Calculate G from the data for the β-transition ^10 C → ^10 B * e^+ ν. The measured half-life i

Calculate G from the data for the β-transition ^10 C → ^10...

Key Concepts

Half-life and Decay Constant Relation

The half-life of a radioactive process is inversely related to the decay constant, which quantifies how rapidly the process occurs. In beta decay, the relationship t_1/2 = (ln 2)/? is used, where ? is the decay constant expressed in terms of the weak coupling constant and the phase space factor. This relationship is essential when using experimental measurements of the half-life to determine underlying parameters such as G from theoretical models.

Weak Interaction Coupling Constant

The weak interaction coupling constant, often denoted as G_F, is a fundamental parameter that measures the strength of the weak force. In beta decay, the decay rate is proportional to the square of this coupling constant. Determining G from beta decay data involves relating experimental observables such as half-life and endpoint energy to theoretical decay rate formulas that incorporate G_F and the phase space factor.

Phase Space Factor

The phase space factor in beta decay quantifies the number of final states available to the emitted particles (the beta particle and the neutrino) as a function of energy. It is determined by the endpoint energy of the decay and plays a crucial role in calculating the decay rate. The integration over the phase space factor, which depends strongly on the available energy (Q value), directly affects the observed half-life of the decay process.

Superallowed Fermi Transition

Superallowed Fermi transitions occur in beta decay when the initial and final nuclear states have the same spin and parity (typically 0+ ? 0+), resulting in a very simple, highly symmetric transition. In these transitions, the nuclear matrix element is known and nearly constant, which allows for precise determinations of weak interaction parameters. The simplicity of the transition arises because only the vector part of the weak current contributes, making them an important testing ground for the Standard Model.

Beta Decay

Beta decay is a type of radioactive decay mediated by the weak force in which a nucleus emits a beta particle (an electron or positron) along with an associated neutrino or antineutrino. This process converts a neutron into a proton or vice versa, changing the nuclear composition and often leading to a more stable configuration. The description of beta decay involves the use of Fermi’s golden rule to relate the transition rate to the square of the weak coupling constant and the phase space available to the emitted particles.

Transcript

00:01 So in this question we are going to use the relationship between decay constant and half -life and the exponential decay rate equation to calculate the unknowns.

00:14 So it is given to us that the decay constant of uranium 235 is some number.

00:25 So the decay constant lambda of uranium 235 is given to us as 9.

00:32 8 times 10 to the power minus 10 per year or year inverse and the first part asks us to calculate the half life of uranium 235 so the half life is related to the decay constant by this formula t half the half life equals ln2 log to the base e of 2 divided by the lambda and if you plug in the numbers for lambda, you'll get that the half -life is 7 .07 times 10 to the power 8.

01:24 Here, this is the answer to the first part.

01:36 The second part, what is the disintegration per second of one microgram of uranium 235? so the molecular mass of uranium 235 is 235 grams, implying that.

01:54 235 gram of uranium 235 will have one mole of uranium 235.

02:09 Just writing it out another way.

02:12 So 235 gram of uranium 235 will have 6 .0233 times 10 to the power 23 atoms of uranium 235 the avogadro number and this would imply that one gram of uranium 235 would have 6 .023 times 10 to the power 23 divided by 23 atoms of uranium 235 and we are given that the sample will have one microgram so 10 to the power minus 6 gram is 1 microgram.

03:17 So 10 to the power minus 6 grams of uranium 235 would have 6 .023 times 10 to the power 23 divided by 235 and multiplied by 10 to the power minus 6 atoms of uranium 235.

03:42 And if you do this calculation, you'll get that this number, the right side, is 2 .56 times 10 to the power 15 atoms of uranium 235.

04:04 So one microgram of uranium 235 will have these many atoms of uranium 235...

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