Nuclear Physics Archives - Time Travel, Quantum Entanglement and Quantum Computing

Gamow’s Calculation of Alpha Decay – WKB Method

anuj — Fri, 29 Aug 2025 03:46:55 +0000

Alpha Decay Explained via Gamow and the WKB Method

Alpha decay, the emission of a helium nucleus ($ \alpha $-particle) from a heavy nucleus, was initially mysterious because classically the alpha particle’s energy is insufficient to overcome the Coulomb barrier. Gamow (1928) used quantum mechanics and the concept of tunneling to explain how alpha particles escape.

1. Nuclear Potential and Physical Setup

Consider an alpha particle inside a nucleus of charge $ Z $ and mass number $ A $, leaving a daughter nucleus with charge $ Z_d = Z-2 $. The radial potential is:

Inside the nucleus: attractive nuclear potential $ V(r) \approx -V_0 $.
Outside the nucleus: Coulomb potential $ V(r) = \dfrac{Z_d Z_\alpha e^2}{4\pi \varepsilon_0 r} $, where $ Z_\alpha = 2 $.

The alpha particle energy $ E $ is much less than the barrier, so classical escape is impossible.

2. Schrödinger Equation and WKB Approximation

For radial motion ($ l = 0 $), the Schrödinger equation is:

\[
-\frac{\hbar^2}{2m} \frac{d^2 u}{dr^2} + V(r) u = E u, \quad u(r) = r \psi(r)
\]
\[
k(r) = \frac{\sqrt{2m |E-V(r)|}}{\hbar}, \quad
\text{allowed: } E>V \Rightarrow k = \frac{\sqrt{2m(E-V)}}{\hbar}, \quad
\text{forbidden: } E \]
\[
\text{Turning points: } r_1 \approx R, \quad r_2 = \frac{Z_d Z_\alpha e^2}{4\pi \varepsilon_0 E}
\]

WKB gives the tunneling probability:

\[
T \approx \exp\left[-2 \int_{r_1}^{r_2} \kappa(r) \, dr \right] = \exp[-2G], \quad
G \approx \pi \eta – k R
\]
\[
\eta = \frac{Z_d Z_\alpha e^2}{\hbar v}, \quad
k = \frac{\sqrt{2 m E}}{\hbar}, \quad
R = r_0 A^{1/3}, \quad r_0 = 1.2 \text{ fm}
\]
\[
v = \sqrt{\frac{2 E}{m_\alpha}}, \quad
\lambda = P_\alpha \nu T, \quad
\nu \approx \frac{v}{2 R}, \quad
T_{1/2} = \frac{\ln 2}{\lambda}
\]

3. Geiger–Nuttall Law

The WKB result yields the empirical relation:

\[
\log_{10} T_{1/2} \approx a \frac{Z_d}{\sqrt{E_\alpha}} + b
\]

It shows the strong exponential sensitivity of half-life to alpha particle energy.

4. Numeric Examples

Constants used:

$ \hbar c = 197.327~\mathrm{MeV \cdot fm} $
$ e^2/(4\pi \varepsilon_0) = 1.440~\mathrm{MeV \cdot fm} $
$ m_\alpha c^2 = 3727.379~\mathrm{MeV} $
Assumed $ r_0 = 1.2~\mathrm{fm} $

Derived quantities and half-lives for two isotopes
Isotope	Z	A	E_α (MeV)	Z_d	R (fm)	v/c	η	k (fm⁻¹)	G	T = e^{-2G}	ν (Hz)	λ (Pα=1) (s⁻¹)	T₁/₂ (Pα=1) (s)	Pα required to match experiment
^{238}U	92	238	4.267	90	7.437	0.04785	27.451	0.9038	79.519	8.518×10⁻⁷⁰	9.645×10²⁰	8.215×10⁻⁴⁹	8.437×10⁴⁷	≈ 5.98×10³⁰
^{212}Po	84	212	8.954	82	7.155	0.06931	17.266	1.3093	44.874	1.055×10⁻³⁹	1.452×10²¹	1.531×10⁻¹⁸	4.526×10¹⁷	≈ 1.51×10²⁴

Notes and Interpretation

The simple WKB/Gamow model with P_α = 1 greatly overestimates half-lives compared with experiment.
Experimental half-lives:
- ^{238}U: 4.468×10⁹ yr ≈ 1.41×10¹⁷ s
- ^{212}Po: 2.99×10⁻⁷ s
The preformation probability $P_α$ required to match experiments is extremely large in this naive calculation.
Gamow’s major contribution was explaining the functional form of the Geiger–Nuttall law: $ \log_{10} T_{1/2} \propto Z_d / \sqrt{E_\alpha} $.

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Some facts and consequences of the binding energy curve

anuj — Sun, 01 Sep 2024 13:30:49 +0000

The binding energy curve is a graph that shows the binding energy per nucleon as a function of the mass number (A) of atomic nuclei. This curve has significant implications in nuclear physics and helps explain various nuclear phenomena.

Some striking facts and consequences:

### 1. **Peak at Iron-56 (Fe-56):**
– **Fact:** The binding energy per nucleon reaches a maximum around Iron-56 (Fe-56), which is one of the most stable nuclei in nature.
– **Consequence:** This implies that energy release is possible both through the fusion of lighter nuclei and the fission of heavier nuclei, leading to phenomena such as stellar nucleosynthesis (fusion in stars) and nuclear power (fission in reactors).

### 2. **Fusion of Light Nuclei:**
– **Fact:** For elements lighter than Iron (A < 56), the binding energy per nucleon increases as nuclei fuse together.
– **Consequence:** Fusion of light elements, such as hydrogen into helium, releases large amounts of energy. This is the fundamental process powering stars, including the Sun, and is the principle behind hydrogen bombs and potential future fusion reactors.

### 3. **Fission of Heavy Nuclei:**
– **Fact:** For elements heavier than Iron (A > 56), the binding energy per nucleon decreases as nuclei become more massive.
– **Consequence:** Heavy nuclei, like Uranium-235 or Plutonium-239, can release energy through fission, where the nucleus splits into smaller nuclei. This process is utilized in nuclear reactors and atomic bombs.

### 4. **Nucleosynthesis and the Origin of Elements:**
– **Fact:** The binding energy curve explains why elements heavier than Iron cannot be formed by fusion in ordinary stars; their formation requires supernovae or neutron star collisions.
– **Consequence:** The distribution of elements in the universe, with lighter elements like hydrogen and helium being most abundant, and heavier elements being rarer, can be explained by the processes governed by the binding energy curve.

### 5. **Radioactive Decay and Stability:**
– **Fact:** Nuclei far from the peak of the binding energy curve tend to be less stable and more likely to undergo radioactive decay.
– **Consequence:** This underpins the principles of nuclear decay chains, where unstable isotopes decay until they reach a more stable configuration with higher binding energy per nucleon.

### 6. **Energy Release in Nuclear Reactions:**
– **Fact:** The difference in binding energy before and after a nuclear reaction determines the energy released or absorbed.
– **Consequence:** This allows for the calculation of the energy yield of nuclear reactions, critical for both nuclear power generation and understanding stellar processes.

### 7. **Nuclear Stability and the Valley of Stability:**
– **Fact:** The binding energy curve and related concepts help define the “valley of stability,” where stable nuclei lie.
– **Consequence:** Nuclei outside this valley are prone to various decay modes (alpha, beta, gamma) as they seek a more stable configuration. This explains the existence of radioactive isotopes and their decay pathways.

The binding energy curve is central to understanding the energy dynamics in nuclear reactions, the life cycle of stars, and the origin of elements, making it one of the foundational concepts in nuclear physics.

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Superluminal Potentials in Quantum Physics

anuj — Sun, 30 Jun 2024 05:33:51 +0000

The paper “Quantum Field Derivation of the Superluminal Schrödinger Equation and Deuteron Potential” by E.J. Betinis, published in Physics Essays in 2002, explores the theoretical foundations and implications of superluminal quantum mechanics. Here’s a summary:

Abstract:

The paper builds on the author’s previous work on the superluminal Schrödinger equation, which addresses kinetic energy forms that do not become singular at the speed of light.
It re-derives this equation using quantum field theory approaches, including constructing Lagrangian and Hamiltonian densities.
The paper solves the superluminal Schrödinger equation for eigenfunctions and iteratively finds the superluminal potential for the deuteron.
The iterative method shows convergence, yielding a potential similar to subluminal potentials, supporting the validity of the superluminal theories.
The study implies that particles within the nucleus may exceed the speed of light, challenging traditional physics boundaries.

Key Points:

Introduction:
- The nuclear force is treated analogously to the electrostatic force in Yukawa’s theory, focusing on spherical symmetry to find superluminal eigenfunctions and potentials.
- The study proposes a boson, with mass equal to the deuteron’s reduced mass, is exchanged between nucleons, leading to the nuclear force and potential.
Superluminal Schrödinger Equation:
- The superluminal form of kinetic energy does not become singular at $v = c$ and increases indefinitely as velocity increases.
- The paper shows that Lagrangian and Hamiltonian densities can re-derive this superluminal Schrödinger equation via quantum field theory.
Eigenfunctions and Potentials:
- The spherically symmetric superluminal Schrödinger equation is solved for eigenfunctions.
- An iterative Laplace transform method finds the superluminal potential for the deuteron, with convergence observed after the fourth iteration.
Comparison with Subluminal Potentials:
- The superluminal potentials closely resemble those found by subluminal approaches, like the Reid potential.
- The potentials have a “hard” core nature, indicating nucleons are not point particles but have a finite size.
Implications for Nuclear Physics:
- The superluminal approach suggests the existence of particles moving faster than light within the nucleus.
- If experimentally verified, this challenges the speed-of-light limitation in other physics branches.

Conclusion:

The paper demonstrates the feasibility of constructing a superluminal Schrödinger equation through quantum field theory.
The derived potentials support the concept of superluminal interactions in nuclear physics.
This work opens the possibility for future studies and experimental verification of faster-than-light particles in the nucleus.

Overall, Betinis’ work extends quantum mechanics into the superluminal regime, providing theoretical tools and results that could reshape our understanding of nuclear forces and particle physics.

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