A bit of maths and physics

Tag: atoms

  • A radioactive smoking gun

    A radioactive smoking gun

    As we discussed in a previous article, microwave ovens don’t destroy atoms – their radiation simply isn’t ionising radiation, while it’s exactly the ionising stuff that is bad for your cells, such as gamma rays and cosmic rays. The latter reside at the high-energy end of the electromagnetic spectrum. We also mentioned that microwave oven radiation isn’t the same as the radiation present in Chernobyl. Microwave ovens are not radioactive. But when do we call something radioactive then? What is radioactivity?

    Unstable to the core

    Matter is radioactive when the nuclei of its atoms are unstable enough to decay into different types of nuclei, emitting any of the following ionising radiation in the process:

    • alpha rays, a stream of clumps of two protons and two neutrons;
    • beta rays, a stream of electrons or positrons;
    • gamma rays, the higher-energy form of electromagnetic radiation beyond X-rays;
    • neutrino, a particle with the smallest mass.

    Two well-known examples of radioactive material are plutonium and uranium, mostly associated with nuclear power plants and nuclear weapons. Perhaps less known to be radioactive are, for instance, radon-222, lead-210, polonium-210, and potassium-40. We will explain those numbers in the next section.

    So, again, just to be clear, microwave ovens do not pour any of these particles over your food. Nothing gets ‘nuked’. There’s nothing nuclear going on. Your food, however, might very well be radioactive as it’s likely to contain potassium-40.

    A cartoon of an atom

    Atoms consist of three constituents: electrons, protons, and neutrons. Only the hydrogen atom lacks neutrons, the rest is a composite of all three elements. Figure 1 shows a cartoon of an atom. The vague, yellow band represents the electron cloud. The nucleus has been magnified so that its protons and neutrons become visible.

    Figure 1

    What makes one atom different from the other – say, calcium from potassium – is the number of electrons, protons, and neutrons. The latter two are called nucleons. The number of protons is most important here: it determines which element of the famous periodic table of elements (type of atoms) we’re dealing with. Potassium has 19 protons – this is crucial. If it were to acquire any other number of protons, it stops being potassium. Calcium has 20 protons, for example. The number 40 in ‘potassium-40’ means that its nucleus has 40 nucleons. In other words, it has 40 nucleons – 19 protons = 21 neutrons. Mind you, there are also potassium-39 (19 protons, 20 neutrons) and potassium-41 (19 protons, 22 neutrons).

    All these potassium-versions – same number of protons, different number of neutrons – are called isotopes. So, while an element (type of atom) has a very specific number of protons, its number of neutrons may differ. Potassium-40 is an example of an isotope of potassium. This and the other aforementioned potassium-isotopes all occur naturally. However, humans are capable of synthesizing another 22 isotopes, no less.

    Radioactive decay

    The nucleus of potassium-40 is unstable. It decays, as it’s called, mostly into calcium-40, which is a stable atom. During the decay, the potassium atom gains a proton and becomes a calcium atom. In the process, a beta particle, i.e. an electron, and an antineutrino are emitted. This is why the decay is called radioactive: it actively radiates stuff. The electron flies off at great speeds and is potentially ionising, i.e. it is capable of knocking another electron from its atom, thereby potentially destroying molecules such as DNA.

    About 0.01% of the potassium mass in our bodies, acquired through food, is of the potassium-40 variety. About 5000 of these atomic nuclei decay every second in an average human body. So, humans possess a radioactivity of 5000 Bq (becquerel, named in honour of Henri Becquerel, who shared the Nobel Prize with Pierre and Marie Curie for their work in radioactivity).

    Bananas are also radioactive as they also naturally contain potassium-40. Its decay rate lies at 14 Bq, i.e. 14 decaying nuclei per second. It won’t set off a Geiger counter. A truck full of them probably would.

    Ionising radiation and the human body

    To assess the extent to which exposure to ionising radiation is likely to have medical consequences, we use units of ‘sievert’ (the symbol is Sv), a measure for the effective dose of ionising radiation in humans.

    The International Commission on Radiological Protection, an independent, non-governmental organisation providing recommendations concerning ionising radiation, assessed that 1 Sv represents a 5% chance of developing cancer.

    Fortunately, we evolved to absorb safe and small amounts of ionising radiation on a daily basis. The talented and clever Randall Munroe, famous for his xkcd.com cartoons, made a wonderful chart ‘with help from Ellen, a Senior reactor operator at the Reed Research Reactor’. We have blatantly copied their brilliant concept in Figure 2 to concisely give you a feeling for the various amounts of ionising radiation (click to enlarge). However, by all means, do also have a look at their original chart. Note that all these diagrams mainly indicate the orders of magnitude – not exact values as sieverts may vary a little for various human bodies, and sources and locations mentioned. This should, nonetheless, give you an idea how much radiation professionals in the radiation industry are allowed to be exposed to on an annual basis.

    Figure 2

    Smoking hot

    No, this isn’t going to be pretty. If anything, it’s pretty bad, actually.

    Earlier, we mentioned radioactive elements radon-222, polonium-210, and lead-210. They naturally occur in the soil and air. They are also present in and on tobacco leaves and remain there even after processing. Once inhaled, sticky tar assures that these radioactive elements remain in the small air passageways and lungs indefinitely. Moreover, radon-222 happens to decay into said polonium and lead isotopes. Hence, the latter two build up even more. This is also true for secondhand smoke. Together with toxic substances such as tar, arsenic, nicotine, and cyanide, the ionising radiation emitting decaying nuclei of radon-222, polonium-210, and lead-210 increase chances of developing lung cancer immensely.

    According to Little and colleagues (1965, 1967), polonium-210 accumulates in certain ‘hotspots’ in the lungs. Karagueuzian and colleagues (2012) reported an estimated lung dose of 165 mSv per year. To get a feeling of how much ionising radiation a smoker’s lungs (especially the hotspots) receive annually compared to how much ionising radiation a whole body of a professional radiation worker is allowed to be exposed to, see Figure 3. Mind you, this is just about radioactive decay in the lungs. We haven’t even discussed the other toxic compounds. And so, while someone may worry about Wi-Fi signals, mobile phones, and microwave ovens, which, by physics, they oughtn’t, keep in mind, in this Universe, that same physics tells us their smoking habit is really bad. Like, really.

    Figure 3

    References

    Little, J. B., Radford, E. P., Mccombs, H. L. and Hunt, V. R. (1965) ‘Distribution of polonium-210 in pulmonary tissues of cigarette smokers’, The New England journal of medicine, vol. 273, no. 25, p. 1343 [Online]. DOI: 10.1056/NEJM196512162732501 (Accessed 9 August 2019).

    Little, J. B., Radford, E. P. and Holtzman, R. B. (1967) ‘Polonium-210 in Bronchial Epithelium of Cigarette Smokers’, Science, vol. 155, no. 3762, pp. 606–607 [Online]. DOI: 10.1126/science.155.3762.606 (Accessed 9 August 2019).

    Karagueuzian, H. S., White, C., Sayre, J. and Norman, A. (2012) ‘Cigarette Smoke Radioactivity and Lung Cancer Risk’, Nicotine & Tobacco Research, vol. 14, no. 1, pp. 79–90 [Online]. DOI: 10.1093/ntr/ntr145 (Accessed 9 August 2019).

    Featured photo by Julia Sakelli.

  • The formula that got Albert Einstein the Nobel Prize and should stop us getting sunburn all the time

    The formula that got Albert Einstein the Nobel Prize and should stop us getting sunburn all the time

    A copy of page 5 of the newspaper The Times of 10 November 1922. Near the bottom, a small article is printed. The title is Nobel Prize for Einstein. The text goes as follows. Stockholm, Nov 9.—The Nobel Prize for Physics—1921—has been awarded to Professor Albert Einstein, of Berlin, in recognition of his work in theoretical physics. The 1922 prize for physics has been awarded to Professor Niels Bohr, of Copenhagen, in recognition of his research work into the structure of atoms.—Reuter.
    ‘Nobel Prize for Einstein’, one sentence was spent in The Times of 10 November 1922.

    In 1921, Albert Einstein won the Nobel Prize “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.” Not a word about relativity. So, no, he did not win the Prize with $ E=mc^2 $. Though it is his most famous equation—which, by the way, is not the complete version—it is not his Nobel Prize-winning formula. We will write it down, but first, we describe what this photoelectric effect is.

    Different stuff is made up of different molecules. Different molecules are made up of different atoms. Different atoms are made up of a variety of nuclear composites and different numbers of electrons. So far, nothing new, perhaps, but here’s the thing. If electrons are exposed to particular amounts of energy, they can be ejected away from the nucleus.

    An atom of which one or more electrons have been blasted away is called an ion. The process is called ionisation. Whether this occurs, depends on a few things such as the type of stuff (=the type of atoms and how they are bound together) and the specific energy it is exposed to.

    If ionisation at the surface of a material is achieved by normal light, we call this the photoelectric effect: light (the ‘photo’-part) causing electrons to leave their nucleus (the ‘electric’-part).

    A diagram of the ionisation of an atom (not to scale). (1) The yellow cloud represents an electron’s (probable) whereabouts. The tiny pink core represents the atom’s nucleus. (2) Photons of a specific colour radiate towards the atom. (3) The electron has flown off. The nucleus remains. The atom has become an ion.

    Not about intensity

    One peculiar thing is worth mentioning. In fact, it was this puzzle that led Albert to his equation. It turned out that what matters is the frequency of the light beam, i.e. the colour of the light, not the intensity of it, i.e. the power per square metre, or Joule per second (watt) per square metre.

    Imagine, in the diagram above, that a billion yellow photons would radiate towards the atom and nothing happened; the electron would stay where it was. Now imagine a billion billion billion billion yellow photons approaching the atom. Still nothing would happen as it is not about intensity.

    Yellow light is less energetic than blue light, so if you would replace the light bulb for a source that delivers pure blue light, with even one blue photon, it could happen easily (though you would have to aim impossibly precise, so it makes sense to actually radiate a lot). This puzzled many scientists, but Albert solved it and won the Nobel Prize.

    With his discovery, quantum physics was starting to get momentum. He, and other good physicists of his time, showed that light could be seen as little packets of energy, which scientists started calling photons. A beam of light was now a stream of photons. The intensity, the amount of photons per second per square metres doesn’t matter but the frequency of a photon, or energy per photon does.

    DNA

    While this is all cool and useful for scientific purposes, we certainly do not want any electrons of the DNA molecules of our skin breaking away from their atomic confines. Atomic bonds would be destroyed and our DNA would become mutated. Even though astonishing molecular biological processes in our body repair defects like this in a staggering, basically inconceivable number of cases, some errors might slip through and may even become the start of tumour growth. Therefore, it is important to know what energy domains would cause our beloved bodily electrons to be blasted off so that humanity can learn to avoid those dangerous environments.

    The problem arises when we get into the mid to high-energy electromagnetic radiation, or light, or photons, if you will. We’re talking the dangerous kind of ultraviolet here, the type of UV causing DNA mutation to occur: UVB to be precise. A photon of UVB-light is about 1.8 times more energetic than a photon of the yellowish light in your home and almost a million times more energetic than a mobile phone photon. So, don’t be scared of being home. As soon as you set foot outside, though, be afraid. Not of the dark, but of the light, for ionising UVB-light is emitted by the Sun.

    A diagram of electromagnetic radiation. Far right, we see the dangerous types of radiation: cosmic rays, x-rays, gamma rays, UV-light. In the middle, we see visible light. Far left, we see the lowest energy photons: WiFi, mobile phones, microwave ovens.
    A diagram (not to scale) of electromagnetic radiation, or photons, if you will. The mentioned values are the frequencies of the photons, expressed in gigahertz (GHz). The higher the frequency, the higher the energy of the photon.

    Fortunately, as stated before, our bodies have evolved to repair the damage when necessary. This is why even X-rays are okay and hospitals and dentists make sure not to expose you to doses of energetic photons you wouldn’t survive. Continuous monitoring of its uses and effects is prerequisite.

    It’s partly a question of the law of large numbers, though. If the number of freely whizzing electrons is large enough, they themselves will become the main cause of an increasing number of damaged DNA molecules, and, eventually, some repairs will fail or not even take place. So, while it is not instantly dangerous, we do recommend some reading up on the subject of sunbathing. Use UV protection. Don’t get sunburnt. And give your body a chance to recover from the ruthless blasts of ionising UV radiation. Forget microwaves, the problem is crispy skin.

    The formula

    So, now we finally get to Albert’s Nobel Prize-winning formula. Here it is

    \[ \frac{1}{2}m_ev^2_\text{max} = h\nu – \phi. \]

    It doesn’t look as sassy as the other one, right? And yet, it’s the one that allows us to calculate if, for instance, electrons of our body’s carbon atoms get blasted out by the photons emitted by the lamp in your lavatory (they do not). Or if the laser pointer knocks some electrons out (it doesn’t), which we use anyway, because we need to point at things on our PowerPoint slides as they might well be ill-designed (they are).

    So, $ \frac{1}{2}m_ev^2_\text{max} $ means maximum kinetic energy, which is simply the energy with which an electron flies away from its nucleus. If its value turns out to be smaller than or equal to zero then the electron is not affected at all. It’ll keep stuck to its nucleus. If it is larger than zero then off it goes. The symbol $ h $ is a constant, which we needn’t worry too much about. It’s a number and it’s called the Planck constant. The Greek letter $ \nu $ is the frequency of the photon. In the diagram above, a few have been mentioned. Mind you, $ h\nu $ means $ h \times \nu $ and is the energy of a photon. Mathematicians, physicists, engineers, and other folks, just like to leave out the $ \times $-sign. The Greek letter $ \phi $ is the so-called work function. It is the minimal energy needed for the occurrence of a photoelectric effect. Its value depends on the type of atom, molecule, material, and surface you want to calculate the photoelectric effect of.

    In conclusion

    Notice Einstein’s formula does not have any term relating to the number of photons radiated per second per square metre towards the atom of interest, i.e. the intensity. Only the frequency is important. This means that atoms—such as your body—will be left undisturbed irrespective of the power of the radiation they are exposed to. There may be a bit of heat but there is no ionisation. The potential danger lies in frequency ($ \nu $), such as that of UV light and higher. Here, both dosage and capability of recovery play a crucial role.

    Young Albert Einstein

    The value of the Planck constant is $ h = 6.626070 \times 10^{-34} $ Js (Joulesecond). The value of the work function of carbon, of which our entire body is made, including our DNA, is $ \phi = 8.0108831 \times 10^{-19} $ J. If a WiFi photon has a frequency of 2.5 GHz, you can calculate yourself if it would yank the electrons from a carbon atom. Remember to convert 2.5 GHz to $ 2.5 \times 10^9 $ / s (per second). Thanks to Albert, calculating this has become child’s play. We could do the maths on the back of an envelope. If all the terms on the right hand side of the equal sign turn out to be larger than zero, then sell your router immediately and—based on this diagram—you most definitely ought to refrain from switching on the light while frequenting the lavatory. Good luck with the calculation! (Or check the working out.)

    Featured image: a 14-year-old Albert Einstein, photographed in 1893. Credits EMILIO SEGRE VISUAL ARCHIVES / AMERICAN INSTITUTE OF PHYSICS / SCIENCE PHOTO LIBRARY / Universal Images Group. Source: Young Albert Einstein, physicist. [Photography]. Encyclopædia Britannica ImageQuest. Retrieved 9 Mar 2019, from 
    https://quest.eb.com/search/132_1258083/1/132_1258083/cite

    Smaller image of an even younger Albert Einstein: Credits EMILIO SEGRE VISUAL ARCHIVES / AMERICAN INSTITUTE OF PHYSICS / SCIENCE PHOTO LIBRARY / Universal Images Group. Source: Young Albert Einstein, physicist. [Photography]. Encyclopædia Britannica ImageQuest. Retrieved 9 Mar 2019, from https://quest.eb.com/search/132_1255429/1/132_1255429/cite

    Newspaper article: “Nobel Prize for Einstein.” Times, 10 Nov. 1922, p. 5. The Times Digital Archive. Retrieved 8 Mar 2019 from http://tinyurl.galegroup.com/tinyurl/9Q37o0.