Thinking in Waves Episode 19 ／ Part VIII · Differentiating the effective constants

Measuring a half-life longer than the universe, in one second

A decay constant is an imaginary frequency ──
so no full period is needed Episode 3 said a component with period longer than the age of the universe is indistinguishable from a constant.
So what about $^{232}$Th, with a half-life of 14.05 Gyr? (The universe is 13.79 Gyr old.)
★ The answer ── one gram gives 4057 counts a second. One second is enough.

Prerequisites: Episodes 3, 4, 6, 9, 11, 13Computation: kenshou/jikan.js

This looks like a head-on counterexample to the series' observation-window argument.
It is not ── and why it is not is the interesting part.

01What sits near the age of the universe?

Nuclide	Half-life [yr]	Ratio to the universe's age
$^{235}$U	$7.04\times10^8$	0.051
$^{40}$K	$1.248\times10^9$	0.091
$^{238}$U	$4.468\times10^9$	0.324
★ age of the universe	$1.379\times10^{10}$	1.000
$^{232}$Th	$1.405\times10^{10}$	1.019
$^{187}$Re	$4.16\times10^{10}$	3.02
$^{87}$Rb	$4.97\times10^{10}$	3.61
$^{147}$Sm	$1.06\times10^{11}$	7.69

$^{232}$Th's half-life and the age of the universe differ by a factor of 1.02 ── they essentially coincide.
So are these indistinguishable from constants?

02The crux ── decay constants live on the imaginary axis

Write oscillation and decay in the same form $$\text{oscillation}:\ e^{i\omega t}\quad(\omega\ \text{real}),\qquad \text{decay}:\ e^{-\lambda t}=e^{i(i\lambda)t}\quad(\omega=i\lambda\ \text{purely imaginary})$$

Axis	Behaviour	How you measure it	Observation time needed
Real axis $\omega$	oscillates	watch the phase go round once	★ of order one period
Imaginary axis $i\lambda$	falls monotonically	count the slope (i.e. the decays)	★ as short as you like

The heart of §02 ── the main finding of this episode

An oscillation's frequency is unknown until a full period has passed.
A decay's rate is known in an instant, because you can count.

A decay constant sits on the imaginary axis of the complex frequency plane ── the same family as Episode 11's diffusion ($\omega=-iDk^2$).

★ Episode 3's "$\omega=0$ is out of reach" was a statement about periodic components. It does not apply to monotone decay.

Take one gram of $^{232}$Th $$N=\frac{N_A}{232.04}=2.595\times10^{21},\qquad \lambda=\frac{\ln2}{T_{1/2}}=1.563\times10^{-18}\ \text{s}^{-1}$$ $$A=\lambda N=\mathbf{4057}\ \text{Bq/g}\qquad\text{★ 4057 decays every second}$$

Conclusion of §02

A substance whose half-life exceeds the age of the universe decays 4000 times a second in one gram.
One second of counting fixes the decay constant to 1.6 % ($\sqrt{4057}/4057$); one day gives 0.017 %.

Put the same number on the real axis and it is an oscillation of period $4.4\times10^{17}$ s ＝ 14 Gyr. Not even one cycle is visible.

★ The same $1.6\times10^{-18}\,\mathrm{s^{-1}}$: unmeasurable on the real axis, one second on the imaginary axis.

03Why it works ── a time average replaced by an ensemble average

Sample	Number of atoms	Decays per second	Wait for one
one atom	1	$1.56\times10^{-18}$	20.3 Gyr
1 ng	$2.60\times10^{12}$	$4.06\times10^{-6}$	2.8 days
1 mg	$2.60\times10^{18}$	4.06	246 ms
1 g	$2.60\times10^{21}$	4057	246 µs
1 kg	$2.60\times10^{24}$	$4.06\times10^6$	246 ns

§03 ── the inverse of Episode 9

Episode 9 said: look for a short time and the frequency blurs.
Here we do the inverse ── instead of watching one system for a long time, watch many systems at once.

★ A time average replaced by an ensemble average.
You cannot do this for an oscillation (the phases are scrambled and cancel). You can for decay, because each nucleus is independent and carries no phase.

── ★ The real axis has phase; the imaginary axis does not. That is why only the imaginary axis lets you buy time by increasing numbers.

04The ladder of half-lives ── 65 orders

Decay	Half-life [s]	$\Gamma=\hbar/\tau$ [eV]	Ratio to the universe's age
$\Delta(1232)$ resonance	$5.6\times10^{-24}$	$8.1\times10^{7}$	$1.3\times10^{-41}$
$\pi^0$	$8.4\times10^{-17}$	5.4	$1.9\times10^{-34}$
$\mu$	$2.2\times10^{-6}$	$2.1\times10^{-10}$	$5.1\times10^{-24}$
neutron	878.4	$5.2\times10^{-19}$	$2.0\times10^{-15}$
$^{14}$C	$1.8\times10^{11}$	$2.5\times10^{-27}$	$4.1\times10^{-7}$
$^{232}$Th	$4.4\times10^{17}$	$1.0\times10^{-33}$	1.02
$^{128}$Te ($2\nu\beta\beta$)	$6.9\times10^{31}$	$6.6\times10^{-48}$	$1.6\times10^{14}$
proton decay (lower bound)	$5.0\times10^{41}$	$9.0\times10^{-58}$	$1.2\times10^{24}$

§04 ── the sharpest "line" in nature

Episode 9 called $^{57}$Fe's $\Gamma/E=3.24\times10^{-13}$ extraordinarily sharp.
$^{232}$Th's relative width is ── $\Gamma/Q=2.52\times10^{-40}$. Twenty-seven orders sharper.

★ But that sharpness is unusable. To drive it as a resonance you would need an equally precise source.
So you count instead ── ★ the real axis is for striking; the imaginary axis is for counting.

05Only things near the age of the universe can be clocks

A dating nuclide has a strict requirement ── too fast and none is left; too slow and none has gone.

Computing the optimum half-life $$\delta t=\frac{1}{\lambda}\cdot\frac{1}{\sqrt N}=\frac{e^{\lambda t/2}}{\lambda\sqrt{N_0}}\quad\Longrightarrow\quad \frac{d}{d\lambda}\Bigl[\frac{\lambda t}{2}-\ln\lambda\Bigr]=0\quad\Longrightarrow\quad \lambda=\frac2t$$ $$T_{1/2}^{\rm opt}=\frac{\ln2\cdot t}{2}=\mathbf{4.78\ Gyr}$$

Nuclide	Half-life [Gyr]	Fraction left after 13.8 Gyr	Ratio to optimum
$^{235}$U	0.704	$1.3\times10^{-6}$	0.147
$^{40}$K	1.248	$4.7\times10^{-4}$	0.261
$^{238}$U	4.468	0.118	★ 0.935
$^{232}$Th	14.05	0.507	2.94
$^{87}$Rb	49.7	0.825	10.4

§05 ── $^{238}$U is the ideal clock to within 7 %

$^{238}$U (4.468 Gyr) is 0.935 of the optimum 4.78 Gyr.

★ A coincidence, of course. Nature did not choose it for us.
But it does explain why geology uses $^{238}$U ── what happened to be usable happened to be there.

Verdict: this optimisation is ours and is idealised (real dating measures parent–daughter ratios by mass spectrometry, a different setting).

06Is a decay constant really constant?

Claim / phenomenon	Verdict	Content
Annual modulation (claimed from 2008)	× largely refuted	not reproduced underground or in temperature-controlled setups; environmental effects favoured
Electron capture in $^{7}$Be	○ real	changes by 0.1–1 % with chemical environment (endohedral $\mathrm{C_{60}}$ etc.)
Fully ionised $^{187}$Re	◎ real	★ 41.6 Gyr → 32.9 yr ($1.3\times10^9$×)
Dependence on $\alpha$	◎ theory	★ alpha decay depends exponentially (§07)

§06 ── a decay constant is not a property of the nucleus alone

The most dramatic case is $^{187}$Re. Neutral, its half-life is 41.6 Gyr. Strip every electron and it is 32.9 years ── a factor of $1.3\times10^9$.
Because a new destination opens: bound-state beta decay.

★ In wave language ── change the density of final-state modes and you change the transition rate.
Exactly the same logic as putting an atom in a cavity and changing its spontaneous emission (the Purcell effect).

So a "decay constant" is a property of the whole system, not a label stuck to a nucleus. The same structure as Episode 17's "mass changes with the environment".

07Alpha decay is an ultra-sensitive sensor for $\alpha$

The Gamow factor acts exponentially $$\lambda\propto e^{-2\pi\eta},\qquad \eta=\frac{Z_dZ_\alpha\,\alpha\,c}{v}\qquad\Longrightarrow\qquad \frac{d\ln\lambda}{d\ln\alpha}=-2\pi\eta$$

Parent	Daughter $Z$	$Q$ [MeV]	$v/c$	$\eta$	$-2\pi\eta$
$^{238}$U	90	4.270	0.04787	27.44	$-172.4$
$^{232}$Th	88	4.082	0.04680	27.44	$-172.4$
$^{226}$Ra	86	4.871	0.05112	24.55	$-154.3$
$^{210}$Po	82	5.407	0.05386	22.22	$-139.6$

$-172$ for $^{238}$U ── change $\alpha$ by 0.1 % and the decay rate changes by 17 %.
Episode 4 quoted $|\dot\alpha/\alpha|<10^{-17}$/yr; alpha decay is one of the most sensitive tools for obtaining such bounds.

08Why the weak force is weak ── the cutoff, not the coupling

Compare the couplings first $$\alpha_{\rm em}(m_Z)=\frac{1}{127.95}=0.00782,\qquad \alpha_W=\frac{\alpha_{\rm em}}{\sin^2\theta_W}=0.0338=\frac{1}{29.6}$$

★ The weak coupling is 4.3 times stronger than the electromagnetic one

Momentum transfer $q$	weak / electromagnetic	Where
1 MeV	$6.7\times10^{-10}$	nuclear beta decay
1 GeV	$6.7\times10^{-4}$	inside a nucleon
10 GeV	$6.6\times10^{-2}$
80.4 GeV ($M_W$)	2.16	★ equal strength ── electroweak unification
1 TeV	4.30

§08 ── three stories converge on one formula

What is weak is the propagator, not the coupling. The amplitude $\propto g^2/(q^2-M_W^2)$ collapses to $g^2/M_W^2$ at low energy.

In Episode 6's language ── $M_W$ is a cutoff frequency, with range $\hbar/(M_Wc)=2.46\times10^{-3}$ fm. Below the cutoff the field is evanescent and decays exponentially.

★ And as Episode 18 showed, that cutoff is made by the Higgs condensate.

"The weak force is weak", "a superconductor expels magnetic field" and "the proton is heavy" ──
all three are the same story: an intercept in a dispersion relation.

09Getting the muon lifetime out of $G_F$

Confirming that the weak decay constant really does the work $$\Gamma_\mu=\frac{G_F^2m_\mu^5}{192\pi^3}\quad\Longrightarrow\quad \tau_\mu=2.187\ \text{µs}\qquad(\text{measured}\ 2.197\ \text{µs},\ \mathbf{0.44\,\%\ agreement})$$

Note the strong dependence $\tau\propto1/m^5$. Try it on the tau.

What the tau lifetime tells us $$(m_\tau/m_\mu)^5=1.345\times10^6\ \Rightarrow\ \tau_\tau^{\rm predicted}=1.633\times10^{-12}\ \text{s}\quad(\text{if there were one channel})$$ $$\tau_\tau^{\rm measured}=2.903\times10^{-13}\ \text{s}\qquad\Longrightarrow\qquad \text{ratio}=\mathbf{5.63}$$

§09 ── the tau lifetime gives the number of quark colours

It decays 5.6 times too fast ── so there are 5.6 channels. Count them:

$\tau\to e\nu\nu$: 1　／　$\tau\to\mu\nu\nu$: 1　／　$\tau\to$ hadrons ($u\bar d$): ★ 3, because there are three colours

total $=2+3\times(1+\alpha_s/\pi)=2+3\times1.105=\mathbf{5.32}$　　measured $=\mathbf{5.63}$

★ Agreement to 6 %. The number of quark colours, 3, falls out of the tau's lifetime.
Verdict: a reproduction of a classic argument, not a new claim (ignoring the Cabibbo angle and cuts sets the accuracy).

10The neutron's lifetime set the composition of the universe

Finally, the mark the weak decay constant left on the cosmos. Big Bang nucleosynthesis goes: ① weak reactions freeze out at $T\approx0.7$ MeV ($t\approx1$ s) leaving $n/p\approx1/6$; ② neutrons decay for a few hundred seconds; ③ once the deuterium bottleneck opens, every surviving neutron ends up in $^4$He.

$\tau_n$ [s]	$n/p$ after 250 s	Helium mass fraction $Y_p$
220 (1/4×)	0.0534	0.101
439 (1/2×)	0.0943	0.172
878.4 (measured)	0.1254	0.223
1757 (2×)	0.1446	0.253
$\infty$ (no decay)	0.1667	0.286

§10 ── the weak decay constant fixes the universe's helium

Our rough estimate gives $Y_p=0.223$ against an observed 0.245 ── 10 % off (we rounded the freeze-out temperature and $t_{\rm nuc}$; the real calculation matches to 0.5 %).

★ What matters is the sensitivity ── a quarter of the neutron lifetime gives $Y_p=0.10$, twice gives 0.25.

And that the neutron's 878 s and nucleosynthesis' $\sim250$ s are the same order is what makes the story possible.
Ten times shorter and every neutron would be gone, leaving a hydrogen-only universe.

Exercises

Is "a period longer than the universe is unmeasurable" in conflict with "a half-life longer than the universe is measurable"?
Show answer
No ── you are measuring different things.
Measuring a period $T$ means watching a phase advance by $2\pi$. A system has only one phase, so you must wait.
Measuring a half-life means counting how many disappear per unit time. There are $10^{21}$ nuclei, so numbers buy you time.
★ The real axis has phase and the imaginary axis does not ── that single point separates them.
Then could a "100-Gyr oscillation of $\alpha$" be measured by increasing numbers?
Show answer
No. $\alpha$ has one value everywhere, so you cannot prepare independent samples.
The $10^{21}$ nuclei helped because each decays independently. An oscillation in $\alpha$ would be in phase across every sample.
★ So Episode 4's observation-window wall is genuine for $\alpha$.
★ There is one way around it ── look far away and you look into the past (quasar absorption lines, the Oklo reactor). Trading space for time.
Where does $^{187}$Re decaying $10^9$ times faster when ionised sit in this series?
Show answer
It is the most extreme example of "a constant is a parameter of the environment".
Episode 17 had effective mass varying by four orders with material. Here a decay constant varies by nine.
★ In wave language both are about the final states available ── band shape for effective mass, mode density for decay.
★ Much of what we call "constants" turns out to be a property of a system plus its environment.

Summary　The real axis has phase; the imaginary axis does not

$^{232}$Th's half-life of 14.05 Gyr is 1.02 times the age of the universe. Yet one gram decays 4057 times a second, and one second fixes the constant to 1.6 %.

The reason is simple ── a decay constant is an imaginary frequency ($e^{-\lambda t}=e^{i(i\lambda)t}$). An oscillation needs a full period; a decay only needs its slope counted. Episode 3's "$\omega=0$ is out of reach" was about periodic components.

It works because a time average can be replaced by an ensemble average ── possible only because each nucleus is independent and carries no phase. ★ The real axis has phase; the imaginary axis does not.

The ladder of half-lives spans 65 orders. $^{232}$Th's relative width $2.5\times10^{-40}$ is 27 orders sharper than $^{57}$Fe's ── but you cannot strike it, only count it.

Only nuclides near the age of the universe can be clocks, and against the optimum 4.78 Gyr $^{238}$U sits at 0.935. And decay constants are not constant ── fully ionised $^{187}$Re decays $10^9$ times faster.

The weak force is weak not because of its coupling ($\alpha_W$ is 4.3 times $\alpha_{\rm em}$) but because of the cutoff $M_W$. $G_F$ gives the muon lifetime to 0.44 %, the tau lifetime gives three quark colours, and the neutron's 878 s sets the universe's helium fraction.

★ "Longer than the universe is indistinguishable from constant" was about oscillations.
Decay lives on the imaginary axis, and needs no full period.

Next time We have insisted throughout that $c$ is a defined value and cannot be differentiated.
But define a coordinate system with $c\cdot t=$const and $c\propto1/t$ ── differentiable after all.
★ Write the wave equation in that system and ── Episode 13's critical $1/r^2$ equation comes out verbatim. And "fluctuations freeze when they leave the horizon" appears as the boundary between real and complex exponents.

This document is Episode 19 of the series "Thinking in Waves". The computation is in kenshou/jikan.js. On the numbers: half-lives, Q values and lifetimes are all published values (representative PDG/NNDC figures), not measured here. The "optimum half-life of 4.78 Gyr" is our own calculation, idealised to Poisson statistics on the surviving nuclei alone ── real dating measures parent–daughter ratios by mass spectrometry, with different optimum conditions (so $^{238}$U at 0.935 is an amusing coincidence, not an explanation of why geology chose it). On seasonal modulation of decay rates the literature contains both claims and refutations; we take the current majority view that it is not reproducible. The 32.9 yr half-life of fully ionised $^{187}$Re is a GSI measurement. The alpha-decay sensitivity $-2\pi\eta=-172$ is the barrier contribution only; including the $\alpha$-dependence of $Q$ the literature gives around $10^3$. The muon-lifetime formula is tree level, ignoring phase space and radiative corrections (hence 0.44 %). The tau channel counting is rough and ignores the Cabibbo angle. The BBN value $Y_p=0.223$ assumes a freeze-out ratio of 1/6 and $t_{\rm nuc}=250$ s and is 10 % off the observed 0.245 ── what this episode shows is not the value but the sensitivity to $\tau_n$. ── To print, use your browser's Print → Save as PDF.

Nuclide	Half-life [yr]	Ratio to the universe's age
\(^{235}\)U	\(7.04\times10^8\)	0.051
\(^{40}\)K	\(1.248\times10^9\)	0.091
\(^{238}\)U	\(4.468\times10^9\)	0.324
★ age of the universe	\(1.379\times10^{10}\)	1.000
\(^{232}\)Th	\(1.405\times10^{10}\)	1.019
\(^{187}\)Re	\(4.16\times10^{10}\)	3.02
\(^{87}\)Rb	\(4.97\times10^{10}\)	3.61
\(^{147}\)Sm	\(1.06\times10^{11}\)	7.69

Decay	Half-life [s]	\(\Gamma=\hbar/\tau\) [eV]	Ratio to the universe's age
\(\Delta(1232)\) resonance	\(5.6\times10^{-24}\)	\(8.1\times10^{7}\)	\(1.3\times10^{-41}\)
\(\pi^0\)	\(8.4\times10^{-17}\)	5.4	\(1.9\times10^{-34}\)
\(\mu\)	\(2.2\times10^{-6}\)	\(2.1\times10^{-10}\)	\(5.1\times10^{-24}\)
neutron	878.4	\(5.2\times10^{-19}\)	\(2.0\times10^{-15}\)
\(^{14}\)C	\(1.8\times10^{11}\)	\(2.5\times10^{-27}\)	\(4.1\times10^{-7}\)
\(^{232}\)Th	\(4.4\times10^{17}\)	\(1.0\times10^{-33}\)	1.02
\(^{128}\)Te (\(2\nu\beta\beta\))	\(6.9\times10^{31}\)	\(6.6\times10^{-48}\)	\(1.6\times10^{14}\)
proton decay (lower bound)	\(5.0\times10^{41}\)	\(9.0\times10^{-58}\)	\(1.2\times10^{24}\)

01What sits near the age of the universe?

02The crux ── decay constants live on the imaginary axis

03Why it works ── a time average replaced by an ensemble average

04The ladder of half-lives ── 65 orders

05Only things near the age of the universe can be clocks

06Is a decay constant really constant?

07Alpha decay is an ultra-sensitive sensor for \(\alpha\)

08Why the weak force is weak ── the cutoff, not the coupling

09Getting the muon lifetime out of \(G_F\)

10The neutron's lifetime set the composition of the universe

Summary　The real axis has phase; the imaginary axis does not

Axis	Behaviour	How you measure it	Observation time needed
Real axis \(\omega\)	oscillates	watch the phase go round once	★ of order one period
Imaginary axis \(i\lambda\)	falls monotonically	count the slope (i.e. the decays)	★ as short as you like

Sample	Number of atoms	Decays per second	Wait for one
one atom	1	\(1.56\times10^{-18}\)	20.3 Gyr
1 ng	\(2.60\times10^{12}\)	\(4.06\times10^{-6}\)	2.8 days
1 mg	\(2.60\times10^{18}\)	4.06	246 ms
1 g	\(2.60\times10^{21}\)	4057	246 µs
1 kg	\(2.60\times10^{24}\)	\(4.06\times10^6\)	246 ns

Nuclide	Half-life [Gyr]	Fraction left after 13.8 Gyr	Ratio to optimum
\(^{235}\)U	0.704	\(1.3\times10^{-6}\)	0.147
\(^{40}\)K	1.248	\(4.7\times10^{-4}\)	0.261
\(^{238}\)U	4.468	0.118	★ 0.935
\(^{232}\)Th	14.05	0.507	2.94
\(^{87}\)Rb	49.7	0.825	10.4

Momentum transfer \(q\)	weak / electromagnetic	Where
1 MeV	\(6.7\times10^{-10}\)	nuclear beta decay
1 GeV	\(6.7\times10^{-4}\)	inside a nucleon
10 GeV	\(6.6\times10^{-2}\)
80.4 GeV (\(M_W\))	2.16	★ equal strength ── electroweak unification
1 TeV	4.30

01What sits near the age of the universe?

02The crux ── decay constants live on the imaginary axis

03Why it works ── a time average replaced by an ensemble average

04The ladder of half-lives ── 65 orders

05Only things near the age of the universe can be clocks

06Is a decay constant really constant?

07Alpha decay is an ultra-sensitive sensor for \(\alpha\)

08Why the weak force is weak ── the cutoff, not the coupling

09Getting the muon lifetime out of \(G_F\)

10The neutron's lifetime set the composition of the universe

Summary The real axis has phase; the imaginary axis does not

Summary　The real axis has phase; the imaginary axis does not