How to measure ultra short half life or lifetimes of radioisotopes?

Ultra short half life or life time of radioisotopes in the range of 10^-9-10^-15 sec can be estimated by two techniques using Doppler-shift methods, 

Recoil Distance Plunger Method (RDPM or RDM) and 

Doppler Shift Attenuation Method (DSAM). 

The two techniques are rather different, but 

Both the techniques rely on the Doppler effect.

Doppler effect is the the shift in energy of a γ-ray when it is emitted from a moving source.  

In Gist

• A γ-ray detector was used to view a narrow region of space at a given distance away from a thin target foil.  
  

• γ-Rays from recoiling, excited nuclei that decayed in the region viewed by the detector were recorded as a function of distance from the target to obtain a decay curve.

• A knowledge of the recoil velocity or velocity distribution allowed the nuclear lifetime to be calculated. Spacial resolution and inadequate knowledge of the recoil velocity distribution limited the accuracy and range of lifetimes that could be measured.

Recoil Distance Plunger Method (RDM)

Step 1: Initial requirement

A nuclear reaction has to be identified which can produce the short lived radioisotope. 

Lets take an example

X(x,y)Y*(γ)Y

Here X is the target

x is projectile

Y* is the product nucleus in excited state

y is the ejectile 

Half life / life time of Y* ha s to be determined. 

Step 2: Nuclear reaction

The thin target X is supported by a Nickel foil.

The target X is bombarded with projectile x. .

The target gets recoiled in the back direction to conserve the momentum. 

The excited state product Y*, due to its inherent kinetic energy leaves the Ni foil, recoiled in the front direction and enters in the vacuum zone. 

The Y* may or may not reach the thick metal plunger at the end after travelling the plunger distance s. 

Step 3: Assay

Gamma radiation is assayed from both during transit of Y* and from the thick metal target, when Y* is at rest. 

The gamma energy at rest i.e. from thick metal target is unshifted/ original gamma energy as per the transition during decay (Eγ = E0). However the gamma energy during transit gets shifted. 

$E\gamma \text{'}=E_0(1+\frac{{\mathrm{vt}}_{}}{c}cos\theta )$

Both the gamma energies are assayed using gamma spectrometry using HPGe detector. 

Step 4: Reason for gamma energy shift

The reason of shift in gamma energy due to doppler effect. 

Doppler shift is the change in frequency of sound wave of a moving object in relation to the stationary observer. 

If a sound causing object is moving towards a stationary observer, the observer received the sound with low to high frequency as the distance between source and observer decreases. 

In this case if the gamma energy during transit is observed compared to rest (c.f. the observer is at metal plunger) the frequency (ν) of the gamma radiation shall be highest near the metal plunger. As we know, E = hν, 

(Eγ)rest > (Eγ)transit. 

Step 5: Actual calculation

the number of nuclei decaying in a flight time tF is 

$I_S=\underset{0}{\overset{t_F}{\int }}(-\frac{dN}{dt})dt =-\underset{0}{\overset{t_F}{\int }}{N}_0.e^{-\lambda t }dt = -N_0 (e^{-\lambda t_F}-1)$ ${ }_{}$

those decaying in the plunger is 

$I_{US}=\underset{t_F}{\overset{\infty }{\int }}\left(-\frac{dN}{dt}\right)dt =\underset{t_F}{\overset{\infty }{\int }}{N}_0.e^{-\lambda t} dt = N_0.e^{-\lambda t_F}$ 

$\frac{I_{US}}{I_S+ I_{US}} = \frac{ N_0.e^{-\lambda t_F}}{-N_0 \left(e^{-\lambda t_F}-1\right) + N_0.e^{-\lambda t_F} }=\frac{ e^{-\lambda t_F}}{-\left(e^{-\lambda t_F}-1\right) + e^{-\lambda t_F} }= e^{-\lambda t_F}$

${{\mathrm{v\; can\; be\; calculated\; from }}_{}}^{}$

$△E=E_{0 }\frac{v_t}{c}cos\theta$ 

△E = Eγ - E0

i.e. △E = (Eγ)rest - (Eγ)transit. 

tF = s/v

So from the equation  $\frac{I_{US}}{I_S+I_{US}}=e^{-\lambda t_F}$  λ is unknown. From λ we can easily calculate the value of half life/ life time of Y*. 

In summary

After traveling a distance s in vacuum, recoiled radioisotope are brought to rest in a thick metal plunger.

The recoil nucleus decays 

either during the flight (i.e. before metal plunger) or 

decay in the metal plunger (i.e. after traveling the distance between thin meta target and thick metal plunger).

γ-Rays from those decaying in flight are Doppler shifted.  

Those excited nuclei that survive the time of flight (tF), s/v, in the vacuum and decay in the metal plunger emit γ-rays unshifted in energy.

Doppler-shifted and unshifted peaks can be resolved using HPGe detector. Their intensities are estimated and is used to calculate the half life/ life time of a radioisotope. 

Life times in the range of 10^-9-10^-12  sec can be measured by Recoil Distance Plunger Method (RDPM or RDM). 

To measure the lifetimes lower than 10^-12 sec, Doppler Shift Attenuation Method (DSAM) should be used, which can measure life times ranging from 10^-12-10^-15 sec. 

Reference

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