Gfer: Reference manual

Introduction

The code of this library is stored in the file Gfer.f08 in the form of Fortran 2008 source. This file can be compiled using GFortran free compiler or other alternatives. For Linux operating systems, it is usually simpler to install the compiler from the official distribution repository. The code has been designed to not require further dependency and can be compiled using the command line reported in the source code.

Alternatively, the library is distributed in the form of compiled shared library Gfer.dll and Gfer.so for Windows™ 64 bit and Linux 64 bit operative systems. A 32 bit version is available only for Windows™.

The name of the functions, exposed by the library, are reported in the List of function session. It has been chosen to use the "C" format for the compiled code so that the functions can be called by almost all the development environment and software for numerical analysis.

All the quantities are represented using double precision floating point with a length of 8 bytes while integer indexes, when necessary, are implemented using signed integer of 4 bytes. Arrays are pointers to double.

All values are passed by reference in the form of pointers.

For a sake of testing, a LabView™ 16 bit library has been implemented to link Gfer functions in a form of a set of virtual instruments. LabView™ Library is contained in the Gfer.llb file. A Maple™ 16 bit version is already available if requested.

It has been chosen to leave to the end user the freedom to provide the properties needed to calculate the corrections.

Sometimes corrections depend on mechanical dimensions of the resonator, ducts, tubes and so on. These quantities are expressed in millimeters while thermodynamic quantities are expressed in SI fundamental units. Internally, the conversion from millimeters to meters is applied when necessary.

An implementation for the Bessel function, optimized for the corrections applied to radial modes, has been included in the library and are called internally when necessary. In this way, Gfer library doesn’t depend from external math libraries.

Installation

Please check the terms of use described in the license before installing this software.

Gfer library is not provided with an automatic installer because its installation depends on the choices of the end-user. For LabView™, the shared library, being Gfer.dll or Gfer.so, should be saved in the same folder of the Gfer.llb file. Other development platforms and software for numerical analysis might request to save libraries in determined places. Please refer to the manual of the software connecting to the Gfer libraries to get the necessary information.

For 32 bits platforms (only Windows™), it is possible use Gfer_32bit.dll. To this end, rename the file to Gfer.dll before using.

Acoustic modes

The following functions will consider only radial acoustic modes with n<11. Modes are labeled as (0,n) with n=2,3,… 10.

Ideal frequencies

f_id(n, w, req)

n	Integer(4)	none	Index of the acoustic mode
w	Real(8)	m/s	Speed of sound
Req	Real(8)	mm	Radius of the equivalent sphere
Return	Real(8)	Hz	Frequency of the mode

Reference:

J. B. Mehl, M. R. Moldover and L. Pitre, Designing quasi-spherical resonators for acoustic thermometry, Metrologia 41 (2004) 295–304; https://doi.org/10.1088/0026-1394/41/4/011

Boundary layer

Thermal penetration length: d_t

d_t(K, rho, cp, f)

K	Real(8)	W/(m K)	Thermal conductivity of the gas
rho	Real(8)	kg/m³	Density of the gas
cp	Real(8)	J/(kg K)	Constant pressure specific heat capacity of the gas
f	Real(8)	Hz	Frequency of the acoustic mode
Return	Real(8)	m	Thermal penetration length

Reference:

M. R. Moldover, J. B. Mehl and M. Greenspan, Gas-filled spherical resonators: Theory and experiment, J. Acoust. Soc. Am. 79 (2}, February 1986; https://doi.org/10.1121/1.393566

Viscous penetration length: d_v

d_v(Nu, rho, f)

Nu	Real(8)	Pa s	Viscosity of the gas
rho	Real(8)	kg/m³	Density of the gas
f	Real(8)	Hz	Frequency of the acoustic mode
Return	Real(8)	m	Viscous penetration length

Reference:

M. R. Moldover, J. B. Mehl and M. Greenspan, Gas-filled spherical resonators: Theory and experiment, J. Acoust. Soc. Am. 79 (2}, February 1986; https://doi.org/10.1121/1.393566

Thermal accommodation length: I_t

I_t(T, p, cv, K, m, h)

T	Real(8)	K	Temperature of the gas
p	Real(8)	kPa	Pressure of the gas
cv	Real(8)	J/(kg K)	Specific heat capacity of the gas
K	Real(8)	W/(m K)	Thermal conductivity of the gas
m	Real(8)	kg/mol	Molar mass of the gas
h	Real(8)	none	Accommodation coefficient
Return	Real(8)	m	Thermal accommodation length

Reference:

G. Benedetto, R. M. Gavioso, R. Spagnolo, P. Marcarino and A. Merlone, Acoustic measurements of the thermodynamic temperature between the triple point of mercury and 380 K, Metrologia 41, 2004, 74–98; https://doi.org/10.1088/0026-1394/47/4/005

Frequency shift

Thermal boundary layer frequency shift

df_th(g, dt, dt_Cu, I_t, K, K_Cu, Req, f)

g	Real(8)	cp/cv	Specific heat capacities ratio
dt	Real(8)	m	Thermal penetration length of the gas
dt_Cu	Real(8)	m	Thermal penetration length of the copper
I_t	Real(8)	m	Thermal accommodation length
K	Real(8)	W/(m K)	Thermal conductivity of the gas
K_Cu	Real(8)	W/(m K)	Thermal conductivity of the copper
Req	Real(8)	mm	Radius of the equivalent sphere
f	Real(8)	Hz	Frequency of the acoustic mode
Return	Real(8)	Hz	Shift of the resonant frequency

Reference:

L. Pitre, M. R. Moldover and W. L. Tew, Acoustic thermometry: new results from 273 K to 77 K and progress towards 4 K, Metrologia 43 (2006) 142–162; https://doi.org/10.1088/0026-1394/43/1/020

Separated contributions

Next three functions reproduce the three terms used to calculate df_th. Sometimes researchers prefer to keep the contributions separated to investigate them.

Frequency shift by d_t

df_dth(g, Req, d_t)

g	Real(8)	cp/cv	Specific heat capacities ratio
Req	Real(8)	mm	Radius of the equivalent sphere
d_t	Real(8)	m	Thermal penetration length of the gas
Return	Real(8)	df/f	Shift of the resonant frequency

Frequency shift by I_t

df_Ith(g, Req, I_t)

g	Real(8)	cp/cv	Specific heat capacities ratio
Req	Real(8)	mm	Radius of the equivalent sphere
I_t	Real(8)	m	Thermal accommodation length of the gas
Return	Real(8)	df/f	Shift of the resonant frequency

Frequency shift by shell-gas coupling

df_CuTh(g, Req, dt_Cu, K, K_Cu)

g	Real(8)	cp/cv	Specific heat capacities ratio
Req	Real(8)	mm	Radius of the equivalent sphere
dt_Cu	Real(8)	m	Thermal penetration length of the copper
K	Real(8)	W/(m K)	Thermal conductivity of the gas
K_Cu	Real(8)	W/(m K)	Thermal conductivity of the copper
Return	Real(8)	df/f	Shift of the resonant frequency

Ducts

df_ducts(r, L, N, rho, cp,w, g, K, Nu, Req, f, n, df_f, dg_g)

r	Array[Real(8)]	mm	Radius of the tube sections starting from the resonator
L	Array(Real(8)]	mm	Length of the tube sections starting from the resonator
N	Integer(4)	none	Number of sections loaded in the arrays
rho	Real(8)	kg/m³	Density of the gas
cp	Real(8)	J/(kg K)	Constant pressure specific heat capacity of the gas
w	Real(8)	m/s	speed of sound of the gas
g	Real(8)	cp/cv	Specific heat capacities ratio
K	Real(8)	W/(m K)	Thermal conductivity of the gas
Nu	Real(8)	Ps s	Viscosity of the gas
Req	Real(8)	mm	Radius of the equivalent sphere
f	Real(8)	Hz	frequency of the acoustic mode
n	Integer(4)	none	Index of the acoustic mode
df_f	Real(8)	df/f	Return the relative frequency shift caused by ducts
dg_f	Real(8)	dg/f	Return the relative half-width increasing of the acoustic mode caused by ducts

Reference:

J. B. Mehl, M. R. Moldover and L. Pitre, Designing quasi-spherical resonators for acoustic thermometry, Metrologia 41 (2004) 295–304; https://doi.org/10.1088/0026-1394/41/4/011

Microphones

df_mic(r, Req, rho w)

r	Real(8)	mm	radius of the microphone
Req	Real(8)	mm	radius of the equivalent sphere
rho	Real(8)	kg/m³	Density of the gas
w	Real(8)	m/s	Speed of sound in the gas
Return	Real(8)	df/f	Frequency shift of the acoustic mode

Reference:

J. B. Mehl, M. R. Moldover and L. Pitre, Designing quasi-spherical resonators for acoustic thermometry, Metrologia 41 (2004) 295–304; https://doi.org/10.1088/0026-1394/41/4/011

Radial modes eigenvalues correction for triaxial-Ellipsoids

O2dZ2AC(Eps, n)

Eps	Array[Real(8)]	none	Values of epsilon_1 and epsilon_2 encapsulated in an array
n	Integer(4)	none	Index of the acoustic mode
Return	Real(8)	dz²/z²	Shift of the square of the eigenvalue dz²/z²

Reference:

J. B. Mehl, Acoustic Eigenvalues of a Quasispherical Resonator: Second Order Shape Perturbation Theory for Arbitrary Modes, J. Res. Natl. Inst. Stand. Technol. 112, 163-173 (2007); https://doi.org/10.6028/jres.112.013

Half-width of the acoustic modes

Half-width of radial acoustic modes

g_t(T, p,g, dt, dt_Cu, K, K_Cu, Req, f)

T	Real(8)	K	Temperature of the gas
p	Real(8)	kPa	Pressure of the gas
g	Real(8)	cp/cv	Specific heat capacities ratio
dt	Real(8)	m	Thermal penetration length of the gas
dt_Cu	Real(8)	m	Thermal penetration length of the copper
K	Real(8)	W/(m K)	Thermal conductivity of the gas
K_Cu	Real(8)	W/(m K)	Thermal conductivity of the copper
Req	Real(8)	mm	Radius of the equivalent sphere
f	Real(8)	Hz	Frequency of the acoustic mode
Return	Real(8)	Hz	Increment of the half-with for thermal boundary layer

Reference:

K. A. Gillis, I. I. Shinder and M. R. Moldover, Phys. Rev. E 70 021201 (2004); https://doi.org/10.1103/PhysRevE.70.021201

Separated contributions

Next three functions calculate the separated contributions included in g_t.

Half-width increase by thermal boundary layer (first order)

g_th(g, Req, d_th)

g	Real(8)	cp/cv	Specific heat capacities ratio
Req	Real(8)	mm	Radius of the equivalent sphere
dt	Real(8)	m	Thermal penetration length of the gas
Return	Real(8)	dg/f	Half-with for thermal boundary layer

Half-width increase by thermal boundary layer (second order)

g_thO2(g, Req, d_th)

g	Real(8)	cp/cv	Specific heat capacities ratio
Req	Real(8)	mm	Radius of the equivalent sphere
dt	Real(8)	m	Thermal penetration length of the gas
Return	Real(8)	dg/f	Half-with for thermal boundary layer

Half-width increase by shell-gas coupling

g_Cu(g, Req, dt_Cu, K, K_Cu)

g	Real(8)	cp/cv	Specific heat capacities ratio
Req	Real(8)	mm	Radius of the equivalent sphere
dt_Cu	Real(8)	m	Thermal penetration length of the copper
K	Real(8)	W/(m K)	Thermal conductivity of the gas
K_Cu	Real(8)	m	Thermal conductivity of the copper
Return	Real(8)	dg/f	Half-with for thermal boundary layer

Half-width by bulk attenuation

g_b(T, p, g, dv, dt, f)

T	Real(8)	K	Temperature of the gas
p	Real(8)	kPa	Pressure of the gas
g	Real(8)	cp/cv	Specific heat capacities ratio
dv	Real(8)	m	Viscous penetration length of the gas
dt	Real(8)	m	Thermal penetration length of the gas
f	Real(8)	Hz	Frequency of the acoustic mode
Return	Real(8)	Hz	Increment of the half-with

Reference:

M. R. Moldover, J. B. Mehl and M. Greenspan, Gas-filled spherical resonators: Theory and experiment, J. Acoust. Soc. Am. 79 (2}, February 1986; https://doi.org/10.1121/1.393566

Electromagnetic modes

Skin effect

Depth penetration of electromagnetic waves

d_sk(mu_r, s, f)

mu_r	Real(8)	none	Relative magnetic permeability of copper
s	Real(8)	S / m	Electric conductivity of the copper
f	Real(8)	Hz	Frequency of the electromagnetic mode
Return	Real(8)	m	Penetration length in the copper

Reference:

E. F. May, L. Pitre, J.B. Mehl, M. R. Moldover and J. W. Schmidt, Quasi-spherical cavity resonators for metrology based on the relative dielectric permittivity of gases, Rev. Sci. Instrum., Vol. 75, No. 10, October 2004; https://doi.org/10.1063/1.1791831

TE frequency and half-width perturbation

df_skTE(d_sk, R_Eq, df_f dg_f)

d_sk	Real(8)	m	Penetration length in the copper
Req	Real(8)	mm	Radius of the equivalent sphere
df_f	Real(8)	df/f	Return shift of the TE mode
dg_f	Real(8)	dg/f	Return half-width of the TE mode

Reference:

E. F. May, L. Pitre, J.B. Mehl, M. R. Moldover and J. W. Schmidt, Quasi-spherical cavity resonators for metrology based on the relative dielectric permittivity of gases, Rev. Sci. Instrum., Vol. 75, No. 10, October 2004; https://doi.org/10.1063/1.1791831

TM1n frequency and half-width perturbation

df_skTM1n(d_sk, R_Eq, n, df_f dg_f)

d_sk	Real(8)	m	Penetration length in the copper
Req	Real(8)	mm	Radius of the equivalent sphere
n	Integer(4)	none	Index of the TM1n mode
df_f	Real(8)	df/f	Return relative shift of the TM1n mode
dg_f	Real(8)	dg/f	Return relative increase of the half-width of the TM1n mode

Reference:

E. F. May, L. Pitre, J.B. Mehl, M. R. Moldover and J. W. Schmidt, Quasi-spherical cavity resonators for metrology based on the relative dielectric permittivity of gases, Rev. Sci. Instrum., Vol. 75, No. 10, October 2004; https://doi.org/10.1063/1.1791831

Ducts

TE modes frequency perturbation

df_dcTE(r, R_Eq)

r	Real(8)	mm	Radius of the duct
Req	Real(8)	mm	Radius of the equivalent sphere
Return	Real(8)	df/f	Shift of the TE modes due to a duct

Reference:

R. J. Underwood, J. B. Mehl, L. Pitre, G. Edwards, G. Sutton and M. de Podesta, Waveguide effects on quasispherical microwave cavity resonators, Meas. Sci. Technol. 21 (2010) 075103; https://doi.org/10.1088/0957-0233/21/7/075103

TM1n modes frequency perturbation

df_dcTM1n(r, R_Eq, n)

r	Real(8)	mm	Radius of the duct
Req	Real(8)	mm	Radius of the equivalent sphere
n	Integer(4)	none	Index of the TM1n mode
Return	Real(8)	df/f	Shift of the TM1n modes due to a duct

Reference:

R. J. Underwood, J. B. Mehl, L. Pitre, G. Edwards, G. Sutton and M. de Podesta, Waveguide effects on quasispherical microwave cavity resonators, Meas. Sci. Technol. 21 (2010) 075103; https://doi.org/10.1088/0957-0233/21/7/075103

Epsilon calculations

Using TM1n first order approximation model

Eps_TM1n(f1, f2, f3, n, ep1, ep2)

f1	Real(8)	Hz	Frequency of the first component of the TM1n mode
f2	Real(8)	Hz	Frequency of the second component of the TM1n mode
f3	Real(8)	Hz	Frequency of the third component of the TM1n mode
n	Integer(4)	none	Index of the TM1n mode
ep1	Real(8)	1-Rx/Rz	Return value of epsilon_1 of the ellipsoidal resonator
ep2	Real(8)	1-Ry/Rz	Return value of epsilon_2 of the ellipsoidal resonator

Reference:

J. B. Mehl, Second-order electromagnetic eigenfrequencies of a triaxial ellipsoid, Metrologia 46 (2009) 554–559; https://doi.org/10.1088/0026-1394/46/5/020

Using TE1n first order approximation model

Eps_TE1n(f1, f2, f3, n, ep1, ep2)

f1	Real(8)	Hz	Frequency of the first component of the TE1n mode
f2	Real(8)	Hz	Frequency of the second component of the TE1n mode
f3	Real(8)	Hz	Frequency of the third component of the TE1n mode
n	Integer(4)	none	Index of the TM1n mode
ep1	Real(8)	1-Rx/Rz	Return value of epsilon_1 of the ellipsoidal resonator
ep2	Real(8)	1-Ry/Rz	Return value of epsilon_2 of the ellipsoidal resonator

Reference:

J. B. Mehl, Second-order electromagnetic eigenfrequencies of a triaxial ellipsoid, Metrologia 46 (2009) 554–559; https://doi.org/10.1088/0026-1394/46/5/020

Second order shape perturbations (applied to the mean value)

TM1n perturbed eigen values

dzm2_zTM1n( ep1, ep2, n)

ep1	Real(8)	none	Value of epsilon_1 of the ellipsoidal resonator
ep2	Real(8)	none	Value of epsilon_2 of the ellipsoidal resonator
n	Integer(4)	none	Index of the TM1n mode
Return	Real(8)	dz²/z²	Shift of the square of the eigenvalue dz²/z² for the mode TE1n

Reference:

J. B. Mehl, Second-order electromagnetic eigenfrequencies of a triaxial ellipsoid, Metrologia 46 (2009) 554–559; https://doi.org/10.1088/0026-1394/46/5/020

TE1n perturbed eigen values

dzm2_zTE1n( ep1, ep2, n)

ep1	Real(8)	none	Value of epsilon_1 of the ellipsoidal resonator
ep2	Real(8)	none	Value of epsilon_2 of the ellipsoidal resonator
n	Integer(4)	none	Index of the TE1n mode
Return	Real(8)	dz²/z²	Shift of the square of the eigenvalue dz²/z² for the mode TE1n

Reference:

J. B. Mehl, Second-order electromagnetic eigenfrequencies of a triaxial ellipsoid, Metrologia 46 (2009) 554–559; https://doi.org/10.1088/0026-1394/46/5/020

Copper properties

Constant pressure specific heat capacity

cp_cu(T)

T	Real(8)	K	Temperature of the copper
Return	Real(8)	J/(kg K)	Constant pressure specific heat capacity of the copper

Thermal conductivity

k_cu(T, RRR)

T	Real(8)	K	Temperature of the copper
RRR	Real(8)	none	Residual resistance ratio of the copper
Return	Real(8)	W/(m K)	Thermal conductivity of the copper

Acknowledgement

This project (18SIB02-RMG1) has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme.

License

MIT-like license with commercial restrictions.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute and sublicense copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
The Software, or part of it, is not included or used in commercial applications without prior written agreement from Le-Cnam/INRiM.
Except as contained in this notice, the name of Le-Cnam/INRiM shall not be used in advertising or otherwise to promote the sale, use or other dealings in this Software without prior written authorization from Le-Cnam/INRiM.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.