Okay, let's try to figure out this problem. So, the question is about particles produced in a spherical detector. The particles are moving at ultra-relativistic velocities, which means their speeds are very close to the speed of light. The detector has a radius of 30 meters. Initially, with a Lorentz factor of about 20, one-third of these fast-decaying particles reach the detector's inner walls. The question is asking what Lorentz factor is needed so that about two-thirds of the particles reach the detector.

Hmm, Lorentz factor, gamma, is a key concept here. The Lorentz factor is given by gamma = 1 / sqrt(1 - v²/c²), where v is the velocity of the particle and c is the speed of light. Since the particles are moving at ultra-relativistic speeds, gamma is going to be much larger than 1. Here, gamma is initially 20, and we need to find the new gamma that results in two-thirds of the particles reaching the detector instead of one-third.

Wait, so the key here is probably the distance the particles travel before decaying. If the particles are moving at a certain Lorentz factor, their lifetime from our (lab) frame's perspective is longer due to time dilation. But the problem mentions that the particles reach the detector inner walls. So, maybe the distance they travel before decaying is related to the radius of the detector?

Let me think. The particles are produced at the center of the spherical detector. The detector's radius is 30 meters. So, the particles need to travel a certain distance to reach the inner walls. If the Lorentz factor is higher, the particles would be moving faster, but their lifetime is longer. Wait, no, time dilation says that the proper time (time in the particle's rest frame) is shorter, but from our frame, the lifetime is gamma times longer. So, the distance a particle travels before decaying is v * tau, where tau is the proper lifetime. But in our frame, tau is gamma times the proper lifetime. Wait, maybe I need to clarify.

Alternatively, perhaps the proper lifetime is tau, and the distance the particle travels before decaying in our frame is gamma * v * tau. But since v is close to c, that's approximately gamma * c * tau. But how does that relate to reaching the detector walls?

Wait, the problem states that the particles are produced at the center and travel uninterrupted. So, the particles' travel path is radial. The detector is a sphere, so the inner walls are at 30 meters from the center. So, the particles need to travel 30 meters to reach the walls. If they don't decay before that, they reach the walls. If they decay before that, they don't.

But the problem says that with gamma ~20, only one-third of the particles reach the detector. So, two-thirds decay before reaching the walls. So, the decay length (the average distance a particle travels before decaying) must be such that, in the lab frame, the average decay length is 20 * c * tau (since gamma is 20, and v ≈ c). Wait, but tau is the proper lifetime. So, the average decay length in the lab frame is gamma * v * tau ≈ gamma * c * tau.

So, if the average decay length is L = gamma * c * tau, then when L is greater than the detector's radius (30 meters), the particles have a higher chance of reaching the walls. Wait, but the problem states that with gamma=20, one-third reach the walls. So, when L is 30 meters, the probability of reaching is 1/3. Then, when we want two-thirds to reach, we need L to be such that the probability is 2/3. Hmm, maybe the decay length is inversely related to the probability?

Wait, actually, the decay follows an exponential distribution. The probability that a particle decays after a certain time t is given by the exponential decay law. The decay length would then be the average distance traveled before decay, which is gamma * v * tau ≈ gamma * c * tau. But the probability that the particle does not decay (i.e., reaches the detector) is the survival probability, which is exp(-x / L), where x is the distance traveled, and L is the mean decay length.

So, if the survival probability is P = exp(-x / L), then for x = 30 meters, when gamma=20, P = 1/3. So, exp(-30 / L_initial) = 1/3. Then, solving for L_initial, we get L_initial = -30 / ln(1/3) ≈ 30 / ln(3) ≈ 30 / 1.0986 ≈ 27.3 meters. Wait, but L_initial is gamma_initial * c * tau. Since gamma_initial is 20, then L_initial = 20 * c * tau. So, 20 * c * tau ≈ 27.3 meters. Therefore, c * tau ≈ 27.3 / 20 ≈ 1.365 meters. So, the proper lifetime tau is approximately 1.365 / c seconds. Wait, but c is about 3e8 meters per second, so tau ≈ 1.365 / 3e8 ≈ 4.55e-9 seconds. That seems plausible for a particle with such a high gamma.

Now, the question is asking for the gamma that would result in two-thirds of the particles reaching the detector. So, the survival probability needs to be 2/3. So, exp(-30 / L_final) = 2/3. Solving for L_final: L_final = -30 / ln(2/3) ≈ 30 / (-ln(2/3)) ≈ 30 / (-(-0.4055)) ≈ 30 / 0.4055 ≈ 74.1 meters. So, L_final needs to be approximately 74.1 meters.

But L_final is gamma_final * c * tau. We already found that c * tau ≈ 1.365 meters. So, gamma_final * 1.365 ≈ 74.1. Therefore, gamma_final ≈ 74.1 / 1.365 ≈ 54.1. So, approximately 54. Therefore, the correct answer would be A. 54.

Wait, let me check the calculations again to be sure. Starting with gamma=20, P=1/3. So, exp(-30 / L) = 1/3. Taking natural log of both sides: -30/L = ln(1/3) ≈ -1.0986. So, 30/L = 1.0986 → L = 30 / 1.0986 ≈ 27.3 meters. So, L = gamma * c * tau. Therefore, gamma * c * tau = 27.3. Then, for gamma_final, we need L_final = gamma_final * c * tau. And we need exp(-30 / L_final) = 2/3. So, -30 / L_final = ln(2/3) ≈ -0.4055. Therefore, 30 / L_final = 0.4055 → L_final = 30 / 0.4055 ≈ 74.1 meters. Then, since L_final = gamma_final * c * tau, and gamma_initial * c * tau = 27.3, we can write gamma_final = (L_final / L_initial) * gamma_initial. So, gamma_final = (74.1 / 27.3) * 20 ≈ (2.714) * 20 ≈ 54.28. So, approximately 54, which is option A.

Alternatively, another approach: since the decay length L is proportional to gamma. So, if we want L_final = 74.1 and L_initial = 27.3, then gamma_final / gamma_initial = L_final / L_initial. Therefore, gamma_final = gamma_initial * (L_final / L_initial) = 20 * (74.1 / 27.3) ≈ 20 * 2.714 ≈ 54.28. Same result.

So, the answer should be A. 54.
Therefore, the correct answer is $\boxed{A}$.
</think>