Modeling Polonium-212 Alpha Half-life

The incredible range of alpha decay half-lives can be modeled with quantum mechanical tunneling. The illustration represents the barrier faced by an alpha particle in polonium-212, which emits an 8.78 MeV alpha particle with a half-life of 0.3 microseconds. The following characteristics of the nuclear environment can be calculated from a basic model of the nucleus: α衰变的惊人范围可以用量子力学隧道效应来建模。图中表示了钋-212核中α粒子面临的势垒，该核发射出一个能量为8.78 MeV的α粒子，其半衰期为0.3微秒。以下核环境的特性可以从核的基本模型中计算得出： Separation of centers of alpha and nucleus at edge of barrier α粒子与核在屏障边缘的分离 9.1 fm 9.1 霍尔效应 Height of barrier 势垒高度 26.4 MeV Radius at which barrier drops to alpha energy 势垒降至α能时的半径 26.9 fm 26.9 fm（专名或术语） Width of barrier seen by alpha α粒子看到的势垒宽度 17.9 fm 17.9 fm（专名或术语） Alpha's frequency of hitting the barrier Alpha 的击穿屏障频率 1.1 x 1021/s Details of calculation 计算细节 Different isotopes 不同同位素

The shape of the barrier must obviously be taken into account since it drops rapidly. But it is instructive to calculate the half-life for a rectangular barrier of that height and width. Using the barrier penetration calculation gives a half-life of 1.5 x 10⁷ seconds, about 13 orders of magnitude longer than the observed half-life. This gives some insight into the large range of alpha half-lives; the tunneling probability depends very strongly on the nature of the barrier. The probability depends exponentially on the height and width, and in this case the height is dropping like 1/r with distance from the nucleus.

A better tunneling probability can be obtained by breaking the barrier into segments and multiplying the successive tunneling probabilities according to the basic rules for combining probabilities. The calculated tunneling probabilities for the segments are shown in the table. 通过将势垒分成若干段，并根据概率组合的基本规则将各段的隧穿概率相乘，可以得到更准确的隧穿概率。各段的计算隧穿概率如表所示。 Segment height 段高 Probability of tunneling 隧穿概率 21.9 1.20 x10-5 16.4 1.74 x 10-4 13.2 1.43 x 10-3 11.0 .98 x 10-2 9.4 .85 x 10-1 Product of probabilities 概率的乘积 2.47 x 10-15 Details of calculation 计算细节

When this tunneling probability is used to calculate the half-life, a value of 0.25 microseconds is obtained. This is surprisingly close to the measured value of 0.3 microseconds, considering that an incorrect barrier can take you off many orders of magnitude. It should be admitted that the same calculation for other nuclei does not yield such close agreement, but this example shows that the observed half-lives are within the range which can be calculated from quantum mechanical tunneling.

What could you do to change the polonium half-life?

For the modeling of the alpha decay half-life of polonium-212 with the tunneling probability, a value of 0.25 microseconds was obtained. This is surprisingly close to the measured value of 0.3 microseconds, considering that an incorrect barrier can take you off many orders of magnitude. It suggests that the major factors affecting such alpha decay have been accounted for in a reasonable manner.

What could you do to change the polonium half-life? The answer is that there is no way to experimentally change that parameter because it ultimately depends upon the values of fundamental physical constants. The illustration below points out that the characteristics which affect the barrier and the tunneling probability are:

1. The nuclear radius and the depth of the nuclear potential well, which depend upon the coupling strength for the nuclear strong force.
   核子半径和核强相互作用势阱的深度，这取决于核强相互作用的耦合强度。
2. The electric repulsion, which depends upon Coulomb's constant, which in turn depends upon the speed of light c.
   电斥力取决于库仑常数，而该常数又取决于光速c。
3. The tunneling probability, which depends upon Planck's constant.
   隧穿概率，这取决于普朗克常数。

The forces inside the nucleus are balanced on a razor's edge. The constants that determine nuclear decay rate are stitched tightly into the fabric of the cosmos. The decay rate can't be changed without changing essentially all of chemistry and physics. 原子核内部的力处于刀刃般的平衡状态。决定核衰变率的常数紧密地编织在宇宙的织物中。要改变衰变率，就必须根本改变化学和物理学。