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Comments on the moving-source soil density gauge
NUCLEAR
1N57RUMEN75
AND ME7H0D5
C0MMEN75
96 (1971) 483-484; • N 0 R 7 H - H 0 L L A N D
Pu8L15H1N6
c0.
0 N 7 H E M 0 V 1 N 6 - 5 0 U R C E 5 0 1 L DEN517Y 6 A U 6 E E. R. C H R 1 5 7 E N 5 E N
Department 0 f E1ectr0phy51c5, 7he 7echn1ca1 Un1ver51ty, DK-2800 Lyn96y, Denmark Rece1ved 1 Ju1y 1971 R~..ent exper1menta1 re5u1t5 f0r the m0v1n9-50urce 5011 den51ty 9au9e are 1nterpreted 1n term5 0f the pr1nc1p1e 0f 51m111tude. 7 h e 1nterpretat10n 1mp11e5 that the ma9n1tude 0f the 1d 2 peak5 1n the (1d 2 v5 d) character15t1c5 15 determ1ned 6y the e4u1va1ent at0m1c num6er 2~4 1ndependent1y 0f the den51ty p.
An 1ntere5t1n9 new ver510n 0f the 9amma 6ack5catter den51ty 9au9e ha5 6een pr0p05ed 6y L1n et a1.~). 1n th15 5y5tem~-3), the d15tance d 6etween 50urce and detect0r 15 var1ed t0 06ta1n a max1mum f0r 1d ~, where 1 15 the 1nten51ty mea5ured 6y the detect0r, and n 15 an exp0nent 1n the ran9e 1 . 5 < n < 3 . 0 . 7he 5em1emp1r1ca1 f0rmu1a
1 = k,(p]d) exp
(- pdk2)
- -
1)1k2,
(2)
when 1d ~ exh161t5 a max1mum. 1n the5e e4uat10n5 p 15 the ma55 den51ty, and k~ and k2 are exper1menta11y determ1ned c0n5tant5. W1th kn0wn va1ue5 0f n and k2, a den51ty determ1nat10n 15 carr1ed 0ut 6y vary1n9 d t0 9et a max1mum f0r 1d ~. 7he va1ue 0f d 06ta1ned 1n th15 way then y1e1d5 the den51ty p 6y e4. (2). Fr0m e4. (1), the f0110w1n9 5tatement5 are deduceda): 1. F0r n = 2, a11 the (1d ~) peak5 are 0f the 5ame ma9n1tude, 1ndependent 0f the den51ty. 2. F0r n < 2, the (1d ~) peak5 1ncrea5e 1n ma9n1tude w1th 1ncrea5e 1n den51ty. 3. F0r n > 2, the (1d ~) peak5 decrea5e 1n ma9n1tude w1th 1ncrea5e 1n den51ty. 7he5e 5tatement5 may, h0wever, 6e deduced fr0m the pr1nc1p1e 0f 51m111tudea) w1th0ut re50rt1n9 t0 any 5em1-emp1r1ca1 expre5510n 11ke e4. (1). 7h15 pr1nc1p1e may f0r the pre5ent purp05e 6e wr1tten
1a 2 = f ( p d ) ,
af(pd••••) ~d
(3)
w h e r e f ( p d ) 15 a funct10n, wh1ch f0r a 91ven 50urce and a 91ven med1um 0n1y depend5 up0n pd. E4. (3) 15 va11d when the phy51ca1 d1men510n5 and the he19ht5 a60ve the 5011 5urface 0f the 50urce and the detect0r are 5ma11 c0mpared w1th the 5pac1n9 d.
+ ~d2f(pd)
= 0.
(4)
51nce p 15 a c0n5tant f0r any 5et 0f mea5urement5, we may wr1te df(u) + n-2f(u) du u
(1)
15 u5ed t0 5h0w that
pd= (n
7he (1d••) peak5 can 6e f0und 6y 5ett1n9 ~(1d~)/t9d = 0
= 0,
(5)
where u = pd. Fr0m e4. (5) a 501ut10n f0r u may 6e f0und. 7h15 mean5, that aU the 1d n peak5 f0r a 91ven med1um and a 91ven va1ue 0f n are character12ed 6y the 5ame pd-va1ue and thu5 the 5ame funct10nf(pd). 1f n = 2, e4. (3) thu5 ju5t1f1e5 5tatement 1. Let u5 c0n51der the ca5e n < 2. We may then wr1te 1d" = (1d2) • d e, where a < 0.51nce 1d 2 = f ( p d ) 15 a c0n5tant, 1t f0110w5 that 1d ~ 1ncrea5e5 w1th 1ncrea51n9 den51ty. 7h15 pr0ve5 the va11d1ty 0f 5tatement 2. A 51m11ar ar9ument app11e5 t0 5tatement 3. We may theref0re c0nc1ude that the 5tatement5 1, 2 and 3 are va11d whenever the pr1nc1p1e 0f 51m111tude 15 app11ca61e. A ch01ce 0f n = 2 then appear5 natura1 f0r the f0110w1n9 rea50n. A5 5h0wn 6y ca1cu1at10n5 and mea5urement55) the ma9n1tude 0f the 1d 2 peak5 depend5 0n the e4u1va1ent at0m1c num6er 2,4. A 1ar9e va1ue 0f 2,4 91ve5 a 5ma11 peak and v1ce ver5a. Further, the p051t10n5 0f the peak5 0ccur f0r appr0x1mate1y the 5ame pd-va1ue 1ndependent 0f 2e4.7h15 app11e5 1f 2 / A 15 c105e t0 a c0n5tant va1ue, u5ua11y 0.5. 8y pr0per ca116rat10n 0ne m19ht thu5 take the ma9n1tude 0f the 1d 2 peak5 a5 an 1nd1cat10n 0f the e4u1va1ent at0m1c num6er 2e4, and a5 de5cr16ed ear11er1) the p051t10n d 0f the 1d 2 peak5 a5 a mea5ure 0f the den51ty p. 5uch an exten510n 0f the m0v1n9-50urce 5011 den51ty 9au9e w0u1d thu5 a110w 51mu1tane0u5 mea5urement 1n a 91ven mea5ur1n9 v01ume 0f the e4u1va1ent at0m1c num6er 2~4 and the den51ty p.
483
484
E. R. CHR157EN5EN
7h15 w0u1d 5eem t0 6e an 1mpr0vement 0ver the u5ua1 m e t h 0 d 6) 0 f mea5ur1n9 2ca a n d p, wh1ch 1mp11e5 tw0 d1fferent detect10n ener9y ran9e5 a n d thu5 tw0 d1fferent mea5ur1n9 v01ume5.
Reference5 1) K. L1n, E. P1r1e and D. 7ay10r, Nuc1. 1n5tr. and Meth. 72 (1969) 325.
3) 6. Dev11n, 1. A. Hender50n and D. 7ay10r, Nuc1. 1n5tr. and Mcth. 76 (1969) 150. 3) 0. 61ftc1091u and D. 7ay10r, Nuc1.1n5tr. and M¢th. 87 (1970) 329. 4) j. A. C2u6ek and J. 6u1tt0n, Pr0(:. 5ymp. War5aw, 5M-68/32 (1AEA, V1enna, 1966). 5) E. R. Chr15t¢n5en, Nuc1. 1n5tr. and Meth. 91 (1971) 461. 6) j. A. C2u6¢k, Pr0c. 5ymp. War5aw, 5M-68/34 (1AEA, V1enna, 1966).
1N57RUMEN75
AND ME7H0D5
C0MMEN75
96 (1971) 483-484; • N 0 R 7 H - H 0 L L A N D
Pu8L15H1N6
c0.
0 N 7 H E M 0 V 1 N 6 - 5 0 U R C E 5 0 1 L DEN517Y 6 A U 6 E E. R. C H R 1 5 7 E N 5 E N
Department 0 f E1ectr0phy51c5, 7he 7echn1ca1 Un1ver51ty, DK-2800 Lyn96y, Denmark Rece1ved 1 Ju1y 1971 R~..ent exper1menta1 re5u1t5 f0r the m0v1n9-50urce 5011 den51ty 9au9e are 1nterpreted 1n term5 0f the pr1nc1p1e 0f 51m111tude. 7 h e 1nterpretat10n 1mp11e5 that the ma9n1tude 0f the 1d 2 peak5 1n the (1d 2 v5 d) character15t1c5 15 determ1ned 6y the e4u1va1ent at0m1c num6er 2~4 1ndependent1y 0f the den51ty p.
An 1ntere5t1n9 new ver510n 0f the 9amma 6ack5catter den51ty 9au9e ha5 6een pr0p05ed 6y L1n et a1.~). 1n th15 5y5tem~-3), the d15tance d 6etween 50urce and detect0r 15 var1ed t0 06ta1n a max1mum f0r 1d ~, where 1 15 the 1nten51ty mea5ured 6y the detect0r, and n 15 an exp0nent 1n the ran9e 1 . 5 < n < 3 . 0 . 7he 5em1emp1r1ca1 f0rmu1a
1 = k,(p]d) exp
(- pdk2)
- -
1)1k2,
(2)
when 1d ~ exh161t5 a max1mum. 1n the5e e4uat10n5 p 15 the ma55 den51ty, and k~ and k2 are exper1menta11y determ1ned c0n5tant5. W1th kn0wn va1ue5 0f n and k2, a den51ty determ1nat10n 15 carr1ed 0ut 6y vary1n9 d t0 9et a max1mum f0r 1d ~. 7he va1ue 0f d 06ta1ned 1n th15 way then y1e1d5 the den51ty p 6y e4. (2). Fr0m e4. (1), the f0110w1n9 5tatement5 are deduceda): 1. F0r n = 2, a11 the (1d ~) peak5 are 0f the 5ame ma9n1tude, 1ndependent 0f the den51ty. 2. F0r n < 2, the (1d ~) peak5 1ncrea5e 1n ma9n1tude w1th 1ncrea5e 1n den51ty. 3. F0r n > 2, the (1d ~) peak5 decrea5e 1n ma9n1tude w1th 1ncrea5e 1n den51ty. 7he5e 5tatement5 may, h0wever, 6e deduced fr0m the pr1nc1p1e 0f 51m111tudea) w1th0ut re50rt1n9 t0 any 5em1-emp1r1ca1 expre5510n 11ke e4. (1). 7h15 pr1nc1p1e may f0r the pre5ent purp05e 6e wr1tten
1a 2 = f ( p d ) ,
af(pd••••) ~d
(3)
w h e r e f ( p d ) 15 a funct10n, wh1ch f0r a 91ven 50urce and a 91ven med1um 0n1y depend5 up0n pd. E4. (3) 15 va11d when the phy51ca1 d1men510n5 and the he19ht5 a60ve the 5011 5urface 0f the 50urce and the detect0r are 5ma11 c0mpared w1th the 5pac1n9 d.
+ ~d2f(pd)
= 0.
(4)
51nce p 15 a c0n5tant f0r any 5et 0f mea5urement5, we may wr1te df(u) + n-2f(u) du u
(1)
15 u5ed t0 5h0w that
pd= (n
7he (1d••) peak5 can 6e f0und 6y 5ett1n9 ~(1d~)/t9d = 0
= 0,
(5)
where u = pd. Fr0m e4. (5) a 501ut10n f0r u may 6e f0und. 7h15 mean5, that aU the 1d n peak5 f0r a 91ven med1um and a 91ven va1ue 0f n are character12ed 6y the 5ame pd-va1ue and thu5 the 5ame funct10nf(pd). 1f n = 2, e4. (3) thu5 ju5t1f1e5 5tatement 1. Let u5 c0n51der the ca5e n < 2. We may then wr1te 1d" = (1d2) • d e, where a < 0.51nce 1d 2 = f ( p d ) 15 a c0n5tant, 1t f0110w5 that 1d ~ 1ncrea5e5 w1th 1ncrea51n9 den51ty. 7h15 pr0ve5 the va11d1ty 0f 5tatement 2. A 51m11ar ar9ument app11e5 t0 5tatement 3. We may theref0re c0nc1ude that the 5tatement5 1, 2 and 3 are va11d whenever the pr1nc1p1e 0f 51m111tude 15 app11ca61e. A ch01ce 0f n = 2 then appear5 natura1 f0r the f0110w1n9 rea50n. A5 5h0wn 6y ca1cu1at10n5 and mea5urement55) the ma9n1tude 0f the 1d 2 peak5 depend5 0n the e4u1va1ent at0m1c num6er 2,4. A 1ar9e va1ue 0f 2,4 91ve5 a 5ma11 peak and v1ce ver5a. Further, the p051t10n5 0f the peak5 0ccur f0r appr0x1mate1y the 5ame pd-va1ue 1ndependent 0f 2e4.7h15 app11e5 1f 2 / A 15 c105e t0 a c0n5tant va1ue, u5ua11y 0.5. 8y pr0per ca116rat10n 0ne m19ht thu5 take the ma9n1tude 0f the 1d 2 peak5 a5 an 1nd1cat10n 0f the e4u1va1ent at0m1c num6er 2e4, and a5 de5cr16ed ear11er1) the p051t10n d 0f the 1d 2 peak5 a5 a mea5ure 0f the den51ty p. 5uch an exten510n 0f the m0v1n9-50urce 5011 den51ty 9au9e w0u1d thu5 a110w 51mu1tane0u5 mea5urement 1n a 91ven mea5ur1n9 v01ume 0f the e4u1va1ent at0m1c num6er 2~4 and the den51ty p.
483
484
E. R. CHR157EN5EN
7h15 w0u1d 5eem t0 6e an 1mpr0vement 0ver the u5ua1 m e t h 0 d 6) 0 f mea5ur1n9 2ca a n d p, wh1ch 1mp11e5 tw0 d1fferent detect10n ener9y ran9e5 a n d thu5 tw0 d1fferent mea5ur1n9 v01ume5.
Reference5 1) K. L1n, E. P1r1e and D. 7ay10r, Nuc1. 1n5tr. and Meth. 72 (1969) 325.
3) 6. Dev11n, 1. A. Hender50n and D. 7ay10r, Nuc1. 1n5tr. and Mcth. 76 (1969) 150. 3) 0. 61ftc1091u and D. 7ay10r, Nuc1.1n5tr. and M¢th. 87 (1970) 329. 4) j. A. C2u6ek and J. 6u1tt0n, Pr0(:. 5ymp. War5aw, 5M-68/32 (1AEA, V1enna, 1966). 5) E. R. Chr15t¢n5en, Nuc1. 1n5tr. and Meth. 91 (1971) 461. 6) j. A. C2u6¢k, Pr0c. 5ymp. War5aw, 5M-68/34 (1AEA, V1enna, 1966).