Nuclear masses and mass formulas. How to calculate atomic mass Find mass of nucleus formula

Many years ago, people wondered what all substances were made of. The first who tried to answer it was the ancient Greek scientist Democritus, who believed that all substances consist of molecules. It is now known that molecules are built from atoms. Atoms are made up of even smaller particles. At the center of the atom is the nucleus, which contains protons and neutrons. The smallest particles – electrons – move in orbits around the nucleus. Their mass is negligible compared to the mass of the nucleus. But only calculations and knowledge of chemistry will help you find the mass of the nucleus. To do this, you need to determine the number of protons and neutrons in the nucleus. Look at the table values ​​of the masses of one proton and one neutron and find their total mass. This will be the mass of the nucleus.

You can often come across the question of how to find the mass, knowing the speed. According to the classical laws of mechanics, mass does not depend on the speed of the body. After all, if a car starts to pick up speed as it starts moving, this does not mean at all that its mass will increase. However, at the beginning of the twentieth century, Einstein presented a theory according to which this dependence exists. This effect is called relativistic increase in body weight. And it manifests itself when the speeds of bodies approach the speed of light. Modern charged particle accelerators make it possible to accelerate protons and neutrons to such high speeds. And in fact, in this case, an increase in their masses was recorded.

But we still live in a world of high technology, but low speeds. Therefore, in order to know how to calculate the mass of matter, you do not need to accelerate the body to the speed of light and learn Einstein’s theory. Body weight can be measured on a scale. True, not every body can be put on the scale. Therefore, there is another way to calculate mass from its density.

The air around us, the air that is so necessary for humanity, also has its own mass. And when solving the problem of how to determine the mass of air, for example, in a room, it is not necessary to count the number of air molecules and sum up the mass of their nuclei. You can simply determine the volume of the room and multiply it by the air density (1.9 kg/m3).

Scientists have now learned with great accuracy to calculate the masses of different bodies, from atomic nuclei to the mass of the globe and even stars located at a distance of several hundred light years from us. Mass, as a physical quantity, is a measure of the inertia of a body. More massive bodies are said to be more inert, that is, they change their speed more slowly. Therefore, after all, speed and mass turn out to be interconnected. But the main feature of this quantity is that any body or substance has mass. There is no matter in the world that does not have mass!

Isogons. The nucleus of the hydrogen atom - proton (p) - is the simplest nucleus. Its positive charge is equal in absolute value to the charge of an electron. The mass of a proton is 1.6726-10’2 kg. The proton as a particle that is part of atomic nuclei was discovered by Rutherford in 1919.

To experimentally determine the masses of atomic nuclei, they have been and are using mass spectrometers. The principle of mass spectrometry, first proposed by Thomson (1907), is to use the focusing properties of electric and magnetic fields in relation to beams of charged particles. The first mass spectrometers with sufficiently high resolution were designed in 1919 by F.U. Aston and A. Dempstrov. The operating principle of the mass spectrometer is shown in Fig. 1.3.

Since atoms and molecules are electrically neutral, they must first be ionized. Ions are created in an ion source by bombarding vapors of the substance under study with fast electrons and then, after acceleration in an electric field (potential difference V) exit into the vacuum chamber, falling into the region of a uniform magnetic field B. Under its influence, the ions begin to move in a circle, the radius of which G can be found from the equality of the Lorentz force and centrifugal force:

Where M- ion mass. The speed of movement of ions v is determined by the relation

Rice. 1.3.

Accelerating potential difference U or magnetic field strength IN can be selected so that ions with the same masses fall into the same place on a photographic plate or other position-sensitive detector. Then, by finding the maximum of the mass spectrum signal and using formula (1.7), we can determine the mass of the ion M. 1

Excluding speed v from (1.5) and (1.6), we find that

The development of mass spectrometry technology made it possible to confirm the assumption made back in 1910 by Frederick Soddy that the fractional (in units of the mass of a hydrogen atom) atomic masses of chemical elements are explained by the existence isotopes- atoms with the same nuclear charge, but different masses. Thanks to Aston's pioneering research, it was established that most elements are indeed composed of a mixture of two or more naturally occurring isotopes. The exceptions are relatively few elements (F, Na, Al, P, Au, etc.), called monoisotopic. The number of natural isotopes of one element can reach 10 (Sn). In addition, as it turned out later, all elements without exception have isotopes that have the property of radioactivity. Most radioactive isotopes do not occur in nature; they can only be produced artificially. Elements with atomic numbers 43 (Tc), 61 (Pm), 84 (Po) and higher have only radioactive isotopes.

The international atomic mass unit (amu) accepted today in physics and chemistry is 1/12 of the mass of the most common carbon isotope in nature: 1 amu. = 1.66053873* 10 “kg. It is close to the atomic mass of hydrogen, although not equal to it. The mass of an electron is approximately 1/1800 amu. In modern mass snectromefs, the relative error in mass measurement is

AMfM= 10 -10, which makes it possible to measure mass differences at the level of 10 -10 amu.

Atomic masses of isotopes, expressed in amu, are almost exactly integers. Thus, each atomic nucleus can be assigned its mass number A(integer), for example Н-1, Н-2, Н-З, С-12, 0-16, Cl-35, С1-37, etc. The latter circumstance revived on a new basis interest in the hypothesis of W. Prout (1816), according to which all elements are built from hydrogen.

The masses of atomic nuclei are of particular interest for identifying new nuclei, understanding their structure, predicting decay characteristics: lifetime, possible decay channels, etc.
For the first time, a description of the masses of atomic nuclei was given by Weizsäcker based on the droplet model. The Weizsäcker formula allows one to calculate the mass of an atomic nucleus M(A,Z) and the value of the binding energy of the nucleus if the mass number A and the number of protons Z in the nucleus are known.
The Weizsäcker formula for nuclear masses has the following form:

where m p = 938.28 MeV/c 2 , m n = 939.57 MeV/c 2 , a 1 = 15.75 MeV, a 2 = 17.8 MeV, a 3 = 0.71 MeV, a 4 = 23.7 MeV, a 5 = 34 MeV, = (+ 1, 0, -1), respectively, for odd-odd nuclei, kernels with odd A, even-even kernels.
The first two terms of the formula represent the sum of the masses of free protons and neutrons. The remaining terms describe the binding energy of the nucleus:

• a 1 A takes into account the approximate constancy of the specific binding energy of the nucleus, i.e. reflects the property of saturation of nuclear forces;
• a 2 A 2/3 describes the surface energy and takes into account the fact that surface nucleons in the nucleus are weaker bound;
• a 3 Z 2 /A 1/3 describes the decrease in the binding energy of the nucleus due to the Coulomb interaction of protons;
• a 4 (A - 2Z) 2 /A takes into account the property of charge independence of nuclear forces and the action of the Pauli principle;
• a 5 A -3/4 takes into account mating effects.

The parameters a 1 - a 5 included in the Weizsäcker formula are selected in such a way as to optimally describe the nuclear masses near the β-stability region.
However, from the very beginning it was clear that the Weizsäcker formula did not take into account some specific details of the structure of atomic nuclei.
Thus, the Weizsäcker formula assumes a uniform distribution of nucleons in phase space, i.e. Essentially, the shell structure of the atomic nucleus is neglected. In fact, the shell structure leads to inhomogeneity in the distribution of nucleons in the nucleus. The resulting anisotropy of the average field in the nucleus also leads to deformation of nuclei in the ground state.

The accuracy with which the Weizsäcker formula describes the masses of atomic nuclei can be estimated from Fig. 6.1, which shows the difference between the experimentally measured masses of atomic nuclei and calculations based on the Weizsäcker formula. The deviation reaches 9 MeV, which is about 1% of the total binding energy of the nucleus. At the same time, it is clearly visible that these deviations are systematic, which is due to the shell structure of atomic nuclei.
The deviation of the binding energy of nuclei from the smooth curve predicted by the liquid drop model was the first direct indication of the shell structure of the nucleus. The difference in binding energies between even and odd nuclei indicates the presence of pairing forces in atomic nuclei. Deviation from the “smooth” behavior of the energies of separation of two nucleons in nuclei between filled shells indicates the deformation of atomic nuclei in the ground state.
Data on the masses of atomic nuclei form the basis for testing various models of atomic nuclei, so the accuracy of knowledge of nuclear masses is of great importance. The masses of atomic nuclei are calculated using various phenomenological or semi-empirical models using various approximations of macroscopic and microscopic theories. The currently existing mass formulas describe the masses (binding energies) of nuclei near the stability valley quite well. (The accuracy of the binding energy estimate is ~100 keV). However, for nuclei far from the stability valley, the uncertainty in the binding energy prediction increases to several MeV. (Fig. 6.2). In Fig. 6.2 you can find links to works in which various mass formulas are presented and analyzed.

A comparison of the predictions of various models with measured nuclear masses indicates that preference should be given to models based on a microscopic description that takes into account the shell structure of nuclei. It must also be borne in mind that the accuracy of the prediction of nuclear masses in phenomenological models is often determined by the number of parameters used in them. Experimental data on the masses of atomic nuclei are given in the review. In addition, their constantly updated values ​​can be found in the reference materials of the international database system.
In recent years, various methods have been developed for experimentally determining the masses of atomic nuclei with a short lifetime.

Basic methods for determining the masses of atomic nuclei

Let us list, without going into detail, the main methods for determining the masses of atomic nuclei.

• Measuring the β-decay energy Q b is a fairly common method for determining the masses of nuclei far from the β stability limit. To determine the unknown mass undergoing beta decay of a nucleus A

,

ratio is used

M A = M B + m e + Q b /c 2.

Therefore, knowing the mass of the final nucleus B, one can obtain the mass of the initial nucleus A. Beta decay often occurs to the excited state of the final nucleus, which must be taken into account.

This relation is written for α-decays from the ground state of the initial nucleus to the ground state of the final nucleus. Excitation energies can be easily taken into account. The accuracy with which the masses of atomic nuclei are determined from decay energy is ~100 keV. This method is widely used to determine the masses of superheavy nuclei and their identification.

1. Measuring the masses of atomic nuclei using the time-of-flight method

Determination of the core mass (A ~ 100) with an accuracy of ~ 100 keV is equivalent to the relative accuracy of mass measurement ΔM/M ~10 -6. To achieve this accuracy, magnetic analysis is used in conjunction with time-of-flight measurements. This technique is used in the spectrometer SPEG - GANIL (Fig. 6.3) and TOFI - Los Alamos. Magnetic rigidity Bρ, particle mass m, its speed v and charge q are related by the relation

Thus, knowing the magnetic rigidity of spectrometer B, we can determine m/q for particles having the same speed. This method makes it possible to determine the masses of nuclei with an accuracy of ~10 -4. The accuracy of nuclear mass measurements can be improved if time of flight is simultaneously measured. In this case, the ion mass is determined from the relation

where L is the flight base, TOF is the time of flight. Flight bases range from several meters to 10 3 meters and make it possible to increase the accuracy of nuclear mass measurements to 10 -6.
A significant increase in the accuracy of determining the masses of atomic nuclei is also facilitated by the fact that the masses of various nuclei are measured simultaneously, in one experiment, and the exact values ​​of the masses of individual nuclei can be used as reference points. The method does not allow the separation of the ground and isomeric states of atomic nuclei. GANIL is creating an installation with a flight path of ~3.3 km, which will increase the accuracy of nuclear mass measurements to several units per 10 -7.

1. Direct determination of nuclear masses by measuring the cyclotron frequency

For a particle rotating in a constant magnetic field B, the rotation frequency is related to its mass and charge by the relation

Despite the fact that methods 2 and 3 are based on the same relationship, the accuracy in method 3 of measuring the cyclotron frequency is higher (~ 10 -7), because it is equivalent to using a longer span base.

2. Measuring the masses of atomic nuclei in a storage ring

   This method is used on the ESR storage ring at GSI (Darmstadt, Germany). The method uses a Schottky detector. It is applicable to determine the masses of nuclei with a lifetime > 1 min. The method of measuring the cyclotron frequency of ions in a storage ring is used in combination with on-the-fly preliminary separation of ions. The FRS-ESR facility at GSI (Figure 6.4) has made precision mass measurements of a large number of nuclei over a wide range of mass numbers.

   209 Bi nuclei, accelerated to an energy of 930 MeV/nucleon, were focused on a beryllium target with a thickness of 8 g/cm 2 located at the FRS input. As a result of fragmentation of 209 Bi, a large number of secondary particles are formed in the range from 209 Bi to 1 H. The reaction products are separated on the fly according to their magnetic hardness. The thickness of the target is selected to expand the range of nuclei simultaneously captured by the magnetic system. The expansion of the range of nuclei occurs due to the fact that particles with different charges are decelerated differently in the beryllium target. The FRS separator fragment is configured to pass particles with a magnetic rigidity of ~350 MeV/nucleon. Through the system at a selected charge range of detected nuclei (52 < Z < 83) fully ionized atoms (bare ions), hydrogen-like ions having one electron, or helium-like ions having two electrons can simultaneously pass through. Since the speed of particles practically does not change during the passage of the FRS, the selection of particles with the same magnetic rigidity selects particles with an M/Z value with an accuracy of ~ 2%. Therefore, the circulation frequency of each ion in the ESR storage ring is determined by the M/Z ratio. This forms the basis of a precision method for measuring the masses of atomic nuclei. The ion circulation frequency is measured using the Schottky method. The use of the method of cooling ions in a storage ring further increases the accuracy of mass determination by an order of magnitude. In Fig. Figure 6.5 shows a plot of the masses of atomic nuclei separated using this method in GSI. It should be borne in mind that using the described method, nuclei with a half-life greater than 30 seconds can be identified, which is determined by the beam cooling time and analysis time.

   In Fig. Figure 6.6 shows the results of determining the mass of the 171 Ta isotope in various charge states. Various reference isotopes were used in the analysis. The measured values ​​are compared with the table data (Wapstra).

3. Measuring nuclear masses using a Penning trap

   New experimental opportunities for precision measurements of the masses of atomic nuclei are opening up in a combination of ISOL methods and ion traps. For ions that have very low kinetic energy and therefore a small radius of rotation in a strong magnetic field, Penning traps are used. This method is based on precision measurement of the particle rotation frequency

   ω = B(q/m),

   trapped in a strong magnetic field. The accuracy of mass measurement for light ions can reach ~ 10 -9. In Fig. Figure 6.7 shows the ISOLTRAP spectrometer installed on the ISOL - CERN separator.
   The main elements of this installation are the ion beam preparation sections and two Penning traps. The first Penning trap is a cylinder placed in a magnetic field of ~4 T. The ions in the first trap are further cooled due to collisions with a buffer gas. In Fig. Figure 6.7 shows the mass distribution of ions with A = 138 in the first Penning trap depending on the rotation frequency. After cooling and purification, the ion cloud from the first trap is injected into the second. Here the ion mass is measured using the resonant rotation frequency. The resolution achievable in this method for short-lived heavy isotopes is the highest and is ~ 10 -7.

   Rice. 6.7 ISOLTRAP spectrometer

Atomic nucleus is the central part of an atom, consisting of protons and neutrons (together called nucleons).

The nucleus was discovered by E. Rutherford in 1911 while studying the transmission α -particles through matter. It turned out that almost the entire mass of the atom (99.95%) is concentrated in the nucleus. The size of the atomic nucleus is of the order of magnitude 10 -1 3 -10 - 12 cm, which is 10,000 times smaller than the size of the electron shell.

Proposed by E. Rutherford planetary atomic model and his experimental observation of nuclei hydrogen, knocked out α -particles from the nuclei of other elements (1919-1920), led the scientist to the idea of proton. The term proton was introduced in the early 20s of the XX century.

Proton (from Greek. protons- first, symbol p) is a stable elementary particle, the nucleus of a hydrogen atom.

Proton- a positively charged particle whose absolute charge is equal to the charge of an electron e= 1.6 · 10 -1 9 Cl. The mass of a proton is 1836 times greater than the mass of an electron. Proton rest mass m r= 1.6726231 · 10 -27 kg = 1.007276470 amu

The second particle included in the nucleus is neutron.

Neutron (from lat. neutral- neither one nor the other symbol n) is an elementary particle that has no charge, i.e. neutral.

The mass of a neutron is 1839 times greater than the mass of an electron. The mass of a neutron is almost equal (slightly greater) to the mass of a proton: the rest mass of a free neutron m n= 1.6749286 10 -27 kg = 1,0008664902 a.e.m. and exceeds the mass of a proton by 2.5 times the mass of an electron. Neutron, along with proton under the general name nucleon is part of atomic nuclei.

The neutron was discovered in 1932 by E. Rutherford's student D. Chadwig during the bombardment of beryllium α -particles. The resulting radiation with high penetrating power (overcame a barrier made of a lead plate 10-20 cm) enhanced its effect when passing through a paraffin plate (see figure). An assessment of the energy of these particles from tracks in a cloud chamber made by the Joliot-Curie couple and additional observations made it possible to exclude the initial assumption that this γ -quanta. The greater penetrating ability of the new particles, called neutrons, was explained by their electrical neutrality. After all charged particles actively interact with matter and quickly lose their energy. The existence of neutrons was predicted by E. Rutherford 10 years before the experiments of D. Chadwig. When hit α -particles into beryllium nuclei the following reaction occurs:

Here is the symbol for the neutron; its charge is zero, and its relative atomic mass is approximately equal to unity. Neutron is an unstable particle: a free neutron for a time of ~ 15 min. decays into a proton, electron and neutrino - a particle devoid of rest mass.

After the discovery of the neutron by J. Chadwick in 1932, D. Ivanenko and V. Heisenberg independently proposed proton-neutron (nucleon) model of the nucleus. According to this model, the nucleus consists of protons and neutrons. Number of protons Z matches the ordinal number of the element in table of D. I. Mendeleev.

Core charge Q determined by the number of protons Z, included in the nucleus, and is a multiple of the absolute value of the electron charge e:

Q = +Ze.

Number Z called charge number of the nucleus or atomic number.

Mass number of the nucleus A is the total number of nucleons, i.e. protons and neutrons contained in it. The number of neutrons in the nucleus is indicated by the letter N. So the mass number is:

A = Z + N.

Nucleons (proton and neutron) are assigned a mass number equal to one, and an electron is assigned a mass number of zero.

The idea of ​​the composition of the nucleus was also facilitated by the discovery isotopes.

Isotopes (from Greek. isos- equal, identical and topoa- place) are varieties of atoms of the same chemical element, the atomic nuclei of which have the same number of protons ( Z) and different numbers of neutrons ( N).

The nuclei of such atoms are also called isotopes. Isotopes are nuclides one element. Nuclide (from lat. nucleus- nucleus) - any atomic nucleus (respectively, an atom) with given numbers Z And N. The general designation of nuclides is……. Where X- symbol of a chemical element, A = Z + N- mass number.

Isotopes occupy the same place in Periodic table elements, which is where their name comes from. Isotopes, as a rule, differ significantly in their nuclear properties (for example, in their ability to enter into nuclear reactions). The chemical (and almost to the same extent physical) properties of isotopes are the same. This is explained by the fact that the chemical properties of an element are determined by the charge of the nucleus, since it is this charge that affects the structure of the electron shell of the atom.

The exception is the isotopes of light elements. Isotopes hydrogen 1 N — protium, 2 N— deuterium, 3 N — tritium differ so greatly in mass that their physical and chemical properties are different. Deuterium is stable (i.e. not radioactive) and is included as a small impurity (1: 4500) in ordinary hydrogen. When deuterium combines with oxygen, heavy metal is formed water. At normal atmospheric pressure it boils at 101.2 °C and freezes at +3.8 °C. Tritium β -radioactive with a half-life of about 12 years.

All chemical elements have isotopes. Some elements have only unstable (radioactive) isotopes. Radioactive isotopes have been artificially obtained for all elements.

Isotopes of uranium. The element uranium has two isotopes - with mass numbers 235 and 238. The isotope is only 1/140th of the more common one.