Sigma_1977-78

KERNKRAFTWERK UNO
MENSCHLICHES DASEIN

(Nuclear Reactors and Human Existence)

by Dr. V. Joseph
Senior Lecturer, Dept. of Mathematics

It took place one evening wayb1ck in 1)55. p:uticip:i.te in the drama of human e�­
The Cile,nistry Lecture Tbe1tre w.1.; pa,;ked istence. According to the articles on the
with students and me,n�rs of th-! ac.1de.nic Nucl.::aT Reactor which appeared in the
staff. The speaker w.1s no other than the recent issues of the magazine "Der Spiegel"
distinguished Don from Cambridg..!-Pro­ sci,.:ntists are unable to arrive at a consensus
fes,or C. J. Eliezer, who held the chair of on the question "Should Nuclear Reactors
Mat,1e,n.iti..:s ia th'! University of Ceylon. bJ built?". One of the contributors to the
It was a popular lecture on the peaceful magazi.1e ha� recalled what Einstein had
use3 of oucle.ir energy. Ev.:ryb.:>dy listened to say about nucl-!ar energy;
intently to th.:: speaker, as was the custom
in the fifties. At the end of the lecture so:ne­ "Atomenergie kann immer nur dcr
body fro:n the audi.!nce r.i.ised a pertinent Zerst6rung dienen"
question "What are we going to do when (..\to:nic energy can only bring about
nuclear fuel i; exhausted 7". The O;,n had destruction)
a ready reply .. We wo.1't b! there in any
case!" A deaJ sil.;nce enmeJ---everyone in Ei ntei 11 seens to have sensed the pre­
the hall thought of the di:>mal future of the di:;a nent of our times. This remark will
human species. But it wasn't long before co:ne as a rude shock to those who have
they dis.nissed all thoughts of the distant b.::en drea ning of a Utopia with abundant
future as irrelevant and applauded the energy. Human life is at stake. The cartoon
speaker. below, which appeared in Der Spiegel 7-3-77,
speaks fo, itself.
But time has caught up with the human
race. Th! profound changes brought ab::>ut It is up to the younger generation to
by sci!nti as could deeply affect the lifes evolve a new sense of values concerning
of the people. As such science should not human existence.
be left in the hands of a sel�cted few, how­
eve: brillilnt. The German playwright Brecht Here is an English translation of the caption;
has stressed the need for the people to

Safety assured; Nuclear Reactor supplies energy for Eternal Light.

8



























A FUNDAMENTAL SET OF UNITS

by Kavan U. Ratnatanga

A.uistant Lecturer, Dept. ofPhysiC8
Colombo Campus

The exponential expansion of the wealth being defined as inertial mass multiplied by
of scb.1.tific k,1-l .vli!dg¢, coald be attrib.1ted Velocity. Newton took the b.>ld step to
to tile sck1tist; ability to e"Cpress his obser­ define the units of the quantity Force in
vations of the "Physical Universe" to one such a way that the inertial mass of the
another. As Newton remarked, "If I have object is equal to the gravitational mass.
seea a little farther than others, it is b:cause The two quantities were considered to be
I have stood on the sh.>:it.l:rs of giants". di ,tinct properties at that time, for it was
Scienthts in th.:ir attempt to write down only 250 years later, that Ei11stein showed
results of measure.neat, have g:nerated a their equivalence in his General Theory of
nu.nber of syste.ns of units. These units Relativity.
have undergone much revision over the last
400 years, h1 ter:ns of the definitions of the The same approach could be very success­
b1Si:: uaits miss, length and ti ne, and also fully applied to a number of other Funda­
in other re,p�ts, sucn as the utiJnalization mental Laws of Physics, for an appropriate
of the MKS units. This International choice of units.
Systelll of Units (SI), is n:>w being in­
cre.1sinJly used in most branch¢S of scientific The SI system (Rationalised MK.SA units)
work. It is a great improve.neut on all has seven Independent Standard basic unit,
older syste ns, and has fod to much simpli­ of Physical quantity
fic.:1.tion, in the form of the fundamental
e�ruti..Jns of Physics, and the inter-relation­ (1) Metre Unit of length
ships b�t.veen p!tysi,;al quantities. However,
most of these changes have firmly adhered (2) Kilogramme - Unit of mass
to ad:>?ting a system, which leaves un­
changeJ, tb.e magnitudes of the units with (3) Second Unit of time
which. th� Sci.!ntific Corn nunity is most
fa niliar. The only radical change was the (4) Ampere Unit electric current
adoption of the decLnal system of units
for scientific work, replacing the more (5) Kelvin Unit thermodynamic
fa'Ililiar foot-pound-second system, with temperature
all its co npli..;ated numerical relationships
between different orders of units. Even (6) Mole Unit amount of
this change, h!ls still to b;: Internationally substance
accepted, for everyday usage. This is due
to man's inherent opposition to c.-hange (7) Candela Unit luminous
from what he has grown accustomed to intensity.
from his childhood.
The mole which is the number of elemen­
If one overcomes this reluctance, and tary entities in an Unit amount substance
looks for a system of units which are in­ defines the Avogardo Constant. This unit
dependent of any arbitrary conventions of could be dispensed with by using the absolute
the Human Species it is possible to create number of the elementary entities to express
a very simple system of units which makes concentrations.
a number of fundamental constants unity.
The candela depends on the sensitivity of
Hhtorically a move in this direction was the human eye (observer) to electromagnetic
taken by Newton. His 2nd Law states that radiation and is as such arbitrary for any
the Newtonian Force is proportional to system of Universal Units.
the rate of change of Momentum. Momentum

22























BRICKS OF THE COSMOS

by L S. Jt.. Wijewardhana

ASsistant Lecturer, Dept. of Mathematics

Since the dawn of history man has tried decreasing order of strength these are the
to understand the nature of the physical stron6 iilt�raction, el.;ctro magnetic interac­
world. Of all th!: di ,coveries he has made in tion wMkinteraction and the gravitational in­
his intellectual pursuits the discovery of rteartaiocstiol:n.10T-h2 e:ir10r-eulat:iv10e-s3t1r•eTnghtehsstraornegiinnttehre­
sub atonic p:irtbles played a leading role action is an extre:nely short rang.! force, its
in a!upi;1g the dc3tiny of mankind. His range of action �ing i11 the order of I 0-13 c.ns.
excursbns i 1 the microscopic world have Tnis int.!raction is supposed to be responsible
for the attractive force that operates b.:twecn
led hi n to b�li�ve that the study of ele­ n:icl �ons in tlle atomic nucl.!us. All charged
particles p.irticipate in the electro magnetic
mentary parti,;les is not an unnec¢ssary i 1teraction. Weak interaction h responsible
luxury, b.1t aa e:1-;ential ractor, if one hopes for the decay of atomic nuclei. The gravita­
to understand the working of the natural tional interaction is so weak that it's effect
world. can be negt...-cted in considering reactions
b�tween eknentary particles.
The prop:rti:s of elcnentary particles �re
so diverse that at a first glan� a novice To summaril.e the properties of elementary
would definitely be ba..'fioo. B<.1t fro n the particles various classification schemes have
apparent co nplexity of their b.:havio1;1r b:en sug�esled. Lighter particles are classi­
certain sy,n,netries have emerged. Certain fied as Leptons. They include Neutrino (�).
properties of eb-nentary p:irticles like mass Electron (e), Muon (:Sf) and· their anti
and el,;ctric charge are co.nnon to macros­ particles. Photon is the quantum of the
copic syste.ns. B.it they have other properties electro magnetic field. It has zero rest mass
like spin, isotopic spin, hypercltarge, Baryon and ,;pin half and the photon is its own anti
number, Lepton nu1n')er, parity etc., which pa�ticle. Neutrino like the photon travels at
have no cl:1s-,i,;al analog.1es.. Elementary the speed of light. Hence it has zero rest
pl.rticles ob,!y 2 typ:3 of statistics, nanely mass. It has spin ¼ and spins about its
Bose Ein,tein and FcrmiDlrac. If the wave direction of motion like a left handed screw.
function de:;cribing a syste-u of ide:itical Its anti particle spin!l in the opposite sense.
particlei b sy nnetric with respect to parti;le It p.1rticipates only in weak interactions and
exchange, the p.1rticles are sai� to_ oabneh)'. is capable of passing through a sheet of lead
Bose stati,tics. If the wave function ts a ligh.t year thick without getting absorbed.
sym,netric the p:irticles are said to ob�y There arc two types of neutrinos one
connected with electrons and the oth!r with
Fermi statir.tics. If ,_,, (:!:',>�.)is the wave muons. Muon is like a big brother of electron,
function of a system of 2 p:irticles anti and is 207 times more manive. Their anti
symmetric with respect to particle exchange particles are of the same mass but of
opposite charge. Electron and muon have
then -,, (::rt, >�.) - - ,,. (:y1 > ::r ,) both spin half and can participate in electro
magnetic and weak interactions. Muon is
If we put �. ==- :!'r then unstable and decays into electron and two
neutrinos one of each kind.
I I),,, (....- t ) .....,}::=. - '\l' ( :!:' ):Y'
Y.--+ e--+ +"SL + :i,e
Therefore "It (y, >:!:'•)- o,
S'+-+e+ + l'Jc +» v,
implyin� that the particles cannot exist at At\ the Leptons have spin half and arc
the same place. This illustrates the Pauli Fermions
Eltch1iion Principle ob;!yed by Fermi parti­
cles, which states that there can be at most
one Fermi particle in a given quantum
state. Particles with half integral spin obey
Fermi statistics while those with integral
�pin obey Bose Statistics.

There are 4 types of interactions that
operate between elementary particle'!. In the

34



difierent charge states of the same particle. (C) and "The Reversal" {T). Ch the process
For example n�utron and proton are con­ of replacing every partci :le by its anti particle
sidered as dilforent charge states of the and T i1 the op!rati:>n of reversinJ all the
particle nucleon. This charge is called the directions of motion including spin. Con•
isotopic spin :I: . Neucleon is supposed to servation Ol C is kno.vn to be violated in
have :I: -¼. In analogue with the quantum weak interaction'>. 8:.it it i.. proved in
theory of angular mo.nentum the third Relativistic Quantum Theory that C and T
component of iso spin has values ± ¼ should b! con,erved io all situations. That
corresponding to the proton and the is corre,ponding to any process tllere is an
neutron respectively. Pion is as,umed to have anti matter mirror image process occurin,g
ho spin t. It's third component can take tn reverse.
value,-1,0,+ 1 corresponding to 1t -, 1t•, 1:+.
The el�ctric charge of A Meson or A Baryon A theory of ele:nentary particles try to
account for their prc.,perties by asmming
+ +is given by the formula Q - ::I:3 S B . that they are co nbia:1.tions of a kind of
Since Q and B are conserved in all re2actio2ns particlei called Q,rnkes. Three kind� of
t:Ih: a3tisis conserved whenever S i� con,erved­ Quarke'i are assu ne.i to ex.i,t with their anti
in strong or electro magnetic interac­ particles. Two of these are supposei to have
tions. But the total isospin is conserved only clurg�,-le and the other a charg-., ¼c.
in strong interactions. BJ.ry-:>n'> are 'lupp:>se.i to b! co npo,ed of 3
Quarkes, while Me�on,; are con,Uered to be
A wave function is said to be of even for,ned by Qlllrke-ANfl Qurke pairs.
parity if'-\' (z) - �t(-z) and of odd parity Even tho.i_gh Q.1arkes have n!ver b�n
if°"(�) - - �( -z). If P is the operator di3cov<!rej the S-'arch still c.uti.ues.
P"'f'("!")-1.\' (-Z) then P2 - I and P has
eigen values ±I, corresponding to even and Certain classification schemes based upon
odd pariti(!s. Every elementary particle has the theory of group3 have b.:en devi;ed for
au intrin�ic parity and the parity of a syste:n str->n�ly interacting par�icles. One such
of partides is the product of parities of each ache,ne is ba'ied upon the group SU(3), the
particle constituting. the system and the iro.1,> of all u11itary uni nod,Jlar matrices of
parity of the wave function of the system. order 3, and is known as the 8 iold way.
If a system is invariant under mirror reflec­ This sche ne was s.1cces�fully used to predict
tion it can be deduced that it's parity should the existence of the Q-.
be conserved. It has �en found that parity
conservation is 11i0lated in certain reactions Man's present day understanding of the
involvin& weak interactions. Two other micro physi;;al world is far fro:n co.nplete.
symmetry operations that can be carried Y.:t he has won s9 many vi<::tories in hi$
out on a system are "Charge Conjugation" battle �ith ignorance, that he is confident
of his ultimate success.

"A gift for mathemetics i.s one of the most specialized talents, and mathematicians as a
class are not particularly distinguished for general ability or versatility."

-G. H. HARDY

36

RAMANUJAN

by
Gnanalakshmi Gnanali02am

Srinivasa Rarnanujan Aiyangar a brillilnt own theorems and ideas. He first devised
mathe.natician was born on the 22nd of so:ne methods for c.>nstructing magic
D!cemb�r, 1837 in Erode, India. the hone squares. Then, he branched off to Geometry,
of bis maternal grandfather. He was the where he took up the squaring of the circle
elje,t child of a Brahmin petty accountant and succeeded so far as to get a result for
working in Kumbakonam in somewhat the length of the equatorial circumference
p.l.:>r circum'>tanres. There is nothing of the earth which differed from the true
spe:ially noteworthy about his ance'>try to length only by a few feet. Finally, he turned
accaunt for his great gifts. For some time his attention to Algebra and obtained
after her marriage, his mother had no several new series. Ramanujan used to say
cb.ildren, and so her father prayed to the that the goddess of Namakkal inspired
fa.nous g.>jdess Na:nagiri, in the neigh­ him with the formulae in dreams. It is a
b.:iurin15 town of Na:nakkal, to bless his remarkabb fact that frequently, on rising
d:ug.1ter with children. Shortly afterwards from �d. he would note down results and
Ramanujan was born. rapidly verify them, though he wa'i not
always able to supply a rigorous proof.
During the first ten years of his life the These results were embodied in a notebook
only indication that he gave of special which he afterwards used to show to mathe­
ability was that in 1897 he stood first maticians interested in his work.
a nonist the successful candidates of the
Tanj.>re Di,trict in the Primary Examinatbn. In December 1903 he secured a scholar­
This uccess secured for hi,n the concession ship to the University of Madras but lost
of being permitted to pay half-fees in his it the following year because h.e neglected
school. As a young boy he was remarkably all other studies in pursuit of mathematics.
quiet and meditative. It is rememb�red that
he used to ask question'> about the distan�es Ramanujan continued his work, without
of the stars. As he held a high place in his employment and living in the most meagre
class his friends used often to go to his circumstances. In the summer of 1909 he
house, but as he knew that his parents married and wanted to settle down in life.
did not care for him to go out he used Belonging to a poor and humble family,
only to talk to them from a window whi;;h with an unfortunate college career, and
overlooke:i the street. While in the third without influence, he found it hard to
form, when his teacher was explaining to secure some means of livelihood. He began
the class that any qu:intity divided by its:lf a search for permanent employment and
was equal to unity, he is said to have stood this cul:ninated in an interview with Diwan
up and aske:i if zero divided by zero also Bahadue R. Ramachandra Rao. His first
was equal to unity. In the fifth form, he interview with Ramanujan in December
obtained unaided Euler's Theorems for the 1910 is better described in his own words:
sine and the cosine and, when he found
out later that the theore:ns had been already "Several years ago, a nephew of mine
proved, he kept the pa�r containing the perfectly innocent of mathematical know­
results secreted in the roof of his house. ledge said to me, 'Uncle, I have a visitor
who talks of mathematics; I do not under­
In 1903, he obtained a copy of Carr's stand him; can you see if. there is anything
''Syn!>pSi'i of Elem::ntary Results in Pure in his talk?' And in the plenitude of my
Mat!i,;_n:itic:i". This c;,llection of some mathematical wisdom, I condescended to
6,000 theorems (none of the material was permit Ramanujan to walk into my presence.
n�wer th'.ln 185)) aroused hi'i genius. A short uncouth figure, stout, unshaved,
Having verified the results in Carr's book, not overclean, with one conspicuous feature­
Ramanujan went bey.ud, developing his shining eyes-walked in with a frayed

37





SOME RESULTS ON FUNCTIONS
WHOSE RIGHT LIMIT EXISTS
EVERYWHERE

by A. P. N. Weerasinghe

Results: Sfuupnp�toisoea fs:.[ta. .Ibiln-f>exRist·sis a real By taking neN• such that n > 1- ,
c+ finitely eo

ve: a�c<b We have I f (s) - f (t) I� .L

(I(II))ThfTc(peaorhni,ousbewnv] itRsedaediebctedealmennot.oahfptnaerndtodifvsIeicbnsottynhbetaogDiutrnafnubdilteeiidessooannotf[a[mf,ab,o]obsnt] Hence xeEn for some neN•
t>DfC-nUeN• En
X
It follows that Df - nUeN• En

(Ill) F : [a,b] -> R s.t, F(x) = f f(t) dt. Now if Df =- 4> Then f is cts. on [a,bl
So f can b� discts. only at b. in [a,b]
a
for all xe [a,b]. Then F is continuous Then the discontinuities of f on [a,bl is
ooFnnis([aa,r,bbilg[hatnadnddirTtehreenrtiigahbtledeervivearytiwvheeoref c:Ju:itable and then the required result
[a, b[ follows.
F at x =F+ = lim f for all xe
it+ *Consider the case Dr 4>
Then as Df - nUeN*En , En 7 4>
Pf:- Dr -set ofdiscontinuites offon [a,b[, for some n,
Let Hence :IneN• s.t. En ', tf> '

Define En - r xe [a,bl / Vb> o, as, Take cc CE0 ,
l Then lim I exists,
oc+
teN (x,6) n [a,b[ s.t. lf(s)-f(t� 1/n}
So that given, €> 0 3:boc > O. such that,
whcelreearNly(x,ES)n�= {Dylfx-vSne<Ny•<x+&} for&> 0
It follows that ncUN• E0 C Df , oe < s,t <oc+� oo < b =-t>
Jf(s) - lf(t) l < .1.
If -D = 4> then Df -C ocUN• E0 ---(1)
+ nn
E _'....n 'rA.
Suppose that }oc, oc I 1(11
hoc 1

take {3 e Jo,. ,oc + 601 [ nEn •

If D 9 c\> take xeDr, Then as /J€En , v€> 0, 3s,te ti {/J,e)
Then v&> 0, 3e0> 0 such that,
such that I f(s) - f(t) I� ;- ,

as,t e �(x,&) n ]a,b[ and lf(s) - f(t)I� e:o +Now take €o> 0 in such a way N(f3, e)
� l 00 boc [ as {3 e]oc,oc+&oc I

40