CHAPTER 8 : Time: Reality We Live or Illusion We Believe?

 

    What exactly is time? Is it just an imaginary thing or does it exists in real life? Is time just an imaginary part of the complex equation? We must have heard this thing that time is delusional maybe in you tube videos and some ancient scriptures. But it is just the conclusion made like the water surface but unexplored deep inside as light rays can't reach so deep. The light rays is just like a human way of thinking which is incapable to go beyond some depth of questioning ourselves. This chapter will not be philosophical, we will be going to discuss how the most accurate time is measured.

    The atomic clock is the most accurate measurement of time in the modern times. As mentioned in the previous chapter that the protons and neutrons also contains the magnetic field but it is very small as compared to that in electrons, so in the SG apparatus that interaction between of the nucleus spin (proton + neutron spin) with the electron spin is ignored. For measuring time with atomic clock, this interaction  between nucleus spin and electron spin is used. So like the silver atom was used in the SG experiment for the reasons mentioned in the previous chapter, for measuring time with atomic clock, we will use cesium - 133 atom. Again the question arises why choose only cesium-133 atom?

    Cesium atom is the only atom in the universe explored till now which has only one naturally occurring isotope: Cs-133. It basically means that every cesium atom in the universe is identical in nuclear structure avoiding any confusion from multiple isotopes. This ensures consistency and universality for time measurement. What exactly is this number 133 and how does it came? The cesium atom contains: 55 protons, 78 neutrons so its total nuclear mass is 133 units ( in atomic mass units) and 133 - 78 = 55 electrons. 

    As discussed and proved in previous chapter, spin is the quantized state. Like silver atom in SG experiment has spin quantum number as S= +1/2 and/or -1/2, the cesium atom also has the same spin quantum number. It has nucleus spin of I = 7/2. The electron spin value came from the SG experiment and the value of nucleus spin came from other experiments of Hyperfine spectroscopy, Nuclear Magnetic Resonance (NMR). The NMR experiment will be explained in the other chapter. The hyperfine splitting (which will be explained in this chapter) is responsible for atomic clock measurement and the value of I. What is hyperfine splitting?

    As there is interaction between the nucleus spin and electron spin due to magnetic interaction. There is always alignment and anti-alignment between the nucleus and electron spin. Depending on whether they are pointing mostly in the same (aligned) direction or the opposite (anti-aligned) direction, the total energy of the atom changes slightly. This small difference between the two arrangements is called hyperfine-splitting. So, in cesium-133 this hyperfine splitting creates two possible energy states in the ground level: one with total spin F=3 (anti-aligned) and one with total spin F=4 (aligned). The gap or jumping between them corresponds to the clock frequency. What exactly is this Energy and what exactly is F here and how did these values came?

    F refers to the total angular momentum quantum number of the atom in its ground state (electron + orbital + nucleus). The values came from:
F = I+J     &                                              .... a)
J = L+S                                                     .... b)
here J = total angular momentum only due to electrons and orbit 
L = orbital angular momentum
In the equation b) substitute L=0, as the only one valence electron present in cesium is in s orbital and also substitute S = + 1/2 and/or -1/2 in equation b and the result will be J = +1/2 and/or -1/2. Accordingly, F=3 and 4 came by taking J as -1/2 and +1/2 respectively and taking I = 7/2 in the equation a). To calculate the energy for F=4 and F=3, we use the following formula:

i) Energy convention

$$E_F=\frac{A}{2}[F(F+1)-I(I+1)-J(J+1)] \; \; \; ....\; c)$$

• $A$ has units of energy.

• $A$ is called the magnetic dipole hyperfine structure constant.

• It tells us how strongly the magnetic field from the nucleus interacts with the magnetic field from the electron’s spin and orbital motion.

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ii) Frequency convention

$$\nu_F = \frac{A_\nu}{2} \, \big[ F(F+1) - I(I+1) - J(J+1) \big]$$

• $A_\nu$ has units of Hz.

• Relation to energy:

  $$E_F=h{\nu }_{\mathrm{F}}$$

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Cesium-133 hyperfine splitting

For the cesium-133 ground state, the measured splitting frequency difference between $F=\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mord">4\; and\; F=3\; is: </span></span>}$

$$\mathrm{\Delta }\nu =\mathrm{9,192,631,770}\text{ Hz}$$

For an $S_{1/2}$ state (with $J=\tfrac{1}{2}$ and $I=\tfrac{7}{2}$), the two hyperfine levels are separated by:

$$\Delta \nu = 4 A_\nu \quad \Rightarrow \quad A_\nu = \frac{\Delta \nu}{4}$$

The exact value of $\Delta \nu$ is measured experimentally (using the Cesium Atomic Beam Clock or the Cesium Atomic Fountain Clock) which will be discussed in the later chapter.
After substituting the value of $\Delta \nu$, we can determine $A_\nu$, and from there calculate $\nu_F$

If you want $A$ in energy units:

$$A=h{A}_{\mathrm{\nu }}$$

and then you can compute $E_F$

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Symbols

• $E_F$: energy of a given hyperfine level.

• $\nu_F$: frequency equivalent of that energy level (via $E_F = h \nu_F$).

• $\nu$: general frequency symbol, e.g. transition frequency or splitting.

• $\Delta \nu$: difference in frequency between two levels.
  
  
  

    So, from equation c), it will be concluded that when the electron's spin and nucleus spin direction is same (aligned), the atom is in higher-energy state and when they point in the opposite direction (anti-aligned), the atom is in lower-energy state. Microwaves at exactly right frequency (9,192,631,770 Hz) can flip the system back and forth between these two states. So the frequency here basically is the number of flips done by atoms between these two alignments in one second i.e. 9,192,631,770 flips happens in one second. When microwaves of this frequency hit cesium atoms, the atoms switch (flips) back and forth between the two alignments.

    So, from the above experiment it proves that time is just calculated based on the natural behavior of atoms. Here, is the question for the readers, as every object is made up of atoms, so what happens at the quantum level to the atom's flipping behavior when any object's life span expires?( for e.g, when human dies). Share the answer if you have, through my social media links, comments.