Can superconductivity be predicted with the aid of pattern recognition techniques ?

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Can superconductivity be predicted with the aid of pattern recognition techniques ?

F.W. Pijpers, G. Vertogen

To cite this version:

F.W. Pijpers, G. Vertogen. Can superconductivity be predicted with the aid of pattern recognition

techniques ?. Journal de Physique, 1982, 43 (1), pp.97-106. �10.1051/jphys:0198200430109700�. �jpa-

00209387�

97

Can superconductivity ^be predicted with the aid of pattern recognition techniques ?

F. W. Pijpers

Department ^of Analytical Chemistry, ^Catholic University, Toernooiveld, Nijmegen, ^The Netherlands and G. Vertogen

Institute for Theoretical Physics, ^Catholic University, Toernooiveld, Nijmegen, ^The Netherlands (Rep ^{le 13} janvier 1981, ^{révisé le 11} mai, accepti ^{le 11} septembre 1981)

Résumé. 2014 Nous employons ^une technique de reconnaissance de corrélations pour essayer de trouver des carac-

tères communs aux éléments supraconducteurs ^{dont les} propriétés déjà ^{connues servent de base de} départ. ^Les caractéristiques essentielles sont : le travail de sortie électronique ^{et le nombre} d’électrons de valence selon Miede- ma, la chaleur spécifique, ^{la chaleur de fusion et de} sublimation, ^le point ^{de fusion et} le rayon atomique. ^Nos prévi-

sions sont valables à 90 %. Cependant ^{en ce} qui ^concerne les alcalins et les alcalino-terreux d’un groupe donné,

nos prévisions ^sont moins convaincantes.

Abstract.

Pattern recognition techniques ^were employed ^{in order to} investigate ^the possibility ^to find features of the elements of the periodic system that may be relevant for the description of their behaviour with respect ^to

superconductivity. Learning ^{machines were} constructed using ^those elements of the periodic system whose super-

conducting properties have been well studied. Relevant features appear ^to be the electronic work function and the number of valence electrons as given by Miedema, ^the specific heat, the heat of melting, ^{the heat of} sublimation,

the melting point ^{and the} atomic radius. The learning machines have a predicting capability ^{of the} order of 90 %.

The predictive power of these machines concerning ^the superconducting ^{behaviour of the alkali and alkaline-}

earth metals belonging ^{to a} given ^test set, however, appears ^to be less convincing.

J. Physique ⁴³ (1982) ^{97-106 JANVIER} 1982,

Classification

Physics ^Abstracts

74.10

74.20D

1. Introduction.

The phenomenon superconduc- tivity ^is only partly understood. Although ^{the basic}

mechanism is explained by ^the celebrated BCS-

theory [ 1 ], ^{which states that the} electron-phonon

interaction gives ^{rise to an} attractive electron-electron interaction resulting ⁱⁿ Cooper pairs, ^the theory ^{is not} entirely satisfactory. ^The theory ^{lacks the power to} predict ^from physical features whether ^a given element,

say Sc, ^{will be} superconducting ^{or not.}

It is clear that the ability ^to predict superconducting

behaviour of a given material, starting ^{from a number}

of its physical ^{and chemical data, would be of great} importance, both from the technological ^{and theore-}

tical point of view. The selection of the relevant

physical and chemical properties ^could be the first step ^{towards a} really predicting theory. ^{Such a} theory

could facilitate the search for, ^{and the} design ^of, superconducting ^{materials with} higher ^transition temperatures ^{than the known} materials. The selection of relevant physical ^{and chemical} data in order to

predict ^{unknown behaviour} is well known in the field of alloying. ^{As shown} by ^Miedema [2] alloy ^formation

can be almost completely predicted starting ^{from the} compressibility ^{and the} electronic work function of the constituent elements. Because of the evident success

of this approach it is obvious to investigate ^whether

such a procedure ^{could be} successful when super-

conductivity ^is considered. In principle, ^{all well}

defined physical ^{and chemical data are} appropriate.

This means an intricate data processing problem ^{is at}

hand. A suitable technique ^{to deal with such a} problem

is offered by pattern recognition [3]. ^This technique ^is

well known in the fields of chemistry, biology ^{and the}

social sciences [4-9].

Pattern recognition, ^as applied ^to superconductivity,

means that the pattern ^{of a} superconducting ^{element is} positioned ^{in a} multidimensional feature space, that is spanned by ^all physical ^{and chemical} data, ^called features, ^{of that} particular element When a number of elements, positioned in that feature space, group

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198200430109700

together ^{or cluster, it is obvious that their} physical

and chemical behaviour is similar. In pattern ^reco-

gnition it is assumed that such ^a behaviour not only

holds for the known physical ^{and chemical data,}

but also reflects similar behaviour of properties ^that

are not yet ^{measured or cannot} be measured without considerable effort.

In this _paper we restrict the discussion to a selection of the elements of the periodic system. Selecting ^{a set}

of elements, ^{that is} composed ^{of both} superconducting

and non-superconducting elements, ^we investigate

whether two different clusters can be found in the feature _space, corresponding ^{with the} presence ^or absence of the property superconductivity. ^Because

of the comparative unacquaintance ^of pattern ^reco- gnition techniques ^{in solid state} physics ^{we also deal} briefly with the relevant mathematics of this strategy in sections 2 and 3. The main conclusions are given

in section 4.

2. Composition ^{of a} training ^set.

^{In table I the}

elements are given ^{that are} used for the construction ^of the training ^{set or} learning ^machine. Only ^those

elements are selected whose superconducting pro-

perties ^{are well described} in the literature [10]. ^Alkali

and alkaline-earth metals and elements with ^a _super-

conducting behaviour that seems to be limited to thin films of material or only ^{occurs at} high pressure, ^are e£duded from this list. ^In table II a list of physical

and chemical properties, ⁱⁿ pattern recognition ^terms

called features, ^is given ^that describes, ^we hope, ^the

behaviour of the elements being ^called patterns.

The matrix spanned by all features of all patterns ^forms the initial data set. In order to get ^an unbiased feature treatment all feature data are autoscaled according ^to

where xi,j ^denotes feature j ^of pattern i, (1j the standard deviation of feature j ^{for all} patterns, Xj ^{the mean}

Table II.

Features investigated for ^{their eventual} predictive power.

value of feature j ^{over the N} patterns ^and x’,j ^the

autoscaled feature value [3].

In an initial calculation the interfeature correlation matrix is constructed from the autoscaled features

according ^to

The interfeature correlation gives ^an indication about the mutual dependence ^of the various features. It can

be used to reduce the dimensionality of the feature space by removal of those features that are highly

correlated to others. Such a reduction is advantageous

because the eventual relationship between the _pro- perty, ^i.e. superconducting ^or non-superconducting,

and the remaining ^selected features is simplified.

Table I.

The training ^{set has been} composed of the following ^{elements tabulated} according ^{to the} periodic system

of ^{elements. The shaded elements are} non-superconducting.

99

The correlation of a feature i with the parametrized property that is taken to be ^one if the element is non-

superconducting ^{and two if the element} is super-

conducting, is calculated according ^to

where pk denotes the value of the property ^of pattert) k corresponding with the absence or presence of _super-

conductivity, and p ^{the mean} value of that property [3].

These correlations are used to select features out of ^a set of highly correlated ones. The set of selected features listed in table III was constructed in this _way. By ^trial

Table III.

The features ^selected by ^means of ^the feature ^to property correlation.

(*) ^The magnetic susceptibility, being infinite for Fe, Co and Ni, is assumed to be 104 times 106 _c.g.s. units because of computational

reason.

and error it was established that the inclusion of the

remaining features did not enhance the information content of the data matrix significantly in view of the successful prediction of the presence ^or absence of the property superconductivity by ^the learning ^machine.

On the other hand removal of one of the selected features diminished the predictive power of the

training ^{set. In the} following section this conclusion

is illustrated by comparing the results of three different

learning machines that ^are denoted by A, ^{B and C}

and given ^{in table IV.}

Table IV.

Various sets of ^selected features,- AH.,.

denotes the heat of sublimation at 25°C; T m ^{is the} melting .temperature ^{and n is the} number of ^relevant

valence electrons rounded off ^to integer ^values.

3. Results.

The selected elements that _compose the learning ^{machine can} be divided into two cate-

gories. The first category contains the _non-super-

conducting ^elements, while the second category ^holds the superconducting ^ones. Consequently pattern ^reco- gnition techniques ^with supervision ^{for two} categories

can be applied. Two methods _may be applied ^{in order}

to measure the features that are mainly responsible

for the discrimination of both categories, namely (1)

the weighting procedure according ^to Fisher and (2)

the weighting procedure according ^to the variances of the features belonging ^{to each} category.

According ^{to both} procedures 0 *, R and cp ^{are the}

three ^most important features for classification. Substi- tution of cp by cp ^In (Tm ⁺ 273) gives ^a slight improve-

ment of this feature for classification, whereas the

replacement ^of AH,,, by AHs,/(T m ⁺ 273) is immaterial with respect ^to classification _purposes.

3 .1 THE TRAINING SET.

After the features were

selected and their relative importance for classi- fication was determined, ^a classification based _upon the hierarchical clustering ^mode (HIER) ^{and the}

minimal spanning ^tree (TREE) [7, 9] ^was performed by using ^{the set} of selected features, ^variance weighted

for two categories. ^The resulting ^{clusters are} presented

in table V for HIER [7, 9] ^{and in} figure ^{1 for TREE.}

Table Va.

Hierarchical dendrogram for ^the learning ^machines A, B and C. The symbol ^{x* means that element x}

is classified ^{in the} wrong category.

Table Vb.

Table Vc.

101

Fig. ^la.

^Minimal spanning tree for the elements of the training ^{set for} learning machine A. The shaded elements are non-superconducting.

Fig. ^lb.

^Minimal spanning ^tree for elements of the training ^{set for} learning ^{machine B.}

The similarity ^values Sij, ^{given in table V, are based}

upon the mutual distance Dij ^between patterns i and j

and ^are defined as Sij

¹

Dij/Dmax. ^{The distance} Dij is the Euclidean distance in the M dimensional space spanned by ^{the M selected features as} given ⁱⁿ

table III (M equals ^{7, 6 and 3} respectively). ^{The ele-}

ments are grouped ^{on levels of} similarity Sij ^{in HIER}

and are linked in branched chains based _upon the distances Dij ^{in TREE.}

It follows from table V that always ^{more than two}

clusters of elements are formed. The first set of selected features, learning ^machine A, gives ^{rise to three}

clusters. Two of the clusters contain elements with the

« wrong » property : Ag ^{and Sb} being non-super-

conducting ^{are found in a} cluster of 14 superconducting

elements on a similarity level of 0.42 and V being

superconducting is found in a cluster of _non-super-

conducting ^{elements on a} similarity level of 0.60.

On a similarity level of 0.57 a cluster of non-super-

conducting ^{elements is combined with a} cluster of

superconducting elements and the classification loses its meaning.

Learning ^{machine B} shows that superconducting

and non-superconducting ^elements already ^cluster together ^{on a} similarity of 0.79. This indicates a

diminished classifying ability compared ^with learning

machine A.

The third set of features, learning ^{machine C,} gives

a clustering ^of superconducting elements with non-

superconducting ones on a similarity ^{level of 0.89.}

The third cluster, however, consists of a combination

of three superconducting ^{elements and one non-}

superconducting ^{element on a} similarity ^{level of} 0.91,

which means a poor classifying ability.

In TREE all elements of the training ^{set are linked} together ^{as shown in} figure ^{1. The} connecting ^lines

have not been drawn ^on scale. Learning ^{machine A} gives ^{rise to one} cluster of non-superconducting

elements surrounded by three clusters of superconduct- ing ^{ones with the} exception of Sn that is classified as

non-superconducting instead of superconducting. ^The

cluster algorithm denotes Pt as a separate ^element because of its distance from all other elements. Learn-

ing ^{machine B also connects the} non-superconducting

elements with each other ^{with the} exception ^{of Sb}

and U; ^the resulting cluster is here also surrounded

by ^{three clusters of} superconducting elements, ^while

two elements, ^{V and} Zn, ^are classified in the wrong category. The element Pt should be considered again

as being separated ^{from all} other elements because of its large distance from these elements. Since the computer algorithm ^{for the} calculation of the minimal

spanning ^{tree is not} suited for a learning machine with three or less features the minimal spanning ^{tree for} learning machine C has not been given.

Another method to test the predictive value of the

training ^{set is} provided by ^{the nearest} neighbour

method KNN [7, 9]. ^{Here a} particular element is classified according ^{to the} classification of its nearest

neighbours ^{as far as the} mutual distance is concerned.

Learning machine A gives the best results with a

group of six nearest neighbours. Then four elements

are faultily classifield : Rh and Sb being ^non-super- conducting ^are predicted ^{to be} superconducting,

whereas the reverse holds for V and Sn. Learning

machine B gives equally good results with a group

Fig. ^2a.

^The Non-Linear Mapping (NLM) of the elements of the

training ^{set for} learning machine A. The shaded elements are non-

superconducting.

of three or four nearest neighbours. ^{Here six elements}

are wrongly classified, viz. Rh, Zn, V, Sn, ^{Sb and U.}

The predictive ^{value of} learning machine C is consi- derable less. The elements, viz. Rh, V, Sb, Pd, Mo, Ag, W, Hg, Cu, ^{Au and} U, have the wrong property.

The classification based _upon the mutual distance

can be visualized by ^a non-linear mapping ^NLM [7, 9]

on a two dimensional plane. Figure ^2a represents the NLM of the results of learning machine A. All non-

superconducting ^{elements are} found in the _upper right part ^{of the} map with the exception ^of V., ^{The NLM}

of the results of learning ^{machine B looks} slightly better, ^see figure ^2b, although ^{here too V is} positioned

nearer to the non-superconducting ^element Cu, ^than

to the superconducting ^{ones Sn and} Zn, ^{whereas Sb}

and U are in a wrong position. ^The mapping ^obtained

with learning machine C is considerably ^less useful,

here Re, Os, U, ^{Ru and} Ag ^{are found in the} wrong

position.

A final test on the quality ^{of the} training ^{set is} given by ^the procedure ^LEAST [7, 9] ^{that aims at the} pre- diction of the property by ^{means of a} linear combi- nation of the selected features. Learning ^{machine A} gives ^{rise to a} linear combination of the form

where a property ^value equal ^or less than 1.5 means

non-superconducting. ^The property of the elements of the training ^{set is} predicted ^{with an} accuracy of 89 %, only Cu, Rh, ^{Sb and Ru are} predicted uncorrectly.

Reduction of the features to a statistically significant

set, ^a procedure called STEP [3], keeps 03A6 *, ^and AHsg,

Fig. ^2b.

^NLM of the elements of the training ^{set for} learning

machine B.

103

whereas the other five features are skipped. ^The predictive value of this reduced set of features dimi- nished til 83 % ; ^here Sb, U, Ru, Os, ^{Ir and} Hg ^are

misclassified. The linear combination of the reduced set reads

The same procedure applied ^to learning ^{machine B} gives ^{a linear} combination that predicts ^the property of 92 % of the elements of the training ^set correctly : only ^{Cu, Rh and Sb are} misclassified It holds

Reduction of the number of features to a statistically significant ^set keeps 0* ^and AH,,g ^{and gives also}

The number of correctly classified elements diminishes,

now Cu, Ag, Sb, U, Ru and Ir are misclassified.

Application ^{of LEAST to} learning ^{machine C} gives

In this case 20 % ^of the elements are misclassified,

viz. Cu, U, Mn, Ag, Au, Re, Os and Ru. A further reduction of features appears ^to be impossible.

3.2 THE TEST SET.

The predictive power of the

learning ^{machine obtained from the} training set,

mainly consisting ^of transition elements, ^{can be} employed ^{to a number of elements} grouped ^{in a test}

set. These elements are the alkali metals Li, Na, K, ^Rb and Cs, ^the alkaline-earth metals Be, Mg, Ca, ^{Sr and} Ba, ^{and some} other elements whose superconducting

behaviour is unknown or restricted to special ^situa-

tions like thin films of material. The elements of the test set are given in table VI. The results of the various

predicting ^{methods based} upon the learning ^machines A, ^{B and C are listed in table VII.}

Learning ^{machine A} predicts almost all elements of the test set to be superconducting using ^{the nearest} neighbour ^{method KNN} irrespective of the number of neighbours. Only the elements Be and Ge are

predicted ^{to be} non-superconducting according ^{to a}

number of neighbours ^{of one up to} eleven. The linear feature combination LEAST predicts B, Be, As, ^Ge and Mg ^{to be} non-superconducting, whereas the reduced feature set of STEP predicts Mg ^{to be} super-

conducting ^but extends the number of _non-super-

conducting elements with Cs, Sc, Y, Bi, ^{Ca and Ba.}

Learning ^{machine B} gives ^{the same} results for the element B as machine A, using ^{the KNN method.}

However, the alkali metals Na, K, ^{Rb and Cs are now} predicted ^{to be} non-superconducting, ^if only ^one neighbour is consulted. The element Ge is predicted

to be superconducting, ^if learning ^{machine B uses} up to six neighbours; ^{with more} neighbours ^{the reverse}

behaviour is predicted. ^A comparison ^{between the}

results of learning machines A and B concerning ^the procedure LEAST shows that they only differ in the

predicted behaviour of Li; according ^{to machine B Li}

is non-superconducting. ^The procedure ^STEP gives

the same results for the learning ^{machines A and B} except for the element B. Finally learning ^{machine C} gives nearly ^{the same results;} only ^{As is now} predicted

to be non-superconducting ^{for all} neighbours ^con-

sulted. The procedure ^LEAST predicts ^here Li, ^B

and Be to be superconducting, ^{while the} procedure

STEP is not applicable.

The classification of the test set elements can be visualized by ^{means of the} Karhunen-Loeve

(KARLOV) transformation [3]. Purpose ^{of this trans-}

formation is to group the features in linear combi- nations that have the highest variance, ^i.e. highest

information content. The seven features of learning

machine A give ^{rise to seven} linear and mutually

u

Table VI.

The test set has been composed of the following elements tabulated according ^{to the} periodic systems

of elements. ^The shaded elements are non-superconducting, ^the superconducting ^behaviour of ^the remaining ^elements

is unknown or depends ^on special situations.

Table VII.

Comparison of ^the predictions for ^{test set} elements made by ^various learning ^{machines with} appli-

cation of different techniques.

1 : Predicted as non-superconducting.

2 : Predicted as superconducting.

? : Depends ^on the number of neighbours applied

orthogonal combinations (o eigenvectors ^{») ranked} according ^{to their} information contents (« eigen-

values » ). ^A projection ^{of all} patterns ^{on the} plane spanned by ^{the first two} eigenvectors having ^the highest

information content and representing ⁷⁹ % ^{of all}

available information in this case is given ⁱⁿ figure ^3a.

Fig. ^3a.

Karhunen-Loeve (KARLOV) transformation of training

and test set elements for learning machine A. The training ^{set ele-}

ments are denoted by circles, the shaded elements are non-super-

conducting. ^{The test elements are denoted} by squares.

Apart ^{from Ge all test set elements,} represented by

squares, ^are found in the region ^where superconducting

elements are positioned, indicating that these elements

are expected ^{to be} superconducting. Application ^{of the}

KARLOV transformation to learning ^{machine B} yields ^{a set of vectors with the first} two vectors having already ^an information content of 73 %. ^{Now Ge is}

not an exception ^{and the} projection ^{of all} patterns

on the plane ^defined by ^{these two most} important eigenvectors produces ^a diagram ^{where the test set}

elements are found in the region ^of superconducting elements, ^see figure 3b. The KARLOV transformation

applied ^to learning machine C does not give ^{rise to a} plot ^of patterns ^{that can be} interpreted ^{as clusters}

of elements with different properties and therefore

was not reproduced ^here.

4. Discussion and conclusions.

The seven-feature

learning ^machine A and the six-feature learning

machine B give ^{rise to} pattern ^{to feature ratios of 5.1} and 6.0 respectively, ^{which are} quite ^reasonable according ^to pattern recognition ^standards [7, 9].

A two cluster training set, however, ^{is never obtained}

as can be seen from table V and figures 1 a and 1 b.

In figure ^{I a the} group of non-superconducting ^elements

with the exception ^{of Sn} is surrounded by three groups of superconducting elements, ^{while in} figure ^{1 b the}

group of non-superconducting elements with the

exception ^{of V and Zn is flanked} by ^two groups of

superconducting ^elements. Taking ^{into account the}

distances between the elements in the hyperdimen-

105

Fig. ^3b.

^KARLOV transformation of training ^{and test set elements}

for learning ^{machine B.}

sional _space, however, the situation is less simple;

Pt is so far apart ^from all other elements that, although

it is non-superconducting, ^{it seems to} represent ^a class of its own.

A comparison ^between learning machine A and B, the latter operating ^{with the} entropy ^features

shows that both do not differ distinctly. ^{This is not} surprising ^{from the} point ^{of view of} pattern recognition

because Tm, ^the melting temperature, ^is highly ^corre-

lated with AH,,g and therefore cannot be expected ^of high importance. Learning ^{machine C} is obtained from A and B by omitting the features _n, Tm, AH.,, ^and AH,,, ^{which do not appear to be strongly correlated}

with the property. Consequently it should be expected

that learning machine C is not really ^{inferior to the}

other ones. The actual results obtained with this machine therefore are rather disappointing.

All three learning ^machines predict Y, ^{Sc and the}

alkali metals to be superconducting, although ^the

nearest neighbour classification according ^to learning

machine B makes an exception for the alkali metals,

if only ^one neighbour ^{is used} (being Sb in this case).

In a recent article [11] ^{it was} shown that a super-

conducting modification of Y and Sc exists at high

pressure, whereas a superconducting modification of Csv is known as well at high pressure [10].

It should be pointed ^out here that the relevant features include the electronic work function 0* and the number of valence electrons n because’ of their relevance in the theory of Miedema concerning ^the

formation of alloys. Both features _appear to be cor-

related with a correlation coefficient of 0.6. This raises

the question whether these features are the correct

parameters ^{to describe} the elements. This does not

affect the theory ^of alloy formation, however, ^because

here only the differences between the features of the considered elements are important.

The relative importance of various features for the

description ^of superconductivity ^is presented ⁱⁿ

table IV. The important ^{ones seem to} be 03A6*, ^{R and} c, ^{or even better} cp In (Tm ⁺ 273). ^{An eventual} theory ^{should at least account for these} physical properties. ^{This does not exclude of course} features, that have not been taken into account, ^ore features,

that were excluded because of their correlation with others and/or their lack of correlation with the _pro- perty superconductivity, ^{as follows from a} comparison

of learning machine C with A and B.

It seems relevant to make a remark about V, ^which is found near the cluster of non-superconducting

elements in learning machine A (see Fig. la) ^and learning ^{machine B} (see Fig I b). ^{This behaviour seems}

in accordance with a speculation of Rietschel and Winter [12] ^{that V is close to} being magnetic ^{and has a} strongly depressed T c ^{due to} spin fluctuations.

The question may be raised whether the classifica- tion into two distinct categories ^{is correct. It is.} appa- rent from figure ^{1 that} non-superconducting ^elements

seem to have more in common than superconducting

ones, which have widely varying feature values. A trial to describe exclusively ^the superconducting ^elements,

that was based upon their transition temperatures ^as given in the literature [11], ^was unsuccessful.

As already mentioned the important ^{features seem}

to be 0 *, ^{R and} _cp ^In (Tm ⁺ 273) according ^to pattern recognition techniques. Theoretically ^{at least the} following ^{features are} expected ^{to be} important : (1) ^the linear term of the specific ^heat describing ^the density

of states at the Fermi level; (2) ^the magnetic suscepti- bility ; (3) ^the high temperature electrical resistivity hopefully measuring ^the electron-phonon coupling;

(4) ^the Debye temperature. However, ^the correlations appear ^to be disappointingly ^low; the feature to pro-

perty correlations are respectively : - ^{0.2 ±} 0.3,

-

0.4 + 0.3, - ^0.2 + 0.3,

^0.3 + ^0.2. Composite

features _may also be considered. According ^{to Mie-}

dema [13] ^{the linear term in the} specific ^{heat divided} by ^the magnetic susceptibility ^is expected ^{to be} highly

correlated with the property superconductivity. ^It

appears that the correlation is low, ^0.3 ± ^{0.2. This} poor correlation cannot be attributed to the fact that we

took finite susceptibility values for Fe, ^{Co and Ni.}

Other composite features like the wave velocity given by ^the square ^root of the Young ^{modulus divided} by

the density ^{or the} characteric impedance being ^the

square ^root of the Young modulus times the density

were also considered; the correlations are respectively

- 0.3 +0.2 and 0.1 +0.3.

It should be remarked here that inclusion of the

remaining features does not enhance the predicting

capability ^{of the} learning ^machine. Clearly the list of

features is by ^{no means} exhaustive. ^{We limited our-} selves to features given ^{in table II, because not all} required ^data for other features, ^like e.g. the sound attenuation in the normal state and the heat conduc-

tivity, ^{could be obtained from the} literature. We took

only ^{those features that were known for} nearly ^all

considered elements.

Finally it should be noted and stressed that pattern recognition ^{does not} provide ^a theory. ^{In that} respect it cannot be compared ^{with the recent} results of

sophisticated ^{ab initio} calculations [14]. ^{Pattern reco-} gnition only may hint at the relative importance ^of

some physical ^{and chemical data for the} problem

under investigation. Clearly ^a predictive power should be ascribed to the learning ^{machines. However, the} predictive power of ^a learning machine, ^{that is} mainly

built _upon the properties ^of transition elements, ^seems

less convincing ⁱⁿ applying the machine to a test set

consisting ^of mainly ^{alkali and} alkaline-earth metals.

Acknowledgments.

The authors wish to thank : A. J. Dekker of the University ^of Groningen, ^{P. G. de}

Gennes of the Ecole Superieure ^de Physique ^{et de}

Chimie Industrielles de Paris, W. Goossens and A. R. Miedema of the Philips Research Laboratories

Eindhoven, ^{and G.} Kateman, ^{F. M.} Mueller,

B. G. M. Vandeginste, ^A. Weyland and P. R. Wyder

of the University ^of Nijmegen ^{for their critical} reading

of the original manuscript and their useful suggestions.

They ^also thank B. R. Kowalski, University ^of Seattle,

for making ^the computer program ^« Arthur » avai- lable to us.

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