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It is now ten years since it was first convincingly shown that
below 1 K the ther mal conductivity and the heat capacity of
amorphous solids behave in a way which is strikingly different to
that of crystalline solids. Since that time there has been a wide
variety of experimental and theoretical studies which have not only
defined and clarified the low temperature problem more closely, but
have also linked these differences between amorphous and
crystalline solids to those suggested by older acoustic and thermal
experiments (extending up to 100 K). The interest in this somewhat
restricted branch of physics lies to a considerable extent in the
fact that the differences were so unexpected. It might be thought
that as the tempera ture, probing frequency, or more generally the
energy decreases, a continuum de scription in which structural
differences between glass and crystal are concealed should become
more accurate. In a sense this is true, but it appears that there
exists in an amorphous solid a large density of additional
excitations which have no counterpart in normal crystals. This book
presents a survey of the wide range of experimental investigations
of these low energy excitations, together with a re view of the
various theoretical models put forward to explain their existence
and nature."
This volume reviews recent advances in the development and
application of the recursion method in computational solid state
physics and elsewhere. It comprises the invited papers which were
presented at a two-day conference at Imperial College, London
during September 1984. The recursion method is based on the Lanczos
algorithm for the tridiago nalisation of matrices, but it is much
more than a straightforward numerical technique. It is widely
regarded as the most elegant framework for a variety of
calculations into which one may incorporate physical insights and a
num ber of technical devices. The standard reference is Volume 35
of Solid State Physics, which contains all the early ideas of
Heine, Haydock and others, upon which the method was established.
The present volume provides the first review of subsequent
developments. It also indicates where problems remain, or opinions
differ, in the interpretation of the mathematical details or choice
of practical techniques in applications. The field is still very li
vely and much remains to be done, as the summary chapter clearly
demonstra tes. We are grateful to the S. E. R. C. 's Collaborative
Computational Project No. 9 on the electronic structure of solids
and the Institute of Physics's Solid State Sub-committee for their
sponsorship of the conference. We thank Angus MacKinnon for his
help in conference organisation and Jacyntha Crawley for
secretarial assistance. December 1984 David G. Pettifor Denis L.
Weaire v Contents Part I Introduction Why Recur? By V."
Directed primarily at college and university undergraduates, this
book covers at basic level the essential applications of
mathematics to the physical sciences. It contains all the usual
topics covered in a first-year course such as vectors, matrices,
differential equations, basic mathematical functions and their
analysis, and power series. There is a strong emphasis on
qualitative understanding (such as curve sketching) and practical
methods of solution. The latter take due account of the impact of
computers on the subject. The principles of mathematical expression
are illustrated by copious examples taken from a wide range of
topics in physics and chemistry. Each of the short chapters
concludes with a summary and a large number of problems.
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