- Email: [email protected]
Further note on moving-source soil density gauge
NUCLEAR
INSTRUMENTS
AND
METHODS
87 ( I 9 7 o) 3 2 9 - 3 3 I ;
© NORTH-HOLLAND
PUBLISHING
CO.
FURTHER NOTE ON M O V I N G - S O U R C E S O I L D E N S I T Y G A U G E O. G I F T C I O G L U a n d D. T A Y L O R
Department of Electronic Science, University of Strathclyde, Scotland Received 29 M a y 1970 This letter discusses a n u m b e r o f m a t t e r s which arise when using a m o v i n g - s o u r c e soil density gauge. All concern the choice of the index n in the (/d n vs d) characteristic to obtain a (Id a) m a x i m u m , which occurs within the range o f d-values possible with a specific design of the i n s t r u m e n t . Also considered is the choice of n when two or m o r e values are possible,
In an earlier publication, Devlin, Henderson and T a y l o r ' ) have shown that the moving-source gamma backscatter gauge provides a convenient method for the determination of the density of soils. In this system the (Id" vs d) characteristic is employed. In this expression I is the counting rate, d the source-detector separation and n an index (not necessarily integer) greater than unity. Such a characteristic exhibits a (Id") maximum [as may be shown by determining ~(Id")/~d and equating to zero] given by:
pdm = ( n - 1)/k2,
TABLE ] C o m p u t e d values o f din, i.e. d at (Id n) m a x i m a (din in cm). p =- 1.0 g/cm a
1.5 2.0 2.5 3.0
8.3 16.6 24.9 33.2
p = 2.0 g/cm 3 4.15 8,30 12.45 16.60
where k, is a scaling factor decided by the geometrical and detection efficiencies of the system. This on substitution using eq. (1) gives
Im.x = [k,p / {(,,- 1)/(ok2)} ] exp [ - ( n - Ok2/k23 = { k l k 2 p 2 / ( n - 1)} exp (1 --n),
(1)
where d = d m at the maxima, p is the density of the soil and k 2 is a constant, determined by the mean mass absorption coefficient of the soil. A pertinent question to ask is how one should choose the value of n in the (/d" vs d) characteristic to ensure that the d-value occurs within the range of d-values possible with a specific design of instrument. In the publication already referred t o ' ) it was shown that k2 = 0.062 cgs units and using eq. (1) we may substitute different values of the density (p) and compute the values of d~ for different power laws, i.e. values of n. Such a set of figures is given in table I.
n
However, it is necessary, in addition, to consider how the counting rate at the maxima varies with density. It is then convenient to use the empirical form ula: I = kl(p/d ) exp ( - p d k 2 ) , (2)
(3)
which shows that /max, the counting rate at a (/d") maximum varies as the square of the density. It is worth noting that it is this magnitude which decides the statistical error* in the measurement with a given radioactive source, always assuming that the shielding is sufficient so that the direct radiation from source to detector is negligible. It is also of interest to consider the magnitude of Id" at a (Id") maximum. This means considering
Id" = [kl(p/d)exp ( - p d k 2 ) ] d" and using eq. (1) giving:
Id" = k,pd" exp (1 - n ) .
(4)
p = 3.0 g/cm 3
I f we consider a number of specific cases, e.g. n = 2, n = 2, n = { and n = 3, this expression can be simplified to the forms shown in table 2.
2.77 5.53 8.30 11.07
TABLE 2 M a g n i t u d e of ld n for various n-values.
0.428 kl ,/(p/ks) 0.368 kl/ke 0.412 kl/V(k23p) 0.540 kx/(k~2p)
The choice of n therefore provides the instrument with a very useful flexibility. In our measurements we have found it convenient to adjust n by as little as 0.1.
-} 2 ,~ 3
* N o t e that in plotting ld n, only t h e / - t e r m is subject to r a n d o m fluctuations a n d so at the peak it is the statistical fluctuations in this term which decides the error in the m e a s u r e m e n t of Id%
It will be seen from this table that: 329
330
O. G I F T C I O G L U
A N D D. T A Y L O R
a. F o r n = 2, all the (Id") peaks are of the same magnitude, independently o f the density. b. F o r n < 2, the (Id") peaks increase in magnitude with increase in density. c. F o r n > 2, the (ld") peaks decrease in magnitude with increase in density. The final question to ask is, having used eq. (1) to determine the value o f n to give a dm occurring within the range of d-value possible with a specific design of instrument, - what does one do if more than one n-value is possible? In general, it can be said that choosing n = 2, confers the advantage that all the (Id") maxima are o f equal magnitude, and so this is the choice whenever possible. However, what a b o u t the slope o f the line near the (/d") maxima, i.e. when d > dm or d < d m. We then proceed as follows:
id~ 24~:los
I d " = k a p d "-a exp ( - k 2 p d ) ,
17C 16o 15o lqo
/I
'I 11
/f I/
L)09
/I!
zdsA Xlo s PAPAFFI N WAX DENSI1-Y 087
~d t
e3( ~.--e"" ° ~ ° - - - ° ~ ° - - a " - Q
~ tg
2Jo
#
\e
x.%\
/
~cc
/
II°7° I06o
~
~a
~)o80
I Ioso
}oo
t~
II
I~o
~030
dV~ i~c
13 14
1020
IZ1
IOlO
99© /2/ i
IS
g8o 16 17 t% 19 2O 2~ ?Z ?3 ?4 25 26 27 DISTANCE [d]c~
Fig. 2. Experimental measurements (Id n vs d) for two n-values, n = 2 and //--7. -~ ~(m")/~d = k , p ( . - Oa o- ~ e~p ( - ~ p d )
- k ~ p d ' - lk2p exp ( - k2pd) = k t p d ' - 2 ( n - 1 - d p k 2 ) exp ( - p d k 2 ) .
~
©BI
o
0
p-2o9/~-~/ LO
Z'.O
3.0
INDEX n Fig. 1. The negative slope of the (Id n vs d) characteristics and its variation with n for densities (p) of 1.0, 2.0 and 3.0 g/cma.
(5)
We m a y substitute p = 1.0 g/cm a, n = ~, k 2 --- 0.062, givingd~ -- I/(2k2) = 8.3cm and take d - - d ~ + 2 = 10.3 cm. In this case, eq. (5) reduces to a slope* o f - 0.023 kx. Similar computations show that the slope of the characteristic, i.e. the sharpness of the peak increases with the magnitude of n and with the density p. This is shown in fig. 1 for p -- 1.0, 2.0 and 3.0 g/cm 3. Hence, for higher values o f n a sharper peak is obtained, particularly in the case o f higher density materials. However, it must be remembered that the r a n d o m fluctuations are decided by the fluctuations in the measurement o f the counting rate (I) and for this advantage to be gained, the measurement of the source-detector separation (d) must be more precise, as must the computation of the power law (d") and the multiplication stage ( I x d"). din- 2 is substituted a slope of the same magnitude but opposite sign is found.
* If d - -
MOVING-SOURCE
Xde 5
PARAFFIN WAX
SOIL
4oc 38C
38(
37c
36(;
35C
34C
33C 32C
31c
16 17 18 L9 2o ?_i 2?- 23 2q 25 DIS~-ANC£ [ d ] c,-'r,
GAUGE
331
a (ld") maximum is from 8.3 to 24.9 cm. If only the
DENSITY 0 . 3 7
xrd
3o~m ~3 14 15
DENSITY
26 7-7
Fig. 3. Similar measurements to those in fig. 2 for n =~, but with a much shorter time-constant counting circuit.
If we now suppose that there exists the possibility with the range of source-detector separations allowable of including n = 1.5 to n = 2.5, i.e. for a density of 1.0 g/cm 3, the optimum source-detector separation (din) for
statistical fluctuations in the measured counting rate need be considered we have a simplified situation. This is usually the case since there is a limit to the strength of radioactive source which may be employed in a moving-source backscatter gauge, as sufficient shielding must be provided to ensure that direct radiation passing from the source to the detector through the shielding is a negligible proportion of that backscattered via the soil. In addition, the din-factor involves a very precise measurement of the sourcedetector separation at the (ld") maxima, and the computation of Id" is similarly precise. From eq. (5) and the results given in fig. 1 the sharpness of the peak is improved in this case by a factor of 0.169/0.023 = 7.3 times. Whether this is useful or not depends upon the increased uncertainty of individual measurements with the increased magnitude of Id". Fig. 2 gives some experimental results with a block of paraffin wax, p = 0.97 g/cm 3 with the power law increased from n = 2 to n = 2.5. The sharpening of the (Id") peak with the higher power law is very apparent. This measurement was made with a long time-constant counting circuit. Fig. 3 shows the same measurement with a very much shorter time-constant counting circuit. It will be apparent that the poor statistics makes the measurement difficult in this case. We are indebted to Mr. E. Pirie with whom we have discussed the matter contained in this report.
Reference 1) G. D e v l i n , I. A. H e n d e r s o n a n d D . T a y l o r , N u c l . Instr. a n d M e t h . 76 (1969) 150.
INSTRUMENTS
AND
METHODS
87 ( I 9 7 o) 3 2 9 - 3 3 I ;
© NORTH-HOLLAND
PUBLISHING
CO.
FURTHER NOTE ON M O V I N G - S O U R C E S O I L D E N S I T Y G A U G E O. G I F T C I O G L U a n d D. T A Y L O R
Department of Electronic Science, University of Strathclyde, Scotland Received 29 M a y 1970 This letter discusses a n u m b e r o f m a t t e r s which arise when using a m o v i n g - s o u r c e soil density gauge. All concern the choice of the index n in the (/d n vs d) characteristic to obtain a (Id a) m a x i m u m , which occurs within the range o f d-values possible with a specific design of the i n s t r u m e n t . Also considered is the choice of n when two or m o r e values are possible,
In an earlier publication, Devlin, Henderson and T a y l o r ' ) have shown that the moving-source gamma backscatter gauge provides a convenient method for the determination of the density of soils. In this system the (Id" vs d) characteristic is employed. In this expression I is the counting rate, d the source-detector separation and n an index (not necessarily integer) greater than unity. Such a characteristic exhibits a (Id") maximum [as may be shown by determining ~(Id")/~d and equating to zero] given by:
pdm = ( n - 1)/k2,
TABLE ] C o m p u t e d values o f din, i.e. d at (Id n) m a x i m a (din in cm). p =- 1.0 g/cm a
1.5 2.0 2.5 3.0
8.3 16.6 24.9 33.2
p = 2.0 g/cm 3 4.15 8,30 12.45 16.60
where k, is a scaling factor decided by the geometrical and detection efficiencies of the system. This on substitution using eq. (1) gives
Im.x = [k,p / {(,,- 1)/(ok2)} ] exp [ - ( n - Ok2/k23 = { k l k 2 p 2 / ( n - 1)} exp (1 --n),
(1)
where d = d m at the maxima, p is the density of the soil and k 2 is a constant, determined by the mean mass absorption coefficient of the soil. A pertinent question to ask is how one should choose the value of n in the (/d" vs d) characteristic to ensure that the d-value occurs within the range of d-values possible with a specific design of instrument. In the publication already referred t o ' ) it was shown that k2 = 0.062 cgs units and using eq. (1) we may substitute different values of the density (p) and compute the values of d~ for different power laws, i.e. values of n. Such a set of figures is given in table I.
n
However, it is necessary, in addition, to consider how the counting rate at the maxima varies with density. It is then convenient to use the empirical form ula: I = kl(p/d ) exp ( - p d k 2 ) , (2)
(3)
which shows that /max, the counting rate at a (/d") maximum varies as the square of the density. It is worth noting that it is this magnitude which decides the statistical error* in the measurement with a given radioactive source, always assuming that the shielding is sufficient so that the direct radiation from source to detector is negligible. It is also of interest to consider the magnitude of Id" at a (Id") maximum. This means considering
Id" = [kl(p/d)exp ( - p d k 2 ) ] d" and using eq. (1) giving:
Id" = k,pd" exp (1 - n ) .
(4)
p = 3.0 g/cm 3
I f we consider a number of specific cases, e.g. n = 2, n = 2, n = { and n = 3, this expression can be simplified to the forms shown in table 2.
2.77 5.53 8.30 11.07
TABLE 2 M a g n i t u d e of ld n for various n-values.
0.428 kl ,/(p/ks) 0.368 kl/ke 0.412 kl/V(k23p) 0.540 kx/(k~2p)
The choice of n therefore provides the instrument with a very useful flexibility. In our measurements we have found it convenient to adjust n by as little as 0.1.
-} 2 ,~ 3
* N o t e that in plotting ld n, only t h e / - t e r m is subject to r a n d o m fluctuations a n d so at the peak it is the statistical fluctuations in this term which decides the error in the m e a s u r e m e n t of Id%
It will be seen from this table that: 329
330
O. G I F T C I O G L U
A N D D. T A Y L O R
a. F o r n = 2, all the (Id") peaks are of the same magnitude, independently o f the density. b. F o r n < 2, the (Id") peaks increase in magnitude with increase in density. c. F o r n > 2, the (ld") peaks decrease in magnitude with increase in density. The final question to ask is, having used eq. (1) to determine the value o f n to give a dm occurring within the range of d-value possible with a specific design of instrument, - what does one do if more than one n-value is possible? In general, it can be said that choosing n = 2, confers the advantage that all the (Id") maxima are o f equal magnitude, and so this is the choice whenever possible. However, what a b o u t the slope o f the line near the (/d") maxima, i.e. when d > dm or d < d m. We then proceed as follows:
id~ 24~:los
I d " = k a p d "-a exp ( - k 2 p d ) ,
17C 16o 15o lqo
/I
'I 11
/f I/
L)09
/I!
zdsA Xlo s PAPAFFI N WAX DENSI1-Y 087
~d t
e3( ~.--e"" ° ~ ° - - - ° ~ ° - - a " - Q
~ tg
2Jo
#
\e
x.%\
/
~cc
/
II°7° I06o
~
~a
~)o80
I Ioso
}oo
t~
II
I~o
~030
dV~ i~c
13 14
1020
IZ1
IOlO
99© /2/ i
IS
g8o 16 17 t% 19 2O 2~ ?Z ?3 ?4 25 26 27 DISTANCE [d]c~
Fig. 2. Experimental measurements (Id n vs d) for two n-values, n = 2 and //--7. -~ ~(m")/~d = k , p ( . - Oa o- ~ e~p ( - ~ p d )
- k ~ p d ' - lk2p exp ( - k2pd) = k t p d ' - 2 ( n - 1 - d p k 2 ) exp ( - p d k 2 ) .
~
©BI
o
0
p-2o9/~-~/ LO
Z'.O
3.0
INDEX n Fig. 1. The negative slope of the (Id n vs d) characteristics and its variation with n for densities (p) of 1.0, 2.0 and 3.0 g/cma.
(5)
We m a y substitute p = 1.0 g/cm a, n = ~, k 2 --- 0.062, givingd~ -- I/(2k2) = 8.3cm and take d - - d ~ + 2 = 10.3 cm. In this case, eq. (5) reduces to a slope* o f - 0.023 kx. Similar computations show that the slope of the characteristic, i.e. the sharpness of the peak increases with the magnitude of n and with the density p. This is shown in fig. 1 for p -- 1.0, 2.0 and 3.0 g/cm 3. Hence, for higher values o f n a sharper peak is obtained, particularly in the case o f higher density materials. However, it must be remembered that the r a n d o m fluctuations are decided by the fluctuations in the measurement o f the counting rate (I) and for this advantage to be gained, the measurement of the source-detector separation (d) must be more precise, as must the computation of the power law (d") and the multiplication stage ( I x d"). din- 2 is substituted a slope of the same magnitude but opposite sign is found.
* If d - -
MOVING-SOURCE
Xde 5
PARAFFIN WAX
SOIL
4oc 38C
38(
37c
36(;
35C
34C
33C 32C
31c
16 17 18 L9 2o ?_i 2?- 23 2q 25 DIS~-ANC£ [ d ] c,-'r,
GAUGE
331
a (ld") maximum is from 8.3 to 24.9 cm. If only the
DENSITY 0 . 3 7
xrd
3o~m ~3 14 15
DENSITY
26 7-7
Fig. 3. Similar measurements to those in fig. 2 for n =~, but with a much shorter time-constant counting circuit.
If we now suppose that there exists the possibility with the range of source-detector separations allowable of including n = 1.5 to n = 2.5, i.e. for a density of 1.0 g/cm 3, the optimum source-detector separation (din) for
statistical fluctuations in the measured counting rate need be considered we have a simplified situation. This is usually the case since there is a limit to the strength of radioactive source which may be employed in a moving-source backscatter gauge, as sufficient shielding must be provided to ensure that direct radiation passing from the source to the detector through the shielding is a negligible proportion of that backscattered via the soil. In addition, the din-factor involves a very precise measurement of the sourcedetector separation at the (ld") maxima, and the computation of Id" is similarly precise. From eq. (5) and the results given in fig. 1 the sharpness of the peak is improved in this case by a factor of 0.169/0.023 = 7.3 times. Whether this is useful or not depends upon the increased uncertainty of individual measurements with the increased magnitude of Id". Fig. 2 gives some experimental results with a block of paraffin wax, p = 0.97 g/cm 3 with the power law increased from n = 2 to n = 2.5. The sharpening of the (Id") peak with the higher power law is very apparent. This measurement was made with a long time-constant counting circuit. Fig. 3 shows the same measurement with a very much shorter time-constant counting circuit. It will be apparent that the poor statistics makes the measurement difficult in this case. We are indebted to Mr. E. Pirie with whom we have discussed the matter contained in this report.
Reference 1) G. D e v l i n , I. A. H e n d e r s o n a n d D . T a y l o r , N u c l . Instr. a n d M e t h . 76 (1969) 150.