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Calculation of the BRT function with a programmable pocket calculator
The capability of currently available programmable pocket calculators is discussed, and a typical application illustrated.
Calculation of the BRT function with a programmable pocket calculator R. L. Fagaly
Ramos and Sanchez ~have tabulated the electronic thermal conductivities of superconductors as described by Bardeen, Rickayzen and Tewordt (BRT). 2 The BRT result of Kes/Ken, where Kes and Ken refer to the superconducting and normal state electronic thermal conductivities respectively, is a function of
y = A(T)/kBT where A(T) is the energy gap s at a temperature T, and kB is Boltzmarm's constant. Bardeen, Cooper and Schrieffer (BSC) 3 have shown that the temperature dependence of A(T) is expressed by: dw
h~ °
gN(0)
J 0
6o2 + A2(T)
~o2 + A2(T) tanh
2kBT
- 1
(I)
where h is Plancks constant/21r, Wo the Debye cut-off frequency, N(0) is the density of states, and g is the interaction energy. A(T) is usually obtained from tabulated results 4 as no solution to (1) exists in closed form. s Programmable pocket calculators offer an alternative to numerical tabulations of such calculations. For example, it is possible to approximate A(T) by a piecewise solution within a desired accuracy. Utilizing least squares regression analysis ~ with a Hewlett-Packard HP41C calculator, we can express A(T), within 0.01% as: (1 - t2"Ts)u2 (0.9847 + 0.I 577t - 0.0953t 2) (2a)
I >t >0.7
A(T!=
( 1 - ts'3)v2(0.9710 + 0 . 1 7 8 6 t - 0.2035t 2)
A(O)
0.7 > t 1 - 1.89(t) 1/2 exp ( - 1.76/t)
0.36 > t
We find: - - =
1 + 12~ -2
Ken
Using standard programming techniques, 6 we have written a short program 7 to determine Kes/gen. A 16 point gaussian quadrature s is used to evaluate the integral in (4). The advantage of using a calculator includes the ability to evaluate ges/Ken for A(0) different from the values listed in Ramos and Sanchez [2A(0)/kB Tc = 2.8, 3.53, 4.2 ]1 without interpolation, accurate values of A(T), and reasonable programme execution times ( ~ 15 s). This programme has been used to calculate Kes/Ken ratios for a superconducting heat switch. 9 Obviously there are problems that programmable calculators are capable of solving, but which are unsuited for them, eg the reduction of a 16 x 16 matrix or a 10 000 point integration using a trapezo'idal rule. However, for a relatively short programme (in this case 127 steps) 7 or as shown above, the small programmable pocket calculator has a significant advantage over larger computing devices in the areas of convenience and cost compared with desk-top calculators just a few years old, the pocket calculator has faster execution times and more programme steps (up to 2200 for some units). 6
(2b)
(2c) References
Using the above expression for A(T) and rearranging the BRT expression for the thermal conductivity2:
1 2
Kes = 2 F ~ ( - y ) + 2yln [1 + e x p ( - y ) +y2/[1 + exp(y)]
where
(3)
2F1(0) o~
f Fn(-Y) = o
zn 1 + exp(z + y )
4
Ramos, E. D., Sanchez, D. H., Cryogenics 14 (1974) 341 Baxdeen,J., Rickayzen, G., Tewordt, L., PhysRev 113 (1959) 982 Baxdeen,J., Cooper, L. N., Schrieff~, J. R., Phys Rev 108 (1957) 1175 Miihlschlegel,B., Z Physik 155 (1959) 313
5
Fetter, A. L., Walecka, J. D., Quantum theory of many
3
6 dz 7
The work was performed at the Centre de Recherches sur les Trt~s Basses Temperatures, CNRS, BP 166 X, 38042 Grenoble-Cedex, France. The authors present address is SHE Corp, 4174 Sorrento Valley Blvd, San Diego, California, USA. Paper received 20 May 1980.
644
0011-2275/80/011644-01
(4)
1
An ~0.02% uncertainty in (4) can arise from the ~0.01% uncertainty in A(T), well below the uncertainty of any current experimental technique. However, if higher accuracy is required, all one need do is add more terms in (2a), (2b) and (2c).
where t = T/Te.
Ken
+--(z)-
2{1 + exp(y)}
8 9
particle systems, McGraw-Hill,New York (1971) chapter 13 HP41CProgrammingguide and HP users library programme # 00674D Hewlett-Packard, 1000 NE Circle Blvd, Corvallis, OR 97330, USA Fagaly,R. L., BRT superconducting thermal conductivities/ BSC energy gaps, HP users l~rary programme # 00329C Kennedy, J., PPCCalculatorJ 6 5 (1979) 19 Fagaly,R. L., Weinstoek, H., Sehmidt, F. A., Cryogenics 19 (1979) 58
$ 0 2 . 0 0 © 1980 IPC Business Press
C R Y O G EN I C S . N O V E M B E R 1980
Calculation of the BRT function with a programmable pocket calculator R. L. Fagaly
Ramos and Sanchez ~have tabulated the electronic thermal conductivities of superconductors as described by Bardeen, Rickayzen and Tewordt (BRT). 2 The BRT result of Kes/Ken, where Kes and Ken refer to the superconducting and normal state electronic thermal conductivities respectively, is a function of
y = A(T)/kBT where A(T) is the energy gap s at a temperature T, and kB is Boltzmarm's constant. Bardeen, Cooper and Schrieffer (BSC) 3 have shown that the temperature dependence of A(T) is expressed by: dw
h~ °
gN(0)
J 0
6o2 + A2(T)
~o2 + A2(T) tanh
2kBT
- 1
(I)
where h is Plancks constant/21r, Wo the Debye cut-off frequency, N(0) is the density of states, and g is the interaction energy. A(T) is usually obtained from tabulated results 4 as no solution to (1) exists in closed form. s Programmable pocket calculators offer an alternative to numerical tabulations of such calculations. For example, it is possible to approximate A(T) by a piecewise solution within a desired accuracy. Utilizing least squares regression analysis ~ with a Hewlett-Packard HP41C calculator, we can express A(T), within 0.01% as: (1 - t2"Ts)u2 (0.9847 + 0.I 577t - 0.0953t 2) (2a)
I >t >0.7
A(T!=
( 1 - ts'3)v2(0.9710 + 0 . 1 7 8 6 t - 0.2035t 2)
A(O)
0.7 > t 1 - 1.89(t) 1/2 exp ( - 1.76/t)
0.36 > t
We find: - - =
1 + 12~ -2
Ken
Using standard programming techniques, 6 we have written a short program 7 to determine Kes/gen. A 16 point gaussian quadrature s is used to evaluate the integral in (4). The advantage of using a calculator includes the ability to evaluate ges/Ken for A(0) different from the values listed in Ramos and Sanchez [2A(0)/kB Tc = 2.8, 3.53, 4.2 ]1 without interpolation, accurate values of A(T), and reasonable programme execution times ( ~ 15 s). This programme has been used to calculate Kes/Ken ratios for a superconducting heat switch. 9 Obviously there are problems that programmable calculators are capable of solving, but which are unsuited for them, eg the reduction of a 16 x 16 matrix or a 10 000 point integration using a trapezo'idal rule. However, for a relatively short programme (in this case 127 steps) 7 or as shown above, the small programmable pocket calculator has a significant advantage over larger computing devices in the areas of convenience and cost compared with desk-top calculators just a few years old, the pocket calculator has faster execution times and more programme steps (up to 2200 for some units). 6
(2b)
(2c) References
Using the above expression for A(T) and rearranging the BRT expression for the thermal conductivity2:
1 2
Kes = 2 F ~ ( - y ) + 2yln [1 + e x p ( - y ) +y2/[1 + exp(y)]
where
(3)
2F1(0) o~
f Fn(-Y) = o
zn 1 + exp(z + y )
4
Ramos, E. D., Sanchez, D. H., Cryogenics 14 (1974) 341 Baxdeen,J., Rickayzen, G., Tewordt, L., PhysRev 113 (1959) 982 Baxdeen,J., Cooper, L. N., Schrieff~, J. R., Phys Rev 108 (1957) 1175 Miihlschlegel,B., Z Physik 155 (1959) 313
5
Fetter, A. L., Walecka, J. D., Quantum theory of many
3
6 dz 7
The work was performed at the Centre de Recherches sur les Trt~s Basses Temperatures, CNRS, BP 166 X, 38042 Grenoble-Cedex, France. The authors present address is SHE Corp, 4174 Sorrento Valley Blvd, San Diego, California, USA. Paper received 20 May 1980.
644
0011-2275/80/011644-01
(4)
1
An ~0.02% uncertainty in (4) can arise from the ~0.01% uncertainty in A(T), well below the uncertainty of any current experimental technique. However, if higher accuracy is required, all one need do is add more terms in (2a), (2b) and (2c).
where t = T/Te.
Ken
+--(z)-
2{1 + exp(y)}
8 9
particle systems, McGraw-Hill,New York (1971) chapter 13 HP41CProgrammingguide and HP users library programme # 00674D Hewlett-Packard, 1000 NE Circle Blvd, Corvallis, OR 97330, USA Fagaly,R. L., BRT superconducting thermal conductivities/ BSC energy gaps, HP users l~rary programme # 00329C Kennedy, J., PPCCalculatorJ 6 5 (1979) 19 Fagaly,R. L., Weinstoek, H., Sehmidt, F. A., Cryogenics 19 (1979) 58
$ 0 2 . 0 0 © 1980 IPC Business Press
C R Y O G EN I C S . N O V E M B E R 1980