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 , T_{c}
, 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 T_{c} 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 T_{c} 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}
. H_{2}O ------------ [ca. 200 °C]
------------> Ca(COO)_{2} + H_{2}O
(g) ;

Step 2 : Ca(COO)_{2}
------------ [ca. 500 °C] ------------> CaCO_{3}
+ CO (g) ; and ,

Step 3 : CaCO_{3}
------------ [ca. 800 °C] ------------> CaO
+ CO_{2} (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 , Al_{2}O_{3}) . 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 , CuSO_{4}
. 5 H_{2}O] . 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 T_{c} 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 T_{c} region .
The tip of that peak could be reported as the genuine T_{c}
for the material . Analysis of the peak could also provide the
onset T_{c} , the full zero resistance T_{c} ,
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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