Use of the First Derivative Electrical Resistivity Trace for the Accurate Determination of Superconductor Critical Temperatures



The current practice in the study of the electrical resistivity of metallic materials , and in particular that of superconductors , is to report the sample resistivity over a range of temperatures in graphical form . Generally , the x axis of the graph extends from 0 K (Absolute Zero) to around 300 K (warm room temperature) . The y axis extends from zero resistivity – that is , a superconducting state – to a significantly higher value which is of course highly dependent on the particular material . Experimental results for superconductor critical temperatures , Tc , are likewise shown graphically , focusing on the narrow temperature range of the observed transition temperature . These graphs will be highly familiar to superconductivity researchers and require no illustration here .

Because of the gradually sloping decline of the graph in the transition temperature region there is always some question of where to locate Tc on it . In many cases both the onset temperature and the zero resistance temperature are reported . The aim of this brief web page is to discuss a possible solution to this problem – a solution already used in several analytical chemistry techniques – which should be readily adaptable to electrical resistivity graphical traces , and could thereby provide an accurate determination of superconductor Tc values .


Examples from Thermogravimetry


It should be possible to compute , either directly during the data acquisition phase (by an electronic “black box” computer through which the electrical resistivity data are continuously fed) , or post-experimentally by a mathematical treatment of the data , the first (or time) derivative of the electrical resistivity trace . This is the rate at which the resistivity varies over infinitely narrow increments of time , that is , continuously . The best example of this technique comes from thermogravimetry (TG) , the measurement of the loss of mass from a sample of material that is heated , usually from ambient to a much higher temperature at which it decomposes and emits volatile by-products . The first – or time – derivative of TG is called “derivative thermogravimetry”, or DTG , which is the rate at which the mass of the decomposing sample changes with respect to its changing temperature . The best way to compare TG and DTG is to examine several experimental examples .

The most-cited chemical reaction studied by TG and DTG is undoubtedly the thermal decomposition of calcium oxalate monohydrate , which can be clearly followed by both TG and DTG . This decomposition occurs in three separate , well-defined steps :

Step 1 :   Ca(COO)2 . H2O ------------ [ca. 200 °C] ------------>  Ca(COO)2   +  H2O (g) ;

Step 2 :   Ca(COO)2   ------------ [ca. 500 °C] ------------>  CaCO3  +  CO (g) ; and ,

Step 3 :   CaCO3  ------------ [ca. 800 °C] ------------>  CaO  +  CO2 (g) .

The following is a graph of the TG (red) and DTG (blue) traces of the thermal decomposition of calcium oxalate monohydrate :

This sketch was copied from the Philips Research bulletin , “Thermal Analysis (TA)” [PDF , 519 KB] . My thanks to the copyright holder .

Conversion of the raw data feed of the mass loss (TG) into its time derivative (DTG) trace has the beneficial effect of producing sharp , well-defined peaks of the transitions from the gently-rounded mass loss curves . These look very much like the electrical resistivity traces for superconductors around their transition temperatures , but in a backward sense from them ; the following sketch is the previous TG/DTG graph , but flipped right-to-left (mirror image) :

These TG transitions (red) look almost identical to a typical electrical resistivity trace of a superconductor around its transition temperature !

Here’s another TG/DTG graph for the thermal decomposition of ammonium chromate :

I.-H. Park , Decomposition of Ammonium Salts of Transition Metal Oxyacids . III . “Thermal Thermogravimetric Analysis of Ammonium Chromate”, Bull. Chem. Soc. Jpn. 45 (9) , pp. 2749-2752 (1972) [PDF , 638 KB] . My thanks to the copyright holder .

Again we see how DTG can convert the somewhat vaguely curved TG traces into well-defined peaks with sharp summits . The areas under the various peaks are directly proportional to the mass changes for each of those chemical transitions , so DTG can be used as an accurate analytical method , providing there is a way of integrating the area under each curve .

Another analytical method in chemistry using time derivative curves is gas chromatography , in which a sample of an organic chemical mixture is fractionated on a GLC column , and the separated fractions are detected and displayed quantitatively as various peaks on a flow chart . Again , the areas under the peaks are integrated , and the results are printed out as the weight percentages of the components of the mixture .

A third analytical procedure using time derivative curves is proton nuclear magnetic resonance (NMR) . The NMR instrument produces the peaks (like DTG) on the graph paper ; then the analyst integrates them individually to produce the vertical traces (like TG) whose magnitudes can be read off the calibrated graph paper .

Returning to thermogravimetry , modern thermal analysis equipment , such as is offered by the French company Setaram [PDF , 1965 KB] , can usually generate both TG and DTG traces simultaneously ; the apparatus obviously includes an onboard computer to automatically calculate the first derivative of the TG data stream . If I may relate my personal experience in this area : my Fisher Scientific TG equipment lacked such a capability , although a Cahn time derivative computer – a data converter that could be connected to the heating block thermocouples – was commercially available (at the time , also from the Fisher Scientific Company) . However , a very smart graduate student helping me was able to devise a program on his hand-held Hewlett-Packard scientific calculator to compute the DTG curves from the raw TG data . His supervising professor referred to it as an “algorithm”, but it may have been some sort of sequential process carried out on the calculator , rather than a true mathematical equation . To verify the procedure I ran the classic calcium oxalate monohydrate pyrolysis (see above) as a “dummy experiment”. My undergrad student assistant Hélène carried out the mind-numbing digitization of the TG data and then passed them over to Philip , the post-grad student , for conversion into DTG curves . Both the TG and DTG traces were in full agreement with literature results . Obviously it’s much more preferable to have an electronic black box provide the DTG data stream automatically for you , than to have long-suffering students do all the hard work of analysis and conversion !

The Cahn time derivative computer is no longer commercially available , but in any case I take it (from a cursory Internet search) that strip chart recorders seem to have gone out of fashion , and to have been replaced by data acquisition with a conventional (desktop , laptop) computer . The company Dataq Instruments of Akron , Ohio , U.S.A. offers electronic equipment and software for data acquisition , processing , and display . Their technical note by R.W. Lockhart , “A Closer Look at the WinDaq Derivative Algorithm”, describes how their WinDaq software can process data to generate its first (time) derivative . Dataq offers the data converter and the WinDaq software (and even the patch cord – probably USB 2.0 – for the converter-to-computer connection) that permits both the raw data and its first derivative trace to be displayed graphically . I believe the laboratory instrument (resistivity) is connected to the converter , which amplifies the output and transforms it to a waveform signal which can be “understood” by the computer within the WinDaq application . This waveform file may be somewhat like the WAV format used by all commercial music CDs . Then WinDaq can process the file , producing both the “direct” data trace – resistivity (r) – and its first derivative (dr/dt) . The resulting computer files can be graphically displayed by the WinDaq program (an example is provided in the technical note mentioned above) . The WinDaq software apparently is suitable only for Windows 2000 , XP , and presumably Vista and Windows 7 . Readers who are interested in the Dataq application should of course contact a Dataq technical representative directly for further advice and assistance (their contact information is provided on the technical note) . Two additional references concerning the first derivative traces of physical data are :

A.S. Lubansky et al. , “A General Method of Computing the Derivative of Experimental Data”, AIChE Journal 52 (1) , pp. 323-332 (2006) ; and ,

 X. Shao , C. Pang , and Q. Su , “A Novel Method to Calculate the Approximate Derivative Photoacoustic Spectrum Using Continuous Wavelet Transform”, Fresenius J. Anal. Chem. 367 (6) , pp. 525-529 (2000) .


Differential Scanning Calorimetry


The related thermal analysis method of differential scanning calorimetry (DSC) might also provide clear , sharp peaks for superconducting critical temperatures . DSC measures the difference in energy required to heat a sample of the material studied versus an inert reference material (in the closely related technique , dta – differential thermal analysis – the reference was always calcined alumina , Al2O3) . DSC takes advantage of the different enthalpies , or heat energies , of two different materials , or more commonly , of their different heat capacities . DSC is briefly described , with examples , in the Philips and Setaram brochures cited above . There is also a competent article about DSC in Wikipedia . The following is an example of simultaneous TG–DSC carried out on a sample of blue vitriol [copper(II) sulfate pentahydrate , CuSO4 . 5 H2O] . I copied the graph from the Philips brochure (changing their label at the upper left from its incorrect designation as calcium oxalate monohydrate) :

As explained in the brochure , the transitions for this sample weren’t very well resolved with TG , but the corresponding DSC trace provided clear peaks for them . Apparently at the transition temperature of superconductors there is a radical (nonlinear) change in their heat capacity which should be detectable in a DSC apparatus when compared against a reference material whose heat capacity remains linear at that same temperature . The following is a graph copied from the Wikipedia article about superconductors , illustrating the appreciable change in the heat capacity of a typical superconductor at its transition temperature :

My thanks to the author of this sketch , and Wikipedia , for implied permission to reproduce it here on this web page .

By contrast , the reference material’s heat capacity would be more or less linear throughout the entire Tc range of temperatures . The difference in energy losses (with the cooling toward Absolute Zero of both materials) should appear on the DSC trace as a peak , representing the superconductor’s Tc region . The tip of that peak could be reported as the genuine Tc for the material . Analysis of the peak could also provide the onset Tc , the full zero resistance Tc , and the width (in kelvins) of the phase transition .

Time-derivative resistivity could also provide useful peaks for other interesting phenomena associated with the electrical conductivity of metallic solids , such as spin density wave (SDW) transition temperatures and the Verwey temperatures of semiconductors (eg. magnetite) . It might also be possible to adapt time-derivative analyses to magnetic susceptibility measurements over an extended temperature range , so as to accurately identify and interpret Curie and Néel temperatures for ferro- , ferri- , and antiferromagnetic materials .

I hope this brief discussion of time-derivative analyses of physical data will be of interest to researchers , prompting them to investigate and adopt this valuable and still underappreciated technique .



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