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Reviews of Modern Style Guide

XVI. MATHEMATICAL MATERIAL TABLE II Diacritical marks and associated LATEX codes. Mathematical symbols must be defined immediately Diacritical mark LATEX code where they are introduced. Exceptions are the funda- ↔x \tensor{x} mental constants: the velocity of light c, Planck’s con- x˙ \{x} stant ¯h, the electronic charge e, Boltzmann’s constant k ¨r \ddot{{\bf r}} ˙ or kB, and the electron mass me. θˆ \dot{\hat{{\theta}}} Avoid using the same for two different things. A \underline{A} If you think that a list of symbols would be helpful to your x + y \{\overline{x} + y} readers, you can provide one preceding the references (see Sec. VI.). In what follows, only the style of and sym- 3. Subscripts and superscripts bols will be considered. Authors are urged to consult LAT X 2 and REVT X 4-1 documentation for instruc- E ε E All available characters can be used as subscripts or su- tions on using these packages to display . perscripts. Position of subscript or superscript is dictated by standard notation. In almost all cases you should A. Characters right and left subscripts and superscripts flush against the symbol they accompany (as in the following). 1. Character Examples:

0 The italic is used for mathematical symbols (this x 110 m (N) R 1 P R0 , Ag , ρ0 , 0 , , liml→∞ (in text) is the default font in LATEX’s math mode). In addition to variables and constants, the italic font is used for particle There are, however, some exceptions to this general symbols, symbols of quantum states, and group-theoretic rule. Examples appear below. designations. In general please use the following hierarchy of font z ;α • tensor notation: gµν (φ ) ;α styles for symbols: + − • molecular ions: H2 ,O2 TABLE I Font style for symbols. a • foonotes in tables: Font style Symbol Example lower case variables, constants, x, α, f Presuperscripts or presubscripts are set flush against and ordinary functions the symbols they accompany. In addition, it is advisable upper case matrices and functions S, F script upper case operators H to insert an extra thin between a presuperscript boldface lower case three-vectors r or presubscript and a preceding symbol in cases where boldface upper case matrices J, B clarity is questionable, i.e., three-vectors 1 + 9 z 3 8pσ Σu or d s p P2

The number of levels of subscripts and superscripts at- 2. Diacritical signs tached to a symbol will also affect clarity. Two double levels is generally considered the most complicated com- A diacritical sign is a marking placed directly above bination acceptable, i.e., or below symbols, e.g., the in ~x. It is possible to a2 make multilevel marks – placing several diacritical marks M i b† above or below one letter or symbol – but this quickly k becomes confusing. Restrict the number of such marks to two to avoid confusion. The underline can appear under When additional indices are needed, insert a or any configuration. and keep the added indices on the same line, For your convenience, Table II contains a list of com- i.e., monly used diacritical marks along with the LATEX code M , σ , or σ that produces them. bk,dp r,s+1 r s+1 2

B. Abbreviations in math i.e., Elab, where lab stands for laboratory and EHF, where HF stands for Hartree and Fock, two proper names. Some abbreviations, such as those for mathematical Please that you should always capitalize abbrevi- functions and those used in superscripts or subscripts, ations that represent proper names. require special handling and are discussed below. When you are creating your own abbreviations, do not put periods in acronyms (whether in line or in sub- scripts), but do insert them if you are abbreviating words 1. Abbreviations designating mathematical functions that are headings in a table.

Multiletter abbreviations of mathematical functions are always written in the roman font (i.e., sin). The C. Mathematical expressions standard trigonometric functions are cos, cot, sec, sin, and tan. Hyperbolic trigonometric functions add “h” to 1. When to display the end; the preferred notation for inverse functions is tan−1 rather than arctan. The preferred notation for the Mathematical material that is set apart from the main to the base e (loge) is “ln”; “log” without a text in the traditional manner is referred to as displayed subscript denotes the logarithm to the base 10 (log10). material. In general, authors should observe the follow- The following guidelines are in general use in clear ing guidelines: Display (1) equations of importance, (2) mathematical writing: all equations that are numbered, (3) those that are too long to fit easily in text (over 25 characters), or (4) those (a) roman multiletter abbreviations must be closed up that are complicated (contain built-up fractions, matri- to the argument following and separated from any ces, or matrixlike expressions). Consider, also, displaying preceding symbol by a thin space, that is, math that contains multilevel indices, , summa- tion, and product signs, with multilevel or complex lim- K cos[Q(z − z0)] its, or any other situation in a formula that creates the 2 − 1 need for extra vertical spacing in a text line. K exp[x (b2 + b1) 2 ]

(b) In addition, by convention it is assumed that an 2. argument ends as soon as another function appears, i.e., sin x cos b, or at a plus or minus sign, i.e., sin x+ Even though displayed math is separated by space from y, but if other mathematics is involved or there is the running text it still is a part of that text and needs to any ambiguity you should insert bracketing, as in be punctuated accordingly. The following is an example. the following examples:

2 The final result is sin[−(x + a)], (sin x)/a, and exp[x (bz − 1 + b1) 2 ]

(c) To treat a function of a function enclose it in bold  Ω 2 |J|2 H = eλK·Rij , (1) round parentheses, i.e., ij 1 ∆ Eg + 2 (Wc + Wv) g(f(x)) where (d) e and exp (for exponent) notation follow both of 1 the preceding conventions. Which form to use, e K = (ˆi +ˆj + kˆ) (2) or exp, is determined by the number of characters a and the complexity of the argument. The e form is and appropriate when the argument is short and sim- ple, i.e., eik·r, whereas exp should be used if the 2 argument is more complicated. λ = ln [WlWv/(12Eg) ]. (3) Note the use of the comma at the end of the first equa- 2. Abbreviations in subscripts and superscripts tion, and the period at the end of the third .

Abbreviations in subscripts and superscripts fall into two categories: (1) single letter and (2) multiletter. Most 3. Equation breaking (multilinear equations) single-letter abbreviations are conventionally printed in the italic font, i.e., EC where C stands for Coulomb. Mul- Mathematical expressions often need to be displayed tiletter abbreviations are conventionally printed in the on two or more lines (“broken”) because of the roman font whether they represent one or more words, limitations of the Reviews of Modern Physics standard 3 two- layout. The best place for a break is just be- D. Bracketing fore an operator or sign of relation. These signs should begin the next line of the equation. When it is neces- 1. Grouping sequence sary to break a product, begin the continued line with a (×). Note that the material that For the purpose of grouping, the sequence of bracketing comes after a break can and should be aligned so that preferred for Reviews of Modern Physics articles is {[()]}, its relationship to the material on the first line is math- working outwards in sets ( ), [ ], and {}. If you have used ematically correct. Here is an example: these three sets and need additional bracketing, begin the sequence again in the same order but in bold print:

iθ(r) { [( { [()] } )] } Nx(r) + iN2(r) = e = exp [−ijxu(r)]. (4) For grouping situations that contain built-up material and need larger-sized bracketing, it is preferable to start Equations that are not displayed, but appear in text again at the beginning of the sequence around the built- may also need to be broken. Basically the same rules up material, e.g., apply as when breaking displayed math. Breaking at an operator or sign of relation is best. The operator or sign (a − 2)1/2  (x + 2)1/2  of relation usually begins the next line of text: = 0. (5) α2 β ˆ . .√ . their respective displacement vectors ai/2 ˆ + 2aj/2 . . . 2. Specific notation

Products are broken with a multiplication sign: Bracketing (ordered and special) is also used to create specific notation that defines what it encloses. A list of . . . keep δm = 4 MeV and choose γ (= -5.46 approved specialized notation is included below. When × 10−3) at . . . used in an equation along with ordered bracketing, this special kind of bracketing should not alter the regular In addition, you may break in text at a solidus, leaving sequence of bracketing. The special notation h i in the the numerator and fraction on the top line. The following equation does not interfere with the sequence denominator will begin the continued line as follows: of the equation bracketing:

1/2 ... J is proportional to T0(∆1, ∆3) / ¯h[hE − (a + 1)i]−1 = 0. (6) N2(0) . . . Please note the difference between hi (LATEX: \langle, \rangle) and <>, the usual greater-than, less-than sym- 4. Equation numbering, special situations bols.

Equation numbers are placed flush with the right mar- gin, as in the above numbered equations. Some situations 3. Specialized bracket notation require unique numbering. Please use the forms shown in the following examples when you encounter similar cir- Table III lists more uses of , including crys- cumstances. tallographic notations. These generally follow common usage and are included here for your convenience. (1) A set of equations of equal importance may be num- bered to demonstrate that relationship, e.g., (1a), E. Additional style guidelines (1b), and (1c). 1. Placement of limits and indices (2) A principal equation and subordinate equations (those that define quantities or variables in that In displayed math, limits and indices on sums, in- equation) may be numbered (1), (1a), (1b), etc. tegrals, and similar symbols are handled in traditional ways: (3) If an equation is a variant of a previous equation (it may be separated from the original equation by X Z +∞ Y other equations and/or by text), it may be num- (7) bered with the same number as the original and a −∞ i,j,k n>1 , double prime, etc., as appropriate. i

(e) Be careful not to write ambiguous fractions when TABLE III Specialized bracket notation. using the slashed notation; clearly indicate order of Object Notation operations where necessary. Plane or set of parallel planes (111) Direction [111] 3. Multiplication signs Class (group) of symmetry equivalent directions h111i Class (group) of symmetry equivalent planes {111} designated by coordinates (x, y, z) The primary use of the multiplication sign is to indi- Lattice position in a unit cell (not bracketed) 1 1 1 2 2 2 cate a vector product of three-vectors (e.g., κ × A). Do Vector written in components (Hx,Hy ,Hz ) Commutator [f, g] not use it to express a simple product except Anticommutator {f, g} Nested commutators [H0, [H0,H1]] Functionals F [x] Sets {xi} (1) when breaking a product from one line to another Absolute value, determinant |x|, |A| (described in Sec. .C.3) or Evaluation of a quantity |φ0=0 Norm ||A|| Average or expectation value h i, h iav (2) for other cases such as indicating dimensions (e.g., 3 × 3 × 3 cm3), magnification (40×), symbols in figures (×’s), or numbers expressed in scientific no- tation (1.6 × 10−19 C). In text, however, space limitations require that single- limit sums or use subscripts and superscripts, P∞ R a for example, n=1 and 0 . Multiple-limit large sym- The center dot (·) should not be used to mean a sim- bols, such as the first sum in Eq. (7), should always be ple product. Use the dot to represent inner products of displayed. vectors (k · r).

2. Fractions 4. Mathematical terms a+b Fractions can be “built up” with a fraction bar, c , “slashed” with a solidus, (a + b)/c, or written with a The use of the following standard symbols is recom- negative exponent, (a + b)c−1. In text all fractions must mended. be either slashed or written with a negative exponent. Observe the following guidelines on the use of fractions.

(a) Use built-up fractions in matrices: TABLE IV Standard symbols. ∼ approximately or varies as  ∂2 0 ∂  ' approximately equal ∂x2 2θ0 ∂x → tends to M1 = −   . (8) 2 ∝ proportional to θ ∂ θ2 ∂ 0 ∂x 0 ∂x2 O() of the order A∗ complex conjugate of A (b) Use built-up fractions in displayed equations: A† Hermitian conjugate of A AT transpose of A " # 1  Q 2 c2 kˆ unit vector k/k H (w) = + e πω2d. (9) A 2 πω2 4d

(c) Using slashed fractions in subscripts, superscripts, limits, and indices is preferred:

Z π/2 5. Radical signs and overbars N −1/2 m (10) 3/2 √ −π/2 You may use radical signs (roofed only, e.g., xx) and overbars (xx) when they go over material of six or fewer (d) Use slashed or sized fractions in the numerators and characters that are without superscripts. If the material denominators of built-up fractions except where ex- is longer or has√ superscripts, alternative notation should cessive bracketing would obscure your meaning or be used. For xx use (xx)1/2 and for xx use hxxi or slashing would interfere with continuity of notation: hxxiav if appropriate. If the overbar means complex con- jugate, then (xx)∗ should be used. A radical sign (roofed) (β/6)ϕ ϕ + = 0. (11) should not be used on built-up material, although an γ + [β(β − 1)/12]ϕ2 overbar can be used.