If tunnelling out of nuclei is possible then so is tunnelling in! As a consequence it is possible to trigger nuclear reactions with protons of much lower energy than would be needed to climb over the full height of the Coulomb barrier. This was the principle used by J.D. Cockcroft and E.T.S. Walton in 1932 when they caused lithium-7 nuclei to split into pairs of alpha particles by bombarding them with high-energy protons. Their achievement won them the 1951 Nobel prize for physics. The same p

Session 2 discusses the scattering of a particle using wave packets. We shall restrict attention to one dimension and suppose that the incident particle is initially free, described by a wave packet of the form

This is a superposition of de Broglie waves, with the function

The content acknowledged below is Proprietary (see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

The author of this unit is Peter Sheldon.

Grateful acknowledgement is made to the following sources for permission to reproduce material

4.4 Other Wenlock Limestone fossils

Among the other fossils common in the Wenlock Limestone are brachiopods (Figure 12a and b), gastropods (Figure 12c) and bryozoans (Figure 12d). You may need to reread Section 1.3 to remind yourself about various aspects of these groups.

Figure 13 (the unit image) is a reconstruction of a typical scene from a Wenlock Limestone environment. See

As we've seen, the Cambrian explosion left the seas teeming with a huge variety of animals. In the following activity you will study some of the marine life at one particular time in the Palaeozoic Era â€“ the middle part of the Silurian Period, 430Â Ma ago. You'll look in detail at some fossils which come from a deposit in the UK called the Wenlock Limestone, famous for its many beautiful fossils. The Wenlock Limestone crops out mainly around Birmingham and the borders of Wales.

Figure

Whatever age they are, men, women and children can all do something to try to prevent future cardiovascular diseases in themselves or their families by eating a balanced diet (see Section 4.6), taking more exercise and modifying their lifestyles to reduce any other known risk factors. If cardiovascular diseases are pre-existi

2.10.1 Mean and standard deviation for repeated measurements

In everyday terms, everybody is familiar with the word â€˜averageâ€™, but in science and statistics there are actually several different kinds of average used for different purposes. In the kind of situation exemplified by Table 2, the sort to use is the **mean**
(or more strictly the â€˜arithmetic meanâ€™) For a set of measurements, this is de

Real functions and graphs

Sometimes the best way to understand a set of data is to sketch a simple graph. This exercise can reveal hidden trends and meanings not clear from just looking at the numbers. In this unit you will review the various approaches to sketching graphs and learn some more advanced techniques. First published on Tue, 28 Jun 2011 as

4.2 Least upper and greatest lower bounds

We have seen that the set [0, 2) has no maximum element. However, [0, 2) has many upper bounds, for example, 2, 3, 3.5 and 157.1. Among all these upper bounds, the number 2 is the *least* upper bound because any number less than 2 is not an upper bound of [0, 2).

2.3 Inequalities involving modulus signs

Now we consider inequalities involving the *modulus* of a real number. Recall that if *a* , then its **modulus**, or **abso**

**The set of natural numbers is
the set of integers is
and the set of rational numbers is
Author(s): **

**Despite the list of advantages given, here is a word of warning: a calculator is not a substitute for a brain! Even when you are using your calculator, you will still need to sort out what calculation to do to get the answer to a particular problem. However skilled you are at using your calculator, if you do the wrong sum, you will get the wrong answer. The phrase â€˜garbage in, garbage outâ€™ applies just as much to calculators as to computers. Your calculator is just that â€“ a calculator!<**

**The calculator is very useful for ordinary arithmetic and yet it can also perform many functions commonly associated with a computer and deal with quite advanced mathematics. It is useful for both beginners and experts alike, because it has a variety of modes of operation.**

**The calculator retains numbers, formulas and programs which you have stored in it, even when it is turned off. You can recall them when you need them and so save time by not having to enter the same information again.**

**The calculator does not make mistakes in the way that human brains tend to. Human fingers do, however, make mistakes sometimes; and the calculator may not be doing what you think you have told it to do. So correcting errors and estimating the approximate size of answers are important skills in double-checking your calculator calculations. (Just as they are for checking calculations done in your head or on paper!)**

**You can see the calculations that you have entered as well as the answers. This means you can easily check whether you have made any mistakes.**

**6.2 Getting the feel of big and small numbers **

**Very small and very large numbers can be difficult to comprehend. Nothing in our everyday experience helps us to get a good feel for them. For example numbers such as 10 ^{99} are so big that if Figure 1 was drawn to scale, you would be dealing with enormous distances. How big is big?**

**First express 1â€‰000â€‰000â€‰000 in scientific notation as 10 ^{9}. Next, to find out how many times bigger 10^{99} is, use your calculator to divide 10^{99} by 10^{9Author(s): The Open University}**

**The aim of this section is to help you to think about how you study mathematics and consider ways in which you can make your study more effective.**

**Many people's ideas about what mathematics actually is are based upon their early experiences at school. The first two activities aim to help you recall formative experiences from childhood.**

**Activity 1 Carl Jung's school days**

**Read**

**This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics.**

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In order to complete this unit you will need**

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