Introduction

An accurate, real time method of determining the element contents of the cement raw material (CRM) is important for cement plants to control cement quality. Although chemical method is the most commonly used in cement plants, it will take more than 1 hour to measure one sample. Therefore it cannot direct the production of cement on time. Neutron induced prompt gamma-ray analysis (NIPGA) is a fast analysis technique, which can analyze one multicomponent sample in a few minutes [1–4]. As a result, it is widely used to analyze the element contents of CRM [5].

In NIPGA, the prompt characteristic gamma-rays induced by thermal neutron capture reactions are used to compute the contents of silicon, aluminum, iron and calcium. Compared with the reactor neutron source, radioisotope neutron source and other accelerator neutron sources, D–D neutron generator is an optimal neutron source for this technique because of the advantages of low cost and no radioactivity after being turned off [6]. Therefore, D–D neutron generator is often used in NIPGA and the 2.5 MeV neutron flux can be regarded as a constant value, and the thermal neutron flux is relevant to the component of CRM. But in some papers, the thermal neutron flux is still treated as a constant value [7, 8]. It will increase the measurement error and make the measurement accuracy cannot meet the requirement of cement plants, which makes NIPGA unpromising in measuring the contents of silicon, aluminum, iron and calcium in CRM. In this paper, 50 samples of CRM were designed based on the data obtained from cement plants and simulated by MCNP. The relationship between the thermal neutron flux and element characteristic gamma-rays counts, such as silicon, aluminum, iron and calcium, was obtained by linear regression. Then, the empirical formulas of computing the contents of silicon, aluminum, iron and calcium were also found by linear regression.

Principle

In NIPGA, the energy of the characteristic gamma-ray induced by thermal neutron capture reaction can be used to identify the element type, such as silicon, aluminum, iron and calcium, and the characteristic gamma-ray count can be used to compute the element content. The basic formula is as follows [9]:

$$ w = k(N/\varphi ) + b $$
(1)

Take silicon for example, w is the mass content of silicon in the sample of CRM, N is the characteristic gamma-ray count of silicon, \( \varphi \) is the thermal neutron flux in the sample, k and b are empirical constants determined by the experimental device.

Once the D–D neutron generator has be selected, the 2.5 MeV neutron flux can be regarded as a constant value. In the sample of CRM, 2.5 MeV neutrons will be moderated by the nuclei such as silicon, aluminum, iron, calcium, oxygen and so on. If they become thermal neutrons, the nuclei of silicon, aluminum, iron and calcium may capture them and emit prompt gamma-rays. Because the abilities to moderate neutrons are different, the thermal neutron flux is relevant to the contents of all elements in the sample. In CRM, oxygen is almost in the form of SiO2, Al2O3, Fe2O3 and CaO, accordingly the content of oxygen can be looked as the function of the contents of silicon, aluminum, iron, calcium. Furthermore, the sum content of silicon, aluminum, iron, calcium and oxygen is more than 93 %. As a result, the thermal neutron flux can be approximately looked as the function of the contents of silicon, aluminum, iron and calcium:

$$ \varphi = f_{{1(P_{\text{Si}} )}} + f_{{2(P_{\text{Al}} )}} + f_{{3(P_{\text{Fe}} )}} + f_{{4(P_{\text{Ca}} )}} $$
(2)

where \( \varphi \) is the thermal neutron flux in the sample. P Si, P Al, P Fe, P Ca is the mass content of silicon, aluminum, iron and calcium respectively. Equation (2) can only calculate the thermal neutron flux in the sample whose element contents are known, but it cannot be used to compute the element contents in CRM. In order to compute the element contents in CRM, the characteristic gamma-rays are used to take the place of the element contents and the formula can be expanded by Maclaurin series as [10–12]:

$$ \varphi_{n} = \sum\limits_{j = 1}^{n} {a_{j} N_{{_{\text{Si}} }}^{j} } + \sum\limits_{j = 1}^{n} {b_{j} N_{\text{Al}}^{j} } + \sum\limits_{j = 1}^{n} {c_{j} N_{{_{\text{Fe}} }}^{j} } + \sum\limits_{j = 1}^{n} {d_{j} N_{{_{\text{Ca}} }}^{j} } + R_{{n(N_{\text{Si}} ,N_{\text{Al}} ,N_{\text{Fe}} ,N_{\text{Ca}} )}} $$
(3)

where \( \varphi_{n} \) is the calculated value of the thermal neutron flux when the maximum value of j is n. N Si, N Al, N Fe, N Ca is the characteristic gamma-ray count of silicon, aluminum, iron and calcium respectively, j is the exponent, a j , b j , c j , d j , is the coefficient of \( N_{\text{Si}}^{j} \), \( N_{\text{Al}}^{j} \), \( N_{\text{Fe}}^{j} \) and \( N_{\text{Ca}}^{j} \), \( R_{{n(N_{\text{Si}} ,N_{\text{Al}} ,N_{\text{Fe}} ,N_{\text{Ca}} )}} \) is the Lagrange remainder. In order to find the coefficients and Lagrange remainder, 50 samples of CRM were designed based on the data obtained from cement plants and simulated by MCNP. The scheme of the simulation system is shown in Fig. 1.

Fig. 1
figure 1

Scheme of simulation system

Full size image

In Fig. 1, the cylinder whose radius is R and axis is y-axis is the neutron shield. It is filled by polyethylene except the space occupied by the D–D neutron generator, the BGO detector and the sample of CRM. In order to increase the accuracy, the centers of the sample, the BGO detector and the target of the D–D neutron generator are all located on the y-axis.

Computing the thermal neutron flux

In this paper, 30 samples (from No. 1 to 30) of CRM were firstly simulated by MCNP. The characteristic gamma-ray energies of silicon, aluminum, iron, calcium are 4.934, 3.466, 7.631 and 6.420 MeV, respectively. The contents of silicon, aluminum, iron, calcium in each sample are given in Table 1. The characteristic gamma-rays counts and the thermal neutron flux calculated by F4 tally are also shown in Table 1.

Table 1 Data of the samples
Full size table

In Table 1, P Si, P Al, P Fe and P Ca is the mass content of silicon, aluminum, iron and calcium. Because the values in MCNP code must be constants, P Si, P Al, P Fe and P Ca have no uncertainties. \( \varphi \) is the thermal neutron flux in the sample simulated by MCNP. N Si, N Al, N Fe, N Ca is the characteristic gamma-ray count of silicon, aluminum, iron and calcium respectively when the yield of D–D neutron generator is 106 n/s and the measure time is 5 min.

In this paper, the relationship between the thermal neutron flux and the characteristic gamma-rays counts of silicon, aluminum, iron and calcium was found by linear regression analysis.

When n = 1 in Eq. (3), the relationship is as follow:

$$ \varphi_{1} = 7.608 \times 10^{ - 3} N_{\text{Si}} + 4.248 \times 10^{ - 2} N_{{{\text{Al}}\;}} + 9.447 \times 10^{ - 4} N_{\text{Fe}} + 4.382 \times 10^{ - 3} N{}_{\text{Ca}} - 1.896 \times 10^{ - 3} $$
(4)

where \( \varphi_{1} \) is the calculated value of the thermal neutron flux when n = 1. When n = 2 in Eq. (3), the relationship is as follow:

$$ \begin{aligned} \varphi_{2} = - 3.965 \times 10^{ - 3} N_{\text{Si}} - 1.041 \times 10^{ - 2} N_{\text{Al}} \; + 7.780 \times 10^{ - 4} N_{\text{Fe}} + 1.532 \times 10^{ - 3} N_{\text{Ca}} \hfill \\ \;\;\;\; + 4.721 \times 10^{ - 8} N_{\text{Si}}^{2} + 3.892 \times 10^{ - 6} N_{\text{Al}}^{2} \; + 3.947 \times 10^{ - 10} N_{\text{Fe}}^{2} + 2.036 \times 10^{ - 9} N_{\text{Ca}}^{2} + 0 \hfill \\ \end{aligned} $$
(5)

where \( \varphi_{2} \) is the calculated value of the thermal neutron flux when n = 2. When n = 3 in Eq. (3), the relationship is as follow:

$$ \begin{aligned} \varphi_{3} = 0 \times N_{\text{Si}} + 0 \times N_{\text{Al}} \; + 0 \times N_{\text{Fe}} + 0 \times N_{\text{Ca}} \hfill \\ \;\;\;\; + 1.898 \times 10^{ - 8} N_{\text{Si}}^{2} + 2.509 \times 10^{ - 6} N_{\text{Al}}^{2} \; + 4.174 \times 10^{ - 9} N_{\text{Fe}}^{2} + 5.445 \times 10^{ - 9} N_{\text{Ca}}^{2} \hfill \\ \;\;\;\; + 6.530 \times 10^{ - 14} N_{\text{Si}}^{3} + 5.836 \times 10^{ - 11} N_{\text{Al}}^{3} \; - 5.881 \times 10^{ - 15} N_{\text{Fe}}^{3} - 2.195 \times 10^{ - 15} N_{\text{Ca}}^{3} + 0 \hfill \\ \end{aligned} $$
(6)

where \( \varphi_{3} \) is the calculated value of the thermal neutron flux when n = 3.

In Table 1,

$$ D_{i} = \frac{{|\varphi - \varphi_{i} |}}{{\varphi_{i} }} \times 100 $$
(7)

In order to select a more accurate relationship, Eqs. (4), (5) and (6) were used to calculate the thermal neutron flux in the samples from No. 31 to 50. The results are also shown in Table 1. From Table 1 six average values can be computed: \( A_{D1} = (\sum\nolimits_{i = 1}^{30} {D_{1i} } )/30 = 0.17 \), \( A_{D2} = (\sum\nolimits_{i = 1}^{30} {D_{2i} } )/30 = 0.15 \), \( A_{D3} = (\sum\nolimits_{i = 1}^{30} {D_{3i} } )/30 = 0.15 \), \( A_{D1} ' = (\sum\nolimits_{i = 31}^{50} {D_{1i} } )/20 = 0.18 \), \( A_{D2} ' = (\sum\nolimits_{i = 31}^{50} {D_{2i} } )/20 = 0.53 \) and \( A_{D3} ' = (\sum\nolimits_{i = 31}^{50} {D_{3i} } )/20\; = 0.51 \).

From Eqs. (4), (5) and (6) and the six average values, we can find that Eq. (4) has greater physical significance and higher accuracy.

Improving the element measurement accuracy

In practical application, it is difficult to measure the thermal neutron flux with high-precision, so \( \varphi_{1} \) has to be used in Eq. (1). The equations for calculating the element contents were found by linear regression analysis with the data from No. 1 to No. 50 in Table 1:

$$ w_{\text{Si}} = 0.2079\;N_{\text{Si}} \;/\varphi_{1} + 0.0943 $$
(8)
$$ w_{\text{Al}} = 1.3253\;N_{\text{Al}} /\varphi_{1} - 0.1771 $$
(9)
$$ w_{\text{Fe}} = 0.0323\;N_{\text{Fe}} /\varphi_{1} - 0.0222 $$
(10)
$$ w_{\text{Ca}} = 0.1278\;N_{\text{Ca}} /\varphi_{1} + 0.9840 $$
(11)

where w Si, w Al, w Fe, w Ca are mass contents (computed by their characteristic gamma-ray counts and the thermal neutron flux) of silicon, aluminum, iron and calcium in the sample, N Si, N Al, N Fe, N Ca are their characteristic gamma-ray counts, and \( \varphi_{1} \) is the calculated value of the thermal neutron flux in the corresponding sample computed by the Eq. (4). The deviations between element contents and their calculated values (w Si, w Al, w Fe, w Ca) are shown in Table 2.

Table 2 Deviations between element contents and their calculated values
Full size table

In some papers, the thermal neutron flux is regarded as a constant value. In order to compare the elemental measurement accuracies, the element contents are also computed when the thermal neutron flux is regarded as a constant value. The deviations between element contents and their calculated values (w Si′, w Al′, w Fe′, w Ca′) are also shown in Table 2.

In Table 2, \( \Delta w_{\text{Si}} = |w_{\text{Si}} - P_{\text{Si}} | \), \( \Delta w_{\text{Al}} = |w_{\text{Al}} - P_{\text{Al}} | \), \( \Delta w_{\text{Fe}} = |w_{\text{Fe}} - P_{\text{Fe}} | \), \( \Delta w_{\text{Ca}} = |w_{\text{Ca}} - P_{\text{Ca}} | \), \( \Delta w_{\text{Si}} \prime = |w_{\text{Si}} \prime - P_{\text{Si}} | \), \( \Delta w_{\text{Al}} \prime = |w_{\text{Al}} \prime - P_{\text{Al}} | \), \( \Delta w_{\text{Fe}} \prime = |w_{\text{Fe}} \prime - P_{\text{Fe}} | \) and \( \Delta w_{\text{Ca}} \prime = |w_{\text{Ca}} \prime - P_{\text{Ca}} | \).

In GB/T 176-2008 (Method for Chemical Analysis of Cement), the absolute value of deviations of silicon, aluminum, iron and calcium are 0.20, 0.30, 0.20 and 0.40 %, respectively. From Table 2 we can find: the deviations in columns 2–5, determined by the present method, are all lower than the requirements of GB/T 176-2008; however, many ones in columns 6–9, determined by assuming constant flux, do not meet the requirement.

Results and discussion

In NIPGA, because the thermal neutron flux in the sample, which should be used to compute the element contents, cannot be determined accurately or computed from the known conditions, the measurement accuracy is very low. In this paper, the thermal neutron flux has been computed by the characteristic gamma-ray counts which are simulated by MCNP and the element contents have been calculated with high-precision by the flux and the characteristic gamma-ray counts. But until now, the errors of the experimental data (N Si, N Al, N Fe and N Ca) determined by our equipment are too big to find the relationship between the thermal neutron flux and the elemental characteristic gamma-rays counts. Now we are improving the equipment to increase measurement accuracy of N Si, N Al, N Fe, N Ca and \( \varphi \).