The publications of Boris Davison and some reminiscences of the man

Progress in Nuclear Energy, Vol. 8, pp. 53-75. 0079-6530/81/0801-0053 $05.1)0/0 Pergamon Press Ltd. Prifited in Great Britain

THE PUBLICATIONS OF BORIS DAVISON AND SOME REMINISCENCES OF THE MAN

Edited by

M. M. R. WILLIAMS

Nuclear Engineering Department, Queen Mary College, Mile End Road, London E1 4NS, U.K.

with contributions by

S. A. KUSHNERIUK B. GAY

W. H. WATSON R. E. PEIERLS

(Received 1 May 1981)

Editorial

As one who was weaned on the works of Boris Davison, it is a great pleasure to see reprinted in the pages of this Journal the abstracts of some of his most famous papers. Davison's vision in transport theory is legendary and many is the time that I have received a manuscript to review for one journal or another only to point out that "Davison did this work in 1946". Usually the paper was not readily available and so the putative author could be forgiven for rediscovering Davison's work. However, a number of scientists and nuclear engineers felt that it was time that Davison's complete works, his total contribution to transport theory and related topics, should be given wider dissemination. It is to the great credit therefore of Atomic Energy of Canada Ltd., that they have produced 5 fine volumes covering Davison's work from 1930-1961.

In what follows we find tributes to Davison by people who have worked closely with him; such well known scientists as R. E. Peierls, W. H. Watson and S. A. Kushneriuk have all been able to give their impressions and tell some anecdotal story about that most modest of men.

This tribute to Davison takes the following form: (i) Preface by S. A. Kushneriuk giving the background to the project of producing the 5 volumes. (ii) A list of Acknowledgements to various people and societies who have granted permission for the reprinting

of Davison's papers. (iii) A list of the 5 volumes, their titles and the periods of Davison's life which they cover. (iv) An appreciation of Boris Davison by a friend and co-worker, W. H. Watson. (v) Impression of Boris Davison by Professor Sir Rudolph Peierls who was instrumental in getting Davison a

job at the University of Birmingham, England in 1942. (vi) Note to volume I (Leningrad and Farnborough, 1930-1941), followed by a list of contents. Many of these

papers were in Russian and have not been translated except for the abstracts. (vii) Note to volume II (Birmingham and Montreal, 1942-1946) Davison becomes part of the British team

working on the atomic bomb and later moves to Canada to continue this work. (viii) Note to volume III (Chalk River, 1946-1947). The war is over and Davison remains in Canada to work on

neutron transport in small highly absorbing media. (ix) Note to volume IV (Harwell and Birmingham, 1947-1954) Circumstances lead Davison to return to

England where he joins the Ministry of Supply and works at the then very new Harwell. He also collaborates with the University of Birmingham again.

ix) Note to volume V (Toronto and Chalk River, 1954-1961) For personal reasons, Davison returns to

53

54 M. M. R. WILLIAMS et al.

Canada and works at the computation centre of the University of Toronto. His final work was concerned with spatial oscillations of xenon in a nuclear power reactor.

(xi) A resum6 of Davison's work together with some abstracts selected to show the scope and nature of his work. This section was compiled and written by S. A. Kushneriuk.

M. M. R. WILLIAMS

The publications of Boris Davison and some reminiscences of the man

Preface 55

The idea of reprinting the published works of Boris Davison in collected form was put forward by former colleagues at the Chalk River Nuclear Laboratories soon after his death in 1961. It was felt that the collection would serve not only as a tribute to the man himself, but also as a record of his considerable contribution to transport and reactor theory. Accordingly it was decided to produce a limited number of copies of the collected works and to donate these to the institutions where Davison had worked and the professional associations to which he had belonged.

Davison's working life divides quite naturally into five main periods, depending upon the nature of his work and the institution with which he was associated at the time. It was therefore decided to arrange his work chronologically in five volumes. In his actual work there are two main divisions. His earliest papers, those published while he was at Leningrad and Farnborough, deal mainly with the subjects of ground-water and air flow and are incorporated in Volume I. Volumes II-V contain the later papers, those published while he was at Birmingham, Montreal, Chalk River, Harwell and Toronto. These deal mainly with criticality, neutron transport and reactor theory. Interspersed in all volumes are papers of a purely mathematical nature, together with a few related to other areas of physics and to biology.

Cross-reference and name indexes are provided. Since the subject matter of the papers in Volume 1 is not directly related to that of the later papers and Davison makes no reference in his later work to these early publications, a separate index is provided for Volume I. Most of the papers in Volume I are in Russian and the index to cited papers in that volume is accordingly mainly in Russian ; the name index is given in both the Cyrillic and Roman alphabets. Volume V contains indexes to all volumes.

The original reports were copied using a photo-lithographic process whenever possible. Typographical and other minor errors were corrected, where noticed, without special mention, although one significant error is the subje.ct of a special note. Retyping was resorted to only when necessary and, where this was needed, the fact is indicated in a footnote.

S. A. KUSHNERIUK B. GAY


Acknowledgements

We are indebted to Mrs. Boris Davison and to the various societies, publishers and others listed below who kindly granted us permission to reprint Davison's papers. A full description of the source is given with each paper, as well as in the Bibliography in this volume. We are similarly indebted to Davison's co-authors who, although not listed below, are acknowledged in the full descriptions. Academy of Sciences of the USSR

lzvestiya Gydrologicheskogo lnstituta Matematicheskff Sbornik Monograph in Some New Problems in the Mechanics of Continuous Media (Moscow, Leningrad, 1938) Trudy Gydrologicheskogo Instituta Uspekhi Matematicheskif N auk Zapiski Gydrologicheskogo lnstituta Proceedings, IVth. Hydrological Conference of Baltic States (Leningrad, 1933)

American Physical Society, Physical Review British Journal of Radiology Institute of Physics, Proceedings of the Physical Society, A National Research Council of Canada,

Canadian Journal of Physics Canadian Journal of Research, A reports

Pergamon Press, Bulletin of Mathematical Biophysics Journal of Nuclear Energy

Philosophical Magazine Royal Society of London, Proceedings, A United Kingdom Atomic Energy Authority, Harwell

reports University of Birmingham (Ministry of Supply) reports

University of Birmingham, B. Davison, Doctoral Thesis, Part I University of Toronto Press,

Proceedings of the Second Mathematical Congress, (Vancouver, 1946)

We are also indebted to the publishers of Nature and to W. H. Watson for permission to reprint Davison's obituary notice and, in particular, to Sir Rudolph Peierls for contributing his personal recollections of Davison. We also wish to acknowledge the work ofJ. E. Woolston and H. H. Clayton, the initiators of this project, whose early planning and subsequent assembly of much of the material laid the foundations for the work. Finally, our grateful thanks go to Margaret Carey, Velma Connelly and Margo Schwantz for their painstaking work in preparing the text.

S.A.K. B.G.

The publications of Boris Davison and some reminiscences of the man

LIST OF VOLUMES

57

Volume I. Leningrad and Farnborough (1930-1941) Bibliography Papers 1-20 Indexes to Volume I.

Volume II. Birmingham and Montreal (1942-1946) Papers 21-36

Volume III.Chalk River (1946-1947) Papers 37-45a

Volume IV. HarweU and Birmingham (1947-1954) Papers 46-58

Volume V. Toronto and Chalk River (1954-1961) Papers 59-72 Complete indexes.

page 60

63

65

66

67


Obituary

PROFESSOR B. DAVISON

Physicists and mathematicians connected with the British and Canadian atomic energy project have learned with regret of the sudden death of Boris Davison on January 24, while still a very productive mathematical physicist engaged on problems of neutron transport.

He was born in Gorky on July 10, 1908, and educated in Russia. He graduated from the University of Leningrad in 1931, and was engaged in hydrology for about seven years thereafter. Through his English grandfather he had a tie with England, and when in 1938 he was requested either to renounce British nationality or leave the U.S.S.R. he elected to go to England. He worked for a short period with Prof. L. Rosenhead, but his health broke down. After a considerable period of convalescence he joined the atomic energy research team at the University of Birmingham under Prof. R. E. Peierls, in 1942. His powers as a mathematician enabled him to make many important contributions, particularly to neutron transport theory, a subject which was to occupy most of his attention for the rest of his life. For some of this work he was awarded the Ph.D. of the University of Birmingham in 1944.

His connexion with Canada started when he joined the Montreal Laboratory of the joint United Kingdom-Canadian Atomic Energy Project in 1943. His association with the late Dr. G. Placzek was a very fruitful one, and he accompanied Drs. Placzek and Mark to Los Alamos. He moved to the new Chalk River Laboratory as a member of the United Kingdom staff in the Theoretical Physics Branch and in 1947 returned to England to work at Harwell. In 1954 he joined the staff of the Computation Centre in the

University of Toronto, lectured in numerical analysis and reactor physics and was appointed associate professor of physics in 1960. For the past six years he served Atomic Energy of Canada, Ltd., as a consultant. His knowledge, experience and talent have contributed in many ways to the technical success of the Canadian Atomic Energy programme.

In 1957 his book on Neutron Transport Theory was published • it established him publicly as the authority in this subject to which he brought a remarkable combination of analytical power and precision, an uncommon facility in numerical analysis and physical perception. Not only has he expounded the methods of other mathematicians but he has also delineated carefully the boundary of the region within which they can be effectively applied. In his preface he was generous in acknowledging assistance, but those who knew him know well his patience and the exacting standards he set for himself.

Much of Davison's work was concerned with im- proving mathematical representation of nuclear pro- cesses in reactors and with the associated com- putational problems. He was a ready inventor of algorithms for computing and for avoiding pathologi- cal conditions in large calculations.

He was a gentle, shy man, painstaking, kind and generous in helping his fellows, but intolerant of pretence. His intellectual life was not confined to the subject of his specialty. He had a broad, modern interest in science and he liked poetry.

He is survived by his wife Oiga.

W. H. WATSON (Nature, 190:306-307, 1961)

The publications of Boris Davison and some reminiscences of the man 59

IMPRESSIONS OF BORIS DAVISON

During the Second World War there existed in England a Central Register of scientists, whose purpose was to direct each person to the place in which he could make the most valuable contribution to the war effort. In Birmingham, our small 'Tube Alloys' team (the name being a cover for atomic energy) was chronically short-handed, and had a standing request with the Register for experienced theoretical physicists. Apart from some excellent fresh graduates, they could usually send us only people with the wrong experience for our purpose, or of inadequate standard. This was understandable in wartime conditions, and we had stopped expecting help from that source.

It was therefore surprising to receive one day the papers of a certain Mr. Boris Davison, who seemed to have just about the right experience in applied mathematics, and who seemed to be of high ability, though this could not be assessed too definitely from the papers. His background was also intriguing; he was a graduate of the University of Leningrad, with which I was familiar, and had lived in the USSR until 1938.

He was immediately invited for the interview. There appeared a small man in rumpled clothing, very polite, rather shy and diffident, whose Russian was better than his English. As soon as the conversation turned to mathematical questions, it became clear that he knew what he was talking about, and he talked in a very articulate manner. Obviously we wanted him; would he like to join our group? I could not tell him the purpose of our work until he was cleared, but I tried to explain the nature of the mathematical problems it led to, and to illustrate this I mentioned an integral equation which had come up recently, and which was giving us difficulty. A few days later I had a letter from him written in the train after the interview. The letter said he would like to come if we would have him and enclosed was the solution to my integral equation, on a few sheets of paper.

After this introduction we expected to find him a powerful addition to our strength, but it was some time before we really got to understand how much he had to contribute, both in doing calculations on his own, and in helping others with their difficulties and often with their ignorance.

His style of work is well described by Philip Wallace, with whom he collaborated later in the Montreal Laboratory: 'His manner of working was a delight to observe. Unlike the rest of us, he rarely passed through the stage of rough drafts and back-of-envelope scribbl- ings. He worked neatly and methodically producing, it seemed, finished work from the start. He had an instinct for difficulties and delicate points. For a while he would gallop ahead, formulae filling the pages, then,

sensing trouble, he would slow down to a crawl, cautiously exploring obstacles and making his way through them with painstaking care. Emerging again on open road he would regain full speed ahead. I never knew him to come to a full stop.'

Soon after his arrival in Birmingham I convinced myself that the level of his initial appointment was not commensurate with his contribution and I set about getting him at least a modest increase in salary. When I told him that the increase had been approved, Boris was indignant. He said he had just decided to ask that his salary be reduced, because he felt he was not pulling his weight on the team (I believe these were his exact words). 1 am happy to say that he was in the end persuaded to accept the increase.

Another episode I remember from those days hap- pened when I one day had my car at the University (unusual in the days of petrol rationing). As he lived close to us and normally commuted by a rather inconvenient route using two buses, I asked whether he would like a ride home. He was rather hesitant and asked whether 1 wanted to talk business on the way. I said no, I was just thinking of his convenience, but if he was headed elsewhere, perhaps to do some shopping, he should just say so. No, that was not the trouble. On being pressed he finally explained his hesitation : it was only 5.40 p.m., and he believed his duty was to stay until 6.00, though there was no rule to that effect.

The reason for the poor state of his wardrobe was, we discovered, that he had spent a considerable time in a sanatorium, suffering from pulmonary tuberculosis. When he was discharged, his original clothes were no longer useable, and he was given whatever could be found in the way of castoffclothing. My wife took him in hand and fitted him out as well as possible within the limits of wartime clothes rationing. The result seemed acceptable, and probably was so in the environment of wartime England. However, when he arrived in the Montreal Laboratory, unkind colleagues quipped that the state of his clothes showed that his boat had been torpedoed in the Atlantic, and that he had had to swim the rest of the way.

During the Montreal period he met and married Olga Hansen. His appearance then underwent a spectacular change. Quoting Philip Wallace again: 'The appearance of Olga in his life manifested itself in a very striking way. One could hardly imagine Boris without the rumpled clothes, the baggy pants, the spotted tie. He created a sensation, then, when he appeared in the lab, in a well-pressed new suit, sparkling white shirt, colourfui tie, a n d shiny new shoes. It was evident that he had reached a turning point in his life.' He was no longer the lonely figure of earlier days; no longer in need of the help of kind


colleagues with the difficult problems of everyday life. He had spent a brief time at Los Alamos, where we

met again. George Placzek, who had learned to appreciate Davison's ability at Montreal, pressed that Davison be invited to follow him to Los Alamos, so that they could continue their fruitful collaboration. The doctors were very doubtful about the effect of the altitude of Los Alamos on Boris's health, and when he did go there, their concern proved right, and he had to return to Montreal.

In 1947 he returned to England, to take up a post in the Theoretical Physics Division at Harweil. This was a regular appointment in which his exceptional qua- lities, by now well known, were to some degree recognized. He continued to work on neutron transport theory, on which he was by now an authority, but was interested in the many problems of concern to Harwell, and always willing to help with a challenging difficulty. Sir Brian Flowers recalls this episode : 'Soon after I became head of the Theoretical Physics Di- vision Boris came to me very bothered indeed about the proposal to build HIPPO (a materials testing reactor, as we would call it nowadays) at Harwell, because in the event of an accident it might do untold damage in the Thames Valley. It was my first experience of reactor safety problems, and I backed him amongst the division heads, and HIPPO was not built. The arguments, one now realizes, were naive, but they could not be rebutted at the time because those responsible had not done the work.'

In 1953, at a time when there was great concern with security, it was decided that it was unwise to have at Harwell someone who had been born in Russia and whose parents were still in the USSR. So Boris could not continue at Harweil. He was encouraged to pursue his plan to return to Canada, which had been in his mind for some time, and meanwhile he was seconded to my department at the University of Birmingham. We were delighted to have him with us again, while deploring the circumstances which had brought him there. He stayed, I believe, about a year, and during that time not only made progress with his own work, including the preparation of his book, on which he had started, but made several important contributions to the work of the department, which were in a very different area. One concerned the idea of functional integrals, which was then topical following Feynmann's paper, and whose mathematical respecta- bility was a matter of controversy. Davison's paper proposed a very simple and convincing way of putting this idea on a rigorous mathematical basis. Others do not appear in publications, but I recall that in the problem of understanding the role of complex poles of scattering amplitudes or Green functions and their

connection with decaying states, which I enjoyed clearing up to my satisfaction, it was a source of great comfort to discuss the mathematical implications with Boris.

We were sad when he departed for Toronto, but we knew that was where he and Olga wanted to be, and we had known that his stay with us could only be transient.

He died, prematurely, after less than seven years in Toronto, but that was probably the longest period fate had let him spend in any one place since his childhood. As a result many places profited from his mathematical power and were inspired by his devotion to research, by his high standards, and by his selflessness.

R. E. PEIERLS

NOTE TO VOLUME !

This volume contains Davison's earliest publications, Papers 1-20, covering the period 1930-1941.

After graduating from Leningrad University (where he received the equivalent of a M.Sc. degree in mathematics in 1930) Davison joined the Hydrological Institute of the University. There, Davison's papers were largely related to the hydrological work of the Institute, and dealt principally with problems in hydrodynamics-mechanics, although solutions to problems of a purely mathematical nature were also published. In 1938 he emigrated to England, where in association with Professor L. Rosenhead of the University of Liverpool and with the Royal Aircraft Establishment at Farnborough, he continued his studies in hydrodynamics, and also in air-flow.

Davison's early papers reveal features characteristic of all his investigations; an outstanding capability in mathematical and numerical analysis and a sense of precision and physical insight. Papers 1-3, 7 and 16 illustrate this feeling. In them variations in hydrological models are adopted and precise statements are then made as to the consequences of the variations. Ex- amples of Davison's purely mathematical papers are numbers 4, 5 and 10. Papers 6, 8, 9, 11-15, 17 and 19 represent contributions to hydrodynamical- mechanical studies; steady state and transient flow of fluids for a variety of fluid properties, system boundary conditions and physical shapes. Paper 18 is Davison's part of a three-part book on the mechanics of continuous media written by S. Christianovitch, S. Mich- lin and B. Davison and published by the Academy of Sciences of the U.S.S.R.

As mentioned in the Preface, since Davison makes no reference in his later works to these earliest publications, a cross-reference and name index has


been prepared separately for this volume and is

appended following Paper 20. Mos t of the papers in this volume are in Russian.

BIBLIOGRAPHY OF BORIS D A V I S O N

VOLUME l: LENINGRAI) AND FARNBOROUGH (1930 1941)

1. K Teopnrl MeTO21a .aorapH~Mu,~ecrofi HHTepnOJlStttln I1OCTpOeHnfl KpHBblX noanopa, H3BECTH~I FH~I.PO- 5IOFHqECKOFO HHCTIdTYTA, N2_ 31, 1930. [On the theory of the logarithmic interpolation method for constructing backwater curves, lzvestiya Gydrolo- gieheskogo h~stituta 31, 15 22, 1930.]

2. O 3aMene HOpMaYlbaOfi rny6Hnbl 6blTOBOfi npn no- czpoeHttlt KpHBblX noanopa, H3BECTH,q FH,/:[PO- 21OFHqECKOFO HHCTHTYTA, J~_ 32, 1931. [On the substitution of the normal depth by the natural one for the calculation of backwater curves, Izvestiya Gydrologieheskogo lnstituta 32, 27-30, 1931.]

3. K OUeHKe omu6Kn B pac,~eTerpnBblx no~lnopa, noay- qammefica OT npene6pe;geuna nOfiMOfi, 3AHHCKH FH~POflOFHqECKOFO HHCTHTYTA, TOM 5, 1931. [On the evaluation of error in backwater curve calculations due to neglecting the zone of overflow, Zapiski Gydrologieheskogo lnstituta 5, 39- 52, 1931 .]

4. Inversion of the unicity-theorem, MATEMATHqE- CKHfl CBOPHHK, T. 38, N2_ I 2, 1931. [Matematieheskil Sbornik 38 (1 2), 45- 47, 1931.]

5. On the uniform convergence of the trigonometrical series, MATEMATHqECKHf| CBOPHHK, T. 39, N-_, 3, 1932. [Matematicheskii Sbornik 39, (3), 71 87, 1932.]

6. 06 yCTaHOBHBUleMC~I ~],BH~eHHH FpyHTOBblX BO./I qepe3 3eM.rlSlHble FLI1OTHHlal, 3AHHCKH FH,/IPOSIOFId- qECKOFO HHCTHTYTA, TOM 6, 1932. [On the steady motion of ground-water through a prismatic dam of rectangular cross section, Zapiski Gydrologicheskogo lnstituta 6, 11 19, 1932.]

7. K TeopuH Mero,u.a Kpacra~a rlcc.rleaoaaHua ce~mc- o6pa3nblx ro.ae6aaafi, 3AHHCKH FH,/],PO.JIOFH- qECKOFO HHCTHTYTA, "I OM 6, 1932. [A note on Chrystal's theory of seiches, Zapiski Gydrologieheskogo lnstituta 6, 20 26, 1932.]

8. 0ber die zweidimensionale station~ire Bewegung der Grundgew:,isser mit freier Oberfl:,iche, IV Hydrolo- gische Konferenz der Baltischen Staaten, Leningrad, September, 1933. [A report given at the IVth Hydrological Conference of Baltic States, Leningrad, September, 1933, Vol. 2, pp. I 9.]

9. (l)OpMbl IIflOCKOFO HeyCTaHOBHBtUerOC~I J1BH)KeHH~:I BH3KOH nec~KHMaeMo~ XH~KOCTH, He 3aBHCfllLIHe OT Ba3KOCTH, 3AHHCKH FH~POflOFHqECKOFO HHCTHTYTA, TOM 13, 1934. [On the independent from viscosity forms of the two- dimensional unsteady motion of a viscous, incompressible fluid. Zapi~ki Gydrologichcskogo Instituta 13, 79 84, 1934.]

10. O HpHMeHeHHH MeTozla laycca Hpn6,qH~reHuoro Bbl- '~ncncHua onpeaeneuHb, X nuTerpanoa, 3AHHCKH FHf[POflOFHqECKOFO HHCTHTYTA, TOM 13, 1934.

[On the Gauss method of approximate computation of definite integrals, Zapiski Gydrologieheskogo lnstituta 13, 141-158, 1934.]

1 I. ~OpMbl n~ocKoro ycTauoaunmeroca ilarixeHua as3KOfi HeCXHMaOMOH XH,/].KOCTH, He 3aBHClilIIHC OT Bfl3KOCTH, 3AFIHCKH FH~PO.IIOFHqECKOFO HHCTH- TYTA, TOM 13, 1934. [On the independent from viscosity forms of the two- dimensional steady motion of a viscous incompressible fluid, Zapiski Gydrologieheskogo Instituta 13, 159 164, 1934.]

12. O neKoTopux npo6~eMax TeopeTHqecrofi rn;lpo.aorun. E. B. ~l, eancoH n C. A. XpncTaaHoBnq, YCIqEXH MATEMATHqECKHX HAYK, BblHYCK BTO- POITI, 1936. [On some problems of theoretical hydrology (with S. A. Christianovitch), Uspekhi Matematicheskil Nauk 2, 238-253, 1936.]

13. On the steady two-dimensional motion of ground- water with a free surface, Philosophical Maga:ine S:7, 21,881 903, 1936.

14. On the steady motion of ground-water through a wide prismatic dam, Philosophical Magazine S: 7, 21, 904- 922, 1936.

15. HeKoTopble c.ay,~au naocKoro ycTaaoaHamerocs ~.au- ;~eHHrl rpyHTOBblX 80,a rips rtaJluquH rlcnapeHna, TPY~bl FH~,POYlOFHqECKOFO HHCTHTYTA. BblHYCK 5, 1937. [Some particular cases of ground-water motion, the evaporation from the free surface being taken into account, Trudy Gydrologieheskogo instituta 5, 183 193, 1937.]

16. K aonpocy 06 oJmoM upaeMe ynpomeunoro pacqeTa HanopHoro ~.axxeHHa rpynTonb]x 1tO/1, TPY,/],bl FH~I, PO.,q OFHH ECKOFO HHCTHTYTA, Bbl- IlYCK 5, 1937. [On some simplification used for computing ground- water motion without free surface, Trudy Gydrologi- cheskogo lnstituta 5, 194 199, 1937.]

17. O HeycTaHOBitBtUeMCa ,/].BH)KeHHH rpyHTOSblX nO,/1 6e3 cao6o,anofi noBepxnOCTH, TPY)Ibl FH~POfiOFH- qECKOFO HHCTHTYTA, BblHYCK 5, 1937. [On the unsteady three-dimensional motion of ground- water without free surface, Trudy Gydrologieheskogo Instituta 5, 212 217, 1937.]

18. HeKoTopble HOBb/e aonpocbl MexaHHKtt cn.aoLunofi cpeau, C. A. XpHCTHaHOBHq, C. F. MHX21HH, B. B. ~eancon, FI3~ATE.IIbCTBO AKA~EMHH HAYK CCCP, MOCKBA, flEHI4HFPA,/],, 1938. qACTb TPETb~I: ~BuxXeHHe rpyHTOatax Boa, B. B. II, eartcou. [Theory of ground-water motion. A monograph, forming one of three parts of the book "Some New Problems in the Mechanics of Continuous Media", published by the Academy of Sciences of the USSR, Moscow, Leningrad, 180 pp., 1938.

Chapter I. Equations of motion of ground water. Chapter 2. Steady motion of ground-water without

free surfaces and without seepage surfaces, in the plane case.

Chapter 3. Unsteady motion of ground-water without free surfaces and without seepage surfaces.

Chapter 4. Plane problems of ground-water motion, solved by the method of conformal transformations.


Chapter 5. Motion of ground-water with a free surface in an embankment with boundaries represented by straight line segments.

Chapter 6. Completeness and uniqueness of the solutions.]

19. Some cases of the steady two-dimensional percolation

of water through ground (with L. Rosenhead), Pro- ceedings of the Royal Society of London A, 175, 346- 365, 1940.

20. Wind tunnel correction for a circular open jet tunnel with a reflexion plate (with L, Rosenhead), Proceed- ings of the Royal Society of London A, 177, 366 382. 1941.


NOTE TO VOLUME I!

This volume contains Papers 21-36, covering the period 1942-1946.

Early in 1942, Davison joined an atomic energy research team at the University of Birmingham, under the direction of R. E. Peierls. Davison was thus introduced into a new and, at the time, urgent area of research-neutron physics, neutron transport theory and chain reactions. The Birmingham group (Peierls, K. Fuchs, A. H. Wilson, Davison and others) and theoreticians located elsewhere ( e . g .P .A .M. Dirac and M. H. L. Pryce) did much work on the properties and solutions of the neutron transport equation, particularly in the one-velocity limit with isotropic scattering and the integral equation formulation, as well as the diffusion approximation. The main con- cerns were questions related to neutron multiplication, critical size, time constants and energy release in assemblies containing fissile materials. Much of the work was, and still is classified. Davison was awarded a Ph.D. degree for some of his work at Birmingham; however, only Part I of his Doctoral Thesis pre- sentation (Paper 26) is unclassified and available for reproduction.

In 1944 Davison joined the Montreal laboratories of the National Research Council of Canada as part of a group of British scientists participating in a joint Canada-United Kingdom-United States program of research in atomic energy. Other theoreticians variously associated with the laboratory at that time were G. Placzek, R. E. Marshak, G. M. Volkoff, J. Carson Mark, J. LeCaine, P. R. Wallace, B. G. Carlson, H. H. Clayton, F. T. Adler, W. P. Seidel, E. D. Courant, M. H. L. Pryce, E. A. Guggenheim and O. Bunemann. This group paid much attention to the fundamentals of neutron transport theory, and also examined such matters as fissile material criticality conditions and a large variety of nuclear reactor concepts.

Davison's main efforts while at Montreal were directed towards the specification of boundary conditions at the surfaces of neutron absorbing rods and cavities in predominantly neutron-scattering media (moder- ators) for the diffusion theory solutions of the neutron flux distribution in these media. Knowledge of these conditions is important for assessing the effectiveness of control, shut-off, and fuel rods in reactors and estimating the effects of voids. Idealized problems were posed and elegant perturbation theory solutions leading to the boundary conditions were developed for bodies of large and small size (e.g. Papers 27, 28, 32, 34-36). Davison also studied the more fundamental aspects of the theory (e.g. Papers 29 and 30). In particular, in Paper 29, Davison discussed the elemen-

tary eigensolutions of the homogeneous transport equation in the plane case, involving the use of the distribution function in the solutions, and their completeness.

There were at this time in the United States other groups doing somewhat similar work at the University of Chicago, at Los Alamos, etc. and Davison spent a short time with the group at Los Alamos.

In connection with Davison's earlier papers in neutron transport theory it may be useful to note that Davison uses the word 'flux' to denote what is usually now expressed as 'current'. In fact, readers of some of the early documents reproduced in this volume had crossed out 'flux' and written in 'current', and these changes have been left as they were made.

VOLUME II: BIRMINGHAM AND MONTREAL (1942-1946)

21. Density distribution near a point source (with R. E. Peierls), MS-76, 7 pp., 1943.

22. Boundary conditions in the modified diffusion theory for the neutron density distribution in the presence of a container, MS-82 and MS-82A, 30 pp., 1944.

23. The critical radius and the time constant of a sphere embedded in a spherical scattering medium (with K. Fuchs), MS-97, 35 pp., 1943.

24. Critical radius for a hemisphere with a one-sided infinite container, MS-100, 14 pp., 1943.

25. Critical radius for a hemisphere completely surrounded by a container (with P. D. Preston), M S-112, 20 pp., 1944.

26. Some questions connected with neutron multiplication and with expansion of a U-sphere, Doctoral Thesis, Birmingham University, 44 pp., October, 1944. Part I.

(1) Transport and integral equations for the spatial distribution of neutrons. Flux.

(2) Adiabatic approximation. Time-constant. Impor- tance of the lowest eigenvalue.

(3) Form oftbe integral equation for some particular bodies.

(4) Diffusion approximation. 4.1 Basic ideas and the differential equation of the

diffusion approximation. 4.2 Boundary conditions in the diffusion

approximation. (5) Behavior of the neutron density in the immediate

vicinity of the boundary. 27. Influence of a small black sphere upon the neutron

density in an infinite non-capturing medium, MT-88 (NRC-1549), 42 pp., 1944.

28. Influence of a large black sphere upon the neutron density in an infinite non-capturing medium, MT-93 (NRC-1550), 17pp., 1944.

29. Angular distribution due to an isotropic point source and spherically symmetrical eigensolutions of the transport equation, MT-112 (NRC-1551 ), 58 pp., 1945.

30. Milne's probkm with capture and production (with G. Placzek), MT-II8, 16pp., 1945.

31. Large spherical h°le in a slightly capturing medium' MT" 124 (NRC-1552), 45 pp., 1945.

32. Influence of a large black cylinder upon the neutron density in an infinite non-capturing medium, MT-135 (NRC-1554), 60 pp., 1945.

33. Neutron density at the centre of a small spherical cavity,


MT-136 (NRC-1555), 34 pp., 1945. 34. Influence of a small black cylinder upon the neutron

density in an infinite non-capturing medium (with S. A. Kushneriuk and W. P. Seidel). Proceedings of the Second Mathematical Congress, Vancouver, 172-195, 1949 (NRC-2487; based on MT-207, 1946).

35. Linear extrapolation length for a black sphere and a

black cylinder (with S. A. Kushneriuk), MT-214 (NRC- 1880, 17 pp., 1946.

36. Influence of an air-gap surrounding a small black sphere upon the linear extrapolation length of the neutron density in the surrounding medium, MT-232 (NRC- 1566), 21 pp., 1946.


NOTE TO VOLUME III

This volume contains Papers 37-45a, covering the Chalk River priod, 1946-1947.

With the building of laboratories and a reactor by the National Research Council of Canada at the Chalk River site, the Montreal laboratories were gradually phased out. Davison moved to Chalk River in 1946 as part of the UK attached staff. He continued investigations similar to those carried out in Montreal on the specification of boundary conditions at the surfaces of neutron absorbing bodies, voids and bodies with gaps for use with diffusion theory (e.g. Papers 41 and 42) and on the more fundamental aspects of solutions to the neutron transport equation (Papers 37 and 45). Neutron transformer rods, variational pro- cedures, statistical analysis on counters, nuclear physics, criticality and radiation physics were the other subject areas covered.

Former Montreal laboratory colleagues of Davison's that moved to Chalk River were H. H. Clayton and, for relatively short intervals, G. M. Volkoff, P. R. Wallace, E. A. Guggenheim, F. T. Adler and E. D. Courant. Newer colleagues, some briefly resident in Montreal as well, were C. A. Rennie, V. H. Rumsey and A. S. Lodge. It was while at Chalk River that Davison met W. H. Watson, who had come to the laboratories to take charge of the Physics Division. Dr. Watson was later, in 1954 when he headed the Physics Department at the University of Toronto, to welcome

Davison back to Canada to a post at the Computation Centre of that University.

VOLUME III: CHALK RIVER (1946-1947)

37. Milne problem in a multiplying medium with a linearly anisotropic scattering, CRT-358, 76 pp., 1946.

38. Conversion of thermal neutron flux into fast neutron flux by means of a uranium container, CRT-334, 67 pp., 1947.

39. A remark on the variational method, Physical Review 71, 694-697, 1947. (NRC-1520; based on MT-89, 1944).

40. Probability of a build-up of a large number of small pulses simulating a large pulse (with G. C. Hanna), TPI- 44, 33 pp., 1947.

41. Linear extrapolation length of the neutron density at the surface of a large hollow cylindrical shaft, CRT-319, 56 pp., 1947.

42. Linear extrapolation lengths for a small black cylinder with a small air-gap and a small imperfectly absorbing cylinder with and without a small air-gap (with S. A. Kushneriuk), TPI-45 (AECL-124), 33 pp., 1947.

43. On the distribution of electric charge in nuclei (with W. H. Watson), Physical Review 71, 742, 1947 (NRC-1539).

44. Critical radius of a nearly spherical body in an infinite container, CRT-361 (NRC-1820), 65 pp., 1945; revised 1948.

45. Angular distribution of neutrons at the interface of two adjoining media, Canadian Journal of Research A, 28, 303-314, 1950 (NRC-2136).

45a. Distributions, in a non-absorbing body, of gamma-ray sources giving a uniform distribution of gamma rays; Appendix III, Supplement No. 2, British Journal of Radiology, 197-198, 1950.

66 M. M. R. WILLIAMS et aL

NOTE TO VOLUME IV

This volume contains Papers 46-58, written in the period 1947-1954 when Davison was back in England, first at the Atomic Energy Research Establishment at Harweli and then, for about a year, at the University of Birmingham again.

Davison worked on a broad variety of problems in this period; distribution of fast neutrons in a reactor cell, slowing down and thermalization of neutrons, particularly in hydrogeneous media, criticality, spherical harmonics solutions, deposition of energy in tissue, etc. His purely mathematical papers included one on 'On Feynmann's "integral over all paths"' . It was during this time that Davison worked on his book on neutron transport theory. He also gave a series of lectures at Harweli.

VOLUME IV: HARWELL AND BIRMINGHAM (1947-1954)

46. Non-uniformity of distribution of very fast neutrons in a cell of a pile, AERE T/R 226, 30 pp., 1948.

47. On the spatial distribution of neutrons slowed down in slightly hydrogeneous media, AERE T/R 334, 50 pp., 1949.

48. Critical radius of an intermediate reactor, AERE T/R 528, 40 pp., 1950.

49. Extensions of Feynmann's method for determining neutron energy spectrum and spatial distribution to the case of anisotropic scattering, Part I, Theory, AERE T/R 590, 18 pp., 1950.

50. Preliminary estimate of energy deposition in soft tissue due to a fast neutron flux, AERE T/M 13, 50 pp., 1950.

51. Influence of a black sphere and of a black cylinder upon the neutron density in an infinite non-capturing medium, Proceedings of the Physical Society, A, 64, 881-902, 1951.

52. Some remarks on application ofthe spherical harmonics method in the case of complex geometries, AERE T/R 700, 19 pp., 1951.

53. Influence of a cylindrical channel on a periodic neutron density distribution, AERE T/R 738, 22 pp., 1951.

54. A new method for calculating the critical radius of systems containing hydrogen and fissile materials, AERE T/R 826, 51 pp., 1952.

55. On the spectrum of thermal neutrons in monatomic hydrogen gas, AERE T/M 82, 23 pp., 1953.

56. An upper bound to the effect of a slit in the reflector upon the reactivity of a nuclear reactor (with P. D. Preston), AERE T/M 85, 34 pp., 1953.

57. On Feynmann's 'integral over all paths', Proceedings of the Royal Society of London A, 225, 252-263, 1954.

58. On the neutron spectrum for v -~ absorption cross- section (with M. E. Mandl), Proceedings of the Physical Society A, 67, 967-972, 1954.


NOTE TO VOLUME V

The papers included in this volume, Papers 59-72, were written in the period 1954-1961 when Davison was back in Canada, at the Computation Centre at the University of Toronto. As mentioned by Professor Watson in the Biographical Note, during this time Davison also had a close association with the Chalk River Nuclear Laboratories which he visited frequently.

Davison's power of analysis is particularly evident in this period in the two interesting series of papers published in the Canadian Journal of Physics. In one, Davison examined analytically the rate of convergence of the spherical harmonics method of solution of the transport equation for a variety of applications of the method. In the other, he developed closed form expressions for the constants of integration in the spherical harmonics solution in an arbitrary order of approximation in terms of the spherical harmonic moments of the solution (i.e. the 'Inversion Formula').

In other investigations, Davison examined the definition of the diffusion length in heterogeneous assemblies, evaluated integrals involving Bessel functions, examined the errors generated in the numerical solutions for the spatial oscillations of the neutron flux induced by xenon production in a reactor. In an interesting paper in biology he applied methods similar to those used in neutron transport theory to examine the relation of postulated survival probability strategies (plateau or Gaussian) to the observed highly peaked Gaussian distribution evident in population characteristics.

A complete index to citations in Volumes I-V is appended.

VOLUME V: TORONTO AND CHALK RIVER (1954-1961

59. The compatibility of the survival plateaux hypothesis with Gaussian distribution of population, Bulletin of Mathematical Biophysics 19, 241-246, 1957.

60. The enrichment cost of power increments gained by flattening and by close rod spacing (with H. H. Clayton), CRT-682 (AECL-446), 94 pp., 1957.

61. Multilayer problems in the spherical harmonic method, Canadian Journal of Physics 35, 55-63, 1957 (AECL- 386).

62. Effective thermal diffusion length in a sandwich reactor, Journal of Nuclear Eneroy 7, 51-68, 1958 (AECL-555; based on CRT-715, 1957).

63. Spherical-harmonics method for neutron-transport problems in cylindrical geometry, Canadian Journal of Physics 35, 576-593, 1957 (AECL-438[ [See also 'Note on an inversion formula by B. Davison' by H. H. Clayton, Canadian Journal of Physics 40, 1254-1258, 1962 (AECL-1564).]

64. Spherical-harmonics method for neutron transport theory problems with incomplete symmetry, Canadian Journal of Physics 36, 462-475, 1958 (AECL-563).

65. On the rate of convergence of the spherical harmonics method for a sandwich reactor of small lattice pitch, Canadian Journal of Physics 36, 784-800, 1958 (AECL- 589).

66. On the rate of convergence of the spherical harmonics method in the Milne problem, Canadian Journal of Physics 36, 1323-1335, 1958 (AECL-672).

67. On the rate of convergence of the spherical harmonics method (for the plane case, isotropic scattering), Ca- nadian Journal of Physics 38, 1526-1545, 1960 (AECL- 1065).

68. Multilayer problems in the multigroup spherical harmonics method, Canadian Journal of Physics 37, 1482-1498, 1959 (AECL-915).

69. Simplified derivation of the spherical harmonics moments for cylindrical and other geometries, CRL-63 (AECL-1068), 37 pp., 1959.

70. Some integrals involving the Bcssel function integrals, CRT-856 (AECL-867), 20 pp., 1959.

71. A semi-numerical semi-analytical method for the two- group theory of xenon oscillation calculations for nuclear reactors, CRT-993 (AECL-1193), 43 pp., 1961.

72. Correction for non-zero time step in the numerical simulation of spatial oscillations of xenon in a nuclear reactor, CRT-1033 (AECL-1292), 17 pp., 1961.

ADDITIONAL PUBLICATIONS

In addition to the papers and reports listed above that are included in volumes I-V, there are many items, e.g. personal notes, memoranda, letters as well as certain document~4~;~at are not reproduced. His book Neutron Transport Theory published by Oxford University Press and already mentioned in the biographical note on Davison, is of course one such item. A major document to which Davison often refers in his papers that is not included is LT-18, 'Transport Theory of Neutrons'. This was a writeup by Davison (213 pp.) of a series of lectures that he gave at the Chalk River laboratory in 1946-47. These notes are not included because their content is now largely available in Davison's book.


A BIBLIOGRAPHY OF THE PUBLICATIONS OF BORIS DAVISON

INTRODUCTION

Boris Davison was born in Russia, in 1908, and educated there, graduating from Leningrad University in 1930. He became associated with the Hydrological Institute of the University in the year of his graduation and remained with the Institute until 1938, when he emigrated to Britain. During 1939-1941, he worked at the Royal Aircraft Establishment at Farnborough and thereafter joined a Ministry of Supply atomic energy research team at the University of Birmingham. It was at Birmingham that he began his work on neutron transport theory, for which he is noted.

In the period 1944-1946, Davison was part of a British contingent of scientists located at the Montreal and Chalk River laboratories of the National Research Council of Canada, Division of Atomic Energy; he also spent some time at the Los Alamos Scientific Laboratory in this period. Davison went back to Britain in 1947, to the Atomic Research Establishment in Harwell. In 1953-1954 he once again spent some time at the University of Birmingham, on leave of absence from Harwell, and then he returned to Ca- nada, joining the Computation Centre at the Uni- versity of Toronto, where he remained until his death in 1961. In this latter period he was very closely associated with the Chalk River Nuclear Laboratories, which he visited frequently. At the University he gave courses on neutron transport and reactor theory and on numerical analysis and also lectured at Chalk River. His book, Neutron Transport Theory, published in 1957 by the Oxford University Press, was written in the HarwelI-Birmingham-Toronto period, with the collaboration of J. B. Sykes.

Davison's working life divides naturally into five main periods which are determined by the nature of his work and the institutions with which he was associated. Accordingly, both the more detailed account of his work that follows and the bibliography have been divided to reflect this arrangement. The subject of the papers in the earlier sections is ground-water and air flow while the later papers deal mainly with neutron transport, criticality and reactor theory. The bibliography includes an Appendix which contains the abstracts or summaries of a selected number of these papers, chosen to illustrate in more detail the nature and type of analysis used in the various subject areas.

Boris Davison had an important impact on the development of neutron transport and related theories in Canada, and the proposal (referred to in the Foreword) to print his collected papers is a recognition

of this impact. Many of his results are now well established in the literature. However, detail on their derivation is often lacking, and it is interesting to examine in the original the elegance and style with which these results frequently were derived.

A RESUME OF DAVISON'S WORK

1930-1941--Leningrad University and the Royal Air- craft Establishment, Farnborough.

The papers of this period are listed as Papers 1 through 20. As mentioned in the Introduction, Da- vison was associated with the Hydrological Institute of Leningrad University during 1930-1938 (he received the equivalent of an M.Sc. degree in mathematics from the University in 1930). His papers are thus largely related to the hydrological work of the Institute, principally problems in hydrodynamics-mechanics, although solutions to problems of a purely mathematical nature were also published.

These early papers reveal features characteristic of all of Davison's investigations; an extraordinary capability in mathematical and numerical analysis, a sense of precision and physical insight. His analytical capability is illustrated, for example, in his purely mathematical papers, numbers 4, 5 and 10. Papers 1-3, 7 and 16 indicate his feeling for precision. Here variations in hydrological models are adopted and precise statements are then made as to the consequences of the variations.

Papers 6, 8, 9, 11-15, 17 and 19 represent contributions to hydrodynamical-mechanicai studies; steady state and transient flow of fluids for a variety of fluid properties, system boundary conditions and physical shapes.

Paper 18 is Davison's part of a three-part book on the mechanics of continuous media written by S. Christianovitch, S. Michlin and B. Davison and published by the Academy of Sciences of the USSR. Chapter headings are given.

The original abstracts, in English, accompanying Papers 5 and 10 (mathematical analysis), 1 and 16 (model variations), 9 and 15 (hydrodynamics) and 20 are given in the Appendix.

1942-1944--University of Birmingham (Ministry of Supply )

The published papers relating to this period consist of Papers 21 through 26.

Early in 1942, Davison became a member of an atomic energy research team at the University of Birmingham under the direction of Professor R. E. Peierls and was thus introduced into a new area of

The publications of Boris Davison

research, neutron physics and neutron transport theory. Besides Professor Peierls, members and as- sociates of the team included theoreticians such as P. A. M. Dirac, M. H. L. Pryce, K. Fuchs and A. H. Wilson. Much fundamental work was done on the properties and solutions of the transport equation, particularly in the one-velocity approximation with isotropic scattering and the integral equation formulation. Of primary concern, however, were questions related to neutron multiplication, critical sizes, time constants and energy releases in assemblies containing fissile materials. Much of the work was classified.

While at Birmingham, Davison was awarded a Ph.D. degree (in 1944) for his work on criticality and the specification of boundary conditions for use in the diffusion approximation of the theory. Only Part I of his thesis has been published. The title headings of the various chapters of this part are cited as Paper 26.

1944-1946--Montreal and Chalk River Laboratories, National Research Council of Canada and, briefly, the Los Alamos Scient!fic Laboratory, New Mexico

The publications in this period consist of Papers 27 through 45. As mentioned, in 1944, Davison was part of a British contingent of scientists at the Montreal and Chalk River laboratories of the National Research Council of Canada in a cooperative Canada-United Kingdom-United States program of research in atomic energy. Davison thus became a member of another group of analysts and theoreticians who devoted much attention to the fundamentals of neutron transport theory, and who also examined such matters as fissile material criticality conditions and a large variety of nuclear reactor concepts (homogeneous, heterogeneous; solid, slurry, solution ; natural and enriched uranium, breeder; etc.). The group variously included G. Placzek, R. E. Marshak, G. M. Volkoff, Carson Mark, J. LeCaine, P. R. Wallace, B. Carlson, H. H. Clayton, F. Adler, W. Seidel, E. Courant, M. H. L. Pryce, and E. A. Guggenheim. This group was rather remarkable for the thoroughness and depth of its investigations and its impressive output, for it was only loosely cohesive, as many of its members moved on to academic posts and other subject areas, while others moved in. There were in the USA at this time other remarkable groups doing somewhat similar work at the University of Chicago, at Los Alamos, etc., and Davison spent a short time with the group at Los Alamos.

The period 1944--1946 was a very productive one for Davison. His main efforts were directed towards the specification of boundary conditions at the surfaces of

and some reminiscences of the man 69

neutron absorbing rods and cavities embedded in predominantly neutron-scattering media (moder- ators) for the diffusion theory solution of the neutron flux distribution in these media. Such conditions were important for assessing the effectiveness of control, shut-off and fuel rods in reactors and estimating the importance of voids. Idealized problems were posed and elegant perturbation theory solutions leading to the boundary conditions were developed for bodies of large and small size (e.g. Papers 27, 28, 32, 34-36 and 42).

Studies were also made on the more fundamental aspects of transport theory (Papers 29, 30, 37 and 45). In particular, in Paper 29, Davison discussed the elementary eigensolutions of the homogeneous transport equation in the plane case, involving the use of the distribution function, and the completeness of the solutions. Neutron transformer rods, variational pro- cedures, statistical analysis on counters, nuclear physics and criticality (Papers 38-40, 43 and 44) respectively) were some other subject areas covered.

The author's abstract or summary for Papers 27 and 32 (use of perturbation theory), Papers 29, 30 and 45 (transport theory) and Paper 39 (variational analysis) are included in the Appendix.

1947-1954--The Atomic Energy Research Establish- ment, Harwell, and University of Birmingham

The publications issued in this period include Pa- pers 46 through 58.

Davison worked on a broad variety of problems in this period; distribution of'fast' neutrons in a reactor cell, slowing down and thermalization of neutrons, particularly in hydrogeneous media, criticality, spherical harmonics solutions, deposition of energy in tissue, etc. In mathematics there is his paper, 'On Feynmann's "integral over all paths"'. It was during part of this time that Davison worked on his book on neutron transport theory. He also gave a series of lectures at Harwell.

Abstracts to Papers 52 (spherical harmonics in complex geometries), 57 (Feynmann integral) and 58 (neutron spectra for v -~ absorption) are included in the Appendix.

1954-1961--University of Toronto and Chalk River Nuclear Laboratories

Publications in this period include Papers 59 through 72.

Davison's power of analysis is particularly evident in this period in the two interesting series of papers

70 M. M. R. WILLIAMS et aL

published in the Canadian Journal of Physics. In one, Davison examined analytically the rate of convergence of the spherical harmonics method of solution of the transport equation for a variety of applications of the method. In the other, he developed closed form expressions for the constants of integration in the spherical harmonics solution in an arbitrary order of approximation in terms of the spherical harmonic moments of the solution (i.e. the 'Inversion Formula').

In his other investigations, Davison examined the definition of the diffusion length in heterogeneous assemblies, evaluated integrals involving Bessel functions, examined the errors generated in the numerical solutions for the spatial oscillations of the neutron flux induced by xenon production in a reactor. In a very interesting paper in biology he applied methods similar to those used in neutron transport theory to examine the relation of postulated survival probability strategies (plateau or Gaussian) to the observed highly peaked Gaussian distribution evident in population characteristics.

Abstracts to Papers 59 (biology), 61 (inversion formulae), 67 (spherical harmonic solution convergence rate) and 72 (errors in numerical solution of the xenon oscillation problem) are included in the Appendix.

SUMMARIES OR ABSTRACTS OF SELECTED PAPERS

1. On the theory of a new method (the logarithmical interpolations method)for calculations of back-water curves (Leningrad, 1930)

Summary--The author is treating a new method of the back-water curves calculations given by Eng. Kriv- oshein ; having shown that the logarithmical anamor- phose of the dependence between the modulus of discharge (i.e. ratio of discharge to the square root of the slope) and the depth must be a straight line (the river-bed satisfying Prof. Bakhmetiev's conditions) the author evaluates the magnitude and the sign of the error.

If the back-water depth is not great (for example, for a broad rectangular river-bed, less than 1.36 of the natural depth) the error is negative and the calculated back-water curve lies below the exact one, while in the contrary case the calculated curve lies above the exact one.

5. On the uniform convergence of the trigonometrical series (Leningrad, 1932)

Summary--The convergence problem of a power- series in case of the expanded function being continuous up to the circumference of the convergence- circle was proposed by Pringsheim in 1900.

In 1910 Fejer showed that this problem as stated by Pringsheim had a negative solution, in a memoir in which he gave an example of a function regular in the circle Izl = 1 and continuous including its circumference, the expansion of the function in the power-series being divergent at z = 1.

Later, in 1913, the problem was examined by Fejer, when the Cesrlro-summability of a power-series in- volves its convergence. In this work he proved a very important test, viz. the convergence of the series Znlan] 2, containing all the criteria known before.

We may notice that the above condition being satisfied, the series is convergent with the Ces~iro- summability of any order, and not only with the (C)- summability of the first order. The same holds good for the present writer's tests.

It is to be noticed further, that if only one component of the power-series is summable (C), the above test will involve the convergence of this component.

In his first criterion, obtained in §1 the present writer does not suppose the series 'to be a Fourier's series.

The second criterion, obtained in §2, holds good only for a Fourier's series. The classical convergence- tests for the Fourier's series are much more numerous. We can mention, that the well-known Dini-Lipschitz condition is included in test §2 (see §3).

The last article is of a somewhat geometrical character. In this article the author applies his second test to the expansion of f (z) in a power-series, the domain of F(z) = S~ f(t) dt being known.

There exist examples showing that the author's criteria are not contained in the F6jer's test, and that the suppositions of the first criterion do not involve the representability of the expanded function, or of its conjugate by the Poisson's integral. But the present writer did not succeed in showing that his second test is not equivalent to the Dini-Lipsehitz's condition.

9. On the independent from the viscosity forms of the two-dimensional unsteady motion of a viscous incompressible fluid (Leningrad, 1934 )

Summary--The author considers the two-dimensional unsteady motion of a viscous incompressible fluid in the case when the outer forces possess a potential, and shows that there are only four cases when the velocities are independent from the value of the coefficient of viscosity.

These are: the motion with a constant vortex (case A), and three other cases, for which the author determines the stream-function explicitly--see formulae (B), (C) and (D).

Further the author determines all the unchangeable regions where a motion of the above-mentioned character may take place, this motion having a


variable vortex and satisfying the usual boundary- condition of sucking of the viscous fluid to the solid bordering the region of motion.

Decomposing the motion upon the translative and the relative one and taking for the translative motion the motion of the solid bordering of the region the author assumes the relative velocity to be bounded. Then the region of the motion must be either a streak bounded by two parallel straight lines, or a ring bounded by two concentricai circles. In either case the relative motion will be a steady one, and the translative motion will consist of a progressive motion and a uniform rotation.

10. On the Gauss method of the approximative computation of definite integrals (Leningrad, 1934 )

Summary--The author considers the case when the integral has a polar, iogarithmical or algebraical singularity in the vicinity of the range of integration.

Starting from the valuation of the derivatives of an analytical function by the maximum of the function in a complex region, the author shows that Gauss method may be applied only if the length of the range of integration is less than the fourfold distance between the path of integration and the nearest singularity. Otherwise the range must be divided into portions and the method must be applied to each portion separately.

If the integral has only one singular point, situated upon the continuation of the path of integration then to receive the required accuracy by a minimal quantity of computations one must divide the range so, that for each of the new ranges its length should be equal to the distance between it and the singular point. The remainder will then converge to zero as C • A -ren, A, C being constants and m • n being the full quantity of points where the integrand must be computed.

If the singular point belongs to the range of integration, a small portion of the range, lying in the nearest vicinity to the singular point must be excluded and the rest must be divided into portions. The greater the accuracy required, the smaller the portion to be excluded: thus the rest must be divided into a greater number of portions.

An assay is made to determine which way of dividing we must use if we want to receive the required accuracy with the minimal quantity of computations. The optimal number of points in each portion is found to be proportional to the number of portions under the condition only that there is such a number ~ that the integrals along the excluded portion were less than Ke?, e-being the length of the excluded portion. The remainder then converges to zero as C • A -,/(m .,), A, C being constants and m • n the full quantity of points where the integrand must be computed.

15. Some particular cases of ffround-water motion, the evaporation from the free surface being taken into account (Leningrad, 1937)

Summary--This note gives a method for the solution of the following problem: two dimensional steady ground-water motion takes place in a region bounded by (1) impervious boundaries, (2) boundaries of water basins and (3) a free surface. The impervious boundaries are straight line segments parallel to each other; the boundaries of water basins are straight line segments orthogonal to the impervious boundaries.

The evaporation from the free surface takes place (but not the penetration of waters from without upon the free surface) and this evaporation from any arc of the free surface is assumed to be proportional to the projection of this arc upon a horizontal axis.

The problem is solved by the method of the conformal representation.

Under the above conditions the boundary of the w region ( w = u x - iuy; ux and u~ are the component- percolation-velocities) is composed of arcs and seg- ment of only one circumference and only one straight line. The circumference corresponds to the free surface and the straight line both to the impervious boundaries and the boundaries of water basins. These two lines must intersect. Denote one of the intersection points by Wo. The author shows that the boundary of the region of [dz/dw (w - w0) 3] values (z = x + iy, x and y being the coordinates of points in the region of motion) is composed of segments of straight lines passing through the origin, i.e. arg [dz/dw (w - Wo) 3] is constant for each portion of the boundary. Therefore one can easily fulfill the conformal representation of [dz/dw ( w - Wo)3]--region upon the w-region and obtain an equation of the following form

whereof

dz (w - w0) 3 = F(w), dw

F(w) dw z = j ( w - w 0 ) ~ + Zo.

An example is solved by means of this method. The problem solved in this note is a particular case

of the problem solved by the author in his paper. 'On the steady two-dimensional motion of ground-water with a free surface' (Phil. Mag., May, 1936), but the method given in this note is much simpler, although both methods start from the same assumption.

16. On some simplifications used for computing ground- water-motion without free surface (Leningrad, 1937)

Summary--The author considers steady two-


dimensional motion of ground-water without free surface and seepage surface and shows the following circumstances.

If we divide the region of motion into parts by lines ending on the boundaries of water basins, then the sum of discharges passing through each such part will be always less than the discharge which percolates through the complete region in reality.

If one the contrary, we divide the region of motion by lines ending on the impervious boundaries, and substitute the permeable massive by a series of massives obtained by means of the above-mentioned division, fictive water basins being introduced between the latter massives, the discharge passing through this system of dams will be always greater than the discharge percolating in reality.

20. Wind tunnel correction for a circular open jet tunnel with a reflexion plate ( Farnborough, 1941 )

Summary--It is advantageous from many points of view to make test models as large as possible. One method of doing this is to measure the characteristics of half the model in existing wind tunnels. One half of the aerofoil is mounted horizontally on a vertical reflexion plate and the plate is placed in a suitable position in an open jet which, in the undisturbed state, is of circular section. The contour of the jet is distorted, especially with models of large semi-span, but this distortion is neglected in the analysis. The correcting factor associated with 'uniform' distribution of lift is worked out exactly and that associated with 'elliptic' distribution approximately. The effect of the induced downwash on the distribution of lift is ignored. The results are given in suitagle tables and figures,

Throughout the working range of normal experi- ments the correcting factor is of the same order of magnitude as that obtaining when a full model is tested in a jet of circular section.

27. Influence of a small black sphere upon the neutron density in an infinite non-capturing medium (Montreal, 1944 )

Summary In the present report we examine the neutron density in a uniform non-capturing medium surrounding a small black sphere, in the case when the density at infinity tends to a definite finite limit. 'Small' sphere means small compared to the mean free path in the surrounding medium. The linear extrapolation length 2 for the asymptotic solution for the density in the surrounding medium is determined as a function of the radius of the sphere. The method used is essentially the perturbation method, treating an infinite space with a constant density in the absence of the black sphere as the unperturbed system, and all the effects of

the presence of the black sphere as small perturbations. Orders of magnitude of the successive perturbations were estimated and the first few were determined explicitly. This gave

2 = 3 - - 9 - a - - 3 - 1 a 21oga

- 1 . 4 0 0 2 a 2 + 0Ca a l o g 2 a )

in which a is the radius of the sphere in units of the external mean free path.

29. Angular distribution due to an isotropic point source and spherically symmetrical eigensolutions of the transport equation (Montreal, 1945)

Summary--In sections 2 and 3 we give the angular distribution of neutrons due to an isotropic point sources in an infinite medium, both with and without capture.

The remainder of the paper contains the derivation of certain families of the spherically symmetrical eigensolutions and a preliminary discussion of their properties. It is shown, in particular, that the point source solution cannot be represented as a super- position of eigensolutions.

This remarkable paper was the forerunner of the method of elementary solutions so elegantly developed by K. M. Case into a powerful tool for solving transport theory problems in .finite media. ( Ed. )

30. Milne's problem with capture and production (Montreal, 1945 )

S u m m a r ~ T h e relation between the solution of the inhomogeneous integral equation of the problem containing an arbitrary source term and of the cor- responding homogeneous equation is investigated. It is shown that the former can be derived in an elementary way-- in terms of the latter. The discussion of the special cases of constant and exponential source terms, given in sections 2 and 3 yields, among other results, an elementary derivation of Adler's albedo formulae. The solution for a source which amounts to the determination of the Green function of the homogeneous equation is derived in section 4.

32. Influence of a large black cylinder upon the neutron density in an infinite non-capturing medium (Montreal, 1945 )

Summary--In the present report we examine the neutron density in a uniform non-capturing medium surrounding an infinitely long black cylinder of a large radius, in the case when the density depends only upon the distance from the cylinder. 'Large' radius means large compared to the mean free path of the surround-

The publications of Boris Davison

ing medium. The linear extrapolation length 2 for the asymptotic solution for the density in the surrounding medium is determined and expressed in terms of the radius of the cylinder. The method is essentially as follows. In zero approximation, and not too far from the surface of the cylinder the neutron density should be roughly the same as in a half-space bounded by a vacuum. In more accurate approximations, but still not too far from the cylinder the effects of its curvature can be treated as small perturbations. On distances comparable with the radius of the cylinder the above approach breaks down because all the successive corrections are then of the same order of magnitude. However, it is possible to determine the leading term, the next largest term, etc., in all the successive corrections simultaneously. And adding first all the leading terms in the successive corrections, next all the second largest terms and so on, we obtain an expansion valid on all distances from the cylinder. This expansion can be utilized to determine, to a given accuracy, the constants entering into the asymptotic solution, in particular the linear extrapolation length 2, for which we obtain the expansion

1 2 = 0.7104+0.2524- + 0.0949

a

5 log a 0.0256 a~ + 0 ~ l°g2 a~ 64 a a \ a 4 ]

in which a is the radius of the cylinder in terms of the external mean free path.

39. A remark on the variational method (Montreal, 1947)

Abstract--The nature of the extremum of the functional

q(x)[q(x)- 2 S q(x)K(x, y) dy] dx/[Sq(x)fo(x) dx] 2,

associated with the inhomogeneous integral equation,

qo(x) = 2 ~ qo(Y)K(x, y) dy +f0(x),

is investigated. Two sets of sufficient conditions for the extremum reached for q(x) = qo(x) to be a minimum (and neither a maximum, nor a 'saddle point') are determined.

45. Angular distribution of neutrons at the interface of two adjoining media (Chalk River, 1950)

Abstract--Two media separated by a plane interface and each filling an infinite half-space are considered. Neutrons come in infinite number from infinity in one medium and the neutron density vanishes at infinity in the other medium. Isotropic scattering in the laboratory system of co-ordinates in both media is assumed

and some reminiscences of the man 73

and all neutrons are assumed to have the same speed. The angular distribution of neutrons emerging from either medium into the other is obtained in terms of the angular distribution of neutrons for the Milne problem, that is, the angular distribution of neutrons emerging from a scattering medium into a vacuum. The latter angular distribution is tabulated in the preceding paper by LeCaine.

52. Some remarks on application of the spherical harmonics method in the cases of complex geometries (Harwell, 1951 )

Abstract--A new procedure is given for setting down the spherical harmonics method equations and reducing them to the equations for one unknown at a time, for the cases when the angular distribution depends on both angles characterising the direction of a neutron. This new procedure is noticeably more economical in algebraic manipulation than that given in LT-18, and noticeably more elementary than that given in MT-92. The essence of this new procedure is that in the earlier stages of the calculation one works in the Cartesian coordinates, and treats all the harmonics of any particular order en block ; the subdividing harmonic of order s into its (2s + l ) components and the passage to the system of coordinates appropriate to the particular geometrical arrangement are carried out after the system of equations has been reduced to equations for one unknown at a time.

The new procedure is illustrated by applying it to the case of a finite cylinder.

57. On Feynmann's 'integral over all paths' (Birming- ham, 1954 )

Abstract--A definition of Feynmann's 'integral over all paths' is given in precise mathematical terms, namely, it is shown, subject to some restrictions on the nature of the Lagrangian, that the ratio of the 'integral' to a certain normalization factor is uniquely determined by postulating: (i) the composition law, (ii) the possi- bility of taking any factor independent of the paths outside the integration over paths, and (iii) that the quantity in question should not involve any arbitrary 'universal' constants of the dimensions of inverse time. The nature of the normalization factor, referred to above, is also examined.

Finally, a method of parametrization of paths is introduced, and it is shown that Feynmann's 'integral over all paths' can, alternatively, be defined in terms of this parametrization.

58. On the neutron spectrum for v -~ absorption cross section ( Harwell, 1954 )

Abstract--An approximate expression is found for the

74 M. M. R. WIt.LIAMS et al.

neutron absorption energy spectrum in an infinite medium consisting of atoms of mass number M, for which scattering is assumed isotropic in the centre-of- mass system and trJtr~ ~ v -s, where ct is a positive constant. The accuracy of certain useful types of integral over the resulting spectrum is examined.

59. The compatibility of the survival plateaux hypothesis with Gaussian distribution of population ( Toronto, 195 7 )

Abstract--It is often assumed that a highly peaked Gaussian distribution of population characteristics would imply a similar distribution of the survival probability. Such interpretation is, however, rather arbitrary and unduly restrictive. In this article it is shown, on the contrary, that, where the characteristics depending on many genes are concerned, even a broad survival plateaux will result, under certain additional conditions, in a highly peaked nearly-Gaussian distribution of the population characteristics.

61. Multilayer problems in the spherical harmonics method ( Toronto, 1958)

Abstract--In applying the spherical harmonics method to multilayer problems it is necessary to invert certain matrices. It is shown, for the cases of plane and spherical symmetry, that these inverse matrices can easily be written down explicitly, so that there is no need for numerical matrix inversion.

67. On the rate of convergence of the spherical harmonics method (Toronto, 1960)

Abstract--The rate of convergence of the spherical harmonics method in neutron transport theory, for the plane case with isotropic scattering, is investigated for an arbitrary number of layers of different materials,

and both in the one-group and in the multi-group theory. The attention is centered upon the cases when there is no external supply of neutrons, so that one deals with an eigenvalue problem. The error in the eigenvalue due to the truncation of the spherical harmonics expansion is found to vary essentially as 0(l/N2), where N is the order of the highest harmonics retained. The proof of this result consists of two parts. Firstly it is shown that the use of the spherical harmonics method, under the conditions stated, is strictly equivalent to making a certain approximation in the kernel of the integral equation for the flux ; and then the error due to this approximation is assessed.

72. Correction for non-zero time step in the numerical simulation of spatial oscillations of xenon in a nuclear reactor (Chalk River, 1961 )

Summary--A reasonably complete investigation of xenon instability in a large high-flux nuclear reactor is necessarily numerical, since the basic equations of the problem are non-linear. Concern, then, for reducing computing time may tempt one to use a rather long time step in the numerical integrations, comparable to, or even longer than, the mean life of xenon in the reactor. Before doing so one should know what error one thus introduces.

The present report examines this question in the case of the linear approximation to the actually non- linear problem in the expectation that such a preliminary investigation will at least shed light on the more realistic problem. It shows that, provided in a certain perturbation procedure corrections of order higher than the first are negligible, there exists a rather simple relationship between the true time constant of any mode of instability and the time constant calculated by a numerical procedure with non-zero time step.


Acknowledgement--A major portion ofth© listings in the bibliography were contributed by J. E. Woolston and H. H. Clayton. B. Gay of the Technical Information Branch assisted in many aspects of the assembly and formulation of the report.

AVAILABILITY OF THE COLLECTED WORKS

Atomic Energy of Canada Ltd., have asked me to make it clear that Davison's papers were reprinted in a very limited edition and that these reprints are available in only a few places. A list of libraries where copies are being deposited is given below. The bibliography AECL-5620 was fully published and is fairly widely available.

Leningrad University Academy of Sciences of the USSR Atominform, Moscow Royal Aircraft Establishment, Farnborough University of Birmingham National Research Council of Canada Chalk River Nuclear Laboratories Los Alamos Scientific Laboratory Atomic Energy Research Establishment, Harwell University of Toronto Atomic Energy of Canada Limited, Head Office Whiteshell Nuclear Research Establishment Atomic Energy of Canada Engineering Company National Library of Canada Canadian Association of Physicists Royal Society of London The Institute of Physics The British Institute of Radiology University of Liverpool International Atomic Energy Agency

Documents

The publications of Boris Davison and some reminiscences of the man