may_2009
========

E\. F\. May, M\. R\. Moldover and J\. W\. Schmidt, "Reference measurements of hydrogen's dielectric permittivity", Mol\. Phys\. 107, 1577 \(2009\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/may_2009.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1080/00268970902980052 <https://doi.org/10.1080/00268970902980052>`__

**Source kind:** measurement

**Recorded technique:** The apparatus of may\_2008\: a quasi\-spherical microwave cavity resonator \(four modes, 2\.6\-7\.2 GHz, n^2 from frequency ratios, Eq\. 5\) and a cross capacitor \(epsilon\_r from capacitance ratios at 2 kHz\) in two pressure vessels in one thermostatted stirred\-oil bath, filled from one gas supply and pressure balance\. Normal H2 \(supplier's purity 99\.9999 %\) along two isotherms, 273\.203 and 293\.213 K \(5 mK, ITS\-90\), 0\.1\-6\.5 MPa in rising and falling steps of 0\.5 or 1 MPa, 2\.5 h equilibration; pressure to 0\.1 kPa\. Both instruments calibrated with helium against its ab initio properties \(Schmidt et al\. 2007; densities from Hurly and Mehl's B and Garberoglio and Harvey's C; D refitted\), so the H2 results are relative to helium's ab initio epsilon\_r\. The resonator's n^2 \(ten times more precise than the capacitor, and agreeing with it to \< 2e\-6\) is converted to epsilon\_r with the calculated magnetic term A\_mu rho \(at most 1\.34e\-7\) \(pp\. 1577\-1582\)\.

**Materials in structured entries:** H2

**Dataset quantities:** molar\_refractivity, polarizability

**Contents:** 6 datasets; 2 formulas.

Curation notes
--------------

Identity checked against the first page \(Mol\. Phys\. 107, 1577\-1585 \(2009\), received 24 February 2009\)\. Page numbers are the journal's\: pdf page 1 is the publisher's cover, so p\. = pdf page \+ 1575\. 'All uncertainties reported here are one standard uncertainty' \(abstract, p\. 1577\)\. The paper hohm\_2013 cites as its ref\. \[43\] \(the H2 static polarizability, re\-pointed 2026\-09\-27\)\.

DEFINITIONS \(p\. 1578\)\. Molar polarizability P = \[\(eps\_r \- 1\)/\(eps\_r \+ 2\)\]/rho = A\_eps \(1 \+ b\_eps rho \+ c\_eps rho^2\) \(Eq\. 1\) and A\_eps = \(N\_A/3 eps0\)\[alpha\_e\(T\) \+ mu\_e^2/\(3 k\_B T\)\] \(Eq\. 2\), mu\_e = 0 for H2\. NOTE THE FORM\: b\_eps here \(cm^3/mol\) is the second dielectric virial coefficient RELATIVE to A\_eps, not the B\_eps = A\_eps b\_eps \(cm^6/mol^2\) of bose\_1970 and schmidt\_2003 \(for H2, B\_eps = 2\.031 b\_eps\); it has no vocabulary unit \(refractivity\_virial\_coefficient takes cm^6/mol^2\) and is kept in the formulas and notes\. The static \(2 kHz and 2\.6\-7\.2 GHz, no frequency dependence detected, p\. 1581\) polarizability of H2 is its electronic polarizability averaged over the populated rotational states of normal H2 \(it rises with T by centrifugal stretching\: A\_tau\)\.

RESULTS \(pp\. 1582\-1585\)\. Method 1 \(Eq\. 9, fitted to all of Table 1 with densities from Leachman et al\.'s equation of state in REFPROP 8\.0; c\_eps not significant, set to 0\)\: A\_eps,273K = 2\.03075 \+\- 0\.00001 cm^3/mol, b\_eps = 0\.0636 \+\- 0\.0015 cm^3/mol, A\_tau = 0\.00599 \+\- 0\.00017 cm^3/mol \(statistical uncertainties\); with Kunz et al\.'s \(GERG\-2004\) equation of state instead, 2\.03042, 0\.0650 \+\- 0\.0019 and 0\.00627 \+\- 0\.00017\: the choice of equation of state dominates \(p\. 1583\)\. Method 2 \(no equation of state\: the difference of the H2 and He polynomial coefficients d\_1, Eq\. 8, with Lach et al\.'s A\_eps\(He\) = 0\.51725419 cm^3/mol\)\: A\_eps = 2\.0307 \+\- 0\.0001 \(273 K\), 2\.0311 \+\- 0\.0001 \(293 K\), hence A\_tau = 0\.0061 \+\- 0\.0014; from d\_2 \(Eqs\. 10\-11\) B \- b\_eps = 13\.87 \+\- 0\.03 \(273 K\) and 14\.31 \+\- 0\.03 cm^3/mol \(293 K\), and with Patkowski et al\.'s ab initio B\(T\) the weighted mean b\_eps = 0\.003 \(standard deviation 0\.02\) cm^3/mol\. RECOMMENDED \(the weighted average of the two methods, 'the uncertainty bound we assign \.\.\. encompasses the standard uncertainties obtained by each method', p\. 1584\)\: A\_eps,273K = 2\.03065 \+\- 0\.00017 cm^3/mol, A\_tau = 0\.0061 \+\- 0\.0014 cm^3/mol, alpha\_e = \(8\.9568 \+\- 0\.0008\) x 10^\-41 F m^2 \(3 eps0 A\_eps/N\_A = 8\.95682e\-41, checked\), b\_eps = 0\.03 \+\- 0\.05 cm^3/mol \(273\-293 K\); dalpha\_e/dT = alpha\_e x \(1\.1 \+\- 0\.3\) x 10^\-5 K^\-1 \(abstract; = A\_tau/\(273\.16 K A\_eps\)\), against the calculated 1\.0e\-5 K^\-1\. For representing Table 1's epsilon\_r\(p, T\) the authors recommend Method 1 with Leachman's equation of state \(p\. 1584\)\.

COMPARISONS quoted \(reproduced datasets\)\: Harvey and Lemmon's alpha\_e = \(8\.9566 \+\- 0\.0026\) x 10^\-41 F m^2 at 273\.16 K from Rychlewski's ab initio polarizabilities averaged over J = 0\-11 \(the bound is the omitted non\-adiabatic part, about 0\.03 %\), and their A\_tau = 0\.0056 cm^3/mol \(p\. 1585; notes only\); Rychlewski's 293 K normal\-H2 average 8\.958 x 10^\-41 F m^2 \(A\_eps = 2\.0309 cm^3/mol\), 0\.25 % above Maryott and Buckley's recommended 8\.981 x 10^\-41 \(293 K, 1 atm, from radio, microwave and optical data\) \(p\. 1578\); Orcutt and Cole's b\_eps = 0\.03 \+\- 0\.10 cm^3/mol at 322 K \(p\. 1585\: orcutt\_1967's H2 row prints 0\.03 \+\- 0\.10 as its second dielectric virial coefficient in cm^6/mol^2, i\.e\. B\_eps, which would be b\_eps = 0\.015 in this paper's form; May et al\. quote the number unchanged as b\_eps, notes only\); Harvey and Lemmon's b\_eps 0\.09 from Stewart's para\-H2 data\. Fig\. 4 \(A\_eps against T with Orcutt and Cole's and Newell and Baird's values\) and Figs\. 2, 3, 5 \(deviations, b\_eps estimates\) are not digitised\.

CHECK \(2026\-09\-27\)\: Method 1's Eq\. \(9\) \(formula eq9\_method1\) reproduces Table 1's epsilon\_r \- 1 to 1\-90 ppm at 1\.0 MPa \(273\.203 K\), 225\-240 ppm at 1\.5 MPa \(293\.213 K\) and about 400 ppm at 2\.5 MPa, and to 0\.04\-0\.14 % \(the printed four digits and u\(epsilon\_r\) = 3\.4e\-7\) at 0\.1 MPa, with densities from p = rho RT\(1 \+ B rho\) and B = \(B \- b\_eps\) \+ b\_eps from the paper's Method 2 values\: the growth with pressure is the neglected third density virial coefficient of H2, which the check omits \(the paper uses Leachman's full equation of state\)\.

TABLE 1 \(p\. 1582, 40 points\: epsilon\_r \- 1 of normal H2, x 10^4, against pressure at 273\.203\-273\.204 K \(27 points, 0\.14\-6\.24 MPa, rising then falling\) and 293\.213 K \(13 points, 0\.10\-6\.49 MPa\)\) is NOT extracted\: the vocabulary has no quantity for a gas's permittivity \(or refractivity\) at a stated pressure \(refractivity is n \- 1 against wavelength at one state\), as for may\_2008's Table III\. Uncertainty of epsilon\_r\: \[1\.7 \+ 2\.7 \(293 K/T\)\(p/6 MPa\)\] x 10^\-7 from u\(T\) = 5 mK and 2\.6e\-7 \(293 K/T\) from u\(p\) = 0\.1 kPa, combined 3\.4e\-7 at 293 K and 3 MPa \(p\. 1582\); deviations from Harvey and Lemmon's correlation below 0\.9e\-6, mean \+0\.3e\-6 \(p\. 1582, Fig\. 3\)\. A permittivity\-against\-pressure quantity would let these data be fitted directly \(reported\)\.

AGAINST may\_2008 \(O2\), the same apparatus and methods\: the same two methods \(Method 1 = may\_2008's Eq\. \(19\) regression, Method 2 = its Eq\. \(18\) coefficient differences\) and the same recommendation \(their weighted mean, with a bound encompassing both\)\. Differences\: H2 is diamagnetic, so n^2 and epsilon\_r agree within 2e\-6 and the \(more precise\) resonator values are used, where for paramagnetic O2 the capacitor's epsilon\_r was used and the resonator measured the magnetic susceptibility; the H2 run adds a second\-order coefficient d\_2 analysis for b\_eps \(Eqs\. 10\-11\); O2 was measured at three isotherms \(273, 293, 323 K\), H2 at two \(the 323 K run was lost to a pressure\-transducer failure, p\. 1580\)\.

**Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets
--------

may\_2009/recommended\_H2
~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "recommended_H2",
     "material": "H2",
     "table": "text",
     "quantity": "molar_refractivity",
     "unit": "cm3/mol",
     "abscissa_unit": "nm",
     "use": "primary",
     "uncertainty": {
       "column": true,
       "level": "1 sigma"
     },
     "parameters": {
       "temperature_k": 273.16,
       "ortho_para": "3:1 (normal H2)"
     }
   }

1 inline points in the source record.

p\. 1584\: the recommended A\_eps,273K = 2\.03065 \+\- 0\.00017 cm^3/mol \(one standard uncertainty; 'the uncertainty bound we assign to these recommended values encompasses the standard uncertainties obtained by each method'\), the weighted average of Method 1 \(2\.03075\) and Method 2 \(2\.0307 at the 273 K isotherm\), referred to 273\.16 K\. Static \(2 kHz\-7\.2 GHz\), dielectric\: the electronic polarizability of normal H2 averaged over its rotational populations at 273\.16 K\. Relative to helium's ab initio A\_eps\. Row = temperature in K\.

may\_2009/method2\_H2
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "method2_H2",
     "material": "H2",
     "table": "text",
     "quantity": "molar_refractivity",
     "unit": "cm3/mol",
     "abscissa_unit": "nm",
     "use": "primary",
     "excluded_rows": {
       "273": "averaged into recommended_H2 (the paper's recommended 273 K value)"
     },
     "uncertainty": {
       "column": true,
       "level": "1 sigma"
     },
     "parameters": {
       "temperature_k": 293.213,
       "ortho_para": "3:1 (normal H2)"
     }
   }

2 inline points in the source record.

p\. 1583\: A\_eps from Method 2 \(Eq\. 8, the difference of the H2 and He resonator coefficients d\_1, with A\_eps\(He\) = 0\.51725419 cm^3/mol, independent of the equation of state\) at each isotherm \(273\.203 and 293\.213 K; rows are the temperatures as printed, '273 K', '293 K'\), one standard uncertainty\. The 293 K value carries the temperature dependence \(A\_tau = 0\.0061 \+\- 0\.0014 cm^3/mol\)\. As may\_2008's eq18\_O2; the dataset's temperature\_k is the second isotherm, the rows differ\.

may\_2009/alpha\_e\_H2
~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "alpha_e_H2",
     "material": "H2",
     "table": "text",
     "quantity": "polarizability",
     "unit": "SI",
     "scale": 1e-41,
     "abscissa_unit": "nm",
     "use": "check",
     "uncertainty": {
       "column": true,
       "level": "1 sigma"
     },
     "parameters": {
       "temperature_k": 273.16
     }
   }

1 inline points in the source record.

pp\. 1577, 1585\: alpha\_e = \(8\.9568 \+\- 0\.0008\) x 10^\-41 F m^2 at 273\.16 K, recommended\_H2 through Eq\. \(2\)\: the same value, not independent \(3 eps0 A\_eps/N\_A = 8\.95682e\-41, checked\)\. F m^2 = C m^2/V\. = 5\.4324 \+\- 0\.0005 au \(hohm\_2013 reprints it so\)\.

may\_2009/harvey\_lemmon\_2005\_H2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "harvey_lemmon_2005_H2",
     "material": "H2",
     "table": "text",
     "quantity": "polarizability",
     "unit": "SI",
     "scale": 1e-41,
     "abscissa_unit": "nm",
     "use": "reproduced",
     "reproduced_from": [
       "A. H. Harvey and E. W. Lemmon, Int. J. Thermophys. 26, 31 (2005) (with J. Rychlewski, Mol. Phys. 41, 833 (1980))"
     ],
     "uncertainty": {
       "column": true,
       "level": "bound"
     },
     "parameters": {
       "temperature_k": 273.16
     }
   }

1 inline points in the source record.

pp\. 1578\-1579, 1585\: Harvey and Lemmon's effective alpha\_e = \(8\.9566 \+\- 0\.0026\) x 10^\-41 F m^2 of normal H2 at 273\.16 K, a population\-weighted average of Rychlewski's ab initio polarizabilities over J = 0\-11 \(v = 0\); 'the uncertainty bound represents the small \(~0\.03 %\) non\-adiabatic contributions omitted from Rychlewski's calculation'\. Theory; a bound\. Not in the library\.

may\_2009/rychlewski\_1980\_H2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "rychlewski_1980_H2",
     "material": "H2",
     "table": "text",
     "quantity": "polarizability",
     "unit": "SI",
     "scale": 1e-41,
     "abscissa_unit": "nm",
     "use": "reproduced",
     "reproduced_from": [
       "J. Rychlewski, Mol. Phys. 41, 833 (1980)"
     ],
     "parameters": {
       "temperature_k": 293
     }
   }

1 inline points in the source record.

p\. 1578\: Rychlewski's ab initio effective alpha\_e of normal H2 at 293 K, averaged over J = 0\-5 \(v = 0\), 8\.958 x 10^\-41 F m^2, 'corresponding to A\_eps = 2\.0309 x 10^\-6 m^3 mol^\-1'; relative uncertainty about 0\.03 % \(omitted non\-adiabatic contributions\)\. Not in the library \(rychlewski\_1983 is a different paper\)\.

may\_2009/maryott\_buckley\_1953\_H2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "maryott_buckley_1953_H2",
     "material": "H2",
     "table": "text",
     "quantity": "polarizability",
     "unit": "SI",
     "scale": 1e-41,
     "abscissa_unit": "nm",
     "use": "reproduced",
     "reproduced_from": [
       "A. A. Maryott and F. Buckley, Tables of Dielectric Constants and Electric Dipole Moments of Substances in the Gaseous State (National Bureau of Standards, Washington, 1953)"
     ],
     "parameters": {
       "temperature_k": 293
     }
   }

1 inline points in the source record.

p\. 1578\: 'the value alpha\_e = 8\.981 x 10^\-41 F m^2 recommended by Maryott and Buckley for H2 at 293 K and 1 atm \.\.\. based on measurements at radio, microwave, and optical frequencies'; Rychlewski's value is 0\.25 % smaller\. A compilation value; static or optical not separated\. Not in the library\.

Formulas
--------

may\_2009/eq9\_method1
~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq9_method1",
     "material": "H2",
     "equation": "Eq. (9), p. 1582, with the Method 1 parameters, p. 1583",
     "printed": "(eps_r^calc - 1)/(eps_r^calc + 2) = A_eps,273K rho (1 + b_eps rho + c_eps rho^2) + A_tau rho (T/273.16 K - 1); A_eps,273K = 2.03075 +- 0.00001 cm^3 mol^-1, b_eps = 0.0636 +- 0.0015 cm^3 mol^-1, A_tau = 0.00599 +- 0.00017 cm^3 mol^-1 (c_eps = 0, not significant)",
     "variable": "rho: molar density in mol/cm^3 from Leachman et al.'s equation of state for normal H2 (REFPROP 8.0); T in K; static (2 kHz-7.2 GHz) relative permittivity",
     "range": "273.203 and 293.213 K, 0.1-6.5 MPa (Table 1)",
     "coefficients": {
       "A_eps_273K": "2.03075",
       "A_eps_273K_sigma": "0.00001",
       "b_eps": "0.0636",
       "b_eps_sigma": "0.0015",
       "A_tau": "0.00599",
       "A_tau_sigma": "0.00017",
       "c_eps": "0"
     },
     "check": "Reproduces Table 1's epsilon_r - 1 to 1-90 ppm at 1.0 MPa (273.203 K), 225-240 ppm at 1.5 MPa (293.213 K), about 400 ppm at 2.5 MPa, and 0.04-0.14 % at 0.1 MPa (the four printed digits), with densities from p = rho RT(1 + B rho), B = 13.87 + b_eps and 14.31 + b_eps cm^3/mol (the paper's Method 2 B - b_eps): the rise with pressure is the third density virial coefficient the check leaves out. Checked 2026-09-27."
   }

The authors' recommendation for representing the epsilon\_r\(p, T\) of Table 1 \(p\. 1584\)\. The uncertainties are statistical; the parameters depend on the equation of state \(with Kunz et al\.'s GERG\-2004\: A\_eps,273K = 2\.03042, b\_eps = 0\.0650 \+\- 0\.0019, A\_tau = 0\.00627 \+\- 0\.00017 cm^3/mol, p\. 1583\), on the number of free parameters and on the objective function\. b\_eps is relative to A\_eps \(Eq\. 1\)\: B\_eps = A\_eps b\_eps\.

may\_2009/eq9\_recommended
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq9_recommended",
     "material": "H2",
     "equation": "Eqs. (1), (9), p. 1582, with the recommended values, pp. 1584-1585",
     "printed": "A_eps(T) = A_eps,273K + A_tau (T/273.16 K - 1); A_eps,273K = 2.03065 +- 0.00017 cm^3 mol^-1, A_tau = 0.0061 +- 0.0014 cm^3 mol^-1; b_eps = 0.03 +- 0.05 cm^3 mol^-1 over 273 to 293 K",
     "variable": "T in K; A_eps the zero-density molar polarizability; b_eps the second dielectric virial coefficient relative to A_eps (Eq. 1)",
     "range": "273-293 K",
     "coefficients": {
       "A_eps_273K": "2.03065",
       "A_eps_273K_sigma": "0.00017",
       "A_tau": "0.0061",
       "A_tau_sigma": "0.0014",
       "b_eps": "0.03",
       "b_eps_sigma": "0.05"
     },
     "check": "At 293.213 K gives A_eps = 2.03110, Method 2's 2.0311 +- 0.0001; A_tau/(273.16 K x A_eps) = 1.10e-5 K^-1, the abstract's dalpha_e/dT = alpha_e x (1.1 +- 0.3) x 10^-5 K^-1; with Eq. (9) it reproduces Table 1 as eq9_method1 does to within 60 ppm more (computed 2026-09-27). One standard uncertainty (the bound encompassing both methods)."
   }

The recommended molecular properties \(weighted average of Methods 1 and 2\)\: 'for the purpose of estimating the molecular properties of H2' \(p\. 1584\)\. b\_eps = 0\.03 \+\- 0\.05 cm^3/mol corresponds to B\_eps = A\_eps b\_eps = 0\.06 \+\- 0\.10 cm^6/mol^2 \(computed, not printed\), 'a property that has not yet been calculated' \(abstract\)\.

Corrections
-----------

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the :doc:`review register </register>` for the checks and decisions.