Some notes on a Fibonacci–Lucas identity

Kunle Adegoke
Department of Physics and Engineering Physics,
Obafemi Awolowo University, Ile-Ife
Nigeria
<EMAIL_ADDRESS>

Robert Frontczak111Statements and conclusions made in this article by R.
Frontczak are entirely those of the author. They do not necessarily reflect
the views of LBBW.
Landesbank Baden-Württemberg, Stuttgart
Germany
<EMAIL_ADDRESS>

Taras Goy
Faculty of Mathematics and Computer Science
Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk
Ukraine
<EMAIL_ADDRESS>
2010 Mathematics Subject Classification: Primary 11B39; Secondary 11B37.

_Keywords:_ Fibonacci (Lucas) number, generalized Lucas polynomial, Chebyshev
polynomial, summation identity.

###### Abstract

In 2016, Edgar stated a nice polynomial identity that connects Fibonacci and
Lucas numbers. Shortly after Edgar’s publication, this identity has been
generalized in two different ways: Dafnis, Phillipou and Livieris provided a
generalization to Fibonacci sequences of order $k$ and Abd-Elhameed and Zeyada
extended Edgar’s identity to generalized Fibonacci and Lucas sequences. In
this paper, we present more polynomial identities for generalized Lucas
sequences. We discuss interesting aspects and special cases which have not
been stated before but deserve recognition. Finally, we prove the polynomial
analogues of these identities for Chebyshev polynomials.

##  1 Motivation

As usual, the Fibonacci numbers $F_{n}$ and the Lucas numbers $L_{n}$ are
defined, for $n\in\mathbb{Z}$, through the recurrence relations
$F_{n}=F_{n-1}+F_{n-2}$, $n\geq 2$, with initial values $F_{0}=0$, $F_{1}=1$
and $L_{n}=L_{n-1}+L_{n-2}$ with $L_{0}=2$, $L_{1}=1$. For negative subscripts
we have $F_{-n}=(-1)^{n-1}F_{n}$ and $L_{-n}=(-1)^{n}L_{n}$. They possess the
explicit formulas (Binet forms)

$F_{n}=\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta},\qquad
L_{n}=\alpha^{n}+\beta^{n},\quad n\in\mathbb{Z}.$

For more information we refer to Koshy [6] and Vajda [14] who have written
excellent books dealing with Fibonacci and Lucas numbers.

Our motivation for these notes is the following prominent identity due to
Edgar from 2016 where he generalized two Fibonacci–Lucas identities from the
American Mathematical Monthly [5]:

$\sum_{k=0}^{n}x^{k}(L_{k}+(x-2)F_{k+1})=x^{n+1}F_{n+1}.$ (1)

This identity holds for all $x\in\mathbb{C}$ and the special cases $x=2$ and
$x=3$, respectively, have been proved by Benjamin and Quinn [2] and Marques
[7]. A polynomial variant was given by Sury [13]. Dafnis, Philippou and
Livieris have generalized this identity to Fibonacci and Lucas numbers of
order $k$ and gave two different proofs for their generalization [3, 8].
Finally, in 2020 Dafnis, Philippou and Livieris derived an alternating version
of their generalized Fibonacci–Lucas sum [4], which when applied to Fibonacci
and Lucas numbers becomes [4, Corollary 1]

$\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}L_{k+1}+(x-2)F_{k}\big{)}=(-1)^{n}F_{n+1}.$
(2)

On the other hand, Abd-Elhameed and Zeyada [1], without a reference to Edgar
or Dafnis et al., derived two polynomial identities, one of which generalizes
the polynomial identity of Sury [13].

In this paper, we proceed in the same direction and present more polynomial
identities for generalized Lucas sequences. We discuss interesting aspects and
special cases which have not been stated before. Finally, we prove the
polynomial analogues of these identities for Chebyshev polynomials.

##  2 Polynomial generalizations

In this section, we give generalizations of (1) and (2) to the general Lucas
sequences and prove two another identities of the same nature. For non-zero
complex numbers $y$ and $z$, the Lucas sequences of the first kind,
$\left(u_{n}(y,z)\right)_{n\geq 0}$, and of the second kind,
$\left(v_{n}(y,z)\right)_{n\geq 0}$, are defined (see e.g., Ribenboim [9,
Chapter 1]) through the recurrence relations

$u_{n}(y,z)=yu_{n-1}(y,z)-zu_{n-2}(y,z),\quad n\geq 2,\quad
u_{0}(y,z)=0,\,u_{1}(y,z)=1$

and

$v_{n}(y,z)=yv_{n-1}(y,z)-zv_{n-2}(y,z),\quad n\geq 2,\quad
v_{0}(y,z)=2,\,v_{1}(y,z)=y,$

with

$u_{-n}(y,z)=-u_{n}(y,z)/z^{n}\,,\qquad v_{-n}(y,z)=v_{n}(y,z)/z^{n}\,.$

Denote by $\tau$ and $\sigma$, the zeros of the characteristic polynomial
$x^{2}-yx+z$ for the Lucas sequences. Then

$\tau=\tau(y,z)=\frac{y+\sqrt{y^{2}-4z}}{2},\qquad\sigma=\sigma(y,z)=\frac{y-\sqrt{y^{2}-4z}}{2}\,,$
$\tau+\sigma=y,\qquad\tau-\sigma=\sqrt{y^{2}-4z}\qquad\mbox{and}\qquad\tau\sigma=z\,.$

The difference equations are solved by the Binet formulas

$u_{n}=\frac{\tau^{n}-\sigma^{n}}{\tau-\sigma},\qquad
v_{n}=\tau^{n}+\sigma^{n}\,.$

We need the following two lemmas, of which the first one contains basic
telescoping summation identities.

######  Lemma 1.

If $(f_{k})_{k\in\mathbb{Z}}$ is a real sequence and $c$, $n\in\mathbb{Z}$,
then

$\sum_{k=c}^{n}\left({f_{k+1}-f_{k}}\right)=f_{n+1}-f_{c}$ (3)

and

$\sum_{k=c}^{n}(-1)^{k}\left({f_{k+1}+f_{k}}\right)=(-1)^{n}f_{n+1}+(-1)^{c}f_{c}.$
(4)

######  Lemma 2.

For all $x$, $y$, $r\in\mathbb{C}$, $r\neq 0$ and $k\in\mathbb{Z}$, we have

$\frac{{x^{k}}}{{r^{k}}}\big{(}{rv_{k}(y,z)+(xy-2r)u_{k+1}(y,z)}\big{)}=y\left({\frac{{x^{k+1}}}{{r^{k}}}u_{k+1}(y,z)-\frac{{x^{k}}}{{r^{k-1}}}u_{k}(y,z)}\right)$
(5)

and

$\frac{{x^{k}}}{{r^{k}}}\left({(y^{2}-4z)ru_{k}(y,z)+(xy-2r)v_{k+1}(y,z)}\right)=y\left({\frac{{x^{k+1}}}{{r^{k}}}v_{k+1}(y,z)-\frac{{x^{k}}}{{r^{k-1}}}v_{k}(y,z)}\right).$
(6)

###### Proof.

On account of the recurrence relation

$u_{k+1}(y,z)+zu_{k-1}(y,z)=yu_{k}(y,z)$ (7)

and the identity ([9, Identity (2.7)])

$v_{k}(y,z)=u_{k+1}(y,z)-zu_{k-1}(y,z),$ (8)

we have

$rv_{k}(y,z)+(xy-2r)u_{k+1}(y,z)=y\big{(}{xu_{k+1}(y,z)-ru_{k}(y,z)}\big{)}$

and hence (5). The proof of (6) is similar. We use

$v_{k+1}(y,z)+zv_{k-1}(y,z)=yv_{k}(y,z)$ (9)

and the identity

$(y^{2}-4z)u_{k}(y,z)=v_{k+1}(y,z)-zv_{k-1}(y,z).$ (10)

∎

######  Theorem 1.

For all $x$, $y$, $r\in\mathbb{C}$ and $c$, $n\in\mathbb{Z}$,

$\sum_{k=c}^{n}r^{n-k}x^{k}\big{(}{rv_{k}(y,z)+(xy-2r)u_{k+1}(y,z)}\big{)}=x^{n+1}yu_{n+1}(y,z)-r^{n-c+1}x^{c}yu_{c}(y,z)$
(11)

and

$\begin{split}\sum_{k=c}^{n}r^{n-k}x^{k}\big{(}{(y^{2}-4z)ru_{k}(y,z)+(xy-2r)v_{k+1}(y,z)}\big{)}&\\\
=x^{n+1}yv_{n+1}(y,z)\,\,-\,\,&r^{n-c+1}x^{c}yv_{c}(y,z).\end{split}$ (12)

###### Proof.

To prove (11), sum (5) from $c$ to $n$, making use of (3). The proof of (12)
is similar. ∎

######  Remark 1.

Equation (11) contains one result of Abd-Elhameed and Zeyada [1, Theorem 1] as
a special case at $r=1$ and $c=0$. Identity (12) seems to be new.

######  Remark 2.

Replacing $x$ by $-x$ in (11) and (12) gives the following companion
identities

$\begin{split}\sum_{k=c}^{n}(-1)^{k}&r^{n-k}x^{k}\big{(}{(xy+2r)u_{k+1}(y,z)-rv_{k}(y,z)}\big{)}\\\
&\qquad\qquad=(-1)^{n}x^{n+1}yu_{n+1}(y,z)+(-1)^{c}r^{n-c+1}x^{c}yu_{c}(y,z)\end{split}$

and

$\begin{split}\sum_{k=c}^{n}(-1)^{k}&r^{n-k}x^{k}\left({(xy+2r)v_{k+1}(y,z)-(y^{2}-4z)ru_{k}(y,z)}\right)\\\
&\qquad\qquad=(-1)^{n}x^{n+1}yv_{n+1}(y,z)+(-1)^{c}r^{n-c+1}x^{c}yv_{c}(y,z).\end{split}$

######  Lemma 3.

For all $x$, $y$, $r\in\mathbb{C}$ and $k$, $n\in\mathbb{Z}$, we have

$x^{n-k}r^{k}\left({rv_{k+1}(y,z)+(xy+2rz)u_{k}(y,z)}\right)=y\left({r^{k+1}x^{n-k}u_{k+1}(y,z)+r^{k}x^{n-k+1}u_{k}(y,z)}\right)$
(13)

and

$\begin{split}x^{n-k}r^{k}\big{(}(y^{2}-4z)ru_{k+1}(y,z)+&(xy+2rz)v_{k}(y,z)\big{)}\\\
&=y\left({r^{k+1}x^{n-k}v_{k+1}(y,z)+r^{k}x^{n-k+1}v_{k}(y,z)}\right).\end{split}$
(14)

###### Proof.

Eliminating $u_{k+1}(y,z)$ between (7) and (8) gives

$v_{k}(y,z)=yu_{k}(y,z)-2zu_{k-1}(y,z)$

and hence

$rv_{k}(y,z)+(xy+2rz)u_{k-1}(y,z)=ryu_{k}(y,z)+xyu_{k-1}(y,z),$

from which (13) follows. The proof of (14) is similar; eliminate
$v_{k+1}(y,z)$ between (9) and (10). ∎

######  Theorem 2.

For all $x$, $y$, $r\in\mathbb{C}$ and $c$, $n\in\mathbb{Z}$,

$\begin{split}\sum_{k=c}^{n}(-1)^{k}&x^{n-k}r^{k}\left({rv_{k+1}(y,z)+(xy+2rz)u_{k}(y,z)}\right)\\\
&\qquad=(-1)^{n}yr^{n+1}u_{n+1}(y,z)+(-1)^{c}r^{c}yx^{n-c+1}u_{c}(y,z)\end{split}$
(15)

and

$\begin{split}\sum_{k=c}^{n}(-1)^{k}&x^{n-k}r^{k}\left({(y^{2}-4z)ru_{k+1}(y,z)+(xy+2rz)v_{k}(y,z)}\right)\\\
&\qquad=(-1)^{n}yr^{n+1}v_{n+1}(y,z)+(-1)^{c}r^{c}yx^{n-c+1}v_{c}(y,z).\end{split}$
(16)

###### Proof.

Sum each of (13) and (14) from $c$ to $n$, making use of (4) since the right-
hand side telescopes in each case. ∎

######  Remark 3.

Identity (15) is a polynomial generalization of (2). Identity (16) is its
companion and also presumably new.

##  3 Discussion

This section is devoted to discuss in detail some special cases of Theorems 1
and 2, hereby paying special attention to Fibonacci and Lucas polynomials
$F_{n}(x)$ and $L_{n}(x)$, respectively. We will rediscover some identities
from [1] but also present new relations that we did not find in the literature
and deserve recognition.

First we note that $F_{n}(x)=u_{n}(x,-1)$, $L_{n}(x)=v_{n}(x,-1)$, and
$F_{n}(1)=F_{n}$ and $L_{n}(1)=L_{n}$. Also, $F_{n}(2)=P_{n}$ and
$L_{n}(2)=Q_{n}$, where $P_{n}$ and $Q_{n}$ denote the Pell and Pell–Lucas
numbers, respectively. Other special values of these polynomials will be used
later. With these choices Theorem 1 becomes the following result, which we
state as a proposition.

######  Proposition 1.

For all $x,y\in\mathbb{C}$ and $n\in\mathbb{Z}$, we have

$\sum_{k=0}^{n}x^{k}\big{(}L_{k}(y)+(xy-2)F_{k+1}(y)\big{)}=x^{n+1}yF_{n+1}(y)$
(17)

and

$\sum_{k=0}^{n}x^{k}\big{(}(y^{2}+4)F_{k}(y)+(xy-2)L_{k+1}(y)\big{)}=y\big{(}x^{n+1}L_{n+1}(y)-2\big{)}.$
(18)

###### Proof.

Set $r=1$ and $c=0$ in (11) and (12). ∎

Proposition 1 provides two polynomial extensions of Edgar’s identity. Both
identities connect interestingly Fibonacci and Lucas polynomials. Since
$L_{n}(y)=F_{n-1}(y)+F_{n+1}(y)$ and $(y^{2}+4)F_{n}(y)=L_{n-1}(y)+L_{n+1}(y)$
they may also be disconnected:

$\sum_{k=0}^{n}x^{k}\big{(}xF_{k+1}(y)-F_{k}(y)\big{)}=x^{n+1}F_{n+1}(y)$

and

$\sum_{k=0}^{n}x^{k}\big{(}xL_{k+1}(y)-L_{k}(y)\big{)}=x^{n+1}L_{n+1}(y)-xy.$

######  Corollary 3.

For all $x\in\mathbb{C}$ and $n\in\mathbb{Z}$,

$\displaystyle\sum_{k=0}^{n}x^{k}\big{(}L_{k}+(x-2)F_{k+1}\big{)}=x^{n+1}F_{n+1},$
(19)
$\displaystyle\sum_{k=0}^{n}x^{k}\big{(}5F_{k}+(x-2)L_{k+1}\big{)}=x^{n+1}L_{n+1}-2,$
$\displaystyle\sum_{k=0}^{n}x^{k}\big{(}Q_{k}+2(x-1)P_{k+1}\big{)}=2x^{n+1}P_{n+1},$
(20)
$\displaystyle\sum_{k=0}^{n}x^{k}\big{(}4P_{k}+(x-1)Q_{k+1}\big{)}=x^{n+1}Q_{n+1}-2.$

###### Proof.

Set $y=1$ and $y=2$, in turn, in (17) and (18). ∎

We point out that (19) and (20) are not new. These are Equations (11) and (15)
in [1].

######  Corollary 4.

For all $x\in\mathbb{C}$ and $n\in\mathbb{Z}$,

$\sum_{k=0}^{n}x^{k}\big{(}L_{3k}+(2x-1)F_{3k+3}\big{)}=2x^{n+1}F_{3n+3}$

and

$\sum_{k=0}^{n}x^{k}\big{(}5F_{3k}+(2x-1)L_{3k+3}\big{)}=2x^{n+1}L_{3n+3}-4.$

###### Proof.

Set $y=4$ in (17) and (18), respectively, and use the evaluations
$F_{n}(4)=F_{3n}/2$ as well as $L_{n}(4)=L_{3n}$. ∎

######  Corollary 5.

For all $x\in\mathbb{C}$ and $n\in\mathbb{Z}$,

$\sum_{k=0}^{n}2^{k}\frac{L_{k}(x)}{x^{k}}=2^{n+1}\frac{F_{n+1}(x)}{x^{n}},\qquad
x\neq 0,$

and

$\sum_{k=0}^{n}2^{k}\frac{F_{k}(x)}{x^{k}}=\frac{x}{x^{2}+4}\Big{(}2^{n+1}\frac{L_{n+1}(x)}{x^{n+1}}-2\Big{)},\qquad
x\neq 0,\,\,x\neq\pm 2i.$

###### Proof.

Set $x=2/y$ in (17) and (18), respectively. ∎

The results in Corollaries 6–9 are identities involving only even-indexed
Fibonacci and Lucas polynomials.

######  Corollary 6.

For all $x,y\in\mathbb{C}$ and $n\in\mathbb{Z}$, we have

$\sum_{k=0}^{n}x^{k}\big{(}yL_{2k}(y)+(x(y^{2}+2)-2)F_{2k+2}(y)\big{)}=x^{n+1}(y^{2}+2){F_{2n+2}(y)}$
(21)

and

$\sum_{k=0}^{n}x^{k}\big{(}y(y^{2}+4)F_{2k}(y)+\big{(}x(y^{2}+2)-2\big{)}L_{2k+2}(y)\big{)}=(y^{2}+2)\big{(}x^{n+1}L_{2n+2}(y)-2\big{)}.$
(22)

###### Proof.

We only prove (21). Use Proposition 1 and the relations

$L_{n}(i(x^{2}+2))=i^{n}L_{2n}(x)\quad\mbox{and}\quad
F_{n}(i(x^{2}+2))=i^{n-1}\frac{F_{2n}(x)}{x},$

where $i=\sqrt{-1}$. When simplifying replace $x$ by $x/i$. The other proof is
similar. ∎

######  Corollary 7.

For all $x\in\mathbb{C}$ and $n\in\mathbb{Z}$,

$\displaystyle\sum_{k=0}^{n}x^{k}\big{(}L_{2k}+(3x-2)F_{2k+2}\big{)}=3x^{n+1}F_{2n+2},$
$\displaystyle\sum_{k=0}^{n}x^{k}\big{(}5F_{2k}+(3x-2)L_{2k+2}\big{)}=3\big{(}x^{n+1}L_{2n+2}-2\big{)},$
$\displaystyle\sum_{k=0}^{n}x^{k}\big{(}Q_{2k}+(3x-1)P_{2k+2}\big{)}=3x^{n+1}P_{2n+2},$
$\displaystyle\sum_{k=0}^{n}x^{k}\big{(}8P_{2k}+(3x-1)Q_{2k+2}\big{)}=3\big{(}x^{n+1}Q_{2n+2}-2\big{)}.$

###### Proof.

Set $y=1$ and $y=2$ in (21) and (22), respectively. ∎

######  Corollary 8.

For all $x\in\mathbb{C}$ and $n\in\mathbb{Z}$,

$\sum_{k=0}^{n}x^{k}\big{(}3L_{4k}+(7x-2)F_{4k+4}\big{)}={7}x^{n+1}F_{4n+4}$

and

$\sum_{k=0}^{n}x^{k}\big{(}15F_{4k}+(7x-2)L_{4k+4}\big{)}=7(x^{n+1}L_{4n+4}-2).$

###### Proof.

Set $y=\sqrt{5}$ in (21) and (22), respectively, and use the evaluations
$F_{2n}(\sqrt{5})=\sqrt{5}/3F_{4n}$ as well as $L_{2n}(\sqrt{5})=L_{4n}$. ∎

######  Corollary 9.

For all $x\in\mathbb{C}\setminus\\!\\{0;\pm 2i;\pm\sqrt{2}i\\}$ and
$n\in\mathbb{Z}$,

$\sum_{k=0}^{n}\Big{(}\frac{2}{x^{2}+2}\Big{)}^{k}L_{2k}(x)=\frac{2^{n+1}}{x(x^{2}+2)^{n}}F_{2n+2}(x)$

and

$\sum_{k=0}^{n}\Big{(}\frac{2}{x^{2}+2}\Big{)}^{k}F_{2k}(x)=\frac{x^{2}+2}{x(x^{2}+4)}\bigg{(}\Big{(}\frac{2}{x^{2}+2}\Big{)}^{n+1}L_{2n+2}(x)-2\bigg{)}.$

###### Proof.

Set $x=2/(y^{2}+2)$ in (21) and (22), respectively, and replace $y$ by $x$. ∎

The next identities provide still other interesting relations that nicely
generalize and complement Edgar’s identity (1).

######  Proposition 2.

For all $x\in\mathbb{C}$ and integers $m$ and $n$, we have

$\sum_{k=0}^{n}x^{k}\big{(}F_{m}L_{mk}+(xL_{m}-2){F_{m(k+1)}}\big{)}=x^{n+1}L_{m}{F_{m(n+1)}}$
(23)

and

$\sum_{k=0}^{n}x^{k}\big{(}5F_{m}F_{mk}+(xL_{m}-2)L_{m(k+1)}\big{)}=L_{m}\big{(}x^{n+1}L_{m(n+1)}-2\big{)}.$

###### Proof.

We only prove (23). Let $m$ be odd. Then by straightforward calculation

$L_{k}(L_{m})=L_{mk}\qquad\mbox{and}\qquad F_{k}(L_{m})=\frac{F_{mk}}{F_{m}},$

where we used the identity $L_{n}^{2}=5F_{n}^{2}+(-1)^{n}4$. This immediately
gives the first identity, using (17). Let $m$ be even now. Then we work with
$iL_{m}$, $i=\sqrt{-1}$, as an argument and get

$L_{k}(iL_{m})=i^{k}L_{mk}\qquad\mbox{and}\qquad
F_{k}(iL_{m})=i^{k-1}\frac{F_{mk}}{F_{m}}.$

Inserting these results into (17) and finally replacing $x$ by $x/i$ completes
the proof. ∎

We proceed with a (short) discussion of Theorem 2. Making the choices $z=-1$,
$r=1$ and $c=0$ it becomes the next identities for Fibonacci and Lucas
polynomials.

######  Proposition 3.

For all $x$, $y\in\mathbb{C}$ and $n\in\mathbb{Z}$, we have

$\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}L_{k+1}(y)+(xy-2)F_{k}(y)\big{)}=(-1)^{n}yF_{n+1}(y)$
(24)

and

$\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}(y^{2}+4)F_{k+1}(y)+(xy-2)L_{k}(y)\big{)}=(-1)^{n}yL_{n+1}(y)+2yx^{n+1}.$
(25)

######  Corollary 10.

For all $x\in\mathbb{C}$ and $n\in\mathbb{Z}$,

$\displaystyle\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}L_{k+1}+(x-2)F_{k}\big{)}=(-1)^{n}F_{n+1},$
$\displaystyle\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}5F_{k+1}+(x-2)L_{k}\big{)}=(-1)^{n}L_{n+1}+2x^{n+1},$
$\displaystyle\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}Q_{k+1}+2(x-1)P_{k}\big{)}=2(-1)^{n}P_{n+1},$
$\displaystyle\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}4P_{k+1}+(x-1)Q_{k}\big{)}=(-1)^{n}Q_{n+1}+2x^{n+1}.$

###### Proof.

Set $y=1$ and $y=2$ in (24) and (25), respectively. ∎

We are not interested in repeating the analogous results to Corollaries
(4)–(9). We mention, however, that the symmetry gets lost when passing from
Proposition 2 to its analogue.

######  Proposition 4.

For all $x\in\mathbb{C}$ and integers $m$ and $n$, we have

$\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}F_{m}L_{m(k+1)}+(xL_{m}-2){F_{mk}}\big{)}=(-1)^{n}L_{m}{F_{m(n+1)}},\qquad
m\,\,\mbox{odd,}$

and

$\sum_{k=0}^{n}x^{n-k}\big{(}F_{m}L_{m(k+1)}-(xL_{m}-2)F_{mk}\big{)}=L_{m}{F_{m(n+1)}},\qquad
m\,\,\mbox{even}.$

Similarly, we have

$\sum_{k=0}^{n}(-1)^{k}x^{n-k}\big{(}5F_{m}F_{m(k+1)}+(xL_{m}-2){L_{mk}}\big{)}=L_{m}\left((-1)^{n}L_{m(n+1)}+2x^{n+1}\right),\qquad
m\,\,\mbox{odd},$
$\sum_{k=0}^{n}x^{n-k}\big{(}5F_{m}F_{m(k+1)}-(xL_{m}-2)L_{mk}\big{)}=L_{m}\left(L_{m(n+1)}-2x^{n+1}\right),\qquad
m\,\,\mbox{even}.$

##  4 Extension to Chebyshev polynomials

Let $T_{n}(y)$ and $U_{n}(y)$ be the Chebyshev polynomials of the first and
second kind defined by $T_{0}(y)=1$, $T_{1}(y)=y$ and for $n\geq 2$

$T_{n}(y)=2yT_{n-1}(y)-T_{n-2}(y)$

and $U_{0}(y)=1$, $U_{1}(y)=2y$ and for $n\geq 2$

$U_{n}(y)=2yU_{n-1}(y)-U_{n-2}(y).$

Both sequences $(T_{n}(y))_{n\geq 0}$ and $(U_{n}(y))_{n\geq 0}$ can be
extended to negative subscripts by writing

$T_{-n}(y)=T_{n}(y),\quad U_{-n}(y)=2yU_{-(n-1)}(y)-U_{-(n-2)}(y).$

######  Lemma 4.

For all $x$, $y$, $r\in\mathbb{C}$, $r\neq 0$ and $k\in\mathbb{Z}$, we have

$\frac{x^{k}}{r^{k}}\big{(}rT_{k}(y)+(xy-r)U_{k}(y)\big{)}=y\left(\frac{x^{k+1}}{r^{k}}U_{k}(y)-\frac{x^{k}}{r^{k-1}}U_{k-1}(y)\right),$
(26)

and

$\frac{x^{k}}{r^{k}}\left((y^{2}-1)rU_{k-2}(y)+(xy-r)T_{k}(y)\right)=y\left(\frac{x^{k+1}}{r^{k}}T_{k}(y)-\frac{x^{k}}{r^{k-1}}T_{k-1}(y)\right).$
(27)

###### Proof.

In view of the recurrence relation

$U_{k}(y)=2yU_{k-1}(y)-U_{k-2}(y)$

and the identity

$2T_{k}(y)=U_{k}(y)-U_{k-2}(y),$

we have

$rT_{k}(y)+(xy-r)U_{k}(y)=y\big{(}xU_{k}(y)-rU_{k-1}(y)\big{)},$

from which (26) follows. Similarly, using the recurrence relation

$T_{k}(y)=2yT_{k-1}(y)-T_{k-2}(y)$

and the identity

$2(y^{2}-1)U_{k-2}(y)=T_{k}(y)-T_{k-2}(y),$

we obtain

$(y^{2}-1)rU_{k-2}(y)+(xy-r)T_{k}(y)=y\big{(}xT_{k}(y)-rT_{k-1}(y)\big{)}$

from which (27) follows. ∎

######  Theorem 11.

For all $x$, $y$, $r\in\mathbb{C}$ and $c$, $n\in\mathbb{Z}$, we have

$\sum_{k=c}^{n}{r^{n-k}x^{k}\big{(}{rT_{k}(y)+(xy-r)U_{k}(y)}\big{)}}=x^{n+1}yU_{n}(y)-r^{n-c+1}x^{c}yU_{c-1}(y)$

and

$\sum_{k=c}^{n}{r^{n-k}x^{k}\left({(y^{2}-1)rU_{k-2}(y)+(xy-r)T_{k}(y)}\right)}=x^{n+1}yT_{n}(y)-r^{n-c+1}x^{c}yT_{c-1}(y).$

###### Proof.

Sum each of (26) and (27), using (3). ∎

In particular, we have

$\sum_{k=0}^{n}{x^{k}\big{(}{T_{k}(y)+(xy-1)U_{k}(y)}\big{)}}=x^{n+1}yU_{n}(y),$

as well as

$\sum_{k=0}^{n}{x^{k}\big{(}{(y^{2}-1)U_{k-2}(y)+(xy-1)T_{k}(y)}\big{)}}=x^{n+1}yT_{n}(y)-y^{2}.$

##  5 Conclusion

In these notes we have provided various generalizations of Fibonacci–Lucas
identities first proved by Edgar and Dafnis, Philippou and Livieris,
respectively. We have also presented a detailed discussion of these
generalizations including many charming polynomial identities as special
cases. We conclude noting two observations. First, concerning the results for
Fibonacci and Lucas polynomials we remark that our results can be combined
with the beautiful identities derived by Seiffert [10, 11, 12] to get some
nontrivial sum relations. This is left as a possible future research project.
Second, as it was shown here, the identities introduced and discussed are not
limited to Fibonacci and Lucas, and Chebyshev polynomials. Analogues are
possible for Pell polynomials, Jacobsthal polynomials and others. By way of a
final illustration, we state the identities in Theorem 1 for the Jacobsthal
and Jacobsthal–Lucas sequences $(J_{n})=(u_{n}(1,-2))$ and
$(j_{n})=(v_{n}(1,-2))$: For all $x$, $r\in\mathbb{C}$ and $c$,
$n\in\mathbb{Z}$, we have

$\sum_{k=c}^{n}r^{n-k}x^{k}\big{(}rj_{k}+(x-2r)J_{k+1}\big{)}=x^{n+1}J_{n+1}-r^{n-c+1}x^{c}J_{c}$

and

$\sum_{k=c}^{n}r^{n-k}x^{k}\big{(}9rJ_{k}+(x-2r)j_{k+1}\big{)}=x^{n+1}j_{n+1}-r^{n-c+1}x^{c}j_{c}.$

Interesting identities can be drawn from these relations. We leave them for
the interested readers.

## References

  * [1] W. M. Abd-Elhameed and N. A. Zeyada, _New identities involving generalized Fibonacci and generalized Lucas numbers_ , Indian J. Pure Appl. Math. 2018, 49(3), 527–537.
  * [2] A. Benjamin and J. Quinn, _Fibonacci and Lucas identities through colored tiling_ , Util. Math. 1999, 56, 137–142.
  * [3] S. D. Dafnis, A. N. Philippou and I. E. Livieris, _An identity relating Fibonacci and Lucas numbers or order k_ , Electron. Notes Discret. Math. 2018, 70, 37–42.
  * [4] S. D. Dafnis, A. N. Philippou and I. E. Livieris, _An alternating sum of Fibonacci and Lucas numbers of order k_ , Mathematics 2020, 8, 1487.
  * [5] T. Edgar, _Extending some Fibonacci-Lucas relations_ , Fibonacci Quart. 2016, 54(1), 79.
  * [6] T. Koshy, _Fibonacci and Lucas Numbers with Applications_ , Wiley-Interscience, 2001.
  * [7] D. Marques, _A new Fibonacci-Lucas relation_ , Amer. Math. Monthly 2015, 122(7), 683.
  * [8] A. N. Philippou and S. D. Dafnis, _A simple proof of an identity generalizing Fibonacci-Lucas identities_ , Fibonacci Quart. 2018, 56(4), 334–336.
  * [9] P. Ribenboim, _My Numbers, My Friends_ , Springer-Verlag, New York, 2000.
  * [10] H.-J. Seiffert, _Problem H-492_ , Fibonacci Quart. 1994, 32(5), 473–474.
  * [11] H.-J. Seiffert, _Problem H-500_ , Fibonacci Quart. 1995, 33(4), 378–379.
  * [12] H.-J. Seiffert, _Problem H-518_ , Fibonacci Quart. 1996, 34(5), 473–474.
  * [13] B. Sury, _A polynomial parent to a Fibonacci-Lucas relation_ , Amer. Math. Monthly 2014, 121(3), 236.
  * [14] S. Vajda, _Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications_ , Dover Press, 2008.


(The following article has been accepted by Applied Physics Letters. After it
is published, it will be found at https://doi.org/10.1063/5.0102370)

# Rapid determination of single substitutional nitrogen N${}_{s}^{0}$
concentration in diamond from UV-Vis spectroscopy

T. Luo  L. Lindner  Fraunhofer Institute for Applied Solid State Physics IAF,
79108 Freiburg, Germany  R. Blinder  Ulm University, D-89081 Ulm, Germany  M.
Capelli  School of Science, RMIT University, Melbourne VIC 3001, Australia  J.
Langer  V. Cimalla  F. A. Hahl  X. Vidal  J. Jeske
<EMAIL_ADDRESS>Fraunhofer Institute for Applied Solid State
Physics IAF, 79108 Freiburg, Germany

###### Abstract

Single substitutional nitrogen atoms N${}_{s}^{0}$ are the prerequisite to
create nitrogen-vacancy (NV) centers in diamonds. They serve as the electron
donors to create the desired NV- center, provide charge stability against
photo-ionisation, but also are the main source of decoherence. Therefore,
precise and quick determination of N${}_{s}^{0}$ concentration is a key
advantage to a multitude of NV-related research in terms of material
improvement as well as applications. Here we present a method to determine the
N${}_{s}^{0}$ concentration based on absorption spectroscopy in the UV-Visible
range and fitting the 270 nm absorption band. UV-Visible spectroscopy has
experimental simplicity and widespread availability that bear advantages over
established methods. It allows a rapid determination of N${}_{s}^{0}$
densities, even for large numbers of samples. Our method shows further
advantages in determining low concentrations as well as the ability to measure
locally, which is highly relevant for diamonds with largely varying
N${}_{s}^{0}$ concentrations in a single crystal. A cross-check with electron
paramagnetic resonance (EPR) shows high reliability of our method and yields
the absorption cross section of the 270 nm absorption band, $\sigma=1.96\pm
0.15$ cm${}^{-1}\cdot$ppm-1 (in common logarithm) or $\sigma_{e}=4.51\pm 0.35$
cm${}^{-1}\cdot$ppm-1 (in natural logarithm), which serves as a reference to
determine N${}_{s}^{0}$ concentrations, and makes our method applicable for
others without the need for a known N${}_{s}^{0}$-reference sample and
calibration. We provide a rapid, practical and replicable pathway that is
independent of the machine used and can be widely implemented as a standard
characterization method for the determination of N${}_{s}^{0}$ concentrations.

††preprint: AIP/123-QED

Nitrogen is one of the main impurities in both natural and synthetic diamonds
Ashfold  _et al._ (2020). Related defects, particularly the nitrogen-vacancy
(NV) center, attract a broad interest for its special optical and spin
properties Aharonovich, Greentree, and Prawer (2011); Doherty  _et al._
(2013). It has been extensively investigated for magnetometry Degen (2008);
Acosta  _et al._ (2009); Rondin  _et al._ (2014); Jeske, Cole, and Greentree
(2016); Hahl  _et al._ (2021), bio-sensing Schirhagl  _et al._ (2014); Wu  _et
al._ (2016), nuclear magnetic resonance (NMR) Glenn  _et al._ (2018); Bucher
_et al._ (2019) and scanning probe microscopy Schell  _et al._ (2011); Zhou,
Stöhr, and Yacoby (2017). To create NV centers, single substitutional nitrogen
atoms, denoted as N${}_{s}^{0}$, P1 or C-centers, are the prerequisite. They
are doped during the diamond growth or implanted on the diamond surface,
followed by irradiation and annealing steps to create vacancies and allow NV
centers to form. They play a role as the typical electron donor to charge the
desired NV- state Collins (2002); Fu  _et al._ (2010); Haque and Sumaiya
(2017); Hauf  _et al._ (2011), and they are often present 10-100 times more
than NV centers even in highly doped material. They determine the NV charge
stability, and optimising the N${}_{s}^{0}$ to NV- conversion ratio is crucial
for improving the NV center’s performance and sensitivity Luo  _et al._
(2022). Moreover, at parts-per-million (ppm) levels they can act as the main
decoherence source of the NV center Bauch  _et al._ (2020). Therefore,
knowledge of the N${}_{s}^{0}$ density gives significant advantages in the
choice of material and performance in applications.

To measure the N${}_{s}^{0}$ density in diamond, the established standard is
electron paramagnetic resonance (EPR) spectroscopy Smith  _et al._ (1959); Van
Wyk  _et al._ (1997); Eaton  _et al._ (2010). Field modulation is applied to
record continuous wave EPR spectra, resulting in a spectrum that approximates
the first derivative of the EPR line shape. Double integration of the acquired
spectrum is taken to determine the EPR peak intensity and hence the
N${}_{s}^{0}$ concentration Tallaire  _et al._ (2006). Although this method
has been well established, it is often hard to access and labour-intensive,
and its high requirement for the surface quality of diamond brings technical
difficulties for the measurement. Moreover, the EPR method measures the spin
number in the entire sample volume and averages different areas in the sample.
For diamonds with less homogeneity, e.g. high-pressure, high-temperature
(HPHT) diamonds that usually show sectors containing very different
N${}_{s}^{0}$ concentrations in a single crystal Capelli  _et al._ (2019), the
spatially resolved determination of the N${}_{s}^{0}$ density can be of
interest rather than the averaged value. In this regard, optical methods with
experimental simplicity and widespread availability have significant
advantages for many groups and companies working with NV centers, which makes
the Fourier-transform infrared (FTIR) and UV-Visible (UV-Vis) spectroscopy
favourable in some cases. Depending on the configuration of the spectrometer,
both volume and spatially resolved measurements are possible by these optical
methods.

The FTIR spectroscopy has been used to determine both N${}_{s}^{0}$ (at 1130
cm-1 and 1344 cm-1) Dobrinets, Vins, and Zaitsev (2016) and N${}_{s}^{+}$ (at
1332 cm-1) Lawson  _et al._ (1998) centers. For this method, a good spectral
resolution is required, and different resolutions of the spectrometer have a
significant effect on the concentration estimation Liggins (2010). Moreover,
for diamonds with low nitrogen densities less than a few tenths of ppm, the
conventional FTIR spectroscopy often shows insufficient sensitivity in
detecting these nitrogen-related centers Dobrinets, Vins, and Zaitsev (2016).
In comparison, the UV-Vis spectroscopy enables the detection of single
nitrogen as low as 0.01 ppm De Weerdt and Collins (2008), which is much below
the detection limit of the conventional FTIR method. Furthermore, the UV-Vis
spectroscopy with its experimental simplicity can be applied as a rapid method
to characterize diamonds, especially for low nitrogen concentrations.

In the UV-Vis spectrum, the absorption band centered at 270 nm has been
suggested to determine the N${}_{s}^{0}$ concentration Dyer  _et al._ (1965);
Chrenko, Strong, and Tuft (1971); Walker (1979). Extracting the 270 nm band
from the spectrum is thus a key step for this method. Different fitting
methods have been introduced, but complex spectra and the simplistic nature of
the literature methods leads to difficulties to determine the band reliably.
Early on, Sumiya et al. Sumiya and Satoh (1996) has suggested to subtract the
spectrum at 270 nm with the ‘tail-line’, then calibrate this height with the
EPR result. The ‘tail-line’ is a straight line fitted with the acquired
spectrum at around 600-800 nm. This method provides a convenient approach
which does not require complicated fitting. For diamonds without additional
spectral components at the ‘tail’ region, it enables a quick and rough
estimation of the N${}_{s}^{0}$ concentration. When the diamond spectrum shows
absorption bands that across the range (for example NV-, NVN- centers, or the
nickel-related broad band centered at 710 nm, etc.), the method will be less
accurate and even invalidated. Especially for chemical vapor deposition (CVD)
diamonds, a perfect ‘tail’ without influence by other defects is unusual and
difficult to achieve in nitrogen-doped growth.

A more advanced protocol has been introduced by Khan et al. Khan  _et al._
(2009, 2013), which avoids being dependent on the ‘tail’ region. This method
relied on more complex fitting components: a ‘ramp’ in the form of
$\lambda^{-3}$ well fitting the overall decreasing trend of the absorption
spectrum (which can be related to single vacancies Luo  _et al._ (2022), or
vacancy clusters Mäki  _et al._ (2007); Jones (2009)); a combination of bands
at 360 nm and 520 nm originate from vacancy clusters and NVH0 centers
respectively; and a ‘reference spectrum’ including the 270 nm band and its
absorption continuum taken from a high-pressure, high-temperature (HPHT) type
Ib diamond. The weight of the ‘reference spectrum’ then gives the strength of
the 270 nm band and thus the N${}_{s}^{0}$ concentration. This protocol
improved the fitting accuracy significantly, it broadened applicable spectrum
types as it was independent of absorption bands at 600-800 nm. Nevertheless,
it is still limited by the HPHT reference spectrum that requires a detectable
and clear 270 nm band. Besides, HPHT spectra have their own ‘ramp’ component
and potentially other spectral features, using these as reference will not
only isolate the 270nm peak but also fit the other components of the reference
to the acquired spectrum. This creates a dependency on the utilised HPHT
reference spectrum. Considering the fundamental difference in material
properties between different synthesized diamonds, to avoid using a specific
type of reference spectrum that contains the band of interest (270 nm) helps
to reduce the fitting uncertainties and further improve the accuracy.

In this work, we present a fitting protocol to estimate N${}_{s}^{0}$
concentrations reliably and precisely from the 270 nm band in the UV-Vis
absorption spectrum. This protocol can be more generally adapted for complex
spectra, especially for CVD diamonds. It also avoids the requirement for a
reference spectrum with the band of interest and increases the fitting
robustness. We furthermore calibrate this method with EPR measurements for a
series of CVD diamonds with varing N${}_{s}^{0}$ concentrations. From the
calibration we find the linearity of the two methods and precision of our
measurement. We deduce the absorption cross-section of the 270 nm band for
N${}_{s}^{0}$, enabling the estimation of its concentrations without the
requirement of a further reference sample with a known concentration.

We grew six (100) oriented CVD diamonds with varying nitrogen-doping levels
for the method calibration. We pre-characterized their N${}_{s}^{0}$
concentrations with an EPR spectrometer (Bruker ELEXSYS E580) at room
temperature. The spectrometer was fitted with a Bruker super-high-Q probehead
(ER4122 SHQE), and the microwave frequency was set to 9.84 GHz. N${}_{s}^{0}$
concentrations were determined using the built-in spin-counting feature, from
the acquisition software (xEPR). This measurement for N${}_{s}^{0}$
concentration carries an error of $\sim$6$\%$ (including the statistic error,
the fitting error, and the accuracy of the EPR spectrometer). For details of
the N${}_{s}^{0}$ concentration see Table 1.

To obtain the UV-Vis absorption spectra for the diamond, we measured the
diamond transmission $T$ in the range of 200-800 nm using an UV-Vis
spectrometer (PerkinElmer Lambda 950) at room temperature, then we deduced the
absorption coefficient $A$ following the Lambert-Beer law: $A=$
$-\log_{10}(T)/d$, where $d$ is the sample thickness. Figure 1 shows a typical
UV-Vis absorption spectrum of CVD diamonds, containing three absorption bands
centered at respectively 270 nm, 360 nm and 520 nm. We fitted the spectrum
with five components (Figure 1):

\- Three Gaussian functions for the three typical bands in CVD diamonds
$g_{j}(\lambda)=a_{j}\cdot\exp{(-(\lambda-b_{j})^{2}/(2c_{j}^{2}))}$, where
$a_{j}$, $b_{j}$ and $c_{j}$ are the fitting parameters for each band $j$=270,
360, 520 (nm). Here 270 nm is the band of interest to extract the
N${}_{s}^{0}$ concentration, 360 nm corresponds to vacancy clusters and 520 nm
to NVH0 centers Khan  _et al._ (2013).

\- A ‘ramp’ function $r(\lambda)=R\cdot\lambda^{-3}$ (same as in Khan  _et
al._ (2009)), where the factor $R$ is the fitting parameter for this function.

\- A spectrum ‘El-offset’, $e(\lambda)$, which is given by an electronic grade
diamond (theoretically a ‘pure’ diamond without defect bands in the UV-Vis
range). This reference spectrum is applied as an offset and baseline to the
acquired spectrum, and it is also fitted with a coefficient $d$.

We fitted a sum of the five components to the original spectrum by a non-
linear least squares fit:

$Fit=g_{270}(\lambda)+g_{360}(\lambda)+g_{520}(\lambda)+r(\lambda)+d\cdot
e(\lambda)$

Figure 1: Fitting for the UV-Vis absorption spectrum including three Gaussian
bands at 270 nm, 360 nm and 520nm, a ’ramp’ function and an electronic-grade
diamond spectrum as the offset. The fitting determines weighting factors to
the components. The 270 nm band (red solid curve) can be taken as a direct
indication of the concentration of N${}_{s}^{0}$.

Different from previous works, here the key component, the 270 nm feature, is
extracted as a Gaussian band. For this band (i.e. $g_{270}(\lambda)$
function), a lower- and upper-boundary of the fitting parameter have been
introduced for the band-center $b_{270}$ (268-272 nm) and the Gaussian RMS
width $c_{270}$ (13-27 nm), in order guide the algorithm to optimize the
parameters in a small range. The boundaries were selected based on free
fitting results of a large amount of samples, they also help to examine
whether the fitting outcome exhibits reasonable values: if a final parameter
is one of the boundary values, this indicates a unreliable fitting
performance, i.e. the boundary has ‘forced’ the algorithm to stop while no
optimization has been found in the expected range. On the contrary, if the
final parameter exhibits a free value in the given range, the fitting was
optimized correctly. For all our fits this condition was met.

With extracting the 270nm band, we then compare its height, $\mu_{270}$ (in
cm-1), with the EPR measurements to investigate if the fitting result scales
linearly to the EPR and thus can be validated. In this method, no reference
sample with an ideal 270 nm band is required, the only reference spectrum in
this method is the ‘El-offset’ component measured from the electronic grade
diamond, which is an undoped diamond and thus avoids the variation which
occurs in doped reference spectra. One can easily obtain this spectrum
repeatably from any ‘pure’ diamond, or simply use a literature spectrum as the
offset for well-calibrated spectrometers. 111We can provide the spectrum data
for the ‘El-offset’ and other technical assistance upon request. This reduces
the difficulty for implementing the fitting, and greatly avoids introducing
undesired spectral components that vary from diamond to diamond. The ‘El-
offset’ keeps the diamond intrinsic spectral feature (with an absorption edge
at around 225-235 nm, then being ‘flat’ in the visible range up to 800 nm),
instead of using a straight line as the offset. Importantly, considering that
the 270 nm band is located very close to the absorption edge ($\sim$225-235
nm), an additional parameter describing the sharp drop in this regime is
necessary to supplement the ‘ramp’ function. In this sense, introducing the
‘El-offset’ can considerably improve the fitting performance.

For a spectrum with a weak or undetectable 520 nm band, i.e. for samples
without NVH0 centers potentially, the fitting method can be also adapted to
four components. This can help to improve the fitting accuracy for some
samples, as the fitting parameters for $g_{520}(\lambda)$ should be nearly
zero in this sense, subtracting this band in the fitting function reduces
unnecessary fitting parameters.

Figure 2 shows the fitting result for the six samples, they all show a good
match between the fitting result and original spectrum. Mismatches appear at
high wavelength ($>$650 nm), which is due to higher spectral features that are
not included in the fitting. A possible candidate for these CVD diamonds can
be H2 centers (NVN-) with a zero-phonon line at 986 nm and a broad phonon side
band centered at around 800 nm. For samples with strong features in this
higher-wavelength regime (i.e. Cas-48, Cas-50 and Cas-68), a cut-off at 650 nm
for the acquired data (i.e. fit for 200-650 nm) can improve the fitting
performance. The good fitting result at lower wavelength shows that our method
is independent of higher spectral features. The spectral variation around
650-800 nm illustrates the problem with defining a ‘tail’ as a baseline
reference for the 270 nm peak Sumiya and Satoh (1996).

Figure 2: Fitting result for all six samples. Details of the N${}_{s}^{0}$
concentration and the peak height of 270 nm see Table 1.

As the next step, we compare the height of the extracted 270 nm Gaussian peak,
$\mu_{270}$, to the EPR result (Fig. 3). The good agreement of the two methods
shown by the close-to-linear arrangement of the data points, proves the
reliability of our fitting method. The 270nm band has been assigned only to
N${}_{s}^{0}$ and has been used in several papers to determine only the
N${}_{s}^{0}$ concentration Sumiya and Satoh (1996); Khan  _et al._ (2009,
2013); Edmonds  _et al._ (2021). We note however that an earlier research has
suggested that the 270 nm band can be influenced by both N${}_{s}^{0}$ and
N${}_{s}^{+}$, especially for diamonds with low nitrogen concentrations Jones,
Goss, and Briddon (2009), which has never been fully ruled out. Although our
results can also not fully eliminate the possibility of a contribution of
N${}_{s}^{+}$, the good linear fit of the EPR and UV-Vis results indicates
that N${}_{s}^{+}$ has minor influence if any. The discrepancies between the
two methods in some previous works might have arisen from an inadequate
fitting method for the spectrum. For more details of the band height and the
EPR result see Table 1.

Figure 3: The UV-Vis result by our fitting method is well aligned to the EPR
result with a linear correlation. The linear fit shows a slope of
1.96$\pm$0.15 cm${}^{-1}\cdot$ppm-1 for $\mu_{270}$ given by decadic
absorption coefficients. The error of the slope is given by 95% confidence
interval of the fitting parameter (gray area in the plot).

From the slope in Figure 3, one can deduce the absorption cross-section
$\sigma$ of N${}_{s}^{0}$ at 270 nm , with the relation:

$\mu_{270}=\sigma\cdot[N^{0}_{s}]$ (1)

where $[N^{0}_{s}]$ is the N${}_{s}^{0}$ concentration in ppm, $\mu_{270}$ is
the absorption coefficient of the extracted 270 nm band height (as described
above). The theoretical derivation of the absorption cross-section is
discussed in supplementary material. For common logarithm (i.e. decadic
absorption coefficients), the absorption cross-section $\sigma=$ $1.96$$\pm
0.15$ cm${}^{-1}\cdot$ppm-1, the error is given by 95% confidence interval of
the fitting parameter for the linear fit to the two methods. $\sigma$ can be
also given in cm2, i.e. $\sigma=$ $(1.11\pm$$0.09)\times 10^{-17}$ cm2, which
is converted by multiplying a factor 10${}^{6}m_{C}/\rho_{dia}$, where
$m_{C}=1.99\times$10-23 g is the atomic mass for 12C and $\rho_{dia}=3.51$
g/cm3 is the diamond density.

For the absorption coefficient calculated by natural logarithm, an absorption
cross-section $\sigma_{e}=$ $4.51\pm$$0.35$ cm${}^{-1}\cdot$ppm-1 should be
applied instead. In previous research, we have found a single reference of the
absorption cross-section at 270 nm Dobrinets, Vins, and Zaitsev (2016), which
has been stated as $1/\sigma=0.6\pm 0.1$ ppm/cm-1 (i.e. $\sigma=1.67\pm 0.28$
cm${}^{-1}\cdot$ppm-1). However, the method of its determination, specifically
how the spectrum has been separated was not stated. We thus assume our
measurement is more precise, the values of these two completely independent
measurements are remarkably close.

From our protocol, one can estimate $[N^{0}_{s}]$ directly from their UV-Vis
spectra, without calibrating by other methods. This can be achieved for any
samples with both sides polished in the following way: Firstly measure the UV-
Vis transmission in percentage then convert it into the absorption spectrum in
cm-1; secondly separate the spectrum to extract the 270 nm band and obtain
$\mu_{270}$ according to the method described above; finally calculate the
N${}_{s}^{0}$ concentration using Equation (1) with the absorption cross-
section $\sigma$ or $\sigma_{e}$ (depending on the logarithm type when
deducing the absorption from transmission). The method is setup-independent
and applicable for UV-Vis spectra taken by any machine, whether by volume
measurements or spatially resolved measurements. No further EPR/FTIR
measurements are required, as Equation (1) gives the absolute value of the
N${}_{s}^{0}$ concentration and our value for sigma can be used.

Table 1: Details of the samples and results. The N${}_{s}^{0}$ concentration [N${}_{s}^{0}$] was measured by EPR, which brings an error of around $\pm$6 %. The fitting for $\mu_{270}$ has an error of $\lesssim$1 %. Sample | [N${}_{s}^{0}$] (ppm) |  $\mu_{270}$ (cm-1)
---|---|---
Cas-40 | 3.2 | 5.9
Cas-44 | 9.5 | 17.7
Cas-48 | 5.2 | 10.9
Cas-50 | 19.3 | 37.2
Cas-51 | 11.2 | 24.7
Cas-68 | 7.8 | 13.9

We developed a fitting method for determining N${}_{s}^{0}$ concentration via
UV-Vis absorption spectrum, which is more widely accessible and easily
implementable than EPR measurements. We showed that the fitting of the
identified bands is a reliable way to extract the 270 nm band from the
background, as seen by the good match between measurement and fit, and
confirmed by the linear relationship between EPR result and our method. The
good agreement with EPR furthermore confirms the assumption that the 270 nm
band indeed is mainly caused by N${}_{s}^{0}$. Our fitting method performs
well for diamond spectra without complex components overlapping with the the
fitting components. In other words, it can be widely applied to different
diamond types apart from as-grown CVD diamonds.

Furthermore, we deduced the absorption cross-section $\sigma=$ $1.96\pm$$0.15$
cm${}^{-1}\cdot$ppm-1 (for common logarithm) and $\sigma_{e}=$ $4.51\pm$$0.35$
cm${}^{-1}\cdot$ppm-1 (for natural logarithm), which can serve to rapidly
determine N${}_{s}^{0}$ densities from UV-Vis measurements without the need to
calibrate a setup via EPR. This also enables the determination of
N${}_{s}^{0}$ concentrations in a lower end that is hardly detectable by EPR
or FTIR methods. The detectable range of N${}_{s}^{0}$ concentration is
limited by the sensitivity of the UV-Vis spectrometer. Based on our
spectrometer, we estimated a detectable range for typical 300 $\mu$m diamond
plates from 0.01 ppm to 30-50 ppm (details are discussed in supplementary
material). For thinner/thicker samples, the determination of higher/lower
N${}_{s}^{0}$ concentrations are also possible. The range can be expanded by
orders of magnitude with more sensitive spectrometers. The fitting protocol
combined with the calibrated absorption cross-section provide a rapid, easy
and replicable pathway for the standardized determination of N${}_{s}^{0}$
concentrations for future research.

## Supplementary Material

See supplementary material for the discussion of the detectable range of
N${}_{s}^{0}$ concentration and the theoretical derivation of the absorption
cross-section.

We thank Brant Gibson, Andrew Greentree, Peter Knittel, Christian Giese,
Christoph Schreyvogel and Oliver Ambacher for valuable discussions. We thank
Fedor Jelezko for valuable discussions and supports of the EPR theory. We
thank Michael Ardner, Christine Lell and Michaela Fritz for preparing diamond
plates, Dorothee Luick for the technical support of UV-Vis measurements. T.L
and J.J acknowledge the funding by the German federal ministry for education
and research Bundesministerium für Bildung und Forschung (BMBF) under Grant
No. 13XP5063. M.C. acknowledges funding from the Asian Office of Aerospace
Research and Development (AOARD, funding FA2386-18-1-4056).

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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  * Note (1) We can provide the spectrum data for the ‘El-offset’ and other technical assistance upon request.
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