# Beryl May Dent

Beryl May Dent

MIEE
Dent in 1928
Born10 May 1900
Chippenham, Wiltshire, England
Died9 August 1977 (aged 77)
Worthing, West Sussex, England
Resting placeWorthing Crematorium (ashes interred)
Alma mater
AwardsAshworth Hallett scholarship
1923
Scientific career
Fields
Other fields
Institutions
ThesisSome theoretical determinations of crystal structure (1927)

Beryl May Dent MIEE (10 May 1900 – 9 August 1977) was an English mathematical physicist, technical librarian, and a programmer of early analogue and digital computers to solve electrical engineering problems. She was born in Chippenham, Wiltshire, the eldest daughter of schoolteachers. The family left Chippenham in 1901, after her father became head teacher of the then recently established Warminster County School. In 1923, she graduated from the University of Bristol with First Class Honours in applied mathematics. She was awarded the Ashworth Hallett scholarship by the university and was accepted as a postgraduate student at Newnham College, University of Cambridge.

She returned to Bristol in 1925, after being appointed a researcher in the Physics Department at the University of Bristol, with her salary being paid by the Department of Scientific and Industrial Research. In 1927, John Lennard-Jones was appointed Professor of Theoretical physics, a chair being created for him, with Dent becoming his research assistant in theoretical physics. Lennard‑Jones pioneered the theory of interatomic and intermolecular forces at Bristol and she became one of his first collaborators. They published six papers together from 1926 to 1928, dealing with the forces between atoms and ions, that were to become the foundation of her master's thesis. Later work has shown that the results they obtained had direct application to atomic force microscopy by predicting that non-contact imaging is possible only at small tip-sample separations.

In 1930, she joined Metropolitan-Vickers Electrical Company Ltd, Manchester, as a technical librarian for the scientific and technical staff of the research department. She became active in the Association of Special Libraries and Information Bureaux (ASLIB) and was honorary secretary to the founding committee for the Lancashire and Cheshire branch of the association. She served on various ASLIB committees and made conference presentations detailing different aspects of the company's library and information service. She continued to publish scientific papers, contributing numerical methods for solving differential equations by the use of the differential analyser that was built for the University of Manchester and Douglas Hartree. She was the first to develop a detailed reduced major axis method for the best fit of a series of data points.

Later in her career she became leader of the computation section at Metropolitan-Vickers, and then a supervisor in the research department for the section that was investigating semiconducting materials. She joined the Women's Engineering Society and published papers on the application of digital computers to electrical design. She retired in 1960, with Isabel Hardwich, later a fellow and president of the Women's Engineering Society, replacing her as section leader for the women in the research department. In 1962, she moved with her mother and sister to Sompting, West Sussex, and died there in 1977.

## Early life

Dent's father, Eustace Edward, photographed at Warminster County School in 1926
The family lived at Boreham Road before moving in 1907 to 22 Portway, Warminster

Beryl May was born on 10 May 1900, at Penley Villa, Park Lane, Chippenham, Wiltshire, the eldest daughter of Agnes Dent (1869–1967), née Thornley, and Eustace Edward (1868–1954).[1] She was baptised at St Paul's, Chippenham, on 8 June 1900.[2]: 1  They had married at St Mary's Church, Goosnargh, near Preston, Lancashire, on 27 July 1898.[3] Her mother was educated at the Harris Institute, Preston, passing examinations in science and art.[4] She was a teacher at Attercliffe School, in northeast Sheffield, before moving to Goosnargh School, near her hometown of Preston, where her elder brother and sister, John William and Mary Ann Thornley, were the head teachers.[5][6] In March 1894, she had applied for the headship at Fairfield School, Cockermouth, making the shortlist, but the board decided to appoint a local candidate.[7]

On 18 March 1889, Dent's father was appointed to a teaching assistant position at Portland Road School, in Halifax, West Yorkshire, after completing a teaching apprenticeship with the school board.[8][9] In the same year, Florence Emily Dent, his elder sister, was appointed head teacher at West Vale girls' school, Stainland Road, Greetland, moving from the Higher Board School at Halifax.[10] In August 1889, he obtained a first class pass in mathematics from the Halifax Mechanics' Institute.[11][a] He enrolled on a degree course at University College, Aberystwyth, in the Education Day Training College.[b] In January 1894, he was awarded a first by Aberystwyth, and a first in the external University of London examinations.[13][14][c] His first teaching post was at Coopers' Company Grammar School, Bow, London,[15] before moving to Chippenham, where he was a senior assistant teacher at the Chippenham County School.[16]

In October 1901, Dent's father left Chippenham to become head teacher of the then recently established Warminster County School, that adjoined the Athenaeum Theatre in Warminster.[17][18][d] The family moved to Boreham Road, Warminster, where houses were built in the early 19th century.[21][22][e] In April 1907, they moved to 22 Portway, Warminster, situated a short distance from the County School and the Athenaeum.[20]: 264 [23] He was elected chair of the Warminster Urban District Council from 1920 to 1922,[24][f] and remained as head teacher of the County School until his retirement in August 1929.[25][g]

Dent's father was also a regular cast member of the Warminster Operatic Society at the Athenaeum and other venues. Dent and her younger sister, Florence Mary, would often appear with him on stage in such operettas as Snow White and the seven dwarfs and the Princess Ju‑Ju (The Golden Amulet), a Japanese operetta in three acts by Clementine Ward.[28][29][30] In Princess Ju‑Ju, she played La La, one of the three maidens attendant on the princess, and sang the first act solo, She must be demure.[31][32] In act two of the same musical, she performed in the fan dance, Spirits of the Night.[29][32][h] She also acted in a scene from Tennyson's Princess at the County School's prize giving day on 16 December 1913.[35][i]

## Education

### Warminster County School (1909–1917)

Former Warminster County School, where Dent's father was head teacher

From 1909, Dent was educated at Warminster County School, where her father was head teacher.[37] At school, she was close friends with her neighbour at Portway, Evelyn Mary Day, the eldest daughter of Henry George Day, a former butler to Colonel Charles Gathorne Gathorne‑Hardy, son of Gathorne Gathorne-Hardy, 1st Earl of Cranbrook.[38][39][j] In August 1914, she passed the University of Oxford Junior Local Examination with First Class Honours, and on the strength of her examination result, she was awarded a scholarship by the school.[40][41] In 1915, she passed the senior examination with second class honours and a distinction in French, and subsequently, her scholarship was renewed.[41][42] She then joined the sixth form and won the school prize for French in December 1916.[43][k] In March 1918, she applied for a scholarship in mathematics from Somerville College, one of the first two women's colleges in the University of Oxford. She was highly commended but was not awarded a scholarship nor an exhibition.[44]

### University of Bristol (1919–1923)

Paul Dirac, Dent's fellow student on the honours course in mathematics at Bristol

In 1918, Dent joined the Royal Aircraft Establishment (RAE) at Farnborough, Hampshire.[37] The First World War opened new employment opportunities for women, and RAE was one of the first military establishments to recruit women into engineering, and mathematical and computational research.[45]: 116 [46]: 10  In the same period that Dent was at RAE, Lorna Swain, then mathematics tutor at Newnham College in the University of Cambridge, worked at the establishment on the problem of aircraft propeller vibration.[45]: 84  The Treasury reduced RAE's funding after the end of the war, and consequently, the number of resources and staff available to support research fell significantly.[47]: 58  In 1919, she left RAE after being accepted on to the general Bachelor of Arts (BA) degree course at the University of Bristol.[37][l] In June 1920, she passed her intermediate examination in French with supplementary courses in Latin, history, and mathematics.[49][50]:

In the following academic year, Dent joined the honours course in mathematics and took an intermediate examination in physics.[51][50]:  After spending the summer of 1921 at her parent's home in Warminster,[52] she returned for the start of the 1921 to 1922 academic year to find that Paul Dirac had joined the mathematics course.[51] The course of mathematics at Bristol University normally lasted three years, but because of Dirac's previous training, the Department of Mathematics had allowed him to join in the second year.[51] They were taught applied mathematics by Henry Ronald Hassé, the then head of the Mathematics Department, and pure mathematics by Peter Fraser. Both of them had come from Cambridge;[51] Fraser had been appointed in 1906 to the staff of the Bristol University College, soon to become the University of Bristol, and Hassé joined him in 1919 as professor of mathematics.[53]: 111  Fraser introduced them to mathematical rigour, projective geometry, and rigorous proofs in differential and integral calculus.[54] Dirac would later say that Peter Fraser was "the best teacher he had ever had."[51]

Dent studied four courses in pure mathematics:[55]

There was a choice of specialisation in the final year; applied or pure mathematics. As the only official, registered fee-paying student, Dent had the right to choose, and she settled on applied mathematics for the final year. The department could offer only one set of lectures so Dirac also had to follow the same course.[56][m] Dent studied four courses in applied mathematics:[55]

In 1923, Dent was accepted as a postgraduate research student by at Newnham College, University of Cambridge

### Newnham College, University of Cambridge (1923–1924)

In June 1923, Dent graduated with Dirac, gaining a Bachelor of Science (BSc) degree in applied mathematics with First Class Honours.[57][51][n] On 7 July 1923, she was awarded the Ashworth Hallett scholarship by the University of Bristol and was accepted as a postgraduate student at Newnham College in the University of Cambridge.[56][58][59] On her death in 1922, Lilias Sophia Ashworth Hallett left one thousand pounds each to the University of Bristol and Girton College, University of Cambridge, to found scholarships for women.[60]: 259, 261  The University of Bristol scholarship was open to women graduates of a recognised college or university, and worth £45 at the time (equivalent to £2,600 in 2019).[61] She spent a year at Cambridge, leaving in 1924 without further academic qualification.[59] Before 1948, the University of Cambridge denied women graduates a degree, although in the same year as she left Cambridge, was the first woman to be awarded a PhD by the university.[62][o]

## Career

### High School for Girls, Barnsley (1924–1925)

The former High School for Girls, Barnsley, where Dent taught mathematics after leaving Newnham College

Dent spent the summer of 1924 at her parent's home in Warminster, playing mixed doubles tennis in a tournament organised by the local Women's Unionist Association.[64] In September of the same year, she was appointed an assistant teacher in mathematics at the High School for Girls, in Barnsley, Huddersfield Road, on an annual salary of £250 (equivalent to £14,400 in 2019).[59][37] Annie Rose Nuttall, the school's head teacher,[65] was a former student at Newnham College.[66][p] In the early 1920s, women who had studied university level mathematics faced limited employment prospects, as mathematics and engineering professions, other than perhaps school teaching, were dominated by men.[69] Dent resigned her position on 31 August 1925 after being appointed a demonstrator (research assistant) in the Department of Physics at the University of Bristol,[59][37] with her salary being paid by the Department of Scientific and Industrial Research, the forerunner of the Science and Engineering Research Council (SERC).[70]: 107 [q]

### Department of Physics, University of Bristol (1925–1929)

John Edward Lennard‑Jones, Dent's advisor and co-author at Bristol in the 1920s

In 1924, the University of Bristol Council had set aside a portion of a bequest from Henry Herbert Wills for the Department of Physics where Arthur Mannering Tyndall was building up a staff for teaching and research in the H. H. Wills Physics Laboratory, Royal Fort House Gardens.[72][r] From August 1925, John Lennard-Jones, of Trinity College, University of Cambridge, was elected reader in mathematical physics.[74][72] In March 1927, Lennard‑Jones was appointed Professor of Theoretical physics, a chair being created for him, with Dent becoming his research assistant in theoretical physics.[75][76]: 24 [77][s] Lennard‑Jones pioneered the theory of interatomic and intermolecular forces at Bristol and Dent became one of his first collaborators.[72][51]

Lennard‑Jones and Dent published six papers together from 1926 to 1928, dealing with the forces between atoms and ions, with the objective of calculating theoretically the properties of carbonate and nitrate crystals.[51][78] Dent's thesis for her master's degree, Some theoretical determinations of crystal structure (1927), was the basis of the three papers that followed in 1927; with Lennard‑Jones, "Some theoretical determinations of crystal parameters. CXVI", and with Lennard‑Jones and Sydney Chapman, "Structure of carbonate crystals" and "Part II. Structure of carbonate crystals".[t] On 28 June 1927, she was awarded a MSc degree for her thesis and research work.[79] In 1927, the physics laboratory at Bristol had a surplus of funds, and so it was decided that the funds would be used to provide more technical help.[u] Consequently, Dent was asked to combine her research duties with the post of part-time departmental librarian, the first appointment of librarian in the Department of Physics.[76]: 26 [1][v]

H. H. Wills Physics Laboratory, University of Bristol, where Dent worked as a researcher

In 1928, Lennard‑Jones and Dent published two papers, "Cohesion at a crystal surface", and with Sydney Chapman, "The change in lattice spacing at a crystal boundary", that studied the force fields on a thin crystal cleavage.[80][81] Around this time, quantum mechanics was developed to become the standard formulation for atomic physics.[w] Lennard‑Jones left Bristol in 1929 to study the subject for a year as a Rockefeller Fellow at the University of Göttingen.[83] She wrote one last paper before leaving the physics department at Bristol: "The effect of boundary distortion on the surface energy of a crystal" (1929) examined the effect of the polarisation of surface ions in decreasing the surface energy of alkali halides.[84] In November 1929, she was appointed to the position of technical librarian for the scientific and technical staff in the research department at Metropolitan-Vickers, Trafford Park, Manchester.[iii]:14–15[85]

In December 1929, Dent resigned her position at Bristol and it was accepted with regret by the university council.[86] Marjorie Josephine Littleton, the daughter of a local Bristol councillor and a graduate of Girton College, University of Cambridge, was appointed as her replacement on the 1 February 1930. Littleton was later Sir Neville Mott's co-author and research assistant in the physics department.[87][88]: 517  In 1930, Lennard‑Jones returned to Bristol, as Dean of the Faculty of Science, and introduced the new quantum theories to the Bristol group.[83][72][x]

### Metropolitan-Vickers, Trafford Park (1930–1960)

Metropolitan-Vickers was a British heavy industrial firm, well known for industrial electrical equipment and generators, street lighting, electronics, steam turbines and diesel locomotives. They built the Metrovick 950, the first commercial transistorised computer.[90] In 1917, a Research and Education Department was established at the Trafford Park site, when the care of the library came within the remit of James George Pearce. He made the library the centre of a new "technical intelligence" section.[85]: 193 [y] In the 1920s, the post of librarian was held by Lucy Stubbs, a former assistant librarian at the University of Birmingham,[91] and past member of the first standing committee of ASLIB.[85]: 228  Stubbs did not possess scientific qualifications, maintaining that a librarian, if assisted by other technical staff, did not need to understand science or engineering.[85]: 193  In 1929, James Steele Park Paton reorganised and expanded the section with Dent succeeding Stubbs as technical librarian on 6 January 1930.[iii]:15[85] She joined the scientific and technical staff as was one of only two senior women in the research department,[92]: 314  and in contrast to Stubbs, was employed principally for her technical skills.[85]: 193

Dent was honorary secretary to the founding committee for the ASLIB Lancashire and Cheshire branch from 1931 to 1936.[93]: 204–205  In 1932, the branch had twenty-six members and had organised four meetings, including one addressed by Sir Henry Tizard, the then President of ASLIB. After the war, it formed the basis for the Northern Branch of the association.[94]: 412  Technical librarianship emerged as a new scientific career in interwar Britain and rapidly became one of the few types of professional industrial employment that was routinely open to both women and men.[92]: 301  By 1933, Dent reported that the Metropolitan-Vickers library had three thousand engineering volumes and around the same number in pamphlets and patent specifications.[95] Besides covering electrical subjects, the library covered accountancy, employment questions, and subjects of interest to the sales department. It also issued a weekly bulletin, scrutinised patents, handled patents taken out by research staff, and exchanged information with associated companies.[96]

Dent continued to publish papers in applied mathematics and contribute to papers on emerging computational technologies. In "On observations of points connected by a linear relation" (1935), she developed a detailed reduced major axis method for line fitting that built on the work of Robert Adcock and Charles Kummell.[97][98] In 1937, David Myers, then at the Engineering Laboratory at the University of Oxford, asked Douglas Hartree and Arthur Porter, to calculate the space charge limitation of secondary current in a triode.[99]: 91 [100]: 96  The calculations relied on some initial numerical integrations that were carried out by Dent on a differential analyser. The results corresponded closely to those obtained experimentally by Myers at Oxford.[99]: 91 [100]: 97  Her knowledge of higher mathematics meant that she was asked to check the mathematics in papers for publication by engineers at Metropolitan-Vickers. For example, Cyril Frederick Gradwell, a graduate of Trinity College, Cambridge, asked her to scrutinise the algebraic part of his work in "The Solution of a problem in disk bending occurring in connexion with gas turbines" (1950).[101][z] She would later analyse the problem of stress distribution in a thick disk based on a method devised by Philip Pollock,[103] for Richard William Bailey FRS, the former director of the mechanical, metallurgical, and chemical sections of the research department at Metropolitan-Vickers.[104]: 16 [aa]

Dent was a delegate at the fourteenth International Conference on Documentation and was invited to the Government's conference dinner on 22 September 1938 at the Great Dining Hall of Christ Church, Oxford.[105][ab] In 1939, she was elected to the editing committee of the ASLIB book list.[106] In 1944, she was put in charge of the women working in the research department laboratory at Metropolitan-Vickers, and in 1946, she was promoted to section leader of the new computation section.[59] Her role would bring her into contact with , a materials science researcher in the department, and a graduate of Newnham College, who would later head the physics department at the University of Salford.[107] In 1953, they collaborated on an investigation into the heating effects that occur when a current is passed through a semiconductor that has no barrier layer.[108] Dent suggested methods to solve the equations and computed the numerical integrations.[109] In the following year, she developed the Fourier analysis in "Regenerative Deflection as a Parametrically Excited Resonance Phenomenon" (1954), that calculated the optimal radial oscillations to maintain cyclotron resonance in a synchrocyclotron. The causes of axial spreading of the charged particle beam during extraction were also analysed.[110]

Dent joined the Women's Engineering Society and published papers on the application of digital computers to electrical design.[111][112] With Brian Birtwistle, she wrote programs for the Ferranti Mark 1 (Mark 1) computer at the University of Manchester, that demonstrated that high speed digital computers could provide considerable assistance to the electrical design engineer.[113] Birtwistle would later have an extensive career in the computer industry, working at, amongst others, Honeywell Information Systems and ADP Network Services.[114] In 1958, she carried out computer calculations for the mechanical engineering team at the Nuclear Power Group, Radbroke Hall. Their paper outlined a procedure for calculating the theoretical deflection (bending) of a circular grid of support girders for a graphite neutron moderator in a gas-cooled reactor.[115] A general expression was derived from the central deflection of the grid and the maximum bending moment on the central cross-beam for a range of grid diameters.[116]

In 1959, and a year from retirement, Dent modelled a proposed Zeta circuit on the Mark 1 computer, for Eric Hartill's paper on constructing a high-power pulse transformer and circuit.[117] The cost of the computation was about two thousand pounds (equivalent to £47,000 in 2019), corresponding to around eighty hours of machine time.[118] She retired from Metropolitan-Vickers in May 1960, with Isabel Hardwich, later a fellow and president of the Women's Engineering Society, replacing her as section leader for the women in the research department.[85]: 232 [119]: 243

## Personal life

Clifton Hill House where Dent was resident in the 1920s

In the 1920s, Dent was living at Clifton Hill House, the university hall of residence for women in Clifton.[120] May Christophera Staveley was her warden and tutor at Clifton Hill House, and Dent returned to Bristol on 22 December 1934 for Staveley's funeral.[121] Dent was a member of the Clifton Hill House Old Students Association, and secretary and treasurer of the group of former Clifton Hill House students.[iii]:1, 9 She would later write "I was very sorry indeed to leave Bristol and have many happy memories of my time there. I shall miss living at the [Clifton Hill House] Hall very much."[iii]:15

In 1926, Dent was elected treasurer of the University of Bristol's Convocation, the university's alumni association.[122] In 1927, she was one of eleven people elected to the standing committee of the Convocation[123][124]: 62  She later represented the Manchester branch of the association.[121] Around 1926, Dent was appointed honorary secretary of the Bristol Cheeloo Association.[125] The association's aim was to raise sufficient funds to support a chair of chemistry at Cheeloo University.[120] In an effort to publicise the cause and raise money, she presented to the local branch of the Women's International League in October 1928.[126]

The Ice Palace skating rink in Cheetham Hill where Dent learnt to figure skate in the early 1930s

In July 1929, in Dent's last year at Bristol, she went on holiday to North Devon with friends that included Gertrude Roxbee, known as "Rox", who had graduated with Dent in 1923 with a BSc in botany.[iii]:12–13[124] After moving to Manchester in January 1930, Dent found shared lodgings at 10 Montrose Avenue, West Didsbury, in the same house as Roxbee who, at that time, was a teacher at Whalley Range High School.[iii]:15, 50 At weekends, she would ramble to Hebden Bridge, and with Roxbee, learnt to figure skate at the Ice Palace, a former ice rink on Derby Street in Cheetham Hill.[iii]:54[127]

In September 1930, she returned to Bristol for the ninety-eighth conference of the British Association for the Advancement of Science (British Association), meeting her friends at an alumnae association lunch.[iii]:93 In the afternoon of the 4 September 1930, she toured Avonmouth Docks as a conference member,[iii]:94[128] and in the evening, was invited to a reception held by Walter Bryant, the then lord mayor of Bristol, at the Bristol City Museum and Art Gallery.[iii]:94[129]. On the following day, she visited an aircraft manufacturer at Whitchurch Airport and attended a garden party at Wills Hall.[iii]:94[128]

On the Monday of the conference, Dent was in the audience to see Paul Dirac present his paper on the proton and the structure of matter.[iii]:94[130] She would later comment:[iii]:94

I heard a striking paper by Dirac, who was a student with me, who is now a very famous person, as I always knew he would be ... I now go about boasting that I was in the same class!

Dent's father died on 24 June 1954, at their shared home, 529 King's Road, Stretford, with the funeral service taking place at St Matthew's Church, Stretford.[131] She had close links to St Matthew's; from 1956 to 1962, she served as a school manager for St Matthew's Church of England Primary School at Poplar Road, Stretford.[59] She never married, believing that getting married, and the subsequent pressures of family responsibilities, would be a "wastage" of a woman's training. However, she also believed that women leaving employment to get married would mean promotion opportunities for other women, and that married women would still be able to return to work in mid-life.[132]

## Later life and death

Dent's funeral was held at Sompting parish church and a memorial to her is inscribed on a brass plaque affixed to the bishop's chair at the church. Her ashes are interred at Worthing Crematorium.

In 1962, Dent and her mother moved from Stretford to 1 Cokeham Road, Sompting, a village in the coastal Adur District of West Sussex, between Lancing and Worthing.[133][134] Her mother died on 5 April 1967 and was cremated at the Downs Crematorium on 10 April 1967.[135] Dent's sister, Florence Mary, also lived in the house until her death on 13 September 1986 (aged 84).[136] After a brief period as a teacher at a prep school in Malmesbury, Wiltshire, Florence worked as a secretary for a marine insurance firm attached to Lloyd's of London at 12 Leadenhall Street, commuting into London from Harrow each day.[iii]:57–59

Beryl May Dent died at Worthing Hospital on 9 August 1977 (aged 77).[137][138] She donated her body for medical examination, on the understanding that her remains would be returned for a funeral service at the Church of St Mary the Blessed Virgin, Sompting, followed by cremation.[139] Her ashes were interred at Worthing Crematorium, in the Gardens of Rest, towards the Spring Glades, and her entry in the book of remembrance at the crematorium states:[140][141][ac]

Beryl May Dent 1900 – A real Christian loved by all – 1977

There is also a memorial to Dent at the Church of St Mary the Blessed Virgin, Sompting. The bishop's chair, situated close to the altar, bears a brass plaque with the following inscription:[142][ad]

In loving memory of BERYL DENT 1900 – 1977

Her Christian faith is perhaps not unexpected, given her father's work for the church in Warminster, and the era she grew up in, where religion pervaded social and political life. However, it is notable that she remained a committed Christian while pursuing a scientific career.[143][144]

## Legacy

An archive of Dent's papers, that relate to her life and work in the 1920s in the physics department at the University of Bristol, is held in the Special Collections of the University of Bristol Arts and Social Sciences Library, in Tyndall Avenue, Bristol.[1] Included in that archive is a series of letterbooks, written in the 1930s by members of the Clifton Hill House Old Students' Association, that include news and photographs of Dent, her family, and friends.[iii][iv]

### Atomic force microscopy

An atomic force microscope on the left with controlling computer on the right. Dent's work had direct application to the development of atomic force microscopy

In 1928, Lennard‑Jones and Dent published two papers, "Cohesion at a crystal surface" and "The change in lattice spacing at a crystal boundary", that for the first time, outlined a calculation of the potential of the electric field in a vacuum, produced by a thin sodium chloride crystal surface.[80][81] They gave an expression for the electric potential produced by a system of point charges in vacuum (although not a real cubic sodium chloride ionic lattice).[145]: 796–797  The expression for the potential in vacuum, ${\displaystyle \varphi _{0}\left(r\right)}$, at the point r = {x, y, z}, near the cubic lattice of point ions with different signs, the charge ${\displaystyle e_{k}}$, and the period a (a crystalline solid is distinguished by the fact that the atoms making up the crystal are arranged in a periodic fashion), can be represented in the form:[145]: 797

${\displaystyle \varphi _{0}\left(r\right)={\frac {2\pi }{a^{2}}}\sum _{l,m}\sum _{s}{\frac {\left(-1\right)^{s}\exp \left[-\left({\frac {2\pi }{a}}\right){\sqrt {l^{2}+m^{2}}}\left(z+z_{s}\right)\right]}{\left|k_{l,m}\right|}}\times \sum _{k}e_{k}\cos \left[k_{l,m}\left(r_{\parallel }-r_{k}\right)\right]}$

(1)

${\displaystyle r_{\parallel }=\left\{x,y\right\}}$ is the lateral vector that fixes the observation point coordinates in the sample plane.
${\displaystyle k_{l,m}}$ is the reciprocal lattice vector.
s is the number of planes to be calculated inside the crystal; s set to zero would calculate the surface plane.

The expression sums the set of potential static charges for the surface and lower planes of the crystal lattice. Lennard‑Jones and Dent showed that this expression forms a rapidly convergent Fourier series.[145]: 797  Harold Eugene Buckley, a crystallographic researcher at the University of Manchester until his death in 1959,[146]: 481  had suggested that their results should be treated with caution. For example, the contraction a crystal plane would suffer under the conditions prescribed would not be the same as that of a similar plane with a solid mass of crystal behind it. Another difficulty arises because calculation of crystal surface field force fields are so great that simplifying assumptions have to be made to render them capable of a solution.[147]

Michael Jaycock and Geoffrey Parfitt concurred with Buckley, noting that "an ideal crystal, in which the ionic positions at the surface were identical to those achieved in the bulk crystal ... is obviously extremely improbable." However, they acknowledged that the Lennard‑Jones and Dent model was singularly elegant, and like most researchers working before the advent of modern computers, they were limited in what could be attempted computationally.[148] Nonetheless, Lennard‑Jones and Dent demonstrated that the force exerted on a single ion, by a surface with evenly distributed positive and negative ions, decreases very rapidly with increasing distance.[149]: 14  Later work by Jason Cleveland, Manfred Radmacher, and Paul Hansma, has shown that this result has direct application to atomic force microscopy by predicting that non-contact imaging is possible only at small tip-sample separations.[150]: 543

### Reduced major axis regression

Linear regression attempts to model the relationship between two variables by fitting a linear equation (straight line) to observed data

The theoretical underpinnings of standard least squares regression analysis are based on the assumption that the independent variable (often labelled as x) is measured without error as a design variable. The dependent variable (labeled y) is modeled as having uncertainty or error. Both independent and dependent measurements may have multiple sources of error. Therefore, the underlying least squares regression assumptions can be violated. Reduced major axis (RMA) regression is specifically formulated to handle errors in both the x and y variables.[151]: 1  If the estimate of the ratio of the error variance of y to the error variance of x is denoted by