2.1. The k-ratio method

In 1997, Bretscher of the Argonne National Laboratory introduced the k-ratio method to evaluate the delayed neutron fraction [13]. According to this method, the fraction of effective delayed neutron can be evaluated by the ratio between the prompt and the total multiplication factors, hence the name of k-ratio method,

where υ, υ_d and υ_p are average numbers of the total fission neutrons, delayed fission neutrons and prompt fission neutrons, respectively, produced per fission; χ, χ_d and χ_p are spectra of the three neutron categories, respectively. The approximation in Eq. (1) is based on two assumptions: 1) (χ_d–χ)υ_d is at least two orders of magnitude smaller than the χυ_p, because υ_d is two orders of magnitude smaller than υ_p; and 2) χ_p is almost equal to χ.

The k-ratio method can be performed by Monte Carlo code (MCNP) which can suppress the delayed neutrons in the critical calculation. In this paper, MCNP5 (ver1.51) and ENDF/B-VII are used. The fraction of effective delayed fission neutrons (βⁿ), which are decayed by the fission products, is modeled by MCNP calculation (MODE N) with and without the delayed neutron (TOTNU card) while ignoring photoneutrons. βⁿ can be estimated by Eq. (2)

where the superscript n stands for MODE N, the subscript t and p denote the neutron transportation with and without the delayed neutron, respectively. As an extension, the contribution of photoneutrons (β^ph) can be obtained by coupled neutron-photon transport (MODE N P) with MPN card to model (γ, n) reaction [8],

where the superscript n+p stands for MODE N P.

There is an approximation in the simulation. Because MCNP does not simulate the delayed particles in critical calculations, the prompt gammas emitted in the process of fission and neutron capture are used, instead of delayed gammas which are emitted in the decay process of the fission products and activation products. It is reasonable, as the difference is small between the prompt and delayed photon spectra from ²³⁵U fission represented by empirical Eqs. (4) and (5) [8, 14].

2.2. The neutron-photon coupled point kinetics equations

With the infinite-velocity approximation, we can get the conventional set of point kinetics equations as Eqs.(6–8). The physical interpretation of this approximation is that the photons are transported instantaneously following a change. This approach has been used for fast neutrons in the neutron-only multi-group transportation [14]. Since the speed of photons is at least one order of magnitude greater than the neutron speed, it is more reasonable here.

where n(t) is neutron flux (or power), C_i is concentration of the i^th group of neutron precursors, λ_i is decay constant of the i^th neutron precursors, C_j=ρ^phΛ^γ/(ΛⁿP_j) is concentration of the ^jth group artificially introduced photoneutron precursors, ρ^ph is the photoneutrons reactivity (photoneutron production rate over photo removal rate), P_j is concentration of the j^th group photoneutron precursors, λ_j is decay constant of the j^th photoneutron precursor. Λⁿ is neutron generation time, ρ*≡ ρ + ρ^ph is the total reactivity, β_eff = ∑_i β_iⁿ + ∑_j β_j^ph is the total effective delayed neutron fraction, β_iⁿ is the i^th group delayed fission neutron fraction, β_j^ph is effective fraction of the j^th group delayed photoneutrons, and Λ^γ is photon generation time.

The coefficients of β_eff and Λⁿ are assumed to be independent of time. This is already the case for the exact point kinetic without photoneutron terms. The photoneutron reactivity is contributed by the gamma ray (both prompt and delayed) reactivity. Also, it is no need to calculate the photon generation time (Λ^γ) and the photoneutron precursor’s concentration (P_j) to get the artificial photoneutron precursor group C_j for the dynamic behavior of the reactor [14].

As listed Table 1, the traditional 6 group delayed fission neutrons parameters per ²³⁵U fission [15], and the additional 9 group delayed photoneutrons parameters per ²³⁵U fission in infinite beryllium, are used together with the corrected effective fraction of the photoneutrons.

2.3. Interpretation of reactivity measurements

Because of the photoneutrons in the core, the in-hour equation used to convert a measured stable doubling period T into a reactivity ρ should be modified as Eq. (9) [10].

Similarly, for the interpretation of inverse kinetics measurements, the time dependent reactivity ρ(t) obtained by the evolution of the neutron population are modified as Eq.(10) [16, 17].

where, C_j⁰ and C_i⁰ are the equilibrium concentration of neutron and photoneutron precursors, respectively; BF_i and BF_j are the build-up factors which account for the building up of delayed neutron and photoneutron precursors, respectively, prior to the actual measurement. The building factors can be derived from the power history before the measurement via Eq.(11), where, n₀ is the neutron flux at full power and T_b is the building up time.

3.1. The delayed fission neutron and photoneutron fraction

Using the k-ratio method described in Section 2.1, the delayed fission neutron and photoneutron fraction of TMSR-SF1 were simulated with 10⁹ neutrons, as the (γ, n) reaction rate is very small. The critical eigenvalues were obtained with a statistical uncertainty of about 0.00003. The results are given in Table 2. The effectiveness fraction of the delayed fission neutron is about 1.04 in Mode N, while it is 0.27 for the ²³⁵U fission [15]. The k-ratio method is very time consuming for low statistical errors. Besides, photoneutrons generated in the active core and the reflector zones are of the same importance in the k-ratio method. Considering TMSR-SF1 has much more coolant in the reflector zone than in the active zone, the k-ratio method may overestimate the delayed photoneutron fraction.

3.2. Influence of photoneutrons on real time reactivity

The real time reactivity measurement, i.e. the reactivity meter, has been used widely [16, 17]. In this section, both positive and negative reactivity insertions are simulated. The theoretical basis is inverse point kinetics descried in Section 2.3.

3.2.1. Negative reactivity insertion

The largest impact of the photoneutrons is expected after shut-down of the reactor that has been operated at high power for a long time. According to the design of TMSR-SF1, the total worth of safety rods is about 6 $ [6], and they should be fully inserted in 6 s at emergency. Such a scram was calculated. In the first 10 s, the reactor is operated at full power, then 6 $ negative reactivity is fully inserted in 6 s.

The relative power level as a function of time following a scram from the full power is shown in Fig. 2(a). In the calculation represented by “6 Group”, the photoneutrons were not taken into account. Meanwhile, the one denoted by “15 Group”, the photoneutrons were included in the calculation. The power difference between two cases grew gradually after shut-down. In the first 100 s, they did not differ from each other because the fission neutrons still took the major part. About 500 s latter, the presence of photoneutrons led to a 10 times larger power level. This can be explained by Eqs. (6–8). After shut-down, the number of photoneutrons was still determined by the original power, while the number of the fission neurons depended on the current low power. The relatively long decay time of relevant fission products made photoneutrons play an important role after shut-down.

Fig. 2Full-screen View

Calculation of negative reactivity insertion (15 Group: with photoneutrons; 6 Group: without photoneutrons.)

The time dependent reactivity calculated with and without considering photoneutrons is shown in Fig. 2(b). In this simulation, the relative power obtained with photoneutrons is regarded as the truth (the blue line in Fig. 2a). One sees that, with photoneutrons, the reactivity meter gives great values; whereas it gives false values without photoneutrons. Although the reactor keeps subcritical after the scram, the reactivity meter without photoneutrons shows that positive reactivity is inserted gradually.

3.2.2. Positive reactivity insertion

To evaluate the influence on positive reactivity, 0.1 $ was inserted after full power operation for a long time. The power evolution after the positive reactivity insertion, shown in Fig. 3(a), was simulated with and without the photoneutrons. It can be seen that the effect of delayed photoneutrons is similar to the delayed fission neutrons. The presence of photoneutrons slowed down the power increase. After 200 s of the reactivity insertion, “6 Group” power led to a power level of 10 times larger than the origin power; while the “15 Group” power led to a power level of 9 times larger than the origin level. The power evaluated with photoneutrons (blue line in Fig. 3a) was considered as the real case. The time dependent reactivities calculated with and without the photoneutrons are shown in Fig. 3(b). With photoneutrons, the reactivity meter met the truth; whereas the reactivity was overestimated when photoneutrons were neglected.

Fig. 3Full-screen View

Calculation of positive reactivity insertion (15 Group: with the photoneutrons; 6 Group: without the photoneutrons.)

3.3. Influence of photoneutrons on doubling time measurement

The doubling time method is widely used for the control rod worth measurement, where the reactivity is calculated by the in-hour equation using the doubling time. The doubling time as a function of the positive reactivity inserted is shown in Fig. 4. In this calculation, follow the regulations, the doubling time was 30 s to 180 s. As expected, the doubling time became longer for the same reactivity insertion by considering the photoneutron contribution. The 60 s doubling time corresponds to 0.108 $ reactivity insertion for “6 Group”; whereas to 0.112 $ for “15 Group”.

Fig. 4Full-screen View

Relationship between the doubling time and the positive reactivity inserted

In order to choose an appropriate waiting time before the reactivity measurement with respect to the positive reactivity insertion, calculations were carried out at different reactivities. The waiting time needed for 2% deviation from the doubling period is shown in Fig. 5(a). The discrepancy in the waiting time was almost negligible for doubling periods close to 30 s, but became more apparent for longer doubling time. At 180 s of doubling time, the waiting time was less than 100 s for “6 Group”, but over 250 s for “15 Group”.

Fig. 5Full-screen View

Calculation of period deviation at 100 s and 10⁸ s of building time, as a function of waiting time.

Since long lived photoneutron precursors take extremely long time for saturation, it is not appropriate to always set the precursors’ concentrations as equilibrium condition. In this context, the building time is used to define the holding time at full power level before the shut-down step. The building time effect on the period measurement was calculated with 0.1 $ reactivity insertion. The error in doubling period measurement as the function of waiting time is shown in Fig. 5(b). Obviously, much less waiting time is required in the non-equilibrium cases to reach the same accuracy as the equilibrium cases. When the reactor is at critical without external source, the equilibrium precursors to neutron flux is proportional to 1/λ_i. The equilibrium precursors play an important role for the reactor of a positive period. The ones with non-equilibrium precursors have shorter asymptotic period. It increases more rapidly than the equilibrium case, so less waiting time is needed for the doubling period measurement.

3.4. Influence of photoneutrons on criticality estimation

Criticality measurements play an important role in reactor experiments. It is a usual method to bring the reactor to a steady power representing criticality. However, the limitation is that the effective multiplication factor, k_eff, equals to one only when the delayed neutron precursors have reached equilibrium. Since the delayed photoneutrons have much longer half-life than the delayed fission neutrons, it takes fairly long time to reach the equilibrium or decay from the equilibrium. Two conditions, non-saturated photoneutron precursors and extra precursors compared to the equilibrium, were analyzed.

3.4.1. Non-saturation of the delayed photoneutron precursors

With increasing precursors, more neutrons are being consumed than yielded when the power is steady. Thus, there is a net deficit in the neutron balance and some excess k_eff is required to keep the power steady. For math simplicity, it was assumed that the reactor was brought up to given power instantaneously and all precursors’ concentrations were zero at the origin. Furthermore, the reactivity is constrained to maintain steady power, i.e., dn/dt = 0. A Matlab code was written to find the root of Eqs.(6–8).

The excess k required to keep the power steady after startup is shown in Fig. 6. In this calculation, different effectiveness factors were assumed. It can be predicted that, the fission neutron precursors got equilibrium in 15 minutes while the photoneutron precursors did not reach equilibrium in 2 hours. Hence, the reactor would need excess reactivity, about a few pcm, even after one or more hours. This may lead some trouble for high-precision measurements. As mentioned before, it was assumed that the reactor was brought up to steady power very rapidly, which is impossible in practice. So we made a correction for the finite start-up time by adding the time equal to the period that the reactor spent on bringing up the power. It should be emphasized that this effect applies only when the reactor is brought to power from shutdown. If the rector restores the steady power, the precursor equilibrium would be essentially undisturbed and no waiting time is needed for the criticality measurement.

Fig. 6Full-screen View

Excess k required to hold steady power after startup with and without the photoneutrons