<< Chapter < Page Chapter >> Page >
r = r 0 A 1 / 3 , size 12{r=r rSub { size 8{0} } A rSup { size 8{1/3} } } {}

where r 0 = 1.2 fm and A is the mass number of the nucleus. Note that r 3 A size 12{r rSup { size 8{3} } µA} {} . Since many nuclei are spherical, and the volume of a sphere is V = ( 4 / 3 ) πr 3 size 12{V= \( 4/3 \) pr rSup { size 8{3} } } {} , we see that V A size 12{V prop A} {} —that is, the volume of a nucleus is proportional to the number of nucleons in it. This is what would happen if you pack nucleons so closely that there is no empty space between them.

This figure shows group of small green and blue spherical objects placed very close to each other forming a bigger sphere representing the nucleus. Blue spheres are labeled as protons and green spheres are labeled as neutrons.
A model of the nucleus.

Nucleons are held together by nuclear forces and resist both being pulled apart and pushed inside one another. The volume of the nucleus is the sum of the volumes of the nucleons in it, here shown in different colors to represent protons and neutrons.

How small and dense is a nucleus?

(a) Find the radius of an iron-56 nucleus. (b) Find its approximate density in kg / m 3 , approximating the mass of 56 Fe to be 56 u.

Strategy and Concept

(a) Finding the radius of 56 Fe is a straightforward application of r = r 0 A 1 / 3 , given A = 56 . (b) To find the approximate density, we assume the nucleus is spherical (this one actually is), calculate its volume using the radius found in part (a), and then find its density from ρ = m/V . Finally, we will need to convert density from units of u / fm 3 to kg / m 3 .

Solution

(a) The radius of a nucleus is given by

r = r 0 A 1 / 3 .

Substituting the values for r 0 and A yields

r = (1.2 fm)(56) 1/3 = (1.2 fm)(3.83) = 4.6 fm .

(b) Density is defined to be ρ = m/V , which for a sphere of radius r is

ρ = m V = m (4/3) πr 3 .

Substituting known values gives

ρ = 56 u (1.33)(3.14) (4.6 fm) 3 = 0.138 u/ fm 3 .

Converting to units of kg / m 3 , we find

ρ = (0.138 u/ fm 3 ) ( 1.66 × 10 –27 kg/u ) ( 1 fm 10 –15 m ) = 2.3 × 10 17 kg/m 3 .

Discussion

(a) The radius of this medium-sized nucleus is found to be approximately 4.6 fm, and so its diameter is about 10 fm, or 10 –14 m . In our discussion of Rutherford’s discovery of the nucleus, we noticed that it is about 10 –15 m in diameter (which is for lighter nuclei), consistent with this result to an order of magnitude. The nucleus is much smaller in diameter than the typical atom, which has a diameter of the order of 10 –10 m .

(b) The density found here is so large as to cause disbelief. It is consistent with earlier discussions we have had about the nucleus being very small and containing nearly all of the mass of the atom. Nuclear densities, such as found here, are about 2 × 10 14 times greater than that of water, which has a density of “only” 10 3 kg/m 3 . One cubic meter of nuclear matter, such as found in a neutron star, has the same mass as a cube of water 61 km on a side.

Got questions? Get instant answers now!

Nuclear forces and stability

What forces hold a nucleus together? The nucleus is very small and its protons, being positive, exert tremendous repulsive forces on one another. (The Coulomb force increases as charges get closer, since it is proportional to 1 / r 2 size 12{1/r rSup { size 8{2} } } {} , even at the tiny distances found in nuclei.) The answer is that two previously unknown forces hold the nucleus together and make it into a tightly packed ball of nucleons. These forces are called the weak and strong nuclear forces . Nuclear forces are so short ranged that they fall to zero strength when nucleons are separated by only a few fm. However, like glue, they are strongly attracted when the nucleons get close to one another. The strong nuclear force is about 100 times more attractive than the repulsive EM force, easily holding the nucleons together. Nuclear forces become extremely repulsive if the nucleons get too close, making nucleons strongly resist being pushed inside one another, something like ball bearings.

Questions & Answers

material that allows electric current to pass through
Deng Reply
material which don't allow electric current is called
Deng
insulators
Covenant
how to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask