<PAGE>


                                                                  Exhibit 4.2


                            INVESTOR RIGHTS AGREEMENT

       THIS INVESTOR RIGHTS AGREEMENT (this "Agreement"), made as of the 16th
day of January, 2001, by and among Bruker AXS Inc., a Delaware corporation (the
"COMPANY"), those persons listed on the signature pages hereto as "Current
Shareholders" (the "CURRENT SHAREHOLDERS"), and the entities set forth on the
signature pages hereto as "Investors" (the "INVESTORS" and, together with the
Current Shareholders and those persons who become holders of the capital stock
of the Company, the "SHAREHOLDERS").

       WHEREAS, the Investors are acquiring an aggregate of up to 5,625,000
shares of Series A Convertible Preferred Stock, $0.01 par value per share (the
"SERIES A PREFERRED STOCK"), of the Company, pursuant to the terms of that
certain Series A Preferred Stock Purchase Agreement dated as of even date
herewith among the Company and the Investors (the "PURCHASE AGREEMENT"); and

       WHEREAS, it is a condition to the obligations of the Investors under the
Purchase Agreement that this Agreement be executed by the parties hereto, and
the parties are willing to execute this Agreement and to be bound by the
provisions hereof;

       NOW, THEREFORE, in consideration of the foregoing and, the agreements set
forth below, the parties hereby agree with each other as follows:

       1. DEFINITION OF SHARES. As used in this Agreement, "SHARES" shall mean
and include all shares of the Series A Preferred Stock and Common Stock, $0.01
par value per share (the "COMMON STOCK") of the Company issuable on conversion
thereof now owned or hereafter acquired by the Investors and all shares of
Common Stock now owned or hereafter acquired by any Shareholder.

       2. PROHIBITED TRANSFERS.

            (a) No Current Shareholder shall sell, assign, transfer, pledge,
hypothecate, mortgage, encumber or dispose (collectively, "TRANSFER") of all or
any of his or her Shares except in compliance with the terms of this Agreement.
Notwithstanding anything to the contrary contained in this Agreement, any
Current Shareholder may Transfer without the necessity of prior approval all or
any of his or her Shares by way of gift to his or her spouse, to any of his or
her lineal descendants, ancestors or siblings, or to any trust for the benefit
of any one or more of such Current Shareholder, his or her spouse or his or her
lineal descendants, ancestors or siblings without the necessity of complying
with Sections 2(b), 3 or 4 of this Agreement as if it were a Current Shareholder
so long as such transferee agrees in writing to be bound by all of the terms and
conditions of this Agreement (a "PERMITTED TRANSFER"), provided that such
Transfers shall not exceed in the aggregate, ten percent (10%) of the Company's
outstanding Common Stock on a fully-diluted basis as of the date hereof.

<PAGE>


            (b) The Company shall not be required (i) to transfer on its books
any of the Current Shareholders' Shares which shall have been sold or
transferred in violation of any of the provisions set forth in this Agreement,
or (ii) to treat as owner of such Shares or to pay dividends to any transferee
to whom any such Shares shall have been sold or transferred in violation of any
of the provisions set forth in this Agreement.

       3. RIGHT OF FIRST REFUSAL ON DISPOSITIONS.

            Subject to Section 2 hereof:

            (a) If at any time a Current Shareholder (a "SELLING SHAREHOLDER")
desires to sell or otherwise transfer all or any part of his or her Shares
pursuant to a bona fide offer from a third party (the "PROPOSED TRANSFEREE"),
the Selling Shareholder shall provide written notice (the "NOTICE") to the
Company and the Investors. The Notice shall disclose the identity of the
Proposed Transferee, the number of such Shares proposed to be sold (the "OFFERED
SHARES"), the total number of Shares owned by the Selling Shareholder, the terms
and conditions, including price, of the proposed sale, and any other material
facts relating to the proposed sale.

            (b) The Company may acquire, in accordance with the provisions of
this Agreement, all or any portion of the Offered Shares for the price and upon
the other terms and conditions set forth in the Notice by delivering written
notice to the Selling Stockholder within fifteen (15) business days after the
Notice is deemed to have been received by it.

            (c) In the event the Company does not exercise its option within
such 15 business day period to purchase all of the Offered Shares, each Investor
shall have the right to purchase, for the price and upon the terms and
conditions set forth in the Notice, provided, however, each Investor shall be
allowed to pay cash equal to the fair market value, as determined in good faith
by the Board of Directors, of any non-cash consideration described in the
Notice, that portion of the remaining Offered Shares (the "Available Shares")
equal to the product obtained by multiplying (A) the number of Available Shares
by (B) a fraction, the numerator of which is the number of shares of Common
Stock owned by such Investor (assuming the conversion of all Series A Preferred
Stock), and the denominator of which is the number of shares of Common Stock
owned in the aggregate by all the Preferred Holders (assuming the conversion of
all Series A Preferred Stock).

            (d) If any Investor desires to purchase all or part of such
Investor's PRO RATA portion of the Available Shares, such Investor shall
communicate in writing its election to purchase to the Selling Shareholder,
which communication shall state the number of Available Shares such Investor
desires to purchase (including whether or not it desires to purchase any
Available Shares not purchased by other Investors) and shall be provided to the
Selling Shareholder within fifteen (15) business days after the lapse of the 15
business day period in Section 3(b). Such communication shall, when taken in
conjunction with the Notice, be deemed to constitute a valid, legally binding
and enforceable agreement for the sale and purchase of such Available Shares. To
the extent any of the Investors elects not to purchase the full number of shares
it is entitled to purchase pursuant to Section 3(c), each other Investor's
rights to purchase shall be increased PRO RATA based on the number of shares
each Preferred Holder has elected to

                                      -2-

<PAGE>


purchase hereunder by the number of shares owned by all other Investors electing
to purchase shares hereunder (assuming the conversion of all Series A Preferred
Stock).

            (e) Sales of such Offered Shares to be sold to the Company and/or
Investors pursuant to this Section 3 shall be made at the offices of the Company
within sixty (60) days following the date of the Notice; provided, however, that
such closing may be extended as to any Investor for purposes of obtaining any
necessary governmental approvals.

            (f) If the Company and the Investors do not purchase all of the
Offered Shares (after giving effect to the over-allotment option in Section 3(d)
above), then the Selling Shareholder may sell all or any portion of the Offered
Shares at any time within ninety (90) days after the date of the Notice, subject
to the provisions of Section 4. Any such sale shall be to the Proposed
Transferee, at not less than the price and upon other terms and conditions, if
any, not more favorable to the Proposed Transferee than those specified in the
Notice. Any remaining Offered Shares not sold within such 90-day period shall
continue to be subject to the requirements of a prior offer pursuant to this
Section 3. If Offered Shares are sold pursuant to this Section 3 to any
purchaser who is not a party to this Agreement, the purchaser of such Offered
Shares shall execute a counterpart of this Agreement as a precondition of the
purchase of such Offered Shares and any Offered Shares sold to such purchaser
shall continue to be subject to the provisions of this Agreement as if such
purchaser was a Current Shareholder.

            (g) The exercise or non-exercise of the rights of the Investors
under this Section 3 to purchase Available Shares in one or more transfers made
by a Selling Shareholder shall not adversely affect their rights to participate
in subsequent transfers by a Selling Shareholder subject to this Section 3.

       4. RIGHT OF PARTICIPATION IN SALES.

            (a) If at any time a Selling Shareholder desires to sell any Shares
owned by him or her to a Proposed Transferee, and those Shares to be transferred
have not been purchased by the Company and/or Investors under Section 3, the
Investors shall have the right to sell to the Proposed Transferee, as a
condition to such sale by the Selling Shareholder, at the same price per share
and on the same terms and conditions set forth in the Notice, a PRO RATA portion
of the amount of Shares proposed to be sold to the Proposed Transferee. The "PRO
RATA portion" of Shares which an Investor shall be entitled to sell to the
Proposed Transferee shall equal the number of Shares proposed to be sold to the
Proposed Transferee multiplied by a fraction, the numerator of which is the
aggregate of all shares of Common Stock which are then held by such Investor
(assuming the conversion of all Series A Preferred Stock), and the denominator
of which is the aggregate of all shares of Common Stock which are then held by
the Selling Shareholder and all Investors (assuming the conversion of all Series
A Preferred Stock) entitled to participate in any sale under this Section 4.

            (b) Each Selling Shareholder who wishes to make a sale to a Proposed
Transferee which is subject to this Section 4 shall, after complying with the
provisions of Section 3, give to the Preferred Holders notice of such proposed
sale (which shall include the same information required to be in the Notice),
and specifying the number of Offered Shares not purchased pursuant to Section 3.
Such notice shall be given at least fifteen (15) business days

                                      -3-

<PAGE>


prior to the date of the proposed sale to the Proposed Transferee. The Preferred
Holder's wishing to so participate in any sale under this Section 4 shall notify
the Selling Shareholder in writing of such intention no later than the
expiration of such period (including whether or not it desires to sell
additional Shares if other Investors elect not to participate in such sale). To
the extent one of the Investors elects not to sell the full number of shares it
is entitled to sell pursuant to Section 4(b), each other Preferred Holder's
rights to sell shall be increased PRO RATA based on the number of shares each
Preferred Holder has elected to sell hereunder by the number of shares owned by
all other Preferred Holders electing to sell shares hereunder (assuming the
conversion of all Series A Preferred Stock).

            (c) The Selling Shareholder and the Preferred Holders, electing to
participate under this Section 4, shall sell to the Proposed Transferee all the
Shares proposed to be sold by them at not less than the price and upon other
terms and conditions, if any, not more favorable to the Proposed Transferee than
those in the notice provided by the Selling Shareholder under subparagraph (b)
above; PROVIDED, HOWEVER, that any purchase of less than all of such Shares by
the Proposed Transferee shall be made from the Selling Shareholder and the
Preferred Holders, electing to participate under this Section 4, PRO RATA based
upon the relative number of the Shares that the Selling Shareholder and the
Preferred Holders, electing to participate are otherwise entitled to sell
pursuant to this Section 4.

            (d) Each Investor who elects to participate in the sale pursuant to
this Section 4 shall effect its participation in the sale by promptly delivering
to the Selling Shareholder for transfer to the Proposed Transferee one or more
certificates, properly endorsed for transfer, which represent that number of
shares of Series A Preferred Stock which is at such time convertible into the
number of shares of Common Stock which such Investor elects to sell; PROVIDED,
HOWEVER, that such Investor may convert such Series A Preferred Stock into
Common Stock and deliver Common Stock as provided in Section 4(e) below. The
Company and the Investor agree to make any such conversion concurrent with the
actual transfer of such shares to the Proposed Transferee.

            (e) The stock certificate or certificates that the Investor delivers
to the Selling Shareholder pursuant to Section 4(d) shall be transferred to the
Proposed Transferee in consummation of the sale of the Shares pursuant to the
terms and conditions specified in the notice required by Section 4(b), and the
Selling Shareholder shall concurrently therewith remit to such Investor that
portion of the sale proceeds to which such Investor is entitled by reason of its
participation in such sale. To the extent that any Proposed Transferee or other
purchasers prohibit such assignment or otherwise refuses to purchase shares or
other securities from an Investor exercising its rights of co-sale hereunder,
the Selling Shareholder shall not sell to such Proposed Transferee or other
purchasers any Shares unless and until, simultaneously with such sale, the
Proposed Transferee shall purchase such shares or other securities from such
Investor on the same terms and conditions specified in the notice required by
Section 4(b).

            (f) If any Shares are sold pursuant to this Section 4 to any
purchaser who is not a party to this Agreement, the purchaser of such Shares
shall execute a counterpart of this Agreement as a precondition to the purchase
of such Shares and such Shares shall continue to be subject to the provisions of
this Agreement as if he or she were a Current Shareholder.

                                      -4-

<PAGE>


            (g) The exercise or non-exercise of the rights of the Investors
under this Section 4 to participate in the transfer of shares in one or more
transfers made by a Selling Shareholder shall not adversely affect their rights
to participate in subsequent transfers by a Selling Shareholder subject to this
Section 4.


       5. REGISTRATION RIGHTS

       5.1 CERTAIN DEFINITIONS. As used in this Section 5, the following terms
shall have the following respective meanings:

       "HOLDER" means the person who is then the record owner of Registrable
Securities which have not been sold to the public.

       "INITIATING HOLDERS" means the Holders or their assignees who in the
aggregate are holders of at least fifty percent (50%) of the Registrable
Securities held by the Holders and their assigns.

       "REGISTRABLE SECURITIES" means (i) all of the Common Stock issued or
issuable upon conversion of the Series A Preferred Stock; and (ii) any Common
Stock issued in respect of the Series A Preferred Stock and the Common Stock
issued or issuable upon the conversion thereof, upon any stock split, stock
dividend, recapitalization or other similar event.

       "REGISTER" means to register under the Act and applicable state
securities laws for the purpose of effecting a public sale of securities.

       "REGISTRATION EXPENSES" means all expenses incurred by the Company in
compliance with Section 5 hereof, including, without limitation, all
registration and filing fees, printing expenses, transfer taxes, fees and
disbursements of counsel for the Company, blue sky fees and expenses, fees and
disbursements of up to $25,000 for one counsel and accountants for each
registration for all the selling Holders and other security holders, and the
expense of any special audits incident to or required by any such registration.

       "SELLING EXPENSES" means all underwriting discounts and selling
commissions applicable to the sale of Registrable Securities.

       5.2 DEMAND REGISTRATIONS

            (a) If at any time following the earlier of: (i) six months after
the closing of a public offering of the Company's equity securities pursuant
to a registration statement declared effective under the Securities Act of
1933, as amended (the "Act"), or (ii) January 16, 2006, the Company shall
receive from one or more of the Initiating Holders a written request that the
Company effect the registration of Registrable Securities with a reasonably
anticipated aggregate price to the public of at least $5,000,000, the Company
will:

                 (i) promptly give written notice of the proposed registration
to all other Holders; and

                                      -5-

<PAGE>


                 (ii) as soon as practicable, and in any event within ninety
(90) days of the receipt of such request, use its best efforts to effect such
registration under the Act of all Registrable Securities which the Holders
request to be registered, subject to the limitations of Section 5.2(d).

            (b) If the Initiating Holders intend to distribute the Registrable
Securities covered by their request by means of an underwriting, they shall so
advise the Company as a part of their request made pursuant to Section 5.2(a).
The underwriter will be selected by the Company and shall be a nationally
recognized underwriter reasonably acceptable to a majority in interest of the
Initiating Holders. In such event, the right of any Holder to include
Registrable Securities in such registration shall be conditioned upon such
Holder's participation in such underwriting and the inclusion of such Holder's
Registrable Securities in the underwriting to the extent provided herein. All
Holders proposing to distribute their Registrable Securities through such
underwriting shall (together with the Company as provided in Section 5.6(d))
enter into an underwriting agreement in customary form with the underwriter or
underwriters selected for such underwriting. Notwithstanding any other provision
of this Section 5.2, if the underwriter advises the Company and the Initiating
Holders in writing that marketing factors require a limitation of the number of
shares to be underwritten, then the Company shall so advise all Holders
participating in such registration, and the number of shares of Registrable
Securities that may be included in such registration shall be allocated among
all participating Holders, including the Initiating Holders, in proportion (as
nearly as practicable) to the amount of Registrable Securities owned by each
Holder; provided that all other securities are first entirely excluded from such
registration; provided further, however, that if Thomas Weisel Capital Partners,
L.P. or any of its affiliates ("TWCP") shall be one of the Initiating Holders,
then the number of shares of Registrable Securities to be included by TWCP in
such registration shall not be reduced unless all other securities are first
entirely excluded from the registration. Notwithstanding the foregoing, in the
event that the number of Registrable Securities to be included in any such
underwriting is decreased by forty percent (40%) or more of the shares requested
to be registered by the Holders, such registration shall not constitute an
exercise of a demand registration right for purposes of Section 5.2(d) below.

            (c) Notwithstanding the foregoing, if the Company shall furnish to
the Holders requesting registration pursuant to this Section 5.2, a certificate
signed by the Chief Executive Officer of the Company stating that in the good
faith judgment of the Board of Directors of the Company, it would be seriously
detrimental to the Company and its stockholders for a registration statement to
be filed and it is therefore essential to defer the filing of such registration
statement, the Company shall have the right to defer taking action with respect
to such filing for a period of not more than ninety (90) days after receipt of
the request of the Initiating Holders; provided, however, that if the
effectiveness of a registration statement is suspended pursuant to this
provision, the period of such suspension shall be added to the end of the period
that such registration statement would otherwise be required to be effective
hereunder so that the aggregate number of days that such registration statement
is required to remain effective hereunder shall remain unchanged; provided,
further, that the Company may not utilize this right more than once in any
twelve-month period.

            (d) LIMITATION. The Holders shall not be permitted to require more
than two registrations under this Section 5.2. provided that such registration
statements have either: (A)

                                      -6-

<PAGE>


been declared or ordered effective (and kept effective for the required time
periods) or (B) have been subsequently withdrawn, unless the withdrawal is (1)
based upon material information concerning the Company of which the Initiating
Holders were not aware at the time of such request, (2) made at the request of
the Company and agreed to by a majority-in-interest of the Initiating Holders,
(3) based on a good faith determination by the underwriter for the registration
that adverse market conditions exist that would seriously jeopardize the success
of the registration or (4) at the request of a majority in interest of the
Holders participating in such registration, provided that such Holders have
reimbursed the Company for reasonable out-of-pocket costs and expenses incurred
in connection with the withdrawn registration statement.

       5.3 "PIGGY BACK" REGISTRATIONS.

            (a) If the Company shall determine to register any of its securities
for its own account other than a registration relating solely to employee
benefit plans, or a registration on any registration form which does not permit
secondary sales or does not include substantially the same information as would
be required to be included in a registration statement covering a sale of
Registrable Securities, the Company will:

                 (i) Promptly give to each Holder of Registrable Securities
       written notice thereof (which shall include the number of shares the
       Company or other security holder proposes to register and, if known, the
       name of the proposed underwriter); and

                 (ii) Use its best efforts to include in such registration all
       the Registrable Securities specified in a written request or requests,
       made by any Holder within twenty (20) days after the date of delivery of
       the written notice from the Company described in clause (i) above. If the
       underwriter for such registration advises the Company that marketing
       considerations require a limitation on the number of shares offered
       pursuant to any such registration statement, then the Company may offer
       all of the securities it proposes to register for its own account or the
       maximum amount that the underwriter considers saleable and such
       limitation on any remaining securities that may, in the opinion of the
       underwriter, be sold will be imposed PRO RATA among all Holders who are
       entitled to include shares in such registration statement according to
       the number of Registrable Securities held by each such Holder; PROVIDED,
       FURTHER, that if the Holders of Registrable Securities are subject to the
       aforesaid cut-backs, no other shareholder shall be entitled to sell
       shares in such registration. In no event, shall the shares of Registrable
       Securities to be included in the offering pursuant to this Section 5.3 be
       reduced below twenty-five percent (25%) of the total amount of securities
       included in such offering.

            (b) The Company shall select the underwriter for an offering made
pursuant to this Section 5.3.

       5.4 EXPENSES OF REGISTRATION. All Registration Expenses incurred in
connection with any registration, qualification or compliance pursuant to
Section 5 shall be paid by the Company.

       5.5 REGISTRATION ON FORM S-3. The Company shall use its best efforts to
qualify and remain eligible for registration on Form S-3 or any comparable or
successor form; and to that end the Company shall register (whether or not
required by law to do so) its Common Stock

                                      -7-

<PAGE>



under the Securities Exchange Act of 1934 (the "EXCHANGE ACT") in accordance
with the provisions of the Exchange Act following the effective date of the
first registration of any securities of the Company on Form S-1 or any
comparable or successor form. After the Company has qualified for the use of
Form S-3, in addition to the rights contained in the foregoing provisions of
this Section 5, the Holders of Registrable Securities shall have the right to
request registrations on Form S-3 (such requests shall be in writing and
shall state the number of shares of Registrable Securities to be disposed of
and the intended methods of disposition of such shares by such Holder or
Holders), provided that in no event shall the Company be required to (i)
register shares which would result in a registration with an aggregate market
value of less than $1,000,000 or (ii) effect more than two (2) registrations
pursuant to this Section 5.5 in any twelve (12) month period. Notwithstanding
the foregoing, if the Company shall furnish to the Holders requesting
registration pursuant to this Section 5.5, a certificate signed by the Chief
Executive Officer of the Company stating that in the good faith judgment of
the Board of Directors of the Company, it would be seriously detrimental to
the Company and its stockholders for a registration statement to be filed and
it is therefore essential to defer the filing of such registration statement,
the Company shall have the right to defer taking action with respect to such
filing for a period of not more than ninety (90) days after receipt of the
request of the Initiating Holders; provided, however, that if the
effectiveness of a registration statement is suspended pursuant to this
provision, the period of such suspension shall be added to the end of the
period that such registration statement would otherwise be required to be
effective hereunder so that the aggregate number of days that such
registration statement is required to remain effective hereunder shall remain
unchanged; provided, further, that the Company may not utilize this right
more than once in any twelve-month period.

       5.6 REGISTRATION PROCEDURES. In the case of each registration effected by
the Company pursuant to this Section 5, the Company will keep each Holder of
Registrable Securities included in such registration advised in writing as to
the initiation of each registration and as to the completion thereof. At its
expense, the Company will do the following for the benefit of such Holders:

            (a) Prepared file with the SEC a registration statement with respect
to such Registrable Shares and keep such registration effective for a period of
one hundred twenty (120) days or until the Holder or Holders have completed the
distribution described in the registration statement relating thereto, whichever
first occurs, and amend or supplement such registration statement and the
prospectus contained therein from time to time to the extent necessary to comply
with the Act and applicable state securities laws;

            (b) Use its reasonable best efforts to register or qualify the
Registrable Securities covered by such registration under the applicable
securities or "blue sky" laws of such jurisdictions as the selling shareholders
may reasonably request; provided, that the Company shall not be obligated to
qualify to do business in any jurisdiction where it is not then so qualified or
otherwise required to be so qualified or to take any action which would subject
it to the service of process in suits other than those arising out of such
registration;

            (c) Furnish such number of prospectuses and other documents incident
thereto as a Holder from time to time may reasonably request;

                                      -8-

<PAGE>


            (d) In connection with any underwritten offering pursuant to a
registration statement filed pursuant to Section 5.2 hereof, the Company will
enter into any underwriting agreement reasonably necessary to effect the offer
and sale of Common Stock, provided such underwriting agreement contains
customary underwriting provisions and is entered into by the Holders;

            (e) To the extent then permitted under applicable professional
guidelines and standards, obtain a comfort letter from the Company's independent
public accountants in customary form and covering such matters of the type
customarily covered by comfort letters and an opinion from the Company's counsel
in customary form and covering such matters of the type customarily covered in a
public issuance of securities, in each case addressed to the Holders, and
provide copies thereof to the Holders; and

            (f) Permit the counsel to the selling shareholders whose expenses
are being paid pursuant to Section 5.4 hereof to inspect and copy such corporate
documents as he may reasonably request.

            (g) Notify each Holder of Registrable Securities covered by such
registration statement at any time when a prospectus relating thereto is
required to be delivered under the Securities Act of the happening of any event
as a result of which the prospectus included in such registration statement, as
then in effect, includes an untrue statement of a material fact or omits to
state a material fact required to be stated therein or necessary to make the
statements therein not misleading in the light of the circumstances then
existing. The Company will use reasonable efforts to amend or supplement such
prospectus in order to cause such prospectus not to include any untrue statement
of a material fact or omit to state a material fact required to be stated
therein or necessary to make the statements therein not misleading in the light
of the circumstances then existing.

            (h) Provide a transfer agent and registrar for all Registrable
Securities registered pursuant hereto and a CUSIP number for all such
Registrable Securities, in each case not later than the effective date of such
registration.

            (i) In the event of any underwritten public offering, cooperate
with, and cause the Company's senior management to cooperate with, the selling
Holders, the underwriters participating in the offering and their counsel in any
due diligence investigation reasonably requested by the selling Holders or the
underwriter in connection therewith, and participate, and cause the Company's
senior management to participate, to the extent reasonably requested by the
managing underwriter for the offering or the selling Holders, in efforts to sell
the Registrable Securities under the offering (including, without limitation,
participating in "roadshow" meetings with prospective investors) that would be
customary for underwritten primary offerings of a comparable amount of equity
securities by the Company.

       5.7 INDEMNIFICATION.

            (a) The Company will, and hereby does, indemnify each Holder, each
of its officers, directors and partners, and each person controlling such Holder
within the meaning of the Act, with respect to which registration, qualification
or compliance has been effected

                                      -9-

<PAGE>


pursuant to this Section 5, and each underwriter, if any, and each person who
controls such underwriter within the meaning of the Act, against all claims,
losses, damages and liabilities (or actions in respect thereof) arising out of
or based on any untrue statement (or alleged untrue statement) of a material
fact contained in any prospectus, offering circular or other document (including
any related registration statement, notification or the like) incident to any
such registration, qualification or compliance, or based on any omission (or
alleged omission) to state therein a material fact required to be stated therein
or necessary to make the statements therein not misleading, or any violation by
the Company of the Act or the Exchange Act or securities laws of any state or
any rule or regulation thereunder applicable to the Company and relating to
action or inaction required of the Company in connection with any such
registration, qualification or compliance, and will reimburse each such Holder,
each of its officers, directors and partners, and each person controlling such
Holder, each such underwriter and each person who controls any such underwriter,
for any legal and any other expenses reasonably incurred in connection with
investigating and defending any such claim, loss, damage, liability or action,
whether or not resulting in any liability, provided that the Company will not be
liable in any such case to the extent that any such claim, loss, damage,
liability or expense arises out of or is based on any untrue statement (or
alleged untrue statement) or omission (or alleged omission) based upon written
information furnished to the Company by any Holder or underwriter and stated to
be specifically for use therein.

            (b) Each Holder will, if Registrable Securities held by it are
included in the securities as to which such registration, qualification or
compliance is being effected, indemnify the Company (severally but not jointly),
each of its directors and officers, each other Holder and each underwriter, if
any, of the Company's securities covered by such a registration statement, each
person who controls the Company or such underwriter within the meaning of the
Act and the rules and regulations thereunder, each other such Holder and each of
their officers, directors and partners, and each person controlling such Holder,
against all claims, losses, damages and liabilities (or actions in respect
thereof) arising out of or based on any untrue statement (or alleged untrue
statement) of a material fact contained in any such registration statement,
prospectus, offering circular or other document, or any omission (or alleged
omission) to state therein a material fact required to be stated therein or
necessary to make the statements therein not misleading, and will reimburse the
Company and such Holder's directors, officers, partners, persons, underwriters
or control persons for any legal or any other expenses reasonably incurred in
connection with investigating or defending any such claim, loss, damage,
liability or action, whether or not resulting in liability, in each case to the
extent, but only to the extent, that such untrue statement (or alleged untrue
statement) or omission (or alleged omission) is made in such registration
statement, prospectus, offering circular or other document in reliance upon and
in conformity with written information furnished to the Company by such Holder
and stated to be specifically for use therein; provided, however, that the
obligations of each Holder hereunder shall be limited to an amount equal to the
net proceeds received by such Holder upon sale of his securities.

            (c) Each party entitled to indemnification under this Section 5.7
(the "INDEMNIFIED Party") shall give notice to the party required to provide
indemnification (the "INDEMNIFYING PARTY") promptly after such Indemnified Party
has actual knowledge of any claim as to which indemnity may be sought, but the
failure of any Indemnifying Party to give such notice shall not relieve the
Indemnifying Party of its obligations under this Section 5.7 (except

                                      -10-

<PAGE>


and to the extent the Indemnifying Party has been prejudiced as a consequence
thereof). The Indemnifying Party will be entitled to participate in, and to the
extent that it may elect by written notice delivered to the Indemnified Party
promptly after receiving the aforesaid notice from such Indemnified Party, at
its expense to assume, the defense of any such claim or any litigation resulting
therefrom, with counsel reasonably satisfactory to such Indemnified Party,
provided that the Indemnified Party may participate in such defense at its
expense, notwithstanding the assumption of such defense by the Indemnifying
Party, and provided, further, that if the defendants in any such action shall
include both the Indemnified Party and the Indemnifying Party and the
Indemnified Party shall have reasonably concluded that there may be legal
defenses available to it and/or other Indemnified Parties which are different
from or additional to those available to the Indemnifying Party, the Indemnified
Party or Parties shall have the right to select separate counsel to assert such
legal defenses and to otherwise participate in the defense of such action on
behalf of such Indemnified Party or Parties and the reasonable fees and expenses
of such counsel shall be paid by the Indemnifying Party. No Indemnifying Party,
in the defense of any such claim or litigation, shall, except with the consent
of each Indemnified Party, consent to entry of any judgment or enter into any
settlement which does not include as an unconditional term thereof the giving by
the claimant or plaintiff to such Indemnified Party of a release from all
liability in respect to such claim or litigation. Each Indemnified Party shall
(i) furnish such information regarding itself or the claim in question as an
Indemnifying Party may reasonably request in writing and as shall be reasonably
required in connection with defense of such claim and litigation resulting
therefrom, and (ii) shall reasonably assist the Indemnifying Party in any such
defense, provided that the Indemnified Party shall be entitled to be reimbursed
by this Indemnifying Party for its out-of-pocket expenses paid in connection
with such assistance.

            (d) No Holder shall be required to participate in a registration
pursuant to which it would be required to execute an underwriting agreement in
connection with a registration effected under Section 5.2 or 5.3 which imposes
indemnification or contribution obligations on such Holder more onerous than
those imposed hereunder; provided, however, that the Company shall not be deemed
to breach the provisions of Section 5.2 or 5.3 if a Holder is not permitted to
participate in a registration on account of his refusal to execute an
underwriting agreement on the basis of this Section 5.7(d).

       5.8 INFORMATION BY HOLDER. Each Holder of Registrable Securities included
in any registration shall furnish to the Company such information regarding such
Holder and the distribution proposed by such Holder as the Company may
reasonably request in writing and as shall be reasonably required in connection
with any registration, qualification or compliance referred to in this Section 5
or otherwise required by applicable state or federal securities laws.

       5.9 LIMITATIONS ON REGISTRATION RIGHTS. From and after the date of this
Agreement, the Company shall not enter into any agreement with any holder or
prospective holder of any securities of the Company giving such holder or
prospective holder (a) the right to require the Company, upon any registration
of any of its securities, to include, among the securities which the Company is
then registering, securities owned by such holder, unless under the terms of
such agreement, such holder or prospective holder may include such securities in
any such registration only to the extent that the inclusion of its securities
will not limit the number of Registrable Securities sought to be included by the
Holders of Registrable Securities hereunder or reduce the

                                      -11-

<PAGE>


offering price thereof; or (b) the right to require the Company to initiate any
registration of any securities of the Company.

       5.10 RULE 144 REPORTING. With a view to making available the benefits of
certain rules and regulations of the Commission which may permit the sale of
restricted securities (as that term is used in Rule 144 under the Act) to the
public without registration, the Company agrees to:

            (a) make and keep public information available as those terms are
understood and defined in Rule 144 under the Act, at all times from and after
ninety days following the effective date of the first registration under the Act
filed by the Company for an offering of its securities to the general public;

            (b) use its best efforts to file with the Commission in a timely
manner all reports and other documents required of the Company under the Act and
the Exchange Act at any time after it has become subject to such reporting
requirements; and

            (c) so long as the Investor owns any restricted securities, furnish
to the Investor forthwith upon request a written statement by the Company as to
its compliance with the reporting requirements of Rule 144 (at any time from and
after ninety (90) days following the effective date of the first registration
statement filed by the Company for an offering of its securities to the general
public), and of the Act and Exchange Act (at any time after it has become
subject to such reporting requirements), a copy of the most recent annual or
quarterly report of the Company, and such other reports and documents so filed
as the Investor may reasonably request in availing itself of any rule or
regulation of the Commission allowing the Investor to sell any such securities
without registration.

       5.11 LISTING APPLICATION. If shares of any class of stock of the Company
shall be listed on a national securities exchange, the Company shall, at its
expense, include in its listing application all of the shares of the listed
class then owned by the Holders.

       5.12 DAMAGES. The Company recognizes and agrees that the holder of
Registrable Securities shall not have an adequate remedy if the Company fails to
comply with the provisions of this Section 5, and that damages will not be
readily ascertainable, and the Company expressly agrees that in the event of
such failure any Holder of Registrable Securities shall be entitled to seek
specific performance of the Company's obligations hereunder and that the Company
will not oppose an application seeking such specific performance based on there
being an adequate remedy at law.

       6. LOCK-UP AGREEMENTS. If requested by the Company or the managing
underwriter, each Shareholder agrees to enter into lock-up agreements pursuant
to which they will not, for a period of no more than one hundred and eighty
(180) days following the effective date of the Company's first underwritten
public offering and for a period of ninety (90) days following the effective
date of any subsequent registration statement for a public offering of the
Company's securities (including Registrable Securities), offer, sell or
otherwise dispose of the Registrable Securities of the Company other than to its
members or partners, as the case may be, except the Registrable Securities sold
pursuant to such registration statement,

                                      -12-

<PAGE>


without the prior consent of the Company and the underwriter, provided that the
officers, directors and all holders of more than 1% of the shares of Common
Stock (calculated for the purpose as if all securities convertible into or
exercisable for Common Stock, directly or indirectly, are so converted or
exercised) of the Company enter into such lock-up agreements for the same period
and on the same terms. If the Company and the underwriters agree to waive this
provision for any Shareholder or other security holder, then such waiver shall
extend, on a pro-rata basis, to all Holders owning $250,000 or more of
Registrable Securities.

       7. ELECTION OF DIRECTORS.

            (a) At each annual meeting of the shareholders of the Company, and
at each special meeting of the shareholders of the Company called for the
purpose of electing directors of the Company, and at any time at which
shareholders of the Company shall have the right to, or shall, vote for
directors of the Company, then, and in each event, the Shareholders shall vote
all Shares owned by them for the election of a Board of Directors set forth in
the manner designated below.

                 (i) One (1) director shall be designated by TWCP, who shall
initially be Daniel Dross (which shall represent the director elected by the
holders of Series A Preferred Stock pursuant to the Company's Amended and
Restated Certificate of Incorporation).

                 (ii) One (1) director shall be the President of the Company,
who is currently Martin Haase (which shall represent one of the directors
elected by the holders of Series A Preferred Stock and the holders of Common
Stock pursuant to the Company's Amended and Restated Certificate of
Incorporation).

                 (iii) Six (6) directors shall be designated by the Current
Shareholders (which shall represent six (6) of the directors elected by the
holders of the Series A Preferred Stock and the holders of Common Stock pursuant
to the Company's Certificate of Incorporation), who shall initially be Brandon
D. Andries, Taylor J. Crouch, Jay T. Flatley, Frank H. Laukien, Albrecht Kehr
and Tony W. Keller.

                 (iv) Any other directors shall be designated and elected in
accordance with the Company's Amended and Restated Certificate of Incorporation.

            (b) In the event that TWCP ceases to own either (i) 2.5% of the
Company's outstanding stock or (ii) 50% or more of its initial investment in the
Company, then the right of TWCP to designate a director hereunder shall
terminate.

            (c) The Company shall reimburse any director for all reasonable
travel expenses incurred to attend all meetings of the Board of Directors and
any committees thereof.

            (d) Any vote or action taken to remove any director elected pursuant
to this Section 7, or to fill any vacancy created by the resignation or death of
a director elected pursuant to this Section 7, shall also be subject to the
provisions of this Section 7.

       8. TERM. This Agreement shall terminate upon the five (5) year
anniversary of the closing of the first underwritten public offering of the
Company's stock, provided that Sections 2,

                                      -13-

<PAGE>


3, 4 and 7 hereof shall terminate immediately upon the closing of the first
underwritten public offering of the Company's stock.

       9. SPECIFIC ENFORCEMENT. Each Shareholder expressly agrees that the other
Shareholders and the Company may be irreparably damaged if this Agreement is not
specifically enforced. Upon a breach or threatened breach of the terms,
covenants and/or conditions of this Agreement by any Shareholder, the other
Shareholders and the Company shall, in addition to all other remedies, each be
entitled to apply for a temporary or permanent injunction, and/or a decree for
specific performance, in accordance with the provisions hereof.

       10. LEGEND. Each certificate evidencing any of the Shares now owned or
hereafter acquired by the Shareholders shall bear a legend substantially as
follows:

       "Any sale, assignment, transfer or other disposition of, or the voting
       of, the shares represented by this certificate is restricted by, and
       subject to, the terms and provisions of a certain Investor Rights
       Agreement dated as of January 16, 2001. A copy of said Agreement is on
       file with the Secretary of the Corporation."

       11. NOTICES. Notices given hereunder shall be deemed to have been duly
given on the date of personal delivery or on the date of postmark if mailed by
certified or registered mail, return receipt requested, to the party being
notified at his or its address specified on the SCHEDULE I hereto or such other
address as the addressee may subsequently notify the other parties of in
writing.

       12. ENTIRE AGREEMENT AND AMENDMENTS. This Agreement constitutes the
entire agreement of the parties with respect to the subject matter hereof and
neither this Agreement nor any provision hereof may be waived, modified, amended
or terminated except by a written agreement signed by the parties hereto;
PROVIDED, HOWEVER, that the holders of 66 2/3 of the Shares owned by all
Investors may effect any such waiver, modification, amendment or termination on
behalf of the Investors, and the Current Shareholders owning at least a majority
of the Shares owned by all Current Shareholders may effect any such waiver,
modification, amendment or termination on behalf of all of the Current
Shareholders. Each of the Shareholders represents that he or it is not a party
to any other agreement which would prevent him or it from performing his or its
obligations hereunder. No waiver of any breach or default hereunder shall be
considered valid unless in writing, and no such waiver shall be deemed a waiver
of any subsequent breach or default of the same or similar nature.

       13. GOVERNING LAW; SUCCESSORS AND ASSIGNS. This Agreement shall be
governed by the internal laws of State of Delaware without giving effect to the
conflicts of laws principles thereof and, except as otherwise provided herein,
shall be binding upon the heirs, personal representatives, executors,
administrators, successors and assigns of the parties. The rights of the
Investors under this Agreement may be assigned by such Investor to a transferee
or assignee of the Shares held by such Investor provided that such Investor
shall, (i) within ten (10) days after such transfer, furnish to the Company
written notice of the name and address of such transferee or assignee and the
securities with respect to which such rights are being assigned and (ii) such
transferee shall agree to be subject to all restrictions set forth in this
Agreement as if it were an Investor.

                                      -14-

<PAGE>


       14. SEVERABILITY. If any provision of this Agreement shall be held to be
illegal, invalid or unenforceable, such illegality, invalidity or
unenforceability shall attach only to such provision and shall not in any manner
affect or render illegal, invalid or unenforceable any other provision of this
Agreement, and this Agreement shall be carried out as if any such illegal,
invalid or unenforceable provision were not contained herein.

       15. CAPTIONS. Captions are for convenience only and are not deemed to be
part of this Agreement.

       16. COUNTERPARTS. This Agreement may be executed in two or more
counterparts, each of which shall be deemed an original, but all of which
together shall constitute one and the same instrument.

       17. STOCK SPLITS, ETC. All share numbers used in this Agreement are
subject to adjustment in the case of any stock split, reverse stock split,
combination or similar events.

       18. AGGREGATION OF STOCK. All shares held or acquired by affiliates or
persons (as defined in the Securities Exchange Act of 1934, as amended) shall be
aggregated together for the purpose of determining the availability of any
rights under this Agreement.

       19. RIGHT TO CONDUCT BUSINESS. The Company hereby acknowledges that the
Investors and/or one or more of their respective affiliates invest in numerous
companies, some of which may be competitive with the Company's business. None of
the Investors shall be liable for any claim arising out of, or based upon, (i)
the investment by the Investors in any entity competitive with the Company, (ii)
actions taken by any partner, officer or other representative of the Investors
to assist any such competitive company, whether or not such action was taken as
a board member of such competitive company or otherwise, and whether or not such
action has a detrimental effect on the Company. In no event shall this provision
prevent the Company from pursuing claims based on breach of fiduciary duty or
abuse of confidential information.

                                      * * *

                                      -15-

<PAGE>


                            INVESTOR RIGHTS AGREEMENT


       IN WITNESS WHEREOF, this Agreement has been executed as of the date and
year first above written.


CURRENT SHAREHOLDERS:                   COMPANY:

                                        BRUKER AXS INC.



/s/ Frank H. Laukien                    By: /s/ Martin Haase
---------------------------------           -----------------------------------
Frank H. Laukien                            Name:  Martin Haase
                                            Title: President & CEO
/s/ Dirk D. Laukien
---------------------------------
Dirk D. Laukien

/s/ Isolde Laukien
---------------------------------
Isolde Laukien

/s/ Joerg C. Laukien
---------------------------------
Joerg C. Laukien

/s/ Marc M. Laukien
---------------------------------
Marc M. Laukien


                                      S-1

<PAGE>


                            INVESTOR RIGHTS AGREEMENT

INVESTORS:

THOMAS WEISEL CAPITAL PARTNERS, L.P.
 By:  Thomas Weisel Capital Partners LLC, its general partner
  By:  Thomas Weisel Partners Group LLC, its managing member


By: /s/ Daniel Dross
   -------------------------------
   Name: Daniel Dross
   Title: Attorney-in-Fact


TWP CEO FOUNDERS' CIRCLE (AI), L.P.
 By:  Thomas Weisel Capital Partners LLC, its general partner
  By:  Thomas Weisel Partners Group LLC, its managing member


By: /s/ Daniel Dross
   -------------------------------
 Name: Daniel Dross
   Title: Attorney-in-Fact


TWP CEO FOUNDERS' CIRCLE (QP), L.P.
 By:  Thomas Weisel Capital Partners LLC, its general partner
  By:  Thomas Weisel Partners Group LLC, its managing member


By: /s/ Daniel Dross
   -------------------------------
 Name: Daniel Dross
   Title: Attorney-in-Fact


THOMAS WEISEL CAPITAL PARTNERS (DUTCH), L.P.
 By:  Thomas Weisel Capital Partners (Dutch) LLC, its general partner
  By:  Thomas Weisel Capital Partners LLC, its managing member
   By:  Thomas Weisel Capital Partners Group LLC, its managing member


By: /s/ Daniel Dross
   -------------------------------
 Name: Daniel Dross
   Title: Attorney-in-Fact

                                      S-2

<PAGE>


                            INVESTOR RIGHTS AGREEMENT

THOMAS WEISEL CAPITAL PARTNERS (DUTCH II), L.P.
 By:  Thomas Weisel Capital Partners (Dutch) LLC, its general partner
  By:  Thomas Weisel Capital Partners LLC, its managing member
   By:  Thomas Weisel Capital Partners Group LLC, its managing member


By: /s/ Daniel Dross
   -------------------------------
 Name: Daniel Dross
   Title: Attorney-in-Fact


THOMAS WEISEL CAPITAL PARTNERS EMPLOYEE FUND, L.P.
 By:  Thomas Weisel Capital Partners LLC, its general partner
  By:  Thomas Weisel Partners Group LLC, its managing member


By: /s/ Daniel Dross
   -------------------------------
 Name: Daniel Dross
   Title: Attorney-in-Fact


TWP 2000 CO-INVESTMENT FUND, L.P.
 By:  Thomas Weisel Capital Partners LLC, its general partner
  By:  Thomas Weisel Partners Group LLC, its managing member


By: /s/ Daniel Dross
   -------------------------------
 Name: Daniel Dross
   Title: Attorney-in-Fact


TWP BRUKER INVESTORS
 By:  David Baylor, its general partner


By: /s/ David Baylor
   -------------------------------

                                      S-3

<PAGE>


                            INVESTOR RIGHTS AGREEMENT


ESSEX PRIVATE PLACEMENT FUND IV - A, LIMITED PARTNERSHIP
By: Essex Investment Management Company, LLC its General Partner


By: [Illegible]
   -------------------------------
Its: Principal
    ------------------------------


ESSEX PRIVATE PLACEMENT FUND IV - B, LIMITED PARTNERSHIP
By: Essex Investment Management Company, LLC its General Partner


By: [Illegible]
   -------------------------------
Its: Principal
    ------------------------------


BURRILL BIOTECH CAPITAL FUND


By: /s/ John Kim
   -------------------------------
Name:  John Kim
Title: Managing Director


                                      S-4

<PAGE>


                                   SCHEDULE I

                                 Notice Address

THE COMPANY:

Bruker AXS, Inc.
5465 East Cheryl Parkway
Madison, WI 53711
Attn:  President

with a copy to:

Hutchins, Wheeler & Dittmar,
A Professional Corporation
101 Federal Street
Boston, MA 02110
Attention:  Richard M. Stein

THE PURCHASERS:

Thomas Weisel Capital Partners, L.P.
TWP CEO Founders' Circle (AI), L.P.
TWP CEO Founders' Circle (QP), L.P.
Thomas Weisel Capital Partners (Dutch), L.P.
Thomas Weisel Capital Partners (Dutch II), L.P.
Thomas Weisel Capital Partners Employee Fund, L.P.
TWP 2000 Co-Investment Fund, L.P.
One Montgomery Street, Suite 3700
San Francisco, CA  94104
Attention:  General Counsel


Essex Private Placement Fund IV - A, Limited Partnership
Essex Private Placement Fund IV - B, Limited Partnership
c/o Essex Investment Management Co., LLC
125 High Street - 29th Floor
Boston, MA  02210
Attn:  Cindy Snow

Burrill Biotech Capital Fund
c/o Burrill & Company
120 Montgomery Street, Suite 1370
San Francisco, CA  94104
Attn:  John Kim M.D.