FOREWORD
This
correction formula in the text below is wrong. This text of mine represents my
initial research; the final version is omitted simply because the copyright of
the advanced paper belongs to the Journal of Nuclear Technology. The paper
with correct equations was published in Nuclear Technology
(an international journal of the American Nuclear Society) in January 2007,
vol. 157, Number 1.
The formula (9)
was corrected by Ph.D. Soab Usman,
reviewer of Journal of Nuclear Technology, who noticed an error in sign in an
equation in the line of many, so I suggested him that he should be coauthor
due to his correction, and he accepted. The correct equation is:
This
formula is now known as Radović–Usman formula. The formula itself in its
basic form does not bring any notable advantage over classical formula in
ordinary usage, but it has tremendous aftermaths to the properly rearranged
algorithm of the automatic control of the nuclear reactors, preventing flipping
of the negative feedback into positive one in the critical moments.
The
basic idea of this article was to be a precursor for a subsequent paper dealing
with this particular flaw in the algorithm of the fission reactor’s automatic control, which
seems to be main cause of the most of major nuclear accidents in the world, from
the Chernobyl to the Fukushima disasters, and few submarines malfunction too. There
is a strange behavior of the algorithm for the automatic control of the fission
reactors exposing only in the critical accidental situations where the flux of
gamma radiation suddenly raises excessively causing the algorithm somehow to stop with
emergency procedures and to start with reestablishing of normal state, which is
utterly wrong directly causing disasters. This flaw in the particular algorithm
has never occured in the coal thermal plants therefore I located it in the sensors’
side of the automatic control process, particularly in the sensors related to the
measurement of gamma radiation flux. Our misunderstanding of these sensors operation
causes them to lead the system of reactor’s automatic control into the blunder
of the false efficiency of the executed emergency procedures aimed to decrease
gamma radiation flux simply by sudden flipping of negative feedback into positive one.
The solution for this catastrophic flaw is relatively simple and yet quite
effective, but it requires profound understanding of the dynamics of the nuclear
reactors. Many people will be saved and many real estates will retain their
initial values if there would be people ready to listen about aforementioned
subject. These nuclear accidents that seem to be triggered by aforesaid flaw
are going to appear again and again repeatedly, irreversibly damaging environment
on two ways: directly by nuclear pollution and indirectly by fossil fuels green
gases pollution by further deprivation of anaerobic nuclear energy usage. Nuclear
vessels’ reactors are especially vulnerable to this flaw which may have tremendous
aftermaths to nations with nutrition based mainly on the seafood. It is quite
amazing but strang too that there had been absolute no interest on the subject in
most of the major global companies dealing with the nuclear reactors construction
and designing – they are either arrogant or simply they don’t care… or both…
Abstract
The article intents to show that correction formula for gas-filled detectors has to contain the half-life constant of specimen, i.e. that characteristic of specimen may have influence on measurement in some rear situations. There is given a more accurate formula for correction of dead-time interval for gas-filled detectors and its deriving procedure:
(9)
Above formula is theoretically more acceptable than classic one.
DEAD TIME CORRECTION OF NON PARALYZING X RAY DETECTORS
In the following text the next symbol's definitions are adopted:
- t_{p}
- - detector's time of paralyzing, i.e. dead time interval,
- λ
- - constant of radioactive decay for nucleus,
λ = Ln(2) / T_{1/2} ,N_{0} - - the initial number of non decayed nucleuses,
- N
- - the number of decayed nucleuses,
- M
- - the number of detected decays,
- k
- - paralyzing factor.
The estimation of non detected particle's decays is based on the simple relation of radioactive decay that compute the number of decayed nucleuses for the dead time of detector paralysis t_{p}:
The value of paralyzing coefficient k is obtained by dividing the left and the right side of the above equation with the number of non decayed nucleuses for the time of paralysis:
(2)
Now it is possible to derive the difference equation of non paralyzing detector:
(3)
This equation has to have the next initial condition:
(4)
The
variable M in difference equation is the number of
M-th detected decay.
On the left side of difference equation is the number of really decayed nucleuses.
On the right side is the number of really decayed nucleus before M-th decay, plus a just
detected one, plus the estimation of non detected nucleuses that have come on the just
detected decay. It is necessary to be known the number of non decayed nucleuses after
detection of M-th decay while the esteem of non detected decays is obtained via
multiplication of coefficient k with the number of non decayed nucleuses because the
paralysis begins after detection of that decay after that there is certainly a non
decayed nucleus less.
The solution of that difference equation is:
(5)
After substituting the value for k in previous equation, it is obtained:
(6)
Suppose that the speeds of counting and decay are known
(
(7)
There is also a valid formula:
(8)
(9)
If we assume that are
(10)
The relation between λ and T_{1/2} is given by next formula:
(11)
The classic correction formula for non paralyzing detectors could be simply derived from
this formula by introduction of the linear approximation of the exponential function
Absolute value of relative error of classic formula (13) related on the formula (9) is given by the next
expression:
(12)
The formula (9) shows that half-life constant of specimen may have influence on
measurement performed with the gas-filled detector especially when the decay's chain of specimen contains
extremely short-living isotopes with the similar order of magnitude as the dead-time constant of
detector.
The classic formula
(13)
does not include the specimen half-life constant at all. If we assume that is:
(14)
Then the fist part of equation (10) that involves influence of decay constant of specimen to counting result will be transformed to:
(15)
The following table contains correction values for some characteristic ratios
between
0 |
1 |
¼ |
0.91585771041165242343 |
½ |
0.83670266201424625056 |
1 |
0.69314718055994530941 |
2 |
0.46209812037329687294 |
REFERENCES
G. F. Knoll, Radiation Detection And Measurement, (John Wiley & Sons, Inc., 2nd edition 1989).
N. Tsoulfanidis, Measurement And Detection Of Radiation, (McGraw Hill Book Company, 1983).
Author of this article and all its formulas is Andrija S. Radovic´,
Press the following button to download the article in the PDF format:
Author:
Dipl. Ing. Andrija
S. Radović
Tel: +381 64 1404075