Enter velocity as a fraction of the speed of light (0 = rest, 0.99 = 99% c).
Time measured in the laboratory frame (microseconds).
Lorentz Factor ($\gamma$):–
Proper Lifetime ($\tau$):2.197 μs
Dilated Lifetime ($\tau_{lab}$):–
Survival Probability:–
Remaining Muons ($N(t)$):–
function calculateDecay() {
// Clear errors
var errorDiv = document.getElementById('error-message');
errorDiv.style.display = 'none';
errorDiv.innerText = ";
// Get inputs
var n0 = parseFloat(document.getElementById('initialMuons').value);
var v = parseFloat(document.getElementById('velocity').value);
var t = parseFloat(document.getElementById('timeElapsed').value);
// Validation
if (isNaN(n0) || n0 < 0) {
errorDiv.innerText = "Please enter a valid positive number for Initial Muons.";
errorDiv.style.display = 'block';
return;
}
if (isNaN(v) || v = 1) {
errorDiv.innerText = "Velocity must be between 0 and 0.999999 (less than speed of light).";
errorDiv.style.display = 'block';
return;
}
if (isNaN(t) || t < 0) {
errorDiv.innerText = "Please enter a valid positive number for Time Elapsed.";
errorDiv.style.display = 'block';
return;
}
// Constants
var tau_proper = 2.1969811; // microseconds
// Calculations
// 1. Lorentz Factor: gamma = 1 / sqrt(1 – v^2)
var gamma = 1 / Math.sqrt(1 – (v * v));
// 2. Dilated Lifetime in Lab Frame: tau_lab = gamma * tau_proper
var tau_lab = gamma * tau_proper;
// 3. Decay Logic: N(t) = N0 * e^(-t / tau_lab)
var decay_exponent = -t / tau_lab;
var probability = Math.exp(decay_exponent);
var remaining = n0 * probability;
// Display Results
document.getElementById('res-gamma').innerText = gamma.toFixed(4);
document.getElementById('res-dilated-tau').innerText = tau_lab.toFixed(4) + " \u03BCs";
document.getElementById('res-probability').innerText = (probability * 100).toFixed(4) + "%";
// Round remaining to nearest integer as you cannot have fraction of a muon usually,
// but showing 2 decimals helps visualize the math. Let's do integer for physical count.
document.getElementById('res-remaining').innerText = Math.round(remaining).toLocaleString();
document.getElementById('results-area').style.display = 'block';
}
Understanding Muon Decay and Time Dilation
The Muon Decay Rate Calculator is designed to solve problems related to particle physics, specifically analyzing how the instability of muons interacts with Special Relativity. Muons are unstable elementary particles similar to electrons but much heavier. Because they are unstable, they decay into other particles (usually an electron and two neutrinos) over time.
The Physics of Muon Decay
At rest, a muon has a mean lifetime ($\tau$) of approximately 2.197 microseconds ($\mu s$). This implies that if you have a group of muons at rest, after 2.197 $\mu s$, the population will have dropped to $1/e$ (approximately 36.8%) of its original size. The decay follows the exponential radioactive decay law:
$$N(t) = N_0 e^{-\frac{t}{\tau}}$$
Where $N(t)$ is the remaining number of particles, $N_0$ is the initial quantity, and $t$ is the time elapsed.
Relativistic Time Dilation
The calculation becomes fascinating when muons move at speeds close to the speed of light ($c$). According to Einstein's Theory of Special Relativity, time passes slower for a moving object relative to a stationary observer. This phenomenon is known as Time Dilation.
For a muon moving at velocity $v$, the observed mean lifetime ($\tau_{lab}$) increases by the Lorentz factor ($\gamma$). The formulas used in this calculator are:
Consequently, high-speed cosmic ray muons created in the upper atmosphere can reach the Earth's surface before decaying, something that would be impossible based on their rest lifetime alone.
Example Calculation
Consider a beam of 1,000 muons traveling at 0.99c (99% the speed of light).
Calculate Decay after 10 $\mu s$: If we observe them after 10 microseconds, the remaining fraction is $e^{-10 / 15.57} \approx 0.526$.
Therefore, approximately 526 muons would remain. Without time dilation, the calculation would use the rest lifetime ($2.197 \mu s$), resulting in only about 10 muons remaining ($e^{-10/2.197}$). This drastic difference serves as experimental proof of relativity.
How to Use This Calculator
1. Enter Initial Muons: The starting count of particles in your sample.
2. Enter Velocity: The speed as a fraction of light speed ($c$). For stationary muons, enter 0. For high energy muons, enter a value like 0.98.
3. Enter Time Elapsed: The time duration measured in the laboratory frame in microseconds.
4. Result: The calculator outputs the relativistic factor, the adjusted lifetime, and the number of surviving particles.