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The reaction 10B(d, n)11C as an ion temperature plasma diagnostic
Nuclear Instruments and Methods 175 (1980) 293-296 © North-Holland Publishing Company
THE REACTION l°B(d, n) 11C AS AN ION TEMPERATURE PLASMA DIAGNOSTIC F.E. CECIL, LEK KEAH LEN Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA and R.J. PETERSON Cyclotron Laboratory, University of Colorado, BouMer, Colorado 80302, USA Received 8 January 1980
The thick target yield for the reaction 10B(d, n)l 1C has been measured for monoenergetic deuterons of energies between 75 and 960 keV. This yield is integrated over a Maxwell-Boltzman distribution to obtain a semi-empirical thick target yield as a function of temperature. It is shown that the yield as a function of temperature provides a practical and highly sensitive measurement of ion temperature for deuterium plasmas with temperatures between about 2 × 10 7 and 2 × 108 K.
The measurement of the ion temperature of a deuterium plasma is a crucial element of CTR Tokamak plasma diagnostics. One technique for ion temperature measurements which would be particularly suited to temperatures characteristic of current CTR research (107-108 K) is charged particle activation analysis. In this technique, the activation sample is exposed directly to the plasma and the subsequent activity of the deuteron induced reaction is measured. The activity is compared with independent measurements of the reaction yield and the temperature of the plasma may be deduced. There are several significant advantages of charged particle activation compared to conventional methods [ 1] of high temperature measurements: (1) the ion temperature is measured directly, as compared with techniques which measure, say, electron temperatures under the assumption of thermal equilibrium between the electrons and ions. (2) for energies less than a few hundred keV, the activation yield is an extremely sensitive function of the charged particle energy since the reaction cross section is dominated by the Coulomb barrier penetration probability [2]. (3) for relatively light (A < 20) activation samples, and for temperatures ~<108 K, the activation process and the D - D or D - T fusion process share the property that nearly all of the reactions occcur in a narrow region of the high energy tail of the plasma's
energy distribution; this region corresponds to the maximum overlap of the population distribution which drops off at high energies and the reaction cross section which drops off at low energies and is known, in astrophysical circles, as the "Gamow peak" [3]. In other words, a charged particle activation measurements of plasma ion temperature is sensitive to the high energy tail of the distribution, which is where it counts. Young et al. [4] have discussed charged particle activation as a diagnostic specifically of intense pulsed ion beams. The only two deuteron induced reactions (as noted by Young, et al. [4])utilizing light samples and resulting in a product with an easily measurable lifetime are l°B(d, n) 11C (tl/2 = 20.5 min) and 14N(d, n)lSo (tl/2 = 66 s). (A reaction such as 6Li(d, n)TBe is not a useful diagnostic since the 7Be half-life is 52 d.) However the l°B and 14N reactions have only been measured down to energies of 300 and 500 keV respectively. As such these measurements cannot be directly applied to the measurement of plasma ion temperatures in the region 107-108 K since, as will be shown below, the energy and fwhm of the Gamow peak for a plasma of temperature 108 K are 100 and 60 keV respectively. In order to provide the measurements needed to utilize the reaction l°B(d, n)llC as a plasma diagnostic for temperatures ~<108 K, we have measured the thick target (in which the beam stops in the target) 293
F.E. Cecilet aL / The reaction 10B(d,n)l 1C
294
yield for this reaction down to an energy of 75 keV. Data between energies of 960 and 220 keV were taken with the magnetically analyzed deuteron beam from the University of Colorado (CU) Cyclotron. The data between energies of 75 and 172 keV were taken with the Colorado School of Mines (CSM) Physics Department Cockcroft Walton accelerator. The energy of the CSM accelerator was measured to within about 1 keV by mapping out the thick target yield of the 163 keV resonance in the reaction liB(p, 7)12C (5). The thick target yields for the reaction l°B(d, n) 11C were measured by counting the 511 keV annihilation quanta following the/3 ÷ decay of the l lc. Natural metallic Boron targets were used and standard techniques employed to deduce the thick target yield in terms of the number of detected 511 keV gamma rays. A consistency check on the CSM data was provided by a measurement of the yield of the reaction 12C(p, 7)13N. Bailey and Stratton [6] quote a value of (7.5 -+ 0.5) × 10 -is 13N/p for this reaction at a proton bombarding energy of 168 keV. We repeated this measurement and found (using the same techniques employed in the l°B(d, n ) l l c measurements) a yield of (9.0 -+ 1.0) × 10 -as 13N/p. The thick target yields measured in the present work are shown in fig. 1 where they are compared to the measured yields at energies up to 1 MeV presented by Young et al. [4]. A parameterization of
~
+*
I
= ~
I
y
,
,
I
I
=
*
I
t
I
*
+
///~
i0-10
U
L. t~
the thick target yield is obtained assuming that
o
o(E')
Y(Eo) = f (0.198) dE(E')/~ PB
dE'
(1)
Eo where o is the reaction cross section (cm2), dE/dx is the stopping power (keV cm-1), 0.198 is the isotropic abundance of l°B in natural boron, PB is the density of natural boron atoms (cm-a). The stopping power for deuteron in boron is taken from Andersen and Ziegler [7]. The reaction cross section at energies below the d - l ° B coulomb barrier (~3 MeV) is assumed to be given approximately by the Coulomb barrier penetration probability
o(E) = S(E) exp ( - 27rZIZ2 e2) E hv
(2)
where v is the d - ~ ° B relative velocity and where
S(E), in the absence of any narrow resonances, is a slowly varying function o f E [8]. We assumed
S(E) =A +BE + CEz
(3)
and obtained optimum values of A, B and C by fitting Y(E) as defined by eq. (1) to the measured values of Y(E) between energies of 75 and 240 keV. In addition, two one-parameters fits were obtained in which (a) B and C were constrained to be zero, and (b) A and C were contrained to be zero. The three parameters sets thus obtained are given in table 1. The predicted yields obtained with eq. (1), (2) and (3) and the parameter sets in table 1 are also shown in fig. 1. It should be noted that all three parameters sets give comparable fits to the measured yields for energies below 240 keV, while parameter set 3 (A = C = 0, B :/: 0) provides an excellent fit to the data up to 1 MeV. The thick target yield per incident deuteron of ~lc for natural boron exposed to a deuterium plasma at a given temperature T is given by
Y(T) = ; N(E) Y(E) V d E ( ' I c - d -1 cm s - l ) ,
10t5
(4)
0
,,,111+ 200
I,,
I I,
400 600 ENERGY (KEV)
I,,,
800
I~
Fig. 1. Thick target yields for reaction l°B(d, n) 11C for a monoenergetic deuteron beam on natural boron. The squares are the CSM data, the crosses are CU data and the circles are the data from ref. [4]. The curves are the yields calculated with eq. (1) and with the three parameter sets of table 1.
Table 1 Parameters for eq. (3) Parameter set
A (keV cm2) B (cm2)
C (keV -t cm2)
1 2 3
1.4 X 10-2o 1.2 X 10-20 0
6.0 X 10-2s 0 0
-1.0 × 10-22 0 1.0 × 10-22
F.E. Cecil et al. / The reaction l°B(d,n)l IC
10-5
295
~-
\\
--\
T T _
+ i0-10
i
b-
\,
~ 0
q uJ
\, "\
"!
\\\
-
I0 20
] O0
200 ENERGY (KEV)
):!
300
Fig. 2. The integrand of eq. (4). Essentially the overlap of the thick target yield and Maxwell-Boltzman distributions for deuterium plasmas of temperatures 6 × 10 7 K and 1 × 108 K.
where Y(E) is given by eq. (1), N(E) is the (normalized) population distribution of the plasma incident upon the boron, V is the velocity of the deuteron (the boron is assumed at rest). Assuming
N(E)
_
2zr (TrkT)a/z
Ell2
e-E/KT
(a Maxwell-Boltzmann distribution) the integrand of eq. (4) (analogous to the Gamow peak mentioned above) is plotted in fig. 2 for T = 6 X 107 K and 1 × 108 K. It is noted in this figure that the maxima of the integrand is about 70 and 100 keV respectively. This support our earlier claim that the most important measurements are at energies between about 50 and 200 keV. (The integrands plotted in fig. 2 were calculated assuming parameter set 1 in table 1; similar calculations assuming parameters set 2 or 3 are nearly identical in the region of the maxima to those shown in fig. (2). The thick target yields as a function of temperature calculated from eq. (4) and each of the three parameters sets of table 1 are plotted in fig. 3 between temperatures of 2 X 107 K and 2 X 108 K. The average of the three yields at each of the temperatures given in fig. 3 are listed in table 2. Figure 3, finally, is the calibration which would
4-
10-15
tA
'+ ...... 50
+ .... I00
I +-+-, , 150
TEMPERATURE (°K x 10 ~')
Fig. 3. The thick target yield per incident deuteron as a function of temperature. The three values at each temperature correspond to the three parameter sets of table 1; circles represent set 1, crosses represent set 2 and triangles represent set 3.
allow the l°B(d, n) 11C reaction to be utilized to measure the ion temperature of deuterium plasmas in the temperature range 107-108 K. For temperatures significantly less than 107 K, the yield becomes prohibitively small (for plasma densities ~ 1 0 is cm-a); and for temperatures significantly greater than 10 s K, the reaction loses its sensitivity (compare the two order of magnitude increase in yield between 40 and 60 million K to the factor of two increase between 180 and 200 million K). It should be emphasized that fig. 3 assumes a Maxwell-Boltzmann distribution. If another distribution is assumed the integration required by eq. (4) must be repeated. In order to calculate the total yield of 1 1C when a sample of natural Boron is exposed to a deuterium plasma of density Pd
Y(T, p ) = ~ Y(T)A(11C s -1) where Y(T) is given by eq. (4) and the factor of 4 is derived in a straightforward manner from elementary kinetic theory. A is the surface area of the exposed boron. Thus for a deuterium plasma with p = 10 is (d cm -3) and T = 6 × 107K the yield of 11C will be roughly 103 (s -I cm-2). While the measurement technique proposed herein offers great intrinsic sensitivity to the ion temperature directly, possible drawbacks
F.E. Cecil et aL / The reaction 10B(d,n) 11C
296
Table 2 Thick target yields per incident deuteron as a function of temperature T x 106 K
Y(d -I cm s-l)
20 40 60 80 100 120 140 160 180 200
4.923 X 10-19 2.886 × 10-14 6.393 × 10-12 2.045 × 10- l ° 2.505 × 10-9 1.746 x 10-s 8.435 X 10-8 3.155 X 10-7 9.785 X 10-7 2.626 X 10-6
to its actual application should be noted: (i) measurement of induced 11C radioactivity in a boron sample allows the inference only of a time averaged temperature. (ii) A significant fraction of the 1~C activity may be lost from the sample during the bombardment due to "blow-off" resulting from extreme surface heating
of the sample by the plasma. (This problem is addressed quantitatively in ref. [4] .) (iii) The background radiation levels in the immediate vicinity of an energetic deuterium plasma may be very high and would, accordingly, impose severe constraints on the manner in which the induced activity is counted.
References [1] Equipe, TFR, Nuclear Fusion 18 (1978) 647. [2] C.A. Barnes, in: Advances in Nuclear Physics, eds. M. Baranger and E. Vogt, Vol. 4 (Plenum Press, New, York, 1971) p. 142. [3] Donald Clayton, Principles of Stellar Evolution and Nucleosynthesis McGraw Hill, New York, 1968) p. 302. [4] F.C. Young, et al., Rev. Sci. Instrum. 48 (1977) 432. [5] F. Ajzenberg-Selove,Nucl. Phys. A248 (1975) 1. [6] C.L. Bailey and W.R. Stratton, Phys. Rev. 77 (1950) 194. [7] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions in Matter; Vol. 3, Hydrogen (Pergamon Press, New York, 1977). [8] C.A. Barnes, op. cit., p. 143.
THE REACTION l°B(d, n) 11C AS AN ION TEMPERATURE PLASMA DIAGNOSTIC F.E. CECIL, LEK KEAH LEN Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA and R.J. PETERSON Cyclotron Laboratory, University of Colorado, BouMer, Colorado 80302, USA Received 8 January 1980
The thick target yield for the reaction 10B(d, n)l 1C has been measured for monoenergetic deuterons of energies between 75 and 960 keV. This yield is integrated over a Maxwell-Boltzman distribution to obtain a semi-empirical thick target yield as a function of temperature. It is shown that the yield as a function of temperature provides a practical and highly sensitive measurement of ion temperature for deuterium plasmas with temperatures between about 2 × 10 7 and 2 × 108 K.
The measurement of the ion temperature of a deuterium plasma is a crucial element of CTR Tokamak plasma diagnostics. One technique for ion temperature measurements which would be particularly suited to temperatures characteristic of current CTR research (107-108 K) is charged particle activation analysis. In this technique, the activation sample is exposed directly to the plasma and the subsequent activity of the deuteron induced reaction is measured. The activity is compared with independent measurements of the reaction yield and the temperature of the plasma may be deduced. There are several significant advantages of charged particle activation compared to conventional methods [ 1] of high temperature measurements: (1) the ion temperature is measured directly, as compared with techniques which measure, say, electron temperatures under the assumption of thermal equilibrium between the electrons and ions. (2) for energies less than a few hundred keV, the activation yield is an extremely sensitive function of the charged particle energy since the reaction cross section is dominated by the Coulomb barrier penetration probability [2]. (3) for relatively light (A < 20) activation samples, and for temperatures ~<108 K, the activation process and the D - D or D - T fusion process share the property that nearly all of the reactions occcur in a narrow region of the high energy tail of the plasma's
energy distribution; this region corresponds to the maximum overlap of the population distribution which drops off at high energies and the reaction cross section which drops off at low energies and is known, in astrophysical circles, as the "Gamow peak" [3]. In other words, a charged particle activation measurements of plasma ion temperature is sensitive to the high energy tail of the distribution, which is where it counts. Young et al. [4] have discussed charged particle activation as a diagnostic specifically of intense pulsed ion beams. The only two deuteron induced reactions (as noted by Young, et al. [4])utilizing light samples and resulting in a product with an easily measurable lifetime are l°B(d, n) 11C (tl/2 = 20.5 min) and 14N(d, n)lSo (tl/2 = 66 s). (A reaction such as 6Li(d, n)TBe is not a useful diagnostic since the 7Be half-life is 52 d.) However the l°B and 14N reactions have only been measured down to energies of 300 and 500 keV respectively. As such these measurements cannot be directly applied to the measurement of plasma ion temperatures in the region 107-108 K since, as will be shown below, the energy and fwhm of the Gamow peak for a plasma of temperature 108 K are 100 and 60 keV respectively. In order to provide the measurements needed to utilize the reaction l°B(d, n)llC as a plasma diagnostic for temperatures ~<108 K, we have measured the thick target (in which the beam stops in the target) 293
F.E. Cecilet aL / The reaction 10B(d,n)l 1C
294
yield for this reaction down to an energy of 75 keV. Data between energies of 960 and 220 keV were taken with the magnetically analyzed deuteron beam from the University of Colorado (CU) Cyclotron. The data between energies of 75 and 172 keV were taken with the Colorado School of Mines (CSM) Physics Department Cockcroft Walton accelerator. The energy of the CSM accelerator was measured to within about 1 keV by mapping out the thick target yield of the 163 keV resonance in the reaction liB(p, 7)12C (5). The thick target yields for the reaction l°B(d, n) 11C were measured by counting the 511 keV annihilation quanta following the/3 ÷ decay of the l lc. Natural metallic Boron targets were used and standard techniques employed to deduce the thick target yield in terms of the number of detected 511 keV gamma rays. A consistency check on the CSM data was provided by a measurement of the yield of the reaction 12C(p, 7)13N. Bailey and Stratton [6] quote a value of (7.5 -+ 0.5) × 10 -is 13N/p for this reaction at a proton bombarding energy of 168 keV. We repeated this measurement and found (using the same techniques employed in the l°B(d, n ) l l c measurements) a yield of (9.0 -+ 1.0) × 10 -as 13N/p. The thick target yields measured in the present work are shown in fig. 1 where they are compared to the measured yields at energies up to 1 MeV presented by Young et al. [4]. A parameterization of
~
+*
I
= ~
I
y
,
,
I
I
=
*
I
t
I
*
+
///~
i0-10
U
L. t~
the thick target yield is obtained assuming that
o
o(E')
Y(Eo) = f (0.198) dE(E')/~ PB
dE'
(1)
Eo where o is the reaction cross section (cm2), dE/dx is the stopping power (keV cm-1), 0.198 is the isotropic abundance of l°B in natural boron, PB is the density of natural boron atoms (cm-a). The stopping power for deuteron in boron is taken from Andersen and Ziegler [7]. The reaction cross section at energies below the d - l ° B coulomb barrier (~3 MeV) is assumed to be given approximately by the Coulomb barrier penetration probability
o(E) = S(E) exp ( - 27rZIZ2 e2) E hv
(2)
where v is the d - ~ ° B relative velocity and where
S(E), in the absence of any narrow resonances, is a slowly varying function o f E [8]. We assumed
S(E) =A +BE + CEz
(3)
and obtained optimum values of A, B and C by fitting Y(E) as defined by eq. (1) to the measured values of Y(E) between energies of 75 and 240 keV. In addition, two one-parameters fits were obtained in which (a) B and C were constrained to be zero, and (b) A and C were contrained to be zero. The three parameters sets thus obtained are given in table 1. The predicted yields obtained with eq. (1), (2) and (3) and the parameter sets in table 1 are also shown in fig. 1. It should be noted that all three parameters sets give comparable fits to the measured yields for energies below 240 keV, while parameter set 3 (A = C = 0, B :/: 0) provides an excellent fit to the data up to 1 MeV. The thick target yield per incident deuteron of ~lc for natural boron exposed to a deuterium plasma at a given temperature T is given by
Y(T) = ; N(E) Y(E) V d E ( ' I c - d -1 cm s - l ) ,
10t5
(4)
0
,,,111+ 200
I,,
I I,
400 600 ENERGY (KEV)
I,,,
800
I~
Fig. 1. Thick target yields for reaction l°B(d, n) 11C for a monoenergetic deuteron beam on natural boron. The squares are the CSM data, the crosses are CU data and the circles are the data from ref. [4]. The curves are the yields calculated with eq. (1) and with the three parameter sets of table 1.
Table 1 Parameters for eq. (3) Parameter set
A (keV cm2) B (cm2)
C (keV -t cm2)
1 2 3
1.4 X 10-2o 1.2 X 10-20 0
6.0 X 10-2s 0 0
-1.0 × 10-22 0 1.0 × 10-22
F.E. Cecil et al. / The reaction l°B(d,n)l IC
10-5
295
~-
\\
--\
T T _
+ i0-10
i
b-
\,
~ 0
q uJ
\, "\
"!
\\\
-
I0 20
] O0
200 ENERGY (KEV)
):!
300
Fig. 2. The integrand of eq. (4). Essentially the overlap of the thick target yield and Maxwell-Boltzman distributions for deuterium plasmas of temperatures 6 × 10 7 K and 1 × 108 K.
where Y(E) is given by eq. (1), N(E) is the (normalized) population distribution of the plasma incident upon the boron, V is the velocity of the deuteron (the boron is assumed at rest). Assuming
N(E)
_
2zr (TrkT)a/z
Ell2
e-E/KT
(a Maxwell-Boltzmann distribution) the integrand of eq. (4) (analogous to the Gamow peak mentioned above) is plotted in fig. 2 for T = 6 X 107 K and 1 × 108 K. It is noted in this figure that the maxima of the integrand is about 70 and 100 keV respectively. This support our earlier claim that the most important measurements are at energies between about 50 and 200 keV. (The integrands plotted in fig. 2 were calculated assuming parameter set 1 in table 1; similar calculations assuming parameters set 2 or 3 are nearly identical in the region of the maxima to those shown in fig. (2). The thick target yields as a function of temperature calculated from eq. (4) and each of the three parameters sets of table 1 are plotted in fig. 3 between temperatures of 2 X 107 K and 2 X 108 K. The average of the three yields at each of the temperatures given in fig. 3 are listed in table 2. Figure 3, finally, is the calibration which would
4-
10-15
tA
'+ ...... 50
+ .... I00
I +-+-, , 150
TEMPERATURE (°K x 10 ~')
Fig. 3. The thick target yield per incident deuteron as a function of temperature. The three values at each temperature correspond to the three parameter sets of table 1; circles represent set 1, crosses represent set 2 and triangles represent set 3.
allow the l°B(d, n) 11C reaction to be utilized to measure the ion temperature of deuterium plasmas in the temperature range 107-108 K. For temperatures significantly less than 107 K, the yield becomes prohibitively small (for plasma densities ~ 1 0 is cm-a); and for temperatures significantly greater than 10 s K, the reaction loses its sensitivity (compare the two order of magnitude increase in yield between 40 and 60 million K to the factor of two increase between 180 and 200 million K). It should be emphasized that fig. 3 assumes a Maxwell-Boltzmann distribution. If another distribution is assumed the integration required by eq. (4) must be repeated. In order to calculate the total yield of 1 1C when a sample of natural Boron is exposed to a deuterium plasma of density Pd
Y(T, p ) = ~ Y(T)A(11C s -1) where Y(T) is given by eq. (4) and the factor of 4 is derived in a straightforward manner from elementary kinetic theory. A is the surface area of the exposed boron. Thus for a deuterium plasma with p = 10 is (d cm -3) and T = 6 × 107K the yield of 11C will be roughly 103 (s -I cm-2). While the measurement technique proposed herein offers great intrinsic sensitivity to the ion temperature directly, possible drawbacks
F.E. Cecil et aL / The reaction 10B(d,n) 11C
296
Table 2 Thick target yields per incident deuteron as a function of temperature T x 106 K
Y(d -I cm s-l)
20 40 60 80 100 120 140 160 180 200
4.923 X 10-19 2.886 × 10-14 6.393 × 10-12 2.045 × 10- l ° 2.505 × 10-9 1.746 x 10-s 8.435 X 10-8 3.155 X 10-7 9.785 X 10-7 2.626 X 10-6
to its actual application should be noted: (i) measurement of induced 11C radioactivity in a boron sample allows the inference only of a time averaged temperature. (ii) A significant fraction of the 1~C activity may be lost from the sample during the bombardment due to "blow-off" resulting from extreme surface heating
of the sample by the plasma. (This problem is addressed quantitatively in ref. [4] .) (iii) The background radiation levels in the immediate vicinity of an energetic deuterium plasma may be very high and would, accordingly, impose severe constraints on the manner in which the induced activity is counted.
References [1] Equipe, TFR, Nuclear Fusion 18 (1978) 647. [2] C.A. Barnes, in: Advances in Nuclear Physics, eds. M. Baranger and E. Vogt, Vol. 4 (Plenum Press, New, York, 1971) p. 142. [3] Donald Clayton, Principles of Stellar Evolution and Nucleosynthesis McGraw Hill, New York, 1968) p. 302. [4] F.C. Young, et al., Rev. Sci. Instrum. 48 (1977) 432. [5] F. Ajzenberg-Selove,Nucl. Phys. A248 (1975) 1. [6] C.L. Bailey and W.R. Stratton, Phys. Rev. 77 (1950) 194. [7] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions in Matter; Vol. 3, Hydrogen (Pergamon Press, New York, 1977). [8] C.A. Barnes, op. cit., p. 143.