310} and
{321} reflections were
measured by the balanced filter technique using copper radiation, pulse height selection and 2@ scans.
The powdered tungsten (99.95% pure) was sieved
140
TUNGSTEN
Vol. 4, No. Z
through a 325 mesh sieve and mounted on the cold finger of a Materials Research Corporation low temperature attachment for the General Electric XRD6 Diffractometer.
Liquid nitrogen was used in the dewar, the sample temperature being
measured with an iron-constantan thermocouple.
The reference junction was an
ice-water mixture and the thermo-e.m.f, was measured using a Leeds and Northrup K3 potentiometer. to be ~ I°K.
The precision of the temperature measurements was estimated
The thermal diffuse scattering correction was calculated using
the method of Chipman and Paskin (8). Results and Discussion Equation 6 shows that when £n (I2/I I) is small, very large errors in @M occur even though the precision of the individual integrated intensity measurements might be of the order of 2% (on a relative scale). high angle reflections should be used.
For this reason
The reflections used were the {310}
and {321} reflections. The measured value of @M is 377 i II°K.
This result is compared to pre-
vious work by other methods in Table i. The present result is appropriate in x-ray diffraction experiments in which the Debye-Waller factor must be used to correct the intensity of a Bragg peak for temperature effects. If the x-ray scattering were measured as a function of temperature down to 4°K then some knowledge of the phonon frequency distribution could be gained. The values of @o M and @ M could be used with O's measured by other means (@ C, @oC, @ E and 0 oE) to determine the moments of the frequency distribution.
This
in turn can yield information about how closely the solid approximates to a Debye Solid. References
i.
I n t e r n a t i o n a l T a b l e s f o r X - R a y C r y s t a l l o g r a p h y , Vol. ILl, e d i t e d by K. Lonsdale,
K y n o c k Press,
Birmingham
(1962).
2. F. H. Herbstein, A d v a n c e s in Physics I0, 313 (1961). 3. L. S. Salter, A d v a n c e s in P~ysics 14, 1 (1965). 4. T. H. K. Barron, A. J. Leadbetter, Acta Cryst. Z.0, 125 (1966). 5. C. Z e n e r and S. Bilinsky, Phys. Rev. 6. R. W. J a m e s , (1962). 7. T. R.
(1956).
J. A. M o r r i s o n and L. S. Salter, 50, I01 (1936).
T h e Crystalline State, Vol. II, G. Bell and Sons, L o n d o n
Waite, R. S. Craig and W. E. Wallace, Phys. Rev.
104, 1240
Vol. 4, No. Z
TUNGSTEN
141
8
D. R. C h i p m a n and A. Paskin, J. Appl. Phys. 3__O0,1998 (1959).
9
E. Gruneisen, Ann. Physik.
I._6_6,530 (1933).
i0
W. Meissner and B. Voigt, Ann. Physik 7, 893 (1930).
ii
F. C. Nix and D. MacNair,
IZ
F. H. Featherston and J. R. Neighbours,
13
J. R. Neighbours and G. A. Alers, Phys. Rev.
14. P. W. Bridgeman,
Phys. Rev. 6__O0,597 (1941); 6_.J_l,74 (194Z).
Proc. A m e r .
Phys. Rev.
130, 13Z4 (1963).
l__!_l,707 (1958).
Acad. Arts Sci. 6__O0,305 (1925).
15. S. J. Wright, Proc. Roy. Soc. AIZ6, 313 (1930). 16. W. Eoster, AppI. Sci. Res. A4, 3Z9 (1954). 17. K. Clausius and P. Franzosini,
Z. Naturforsch.
14a, 99 (1959).