Download Basic properties of atomic nuclei

Basic properties of atomic nuclei
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radius of a nucleus R = R0 A 3
where R0 is an experimentally determined constant:
R0 = 1.2 × 10 −15 m = 1.2 fm
A is called the nucleon number.
The nucleon number A is also called the mass number because it is the nearest whole
number to the mass of the nucleus measured in unified atomic mass units (u).
The volume V of a sphere is equal to
V = 4πR 3 / 3 so the volume of a nucleus is proportional to A:
V = 4πR03 A / 3
The most common kind of iron nucleus has a mass number of 56. Find the radius,
approximate mass, and approximate density of the nucleus .
The masses of the particles:
Proton: mp = l.007276 u = l.672622 x 10-27 kg
Neutron: mn = l.008665 u = l.674927 x 10-27 kg
Electron: me = 0.000548580 u = 9.10938 x 10-31 kg
A=Z+N
A single nuclear species having specific values of both Z and N is called a nuclide.
The magnitude of the spin angular momentum S of a nucleon is S =
2
h.
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the z-component is S z = ± h
2
orbital angular momentum J =
j ( j + 1)h
z-component J z = m j h where: m j = (− j ,− j + 1,..., j − 1, j )
When the total number of nucleons A is even, j is an integer; when it is odd, j is a
half-integer. All nuclides for which both Z and N are even have 1 = 0, which suggests
that pairing of particles with opposite spin components may be an important
consideration in nuclear structure.
nuclear magneton µn:
eh
µn =
= 5.05078 × 10 -27 J/T
2m p
Because the proton mass mp is 1836 times larger than the electron mass me, the
nuclear magneton µn is 1836 times smaller than the Bohr magneton µB.
When a nucleus is placed in an external magnetic field B, there is an interaction
energy U = -µ
µ·B = -µzB just as with atomic magnetic moments. The components of the
magnetic moment in the direction of the field µz are quantized, so a series of energy
levels results from this interaction.
Properties of Nuclei
43.1. How many protons and how many neutrons are there in a nucleus of the most
common isotope of (a) silicon, 1428 Si ; (b) rubidium, 3785 Rb ; (c) thallium, 205
81Tl ?
43.2. Consider the three nuclei of Exercise 43.1.
Estimate
(a) the radius,
(b) the surface area, and
(c) the volume of each nucleus.
Determine
(d) the mass density (in kg/m3) and
(e) the nucleon density (in nucleons per cubic meter) for each nucleus.
Assume that the mass of each nucleus is A atomic mass units.
43.3. Hydrogen atoms are placed in an external magnetic fleld. The protons can
make transitions between states in which the nuclear spin component is parallel and
antiparallel to the fleld by absorbing or emitting a photon. What magnetic-fleld
magnitude is required for this transition to be induced by photons with frequency 22.7
MHz?
43.4. Neutrons are placed in a magnetic field with magnitude 2.30 T.
(a) What is the energy difference between the states with the nuclear spin angular
momentum components parallel and anti parallel to the field? Which state is lower in
energy, the one with its spin component parallel to the fleld or the one with its spin
component antiparallel to the field? How do your results compare with the energy
states for a proton in the same fleld?
(b) The neutrons can make transitions from one of these states to the other by
emitting or absorbing a photon with energy equal to the energy difference of the two
states. Find the frequency and wavelength of such a photon.
43.5. Hydrogen atoms are placed in an external 1.6S-T magnetic field.
(a) The protons can make transitions between states where the nuclear spin
component is parallel and anti parallel to the fleld by absorbing or emitting a photon.
Which state has lower energy: the state with the nuclear spin component parallel or
anti parallel to the field? What are the frequency and wavelength of the photon?
In which region of the electromagnetic spectrum does it lie?
(b) The electrons can make transitions between states where the electron spin
component is parallel and anti parallel to the fleld by absorbing or emitting a photon.
Which state has lower energy: the state with the electron spin component parallel or
anti parallel to the fleld? What are the frequency and wavelength of the photon?
In which region of the electromagnetic spectrum does it lie?