Neutron temperature

Figure 7 Intensity lines from (a) neutron (temperature dependent, courtesy Zheludev et al. ) and (b) x-ray elastic scattering experiments (stress dependent, courtesy Martynov et al. ) showing the existence of a satellite at 1/6 [110] corresponding with the modulation wavelength in the ISO s visible in figure 6.

It is not possible to determine k for a hydrogen atom directly from experimental X-ray data, because its value correlates strongly with the temperature parameter due to the absence of unperturbed inner-shell electrons. The use of neutron temperature parameters provides an alternative. Combined analysis of X-ray and neutron data on glycylglycine and sulfamic acid suggests that for X—H (X = C, N) groups, the H atom is more contracted than for the H2 molecule, with a k value as large as 1.4 for both C—H and N—H bonds (Coppens et al. 1979). 

The Khp = 1.4 value seems large, and possibly corresponds to an overestimate of the contraction, as the analysis depends on the reliability of the neutron temperature factors. Nevertheless, it has been used successfully. There is no doubt that the contraction of the hydrogen-atom density must be taken into account in accurate structure analysis. 

Early studies, which did not include many high-order reflections, revealed systematic differences between spherical-atom X-ray- and neutron-temperature factors (Coppens 1968). Though the spherical-atom approximation of the X-ray treatment is an important contributor to such discrepancies, differences in data-collection temperature (for studies at nonambient temperatures) and systematic errors due to other effects cannot be ignored. For instance, thermal diffuse scattering (TDS) is different for neutrons and X-rays. As the effect of TDS on the Bragg intensities can be mimicked by adjustment of the thermal parameters, systematic differences may occur. Furthermore, since neutron samples must be... 

To account for temperature factor differences, a temperature scale factor kT multiplying the neutron temperature parameters may be introduced, as defined by the expression (Coppens et al. 1981)... 

The X-N technique is sensitive to systematic errors in either data set. As discussed in chapter 4, thermal parameters from X-ray and neutron diffraction frequently differ by more than can be accounted for by inadequacies in the X-ray scattering model. In particular, in room-temperature studies of molecular crystals, differences in thermal diffuse scattering can lead to artificial discrepancies between the X-ray and neutron temperature parameters. Since the neutron parameters tend to be systematically lower, lack of correction for the effect leads to sharper atoms being subtracted, and therefore to larger holes at the atoms, but increases in peak height elsewhere in the X-N deformation maps (Scheringer et al. 1978). 

In the determination of the number of fissions in an irradiated sample by the use of flux monitors, account must be taken of the flux depression in the sample due to self-shielding to obtain an effective flux. Also, the capture cross sections of the monitors and the fission cross sections of the sample are neutron energy dependent. It is, therefore, necessary to know the eneigy distribution of the neutrons or the neutron temperature and to determine effective cross sections (Section IV). This can be done by using two monitors such as cobalt and samarium, the one monitor being used to determine the neutron temperature corresponding to the neutron distribution as described by Fritze et al. (35). 

Effects 1 and 2 are functions of the bulk temperature of the plates and HjO,respectively. Effect 3 is a function only of the neutron tenq>erature. To determine the neutron temperature from the H O temperature is not easy it is supposed that the two are equal. ... 

As the absorption becomes stronger, the reactor becomes less thermal, and the spectrum of neutrons in the thermal energy region becomes hardened with respect to an equilibrium Maxwellian distribution. This hardening is frequently described in terms of a neutron temperature which exceeds the moderator temperature, but this description has no theoretical basis. For small l/v absorption, the hardening is best described in terms of the function G v) defined in Equation (13). This function depends on the model used for thermalization calculations, but not on the absorption. The sensitivity of this function to scattering model has not been well explored. 

Another type of purely neutronic effect which is important in thermal reactors arises from the heating of the moderator and consequent rise in neutron temperature. This results in a change in both the thermal diffusion length and the neutron age to thermal, thus a change in migration length and hence in neutron leakage. 

The assumption that the activation cross section varies with neutron energy as 1/v in the thermal neutron region is valid for most (n,y) reactions. The two reactions that deviate the most from the 1/v assumption are Lu(n,y) Lu (typically +0.4%/K) and Eu(n,y) Eu (typically —0.1%/K). The reaction rates for these two reactions, relative to a monitor reaction like Au(n,y) Au, will depend on the thermal neutron temperature in the irradiation channel used. A new set of equations, the Westcott formalism (Westcott 1955), was developed to account for these cases and used the Westcott g T ) factor, which is a measure of the variation of the effective thermal neutron activation cross-section relative to that of a 1/v reaction. In the modified Westcott formalism, the following differences are also included the Qo(a) value of the Hogdahl formalism is replaced by the So(< ) value, and the thermal to epithermal flux ratio, f, is replaced by the modified spectral index, r a) TJTo). To use this formalism with the kg method (De Corte et al. 1994), it is necessary to measure the neutron temperature, r , for each irradiation and a Lu temperature monitor should be irradiated. The Westcott formalism needs to be implemented only when analyzing for Lu and Eu. There are several other non-1/v nuclides Rh, In, Dy, Ir, and Ir, but for these the error... 

In most nuclear reactors, thermal neutrons, that is, neutrons that are in thermal equilibrium with matter at room temperature, have by far the highest density. The velocity of thermal neutrons exhibits a Maxwell distribution known from the kinetic theory of gases, where the number of neutrons having a velocity between v and v + dv is expressed in terms of the total number of neutrons, temperature, and velocity. Neutron density per unit velocity is given by the equation... 

An obvious advantage can be seen to using 4>o since cross section tables usually give a list of (To values and since it is necessary to know the temperature corresponding to the Maxwellian velocity in order to determine other fluxes and cross sections. Often, the neutron temperature is not known, and this lack of knowledge is reflected in the uncertainty in any cross section measurement if the species in question is not known to obey the 1/v law in and near the thermal region. In any case, it is customary to measure the thermal flux with a 1/v absorber of known cross section and then to use the same flux for determining an unknown cross section. 

Values of the epithermal spectrum index, r, and ffie effective neutron temperature, Tn. wei deduced from fa-to-Mn and Lu-177-to-Mn acttvation ratios using the technique of Chidley et aL modified to allow for the variation of the g-value of fa with neutron temperature. 

Detailed parameter measurements made in the SE included determinations of the thermal-neutron distributions in terms of subcadmium foil activation thermal-neutron temperatures in terms of Lu/l/v activity ratios epithermal-neutron fractions in terms of Th capture and u fission cadmium ratios U production in terms of neutron captures in Th and lattice fissions in terms of the Th/ U fission ratios. The HAMMER computations generally agreed with the intracell activation profiles. The spectral index measurements agreed poorly because the computations overestimated the spectral index in all coolants. 

Fig. 1 Criticai mass of as a function of core height for assumed core neutron temperatures of 10, 20,and30 K.

The mafiionatical-criticality analyses were performed using the KENO Monte Carlo computer program together with the 123-groiq> GAM THERMOS neutron cro -s6ction set. Fuel pin cdl k, Calculations were performed to establidi the most reactive lattice pitch as a frmetion of U enrichment and bonm concentrations. Three-dimensional (3-D) KENO keff calculations were used to establish the reactivity of tile reactor under a variety of states. Since the TMI-2 coolant will eventually reach room temperature, all criticality analyses (with the exception of the hot, clean benchmark TMI-2 conjuration) were performed at this most reactive (neutronically) temperature. The moderator density was taken as 1.0 g/cm and the fuel (UOj) density was assumed as 95% theoretical. 

Since most of the nuclear reactions In N Reactor occur at thermal or near-thermal neutron energies the most Important cross sections are those at these energies. The Vestcott formulation Is particularly useful since It defines an effective cross section for a materiel, vhlch when multiplied by the prqper flux gives the total reaction rate over the entire thermal and slowlng-down neutron spectrum. The flux employed Is nvo vhere Vq Is 2200 m/s (the velocity of neutrons corresponding to a neutron temperature of 20 c) and n Is the total neutron density (Integrated over all energies). 

Hheoretical spectral indices and neutron temperatures for the N Reactor fuel element are given in Table 2 3 2.1.1 for equilibrium condition... 

Determine the neutron temperature of the Maxwellian portion of the spectrum. 

Other items of Interest are also given in Table 2.7.I such as physical temperatures neutron temperatures region areas etc. Tiie void regions are cyllndrized void regions to moekup the actual void spaces. 

In general, the neutron temperature will vary over the lattice and its calculation is exceedingly complex. approximation used in the NDFEA and... 

The f-coefficient is not amenable to a simple calculation since the knowledge of the neutron temperature and flux in each part of the lattice cell is required. The calculation of the fine flux structure across the cell requires the solution to the Boltzman transport equation. (The FIfiX and MDFDd codes employ the po approximation to the transport equation.) The same is true for the L -coefficient. 

The variation in the Fermi age, T, vith graphite temperature arises from the fact that the neutrons have to slow down over a smaller energy range as the neutron temperature increases. The coefficient is small and positive and for a hetereogeneous reactor is difficult to calculate. For these reasons it can be neglected. 

Increasing the water temperature increases the neutron temperature in the fuel more than in the graphite This reduces the absozptiveness of the uranium to thermal neutrons which also decreases the cell disadvantage factor ... 

The behavior of ff yith coolant temperature is connected solely vith the effects on the neutron temperature in the fuel, thus... 

Ohe variation in with coolant tenperature is linked vith the decreased moderating effect as the coolant density is reduced. Thus> the age vill Increase as the coolant is heated, ere is also the same effect discussed earlier vith respect to graphite temperaturej and that is that as the vater temperature is raised the general neutron temperature is raised and hence the neutrons have to alow down over a smaller range to reach thermal equilibria This ... 

One way in which this shift in the neutron density may be taken into account is by introducing an effective neutron temperature Tn. Thus we define some fictitious temperature Tn which, when used in the function m of (4.196), gives a new Gaussian that yields a better fit to the distorted density distribution arising from the presence of the absorber. The determination of this parameter is deferred until a later section. For the present, all we require is that such a number can be specified for a given system. 

A useful measure of the flux distortion is provided by the concept of the effective neutron temperature Tn- We define this temperature as that number which when used in (4.197) gives the best least-squares fit of a Maxwell-Boltzmann distribution to the computed flux (solid-line curve) in the range 0 < x < 35. The ratio of the effective neutron temperature to the moderator temperature Tn/Ts is indicated in the figure for the first two cases. The ratio has been omitted from the last case (A = 9, = 2 ) because the flux was so severely distorted from a... 

Maxwell-Boltzmann distribution that the concept of an effective neutron temperature was meaningless. It was found that the linear relation... 

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