# What is the SI unit for matter

## The origin of mass

- How is mass defined in terms of measurement technology?
- What are the problems with this definition and why is it important to define mass so precisely?
- What is the natural system of units and how does it work?
- Why do we use electron volts and not kilograms as the unit of mass in particle physics?

In everyday use, crowds are mostly in the** SI unit** \ (kg \) specified. Like all other SI quantities, this basic quantity was set arbitrarily. A kilogram is a kilogram precisely because it is a handy size in everyday life.

But if we deal with matter in the atomic or even subatomic range, it can be tedious to state masses in \ (kg \), since an electron, for example, has a mass of about \ (9.1 \ cdot 10 ^ {- 31} kg \) Has. First of all, we can't imagine much by this and, secondly, it can be cumbersome to constantly include all the exponents in our calculations. In particle physics we therefore do not use \ (kg \) as a unit for mass. But apart from the fact that we do not use SI units, there is another special feature: The unit used, the electron volt, is actually a unit of energy. We explain below why we find this practical and why it works. But first, let's take a look at the definition of the kilogram.

### Measure = compare

We only can **measure up**, in order we **to compare**, and for this we need a comparison measure. For example, we compare a distance with a ruler and we can imagine more or less how big a distance is when we say a length in the range from \ (mm \) to \ (km \). We have gathered empirical values in these dimensions almost all our lives, which is why it is easy for us to imagine such dimensions. Therefore, it may seem surprising at first glance that in particle physics we specify masses, for example, in electron volts and not in kilograms, but this is only due to the fact that one is not used to handling this unit.

The unit kilogram was established in 1889 via the **Original kilogram **Are defined. This is a cylinder with \ (39 \, mm \) height and \ (39 \, mm \) diameter, which consists of a platinum-iridium alloy. In 2019, however, this definition was replaced by a more modern one, which defines the unit kilogram only using natural constants. Problems that a definition on the basis of an object entails have thus been eliminated.

The original kilogram.

Source: https://www.bipm.org/en/bipm/mass/ipk/, with permission from BIPM

The current definition of the unit kilogram is based on Planck's constant, the value of which was set to \ (6.626 070 15 × 10 ^ {- 34} J \, s \). The unit \ (J \, s \) corresponds to \ (kg \, m / s \), whereby the meter and second are themselves defined by natural constants. Specifically, this is the speed of light or the duration of a certain process in a cesium-133 atom. To apply this definition, the Planck constant has to be measured very precisely. This is done, for example, with a silicon ball, which is the roundest object in the world. Only a specific isotope of silicon was used to form this sphere. A second method uses what is known as a kibble balance, which compares weight, i.e. gravitational force, and electromagnetic force.

Exactly defining the kilogram is important because in the truest sense of the word all masses and all measures dependent on these masses, for example Newtons or Joules, depend on it. For example, to measure the gravitational force of the earth, we need a very precise value for the kilogram, since the gravitational force of the earth varies slightly depending on the location. In particle physics, too, we depend on an exact definition of the kilogram, since elementary particles are very light.

### Natural units

In particle physics, we do not use the SI system of units, but the so-called natural system of units. In this system of units, one selects the minimum number of units that need to be set and expresses all other units above it. It was already mentioned above that the meter is defined by the distance that the light travels in a certain time. The speed of light, which is usually abbreviated to $ c $ in the literature, is a natural constant. So we can express length over time multiplied by the speed of light. This is the case, for example, when we speak of a light year, which is a unit of length. It becomes particularly easy if you define a system of units in which the speed of light is \ (c = 1 \). Then it no longer appears in the unit information. For example \ (1 \, m = 1/299 792 458 \, s \ times c = 1/299 792 458 \, s \). Only when converting to SI units do you then have to use the corresponding value of $ c $ in SI units. Thus the speed of light is simply our "ruler", our new unit of measure for speeds. This is practical, because the speed of light appears as a constant in many other equations as well, and when this \ (1 \) is set, we no longer need to worry about it.

Masses can be related to energy using Einstein's famous formula \ (E = mc ^ 2 \). So mass can be given as energy divided by \ (c ^ 2 \). And that's exactly how we do it in particle physics. What we still need now is a suitable unit of energy, because the common unit of energy, the joule, is too large for the energy scales with which we are concerned in particle physics. We use the unit instead**Electron volts**, abbreviated as \ (eV \) and given by \ (1eV = 1.602 \ cdot 10 ^ {- 19} J \). An electron volt is defined as the kinetic energy that an electron receives when it is accelerated by an electric field with a voltage of \ (1 \, volt \).

If we have now completed our mass calculations in \ (eV \) and want to look at the calculated value in \ (kg \), for example, we have to convert the electron volts through the value of \ (c ^ 2 \) in our SI units ( So divide \ ((299792458 \, m / s) ^ 2 \)) and then convert the electron volts into joules. In summary this means: In particle physics we write \ (eV \) as a unit for masses. But actually it means \ (eV / c ^ 2 \). The conversion factor is \ (1eV = 1.782 \ cdot 10 ^ {- 36} kg \).

\ (1 \, eV \) (mass in natural units with $ c = 1 $) = \ (1 / c ^ 2 \, eV \) = \ (1.782 \ cdot 10 ^ {- 36} kg \) (mass in SI units)

The usual prefixes are also used. So kilo electron volts for \ (1000 \, eV \), megaelectron volts for \ (1,000,000 \, eV \) etc. Depending on the application, one then calculates with the appropriate unit. For example, electrons have a mass of \ (511 \, keV \), while a proton has a mass of \ (938 \, MeV \).

symbol | Surname | value |

... | ... | ... |

T | Tera | 10^{12} |

G | Giga | 109 |

M. | Mega | 10^{6} |

k | kilo | 10^{3} |

H | Hecto | 10^{2} |

there | Deca | 10^{1} |

--- | --- | 10^{0} |

d | Deci | 10^{-1} |

c | Centi | 10^{-2} |

m | Milli | 10^{-3} |

μ | Micro | 10^{-6} |

n | Nano | 10^{-9} |

p | Pico | 10^{-12} |

... | ... | ... |

In summary, we can say that in principle we can use any system for physical units. For example, the American system works just as well with pounds (\ (lbs)) and ounces (\ (oz \)) for masses. However, you have to be careful that you use the selected system of units consistently and not mix it with other systems. As a warning example, the crash of the Mars Climate Orbiter, which cost 125 million dollars in 1999, can be traced back to a mixture of pounds and kilograms in the computer code.

**Video:** Brief historical outline of the definition of mass including temporal fluctuations in the copies of the original kilogram, explanation of the importance of an exact definition and attempts at an object-independent definition

**Podcast:** Definition of original meter and original kilo and a Nobel Prize winner: methodologically incorrect, episode 97, from 9:00 p.m., original kilo from 33:00 p.m., Avogadro project from 38:50 a.m., Nobel laureate and diamond from 45:00 p.m. to 04:00 p.m.

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