Manipulate core radius, reflector thickness, diffusion length, and k-infinity to visualize neutron flux distribution and calculate effective multiplication factor keff.
Reactivity $\rho=\dfrac{k_{\text{eff}}-1}{k_{\text{eff}}}$. With $\rho>0$ the power rises, with $\rho=0$ it is steady and with $\rho<0$ it falls. Delayed neutrons (fraction $\beta$) lengthen the reactor period and make control possible. As long as $\rho<\beta$ the reactor is delayed-critical and can be controlled safely.
What is One-Group Neutron Diffusion?
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What exactly is "neutron flux" in a reactor, and why is it so important to calculate?
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Basically, the neutron flux, often written as $\phi$, is a measure of how many neutrons are zipping around in a given volume per second. It's the lifeblood of a nuclear chain reaction. In practice, we need to know its shape—how it's distributed across the reactor core—to ensure we have a stable, controllable power output. Try moving the "Core Radius" slider above; you'll see the flux profile change shape dramatically.
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Wait, really? So the shape changes with size. What's that "keff" value the simulator calculates, and why is it always aiming for 1.0?
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Great question. $k_{eff}$, the effective multiplication factor, is the most critical number in reactor physics. If it's exactly 1.0, every fission generation produces exactly enough neutrons to sustain the next one—that's a steady-state, critical reactor. A common case is a reactor starting up, where designers adjust materials to get $k_{eff}$ to 1.0. In this simulator, when you change "k-infinity" (a material property), you're directly influencing whether $k_{eff}$ is above, below, or at that magic number.
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I see the graph has a "reflector" region too. What does that do, and why would I want to adjust its thickness?
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The reflector is like a neutron mirror! It's usually a material like graphite or water that scatters neutrons back into the core instead of letting them escape. For instance, in small research reactors, a good reflector allows you to use less fissile fuel. When you increase the "Reflector Thickness" parameter here, watch how the flux in the core becomes flatter and higher—that means more neutrons are being saved, which can make the reactor more efficient or allow for a smaller core size.
Physical Model & Key Equations
The core behavior is governed by the one-group neutron diffusion equation. It balances neutron production from fission against losses from absorption and leakage.
Where for the core region: $D_c$ is the diffusion coefficient, $\Sigma_{a,c}$ is the macroscopic absorption cross-section, $\phi_c$ is the neutron flux, and $k_{\infty}$ is the infinite multiplication factor (a key material property you adjust in the simulator). The term $D_c \nabla^2 \phi_c$ represents neutron leakage.
In the reflector, there is no fission source, so the equation simplifies. The solutions in both regions must match at the boundary.
$$D_r \nabla^2 \phi_r - \Sigma_{a,r}\phi_r = 0$$
Here, $D_r$ and $\Sigma_{a,r}$ are for the reflector. The critical condition—solving these equations with the correct boundary conditions—yields the value for $k_{eff}$. This is the eigenvalue the simulator calculates in real-time as you change parameters.
Frequently Asked Questions
Increasing k∞ (fission cross section) raises keff, and enlarging the core radius reduces leakage, increasing keff. Conversely, increasing reflector thickness raises keff due to the reflector savings effect. First, fine-tune k∞ or the core radius.
It is an indicator of how much the core radius required for criticality can be reduced by installing a reflector compared to a bare core. Since the reflector returns neutrons to the core, a smaller core can achieve the same keff. You can intuitively understand this by comparing different reflector thicknesses in the simulator.
The core center experiences less neutron leakage than the periphery, and neutrons generated by fission accumulate more easily. In one-group diffusion theory, for a spherical core, the neutron flux follows a cosine or Bessel function, with the maximum at the center. With a reflector, the gradient near the core boundary becomes gentler.
The diffusion length L is a measure of the average distance neutrons diffuse before being absorbed. A larger L means neutrons are more likely to leak far away, requiring a larger core or higher k∞ for criticality. Conversely, a smaller L means neutrons are more easily absorbed, allowing a smaller core size.
Real-World Applications
Reactor Core Design: Engineers use this exact one-group diffusion theory for preliminary sizing of new reactor cores. By adjusting core radius and material properties, they can estimate the minimum amount of fuel needed to achieve criticality, saving millions in prototyping costs.
Safety Analysis: Understanding flux distribution is crucial for predicting "hot spots" where power and temperature could be highest. A peaked flux profile might require different cooling strategies than a flat one, influencing the placement of coolant channels and safety systems.
Reflector Optimization: In small modular reactors (SMRs) and naval reactors, space is at a premium. Analyzing how reflector thickness impacts core size and $k_{eff}$ helps designers create compact, efficient power units without compromising safety.
Fuel Cycle Management: As fuel burns up in a reactor, $k_{\infty}$ of the core material decreases. Operators use diffusion models to predict how control rods need to be adjusted over time to compensate and maintain $k_{eff} = 1.0$, ensuring steady power until the next refueling.
Common Misconceptions and Points to Note
First, keep in mind that the "one-group" model is not a panacea. It is an ultra-simplified model dealing with an "average neutron." For instance, fast neutrons and thermal neutrons diffuse and are absorbed in completely different ways, yet they are lumped together here. Therefore, it is optimal for qualitative trends and sensitivity analysis but cannot be used for detailed design of an actual reactor. As a next step, you need to use a multi-group diffusion code.
Next, a common pitfall in parameter setting is confusing the "Diffusion Length L" with the "Migration Area M²". The simulator has sliders for both, but L² is the mean square distance a neutron travels "from the end of slowing-down until its absorption." On the other hand, M² is the mean square distance "from birth in fission to absorption (including the slowing-down process)." In many cases, the relation M² ≒ L² + τ holds (where τ is the Fermi age). For example, in a light water reactor, L is on the order of a few cm, while M is on the order of tens of cm. If you are not aware of this difference, you will misinterpret the physical meaning of changing these parameters.
Finally, beware of the misconception that "if you just increase k∞ enough, anything will go critical." While k∞ indeed represents material properties, if the reactor size is too small, leakage becomes dominant, and you might not be able to bring keff to 1 no matter how much you increase k∞. Conversely, once you exceed a certain size (the critical size), keff asymptotically approaches k∞. Experiencing this tug-of-war between "leakage" and "material" is a key strength of this tool.
Select "Manual control" and move the control-rod insertion (at 50% the reactivity is ρ = 0, critical; less insertion gives ρ > 0 and rising power, more insertion gives ρ < 0 and falling power)
The total control-rod worth is the total change in reactivity when the rods move from 0% to 100% insertion. Reactivity versus insertion follows an S-curve (the rods are most effective near the middle)
Changing the delayed-neutron fraction β (about 650 pcm for U-235) and the neutron generation time Λ changes how fast the power rises (the reactor period) for the same reactivity
In "Auto demo" the rods move automatically: hold critical → withdraw → critical → insert → critical → slight withdrawal
Worked Example
With a total rod worth of 2000 pcm, β = 650 pcm and Λ = 50 μs, setting the insertion to 40% gives a reactivity ρ = 2000 × (0.5 − f(0.4)) ≈ +387 pcm (f is the integral worth x − sin(2πx)/(2π) for insertion x) and k_eff = 1/(1 − ρ) ≈ 1.0039. Since ρ < β the reactor is supercritical but delayed-critical; the one-group inhour equation gives a stable period of about 8.5 s, so the power grows by a factor e roughly every 8.5 s. At 60% insertion ρ ≈ −387 pcm and the power falls. If instead the total worth is set to 6000 pcm and the rods are withdrawn to 25%, ρ ≈ +2450 pcm > β and the prompt-critical warning appears (a dangerous state in which the power surges without waiting for delayed neutrons).
Practical Notes
This tool is a point-kinetics model with the delayed neutrons lumped into one group. Real reactor analysis uses six delayed groups and the spatial flux distribution
Temperature feedback (Doppler effect, moderator temperature coefficient) is not included, so a power rise is not self-limiting here. In real reactors negative reactivity coefficients restrain power increases
Prompt criticality (ρ ≥ β) makes the reactor period extremely short; plants are designed so that it can never be reached, for example by limiting the rod withdrawal speed
β depends on the fuel (about 650 pcm for U-235, about 200 pcm for Pu-239); in cores with a small β, such as MOX cores, the same reactivity gives a shorter period