Random number generator

The random number generator used was the internal procedure in XL For­ tran. In order to ensure that the generator was suitable for simulation work, a number of statistical tests were performed. These were taken from Knuth’s book [76], and are presented here with only brief details.

The desirable criteria for the random number generator are that the num­ bers produced are equidistributed between 0 and 1, and that they are inde­ pendent on successive requests. The requirement of independence cannot be strictly met with any algorithm on a computer, as it is deterministic with finite storage space, thus must eventually repeat. The purpose of the tests, then, is to give some assurance that the values are independent enough, for the purposes desired.

The tests work by collecting statistics, and then applying the chi-squared 57

test, for discrete categories, or the Kolmogorov-Smirnov test for a continuum of possible values. These tests then give a number, describing the likelihood of an equidistributed random sequence producing that set of values. Both tests rely on the number of observations being large. It is not possible to quantify how large, however, because as the number of observations, n, increases, globally non-random behaviour is more apparent, but locally non-random behaviour is less apparent. Each test should therefore be run with a series of values of n. Passing these tests is a necessary, but not sufficient test of randomness.

The equidistribution test examines the spread of numbers to see if they generated uniformly between 0 and 1. This test was performed by choosing a convenient integer, d, say 100, and multiplying the sequence by this number, and taking the integer part, producing a sequence of integers in the range 0 to d — 1. The frequency of each number was counted and a chi-squared test with d degrees of freedom, with the probability ^ for each category was applied. This can also be done by applying a Kolmogorov-Smirnov test to

the reals produced with the distribution function F{x) = x.

In addition to testing that the sequence generates the full spectrum of numbers, a random sequence has no relation between successive pairs of numbers. To test for this, the sequence of real numbers is, again, converted to integers in the same way as above. The frequency of successive pairs of integers are counted, noting that each number can be counted only once. A chi-squared test is then applied to these categories, with cf degrees of freedom, and probability of ^ of each category. It is apparent that this test, the serial test, can be generalised to 3, 4, and higher sets. As the purpose

of the tests is to ensure that the simulation will be ergodic, it is sufficient to test triples, on the basis that there is one number used per site, and that any short range correlation shorter than three will impact on the overall results.

All tests were reduced to a chi-squared test, where 1000 and 10000 ob­ servations on average per category were recorded. Each test was run 100 times at each level of observation. Table 2.5.1 gives the results for 1000 observations, and table 2.5.1 for 10000 observations.

These tables consist of a banding of the results of the chi-squared test, into likely hood bands. These represent the probability of sequence of independent values evenly distributed between 0 and 1 giving rise to the observations noted. For a good generator, the probability of any given chi-squared value should be proportional to the size of the category, although for a truly random number generator, there will be variance from that.

The meaning of the codes in the tests column is given in table 2.5.1.

Test Categorised Results >99% 99-95% 95-75% 75-50% 50-25% 25-5% 5-1% <1% Eq(lO) 0 0 16 35 33 16 0 0 Eq(20) 0 1 13 28 37 20 1 0 Eq(30) 0 1 11 30 43 14 1 0 Eq(40) 0 1 10 39 34 16 0 0 Eq(50) 0 18 35 26 19 2 0 Eq(lOO) 0 1 20 29 30 19 1 0 Eq(150) 0 1 12 33 37 15 2 0 Eq(200) 0 14 28 38 17 0 0 Pair(lO) 0 1 12 25 45 16 1 0 Pair(20) 0 1 19 33 29 18 0 0 Pair(30) 0 1 13 31 31 24 0 0 Pair(40) 0 14 36 30 19 1 0 Pair(50) 0 1 13 33 36 14 3 0 Pair(60) 0 1 12 28 37 21 1 0 Pair(70) 0 1 17 30 34 17 1 0 Pair(80) 0 0 8 22 43 21 5 1 Pair(90) 0 0 19 29 33 17 2 0 Pair (100) 0 0 15 26 37 20 2 0 TYip(lO) 0 0 16 30 38 12 4 0 Trip(15) 0 2 7 40 27 23 1 0 Trip(20) 0 2 19 35 23 21 0 0 Trip(25) 0 6 18 36 36 4 0 0 Trip(30) 0 0 12 28 31 24 5 0

Test Categorised Results >99% 99-95% 95-75% 75-50% 50-25% 25-5% 5-1% <1% Eq(lO) 0 0 21 32 32 15 0 0 Eq(20) 0 1 22 33 27 17 0 0 Eq(30) 0 0 19 34 32 15 0 0 Eq(40) 0 0 17 35 31 16 1 0 Eq(50) 0 1 20 27 30 21 1 0 Eq(lOO) 0 2 27 31 25 14 1 0 Eq(150) 0 5 24 33 30 8 0 0 Eq(200) 0 3 25 33 36 3 0 0 Pair(lO) 0 0 12 30 38 18 2 0 Pair(20) 0 0 14 37 26 19 4 0 Pair(30) 0 0 15 33 36 16 0 0 Pair(40) 0 1 17 28 32 20 2 0 Pair(50) 0 0 14 34 32 20 0 0

Table 2.4: Chi-squared results of randomness test for 10000 observations

Code Description

Eq(n) Pair(n) Trip(m)

Equidistribution test for n categories Serial test, on pairs, for n categories Serial test, on triplets, for n categories Table 2.5: Description of the test codes given in tables

Chapter 3

(111) layers

3.1 Introduction

Experimental results for FeO (lll) on Fe [77] found the surface to have ferro­ magnetic order, at temperatures above the bulk Neel temperature. A num­ ber of possible explanations have been given, most considering some surface reconstruction to be responsible. Phase diagrams of Hjsmg projected onto (100) layers of FCC lattice by Mackrodt and Noguera [78] found multicrit- ical behaviour, and a quantitative description of the reduction of the Neel temperature in films found by Alders et al [69].

Where the (100) film ordered with AF2 ordering, as most magnetic top row transition monoxides do, if there are an odd number of layers in the film, the outermost layers have a distinct, and slightly lower, transition tempera­ ture from the subsurface layers. Surprisingly, where there are an even number of layers in the film all the layers share a single critical temperature. The position of the phase boundary between AF2 and the ferromagnetic phase

was also predicted to move slightly, although this was quite a small change. The agreement with Alders et al experimental data was good, confirming that the Ising model is a good predictor of the observable change in Neel temperature with film thickness.

In light of these studies, the Ising Hamiltonian projected onto the (111)

layers was studied. One aim was to examine the behaviour for any behaviour j

not seen in the bulk, such as the multicritical behaviour for the (100) layers, ;

and also to provide a foundation for examining the effect of any surface re­

constructions on the magnetic order. As a first step towards understanding ;

I

the behaviour of real films, unreconstructed, stoichiometric films were exam­ ined. For the thinest films these may correspond to physically meaningful cases, and serve as an initial approximation for other cases.

In the (111) orientation rocksalt type structures present a polar surface [20], which has the direct consequence that, in general, it must be stabilised to avoid large surface energies. Nevertheless, examining the behaviour of the perfect structure is an instructive first approximation to the real world behaviour of these systems. Additionally, as the dipole moment is dependent on the thickness of a film, it is possible to construct thin, stoichiometric films of the (111) surface.

For NiO (111) surfaces on A u(lll) [22] and N i(lll), low energy electron diffraction (LEED) experiments show a p(2x2) reconstruction, where recent [79] grazing incidence x-ray diffraction (GIXD) experiments have confirmed the octopolar reconstruction, with Ni termination of a single crystal, and both terminations observed on a thin film. This was kept in mind with the code developed here, so as to allow maximal code reuse.