The neutron
One characteristic of the atomic nuclei is their relatively great mass.There is only one known particle of the comparable mass which is not an atomic nucleus, namely, the neutron. Its mass is closely equal to the mass of the proton. (Actually the neutron is about 0.1 per cent heavier than the proton). The property that sets the neutron apart from the atomic nuclei is that it does not carry any charge and therefore does not attract electrons and does not surround itself with an electronic shell. Neutrons are produced in some close nuclear collisions, that is, in collisions in which nuclei get into contact with each other.
The only interaction of neutrons with atomic nuclei is one of short range which is of the same type as the forces giving rise to anomalous alpha-particle scattering. Thus a neutron must as a general rule get to the surface of a nucleus in order that it should be deflected. The only established interaction of neutrons with electrons is a weak force of the magnetic type. Further short-range interactions do not exist or are extremely small. It can be shown that owing to the small mass of the electrons these weak forces are particularly ineffective in the interaction of free electrons with neutrons. The probability of electronic excitation by neutron impact is very small. But electrons in atomic orbits can influence the path of a neutron with higher probability if during a collision the electrons do not become excited. In this case the electrons act as parts of the atom and can be said to possess effectively the mass of the whole atom. In such collisions the weak magnetic interaction was detected. All interactions of neutrons with electrons seem, however, to be of small importance in our discussion.
The most important interaction of neutrons with matter remains the collisions with atomic nuclei. According to a geometrical picture these collisions ought to have a cross section of the order of 10-24 cm2. This could lead to a mean free path of a neutron in a solid which may be longer than a centimeter. As a general rule neutrons do penetrate solids as is suggested by this long free path. This fact is a very direct illustration of the small extension of nuclear particles.
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Algebraic language
Use of letters. - Letters are used to express the general properties of numbers. Suppose that we want to express briefly in a written form that the product of two numbers remains unaltered when we interchange the position of the multiplic and the multiplier. Then, representing one of the numbers by the letter a and the other by the letter b, we shall be able to write the equality: a×b=b×a, or, shortly: ab=ba, having agreed once and for all that the multiplication sign is understood between any two letters, written side by side, if no other sign is indicated. Consequently, letters are always used to express that a certain property is peculiar to numbers in general and not to any particular numbers.
Letters of the Latin alphabet are generally used to represent numbers.
Algebraic expression. - An algebraic expression is an expression in which several numbers represented by letters (or by letters and figures) are connected by means of signs indicating the operations to which the numbers must be subjected and the order of these operations.
Such are, for instance, the expressions:
For the sake of brevity we shall often simply say “expression” instead of “algebraic expression”.
To evaluate an
expression when the numerical values of
the letters are given, is to substitute the numerical equivalents for
the letters and perform the operations indicated in the expression.
The number obtained is known as the numerical value of the algebraic
expression for the given numerical equivalents of the letters. Hence,
the numerical value
of the expression
whenp=3
and a=520
is
=5.2×3=15.6
Order of operations. - With regard to the order in which the operations indicated in an algebraic expression should be performed, it was agreed upon to perform the operations of the higher order first, i.e., involution and evolution, then multiplication and division, and, finally, addition and subtraction.
Thus, if we have the expression
we must, when evaluating it, first perform the involution (square the
numbera
and cube the numbet b),
then the multiplication and division (multiply 3 by a2
and the result obtained by b;
divide b3
by c) and,
finally, the subtraction and addition (subtract from 3a2b
and add d
to the result).
Notion of Values which may be taken in two opposite senses. - Problem.- At midnight a thermometer read 2° and at noon 5°. How many degrees did the temperature change between midnight and noon?
The conditions of this problem are not sufficiently clear; we must know whether the reading at midnight was 2° below or above 0°, the same must be mentioned for the noon reading. If e.g. both at midnight and at noon the temperature was above 0°, then during the given period of time the temperature rose from 2° to 5°, i. e. 3°; while if the temperature at midnight was 2° below zero and at noon it was 5° above zero, the temperature rose 2 + 5, i. e. 7°, and so on.
In this problem we deal with a quantity having a direction: the number of degrees may be read either upwards or downwards from zero. The temperature above 0° (heat) is known as positive and is recorded as the number of degrees preceded by the + sign, and the temperature below 0° (cold) is known as negative and is recorded as the number of degrees preceded by the - sign (there will be no misunderstanding if the first reading is taken without any sign at all).
Now let us formulate our problem, for instance, as follows: At midnight a thermometer read -2° and at noon it read + 5°. What is the change in temperature between midnight and noon? As it is, the problem has a definite answer: The temperature rose 2 + 5, 1. e. 7°.
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